Advances in Electronics and Electron Phisics. Vol. 68

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 68 EDITOR-IN-CHIEF PETER W. HAWKES Laboratoire d Optique Electr...

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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 68

EDITOR-IN-CHIEF

PETER W. HAWKES

Laboratoire d Optique Electronique du Centre National de la Recherche Scientifique Toulouse, France

ASSOCIATE EDITOR-IMAGE

PICK-UP AND DISPLAY

BENJAMIN KAZAN Xerox Corporation Palo Alto Research Center Palo Alto, Calijornia

Advances in

Electronics and Electron Physics EDITEDBY PETER W. HAWKES Laboratoire d Optique Electronique du Centre National de la Recherche Scientifique Toulouse, France

VOLUME 68 1986

ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers Orlando San Diego New York Austin Boston London Sydney Tokyo Toronto

C O P Y R I G H T 0 1986 BY ACADEMIC PRESS. INC A L L RIGHTS RESERVED NO PART O F T H I S PUBLICATION MAY BE R E P R O D U C E D O R T R A N S M I T T E D IN ANY FORM O R BY ANY M E A N S , ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY. RECORDING. OR ANY INFORMATION STORAGE A N D RETRIEVAL S Y S T E M , W I T H O U T PERMISSION IN WRITING FROM T H E PUBLISHER

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CARD NUMBtR

(alk. paper)

P K I \ I I L) IN 1116 I N I T F C FT4TES 01 4 M t K I C A

9 8 7 6 5 4 1 2 1

LTD

49-7504

CONTENTS PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Distance Measuring Equipment and Its Evolving Role in Aviation ROBERTJ . KELLYand DANNYR . CUSICK I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. General Background to Distance Measuring Equipment Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. The DME/N System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Systems Considerations for the New DME/P International Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Einstein-Podolsky-Rosen Paradox and Bell’s Inequalities W . DE BAERE 1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. The Einstein-Podolsky-Rosen Argumentation . . . . . . . . . . . . . . . 111. The Bell Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Experimental Verification of Bell’s Inequalities . . . . . . . . . . . . . . V . Interpretation of Bell’s Inequalities and of Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

1 3 118

171 236 237

245 246 273 304 315 325 327

Theory of Electron Mirrors and Cathode Lenses E . M . YAKUSHEV and L . M . SEKUNOVA I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Electron-Optical Properties of Axially Symmetric Electron Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Mirror Electron Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.CathodeLenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

350 372 397 414 415

INDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

417

V

337 338

This Page Intentionally Left Blank

PREFACE

In Volume 57 of these Advances, an article by H. W. Redlien and R. J. Kelly was devoted to “The microwave landing system: A new international standard,” and I am assured that this has been heavily used by the aviation community. Since then, precision distance measuring equipment (DME/P) has come into widespread use as an integral and critical part of the MLS, a development in which R. J. Kelly has been deeply involved. I am therefore delighted to be able to publish his detailed account with D. R. Cusick of this new addition to the instrumentation that enables modem aircraft to land and take off so safely. In view of the importance of the subject and the authors’ concern to make their account at once complete, self-contained, and yet readable, this article dominates the volume and I have no doubt that members of the aviation world will be grateful for so clear and full a document. In 1935 A. Einstein, B. Podolsky, and N. Rosen published a paper in Physical Review entitled “Can the quantum-mechanical description of physical reality be considered complete?” This analysis of the orthodox interpretation of the quantum theory revealed a paradox, now universally known as the EPR paradox. This in turn has generated a voluminous literature, among which a paper by J. S. Bell is something of a landmark, adding Bell’s inequalities to the scientific vocabulary. The fiftieth anniversary of the EPR paper seemed a good occasion to survey this work, and I am extremely happy that W. De Baere, himself an active contributor to the debate, has agreed to put all these complex ideas into perspective. The point at issue is a difficult one, but is vitally important for the understanding and future development of quantum mechanics. The final article is devoted to two common elements of systems of chargedparticle optics that have been neglected in comparison with round lenses, prisms, and multipoles. These elements, mirrors and cathode lenses, have been regarded as difficult to analyze owing to the fact that some of the usual mathematical approximations that are fully justified in lenses and prisms can no longer be employed. Considerable progress with this problem has been made in the group led by V. M. Kel’man, co-author of the standard Russian book on electron optics, in the Nuclear Physics Institute in Alma-Ata. This work is available in translation since most of it appeared in the Zhurnal Tekhnicheskoi Fiziki (Soviet Physics: Technical Physics), but it is not widely known and I hope that this connected account in a widely accessible series will help to remedy the situation. The authors have taken considerable trouble to make their article self-contained, and thus reference lists of key formulas, for aberration coefficients in particular, are included. vii

...

Vlll

PREFACE

As usual, I conclude by thanking all the authors and by listing forthcoming articles.

Monte Car10 Methods and Microlithography Invariance in Pattern Recognition

K . Murata H. Wechsler

PETERW. HAWKES

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS, VOL. 68

Distance Measuring Equipment and Its Evolving Role in Aviation ROBERT J. KELLY AND DANNY R. CUSICK Bendix Communications Division Allied Bendix Aerospace Corporation Towson, Maryland 21204

I. INTRODUCTION For almost 40 years, distance measuring equipment (DME) has been a basic element in world air navigation. Increasing needs of automation in aviation-and the increasing levels of sophistication that resulted-fueled its evolution, significantly improving its capabilities, while the soundness of its principles guaranteed its wide use. As a result of this continuing evolution, a new version of DME is now part of the new international standard for landing systems, the microwave landing system (MLS). Because of its persistent presence, DME has played a dominant role in the development of air navigation, and its principles are common to other parts of the air traffic control system (ATC). This article discusses these basic principles by placing them in perspective with other types of air navigation systems as part of a broader, integrated discussion of navigation requirements. Specifically, the discussions relate to the needs of the air traffic control system and to the dynamic flight characteristics of the aircraft operating within that system. To expose the diverse requirements that the DME element must satisfy in the modern and evolving National Airspace System, substantial background information is provided on the elements of the ATC system and on the implementation of automatic flight control systems. Two main types of DME have been standardized by the International Civil Aviation Organization (ICAO). The DME that was standardized ;n 1952 and that has been in general use during the past 40 years is referred to as the DME/N. The new precision DME, to be used with MLS, is known as DME/P. The ICAO standardized it in 1985.

1 Copyright @ 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.

2

ROBERT J. KELLY A N D D A N N Y R. CUSICK

The DME principle is simple. Distance information is computed in the aircraft by measuring the round-trip time between interrogations from an airborne transmitter and replies to those interrogations from a gound beacon. Because the DME has proven itself to be a robust technique, its use and implementation will continue into the next century. This chapter views both DME/N and the new DME/P from a broad systems engineering perspective. Like many other navigation aids DME is characterized in terms of a signal format, accuracy (error budgets), coverage (power budgets), radio frequency assignment(channel plan), the measurement technique used to extract the DME navigation parameter (range), and the output guidance information rate. A DME must also satisfy certain levels of signal integrity and availability. The signal format is the blueprint of the system; it ensures that the DME ground and airborne elements will be compatible. Range accuracy must facilitate safe arrival of aircraft at their destinations. The ground facility must provide a usable signal that covers the volume of airspace where navigation information is required. Well-conceived radio frequency (rf ) channel plans assure that ground facilities can be geographically located so that their individual service volumes unambiguously “combine” to provide navigation guidance throughout the navigable airspace. The techniques used to make navigation parameter measurements must be simple so that ground and airborne equipment designs are reliable and economical while achieving the necessary accuracy and coverage. Finally, the guidance output information rate must be consistent with the aircraft flight control system’s (AFCS) dynamic response so that it will not limit the aircraft’s maneuvers. The DME system discussions in this article will follow the functional theme outlined herein. Emphasis will be directed toward systems considerations rather than toward hardware/software implementations or flight test data analysis. Section I1 provides the background and historical perspective necessary to view DME in the broader context of air navigation and its specific role in the National Airspace System (NAS). Section I11 is devoted to DME/N, and Section IV addresses the most accurate version of DME, the DME/P. The navigation applications discussed in this article are restricted to the air traffic control system as it is defined by the International Civil Aviation Organization and as it is implementedin the United States.Therefore,only the domestic enroute, terminal, and approach/landing phases of flight are addressed. Enroute and approach/landing scenarios are defined using only the conventional takeoff and land (CTOL) aircraft type. However, DME is not restricted to these applications or to CTOL aircraft. Both helicopter and remote area operations, for example, come under the broad envelope of DME applications.

DISTANCE MEASURING EQUIPMENT IN AVIATION

3

11. GENERAL BACKGROUND TO DISTANCE MEASURING EQUIPMENT APPLICATIONS A . Basic Operational Scenario and Introduction to the Navigation Services

A commercial jet aircraft leaving Chicago’s O’Hare International Airport for New York’s LaGuardia Airport will fly along a national airway system divided into enroute and terminal areas. During the enroute portion of the flight, the aircraft flies airways defined by a network of straight-line segments or radials. These radials are generated by very-high-frequency omni range (VOR) and DME navigation aids that provide direction and distance information to known ground locations. Using these radials, the pilot can navigate to the destination terminal area. Arriving in the LaGuardia terminal area, the aircraft enters a region of converging tracks and high-density traffic. Position accuracy becomes increasingly more important as the aircraft approaches its runway destination, particularly when visibility is reduced. Here, the enroute guidance supplied by the VOR/DME radials is replaced with approach and landing guidance such as that generated by the MLS. The three-dimensional guidance provided by MLS is of sufficient .accuracy to guide safely the aircraft to a landing, even under zero-visibility conditions. Accurate range data are essential during both phases of flight. The guidance aid which generates this ranging information is DME. It provides range data accurate to 0.2 nautical miles (nmi) along the enroute airways. Within the terminal area, its range error decreases to 100 feet or less for the landing guidance applications. DME has evolved from the radar beacon concept to satisfy the navigation needs of both the civilian and the military aviation community. This evolution is reviewed in Subsection B. The general principles of aircraft position estimation, as applied to the radio navigation aids used in the NAS, are summarized in Subsection C. Both bearing and range measuring systems are treated. In order to place the DME in its proper operational context, other navigation aids that operate in conjunction with it must be discussed. In Subsection D the role of VOR, ILS, MLS, and the ATC surveillance system (secondary radar) in the overall ATC system are briefly discussed. Also included is the Tactical Air Navigation System (TACAN) upon which the military currently depends for its enroute navigation. In this system, DME plays a central role in a signal format that is arranged to contain both range and bearing information. Both military (TACAN) and civil (VOR/DME) enroute navigation

4

ROBERT J. KELLY AND DANNY R. CUSICK

systems are combined on the ground in the configuration called VORTAC. For the United States NAS and many of the European ICAO member states, the principal ATC enroute airways are based on range and bearing information from VOR/DME or VORTAC facilities. For the precision phases of flight conducted in the terminal areaapproach, landing, missed-approach, and departure- the principal aids are ILS and MLS. A high-precision DME is the range-providing element of the microwave landing system. When used with the instrument landing system (ILS), another version of DME/N provides the range information normally generated by the ILS marker beacons (which "mark" a few specific points along the straight-line approach path). DME, VOR, ILS, MLS, and TACAN are navigation aids (navaids) that provide the primary navigation parameters of range and bearing. These navaids can be used directly for navigation, or they can be integrated into the aircraft by an area navigation system known as RNAV. Navigation derived from RNAV is becoming increasingly important and has led to the development of RNAV routes in addition to the VORTAC routes. Subsection D briefly describes RNAV. Table I summarizes the navigation services. With the exception of MLS, they have been in use for the past three or four decades. It is essential to understand the interaction of navigation signals with the Aircraft Flight Control System (AFCS) in order to comprehend a navigation system. To assist in this understanding, background information on aircraft guidance and control is given in Subsection E. Accuracy performance comparisons between VOR/DME, DME/DME, and TACAN, using RNAV to calculate the position fixes, are given in Subsection F. TABLE I PRESENT NAVIGATIONAL SERVICES Service VHF omnidirectional range (VOR) Distance measurement, DME/N Air traffic control surveillance system" Interrogator Transponder Instrument landing system Localizer Glide slope Marker MLS Azimuth and elevation DME/P

Frequency band (MHz) 112-118 960-121 5 1030 1090 108-112 328-335 15 5031-5091 (Same as DME/N)

Includes mode S and the traffic alert collision avoidance system (TCAS).


5

B. History of the DME Principle

This section introduces the principal elements of DME, the airborne interrogator, and the ground transponder, by tracing the early history of their development. Both the DME and the ATC Secondary Radar were derived from the radar beacon concept developed during World War 11. All of the techniques described herein were operational in the early 1940s (Roberts, 1947). Conventional radar operates by sending out high-energy pulses of radio waves. The radar receiver detects the echo of the transmitted energy that is reflected back from a target. The elapsed time between the transmission of the pulse and the reception of the echo is a measure of the range (or distance) to the reflecting target. Further, by use of a suitable antenna design, the radio energy is concentrated into a narrow beam so that the echoes are received only when the radar is ‘‘looking’’ at targets residing within the antenna’s beamwidth. By proper coordination of the motion of the antenna and the sweep of an intensity-modulated cathode-ray tube (CRT) display device, a plan view (like a map) of the reflecting targets in the region of the radar set is traced out. This map-type display, which is illustrated at the top of Fig. 1, is called the plan-position indicator (PPI). Thus the target’s position is identified by a “bright spot” on the PPI. The radial trace through the spot is the target’s bearing relative to the radar’s reference. The distance from the center of the PPI to the bright spot is proportional to the target’s range. I . The Beacon Concept Conventional radar pulse transmissions are reflected by targets of different sizes regardless of their individual importance to the radar user. Sometimes echoes are too weak to be observed, as are those from a small airplane at a great distance. Under other circumstances, strong echoes from buildings or natural features such as mountains may mask weaker echoes from aircraft. Furthermore, the exact location of a geographical point on the ground may be of importance to a radar-equipped aircraft even though the terrain gives no distinguishing echo. In all such cases, radar beacons find their use as cooperative aids in pinpointing specific targets. A radar beacon is a device that, upon reception of the original radar pulse, responds with its own transmitter to radiate a strong signal of one or more pulses which is easily distinguished from the weaker radar echoes from surrounding objects. The beacon, therefore, is a device which amplifies the echo. The beacon transmitter need not be very powerful to be able to give a reply much stronger than the target echo itself. Most importantly, the beacon, when placed upon the ground, can provide an accurate reference point which can be used to determine an aircraft position relative to the reference point.

6

ROBERT J. KELLY AND DANNY R. CUSICK NORTH

INTERROGATOR

e

= BEARING R = RANGE

DIRECTIONAL ANTENNA

RECEIVER

TRANSMITTER BEACON

FIG. 1. The beacon concept.

In a typical beacon application (Fig. l), the scanning narrow beam emanates from the aircraft. The position of the beacon on the ground (its range and bearing relative to the aircraft) can be determined by viewing the airborne PPI display. The ground beacon’s antenna is designed to be omnidirectional, that is, to receive signals equally well from all azimuth directions. The beacon


7

bearing is derived from the narrow scanning beam while the “round-trip” delay of the radar transmission and its reply from the beacon is proportional to the beacon’s range to the aircraft, resulting in a unique measurement of position. The measurement of range and bearing constitute one of the most fundamental procedures in determining a position “fix” by a radio navigation aid. It is the basis, for example, by which guidance is obtained to fly the VORTAC routes. The process by which a radar set transmits a signal suitable for triggering the beacon is known as interrogation; the corresponding beacon transmission is termed the reply. Radar beacons which reply to interrogations are called transponders and the radar set used to interrogate a beacon is called an interrogator. The provision of a special character to the pulsed signals of either the interrogator or the transponder is called encoding. A device that sets up such a coded signal is called an encoder; a device that deciphers the code at the other end of the link is called a decoder. Although the transponder reply is like an echo, it differs from one in several significant respects. The strength of the response is independent of the intensity of the interrogating signal, provided only that the interrogating signal exceeds a predefined minimum threshold intensity at the beacon receiver. Also, the response frequency is different from that of the interrogation frequency. The response signal pulse may also differ from the interrogation signal in form. It may even consist of more than one pulse, the duration and spacing of the pulses being an arbitrary design choice. Thus echoes from a desired beacon on the ground can be distinguished from others on the ground not only in terms of pulse amplitude (nearer beacons have larger amplitude returns) but also in terms of radio frequency and pulse spacing (code). All of the important components necessary to form a beacon system are illustrated in Fig. 1. Roberts (1947) is a complete treatise on the systems-engineeringconsiderations of radar beacon designs. The complete downlink/uplink cycle starts with the generation of a trigger pulse in the airborne radar interrogator. This trigger initiates the transmission of an interrogating signal and, after a predetermined delay (called the zero mile delay),starts the cathode-ray tube sweep on the indicator. On the ground, the beacon transponder receiver detects the interrogating signal and generates a video signal that is passed to the decoder. The decoder examines the video signal to see if it conforms to an acceptable code. If not, it is rejected. If accepted, a trigger pulse is passed to the blanking gate that prevents the receiver from responding to any further interrogations for a time sufficient to permit the complete coded reply to be transmitted. This time, called the reply dead time, varies generally from about 50 to 150 ys. Because the beacons require a reply dead time, there is a limited number of interrogations (called

8


the traffic load) that can be accommodated by the transponder (see Section 111,C). The transponder decoded output, a single pulse, is delayed for a time which when added to the reply code length equals the “zero mile delay” used in the interrogator. The delayed signal is then encoded with a reply code and transmitted equally in all directions on a different rf frequency. At the airborne radar antenna the reply is detected by the interrogator receiver, decoded, and displayed to the operator on the PPI indicator. If the interrogator is “on top of” the transponder, i.e., if the range were zero miles-there would be no propagation delay through space, and the transponder reply signal’s arrival time would equal the zero time delay. The PPI trace would then show a “spot” at the center of its display. This system was developed almost 50 years ago and forms the basis of the secondary surveillance radar (SSR) system and the DME. If the airborne element is the transponder with nondirectional antenna and the interrogator with its directional antenna is placed on the ground, the configuration is the basis of the SSR system. Range and bearing and, thus, a position fix on the aircraft are made available to the air traffic controller on the ground. On the other hand, if the beacon is located on the ground and the PPI scope in the aircraft is replaced by an automatic range tracking circuit, the range to the fixed beacon is displayed to the pilot. This is the DME concept (see Fig. 2). Interestingly enough, the principal components-interrogator transmitter and receiver, encoder, decoder, transponder dead time, echo suppression, transponder receiver, and transmitter -remain the principal elements of today’s DME system. Only elements of performance such as accuracy have been improved over the years in response to more demanding applications. Since range is the only parameter of interest in DME, bearing information is sacrificed in exchange for use of simple omnidirectional ground and airborne antennas. In addition, a higher data rate is possible since the ground and airborne equipments are always “looking” at each other. An aircraft “range-only’’ position fix can be obtained by measuring the range from two separated ground transponders. See Subsections C and D,5 on scanning DME. 2. History of the Beacon Concept

The radar beacon was invented in 1939 by a group at the Bawdsey Research Station of the Air Ministry in the United Kingdom (Williams et al., 1973). It was developed in response to a military need, and its invention was not immediately made public for reasons of security. During the war, radar beacons were used by Germany, Japan, and the Allies. Because these countries

9


TRIGGER AIRBORNE INTERROGATOR

4 ENCODER

DECODER

4

t

TRANSMITTER

RECEIVER

a

\

AIRCRAFT SKIN

4

El GROUND BEACON

FIG.2. DME operation.

veiled the invention in secrecy, there were no pre-World War I1 nonmilitary uses of beacons. The initial purpose of the beacon was for Identification of Friend or Foe (IFF), where appropriate transponder responses to coded interrogations indicated that a particular military vehicle was either a “friend” (correct code response) or “foe” (incorrect response). It was soon discovered that beacons could be used for offensive purposes by helping to locate target areas. That is, they were used not only for identification but for navigation as well. One such system which employed lightweight ground beacons was the Rebecca-Eureka system. It included an airborne interrogator called “Rebecca” and a ground beacon called “Eureka.” Originally designed by the Telecommunications Research Establishment (TRE) for British use, the Rebecca-Eureka system was later adopted by the United States Army and used in many operations. The Rebecca-Eureka system operated at a radio frequency of 200 MHz. Table I1 shows its channel plan, which is the forerunner of the DME channeling concept (Roberts, 1947).

10

ROBERT J. KELLY AND DANNY R. CUSICK TABLE I1 REBECCA-EUREKA CHANNEL PLAN Interrogation channels (MW

Response channels (MW

A 209 B 214 C219 D 224 E 229

209 214 219 224 229

Characteristics Only cross-channel used; e.g., interrogation on A, response on B, C, D, or E

3. Development of the D M E Concept

As remarked in Dodington (1980), “it must have occurred to many people that the addition of a relatively simple automatic tracking circuit would enable an interrogating aircraft to constantly read its distance with respect to a ground beacon.” However, it was apparently not until 1944 that such a device was actually built (by the Canadian Research Council), operating in the Rebecca-Eureka band of 200 MHz. In November 1945, the Combined Research Group at the U.S. Naval Research Laboratory built and demonstrated such a system operating at lo00 MHz. It was largely based on work that the Hazeltine Corporation had been doing on the Mark V IFF, at 9501150 MHz. This system, however, did not use precise frequency-control techniques. A crystal-controlled DME system was developed at ITT Nutley Laboratory in 1946 (Dodington, 1949) and had a large impact on the final DME channel plan selected by ICAO. Whether the DME had crystal control or not was not simply a hardware question. It was most importantly a signal format question involving the efficiency of radio frequency spectrum use. The use of precise frequency control permitted more DME channels to be accommodated within a specified frequency bandwidth. The development of DME from 1946 to 1959 was very complex, involving the U.S. military, Congress, industry, and national pride and will receive only a cursory review. The reader is referred to several excellent references (Dodington, 1980; Sandretto, 1956;Rochester, 1976)for details on this subject. As described in Sandretto (1956), the air-navigation and traffic-control system in the United States grew as the product of necessity and without being planned. It began about 1919 with the installation of 4-course radio ranges for furnishing guidance to aircraft along defined routes. Later, other navigational aids were added, such as markers (both low and high frequency), to indicate points along the airways. Still later, low-approach equipment was added. When a controlling agency was established on the ground, it was equipped with slips of paper to designate aircraft, and utilized limited knowledge of airspeed, heading, and winds to


11

predict their future positions. Reliance was placed on communicationsfrom the aircraft to the ground controller to learn of the aircraft’s actual positions. Communications were furnished largely through the high-frequency equipment owned and operated by the airlines.Teleprinter communicationswere later added between the various control centers.

In reviewing this system, it is clear that it was slow and inefficient. Because of this situation, several companies (ITT, Hazeltine, RCA, and Sperry) proposed schemes to alleviate this problem through implementation of a comprehensive navigation and traffic control system. An ITT system called NAVAR was based upon the following concepts developed in 1945 (Busignies et al., 1946): (1) means whereby the pilot can know his position in three dimensionsso he can navigate to his destination; (2) means whereby a ground authority, capable of regulating the flow of air traffic, can know the position of all aircraft; (3) means whereby the ground authority can forecast the future positions of all aircraft; (4) means whereby the central authority can issue approvals to pilots of aircraft to proceed or to hold. When the present ATC system is discussed later, it will be interesting to note how closely it approached ITT’s original planning concepts. The first navigation elements of NAVAR were the DME and an associated bearing device. They were demonstrated for three weeks in October, 1946, at Indianapolis where the U.S. played host to the Provisional International Civil Aviation Organization (PICAO). A major result of the Indianapolis demonstrations was that ICAO later decided that the future short-range navigation system would be p-8, i.e., range-bearing [as opposed to GEE, which was the British VHF hyperbolic navigation system (Colin and Dippy, 1966)l. Further, it was decided that the DME part would be located at 1000 MHz, not 200 MHz. It was argued that the p-8 system was the natural choice since man is born with the knowledge of left and right and learns about distance when he learns to walk (Sandretto, 1956). Meanwhile, subcommittees for both the Radio Technical Commission for Aeronautics (RTCA) and the ICAO began working on standards for a coordinated navigation system, including DME. During 1946, RTCA Special Committee SC-21 developed a DME channeling plan at L band (the 1000-MHz region) that ignored crystal control (RTCA, 1946).Its 13interrogation channels and 13 reply channels were spaced at a relatively wide separation of 9.5 MHz based on using free-running oscillators whose stability was to be k 2 MHz. Later in the summer of 1947, RTCA organized committee SC-31 on Air Traffic Control. This committee’s

12


report (RTCA, 1948a) stated that The Navigation Equipment is a transmitter-receiver having multiple channels. This equipment, with associated ground equipment:

(1) Provides distance and bearing information for navigation. These data, when used in conjunction with a computer, will allow the pilot to fly any desired course. (2) Provides precise slope, localizer, and distance information for instrument approaches. (3) Provides information for airport surface navigation to enable the pilot to taxi his aircraft. (4) Provides air-ground aural communication of a reliable and status-free type. ( 5 ) Provides a situation display in pictorial form which enables the pilot to monitor traffic conditions in his vicinity or receive other pertinent data such as holding areas, airlane locations, and weather maps from the ground. (6) Provides suitable output to allow the aircraft to be automatically flown, either enroute or during final approach and landing.

In retrospect this appears to be the first attempt to define the ATC system as it exists today. The SC-31 committee’s navigation system is further described to have “the airborne navigation equipment and its directly associated ground elements wholly contained in the 960-1215 MHz band. The airborne navigational equipment for the ultimate system will have accuracy of k0.6“ in bearing and & 0.2 nautical mile or 1% of the distance, whichever is greater.” As will be discussed in Subsection F, none of the bearing systems finally placed in operation achieved this & 0.6” accuracy. However, DME/N did achieve a performance of 0.2 nautical mile and-with the recent development of DME/P-has surpassed that goal by almost twentyfold. Although the equipment proposed by the SC-31 committee for the ultimate or target system was very dissimilar to the NAVAR system, it constituted a coordinated system with essentially the same elements that had been described by ITT. Committee SC-31 had brought together the thinking of the Air Force and the Navy as well as the Civil aviation industry. By participating in the committee’sactivities, each group learned of the work that had been done in the other groups. As proposed by SC-31, only the DME and the Secondary Surveillance Radar remained in the 960-1215 MHz band. The bearing information (VOR) was assigned to the VHF band while the aircraft landing systems ended up at VHF for ILS and C-band (in the 5000-MHz region) for MLS (see Table I). That is, through the consensus of committee participation, the NAVAR and other proposals were discarded. In early December, 1947, another RTCA Committee (RTCA, 1948b),SC-40, was convened for the purpose of studying two channeling techniques under development for DME. One of these was the multiple radio frequency channel, crystal-control technique, and the other the pulse-multiplex system which was similar to that proposed by SC-21. The committee made its report in December of 1948 with a recommendation that


13

both crystal-control and pulse-multiplex techniques be included in the channel plan. The envisioned DME system would interrogate on ten crystal-controlled channels at the low end of the band and reply on ten crystal-controlled channels at the high end of the band. Channels were spaced 2.5 MHz apart and used 10 pulse codes in multiples of 7 ps, starting with 14 ps. Thus, there were 10oO combinations of frequency pairs and codes, of which it was proposed to pair 100 with the VOR and the ILS localizer (see Table VI). 4. DME Versus TACAN

The U.S. Civil Aviation Agency (CAA)-the forerunner of the Federal Aviation Administration-adopted the SC-40 plan as its official position and convinced ICAO to adopt it. The first ICAO Channel Plan, given in Tables V and VI, is essentially the same as that proposed by SC-40. However, as noted in Dodington (1980), there was little enthusiasm for this system outside the CAA, both within and outside the US. The main reason was that the US. military was developing an entirely different system in the same frequency band. When the SC-40 system was finally abandoned in 1956, there were only 340 airborne sets in existence. The trouble started shortly after World War 11, when the CAA began to install the recently developed VOR and DME units that were expected to form the nucleus of the emerging “common” system. The Navy found VOR/DME to be unsatisfactory for aircraft carrier operations because of the large size and complex siting requirements of the relatively low-frequency VOR antenna. They contracted with the ITT’s Federal Telecommunications Laboratories in February, 1947, to design an alternative system that could be used effectively at sea. By 1951,the Navy project had not only acquired the name TACAN but also the support of the Air Force, which had originally endorsed VOR/DME. The Air Force was converted to TACAN by its promise of greater flexibility and accuracy and its integration of the bearing and range elements into a single unit and, accordingly, a single frequency band. Thus, scarcely before the SC-40 “common system” document had circulated, civil and military agencies were following separate courses of action. Rochester (1976) provides a complete history of the VOR/DME/TACAN controversy. It was resolved in August, 1956,by discarding the SC-40 system and replacing it with a TACANcompatible DME. Part of the solution involved a fixed pairing of 100TACAN channels one-for-one with 100 VOR/ILS channels. This proposal by Dodington (1980) thus allowed a single-channel selector for VOR/ILS/DME and removed the objections of the airlines which were concerned that blunders would result from VOR bearing information received from one station and DME from another.

14


The new enroute navigation system, christened VORTAC, retained the VOR bearing component but replaced civil DME with the distance component of TACAN. Each VOR station, instead of being collocated with a DME, was now collocated with a TACAN beacon (which, of course, also provides the DME service). Figure 3 shows the general VORTAC arrangement. At the ground station, the VOR central antenna is housed in a plastic cone that supports the TACAN antenna. Leads to the TACAN antenna pass through the middle of the VOR antenna, along its line of minimum radiation, and thus do not disturb the VOR antenna pattern.

1

I

I

I I -

i'

I

CHANNEL SELECTOR

TACAN

I I

I

I

L - _M l L l-T l 7

CHANNEL

- - - -----

1 SELECTOR

CHANNEL SELECTOR

TACAN ANTENNA

'

VOR

I VOR

ANTENNAS

NORTH NORTH

BEARING

70 m.BEARING TO BEACON NOTE: W E AND VOR CHANNELS ARE PAIRED

FIG.3. The VORTAC system.


15

Civil aircraft read distance from the TACAN beacon and bearing from VOR transmitter. Military aircraft read both distance and bearing from the TACAN beacon. Both types of aircraft therefore fit into the same ATC network. Although the CAA and DOD had thus resolved their differences, the system defined by ICAO in 1952 was still the SC-40 format. Most of ICAO’s member countries did not implement it and, in fact, some took the opportunity to implement a hyperbolic system such as DECCA, the British low-frequency navigation system. After more years of debate, ICAO finally adopted the TACAN-compatible DME in 1959, and the VORTAC system has remained intact since that time. The evolution of the international DME standards will be detailed in Section 1II.A.

C. Principles of Air Navigation Systems

In this section basic concepts of navigation, position fixing, and dead reckoning are defined and applied to air navigation. I . Dead Reckoning, Fixing, and Navigation Systems

The fundamental task of a navigator is the selection of a route to be followed by his aircraft. To do this the navigator must know his aircraft’s present position and destination before he can choose a route and, from that, a course and heading. All position-determination schemes can be classified as dead reckoning, position fixing, or a combination of both. A position, once known, can be carried forward indefinitely by keeping continuous account of the velocity of the aircraft. This process is called dead reckoning. Because velocity v is a vector, it has a direction and magnitude (speed)u. From a given starting point p o and a velocity vo, one can determine a new position p after time Ato by stepping off a distance Ad = vot, in the direction of the velocity vo. At position p l , the process is repeated to find the next position p z using v1 and Atl, and so on. Unfortunately, direction and speed cannot always be precisely measured, particularly if direction is derived from aircraft heading only and effects of winds are approximated. Thus, on the average, the errors of dead reckoning increase in proportion to the length of time it is continued. Ordinarily, such an elementary dead reckoning process is seriously in error after an aircraft has traversed a few hundred miles without any indication of position from some other source. The Doppler radar navigator is an example of a good and more complex dead reckoning system (Kayton and Fried, 1969). It can guide an aircraft across the Atlantic Ocean and have a position error at its destination of less than 10 nautical miles. In

16

ROBERT J. KELLY A N D DANNY R. CUSICK

recent years, the Doppler radar navigator has been replaced by the inertial navigation system (INS) for transatlantic flight navigation. A position fix, in contrast to dead reckoning, is a determination of position without reference to any former position. The simplest fix stems from the observation of a recognizable landmark. In a sense, any fix may be thus described; for example, the unique position of stars is the recognizable landmark for a celestial fix. Similarly, some radio aids to navigation provide position information at only one point, or relatively few points, on the surface of the earth. In this case the landmarks are the physical locations of the ground stations. To obtain a position fix using radio navaids such as DME or VOR requires measurements from at least two different ground facilities;for example, the use of two differentDME stations (DME/DME) requires measurementsfrom two different ground stations or, for DME/VOR, two different measurements (range and bearing) from the same location. A single measurement yields only a locus of position points; two measurements are necessary to identify the point corresponding to an aircraft’s position with respect to the surface of the earth. Since a fix is based upon essentially “seeing” landmarks whereas dead reckoning is more associated with a computational process, it is “natural” that most electronic aids to navigation are based on the determination of fixes. Dead reckoning can be combined with position fixing to realize a more accurate navigation system. These configurations are referred to as multisensor navigation systems in the literature (Fried, 1974; Zimmerman, 1969). As will be shown in Subsection E, dead reckoning and position fixing complement each other; each provides an independent means of checking the accuracy of the other. Where position fixing is intermittent, with relatively long intervals between fixes, dead reckoning is appropriately considered the primary navigation method. In these systems, position fixing constitutes a method of updating the dead reckoning calculations such as in the “aiding”of an INS by a VOR and DME position fix, as shown in Fig. 4. Notice how the position fix corrected the dead reckoning measurement. Britting (1971) gives an in-depth discussion of INS. This correcting combination is extremely important in modern air transports which may have the navigation guidance autocoupled to the flight control system (Subsection C,3).A major concern is controlling the aircraft so that it maintains its desired course in the presence of wind shears and wind gusts. Small tracking errors are achieved by having the aircraft tightly coupled to the guidance system. Forcing the aircraft to follow the guidance signal closely requires low-noise navigation sensors. The INS is such a low-noise device, but, as mentioned above, it has a tendency to drift from its initial setting with time (about 1 nmi/h). These drift errors are corrected periodically by


17

NORTH

t

NORTH

\

$ = HEADING

GT

--

-TRACK ANGLE R RANGE V AIRCRAFT GROUND VELOCITY

FINAL APPROACH WAY POINT

-

-

FLIGHT PATH CONSTRUCTION DESTINATIONS .WAY POINTS

position-fixing radio navaids. With position-fix aiding, the INS becomes the desired accurate windproof navigation system. If fixes are available continuously or at very short intervals, position fixing becomes the primary navigation method. Rapid position fixing constitutes the most common primary navigation system currently in use over the continental US. (see Subsection D,5 on RNAV). Its popularity arises from the low acquisition and maintenance cost of the airborne element. In some cases, dead reckoning measurements are necessary to augment the rapid position-fix information. Such a case arises when excessive noise from the VOR/DME sensors causes the aircraft flight control system to have undesirable pitch and

18


roll activity. The aircraft may also follow the sensor noise because it is tightly coupled to the guidance signal (high signal gain). Unfortunately, reducing the noise by uncoupling from the guidance signal (low signal gain) can degrade the aircraft’s transient response so that it cannot closely follow the course in the presence of wind turbulence.’ On the other hand, a stable aircraft transient response requires accurate rate information (velocity) to generate the necessary dynamic damping so that the aircraft will not oscillate about the desired course. Since low-noise rate information cannot always be derived from the position-fix information, accurate rate information may be derived from dead reckoning measurements. An accelerometer output, for example, can be integrated once to provide the velocity. In this case dead reckoning measurementsfrom inertial sensors aid the VOR/DME position-fix guidance. This augmentation is called inertial aiding. In the context of the above discussion, both velocity data as well as position data are necessary to control an aircraft accurately along its desired flight path. Complementing position-fix data with dead reckoning information and vise versa is a notion which will be encountered again in detail in Section V. It is one of the most important solutions to the problem of obtaining accurate navigation, particularly during the approach/landing applications. As stated in Pierce (1948), navigation does not consist merely of the determination of position or the establishment of a compass heading to be followed. Navigation requires the exercise of judgment; it is a choice of one out of many courses of action that may lead to the required result based on all available data concerning position, destination, weather, natural and artificial hazards, and many other factors. Therefore, there can be no electronic navigational system-only electronic aids to navigation. An aircraft may be automatically guided along a predetermined course by the use of equipment that performs the dead reckoning function, or may be made to follow a line of position known to pass through a desired objective. Neither of these achievements constitutes navigation by equipment. The automatic devices simply extend the control exercised by the navigator in time and space. A navigation system comprises: (1) Sensors or navigation aids that generate primary guidance parameters; (2) a computer that combines the guidance parameters to obtain the aircraft position; (3) flight path commands (the path to be followed) that, when contrasted with the aircraft’s present position, generate the steering commands that define needed corrections in flight path; and (4) a flight control system, which causes the aircraft to correct its flight path while remaining stable. Figure 4 functionally describes the principal sensor and position calculation elements for a two-dimensional(2D) lateral guidance navigation system. Three-dimensionalflights would include Alternatively, filtering the sensor output data is not always a viable solution because the added phase delays may reduce the aircraft flight control system’s stability margin.


19

the aircraft’s altitude which would be used to derive the vertical guidance component. A navigation system provides output information in a variety of forms appropriate to the needs of the aircraft. If the information is to be directly used by the pilot, it involves some type of display; other outputs can be steering signals sent directly to the autopilot. In some applications such as manual flights along the VORTAC radials, the sensor outputs-range and bearing-are sent directly to the displays without additional data transformations and computations. Subsection C,3 will describe how the outputs of the navigation system are used to “close the loop” with the aircraft to form the total aircraft flight control/guidance system. Only position-fix navigation systems will be treated. Bearing “only” navaids, such as the airborne direction finder (ADF) using nondirectional beacons (NDB),and not discussed in this article (see Sandretto, 1959). 2. Principles of Navigation Parameter Measurement: Range and Bearing

Electronic navigation aids are defined as those systems deriving navigationally useful parameters-for example, range and bearing- through the application of electronic engineering principles and electronic technology.’ Following the partition developed in Sandretto (1959), these systems are of two general classes: the path delay and the self-contained. Path delay systems are characterized by employment of at least one radio transmitter and at least one radio receiver. The transmitter emits energy that travels to the receiver. The navigational parameter is derived by measurement of the delay incurred in the transmission. All path delay systems are based on the assumption that radio wave propagation is rectilinear and that its velocity of propagation is constant. Therefore, parameters such as phase delay or pulse arrival time delay between a reference and a distant object can be measured and made proportional to range or bearing. The self-contained systems consist of devices that sense certain natural phenomena and, with the aid of computers, derive navigational parameters. The INS is such a system. Path delay systems are characterized by the various types of position lines which they produce. There are only two fundamental types: the single-path and the multiple-path system. Figure 5 notes the two main variations of each of these types. The single-path system measures absolute transmission time and produces circular “lines of position”- that is, a single-path navigation system determines aircraft range. DME is such a system. The terms navigation aid (navaid)and navaid sensor are used synonymouslyin this article. In the system context, a radio navaid is a sensor which includes both the ground and airborne elements and thus includes cooperative techniques such as DME. Traditionally,MLS, ILS, and VOR are characterized by a signal in space (ground element) and a sensor (airborne element).

20

ROBERT J. KELLY AND DANNY R. CUSICK NAVAIDS USED I N ATC SYSTEM

SELF-CONTAINED SYSTEMS MEASUREDELAYOVER TRANSMISSION PATH POSITION FIX

I SATELLITE

SINGLE PATH (CIRCULAR LOP)

4

FI

I

OBSERVES NATURAL PHENOMENA & COMPUTES

FlFl RECKONING

I + ’

MULTIPLE PATHS PATH DELAYS COMPARED

.CELESTIAL FIXES (NOT USED I N SYSTEM)

LOP

-

LINES OF POSITION

.INS

mzri:R

RNAV

NAVIGATOR

.AIRDATA

TRUE HYPERBOLIC

ONE-WAY

DME

I

FOR EQUAL DELAY (RADIAL LOP)

(HYPERBOLIC LOP)

VOR RADAR BEARING ILS MLS ANGLE .TACAN BEARING

‘UNDER DEVELOPMEN7

FIG.5. ATC navaid family tree.

The multiple-path system measures difference in, or compares, transmission times. Since the locus of all points having a constant difference from two other points is a hyperbola, these systems should produce lines of positions which are hyperbolas. Many of these mutliple-path systems are so instrumented that they can only determine when the differencein transmission time is zero (that is, when the times are equal). In this case, the line of position is a hyperbola of zero curvature, or a striaght line. That is, the locus of all points that are equidistant from two other points is a straight line (the bisector). These simplified multiple-path systems are known as radial systems; those with nonzero time delays are called true hyperbolic systems. Radial systems measure aircraft bearing. Examples are VOR and the TACAN bearing signal. True hyperbolic systems include DECCA, LORAN, and OMEGA (Kayton and Fried, 1969). In summary, multiple-path systems must use transmissions which differ in either time or frequency. a. Single-Path Systems. The simplest system in concept is the single-path,

one-way system; it employs a transmitter at one point and a receiver at


21

another. Transmission occurs over a single path between the two points. The delay in transmission over this path is the important element in securing the navigational parameter. The parameter which is delayed, for example, could be the phase of a subcarrier whose frequency is maintained very accurately. The receiver similarly employs an oscillator matched in phase to the transmitter subcarrier. The transmitter-receiver phase difference, therefore, is a measure of the time and, therefore, the distance taken by the wave to traverse the transmitter-to-receiver distance. Such systems are called coherent systems. They require very stable references such as atomic clocks and are not used in the navigation systems discussed in this article. A second type of single-path system employs a round-trip path as shown in Fig. 6. Energy from the transmitter illuminates a “reflector.” A reflector may be passive, such as an airplane hangar, or active, such as a radio beacon. The reflected energy is returned to the originating transmitter location where its delay (time of arrival) is compared with the initial reference time. Conventional radar applies the passive reflector concept, while DME and the ATC secondary radar are based on the beacon idea. The two-way, single-path system overcomes the costly reference problem associated with the one-way, single-path system because only one reference oscillator with greatly reduced frequency stability characteristics is necessary to achieve a coherent system. Thus, the stringent frequency stability requirements for coherent signal processing need be maintained only over the equivalent of several round-trip delay periods. Coherent signal processing is desirable when high accuracy performance is necessary. It permits both the amplitude and the phase information in the reflected echo to be used in determining the target range and velocity. However, in systems where the accuracy requirements are modest, such as the DME application, noncoherent signal processing is sufficient. Frequency stability requirements can then be reduced even further. Noncoherent systems measure only the time delay of the return signal’s pulse envelope, as illustrated in Fig. 6. In these systems the frequency spacing of the rf channel usually dictates the frequency stability requirements. As mentioned in Subsection B, the conventional airborne radar is not an effective navigation aid because it cannot easily identify ground reference points. Position-fix navigation using beacons can easily identify the ground reference landmark via the channel plan using a unique radio frequency and/or pulse code. Because radar-to-beacon and beacon-to-radar transmissions are each one way, the signal power varies as the inverse square of the range, rather than as the inverse fourth power, as do ordinary radar echoes. This means that the range of a radar beacon may be doubled by increasing the radar transmitter power or the receiver sensitivity fourfold, whereas a 16-fold increase would be

22


+ d = DISTANCE BETWEEN RADAR AND REFLECTOR c = VELOCITY OF LIGHT

At =

9 ;PULSE LEADING EDGE DELAY MEASUREMENT

TRANSMITTED PULSE ENVELOPE

ECHO OR

W E IS NON COHERENT RANGE SYSTEM (PULSE ENVELOPE)

COHERENT RADAR MEASURES BOTH AMPLITUDE AND PHASE TO OBTAIN TARGET RANGE AND RANGE RATE

MEASUREMENT

PHYSICAL REFLECTOR

RECEIVER

4 - - - - d d

-

MEASUREMENT RECEIVER

REFLECTOR (BEACON) TRANSMITTER

FIG.6. Two-way single path systems

required to double the range of ordinary echoes. Consequently, beacon systems are relatively modest in size and require less transmitter power than conventional radar. b. D M E : A Single-Path Two-way Navaid. For the above reasons, the DME is based upon the beacon principles. Ground transponders are used to mark and identify known positions on the ground so that the aircraft can determine its position relative to them. Such a system is cost-effective because the user requires only the purchase of the airborne unit, while the ground equipment is


23

purchased and maintained by local, state, or federal governments. The system design permits low-cost airborne equipment having small weight and size for two reasons. One, the codes and different radio frequencies permit omnidirectional antennas, and two, the transmitter power requirements have to overcome only the inverse square of the range propagation loss as noted above. System costs are further reduced because the two-way, single-path system does not require a coherent reference. That is, the operational accuracy (0.2 nautical miles for DME/N and 0.017 nmi for the DME/P) can be achieved with simple noncoherent time-of-arrival “leading edge” measurements as indicated in Fig. 6. In addition, since the system does not have to be coherent, the uplink radio frequency can be different than the downlink without degrading the system, thus permitting a flexible channel plan while contributing to the low cost of the airborne equipment. Finally, the beacon concept permits the use of an omnidirectional antenna on the ground, allowing 360” azimuth coverage in the horizontal plane with low transmitter power at a sufficient data rate. Aircraft can then obtain range guidance at all points necessary for enroute, terminal, and landing applications. c. Radial Navaids. Radial systems used in the ATC are radio navaids providing bearing information. They generate a zero delay time by having coincident paths (same ground antenna). This means that the two path signals must be coded differently: A reference signal is required to establish zero time delay while a second signal provides the variable delay. Figure 7 illustrates the principle of the radial system used in VOR and TACAN. The rotating cardioid pattern generates a sinusoidal signal, which is a delayed version of the reference signal. In order to separate the reference from the signal, it must vary in time or have a different subcarrier modulation f ~ r m a t . ~ Radial systems used in ATC system are VOR, TACAN (bearing), ILS, and MLS (angle).The VOR and TACAN derive bearing by measuring phase delay between the reference and the signal, which are diverse in subcarrier modulation formats. In MLS (angle) this bearing information is proportional to a time delay. In ILS the signals are diverse in frequency, and bearing is derived from the amplitude differences of the two signals.

d. Radio Navaid Position-Fix Conjigurations. Radio navaids whose locus of points define a circle are called p systems, while those whose locus of points define radials are called 8 systems. The four systems used in the ATC system are summarized in Fig. 8 in terms of the p/8 navigation parameters. Not In terms of performance,it is noteworth that two-way systems have their reference signal in the aircraft. The reference clock (delay) is “turned on” with the interrogation transmissionand is “turned off” with the reply signal. The elapsed time on the clock is proportional to the range. Unlike radial systems, the reference element in two-way range systems is not corrupted by propogation effects and receiver noise.

24

ROBERT J. KELLY AND DANNY R. CUSICK NORTH

GROUND STATION RADIATES BOTH REFERENCE AND ROTATING CARD10

CARDIOID ANTENNA PATTERN ROTATES AT FREQUENCY f

INFORMATION SIGNAL (CARDIOID)

FIG.7. Principle of radial navaid for VOR and TACAN.

shown in Fig. 8 is the VOR/DME configuration, which is not widely implemented in the enroute application. The ATC enroute and terminal navaids can determine position fixes by the intersections of radials and circles (see Fig. 9). In the 2 D navigation application, the slant range from the ground station (DME or VOR) is projected onto a plane tangent to the earth. This plane is illustrated later in Fig. 10. Thus the lines of position shown in Fig. 9 are assumed to also lie in the tangent plane. In ILS, the position fixes are determined by the intersection of a sphere (DME), a vertical plane along the runway center line (localizer),and a lateral plane perpendicular to the localizer plane (glide path). For MLS, the point in space is the intersection of two cones (azimuth and elevation scanning

25

DISTANCE MEASURING EQUIPMENT IN AVIATION AIR -

DMEIP IIA)

DMEIP

GROUND NOTE: NOT SHOWN IS A VORlDME CONFIGURATION

FIG.8. Air-derived p/B navaids in the ATC system.

beams) and a sphere (DME/P). Wax (1981) demonstrates how several different position-fix techniques can be integrated into an algorithm to estimate an aircraft’s position. All the navigation aids described above are air-derived systems; that is, the relevant navigation parameter is calculated in the aircraft. The Secondary Surveillance Radar is an example of a ground-derived system; that is, the relevant information is derived on the ground and then transmitted to the aircraft via a communications link. The global positioning system (GPS) is a new position-fix system which provides three-dimensional position and velocity information to users anywhere in the world. Position fixes are based upon the measurement of transit time of rf signals from four of a total constellation of 18 satellites (Milliken and Zoller, 1978). The exact role that GPS will play in the future NAS has not yet been defined. One of the tasks of RTCA Special Committee 159 is to define that role (RTCA, 198%). As stated by the Department of Defense/Department of Transportation (DOD/DOT) (1984), the DOD intends to phase out its Air Force TACAN systems beginning in 1987. They will be replaced with GPS. 3. Additional Navigation System Considerations

The purpose of this section is to complete the general discussion of navigation principles by describing how a navigation system is “connected” to the aircraft to form a closed-loop control system. The central notion represents

26


n VO R IDME OR TACAN

VOR/VOR

FIG.9. Position fixes determined by p / B intersection.

an aircraft in flight as a time-varying velocity vector. It has a direction in which it is going and a speed at which it is going there. The time integral of this velocity vector is its flight path. The navigation system’s role is to generate position and velocity corrections so that the aircraft’s velocity vector can be altered to follow the desired flight path in an accurate and timely manner. To continue the conceptual development of navigation principles further requires a coordinate system to reference the desired flight path and the navigation system’s output signal. Since the aircraft’s guidance system is useful only if the aircraft closely tracks the desired flight path, accuracy performance definitions are necessary to compare the AFCS’s tracking performance. This subsection briefly describes coordinate reference frames and accuracy considerations as they apply to a closed-loop navigation system. An aircraft in flight has a position and velocity at some altitude above the earth at some instant in time. In order to specify these aircraft states, a reference coordinate system needs to be defined. The subject of navigation coordinate frames is a vast and difficult topic and will not be pursued in any depth here. For example, seven reference systems, useful to navigation are defined in Chapter Two of Kayton and Fried (1969). One of these systems, geodetic spherical coordinates, has wide application in the mechanization of dead reckoning and radio navaid systems. It gives the aircraft lateral position in terms of longitude I and latitude CD on an ellipsoidal (earth), where I and @ are measured in degrees. The coordinate system is used

a. Coordinate System.


27

in current RNAV equipment with the waypoints also being defined in A and (D. The vertical dimension is given by altitude h above the reference ellipsoid. See Appendix D of RTCA (1982). Navigation system designers assume that the earth‘s surface can be approximated by an ellipsoid of rotation around the earth’s spin axis. In this representation it is postulated that, when the earth cooled from its molten mass, its surface assumed contours of equipotential gravity. The resulting gravitation field g is very nearly perpendicular to the earth’s surface. It is called the “apparent” gravity and is the vector difference between the Newtonian field of gravitational attraction and the centrifugal forces due to the earth‘s rotation. The reference ellipsoid is chosen to minimize the mean-square deviation between the normal to the ellipsoid and g. In order to keep the narrative simple and to focus on the essential ideas, the normal to the ellipsoid will be assumed to lie along g (the direction that a “plumb b o b falls). Use of the geodetic coordinates which are given in degrees allows the lateral position (say A. and (Do) of the aircraft above the earth to have the same coordinates (Ao, (Do) on the earth’s surface. The aircraft’s altitude is its height above the reference ellipsoid as measured along the ellipsoid’s normal (i.e., approximately along g). Enroute navaids provide guidance so that the pilot can fly between two waypoints specified in terms of A and (D. For example, assume that the waypoints were at the same height. Steering commands would be generated by the navigation system to guide the aircraft along the geodetic bearing and distance. The aircraft’s trajectory would trace an arc following the earth’s curvature at a constant attitude, as shown in Fig. 10(a). For short-range navigation (50 to 100 nmi for enroute and 20 nmi for approach and landing), the lateral position can be referenced to Cartesian coordinates instead of to the geodetic coordinates. As a result both the calculations and the geometric picture of the flight path are greatly simplified. This is accomplished by using a tangent plane to approximate locally the ellipsoid, as shown in Fig. lqa). It can be shown that, for short-range navigation, the length of the aircraft’s ground track on the earth’s surface will not deviate significantly from the length of the flight path’s projection onto a plane tangent to the earth’s surface near the derived waypoints. At a distance of 50 nmi from the tangent point the difference in length is about 16 ft. Measuring the aircraft’s height as the perpendicular distance to the tangent plane would, however, deviate significantly in an operational sense from that height measured along the normal to the earth’s surface. At 50 nmi the altitude difference is about 1300 ft, which is unacceptable when compared with the lo00 ft aircraft separation distance defined for the ATC route structure. Thus, for short-range enroute and terminal area navigation, lateral guidance can be defined with respect to the tangent plane. However, the


28

A/C FLYING CONSTANT ALTITUDE

\ _c--

TANGENT PLANE

9

(a)

AIRCRAFT CM

a

i 9

FIG.10. Earth reference coordinate system.


29

vertical guidance must be measured along the normal to the earth’s surface using, for example, a barometric altimeter. For enroute navigation it is the aircraft’s altitude directly above the earth‘s surface which is operationally important, not the aircraft’s height above the tangent plane reference point. In the final approach landing application the operationally significant coordinate system is referenced to the tangent plane which contains the runway. Thus the aircraft’s position can be given in xyz Cartesian coordinates. The decision height at the terminus of the final approach is measured with respect to the glide path intercept point (GPIP) on the runway (see Subsection D,4). The elevation angle glide path is chosen so that the aircraft will be safely above the hazardous obstacles along the approach path. This article, then, will define a very simple Cartesian coordinate system that permits description of the most essential concepts of navigation, guidance, and control. The Cartesian reference frame is defined at some point on the earth’s surface (e.g., airport runway). As shown in Fig. 10(b) this fixed Cartesian reference frame is called an earth-fixed reference. A tangent plane is defined through this point. Two of the Cartesian coordinates lie in the tangent plane, one of which lies parallel to the meridian (North). A third coordinate lies along the earth’s gravity vector and is normal to the tangent plane. It defines the aircraft altitude above the tangent plane. However, as long as the aircraft’s altitude is determined consistent with the enroute, terminal area, and approach/landing, then the earth-fixed reference will accurately describe the aircraft flight path. In other words, the tangent plane altitude calculation must be corrected for the earth’s curvature during enroute flight missions. The earth-fixed reference,just like the geodetic reference, permits the navigation problem to be separated into aircraft lateral positions and altitudes. Making a second assumption that the earth is fixed in space allows an inertial frame of reference to be defined on the earth’s surface; that is, unknown acceleration effects are neglected (McRuer et al., 1973). This inertial frame, which is accurate for relatively short-term navigation, guidance, and control analysis purposes, permits a description of aircraft motion in terms of inertial sensor^.^ It does have practical limitations for very long-term navigation (worldwide). With respect to this earth-fixed coordinate system, the aircraft’s center-ofmass (c.m.) can now be specified by a position vector with components (Xa,x,Za).In addition, three orthogonal axes exist within the aircraft. They are the pitch, roll, and heading axes of the airframe. The origin of this bodyfixed triad “sits” on the tip of the aircraft position vector, which is coincident with the aircraft’s c.m. Aircraft equipped with INS can establish a local vertical using Schuler tuning (Schneider, 1959), thereby forming the required inertial reference. Accurate position data can then be obtained from the platform stable accelerometers.

30


The attitude of the aircraft is defined by the Euler angles (a, /?,and y ) through which the earth-fixed axis must be rotated to be coincident with the airframe body-fix axis. Understanding aircraft navigation techniques cannot be confined to just the motion of the aircraft’s c.m. Attitude must also be considered, because the flight path traced out by the motion of the aircraft c.m. is controlled by changes in the aircraft’s heading, roll, and pitch. The navaid guidance signal is also referenced to the earth-fixed system so that the steering commands will be in the correct form to modify the aircraft’s velocity vector when necessary. Subsection E will show that the AFCS for conventional fixed-wing aircraft has two orthogonal channels: a lateral channel and a longitudinal (vertical) channel. Since the earth-fixed reference frame is an orthogonal system, as described above, the lateral navigation problem can be separated from the vertical navigation problem as a consequence of this simplification. This is an important distinction because later it will be shown that the lateral enroute navigation control laws are similar in form to the lateral (azimuth) approach/landing control laws. With this simplifying division of the navigation problem, the narrative and the examples given in the balance of this article will be selected with respect to the lateral navigation application, wherein the aircraft’s altitude is assumed to be constant. When the aircraft’s altitude and TRUE N O R T H

4

= HEADING

e= BEARING SIDE SLIP ANGLE = 0

FIG.11. Definition of true track angle during coordinated flight.


31

time are included as steering commands, the navigation problem becomes a 3D or a 4D problem, respectively. The following terms-bearing, heading, ground track velocity, and track angle-are important for the understanding of lateral navigation discussions. As shown in Fig. 11,the bearing 8 is the angular position of the aircraft c.m. to its intended destination as measured clockwise from a reference such as true North. Track angle tjT is the true angular direction of the aircraft’s ground velocity. The heading tj is the angular direction that the longitudinal axis of the aircraft is pointing. Air speed is the speed of the aircraft in the direction of its heading through the air mass. From these definitions, three distinctions in the aircraft flight path can be given. If range to a destination can also be determined, then a desired flight path (course) can be defined as a connected time series of intended bearing and range points. The aircraft’s actual flight path would be a connected time series of true track angles and true ranges, while the indicated flight path is a time series of range and track angle measurements. As part of a recurring theme in this article, track angle or ground velocity is the critical parameter for enroute navigation. When accurate navigation is necessary, it can be determined from a series of position measurements or it can be derived from accelerometers on board the a i r ~ r a f t Guidance .~ and control are simply the corrections of the velocity vector in such a manner that the aircraft follows its intended flight path. Clearly, if air speed and heading are known, then, as indicated in Fig. 11, a simple but less accurate method is available; one uses the wind vector and the air speed vector to obtain the ground speed velocity.

b. The Aircraft Flight Control/Guidance System. The task of a navigation system is not only to derive the aircraft’s position, but also to generate commands to change its velocity vector. In order to accomplish these changes, its output signals are fed into the aircraft flight control system, thereby forming a “closed loop” composed of the aircraft and the navigation system. This “closed loop” is called the aircraft flight control/guidance system. If the navigation system correction signals are sent to a display, then the loop is

’

The central concept of INS consists of a platform suspended by a gimbal structure that allows three degrees of rotational freedom (Maybeck, 1979).The outermost gimbal is attached to the body of the aircraft so that the aircraft can undergo any change in angular orientation while maintaining the platform fixed with respect to, for example, the earth-fixed coordinate frame. Gyros on the platform maintain a desired platform orientation regardless of the orientation of the outermost gimbal. Thus, the platform remains aligned with respect to a known reference coordinate system. Accelerometers on this stabilized platform can then provide aircraft acceleration with respect to that known set of reference coordinates. After local gravity is subtracted, the resulting signals can be integrated to yield aircraft velocity and acceleration. An alternative to the gimbaled implementation is the strap-down inertial system (Huddle, 1983).

32


-

(a) MANUAL-COUPLED FLIGHT

-

1

AIRCRAFT POSIT1ON

AIRCRAFT A

COURSE DEVIATIONS ~

NAVIGATION SYSTEM

.

DESIRED FLIGHT PATH

NAVAID SENSOR ENROUTE DME VOR APPROACH & LANDING I LS M LS

(b) AUTOCOUPLED FLIGHT AIRCRAFT POSITION STEERING

DESIRED FLIGHT PATH

FIG.12. Principal aircraft flight control/guidancesystems.

closed through the eyes and hands of the pilot, as shown in Fig. 12(a). When the pilot closes the loop, the configuration is called a manual-coupled flight control system (FCS). Similarly, if the guidance signal is sent to an autopilot, then the configuration is called an auto-coupled FCS, as shown in Fig. 12(b). Guidance then involves the instrumentation and control of the six aircraft coordinates. For the 2D problem, the guidance system provides corrections to the aircraft’s velocity vector by changing its bank angle and heading, such that the aircraft’s c.m. follows the desired 2D flight path. The aircraft’s c.m. velocity is called the ground track velocity. When altitude corrections are included, the aircraft velocity is a 3D vector composed of the ground track velocity plus its vertical speed. The navigation system outputs for the autocoupled 2D enroute application are lateral steering commands. For manual-coupled flight, the visual display will present bearing and range guidance. To maintain the intended


33

flight path the guidance corrections are displayed as (1) range along the straight-line segments between selected waypoints, and (2) bearing or crosscourse deviations (left/right) normal to the straight-line segments. The guidance correction signals for autocoupled flight are steering commands expressed in terms of cross-course deviations. The cross-course corrections are coupled to the lateral channel of the autopilot. Some aircraft are instrumented to adjust their power (thrust) to maintain a range schedule along the flight path. For the straight-in final approach and landing application, the aircraft velocity vector lies in and points down an inclined plane called the glide path. The navigation system, or more accurately, the landing guidance system, shown in Fig. 12, is now an ILS or MLS. Both use DME and both are described in Subsection D,4. For manual flight on final approach, landing guidance is displayed as left/right indications on the course deviation indicator (CDI),which are crosscourse deviations about the extended runway center line. Vertical deviations about the inclined plane are displayed on up/down indications on the CDI. The CDI is also used in enroute applications. Lateral steering corrections in autocoupled flights are sent to the lateral channel of the autopilot, and vertical corrections are sent to the autopilot’s longitudinal or vertical channel. Fixed-wing, conventional take-off and land aircraft will be considered in the closed-loop aircraft/flight control system discussions and analyses. The other major aircraft types, short takeoff and land (STOL) and vertical takeoff and land (VTOL), will not be considered. However, the ATC navaids summarized in Subsection D (especially MLS and its DME/P) can and do provide navigation guidance to both the STOL and VTOL aircraft types. c. Navaid Accuracy Considerations. In navigation, the accuracy of an estimated or measured position of an aircraft at a given time is the degree of conformance of that position with the true position of the aircraft at that time. Because of its statistical basis, a complete discussion of accuracy must include the following summary of the statistical assumption made and the measuring techniques used. A navigation system’s performance must be evaluated in the context of the total system, aircraft plus navigation system. Several measures of a navigation system’s performance require that the aircraft flight path remain inside its designated air lanes, and that the pilotage error of the aircraft about the indicated navigation signal as well as its attitude be maintained within prescribed limits. Keeping the aircraft within its air lane is directly related to the navigation system’s bias accuracy; while maintaining a small tracking error and prescribed attitude are direct functions of the closed-loop systems’ bandwidth, autopilot gain, and the navigation system’s noise error. Understanding the performance of the closed-loop guidance system requires an

34


understanding of the navigation system’s error mechanisms. The navigation system, in turn, is a configuration of navaid sensors, a computer, and output signals that drive an autopilot or a cockpit display, as shown in Fig. 4. Determining the performance of the navigation system, therefore, also entails determining the performance of the navaid. This section discusses the necessary considerations. The navaid sensor measures some physical phenomena such as time delay of the electromagnetic radiation. As indicated in Fig. 4, the sensor measurements are then transformed to a coordinate system where they are combined by the navigation system to estimate the aircraft’s position and velocity vector. The aircraft’s position, as determined by the navigation system, will differ from its true position due to errors in the sensor measurements, assumptions in the earth coordinate system reference model, errors in computation, and, finally, departures in the mathematical model implemented and the real world. This difference is called the RNAV system error, as shown in Fig. 13. The figure also indicates an additional error, namely the flight technical error, which combines with the previously addressed Navaid system error to produce the total system error or tracking error. The flight technical error (or autopilot error for autocoupled flight) is the difference between the indicated flight path and the desired flight path. The “flight technical error” (FTE) refers to the accuracy with which the pilot controls the aircraft as measured by his success in causing the indicated aircraft position to match the indicated command or desired position on the display. FTE error will vary widely, depending on such factors as pilot experience, pilot workload, fatigue, and motivation. Blunder errors are gross errors in human judgment or attentiveness that cause the pilot to stray significantly from his navigation flight plan, and are not included in the navigation system error budget. Blunder tendency is, however, an important system design consideration; their effects are monitored by the ATC AIC TRUE /FLIGHT PATH

A/C TRUE

/ POSITION

ERROR

ERROR

OESl RE D COURSE

RNAV ROUTE WIDTH

RNAV FLIGHT PATH

WHERE RNAV SAYS A/C IS

FLIGHT TECHNICAL ERROR

DESIRE0 A/C POSIT ION

FIG. 13. RNAV position error.


35

controller. Roscoe (1973) discusses flight test results which attempted to measure flight technical errors and blunder distributions. To complete the picture, the total system error (tracking error) consists of the navigation system error and the flight technical error; it is the difference between the aircraft’s actual position and the desired flight path. An example of a navigation system error budget is given in Table V and illustrated in Fig. 40 in Subsection F. A navigation system can be the sophisticated configuration of sensors, as shown in Fig. 4, or it can be simply the navaid measurements themselves, as in VOR/DME, where bearing and range are directly displayed to the pilot. As discussed previously, the intended aircraft’s position (c.m.)as well as its measured position are determined relative to the earth-fixed coordinate system. The position errors can be defined in at least two differentways: They can be referenced directly to the earth-fixed coordinates or they can be referenced to the aircraft’s intended flight path, which, in turn, is referenced to the earth-fixed system. To keep the discussion focused on the important concepts, only the two-dimensional plane used in lateral navigation will be treated. (The vertical component, altitude, is simply a one-dimensional measurement, orthogonal to the lateral plane.) In the first method, the error components are traditionally defined in terms of a location error (bias) and a dispersion error (variance). When the error components are defined along the coordinates of a Cartesian system, the position error dispersion (variance) is given in terms of the circular error probability (CEP). The CEP is the radius of a circle such that the probability is 0.5 that an indicated position will lie within the circle (Burt, et al., 1965). The center of the circle is chosen at the center of mass of the probability distribution, which is the desired destination point of the aircraft. For a biased system, the center of the circle will be displaced from the aircraft’s desired destination. The second method measures 2D navigation system performance in terms of its error about the intended flight path. It is the method used in this article to evaluate the accuracy performance of navigation systems. As shown in Fig. 13, the error normal to the indicated flight path is called the cross-track (XTRK) error, while the error along the indicated flight path is called the along-track (ATRK) error. These error components are thus total system errors or tracking errors. Referencing the aircraft’s true position against the desired flight path is the natural method of specifyingnavigation errors. The cross-track error must be consistent with the width of the enroute air lanes or RNAV routes, while the along-track error must not degrade the aircraft longitudinal separation criteria from other aircraft. This cross-track/along-track theme is also present in the straight-in approach and landing application where, for example, the

36


cross-track errors are the lateral azimuth errors. In this case, the allowed errors must be consistent with the runway width, while the along-track errors must be consistent with the runway touchdown zone. In the simple VOR/DME navigation application, the VOR error is the cross-track error, while the DME error is the along-track error. The aircraft path deviations-cross-track (lateral), along-track, and vertical-are assumed to be orthogonal random variables. Since most of the operational accuracy questions in this article address only one of these components, the statistical problem is one dimensional. For example, the 2D navigation application usually addresses only the cross-track corrections. Therefore, the 95% confidence limits are one-dimensional statistics using the variance and probability distribution of the cross-track deviation measurement. On the other hand, when analyzing a 3D navigation flight path, additional considerations arise because the error surface can be envisioned as a tube in space. In this case, the 95% probability error contour corresponding to the tube’s cross section is determined using a bivariate distribution of the lateral and vertical flight path deviations. [See Table 2.1 in Mertikal et al. (198511. To summarize the tracking error or total system accuracy is a combination of the autopilot accuracy and the navigation system accuracy. The accuracy performance of navigation systems is specified in terms of 95% probability. This means that for a single trip along a defined flight path (e.g., between two waypoints) 95% of all position measurements must lie within the specified XTRK error allowance. This implies that out of 100 flights, 95 of them must satisfy the specified XTRK error allowance. In other words, the measurement defines a confidence interval and its confidence limits are compared to the accuracy standard. It is a confidence interval because the desired flight path is an unknown constant parameter which the navaid is estimating. The computed 95% confidence interval is a random variable which covers the desired flight path 95 out of a hundred times. The confidence interval is used to evaluate one-dimensional guidance signals (e.g., XTRK deviation). Discussion will now center on the navaid error components which are a subset of the navigation system components. The navaid sensor measurement differs from the true navigation parameter due to variations to the phenomena (e.g., time delay errors induced by multipath) and due to errors in measurement and computations (instrumentation errors).Navaid errors fall in three separate categories: outliers, systematic, and random. Outliers are the results of miscalculations, electrical power transients, etc., which can be detected by comparing a new measurement with respect to the three or four standard deviation values of past position estimates (see Section IV,D). Systematic errors obey some deterministic relation, such as a circuit temperature dependency, or are the result of faulty calibration procedures. They


31

usually can be removed when the deterministic relation is known or the calibration procedure is corrected. In other words, if an ensemble of navigation instruments were known to produce nonzero average errors, simple recalibration or computation could nullify these effects. No generality is lost, therefore, in assuming zero mean ensemble errors. Random errors are unpredictable in magnitude and they are best described by statistical methods. It is these errors which are addressed by the navaid accuracy performance standards. Other accuracy types, such as predictable, repeatable, and relative accuracy are defined in DOD/DOT (1984). Predictable and repeatable accuracies are included in the error components described next and in Subsection F. Relative accuracy does not apply to the applications addressed in this article. In summary,it is assumed that both the outliers and systematicerrors have been removed and that only the random errors remain. This means that although any particular navaid can have a bias AxB,the average bias of the ensemble of navaids is zero, i.e., E[AxB] = 0, where E is the expectation operator. Further, each navaid has an equipment variance (noise)component a; that is assumed to be identical for each member of the class of navaids. Therefore, as with the navigation system as a whole, the navaid errors may be specified in terms of bias and noise (variance). The assumptions above are important, and they are discussed further in the next section. Once equipment errors have been accounted for, the only remaining major error component is site-dependent errors such as multipath. Unlike equipment noise errors, which vary with time but are essentially independent of aircraft motion or position, site-dependent errors remain unchanging for long periods of time, while exhibiting extreme variations as the aircraft changes position. Thus site-dependent errors are said to be spatially distributed. A time-varying error is generated when the aircraft flies through the spatially distributed interference fields. It is implicitly assumed that the variance component associated with site-dependent errors and with aircraft motion is detected by all radio navaids having simple signal processing algorithms. The degree to which this variance component is reduced depends upon the level of output data filtering applied.

d. Error Budget Considerations. The purpose of an error budget is to allow facility planning by local, state, and federal governments, including the obstacle clearance needed to define enroute airways, terminal approaches, and runway clearance areas. They are also used to prepare equipment procurement specifications.Error budgets must be simple if they are to be useful. Also they must be simple because there usually is no database to substantiate anything more than simple statistical procedures. It is for this reason that the root-sum-square (RSS) calculation procedure is used extensively throughout

38


the navaid industry to estimate system performance. The purpose of this section is to identify the assumptions upon which RSS calculations are valid. Even though these assumptions may only be partially justified, the RSS procedure is applied anyway because there is no other alternative short of directly adding the maximum excursion of the error components. Direct addition of the error components is not statistically or economically defensible. The true navaid system accuracy lies somewhere between these two calculation procedures. The problem is how to use ensemble statistics which were obtained by averaging over all terminal areas, equipment classes, and environmental conditions to predict the performance of a given aircraft on a single mission to a given runway. An estimate of the total system error is the mean square error (MSE) or second moment. If uncorrelated ensemble statistics can be used and the ensemble average of each error component is zero, then the value of each component in the error budget (including the bias errors) can be given by their respective variance. Under these conditions, the MSE can be easily calculated by simply summing the variances and extracting the positive square root. The MSE is thus the variance for the total system. In the literature, this calculation is known as the root-sum-square. The difficulty with the RSS approach is that the random variables which represent the slowly changing drift errors may be correlated during flight missions to the same airport. To understand the problem, consider the error due to temperature effects. The equipment design engineer can determine the maximum temperature drift errors from the specified operating temperature range. Civil navigation equipments are usually designed to operate over temperature ranges between 70 and - 50 "C. These excursions usually have daily and seasonal cycles whose average values would be nearly zero when the equipments are calibrated for the average temperature of a geographic area. This means that, although the drift errors are a nonstationary process, they will have a near zero average over a long time interval. This point can be used to advantage because it implies that repeated flights by a given aircraft to the same runway (for example) will at worst only be partially correlated. In any event, the ensemble average over all conditions and missions will be nearly zero. Although the value of the ensemble variance is less than the peak drift error, it is common practice to use the peak drift value for the drift error variance component. Thus the error budgets for some geographic locations may be pessimistic when the ensemble variance of the temperature drift errors are added with the other error component variances. There is generally no statistical correlation computation problem with the noiselike errors. Physically the bias error component and the noise errors are associated with separate error source mechanisms. They can be viewed in the frequency


39

domain as separate spectral components and thus as separate random variables. The low-frequency component when grouped together are called the bias error, while the sum of the high-frequency components is called the noise error. Both spectral groups, bias and noise, are defined as random variables having zero means and variances equal to otiasand 0.; In principle, each flight mission can be considered as a statistical experiment where each biaslike error component AxB is selected from separate batches, each having a zero ensemble mean E [ A x B ]and variance otias.Before each flight mission, one error component is drawn from each batch and is directly added with samples from the noise component (a;) batches. The result should, with high probability, be within the total specified system error allowance. In the next section the discussions above will be further refined to include quantitative definitions of the time average, ensemble averages, and the RSS calculations. e. Statistical Dejnitions and Root-Sum-Square Applications. Navaid output data can be viewed as a combination of a static or quasi-time-invariant component (bias) and a dynamic or fluctuating component (variance). The dynamic component or noise term can be further partitioned into spectral components, corresponding to errors, which (1) aircraft can follow and are called scalloping course errors or “bends”; (2) can affect the aircraft’s attitude; or (3) have no effect on the AFCS. (The degree of data filtering performed determines whether the noise component of a navaid disturbs the aircraft position or its attitude or both.) DME/P and MLS angle guidance systems define the noise error components in terms of these spectral components. ILS defines the course errors in terms of bias and noise with no distinction made between the spectral groups comprising the noise. Most navaids, such as VDR, DME/N, and TACAN, relax the definition error further and make no distinction between bias and noise. For these navaids the accuracy is described by a single value equivalent to the mean square error. The MSE describes in a rudimentary way the general intensity of random data. Before defining MSE, two types of averaging processes, time averages and ensemble averages, will be defined. The accuracy performance of a class of navaids is determined by first calculating the time average on one member of the class. To do this M time sample measurements x ( t i )are taken over a short period of 10 to 40 seconds, where i = 1, 2,. . .,M. From these measurements, M error terms A x ( t i )are calculated. Using the M error terms, the time-average sample error AX^, the time sample variance S i , and the time sample mean square error

40


are determined. Figure 14(a) illustrates these concepts for a series of range measurements from a s i n g l a M E interrogator/transponder combination. The positive square E is called the RMS error (Bendat and Piersol, 1971) and is = ,/when the measurements are uncorrelated. To obtain the ensemble average of the short interval time averages, time measurements are repeated on (N - 1)additional units. The ensemble average is then determined from the N sets of measurements. Let

where j is the jth navaid and the bar indicates the ensemble average calculations. The sample variance of the bias measurements is I

N

In this discussion, both N and M are sufficiently large that the sampling errors are negligibleand thus represent the population statistics for a given type of 2 navaid, i.e., AxB N E[AxBj], S;,, 2: dbiasr S i s' o i , and MSE = E[Axs]. Further, it is assumed that all the bias measurements are uncorrelated, i.e., E[AXB~,AXB~] = E[AXBj12. AS assumed above, E[AxBj] = 0 and the noise component of each navaid in the ensemble is equal, that is, a i j = dj?,k = o i . Therefore, with these assumptions and as shown in Fig. 14(b), the ensemble mean-square error is simply MSE = nj?,+ otias.Its positive square root is defined as the root sum square, RSS = ,/=. The distinction in this article between RMS and RSS is that when E[AxBj] = 0, the RSS is only a sum of variances. With the above-defined measurement procedures, it is now possible to illustrate how both a navaid error budget and a navigation system error budget are constructed using the RSS procedures. Let A X AR, and A F be different error components of a navigation system such as RNAV. The XTRK component of the sensor error could be represented by AYwhile AR and A F could represent the RNAV computation error and the XTRK flight technical error, respectively. Assuming a linear model, the system error AS is given by AS = A Y + AR + AF. Assume further that these error components are zeromean, uncorrelated random variables; then the ensemble mean square is

BIAS

NOISE

which itself is the standard The total system error is simply the RSS = deviation for the total system as indicated in Fig. 14(b).

41

DISTANCE MEASURING EQUIPMENT IN AVIATION X AXIS PROBABILITY DENSITY FUNCTION

b

DME MEASUREMENTS

ARIANCE (NOISE)

DME GROUND STATION

@

2 r

Y AXIS MEASUREMENTS ON A SINGLE INTERROGATOR /TRANSPON DER COMBINATION

(a) TIME AVERAGE MEASUREMENT

E [ A X j ] = O MSE = E [ A X j

=

1.2 '

' ' ' '

N NAVAID UNITS

DME ENSEMBLE BIAS ERRORS

j

=

f]=

UN2+ UBfAs

1, 2 ' . ' ' . N NAVAID UNITS

RSS = DME ENSEMBLE MEAN SOUARE ERRORS

(b) ENSEMBLE AVERAGE MEASUREMENT

FIG.14. Range error definitions.

42


For the DME/N, VOR, and TACAN area navigation systems, the total system error is the single number, RSS = (a: + ... + a;)”’, where a:, a:, . ..,a: are the individual variances for each of the P error components. In ILS system error analysis, the RSS of the bias variance components are calculated separately from the RSS of the noise variance components, as indicated in Table 111. For DME/P and the MLS angle guidance, the error components are defined in Sections E,7 and IV,C,1 and in Table 111. Combining error components on an RSS basis is almost universally adopted in the literature (RTCA, 1984, 1985a; DOD/DOT, 1984). They implicitly assume that the error components satisfy a linear model and that they are uncorrelated, zero-mean, random variables. The linear model assumption further implies that there is no coupling between the navigation system error (NSE) and the flight technical error (FTE). In fact, there is a small nonlinearity between these terms because of the NSE changes with aircraft position, which, in turn, is a function of the FTE. This nonlinearity is negligible, and the NSE and FTE can be combined on an RSS basis if the pilot has no a priori knowledge of the NSE (e.g., severe VOR course bends). The validity of the uncorrelated error component assumption depends upon several factors. Clearly the error components are uncorrelated over the ensemble of all flight routes. For repeated flights over the same route the sitedependent errors may be correlated while the equipment bias errors depend upon the nature of the drift mechanism. Bias errors are assumed to be constant over the duration of a single mission, however. In any event for a given facility installation the site and equipment are selected and configured such that the system accuracy for a single mission is achieved. In addition, the FAA periodically has flight inspections to check a facility’s performance. Since the exceptional sites are treated on an individual basis to ensure flight safety, and since the statistical distributions are not known anyway, the uncorrelated assumption is retained because of the simplicity of the RSS calculation. Simple addition of the error components would unrealistically limit the approach path and skew the facility implementation economics. Error budgets are defined in terms of 95% confidence limits, rather than 20 limits, because the two specifications are essentially equivalent only for Gaussian random processes. The probability that a given event will occur is the important quantity not the 20 value. The probability of exceeding specified limits is the only meaningful measure by which air-lane route widths, decision windows, touchdown foot prints, and obstacle clearance surfaces can be operationally defined. Although each error component can be given in terms of a sample time variance, S i , the calculation of the 95% confidence interval requires knowledge of the underlying probability distribution function which is, in general, not known. In practice this lack of knowledge is circumvented by


43

estimating the confidence intervals using the 2.5 and 97.5 percentile limits as determined by the measured data. The overall system error is then determined by the RSS of the 95 percentile value of each error component. The confidence limits of the combined measured error samples is then compared to the appropriate accuracy standard. Figure 83 illustrates the measurement methodology for the DME/P. In particular, it shows how the 95% confidence limits are determined from the system flight test results and the individual navaid component test results. Caution should be exercised when invoking the central limit theorem to approximate the total navigation system error distribution. [See Fig. 2 in Hsu (1979).] Based upon the above definition, it should be emphasized that the term “bias with 95% probability” means that, over the ensemble of navaids, 95% of the equipment bias measurements lie between the 2.5 and 97.5 percentile limits. The specified bias error value is the upper and lower percentile limits, as noted in Fig. 14. Figures 43 and 44 in Subsection F show how errors are combined to yield position-fix RSS accuracy estimates. All error components defined in Sections I11 and IV of this article are 95% confidence level specifications. One operational basis for using the single RSS value is that guidance obtained from enroute navigation sensors for manual flight is usually filtered extensively by the flight control system. Errors varying at a very high frequency are readily filtered out in the aircraft equipment, leaving only lowfrequency error components. Since this error component can be tracked by the aircraft, it can, when combined with the bias component, displace the aircraft from its intended flight path. Thus, the single RSS value is a useful measure of how well the aircraft will remain in its designated air lane. This rationale does not apply to autocoupled approaches. Biaslike errors again must be controlled so that the aircraft will remain in its air lane. The variance of the noise errors cannot always be reduced by heavy filtering if certain AFCS stability margins are to be maintained. Therefore, the unfiltered portion of the noise may induce aircraft pitch and roll motions (control activity) that may be unacceptable to the pilot (see Subsection E). As noted earlier, the single RSS specification is not adequate for the approach and landing application. Two numbers are defined for the ILS accuracy standard, the mean course error, crbias, and the “beam bends” noise, crN. The noise term is specified separately to limit the control activity and to limit, for example, lateral velocity errors away from runway centerline. (Note that an ILS localizer or MLS azimuth bias error always directs the aircraft toward the runway centerline, whereas a low-frequency noise error component may direct the aircraft away from runway centerline.) With the advent of the MLS, accuracy standards made a significant departure by expanding the traditional notions of bias and noise to include the spectral content of the navigation signal. With these new definitions it is

44


intended that the MLS angle and DME/P accuracy standards will more accurately represent the qualities needed to achieve successful landings. The lateral error components for the enroute, terminal, and landing phases of flight are summarized in Table 111, Subsection D,2. The table indicates the increasing level of sophistication in the error component definitions. Clearly, the magnitude, nature, spectral content, and distribution of errors as a function of time, terrain, aircraft type, aircraft maneuvers, and other factors must be considered. The evaluation of errors is a complex process, and the comparison of systems based upon a single error number could sometimes be misleading. As stated earlier, the purpose of an error budget is to estimate the accuracy performance of a proposed navigation system hardware (and software) implementation. The idea is to select the 95% confidence interval for each component of the navigation system, such that the RSS of the total combination is within the 95% probability error limits specified for a given application. If the assumed conditions are correct, then there is high confidence that 95% of the aircraft’s position measurements will be within the allowed error limits during a specified segment of the aircraft’s Bight mission.

4 . Summary The basic notions of a navigation system were defined with particular emphasis on the DME navaid. DME, a position-fix navaid, was contrasted with dead reckoning techniques. DME can be used in a navigation system in two ways. If it is the primary method of navigation, then its data may require inertial aiding so that its guidance corrections do not affect the AFCS dynamic response. However, if dead reckoning is the primary navigation aid, then DME may assist it by periodically correcting its drift errors. It was shown that the navigation parameters follow a common theme whether the application is enroute navigation or aircraft approach and landing guidance. The desired flight path and indicated flight path coordinates can be defined for both applications with DME playing a central role. Error components for the approach and landing application are smaller and are more refined than in the enroute navigation application. D. The Air TrafJic Control System and Its Navigational Aids

This section describes the principal elements of the Air Traffic Control System, thereby helping to illuminate the operational role of DME in the context of the enroute, approach, and landing phases of flight.


45

1. Goals of the National Airspace System

The National Airspace System (NAS) is a large and complex network of airports, airways, and air traffic control facilities that exists to support the commercial,private, and military use of aircraft in the United States [Office of Technology Assessment (OTA), 19823. The NAS is designed and operated to accomplish three goals with respect to civil aviation: (1) safety of flight; (2) expeditious movement of aircraft; and (3) efficient operation.

These goals are related hierarchically with safety of flight as the primary concern. The use of airport facilities, the design and operation of the ATC system, the flight rules and procedures employed, and the conduct of operations are all guided by the principle that safety is the first consideration. In the U.S., the FAA has defined a single nationwide airway system shared by all users of the NAS based on strategically located elements of VOR, DME, and TACAN navaids (Fig. 15). These facilities are geographically placed so as to provide for continuous navigation information along the defined airways. The airways serve as standard routes to and from airports throughout the U.S. DME provides the means to determine the position of the aircraft along a given route and to aid in verifying arrival at intersecting airways. The airways are divided into two flight levels. Below 18,000 feet mean sea level (MSL), the airways that run from station to station make up the system of VOR, or “VICTOR,” airways. These airways can be successfully navigated utilizing VOR bearing information only. The use of DME is not required but

FIG.15. Depiction of how air routes are constructed from VOR (and TACAN) defined radials.

46


I

FIG.16. Excerpt from low-altitude enroute chart depicting the VICTOR airway system in Southern Virginia.

certainly simplifies the process of position determination along an airway. VICTOR airways are depicted on low-altitude navigation charts (Fig. 16), marked with a letter V and a number, e.g., V23. VOR stations in these airways are located an average of 70 miles apart. The high-altitude routes, or jet routes, are defined for navigation at altitudes of 18,000 to 45,000 ft and are utilized by high-performance turbine powered aircraft. These routes are identified on high-altitude navigation charts by the letter J followed by a number, e.g., J105. Unlike the VICTOR airways, DME, or its equivalent, is required for navigation along the jet routes. In addition to the VICTOR and JET airways the FAA (Federal Aviation Administration, 1983d) recently authorized random route using area navigation (RNAV). VOR, DME, and TACAN derived position information is also utilized for terminal area navigation (Fig. 17).


47

VORlDME or TACAN 1 RWY 14

FIG. 17. A terminal instrument approach procedure utilizing a DME defined arc with position fixes along the arc defined by VOR radial intersections. Missed approach procedure (to BOAST intersection)is defined by a DME arc.

The FAA provides planning and advisory services to guide the aviator in making use of the NAS under either of two basic sets of rules-visual flight rules (VFR) and instrument flight rules (1FR)-which govern the movement of all aircraft in the United States. Similar visual and instrument flight rules are in force in foreign countries that are members of ICAO. In many cases, ICAO rules are patterned on the US.model. In general, a pilot choosing to fly VFR may navigate by any means available to him; visible landmarks, dead reckoning, electronic aids such as the VORTAC system, or self-contained systems on board the aircraft. If he intends to fly at altitudes below 18,000 ft, he need not file a flight plan or follow prescribed VOR airways, although many pilots do both for reasons of convenience and safety. The basic responsibility for avoiding other aircraft rests with the pilot, who must rely on visual observation and alertness-the “see-and-avoid” principle.

48


In conditions of poor visibility or at altitudes above 18,000 ft, pilots must fly under IFR. Many also choose to fly IFR in good visibility because they feel it affords a higher level of safety and access to a wider range of ATC services. Under IFR, the pilot navigates the aircraft by referring to cockpit instruments (e.g., VOR and DME for enroute flights) and by following instructions from air traffic controllers. The pilot is still responsible for seeing and avoiding VFR traffic when visibility permits, but the ATC system will provide separation assurance from other IFR aircraft and, to the extent practical, alert the IFR pilot to threatening VFR aircraft. The distinction between VFR and IFR is basic to ATC and to the safe and efficient use of airspace since it not only defines the services provided to airmen, but also structures the airspace according to pilot qualifications and the equipment the aircraft must carry. Much of the airspace below 18,000 ft is controlled, but both VFR and IFR flights are permitted. The altitudes between 18,000 and 60,000ft are designated as positive control airspace; flights at these levels must have an approved IFR flight plan and be under control of an ATC facility. Moreover, the aircraft must be equipped with VOR and DME in order to fly the VORTAC routes or other approved navigation aids if using RNAV. Airspace above 60,000 ft is rarely used by any but military aircraft. The airspace around and above the busiest airports is designated as a terminal control area (TCA), in which only transponder-equipped aircraft with specific clearances may operate regardless of whether using VFR or IFR. All airports with towers have controlled airspace to regulate traffic movement. At small airports without towers, all aircraft operate by the see-and-avoid principle, except under instrument meteorological conditions. Figure 18 is a schematic representation of the U.S. airspace structure. Aircraft flying under IFR, on the other hand, are required to have a radio and avionics equipment allowing them to communicate with all ATC facilities that will handle the flight from origin to destination. Figure 19 illustrates a hypothetical VORTAC and RNAV route from takeoff to touchdown. 2. The ATC System

The third major part of the National Airspace System offers three basic forms of service: navigation aids (including landing), flight planning and inflight advisory information, and air traffic control. The essential feature of air traffic control service to airspace users is separation from other aircraft. The need for this service derives from the simple fact that under IFR conditions the pilot may not be able to see other aircraft in the surrounding airspace and will therefore need assistance to maintain safe separation. Figure 20 represents the four functional elements of the ICAO

z=


4

60,OOOFT.

A

(FL600)

:ONTINENTAL CONTROL AREA

t

49

45,000 FT. (FL450)

TRANSPONDER WITH ALTITUDE ENCODING PoS'TIVE REQUIRED

18,000 FT. MSL

14.500 FT MSL

FIG.18. (FAA).

Airspace structure. AGL,above ground level; MSL, mean sea level; FL,flight level

standardized air traffic control system: radar, VHF/UHF communications, enroute navigational aids, and landing aids. The ATC navigation service is, of course, a subject of interest in this article because it intimately involves DME. In the U.S. the Department of Transportation (DOT), through the Federal Aviation Administration (FAA), operates these four systems plus a primary radar system to fulfill its statutory responsibilities for airspace management. Controllers use these systems to provide navigation services, separation between aircraft, and ground-proximity warnings. Pilots require these systems for navigation and to receive ATC services. Specifications for these systems

50


ZNM SECONDARY AREA

4NM

I

t

IAWP VORIDME INWP

FAWP

IAWP = I N I T I A L APF’ROACH WAVPOINT INWP = INTERMEDIATE APPROACH WAVPOlNl FAWP = F I X E D APPROACH WAYPOINT MAWP = MISSED APPROACH WAVPOINT

0

RUNWAY THRESHOLD

_ _ ---_ I ,

DEPARTURE AIRPORT

.C..I.....^. AZIDMEIP

AREA

AREA

I AZIDMEIP

FIG. 19. Hypothetical VORTAC and RNAV f l i g h t paths and route structure.

such as frequency and power output are determined by both national and international standards. These “service volume” standards are absolutely necessary. Unacceptable interference in the air-ground or ground-air links of these systems could deny or distort information essential to the controller or pilot in providing or receiving safe ATC service. Sections III,C,2 and 3 describe the procedure for ensuring interference-free DME service volume. The systems presently in use for ATC are: (1) A means for the controller to detect and identify aircraft using the primary or secondary radar;


51

OTHER AIRCRAI

USED BY CONTROLLER

FIG.20. The ATC system.

(2) A common enroute navigation system (VORTAC); (3) Precision approach and landing guidance (MLS/ILS); (4) Nonprecision approach guidance (e.g., VOR/DME); (5) Facilities for direct controller-to-pilot communications.

Note how closely this ATC system satisfies the four requirements stated by ITT in 1945.See the Institute of Electrical and Electronics Engineers (1973)for an in-depth review of the ATC system. a . Enroute and Terminal Area IFR Route Structures. In addition to aircraft separation, the pilot under IFR conditions requires navaid guidance to avoid obstacles as he descends from the enroute structure to the terminal area and executes a precision or nonprecision approach. The following discussion illustrates how the navaid accuracy performance requirements are consistent with the airway route structure. The narrative will be in the context of area navigation (RNAV) because the VOR/DME radials are really special cases of the more general RNAV route. RNAV and its present status in the NAS is reviewed in Subsection D,6. It will be assumed that the primary navigation guidance is derived from position-fix radio navaids such as VOR/DME, TACAN, or DME/DME. In addition to the navaid guidance, flight procedures and obstacle clearance surfaces must be defined so that aircraft can safely fly along the

52


enroute airways and maneuver in the terminal area. Enroute procedures are given in FAA (1975b, 1984)while terminal area procedures are documented in FAA (1976) and ICAO (1982). Enroute airways and terminal area approach paths are defined by a series of waypoints connected by straight-line segments. The enroute airways are usually specified by VORJDME radials with waypoints defined directly over the ground stations or at the intersection between the radials. A waypoint may be identified in several ways, i.e., by name, number, or location. Waypoint location is necessary in the computation of navigation information and to minimize pilot workload during time-critical phases of flight. At a minimum, enough waypoints are provided to define the current and next two legs in the enroute phase of flight, and to define an approach and missed approach. In some mechanizations, waypoints are entered into the equipment in terms of their latitude and longitude. In other equipment a waypoint’s location may be specified in terms of a bearing and distance from any place whose position is itself known to the pilot. Recalling earlier discussions in Subsection C, lateral guidance using path deviation correction signals is the principal means of enroute and terminal area navigation. Vertical guidance is typically not derived from path deviation signals; aircraft separation and obstacle clearance is achieved by measuring height with a barometric altimeter. Although the most inexpensive and therefore most implemented form of lateral guidance is essentially bearing information using NDB or VOR, the trend is toward bearing and range position-fix navigation. From the position fixes, deviation signals about the desired course called steering commands are input to the aircraft flight control system. In particular, for the RNAV application, the lateral navigation information displayed to the pilot commonly takes the form of bearing and distance from present aircraft position to the defined waypoint and deviation from a desired track proceeding to or from the waypoint. As noted in Subsection C,2, when the displayed guidance permits lateral position-fix navigation the process is called 2D navigation. In summary, the most utilized form of area navigation is 2D lateral guidance. The role assigned to vertical information as derived from the barometric altimeter is minimum descent altitude alerts and aircraft descent rate. When vertical deviation signals are derived from the altitude data, then RNAV guidance is called 3D navigation. To avoid collisions with other aircraft, precipitous ground terrain, and airport obstacles, the enroute airways and terminal approaches are separated both laterally and vertically. Lateral separation is defined by primary and secondary protection areas, as shown in Fig. 19. The full obstacle clearance is applied to the primary area, while for the secondary area the obstacle clearance is reduced linearly as the distance from the prescribed course is


53

increased. The widths of the air route are based upon the accuracy of the navaid guidance, the flight technical error, and the RNAV equipment errors. Blunder errors are monitored by enroute and airport surveillance radars and are not included in the flight technical error component. The minimum authorized flight altitude is determined by the obstacle clearance, which is the vertical distance above the highest obstacle within the prescribed area. For precision approaches this distance is based upon a collision risk probability of lo-’ over the entire flight mission. The minimum flight altitude may equal the obstacle clearance or it may be raised above it under certain conditions. If there are no obstacles then the obstacle clearance is the height above the ground. A displacement area is defined about each waypoint. It is a rectangular area formed around the plotted position of the waypoint. The dimensions of this area are derived from the total navaid system alongtrack and across-track error values, including the flight technical errors (see Subsection F). In other words, the waypoint displacement area defines the accuracy with which an aircraft can reach a waypoint and is directly related to the airway route width or terminal area approach path width. In the final analysis, it is the displacement area of the approach path which defines the minimum descent altitude. For the enroute airways, the lateral guidance protected area is k4 nmi about the route center line and the secondary area extends laterally 2 nmi on each side of the primary area, as shown in Fig. 19. The required vertical separation is lo00 ft below 29,000 ft altitude and 2000 ft above 29,000 ft altitude. The minimum required altitude (MRA) is lo00 ft and is equal to the obstacle clearance. In general, the minimum enroute altitude (MEA) is equal to the MRA. However, as noted earlier, the MEA may be raised above the MRA when, for example, the air route is over mountainous terrain. For the mountains in the eastern United States MEA = MRA 1000 ft; over the Rocky Mountains, MEA = MRA + 2000 ft. Terminal area approach paths are also protected by lateral and vertical separation distances. When in the terminal area, a series of position fixes define the approach path. They are initial approach fix (IAF), intermediate approach fix (INAF), final approach (FAF), missed-approach fix (MAF), and the runway fix (RWYF). The path between each pair of fixes is called a segment; e.g., the intermediate segment is between the INAF and the FAF. About each segment there is defined a protected primary area and a secondary area. Terminal protected areas are 2 nmi about the route center line and the secondary obstacle clearance areas extend laterally 1 nmi beyond the primary area. FAA Handbook 8260.3A, called the TERPS (FAA, 1976), prescribes criteria for the design of instrument approach procedures. Although it does not at the present time contain special criteria for the design of RNAV

+

54


procedures, most of the TERPS criteria are applicable to RNAV procedures with minor modifications. Guidelines for implementing RNAV within the NAS are given in AC 90-45A (FAA, 1975), which is oriented toward VOR/DME. Future revisions to AC 90-45A will reflect the more general configurations of multisensor RNAV as defined in RTCA (1984).In applying TERPS to a RNAV procedure, the term “fix”isequivalent to “waypoint”;thus “final approach fix (FAF)” becomes “final approach waypoint (FAWP).” Therefore, an RNAV instrument approach procedure may have four separate segments. They are the initial, the intermediate, the final, and the missedapproach segments. The approach segments begin and end at waypoints or along-track distance (ATD) fixes, which are identified to coincide with the associated segment as shown in Fig. 19. The RNAV instrument approach procedure commences at the Initial Approach Waypoint (IAWP).In the initial approach, the aircraft has departed the enroute phase of flight and is maneuvering to enter the intermediate segment. The purpose of the intermediate segment is to blend the initial approach into the final approach and provide an area in which aircraft configuration, speed, and positioning adjustments are made for entry into the final approach segment. The intermediate segment begins at the Intermediate Waypoint (INWP) and ends at the FAWP. For a standard procedure, the INWP is about 8 nmi from the runway with a minimum segment length of 3 nmi. During the final approach segment, the aircraft aligns itself with the runway and begins the final descent for landing. The final approach segment beings at the final approach waypoint or along-track distance fix and ends at the missed-approach point, normally the runway threshold waypoint. When it is not the runway threshold, it is an ATD fix based on a distance to the runway waypoint. The optimum length of the final approach segment is 5 miles. The maximum length is 10 miles. The final approach primary area narrows to the width of the FAWP displacement area at the runway threshold. A missed-approach procedure is established for each instrument approach procedure. The missed approach must be initiated no later than the runway threshold waypoint. The obstacle clearance for the initial approach segment is 1000ft and thus matches the MEA of the enroute airway. Its secondary area obstacle clearance is 500 ft at its inner edge, tapering to zero at the outer edge. The optimum descent gradient to the intermediate fix is 250 ft/mi. Obstacle clearance for the intermediate segment is 500 ft. Because the intermediate segment is used to prepare the aircraft speed and configuration for entry into the final approach segment, the gradient should be as flat as possible. The optimum descent gradient in this area for straight-in courses should not exceed 150 ft/mi. The


55

obstacle clearance for the final approach segment begins at the final approach waypoint and ends at the runway or missed-approach waypoint, whichever is encountered last. Associated with each final approach segment is a minimum descent altitude (MDA). It is the lowest altitude to which descent shall be authorized in procedures not using a glide slope. Aircraft are not authorized to descend below the MDA until the runway environment is in sight, and the aircraft is in a position to descend for a normal landing. As mentioned earlier, sometimes the MDA must be raised above the obstacle clearance. Conditions which necessitate raising the MDA are precipitous terrain, remote altimeter setting sources, and excessive length of final approach. The required obstacle clearance for a straight-in course is 250 ft. The decision height (DH) minimum is also 250 ft when the visibility is 1 nmi. The above final approach discussions are directed toward nonprecision approaches, as contrasted with precision approaches. Precision approaches require a descent path from which vertical deviation correction signals can be derived, as described in Subsection D,4. The MDA and DH are determined by the size of the final approach displacement area and the obstacle clearance. Although typical nonprecision approach heights are 400 ft and above, the lowest authorized minimum is 250 ft. Precision approaches routinely achieve decision height at 250 ft and below because the final approach displacement area is smaller. Precision navaids and cockpit displays achieve smaller displacement areas because (1) positive vertical guidance is required (i.e., a glide path as well as azimuth guidance); (2) the navaid guidance errors are smaller; and (3) the display sensitivities are increased to reduce the flight technical errors. For example, the full-scale sensitivity of the CDI is L 1.25 nmi at 20 nmi and is +350 ft at runway threshold. The full scale sensitivity is always designed to be within the obstacle clearance surface. With smaller displacement areas the obstacle clearance for precision approaches can be satisfied more easily because the projected footprint of its displacement area is smaller. Consequently, with precision approaches, lower decision height minima are more readily achievable to airports or runways with smaller obstacle-free surfaces. The principal distinction between nonprecision and precision approaches as used by the FAA is that a combined azimuth and elevation guidance (glide slope) is required for precision approaches (FAA, 1976). The term nonprecision approach refers to facilities without a glide slope, and does not imply an unacceptable quality of course guidance. It is the reduction of the final approach displacement area which is important to achieving lower minima. Positive vertical guidance is not necessary down to 250 ft because adequate obstacle clearance is provided. At that altitude the pilot has visible cues and is in the region of “see and avoid.” He can correct the attitude and position of the

56


aircraft to accomplish a manual flare and have a safe landing. Nonetheless, the pilot is more comfortable executing nonprecision approaches down to higher minima such as 400 ft. It is clear that precision approaches, with their accurate descent path, provide additional margins of integrity and pilot confidence. Table 111 summarizes the lateral route width accuracy requirements for the radio navaids servicing the enroute, terminal area, and the approach/landing application. TABLE I11 LATERAL ACCURACIES IN NAS"

Route type

XTRK system error (95% prob.)

Navaid XTRK error (95% prob.)

Includes FTE

No FTE

Route width

+ 4.0 nmi

Random

k 3.8 nmi

f3.0 nrni

J/V

& 2.8 nmi

+4 nmi + 4 nmi

Random JIV

+ 2 nmi Runway displacement area

k 2 nmi

f1.7 nmi

(k0.7 nmi) Runway displacement area

kO.5 nmi

ILS Cat I

k455 ft

& 455 ft

Bias 45 ft Noise 46 ft

ILS' Cat I1 approach

f400 ft Obstacle clearance

75 ft bias (runway width)

25 ft Noise

k27.5 ft 12.7 ft

ILSd Cat 111 landing MLS' Cat I1 approach

k 7 5 ft (Runway width) +400 ft

k 27.5 ft

Bias Noise

+ 11.5 ft

+75 ft

+ PFE +PFN +BIAS kCMN

21.7 ft

MLS" Cat 111 1anding

+75 ft (Runway width)

+ PFE +PFN +BIAS *CMN

20.0 ft 11.5 ft 1O.Oft 10.4 ft

Enrouteb Random JJV Terminalb RNAVb Non precision approach

+

95% Prob. (runway width)

+

27.5 ft About runway centerline

a J = Jet routes, V = VOR routes, random error; runway length 10,000 ft RTCA (1984). ICAO (1972b); FAA (1970). ICAO (1972b); F A A (1971, 1973). ICAO (1981a).

=

RNAV; FTE

+

+loft

12.5 ft 10.8 ft 11.3 ft

= flight

technical


57

Adding a third dimension of vertical guidance to the two-dimensional RNAV systems can achieve significant operational advantages. Briefly, a 3D RNAV capability permits altitude change by following vertical routes (tubes) of known dimensions; thus vertical guidance is available for stabilized descent in instrument approach procedures using computed glide path information. In some cases, the computed glide path can make it possible to safely eliminate obstacles from consideration. To provide vertical guidance during ascent or descent, the 3D RNAV equipment compares the indicated altitude with the desired altitude and presents the computer correction instrumentally. In prescribing obstacle clearance for 3D RNAV, it is useful to think of the vertical route as being the center of a tube of airspace. The lateral dimension of the tube is the width of the RNAV route as described in Table 111. The vertical dimension of the tube is sufficient to contain the combined 3D RNAV vertical errors. The longitudinal dimension of the tube is limited only by its operational use. For example: The distance required to climb 10,000 ft at a climb angle of 2", or the descent from the final approach waypoint to the missed approach waypoint at a descent angle of 3". For obstacle clearance, it is necessary to consider only that portion of the tube which is at and below the designed vertical flight path. An aircraft is protected from obstacles when no obstacles penetrate the tube from below. The along-track error also has significance in vertically guided flight. When an aircraft is ahead or behind its assumed position, it will be either above or below its intended path. The possibilities for a hazardous incident are analyzed using the total navigation system error and the distance to an obstacle. For the approach/ landing application, the hazard probabilities are small and the obstacles are known fixed objects, e.g., towers, hangars, etc. Although the probability of exceeding the route width of an enroute airway is only 95%, the lateral obstacle is in effect another aircraft in a parallel route; thus the joint probability of a collision is very small, especially when ATC aircraft separations are maintained using ground radar control. The longitudinal separation using the ground radar (ARSR) is 3 to 5 nmi. Without radar it is 5 nmi with DME and the faster aircraft is in front of the slower aircraft (Kayton and Fried, 1969). The principal ideas associated with terminal approaches are worth restating. They are: (1) The collision risk probability establishes the obstacle clearance; (2) the total navigation system error determines the approach displacement area; and (3) the height of the displacement area for a given approach path is determined by the obstacle clearance. Thus the displacement area is lowered until it equals the obstacle clearance. For a nonprecision approach the displacement area is larger than the precision approach displacement area. Therefore, its MDA or DH will generally be higher than that required for a precision approach.

58


6. Surveillance Radar. The primary radar is intended to detect all aircraft by processing echoes of ground facility transmissions-no airborne equipment is required. There are two types. The first, air route surveillance radar (ARSR), operates in the 1300-1350 MHz band and normally has a range of 200 nmi. The airport surveillance radar (ASR),in the 2700-2900 MHz band, has a range of 60 nmi. ARSR systems have peak power outputs of 1.0 to 5.0 MWatts, and ASR systems from 400 kW to 1.0 MW. The power output for a particular site is determined by site conditions, i.e., terrain, elevation, and the proximity of adjacent sites. Bearing and range of the aircraft are displayed to the controller on a plan position indicator (PPI)called a plan view display (PVD).The primary radar permits the controller to see any aircraft within the coverage area, regardless of whether or not the pilot is using the ATC system. The ATC primary surveillance radar is a self-contained system whose purpose is to maintain aircraft separation by dicect communication with the pilot when the aircraft is carrying a radio. Since the ARSR and ASR do not involve equipment on board the aircraft, international standardization is not required. The secondary radar, as noted in Subsection B, or air traffic control radar beacon system (ATCRBS), is a system wherein the airborne element is the transponder and the ground element is the interrogator. The ATCRBS power output is 200-2000 W and has a nominal 250 nm range when used in enroute applications. The secondary radar ground-air link, or interrogator, operates at 1030 MHz and activates the 1090 MHz aircraft air-ground link, or transponder. The transponder reply among the 4096 available codes generally includes an identification code and an altitude code. Transponder replies are received by the ground interrogator, and heading and ground speed are calculated by tracking devices. Heading and ground speed are displayed on the PVD along with the aircraft identification number and altitude. A rotating directional ground-based interrogator transmits approximately 400 interrogations per second. The cooperative ATCRBS is one of the keys to advanced ATC automation. The ATC system is devloping additional automated features to economically meet the high demand for services in the 1990s. In the U.S. there are about 300 ATC radars in use, over 100 ARSRs and about 200 ASRs. All ATC primary radars are equipped with the ATCRBS secondary radar system. Unlike the primary radar system, the secondary radar is an ICAO standard; its “Standards and Recommended Practices”(SARPs) are given in ICAO, (1972b, para. 3.8). 3. Navigation Aids

Aircraft users require short-range navigation aids to probide enroute guidance over the defined airways under IFR conditions. The VORTAC


59

routes were established to provide a simple, common system that is easy to use, thereby reducing pilot workload. DME/N and VOR provide the guidance with RNAV augmenting the VORTAC route structure by providing random routes. Table IV places the VORTAC system in perspective by comparing the number of units in use with all the air and marine navigation aids in use throughout the world. The table was excerpted from Dodington (1984). The accuracy and coverage of navigational aids are determined by the enroute and terminal area operational requirements. Systems deriving bearing information (VOR and TACAN) must offset the effects of multipath and siting,which are the major sources of error (Subsection F). Signal coverage is a design problem only because adjacent channel effects must be considered (see Section III,C,3). In other words, it is easy to obtain coverage with a single facility, but it takes special care to eliminate spillover into other channels so that coast-to-coast service can be provided using a limited number of channel frequencies. The next several subsections describe the VOR, TACAN, ILS, and MLS angle systems. The DME/N and DME/P are treated in Sections 111and IV of TABLE IV MAINRADIONAVIGATION SYSTEMSUSEDTHROUGHOUT THE WORLD' Users System

Frequency

Omega Loran-C Decca ADFb ILSb

10-13 kHz 90-1 10 kHz 70-130 kHz 200-1600 kHz 75,108-1 12, 329-335 MHz 108-118 MHz 150,400 MHz 960-1215 MHz 960-1215 MHz 1030, 1090 MHz 5031-5091 MHz

VORb Transit Tacan DMEb SSRb MLSb

Number of stations 8

33 chains 50 chains 5000 1600

2000 5 satellites 2000 1000

Air

Marine

10,600 2000 1000 200,000 70,000

7000 60,000 30,000 500,000

200,000

-

-

38,000

17,000 80,000 100,000 Implementaticm about to' start

Estimated number of U.S. aircraft: 224,000 General aviation (Quinn; 1983) 2500 Air carrier 20,800 Military aircraft Dodington, 1984. Signal format standardized by the International Civil Aviation Organization. a

60


this article. These navaids are viewed from a system level viewpoint. Each system is characterized by its signal format, accuracy, coverage, channel plan, and data rate. Two other system level characteristics are needed before the navaids used in the ATC system are complete. They are integrity and availability. Integrity is a measure of the truthfulness of the signal in space. It is ensured by a ground monitor system, which shuts the facility down if certain critical signal characteristics exceed predetermined limits. Integrity is quantified by the reliability (mean time between outages, MTBO) of the ground equipment and its monitor. A navigation system must be available to the pilot, especially under IFR conditions. Although a facility which has been shut down has good integrity, it is not available. The system availability A is given by A = MTBO/(MTBO + MTTR), where MTTR is the mean time to repair. Integrity and availability are critical to the successful operation of approach/landing systems. Radio navaid accuracy performance comparisons are made in Hogle et al. (1983). Integrity and reliability comparison of civil radio navaids are made in Braff et al. (1983). a. VOR.

The very-high-frequencyomnidirectional range is the ICAO international standard for providing short-range enroute bearing information. Paragraph 3.3. of ICAO (1972b) is the ICAO SARPs for VOR. As noted in Subsection C , it is a radial determining navigation aid, and along with the automatic direction finder (ADF) (see Table IV) is the most popular system currently in use. Since it is a radial system, it must have a reference signal and a signal which carries the bearing information. The reference signal is frequency modulated (FM) at 30 Hz on a 9960 Hz subcarrier. A cardioid antenna pattern rotates 30 times per second (Fig. 21), generating 30 Hz amplitude modulation (AM)in the receiver, and thus provides the bearing information. The airborne receiver then reads bearing as a function of the phase difference between the FM reference signal and the AM modulated signal. The system is graphically described in Fig. 2 1. The VOR estimates the aircraft bearing in the same manner as the TACAN; it employs a phase detector. Early implementations of phase detectors were electromechanical devices using resolver phase shifters which were rotated by a servo motor until a peak or null was obtained (Hurley et al., 1951). The bearing display was connected directly to the phase shifter shaft. Today’s systems are all-electronic digital implementations. The VOR ground transmitter radiates continuous-wave signals on one of 20 channels between 108 and 112 MHz (interleaved with ILS localizer frequencies) and 60 channels between 112 and 118 MHz. Transmitter power is from 50 to 300 W.

61

DISTANCE MEASURING EQUIPMENT IN AVIATION CHANNEL A f, = 9960 Hr

AIRBORNE RECEIVER

PHASE DETECTOR

e^

CHANNEL B CHANNEL B CHANNEL A

PAlTERN ROTATES

FIG.21. Principles of VOR.

Because of the frequency used, it is difficult to get an antenna with good vertical directivity. Consequently, significant radiation strikes the ground in the vicinity of the transmitter which, upon reflection, interferes with the desired radiating signal causing a bearing error. This error, which is sometimes 2" to 3" in magnitude, appears as course variations along the radials of the VOR station. Several approaches to fixing this problem have been tried, including elevating the antenna to get it farther away from reflecting objects and using a large counterpoise (ground system) under the antenna. Both of these methods have been somewhat successful, but the development of Doppler VOR (DVOR) and Precision VOR (PVOR) offered better solutions. Severe constraints, however, were placed upon these solutions. They were not to affect the service of the already existing airborne equipment and, if possible, should improve the system's overall performance. Doppler VOR accomplished this by simply reversing the roles of the reference signal and the information signal. Because their phase relations remain the same, they allow a standard airborne receiver to operate without

62


modification. With this modification, the effective aperture of the bearing information signal increased more than 10:1 (antenna diameter of 44 ft); thus, the effects of site error could theoretically have a tenfold reduction. Experiments have confirmed this; site errors of 2.8" were reduced to 0.4". Steiner (1960)describes DVOR. The second solution used the same idea as TACAN; that is, it modified the ground antenna to produce multiple lobes, thereby creating a two-scale system. The coarse information was 30 Hz and the multilobe antenna (11 lobes) generated a 330 Hz harmonic for the fine information. It could potentially obtain and 11:l reduction not only in site errors but also in airborne receiver instrumentation errors (Winick, 1964; Flint and Hollm, 1965).The primary reasons the FAA selected the DVOR solution and did not pursue the PVOR further were that the conventional VOR receivers could not decode the multilobe harmonic and DVOR sufficiently reduced site errors to less than 0.5". Since a DVOR installation is a rather extensive and expensive process, it is reserved for the most difficult sites, where the simpler techniques do not suffice. Given the deficiencies of VOR, many airlines are making use of the scanning DME as their primary enroute aid to form a p-p navigation system based on RNAV principles. VOR/DME is then used as an alternative if acceptable DME/DME geometric configurations cannot be found (see RNAV, Subsection D,6). b. TACAN. TACAN range and bearing information defines a natural p-8 system. Unlike VOR/DME, which uses two different rf transmissions, TACAN provides range and bearing information using a single rf emission. TACAN is not an ICAO standard. Range information is derived within the TACAN system in the same way as for DME: A transponder responds to an interrogation. It uses the same frequencies, pulse characteristics (coding, form), and interrogator pulse rates as DME/N. Bearing information is determined at the interrogator in a manner that is functionally identical to the VOR phase detector. That is, it generates both a reference signal and bearing information signal and is, therefore, a radial system. Because DME is a pulse system, the TACAN bearing information is equivalent to a sampled data VOR system. Moreover, TACAN bearing information is a two-scale system similar to precision VOR. The coarse bearing is 15 Hz and the fine bearing is the ninth harmonic or 135 Hz. Bearing information is obtained with the following additions to the DME signal format:

(1) A circular antenna array replaces the simple omnidirectional DME antenna. It generates a cardioid pattern in space and sequentially rotates it at 15 Hz. At a fixed point in the coverage sector, the DME pulses appear to be


63

amplitude modulated at 15 Hz. A second pattern composed of 9 lobes spatiallymodulates the cardioid pattern, which generates a 135 Hz sinusoid or ninth harmonic at a fixed receiver in space. (2) The 15 Hz reference signal is generated by a North reference burst which is composed of 24 pulses, the spacing between pulses being alternately 12 and 18 ps. When decoded in the airborne equipment they become 12 pulses spaced 30 ps apart. This pulse train is referenced once per revolution of the cardioid antenna pattern. (3) The 135 Hz reference signal is generated 8 times per revolution of the cardioid antenna pattern. It consists of 12 pulses spaced 12 ps apart. (4) The transponder operates at a constant duty cycle of 2700 pps. When interrogations are lacking, pulse pairs with random spacing are generated (called squitter pulses) so that the output rate is kept constant at 2700 pps. This constant duty cycle permits the sampled 15 and 135 Hz sinusoids to be reconstructed by zero-order hold (ZOH) circuits in the aircraft with complete fidelity. It also permits AGC to be generated in the aircraft such that the 15 and 135 Hz AM modulated signals can be detected without distortion. In the airplane the 15 Hz reference burst provides a time reference mark from which the phase of the 15 Hz information signal can be determined. Similarly, the 135 Hz reference burst provides a time reference mark from which the phase of the 135 Hz information signal can also be determined. After reconstruction from the samples, the information signal is sent to the 15 and 135 Hz phase detectors where the operations are functionally identical to that used in the VOR receiver. The 135 Hz phase detector accurately estimates the aircraft's bearing; it is, however, ambiguous over each 40" sector (there are nine of them). The 15 Hz phase detector resolves the ambiguity by determining in which of the 9 sectors the actual bearing angle lies (see Fig. 22). Because TACAN utilizes a multilobe spatial antenna pattern, it can employ a two-speed phase detection system which is less susceptible to siting effects and equipment error than is conventional VOR. TACANs accuracy is comparable to the PVOR. The ninth harmonic reduces the bearing error by about 4 of those for the 15 Hz signals. TACAN's accuracy is discussed further in Subsection F. 4 . ApproachfLanding Aids

A guidance system for approach and landing is simply a precise, lowaltitude navigation aid with the additional accuracy and reliability needed for landing aircraft under reduced visibility. The standard system now in use is the Instrument Landing System; its SARPs are defined in paragraph 3.3 of ICAO (1972b). The Microwave Landing System SARPs are defined in paragraph 3.11 of ICAO, (1981a);it is scheduled to replace ILS in 1998. ILS and MLS permit precision approaches at or below a 250 ft weather minimum

64

ROBERT J. KELLY AND DANNY R. CUSICK HORIZONTAL ANTENNA PATTERN MAGNETIC NORTH

15 RESOLUTlONSlSECOND

WEST 270'

SOUTH 180'

---

HORIZONTAL PATTERN COMPONENT THAT GENERATES 15 Hz SIGNAL COMPOSITE PATTERN THAT GENERATES 15 Hz AND 135 HZ SIGNALS

TIME WAVEFORM AND REFERENCE BURSTS

15 Hz MODULATION

-b

1/15 SECOND

. .

I.

.ANTENNA PATTERN ORIENTATION SHOWN AT THE TIME OF NORTH REFERENCE BURST LET THE NORTH REFERENCE BURST DEFINE TIME 0. THE RECEIVED SIGNALS ARE THEN 15Hz : -SIN r m o n t - 8 ) 1 % ~ -SIN ~ : rz701~t-se) WHERE IS THE BEARING FROM THE STATION. AUXILIARY REFERENCE BURSTS ARE TRANSMITTED EACH TIME A 135 Hz PATTERN PEAK POINTS EAST

e

FIG.22. Principles of the TACAN bearing measurement.


65

using a barometric altimeter. If in addition a middle marker or DME measurement is included, the minimum can be reduced by 50 ft to 200 ft. Nonprecision approaches do not require positive vertical steering commands; minimum descents down to 250 ft are permitted depending upon the obstacle clearance. Before describing ILS and MLS, this section will discuss the landing operation itself, using MLS as an example. The landing process consists of curved or segmented flight paths for noise abatement and the transition to the final centerline approach (Fig. 23). The decision heights where the pilot must be able to “see to l a n d are 200 and 100 ft ceilings-Categories I and 11, respectively. The flare maneuver, touchdown, and rollout complete the landing. MLS must support each of these maneuvers under IFR conditions. The elevation element, typicallly located 861 ft from threshold and offset from the centerline by 250 ft, provides elevation guidance to the decision height of Category I and I1 operations and to threshold in Category I11 (zero ceiling). The approach-azimuth element supports lateral guidance to the decision height in Categories I and I1 operations and to touchdown and rollout in Category 111. The flare maneuver is typically performed manually for

FIG.23. The landing operation with MLS.

66

ROBERT J. KELLY AND D A N N Y R. CUSICK

Categories I and I1 by visual reference, but, for Category 111, positive azimuth and vertical control are required to touchdown. The Category I1 “window” is centered at the 100 ft decision height. Its height is f 12 ft around the indicated glide path and its width is the lateral dimensions of the runway (typically f 7 5 ft). If the aircraft is not in the window, the pilot will execute a missed approach. If the aircraft passes through the Category I1 window within specified variations of pitch, roll, speed, and lateral drift velocity, then the probability is high that a successful landing can be achieved when the pilot commences the flare maneuver. The decision height must be determined within f 5 ft. On the other hand, no Category I window is necessary because the pilot can maneuver his aircraft to the correct position and attitude before he executes his flare maneuver (FAA, 1970). MLS facilitates the autoland maneuver with its increased accuracy, which is extended to lower elevations further down the landing path than provided by ILS. Flare to touchdown requires the height above the runway and, in MLS, it may be provided by a combination of the approach elevation angle and DME prior to runway threshold and then by transitioning to the radar altimeter in the vicinity of runway threshold. Terminal area IFR procedures for MLS are currently under development by the FAA. a. ZLS.

The Instrument Landing System is a radial system providing guidance for approach and landing by two radio beams transmitted from equipment located near the runway. One transmitter antenna, known as the localizer, forms a single course path aligned with the runway center line. The other transmitter, the glide slope, provides vertical guidance along a fixed approach angle of about 3”.These two beams define a sloping approach path with which the pilot aligns the aircraft, starting at a point 4-7 mi from the runway. Instrument landing systems operate on frequencies between 108.1 and 111.9 MHz for localizers, and between 328.3 and 335.4 MHz for glide slopes. Both localizer and glide slope are paired on a one-to-one basis. The service volumes for an ILS are bounded by wedges of an 18 or 25 nmi radius from the localizer antenna. Lateral guidance (Fig. 24) is provided by the localizer located at the far end of the runway. Two identical antenna patterns, the left-hand one modulated by 90 Hz and the right-hand one by 150Hz are provided. The vertical needle of the airborne display is driven right by the 90 Hz signal and left by 150 Hz signal, and centers when aircraft is on course. Vertical guidance (Fig. 24) is generated by the glide slope antenna located at the side of the approach end of the runway. A 150 Hz amplitude-modulated signal is provided below course and 90 Hz modulation is provided above


67

LOCAL1ZER

A

A\

-

ANTENNA

RUNWAY

FIG.24. Principles of ILS.

course. The horizontal needle of airborne display is driven up when the amplitude of the 150 Hz signal exceeds that of the 90 Hz signal and is driven down when the reverse occurs. The needle is horizontal when the aircraft is on course. (The “cross-pointer” is a course deviation indicator combining a vertical localizer needle and a horizontal glide slope needle.) Along-course progress is provided by fan markers that project narrow beams in the vertical direction and operate at 75 MHz. The middle marker is placed at the 200 ft decision height; the outer marker about 5 miles out. The outer marker corresponds to the final approach fix. The outer marker can be viewed as the final approach fix. Since DME may be used as a replacement for the markers, 20 DME channels are provided for hard pairing with ILS localizer frequencies. ILS’s limitations are detailed in Redlien and Kelly, (198 1). b. MLS. Like ILS, the angle portion of MLS is an “air-derived’’ radial system where signals are radiated from ground antennas in a standard format and then processed in an airborne MLS receiver. The angle information is derived by measuring the time difference between the successive passes of highly directive narrow fan-shaped beams, as shown in Fig. 25. Range information, required to obtain the full operational benefits of the MLS angle data, is derived from precision DME, which Section IV describes in detail. MLS operation is detailed in Redlien and Kelly, (1981).

68


m .-dI

TIME-ANGLE

4-

AIRCRAFT RECEIVER

FIG.25. MLS angle measurement technique.

MLS is a highly modular system and it may be implemented in simple configurations. The ground systems may contain approach azimuth, the “bearing” facility, approach elevation, back azimuth, and DME/P, as shown in Fig. 26. The signal format is time multiplexed; that is, it provides information in sequence on a single-carrier frequency for all the functions(azimuth, elevation, basic, and auxiliary data). The format includes a time slot for 360” azimuth guidance with provisions for growth of additional functions. The angle guidance and data channel plans provide 200 C-band channels between 5031 and 5091 MHz. Narrow fan-shaped beams are generated by the ground equipment and scanned electronically to fill the coverage volume. In azimuth, the fan beam scans horizontally and has a vertical pattern that is shaped to minimize illumination of the airport surface. In elevation, the arrays are designed to minimize unwanted radiation towards the airport surface, thereby providing accurate guidance to very low angles. Unlike ILS, the MLS conforms to a single accuracy standard equivalent to that required to perform fully automatic landings. A ground-to-air data capability is provided throughout the angle guidance coverage volume by stationary sector coverage beams that are also designed to


69

77ms SEQUENCE

BASIC DATA NO. 2

FIG.26. MLS signal format.

have sharp lower-sidecutoff characteristics. This capability is used to transmit the identity of each angle function and to relay information (basic and auxiliary data) needed to support simple and advanced MLS operations. The wide angle capability of MLS is graphically displayed in Fig. 27. Because proportional guidance is available over a +40" azimuth sector, RNAV curved and segmented approaches down to the 200 ft decision height can be executed. Shown in the figure are the airborne elements which comprise the MLS RNAV equipment configuration. RTCA Special Committee 151 is currently defining the minimum operational performance expected of the airborne equipment intending to perform MLS/RNAV approaches (RTCA, 1985a). For approaches below the 200 ft decision height, the MLS data will bypass the RNAV box and be utilized directly by the aircraft flight control

70

ROBERT J. KELLY AND DANNY R. CUSICK MLSRCVR

I

A2 DMEP FLIGHT DIRECTOR

I

RNAV

CONTROL PANEL

I

W

FIG.27. MLS avionics for segmented approaches

system. RTCA (1985a) recommends that the RNAV error contribution be limited to 35 ft (95% probability) for MLS area navigation applications. c. Nonprecision Approach. Aircraft equipped with the appropriate enroute navaids are permitted to fly final approaches down to MDAs which are consistent with the obstacle clearance.The nonprecision approach procedures additionally require that the final approach displacement area be 20.7 nmi, as indicated in Table 111. Over 30% of the runways with nonprecision approaches use on-airport VOR (DOD/DOT, 1984). See the discussion in subsection D.5, which addresses the role that the scanning DME may play in nonprecision approaches. 5. Communications

Radio communications between controllers and pilots take place on VHF frequencies between 118.0 and 136.0 MHz, and on UHF frequencies between 225.0 and 400.0MHz. ATC communications are necessary to ensure the safe, orderly, and expeditious flow of air traffic.Chapter 4 of ICAO (1972b)defines the ICAO standard for communication systems.


71

6. Special Radio Navaid Techniques

This section discusses two techniques which enhance the operational performance of DME and other navaids in the ATC system. They are RNAV and the “scanning DME.” RNAV enhances enroute navigation and nonprecision approaches by providing point-to-point navigation along line segments other than the VORTAC radials. Scanning DME is a single interrogator that obtains the range to several different ground transponders by using frequency hopping or scanning frequency techniques. From these data a very accurate position fix is determined. a. RNAV. Since the introduction of VORTAC, air navigation has been improved by providing a means for unlimited point-to-point navigation. This improved method of navigation, which utilizes a computer to determine courses that need not lie along VORTAC radials, is called Area Navigation. It permits aircraft operations on any desired course within the coverage of various stations providing referenced navigational signals, i.e., DME or VOR, or within the limits of self-contained airborne systems, such as the INS. Sensor inputs to the RNAV may be p-p, 8-8, or p-8. As stated in Quinn (1983), there are three general operational concepts in area navigation: (1) two dimensional (2D), involving only horizontal movements; (2)three dimensional (3D), in which vertical guidance is combined with horizontal; and (3) four dimensional (4D), in which time of departure and arrival are combined with horizontal and vertical movements for complete aircraft navigation. At present, 2D area navigation is the most widely used version. Many pilot-related operational issues regarding 3D and 4D approach procedures have been and still are under study (Jensen, 1976; RTCA, 1985a). The conventional implementation of2D RNAV divides the flight course into a series of waypoints connected by straight-line segments (see Fig. 19). The waypoints are stored in the RNAV computer, and are identified by suitable coordinates such that DME/DME or VOR/DME measurements can generate the steering commands to guide the aircraft along the straight-line segments.In the more simple implementations, the position fixes derived from VOR and/or DME measurements are sometimes aided by inertial sensors so that a waypoint turn can be anticipated and thus minimize the error incurred in capturing the next flight path segment (Tyler et al., 1975). In the early 1970s, the FAA established a high-altitude RNAV airway structure. As stated in RTCA (1984) these airways were generally aligned to avoid all special-use airspace and coincide with regional traffic flows. They consequentlyoffered little in mileage savings over VOR airway structures. The RNAV structure did not take into consideration the center-to-center traffic flow that had evolved. Since preferential procedures are usually tied in to the VOR system, the high-altitude RNAV airway structure was not used to any

72


great degree after a short period of interest in the early 1970s. In 1978, the RNAV airway structure was significantly reduced, and in early 1981 all published high-altitude RNAV routes were revoked. By the end of the 1970s, it became apparent that the potential economy and utility of RNAV could best be realized by random area navigation routes rather than by the previous grid system. Without a fixed route structure to use in normal flight planning, pilots and air traffic controllers developed an informal system in which pilots requested “RNAV direct destination” routing from controllers after they were airborne and beyond the airport terminal area. Such direct clearances are granted when possible. Figure 19 illustrates such a requested direct destination route. These procedures reflect the original intent of the RNAV concept wherein the “R’in RNAV signifies “random.” An operational evaluation by the FAA showed that there were no adverse effects in using latitude and longitude coordinates for domestic routes having direct random route clearance (FAA, 1981). Advisory circular 90-82 (FAA, 1983d) was issued, authorizing random routes for RNAV equipped aircraft having 1and (D coordinates and flying above 39,000 ft. The evident benefits in the use of RNAV are a reduction in flight time obtained by following the shortest or best route between origin and destination, and reduced fuel consumption. Other benefits include more efficient use of airspace, reduced pilot workload, and weather avoidance. A status review of RNAV is given in Quinn (1983),where an informal 1981 FAA study estimated that the largest number of RNAV avionics units was VOR/DME-based (over 33,000 units). About 20,000 of these units accepted 4 or 5 waypoints. Costs range from $2,000 to $20,000. The balance of the units included Omega/VLF (6,400 units) with costs ranging from $20,000 to $80,000; LORAN-C (700 units) with costs ranging from $2,000 to $20,000;and expensive INS units ($100,000 and above). It is estimated that today about 20% of the civil fleet has RNAV. The RTCA minimum standards on airborne RNAV equipment using VOR/DME imputs are documented in RTCA (1982). As shown in Fig. 4,RNAV outputs can be displayed to the pilot for manual coupled flight or they can be coupled directly as steering commands to the autopilot. Autocoupledflightsusing RNAV steeringcommands result in small tracking errors (Fig. 13).This means that the optimal maneuvers necessary to fly the desired course can be chosen using, in some cases, a flight management system (DeJonge, 1985), which minimizes fuel consumption. Moreover, since the pilot has confidence in his RNAV coupled AFCS, he can reduce his workload. He is free to fix his attention on other cockpit concerns. For manual coupled flights, the RNAV inputs can be as simple as a position-fix VOR/DME or DME/DME, with little or no inertial aiding (Bryson and Bobick, 1972). Autocoupled systems use VOR/DME or DME/DME with inertial aiding for dynamic damping to reduce control


73

activity (ROLL) and tracking errors. In the most sophisticated RNAV implementations, INS is used as the primary navigation input with radio aiding by VOR/DME or DME/DME [Karatsinides and Bush, 1984; Bobick and Bryson, 1973; Zimmerman, 1969). Minimum operational peformance standards for these multi-sensor RNAV systems have recently been published in RTCA (1984). With over 750 VORTAC stations currently in service in the continental U.S., VOR/DME signal coverage is not a problem in flying defined air routes above the minimum enroute altitude. It is, however, incomplete for aircraft that regularly operate at low altitudes (above 2000 ft) in offshore and mountainous areas in the eastern part of this country. For direct requested RNAV routes, signal coverage is a factor which must be considered. According to Goemaat and Firje (1979),RNAV coverage throughout the 48 states above 10,OOO feet using VOR/DME is essentially complete. Coverage criteria were based upon the FAA Standard Service Volume (SSV) (see Section 111,C). RNAV coverage using DME/DME was more complete than VOR/DME because a line-of-sight criterion could be employed instead of the more restrictive SSV. For example, 86%of the continental U.S. (CONUS) had dual DME coverage at 3000 ft above ground level. Phasing out the current airway structure and converting to a more flexible system of area navigation is a process that will require many years to complete. At present, the FAA is upgrading VORTAC stations to solid-state equipment at a cost of roughly $210 million (fiscal year 1980 dollars). At the same time, ihe question of adopting a new navigation technology to conform to new international standards is scheduled for consideration by the ICAO. The issue is not so much selection of a single new navigation system to replace VORTAC as it is a question of adopting procedures for worldwide navigation using RNAV, which will be compatible with several possible technologies. Nevertheless, the national airspace plan has been updated to ensure the continuation of VORTAC service until 1995. Present indications are that these systems will remain prime elements in a mix of future navigation system plans which extend to beyond the year 2000. Subsection F compares the accuracy performance of DME/DME, VOR/DME, and TACAN, using an RNAV computer, and discusses additional RNAV system considerations.

b. Scanning DME. A position fix can be determined from the range measurements obtained from two DME ground stations. The intersection of the circular locus of points representing each DME range measurement defines the airborne receiver position. This position fix technique is referred to as DME/DME. In areas that have suitable DME/DME coverage the potential accuracy of dual DME offers a significant improvement over

74


VOR/DME. The degree of accuracy improvement is dependent upon the geometry between the interrogator and the selected DME ground stations. In principle these measurements can be made using two or more interrogators, requiring provisions for channel selection and synchronization of the range measurements from the individual interrogators. In one implementation which complements an INS (Karatsinides and Bush, 1984), two DME measurements are processed every 5 s. The “scanning” DME, however, performs all these tasks in a single interrogator. The “scanning” DME innovation provides the potential for rapid position fixes (position update rate) consistent with the AFCS control laws used in many jet transports. The idea is to use rf frequency-hop techniques to scan the spectrum of usable ground transponders in a given geographic area. The scanning DME, as defined in Aeronautical Radio, Inc. (ARINC) (1982), has two frequency scanning modes, a directed frequency foreground mode and a background mode. The tuning source may designate from one to five stations for geometry optimization, with one of these stations available for cockpit display. These stations are designated as foreground stations and are placed in a loop which cycles through the full list at least once every 5 s. All five selected stations are identified by separate digital words containing both the rf channel frequency and the distance information. A second loop, called the background loop, provides the acquisition and output of data obtained during the background scan mode. When in the background mode the interrogator scans through the entire DME channel spectrum, excluding those channels already designated in the foreground loop. For any station where squitter pulses are received during the background scan, the interrogator computes the distance to that station. Both the distance measurement and the channel frequency are given to the aircraft navigation systems utilizing the DME data. The ARINC characteristic states that the initial scan of the background channels should be completed in a maximum of three minutes. This full spectrum scan time is based upon 20 channels, occupied by a ground transponder with a 70%reply efficiency and a minimum reply signal strength of - 87 dB m. In effect, the scanning DME has six time slots available every second. Five are utilized for obtaining the range to the designated foreground stations while the sixth, or “free scan” time slot, obtains range data on channels in the background loop whenever stations are present. As required by ARINC (1982), the data output rate (on the ARINC 429 bus) shall not fall below six outputs per second and the measurement age of the data output should not exceed 0.2 s. This means that each time slot (or data output) per frequency can occur at least once per second. Since the interrogation PRF is limited to 30 pulses per second, this implies that each time slot has 3 to 4 replies available for


75

filtering before being sent to the 429 data bus, assuming a 70% reply efficiency. As the aircraft proceeds along its desired route, foreground stations are discarded and replaced by background stations having better position fix geometries. Algorithms have been developed that select foreground channels to be used in determining a position fix. The advantage of the ARINC 709 characteristic for scanning DME is its short dwell time per channel (0.067 s.), which permits rapid position fixes. Because of the short dwell time, the range to 5 or more ground transponders can, in principle, be used by an RNAV computer to determine the aircraft’s position. Using special signal processing algorithms (e.g., regression analysis (Latham, 1974) demonstrated achievable 100 ft CEP. However, since RTCA (1984) requires only a cross-track error of 3.8 nmi for enroute RNAV systems, there is no motivation to reduce the errors to less than that obtained with a two-station DME fix. In general, the noise is reduced only about 50% for a five-station fix. Most of the error reduction is achieved by removing the bias contribution from each ground station. Because of the above considerations some airframe manufacturers use the five foreground stations as follows: position fix (2),ILS (l), display (2). They then use the background stations to ensure that the facilities chosen for their planned navigation flight route are operating correctly. As the geometry of the foreground stations used to obtain the position fix deteriorates (geometric dilution of precision, GDOP), these foreground stations are replaced by the appropriate background channels. Station selection algorithms are under development that pick not only stations within signal coverage but also provide acceptable position-fix geometries whenever they exist (Ruhnow and Goemaat, 1982).This means the stations should be offset to the left and right of the flight path. Subsection F discusses the DME/DME GDOP considerations. Goemaat and Firje (1979) analyzed the number of major airports within CONUS in which an aircraft can make a landing approach down to 400 ft. (nonprecision)using scanning DME. They concluded that 22% of the airports examined provided DME/DME approach capability on at least one major runway (5000ft in length or more). This coverage will increase with the implementation of DME/P as the FAA begins its deployment of the planned 1200 MLSs. In some systems a flight management computer system (FMCS) contains a navigation database for all navaids (DME and VOR) in the flight area. The selection process used in the Boeing 737-300 FMCS is detailed in Karatsinides and Bush (1984). Failure modes for navigation systems which use INS to aid DME/DME position-fix navigation are still under investigation. For example, the Boeing 737-300 FMCS, in the absence of inertial inputs, switches into a second mode and works on DME information only, with some

76


degradation of the navigation performance. When DME fails temporarily, the system continues in a crude inertial dead-reckoning mode. 7 . Summary In this section, today’s ATC system was described, both from the operational and navaid equipment viewpoint. The narrative highlighted the roles of the conventional DME/N and of the new DME/P. For enroute navigation the DME/N is one of the central elements to the VORTAC system of air lane radials. Today it participates as one of the principal inputs to RNAV. Tomorrow, it is expected that the DME/N will have an even bigger role in the enroute air structure and in nonprecision approaches where the scanning DME, because of its high accuracy, can replace the VOR bearing information as a principal input to the RNAV. With respect to the approach/landing applications, DME/Ns role is expanding, where it will supplement the ILS middle markers and will provide three-dimensional guidance at those locations where middle markers cannot be installed because the necessary real estate is not available. Today and in the future, DME/P plays a critical role in MLS. This role will become even more visible and necessary as all the MLS operational features are exercised (1) by increasing demand for more certified Category I1 runways, (2) by coordinate conversionsto achieve curved approaches and (3)by providing MLS service to more than one runway using a single ground facility. E. Principles of Air Navigation Guidance and Control

The purpose of this section is to present the principles upon which the guidance and control of aircraft are based, with particular emphasis on the DME as it is used in the enroute (VOR/DME) and the approach/landing phases of flight (MLS). For the most part, the material will be qualitative with emphasis on the four or five key concepts associated with aircraft control systems. It is not intended that the material represent, even in summary form, the current thinking of today’s AFCS engineers. These concepts are necessary to understand the requirements imposed upon the output data characteristics of a radio navaid. An aircraft follows the desired flight path when the AFCS control law, which requires position and velocity information as inputs, is satisfied. The ideal or natural solution is to directly measure the position and velocity deviations of the aircraft from its intended path. Radio navaid and referenced accelerometers make these measurements directly; other inertial sources are indirect measurements. Radio navaids are desirable sources for this information because they are less expensive and more easily maintained than referenced accelerometers.


77

There are, however, constraints imposed upon the radio navaid when velocity data as well as position displacement information are derived directly from the navaid guidance signal. The two applications of interest in this article, enroute navigation and approach/landing guidance, have widely different requirements in terms of output signal noise and information bandwidth, the most restrictive being the approach/landing application down to 100 ft decision height and lower. At these weather minima imperfections in the approach/landing guidance signal usually require that the velocity or the rate information be complemented by inertial sensors. This article develops the background necessary to explain the complementation process, and extends it to include the concepts of path following error (PFE) and control motion noise (CMN) which form the basis of the DME/P and MLS angle accuracy specifications.

I . Feedback-Controlled Aircraft- General Principles Aircraft use enroute navaids to fly between terminal areas and, once in the terminal area, landing guidance aids are used to complete their flight. These navaids can be coupled to the AFCS manually (through the pilot) or automatically via an autopilot. In either case, the guidance loop is closed and a feedback control system is created. Feedback control systems tend to maintain a prescribed relationship between the output and the reference input by comparing them and using the difference as a means of control. Such systems play a dominant role in flight control because they suppress wind disturbance and reduce the phase delay of the corrective signals and the effects of nonlinearities. For example, in the presence of disturbances such as wind gusts, feedback control tends to reduce the difference between the reference input (the desired flight path) and the system output of the actual aircraft position. When the pilot is operating in clear air, information about the attitude of his aircraft, its position, and velocity with respect to objects on the ground and other aircraft are fed back to him through his eyes. Under these VFR conditions, a navaid is not necessary because the pilot navigates using visual “landmarks.” The feedback system is then a special case of that shown in Fig. 12, where the block marked nauaid is replaced with a straight line and the display is eliminated. When visual information is unavailable, the pilot can use his cockpit instruments to obtain pitch, roll, heading, air speed, rate of turn, altitude, and rate of descent and, when combined with considerable skill, may reach his destination. In general, however, he does not have the information to obtain the guidance needed to follow a prescribed path over the ground, much less for approach/landing maneuvers in mixed air traffic.


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It is the purpose of the navigation system to help the pilot follow his desired flight path by providing electronic “landmarks” under IFR conditions. This desired flight path can be stored, for example, in computer memory. In some cases, only several numbers need be preprogrammed; this is possible when the flight path is simply a straight-line segment connecting two known enroute waypoints or is a line segment connecting the ILS outer marker with the decision height during a landing approach. Information from the navaid “tells” the pilot where the aircraft is. The difference between the desired flight path and the actual aircraft position is the correction (or error) signal. The pilot flies along his desired flight path by maneuvering his aircraft toward an indicated null position on his display. This null seeking process is mechanized using a negative feedback control system, as shown in Fig. 28. In accordance with control system terminology, the aircraft is the controlled element, the navaid generates a measurement relative to a known reference, and the controller is a human pilot or autopilot which couples the correction signal to the aircraft. The correction signal is maintained near its null position by changing the direction of the aircraft’s velocity vector by controlling the attitude of the aircraft. Inducing a roll angle command causes the aircraft to turn left or right from its flight path. Similarly, a change in its pitch angle causes the aircraft to fly above or below its intended flight path. An early paper which discusses the general problem of automatic flight control of aircraft with radio navaids is by Mosely and Watts (1960). As stated in Graham and Lothrop (1955), “the human pilot himself represents an equivalent time lag of 0.2-0.5 second, therefore, he is seldom able to operate at ‘high‘ gain without instability (overcontrolling). For this reason the pilot is generally unable to achieve the accuracy and speed of response of an automatic system. He represents, however, a remarkable

-

STEERING CONTROLLER (AUTOPILOT)

AIRCRAFT (CONTROLLED ELEMENT)

a

POSITION ESTIMATE A

-

r(t)

t

AIRFRAME ATTITUDE SENSORS

c

+

OBSE&VATlON

at)

RADIO NAVAID (SENSOR)

.

‘(t?;A;pdN’


79

variable-gain amplifier and nonlinear smoothing and predicting filter. He can stabilize and control a wide variety of dynamics. Memory and rational guesswork often play an effective part in deciding his control actions.” An autopilot can be used to overcome the limitations of the human pilot. Autopilot feedback control provides speed of response and accuracy of control. These particular advantages are enhanced by high gain. Unfortunately, the combination of fast, high-gain response will adversely affect dynamic stability. If time lags are present in the control system, high gain can cause the system to hunt or oscillate. It also usually increases the susceptibility of the system to spurious signals and noise. For these reasons, an autopilot, unlike the human pilot, usually gives satisfactory performance in only a narrow range of system parameters. As noted in Graham and Lothrop (1955), design of an automatic feedback control system for an aircraft begins with a thorough understanding of the inherent mechanics of aircraft response to its controls. The conventional airplane has at least four primary flying controls-ailerons, rudder, elevator, and throttle-which produce forces and moments on the airframe.6Actuation of these controls gives rise to motions in the six output variables: roll, pitch, heading, forward velocity, sideslip, and angle of attack. These motions (in particular the aircraft attitude change) redirect the velocity vector (whose time integral is the flight path), as shown in Fig. 29. Qualities of an aircraft which tends to make it resist changes in the direction or magnitude of its velocity vector are referred to as stability, while the ease with which the vector may be altered to follow a given course are referred to as the qualities of control. Stability makes a steady, unaccelerated flight path possible, and maneuvers are made with control. 2. Aircraft Guidance and Control Using Navigation Systems

As shown in Fig. 29, the aircraft’s center of mass (c.m.) is positioned on the tip of a position vector r(t). Its time derivative dr(t)/dt is the aircraft velocity v. Its components are measured relative to the same earth reference coordinate system as the position vector. The aircraft’s attitude (pitch, roll, and heading)is measured relative to a translated replica of the ground coordinate system which rides on the tip of the moving position vector. That is, the rotation of the aircraft’sairframe coordinate system (Fig. 29) is defined by the Euler angles relative to the translated earth-fixed coordinates. Similarly, the guidance signal after a suitable coordinate transformation in the airborne navaid is defined with respect to the earth-fixed coordinate frame. Since the aircraft’s position as measured by the navaid and the pilot’s desired position are in the In the cockpit, rotation of the pilot’s wheel moves the ailerons.Pushing the wheel’s column back and forth raises and lowers the elevators.

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ROBERT J. KELLY A N D DANNY R. CUSICK A/C ATTITUDE COORDINATE SYSTEM PITCH =

ep

ROLL = 0 HEADING = 0

AIC POSITION VECTOR

REFERENCE SYSTEM

LATERAL CH-ANNE L COMMAND DEVIATION SIGNAL

FIG.29. Aircraft attitude, position, and velocity coordinate definitions.

same coordinate system, then their difference generates the correct deviation signal. The AFCS for a CTOL aircraft is a two-axis system having lateral (roll) and longitudinal channels. The longitudinal channel is further partitioned into a vertical mode (pitch) and a speed control (thrust) mode. The input signals to these channels depend upon the deviation signal’s relation to the aircraft’s velocity vector. For example, assume as shown in Fig. 29 that the aircraft’s velocity vector v(t,) is in the horizontal plane (tangent plane). Then the component of the deviation signal which is assumed to be also in the horizontal plane and perpendicular to v(t2) will enter the lateral channel of the autopilot. It is important to note that the aircraft, for the practical reasons given below, does not maintain its commanded course using angular correction signals. Each navaid sensor must eventually have its output signal transformed to the rectilinear coordinates of the earth-fixed reference. Although an AFCS can be designed to receive guidance signals in any coordinate system, rectilinear coordinates are preferred because the flight mission is always defined in terms of linear measure (feet, meters). In the aircraft landing application, the aircraft must ultimately fly through a decision window having linear dimensions so that it will land in the touchdown zone located on the runway. This is why the MLS angle accuracy standards (ICAO, 1981a) are given in linear measure and not in angular measure (degrees).


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Similarly, in the enroute and terminal area application, the flight paths (air lanes) are defined and separated in terms of linear measure such that the aircraft cannot “bump” into each other. In both cases, it is the linear deviations from the intended course in feet or nautical miles at each point in space, which are important for aircraft navigation. Guidance data can be derived from any navigation sensors as long as they are transformed into the appropriate rectilinear coordinate system of the autopilot channels. Although not immediately obvious, even straight-in MLS (or ILS) approaches, as illustrated in Fig. 30, require this transformation. In the literature the process is usually called course softening or desensitization. Lateral conversion is obtained by simply multiplying the azimuth guidance angles by the range to the antenna site, using information provided by the DME/P element of the MLS ground equipment. In Fig. 31, the aircraft and the navigation aids are placed in their automatic feedback control system configuration. The block diagram functionally represents one of the two autopilot channels. The intended flight path is the input signal command with the output being the actual aircraft path. The outer loop, also called the guidance loop, contains the aircraft and flight control systems, the ground reference navigation aid (MLS, ILS, DME, TACAN), the airborne sensor, and coordinate transformer. A display device which allows the pilot to monitor or use the measured deviation of the aircraft from the selected flight path completes the system. Unlike today’s ILS, which supports straight-in approaches only, the MLS guidance loop must be prepared to accept any of several types of input signals as itemized in Fig. 31. In addition to the guidance loop, there is a series of inner loops involving the feedback of airframe motion quantities such as attitude (roll and pitch) and attitude rate which serve to control the aircraft. These are called control loops in Fig. 31. As with any closed-loop control system, the AFCS guidance loop has specific gain, bandwidth, and stability properties based on the components of the loop. Separate guidance loops exist for each of the aircraft channels

5

APPROACHING AIRCRAFT WITH ITS COORDINATE SYSTEM

I-:tf(

R = DMEIP

AY

R A 6 AZ

FIG.30. Simple straight-in approach coordinate system.

82

ROBERT J. KELLY AND DANNY R. CUSICK CURVED APPROACH PATH WAYPOINTS FOR SEGMENTED APPROACH PATH mSELECTEDGLIDE PATH AND RADIAL

A I R C R A F T. -[NEAR ._...-..-. I

FROM - DEVIATION COMMAND

Y

INPUT SIGNAL

4-

(COMMANDED @OEVIA!IOtFROM POSITION) -f

FLIGHT GAIN +CONTROL SYSTEM

-

POSITION (FEET)

AIRCRAFT 3

COORDINATE TRANSFORM

t

AIRBORNE ANTENNA L

-

RADIO NAVAID

mentioned above. For CTOL aircraft the guidance loop bandwidth is less than 0.5 rad/s for the lateral autopilot channel and less than 1.5 rad/s for the vertical (pitch) mode of the longitudinal channel. 3. Qualitative Description of the Aircraft Control Problem

It is important in the following discussions to understand the distinctions between the navaid output observations, the navigation system steering commands, control law, and control action. These distinctions will be clarified below using as an example the system shown in Fig. 28. Illustrated in the figure is a functional description of an aircraft/guidance system which uses inertially aided position-fix measurements from a radio navaid. The navigation system is composed of the radio navaid sensors, airframe attitude sensors, aircraft position estimator, and the steering command summer. The problem is to control the aircraft such that its flight path r(t) follows the commanded path c(t).How closely r(t) follows c(t)is determined by how timely and accurately the aircraft velocity vector d r / d t can be changed. Automatic flight control systems are easily comprehended using the notation and bookkeeping of the state-variable formalism (Schmidt, 1966).In


83

this formalism, drldt is related to r and the airframe attitude states through a system of first-order differential equations. Integrating dr/dt then yields the aircraft flight path. Controlling the aircraft trajectory proceeds according to the following sequence: (1) Radio navaid sensors (e.g., VOR/DME) measure the aircraft position state observable z(t), which contains the aircraft true position states r ( t ) plus random disturbances. (2) Estimating techniques on board the aircraft attempt to extract r ( t ) from z(t). The resulting estimate r ( t ) can be obtained by suppressing the random disturbances using a simple low-pass filter or a sophisticated technique such as a Kalman filter. ( 3 ) The estimate r(t) is compared with c ( t )to determine if control action is required. If they are equal, no added control is necessary. That is, the steering command remains unchanged, which, in turn, does not change the aircraft control surfaces (e.g., ailerons or elevators). (4) A “control law” is used to calculate the control action necessary to make the actual flight path r ( t ) at time t + At equal the command c(t + At). The “control law” must also use the velocity vector dr/dt to ensure that r(t) is stable and does not oscillate around c(t). The velocity input dr/dt can be obtained from r ( t ) directly, or it can be derived from the airframe attitude sensors. ( 5 ) Application of control action in accordance with the “control law” is achieved by changing, for example, the aileron position to bank the aircraft SO that its c.m. position r ( t ) will move to follow the command c(t).

Using the above sequence to fix the ideas contained in Fig. 28 permits the terms guidance and control to be succinctly defined. Guidance (McRuer et al., 1973) is the action of defining the velocity relative to some reference system to be followed by the aircraft. Control is the development and application of appropriate forces and moments that (1) establish some equilibrium state of the aircraft and (2) restore a disturbed aircraft to its equilibrium state. Control thus implicitly denotes that r ( t ) will be a stable flight path which is achieved using both position displacement feedback, r(t), and rate feedback, dr/dt. When performing an analysis, it is the deviations about the intended flight path c ( t )which are important. In this case, a perturbation model can be used which, in many applications, may reduce a nonlinear AFCS to a system of linear first-order differential equations. The following discussions assume that the controller is an autopilot. The key issues associated with a radio navaid and its interaction with AFCS can be clearly presented using the lateral guidance loop as a discussion

84


example. In the following the terms A+, A$, and Ay are deviations from the desired attitude or course. Further the notation A j will represent the velocity of the deviation signal at Ay. The detailed mechanism for turning an aircraft onto a desired flight path proceeds as follows. A lateral position correction signal Ayc (ft) is transformed into a roll command A4,(deg) which causes the aircraft ailerons to deflect a given number of degrees. Relative to position changes in the aircraft c.m., this action occurs rapidly (< 1 s). The moment generated by the air flow across the deflected ailerons causes a change A 4 in the aircraft attitude about its roll axis. After about 1 s A 4 = A$c. Slowly the aircraft velocity vector begins to rotate such that the aircraft’s c.m. departs laterally from its previous flight path until Ay = Ayc. During this time the aircraft heading $ is also changing. A widebody jet will reach this new equilibrium state in about 10seconds. This process continues until the aircraft is on the desired flight path as represented by a new velocity vector with track angle $T. The feedback loop which tracks attitude changes A$ are called control loops; those which track the c.m. changes Ay are called guidance loops. Thus the time constants of the control loops are faster than the time constants for the guidance loops. The accuracy with which the null-seeking loops track the attitude and position commands is dependent on loop gain, which, in turn, must be chosen to ensure that they remain stable over the range of input parameter changes. An unstable control loop would lead to uncontrolled attitude variations, which, unfortunately, would not be related to the steering input command signals. Under such conditions, the aircraft cannot be controlled to stay on the flight path. In the following discussions it will be assumed that the control loops are stable. Attention will now focus on the requirements for a stabilized guidance loop. A small tracking error, sufficient guidance loop stability margin, and good pilot acceptance factors are the critical performance indices necessary to achieve a successfulautocoupled guidance. Each of these points will be treated separately. Clearly, an autocoupled aircraft must accurately follow the desired track angle in the presence of steady winds, gusts, and wind shears. To correct the differencebetween the desired course (input) and the aircraft track (output), the guidance system must obtain the position and velocity of the aircraft and determining its heading. Assume a pilot wants to turn the aircraft to follow a new course which intersects his present course to the right. His navigation system will provide a displacementcorrection signal Ay to the autopilot which in turn generates a roll command to bank the aircraft. If only the position displacement signal were used, it is apparent that a stable approach could not be achieved, because the aircraft would continuously turn as it approaches the new course and overshoot it. That is, the air-


85

craft’s c.m. would be accelerated toward the new course in proportion to the distance off course as if it were constrained to the course by a spring. Without damping, the c.m. would oscillate continuously about the new course. To prevent this unstable maneuver, a term related to the rate of closing toward the new course must be introduced. That is, to use the above analogy, the spring requires damping. Such a “rate” term A j is available either by differentiatingdirectly the position displacement A y with respect to time or by using the relative angle between the aircraft track angle and the new course bearing. In either case, a signal proportional A j can be derived to provide dynamic damping. An equally important situation arises when the aircraft is subject to steady winds and wind turbulence. Because an aircraft has a response time inversely proportional to its guidance loop bandwidth, presence of wind should be detected as soon as possible. Inserting a rate signal A j in addition to Ay into the autopilot anticipates the wind disturbance so that the aircraft’s attitude can be adjusted to counter the effects of wind force and moments which may push it off course. These examples qualitatively demonstrate why both position and velocity information are necessary for stable flight path control. To achieve the tracking accuracy necessary to guide the aircraft through the Category I1 window on a final approach places additional demands on the feedback signals. The aircraft must be tightly coupled to the guidance signal, which, in engineering terms, means that its guidance loop gain must be large. High guidance loop gain, however, is synonymous with high guidance loop bandwidth. A wide guidance loop bandwidth is necessary so that the highfrequency components of the wind gusts can be counteracted by the AFCS. This is desirable to decrease the influence of wind, wind shears, and gusts which tend to cause the aircraft to depart from its intended approach and landing path. On the other hand, tight coupling is undesirable if the guidance signal is noisy or has guidance anomalies caused by multipath. In this case, the tightly coupled aircraft will tend to follow the anomalies and depart from its intended approach and landing path. Regardless of whether the departures from the intended path are because of winds, shears, and gusts or due to anomalies in the received signals, they are tracking errors. With this background the discussion will move on to examine the performance of the radio navaid. Unlike wind disturbances, which can be attenuated by increasing the loop gain, guidance signal noise will not be attenuated because its source is not in the forward portion of the loop. Increasing the loop gain (or bandwidth) amplifies the effects of the noise. The situation is further compounded when the rate feedback signal is derived from the guidance signal, since the differentiating process accentuates the high-frequency components of the noise.

86


Reducing the guidance loop gain is not always a solution because the effects of wind gusts and shears may increase. Removing the higher-frequency components from the wind disturbance means that they are not available as correction signals. The consequence is larger tracking errors. Narrow loop bandwidths (low gain) will also degrade the aircraft's transient response, perhaps to a point which is unacceptable to the pilot. Heavy filtering of the navaid output signal does not alleviate the above problems. The data filtering process introduces phase delays on the guidance signal that may erode the AFCS stability margin, which is only about 60" to 90" to begin with. That is, an additional phase delay of 60" to 90" will destabilize the system. There is yet a third problem. To maintain a sufficient phase margin, the navaid data filter bandwidth must be much wider than the guidance filter bandwidth. Guidance signal noise may disturb the roll and pitch control loops and also the control surfaces (ailerons and elevators). Since the wheel is coupled to the ailerons and the column is coupled to the elevators, unwanted control activity may also induce unwanted wheel and column motion. During an autocoupled approach, excessive control activity can cause the pilot to lose confidence in the AFCS. Nevertheless, although these noise components cause airframe attitude fluctuation and control surface activity, their spectral content lies outside the guidance loop bandwidth. The induced motions are too rapid to influence the direction of the aircraft's c.m. and therefore do not afect the aircraf's Jlight path. How are the above noisy navaid issues resolved? For some autocoupled enroute applications using VOR/DME and for some ILS approaches, the solutions include (1) doing nothing and accepting the unwanted control activity, (2) decreasing the loop position gain and accepting larger tracking errors, (3) using inertial aiding for dynamic damping, (4)reducing the control activity by reducing the autopilot gain and the navaid filter bandwidth. Clearly the solutions involve trade-offs between tracking error, stability margin, windproofing, control activity, and costs. A more sophisticated solution is the complementary filter.' Subsection E,6 describes the complementary filter approach used for more demanding ILS applications. MLS system designers took the approach that the radio navaid designs must begin with a low-noise output signal. The MLS angle and the DME/P standards, which are the most recent navaid standards approved by ICAO,

' A complementary filter blends the good high-frequency characteristics of the inertial sensors with the good low-frequency characteristics of the radio navaids. The overall combination will then have the desired wideband frequency response with low sensor noise and minimum data phase lag. A complementary filter permits the radio navaid output data to be heavily filtered if necessary.


87

attempt to correct the deficiencies of the previous systems by purposely defining low-noise accuracy standards even in the presence of multipath. Despite these more stringent requirements, the MLS signal-in-space may require some degree of inertial aiding or complementary filtering to provide the high dynamic damping and the position accuracy necessary for achieving successful Category I1 approaches, Category I11 landings, and automatic landings under VFR conditions.

4. Radio Navaid Data Rate In all AFCSs the data sampled guidance signal must eventually be reconstructed back into an analog signal since, in the final analysis, the aircraft moments and wind forces are analog by nature. For sampled data systems such as MLS, DME, and TACAN, the position samples must be provided at an adequate data rate so that the signal can be faithfully reproduced. Undersampling can generate an additional tracking error component and can induce an additional phase delay after the signal is reconstructed. For example, guidance deviations caused by wind gusts must be accurately reconstructed so that the control loops can apply the appropriate compensating corrections in a timely manner. Reconstruction circuits (e.g., a zero-order hold, ZOH) introduce phase delay in addition to the data smoothing filter. In order to provide a low-noise guidance signal with a bandwidth that satisfies the requirements developed in Subsection E,3, the navaid data rate must be high enough to (1) minimize the signal reconstruction errors and the reconstruction circuit phase delays, and (2) allow data smoothing while still providing enough independent samples to reconstruct the signal. It can be shown that if thebutput data filter satisfies a six-degree phase delay criteria, then it will also satisfy the data reconstruction requirements for independent samples (Ragazzini and Franklin, 1958).Therefore, only the six-degree phase delay filter criterion need be satisfied, as discussed in Subsection E,7. The second requirement is that the data rates must be high enough to permit filtering of the raw data samples if the CMN is excessive. For a given noise reduction factor g, the second data requirement can be derived from Eq. (40)in Section IV,D,l. It is data rate = f , = 29' (noise BW), where (noise BW) = (a/2)wfi, for a single-pole filter. For example, if the raw CMN is 0.14, then to achieve 0.05" CMN at the input to the AFCS requires that g 2 = 8. Assuming mfi, = 10 rad/s, then the data rate must be at least 40 Hz. Figure 37 illustrates the concepts described above. The data rate relation applies when the noise source has a broad power spectral density such as the MLS azimuth function. If the radio navaid sensor cannot provide an adequate number of independent samples, some form of inertial ordering will

85


be necessary. Example implementations are given later in Fig. 38.’ The need for small tracking errors and wide-bandwidth navaid filters that minimize phase delay points toward data rates which are about 20 times the loop guidance bandwidth. 5 . Functional Description of AFCS The conclusions in Subsections E,3 and E,4 placed qualitative accuracy, bandwidth, and data rate constraints on the output data of radio navaid sensors. With these constraints an aircraft guidance system can be designed that is accurate, responsive, and stable. In this section, the key issues will be revisited using the mathematical formalism of control theory. This elementary review will serve to fix clearly the ideas treated earlier and to indicate how the aircraft and its navigation system can be analyzed quantitatively. For the moment, the AFCS functional description will be general, applying either to the lateral or to the vertical channels. Figure 32 is the basic control system, where K is the autopilot gain, H,(s) and H2(s) represent the transfer functions of the autopilot, aircraft, and the aircraft dynamics. F(s) is the transfer function of the sensor, s is the Laplace transform variable, and the input/output functions, c(t) and r(t), are scalar quantities. In an ideal control system the controlled output r ( t ) should followthe commanded input c(t) with the smallest mean-square tracking error e(t)’, where e(t) = c ( t ) - r(t). This is the optimum meaning of accuracy, fast response, and stability. The intended purpose of the AFCS is to keep the aircraft as close as possible to its intended course c(t), even in the presence of wind gusts d(t) and with an imperfect radio navaid having error n(t). Let R(s) and C(s) be the Laplace transforms of r ( t ) and c(t), respectively. For the time being, let the sensor error source n(t) = 0 and the disturbance d ( t ) = 0; then, from Fig. 32,

Equation (1) is the closed-loop transfer function of the aircraft’s guidance loop. The 3 dB bandwidth of R(s)/C(s)is the loop guidance bandwidth w 3dB; the forward transfer function is KH,(s)H,(s), and the open-loop transfer function is KF(s)H,(s)H,(s). For radio navaids, F(s)is equivalent to a low-pass filter. As stated in Subsection E,3, the 3 dB bandwidth of the navaid sensor transfer function F(s) should be much larger than the loop guidance bandwidth w~~~ for two reasons. One, the phase delays of the corrective signal’s components should not erode the phase stability margin of R(s)/C(s).

* Accuracy and data rate limitations are among other considerations that may preclude GPS from being used as a primary navaid in approach/landing applications.


89

WIND DISTURBANCE

d(t)

GAIN K

AIRCRAFT

FIG.32. Generalized block diagram of attitude controlled A/C and navaid.

Two, the spectral component of c ( t ) and d ( t ) should be faithfully reproduced so that e(t)2 is maintained within design limits. When F(s) cannot satisfy these requirements, the radio navaid may require some form of inertial aiding. An important design consideration is the effect of sensor disturbances upon the tracking error. The AFCS should respond to the desired signal c(t), but not respond to extraneous noise sources. Let N(s)be the Laplace transform of n ( t ) the error term of the navaid sensor. In Fig. 32, let c(t)=O and d ( t ) = O ; then the closed-loop transfer function R(s)/N(s)for the noise source is identical to Eq. (1). Again, the navaid sensor determines the guidance system performance. Unfortunately, the characteristic that F(s) should possess for small tracking errors, when combined with an imperfect navaid, is in conflict with those required for the command signal c(t). For the command signal, the bandwidth of F(s) should be greater than co3dB, while, for the navaid error source n(t), the bandwidth of F(s) should be small. In other words, the spectral content of c ( t ) should be unaltered while that of n(t) should be suppressed when the variance of n(t) is excessive. An important property of a closed-loop AFCS transfer function is its capability to suppress wind disturbances. Again the essential design consideration is high loop gain. In Fig. 32 let D(s)be the Laplace transform of d ( t ) and let c(t) = n ( t ) = 0. Using Fig. 34 the closed loop transform for a wind disturbance is R(s) - D(s)

H,(s) 1 + KF(s)H,(s)H*(s)

(2)

The effects of negative feedback on d ( t )can be compared with c ( t )and n ( t ) by inserting the wind disturbances on the input side of H(s). The transfer functions for all three sources are now equal to Eq. (1) if D(s) is transformed

90


into D(s)/KH,(s).Wind disturbances are therefore reduced by large autopilot gain. In summary, the loop bandwidth, w 3 d B , must be sufficiently large to reproduce c ( t ) faithfully and attenuate d ( t ) , but not to the extent that n(t) introduces excessive tracking errors. After choosing w3 dB, the navaid noise can be filtered to a level consistent with the desired control activity and windproofing using a complementary filter. The formalism above implicitly assumes the control loops needed for dynamic damping. The tracking error e ( t ) thus contains three components: the effects of the band-limited signal c(t), the wind disturbance d(t), and the radio sensor error n(t). a. Lateral Channel Control Law. The control law for the lateral channel will now be developed. The lateral channel is chosen because essentially the same control law is used in the enroute, terminal, approach, and landing phases of flight. As stated in Subsection E,3, it is necessary that the bank angle command be proportional to the lateral deviation from the desired flight path. Moreover, in order for the flight path to be stable, the bank command must also be proportional to the rate of change of the path deviation. Consequently, the bank angle control law assumes the form (3) A 4 c = K , A y +- K + A j Not included in Eq. (3) is integrated feedback, which removes any dc offset errors. Control law gain coefficients are chosen to ensure a given stability margin while maintaining an operationally acceptable tracking accuracy. It is an involved analysis using root locus techniques (see Bleeg, 1972,Appendix B). Although the derivation of these coefficients requires extensive analysis, once they are determined, 03dB can be derived using simple relations such as those given in Fig. 33(b). This is fortuitous because the guidance loop bandwidth (03 dB) is critical to the navaid filter design. The control loop's performance, on the other hand, will not succumb to a simple analyses; a full-blown computer simulation is necessary. At this point, the design engineer must make several decisions. From what sensor should the lateral path deviation A y and its velocity A j be measured? Ideally, one would want to obtain A y from the ground reference navaid, and using that position measurement, derive the velocity data needed for dynamic damping. Relative to inertial aiding this is the most economical approach. However, as noted in Subsection E,3 there are several important considerations depending upon which phase of flight is being addressed. As expected, the autocoupled final approach/landing phase has the most stringent requirements. Anomalies in A y and A j must be limited such that unwanted roll commands A$c as expressed by Eq. (3) are limited to less than 2" (95% probability) (paragraph 2.1.4, Attachment C, ICAO, 1972b; Kelly, 1977) and Fig. 38. It is the opinion of many fight control engineers that the 2" roll


91

requirement for autocoupled flight also applies to the terminal area and enroute applications. Control law considerations for MLS are discussed in Section II,E,6. For the enroute application, a wide variety of control law implementations are available to the user, depending upon whether he flies the VORTAC radials or uses RNAV routes, and whether he files manual or has his RNAV outputs coupled directly to the autopilot. When flying manual, the displayed VOR and DME raw data may be adequate because the pilot is more forgiving when he is in control of the aircraft. However, when the aircraft is in the autocoupled mode, autopilot tracking errors are viewed more critically by the pilot because he is not in direct control of the aircraft. Thus, DME and VOR position data for the autocoupled mode may require velocity inertial aiding. Anomalous attitude or control surface activity under no wind conditions is disconcerting to the pilot and may result in negative pilot acceptance factors. On the other hand, the pilot expects control activity when he encounters air turbulance. Small tracking errors in the presence of wind and good pilot acceptance factors are desirable because pilot work loads can be reduced and fuel consumption can be minimized. Only when the pilot can use the autocoupled mode with confidence will these goals be achieved. Although some ILS landing operations require the position data to be complemented with inertial aiding (see Fig. 38c), the major concern is the accuracy of the velocity information. Only the velocity information considerations will be treated below. In order to address explicitly which variable can be used to provide the velocity feedback components, the physics and geometry of how an aircraft negotiates a lateral turn must be analyzed. Figure 33 illustrates such an analysis, where a small-signal perturbation model has been developed, i.e., c ( t ) - r ( t ) = Ay when the desired path has changed from c ( t ) = 0 to c(t) = Ay. The necessary variables can be determined using Eqs. (1)-(5) in Fig. 33(b). They are derived by successive integrations beginning with the equation of motion for a coordinated turn [Eq. (I)]. Equations (1)-(5) illustrate how the velocity vector is changed so as to achieve a desired flight path. For ease of exposition the equations of motion in Fig. 33, assume that the direction of the aircraft’s c.m., ICIT, equals $; i.e., the effects of wind are negligible. Also assumed is the complete decoupling between the roll and directional axes. Since o3dB is small compared with the natural frequency of the bank angle transfer function 6/OCN 1. The actuator dynamics have also been neglected. The variables available for dynamic damping are: (1) A j as derived from Ay [Eq. (4)] (2) bL as integrated from 4, where CEq. (113

6 is

measured by a vertical gyro


92 a

AIRBORNE RECEIVER

P. 0

e.g VORIDME

i

”,

@ ‘BANK ANGLE (DEGREES) $ = AIC HEADING IDEGREESI

GROUND REFERENCE NAVAIDS e.9. VORIDME

p w =WIND GUST

V, = AIC AIRSPEED (FEETISECOND)

9

=

GRAVITY I32 FEETISECOND’)

$

=

LAPLACE INTEGRATION OPERATOR

W

=

-

IDEGREESl

WIND SPEED IFEETISECOND)

S = LAPLACE DIFFERENTIATION OPERATOR

NOTE: A y IS I N THE HORIZONTAL PLANE NORMAL TO c(tl

FIG.33. (a) Lateral channel perturbation model.

(3) $ (heading), as measured by the directional gyro or radio compass rEq. (3)1 (4) $T (track angle), as determined from v and measured by referenced accelerometers. Under wind conditions $T # $ at point A in Fig. 33. $T = tan-’(j/i), where v = From v the feedback correction term A$T is calculated, where is the difference between the commanded track angle and the true track angle $T. (See Fig. 11.)

Jm.

Assuming that A j cannot be derived accurately from the radio navaid, A$T is the ideal input for dynamic damping, since it is based upon the true tracking angle of the aircraft. However, the determination of At)T requires an expensive INS platform or its equivalent in strap-down inertial sensors. Less accurate, but perhaps adequate, tracking performance can be obtained from the heading $ and the bank angle 4 to obtain the damping signals. In the 1950s and 1960s, heading feedback was the solution used in many autopilots for approach and landing operations. However, when the aircraft is subjected to a low-frequency crosswind, the tendency of the airplane is to


ASSUME

(1)

va

=

mv2 =

Ra

93

7 mg Tan A@

BUT V = A $ X R,

THEREFORE

t mg

FIG.33. (b) Physics of a lateral turn.

"weathercock" in the wrong direction. This produces a heading change which is not related to +T (Bleeg et al., 1972). In general

$*=*,+*=*,+-

:sd

4dt

(4)

where $, is the initial wind heading transient, 4 is a function of time, and the wind disturbance is injected as indicated in Fig. 33. Thus, using $ as a damping term, is not an ideal solution in the presence of steady wind and low-frequency wind shears. Before the advent of INS, the means to obtain dynamic damping and windproofing was to measure the bank angle directly and to perform a long-term constant integration, as indicated by

94


Eq. (4). A solution was to incorporate the operation called lagged-roll, as indicated in Fig. 33. However, pure integration was not desirable because the airplane may be mistrimmed; i.e., it may have a steady roll angle which could prevail throughout the approach.’ Another approach to obtaining dynamic damping is to combine heading with a rate signal derived from the imperfect navaid sensor in a complementary filter. The heading signal has good highfrequency response and correspondingly poor low-frequency wind response while the navaid rate signal has good low-frequency characteristics. Such an example is given in Brown (1983) for the ILS application. Because the exposition which follows will be simpler and clearer, $ will be used to obtain A j with the full understanding of the limitation mentioned above. The use of K , A$ = K+A j will allow an estimate to be made in the tracking error as generated by the sensor noise errors. Equation (3) now becomes

A4= = K , A y

+

(5)

The block diagram in Fig. 33 is obtained by starting with Eq. (5) and using Eqs. (2)-(5) of Fig. 33. The blocks with dotted inputs and outputs are the alternate feedback terms A j and A&. Wind effects which push the aircraft c.m. off course can be included by a summing junction in Fig. 33. For a steady crosswind, the input to the summing junction is a step function. The resulting path deviation A y would be a ramp function. A wind shear would be represented by a ramp function at the summing junction input. The composite transfer function using only heading feedback is given in Fig. 34. The analysis assumes that the sensor output data filter bandwidth is large compared with the guidance loop bandwidth (look ahead to Fig. 38). As shown in the figure, the transfer function is a low-pass filter of second order, which is given in standard form in terms of the damping factor t and the natural frequency wo to emphasize the important performance factors. Also given is the expression relating the loop guidance bandwidth w3dB to w o . By choosing K , and K,, the loop guidance bandwidth wJdBand t can be adjusted to suit the enroute, terminal, approach, and landing phases of flight. For the approach/landing phase 5 z 0.5-0.7 and w3dB2: 0.1-0.2 rad/s. Note that if K , = 0 (no damping, t = 0) then the guidance loop is conditionally stable; i.e., it will become unstable with any input. This condition dramatizes the need for velocity information in the aircraft guidance loop. As shown in Fig. 34, increasing or decreasing K y changes w 3dB as mentioned several times in Subsection E,3. If rate feedback were incorporated as shown in Fig. 33(a), then wo = 4and 5 = 3(Kj/K,)wo. Using I) in the control loop to maintain a constant aircraft heading will cause the aircraft c.m. to drift off course in the presence of wind. To avoid drift errors in the approach/landing application the aircraft “decrabs”just before touchdown.


95

NATURALFREQUENCY OF THE GUIOANCE LOOP

STANOAROIZEO TRANSFER FUNCTION

Jc

LOOP GUIDANCE BANOWIOTH w3db =wo

KII, IS IN '$)/

t 2,

+

2-4

t 2+ 4

OJ/

NOTE: THE COEFFICIENTS VARY SLOWLY WITH TIME AS A FUNCTION OF THE AFCS GAIN SCHEDULING

KylSIN O@/FT W o IS

1-2

I N RAOIAN / SEC

FIG.34. Transfer function of lateral channel perturbation model.

Using certain simplifying assumptions, the desired tracking error can be easily calculated. The mean-square tracking error e ( t ) = c ( t ) - r ( t ) is defined by

S'

-

e(t)2 = lim e 2 ( t ) d t= T - ~ -~T T

where E(s) is the Laplace transform of e(t). If the input command signal c(t), the wind distance d(t), and the radio navaid errors n ( t ) are zero mean uncorrelated random variables, then 3

2

=

C(t)++ d(t)++ n(t)+

(7)

96


where each component on the right is the tracking error contribution due to c(t),d ( t ) , and n(t), respectively. Data sampled reconstruction effects have not been included. (See Subsection E,4.) Note that inertial aiding component errors are assumed to be negligible. Assume in the following that the bandwidth of c(t)is well inside w 3dB, then c(t)gwill be small. Assume also that the autopilot gain is sufficient to suppress the effects of d(t), then under these assumptions, e(t)’ z n(t)+. Since the guidance loop transfer function H ( s ) has been derived in Fig. 34 using heading as the damping term, the tracking error due to sensor error perturbations can now be calculated. Using Eq. (6)

where E [ s ] = H ( s ) N ( s )and N ( s ) is given by (9) below. Let the sensor noise be represented by a Gauss-Markov process (Brown, 1983), then its power spectral density is

where the sensor filter bandwidth 0, >> W j d B and 0,’is the noise power at the output of the sensor. Then N(s) = m

/

(

S

+ w,)

where, for example, in the autocoupled approach/landing application w , = 10 rad/s. Performing the integration in Eq. (8) using the integrals evaluated in James et al. (1947), the tracking error is

-

e(t)2= o ~ K , u / w , K , ( 5 7 . 3 )

(10)

Thus, although the tracking error due to a sensor with noise errors is proportional to autopilot gain K,, it is also inversely proportional to K,. For a given noise power spectral density S,(w) and aircraft speed u, the tracking error can be reduced by decreasing the ratio K,/K,. Since there are two degrees of freedom, K , is selected to achieve the desired dynamic damping, while K, is adjusted to obtain the desired tracking error. The above analysis assumed that the noise error of the heading feedback was negligible. If the navaid error n ( t ) has a bias component, it would be added to Eq. (10) on a root-sum-square basis as indicated in Subsection C,3. Equation (10) has a simple physical meaning when w, >> w J d B . The noise power spectral density given in Eq. (9) becomes S , N 2a,’/w,, a constant. Using the relations for K , and K, in Fig. 34 and letting = 1/& then


wo = w3dB and Eq. (10) becomes = Snw3 dB/(2

97

a>

The tracking error is simply that portion of the noise power residing inside the guidance loop bandwidth which is intuitively satisfying. A similar analysis could be performed to determine the effects of wind gusts on tracking accuracy. Equation (9) can also be used to simulate wind gusts. The above discussion was meant only to introduce the basic uses of AFCS design and to show why it is necessary to have radio navaids which can provide wide-band guidance (about 10 x w3dB)with a low-noise output. A more complete discussion of the lateral channel as it applies to the approach and landing maneuver is given in McRuer and Johnson (1972). In the next section it is shown how the complementary filter permits the designer to achieve system stability without regard to the time constant introduced by heavily filtering the noisy radio navaid output. 6. The Notion of the Complementary Filter

In the preceding sections, the problem of using a radio navaid sensor with a noisy output continually figured highly in all the discussions. The problem was how to filter the noise without introducing excessive data delays. It was suggested in the earlier narrative that the solution involved the use of inertially derived position data which would augment the filtered radio navaid data. Also discussed were ways of providing dynamic damping using inertial sensors such as vertical gyros to replace rate signals derived from the radio navaid position fixes. The problem now is how to augment the position data fixes with dead reckoning position data obtained from an inertial sensor. The idea is that the radio navaid can obtain good low-frequency data after filtering while the inertial sensors (accelerometers) have good high-frequency characteristics (i.e., small high-frequency noise components). Drift errors are typically the source of the poor low-frequency performance exhibited by inertial sensors. Radio navaids have goods-frequency response because ground field monitors ensure that the bias errors are maintained within acceptable limits. The technique of combining the two sensor outputs to yield a wide-band filter without phase delays, signal distortion, or excessive noise error is called the complementary filter. The concept assumes each source has the same signal, but the measurement noise from each source is different. That is, both the radio navaid and the inertial sensor are measuring the same navigation parameter, only their measurement noises are different. The development below follows Brown (1973). Let s(t) be the position of the aircraft in flight, and let nl(t) be the inertial sensor noise and n2(t)be the radio navaid noise. Configure the filter as shown


98

in Fig. 35(a). The output is the estimate $(to = z(t). The Laplace transform of the output is

Z ( S )=

S(S)

+ N,(s)[l - G(s)] + N ~ ( s ) G ( s )

Signal term

Noise terms

(1 1)

From Eq. (11)the signal term is not affected by the choice of G(s) while the noise terms are modified by the filter. The point here is that the noise can be reduced without affecting the signal in any way. If, as in the navigation problem, the two sensors have complementary spectral characteristics, then G(s) can simply be a low-pass filter and the configuration in Fig. 35(a) will attenuate both noise sources. Finding the optimum cutoff point of the lowpass filter G(s) is an optimization problem which can be solved using the Wiener filter theory. The filter configuration in Fig. 35(a) can be rearranged into the feed-forward configuration shown in Fig. 35(b). An equivalent feedback configuration is discussed in Gelb and Sutherland (1971). The feed-forward implementation of Fig. 5(b) provides some insight into how the complementary filter works. The goal is to eliminate the signal in the lower leg by substitution thereby producing a new input signal nl(t) - n2(t).A filter design is chosen to obtain the best estimate of nl(t) in the presence of a distrubing signal n2(t).The estimate n,(t) is then subtracted from the sensor output s(t) nl(t), yielding an estimate of s(t). Intuitively, the filter mechanism is clarified if n,(t) is predominantly high-frequency noise and n2(t) is predominantly low-frequency noise. In this case, 1 - G(s) is a high-pass filter which passes nl(t) while rejecting n2(t). The Kalman filter gain optimization for the case where aircraft range is determined from a radar and the double

+

FIG.35. (a) Basic complementary filter. (b) Feed-forward complementary filter configuration.

99


integration of a strap-down accelerometer is noted in Maybeck (1979). The Kalman filter overcomes the limitations of the Wiener filter which is only optimum for stationary random processes. The configuration in Fig. 35(d) forms the basis of the inertially aided navigation system, which is the cornerstone of many modern flight management systems currently in use on new jet transports. Figure 36 (Brown, 1983; Bobick and Bryson, 1970; Gelb and Sutherland, 1971; Karatsinides and Bush, 1984) is the state vector representation of Fig. 35(b), where multiple sensors are used as inputs. It employs a linear Kalman filter which assumes that the difference between the inertial and the aiding sensor errors is small and therefore justifies the use of a linear filter. Thus the aided INS is a dead reckoning system which uses redundant navigation information from other sensors to compensate for its own error sources. The comparison of the complementary filter with the Kalman filter is given in Brown (1973) and Higgins (1975). It should be noted that the complementary filter can also be used to obtain the dynamic damping feedback by complementing radio navaid rate data with an inertial source such as the track angle error The output of the complementary filter is multiplied by the appropriate gain constant and inserted into the control law defined by Eq. (3). For example, a velocity complementary filter output would be A j c . The state vector configuration of the complementary filter is known in the literature as a “multisensor navigation system” (Fried, 1974; Zimmerman, 1969). The concept is very general. Although the example in Fig. 36 used the INS as the primary sensor, depending upon the application, radio navaid can also serve as the primary sensor. 7. Concepts of PFE and CMN

In this section the concepts of path following error (PFE) and control motion noise (CMN) are introduced together with the rationale for defining the guidance errors in terms of these concepts. They are generalizations of the traditional measures of error, namely, bias and noise. These concepts, TRUE VALUES + INERTIAL ERRORS BEST ESTIMATE OF POSITION & VELOCITY

INS

I

I\

AIDING SENSORS

KALMAN FILTER

INERTIAL ERRORS

TRUEVALUES MEAS ERROR

t

FIG.36. State vector complementaryfilter configuration.

100


originally defined in Kelly (1974), are the accuracy terminology used in specifyingthe MLS angle and the DME/P signal in space. These concepts are very general and apply to all navaids as well as MLS. The MLS angular error or DME/P range error is the difference between the airborne receiver processed sampled data output and the true position angle at the sampling time. The MLS guidance signal is distorted by ground/airborne equipment imperfections and by propagation-induced perturbations such as multipath and diffraction. To fully delineate the signal-inspace quality, these perturbations are viewed in the frequency domain, that is, as an error spectrum." This representation allows the interaction between MLS errors and the AFCS to be understood in terms of the parameters which define a successful landing. Associated with the signal error spectrum is the idea of the AFCS frequency response. The frequency response of the aircraft dynamics can be divided into three major spectral regions-a low-, middle-, and high-frequency region. Guidance error fluctuations in the low-frequency region fall within the aircraft's guidance loop bandwidth and, since they can be tracked, cause physical displacement of the aircraft. The effect of these lateral or vertical displacements is measured in terms of unsuccessful landings. The middle frequencies result in attitude changes (roll and pitch) and in control surface activity whose motions are too rapid for the aircraft to follow. They also include induced wheel (lateral channel) and column motions (longitudinal channel). Control surface motions (ailerons and elevator) can lead to component wear-out and fatigue, whereas the column and wheel motion affect pilot acceptance factors. Figure 37 summarizes the azimuth control activity criteria used by the airframe industry. One measure of these pilot acceptance factors is the Cooper Rating (Hazeltine, 1972). Although subjective and not universally accepted, it is the only test currently in wide use for measuring these effects.The high-frequency regime, which begins around 10 rad/s, does not affect the aircraft guidance system or pilot and passenger comfort measures. Figure 37, which was taken from Neal (1975),illustrates the lateral channel spectral response of a typical transport aircraft. Note that the roll and aileron responses cannot be approximated by a simple filter as can the guidance loop response (Fig. 34). Based upon the above considerations, ICAO defined PFE and CMN in the MLS angle and DME/P SARPs (ICAO, 1981a, 1985). The SARPs definitions for DME/P are: PFE-that

portion of the guidance signal error which could cause aircraft

l o The error spectrum is defined as the Fourier transform of the flight test error trace. The flight test error trace is the difference between the MLS measurement and the actual position of the aircraft.


NAVAIO

OUTPUT OATA

SENSOR

OlGlTAL FILTER

ZERO OROER ANALOGUE

BW 'WF IL

BW-fJZ

101

OELAV

-

OUTPUT :WJdbT,

w3dB = GUIDANCE LOOP BANOWIOTH

'

W3db

wFIL

fr

fJz

AZIMUTH CONTROL ACTIVITY CRITERIA COOPER RATING: LATERAL ACCELERATION: ROLL ATTITUDE VARIATION: CONTROL WHEEL VARIATION: AILERON DEFLECTION: CONDITION:

0.01

1

COOPER 11, PLEASANT TO FLY 0.04 0

2 1'

i2'

2 20

+5 2'

+5 So

15.OOO FT RUNWAY, AT THRESHOLD

0.10 0.8 1.0 SIGNAL E ~ O R SPECTRUM FREOUENCY IRADISECI __*

2.0

10

FIG.37. (a) Output data filter and data reconstruction unit. (b) Typical transport aircraft power spectral response.

102


displacement from the desired course and/or glide path. PFE frequency regions are 0-1.5 rad/s for elevation and 0-0.5 rad/s for azimuth. For the angle SARPS the PFE is partitioned into 2 additional components, pathfollowing noise (PFN) and bias. PFN is the PFE component with the bias removed. CMN-that portion of the guidance signal error which could affect aircraft attitude angle and cause control surface, wheel, and column motion during coupled flight, but which does not cause aircraft displacement from the desired course and/or glide path. The frequency regions for CMN are 0.5-10 rad/s for elevation and 0.3-10 rad/s for azimuth. Clearly, an excellent guidance signal would result if all frequency components above the path-following error regime were suppressed by a smoothing filter placed at the output of the navaid receiver. However, this cannot be accomplished without introducing a large phase lag in the higher path-following frequency components with the resulting adverse effects on AFCS stability discussed in Subsections E,3 and E,4. The MLS community chose a 10-rad/s, single-pole filter that induces less than a 6" phase lag for a 1-rad/s path-following signal component. In other words, the navaid filter should be 2 10 times the guidance loop bandwidth. The penality is, of course, that the higher nonguidance (i.e., control motion) frequency components become inputs to the AFCS:As shown in Fig. 37, the MLS and DME/P SARPs permit all MLS angle receivers and DME/P interrogators to band limit their output data to 2 10 rad/s using a single-pole filter or its equivalent. Based on the 0.2 rad/s guidance loop bandwidth shown in Fig. 37, the MLS data can be filtered (down to 2 rad/s) while still maintaining the I6" phase lag. A measurement methodology is also defined to determine if the MLS angle and DME/P equipment satisfy the PFE and CMN error specifications(Kelly, 1977; Redlien and Kelly 1981). Using the definitions of PFE and CMN above, the question of obtaining rate information from the radio navaid can be revisited. The control law in Eq. (3) requires knowledge of the bank-angle transfer function, the pseudoderivative transfer function, and most importantly the power spectral density of the MLS azimuth signal before the unwanted control activity can be estimated (see Fig. 37). An example of an MLS control law proposed for autocoupled approaches in the terminal area (Feather, 1985) is

A4c = 0.026 Ay + 0.49 A j Complementary filters using simple inertial sensors (i.e., non-INS) were used to ensure that the anomalies in Ay and A j did not cause A4c to exceed the 2" limit. MLS must be operational for several years before a statistical representation of the MLS signal can be defined. Only then will the required degree of inertial aiding for each aircraft type be accurately known. Based upon flight


103

test data collected during the MLS development program, there is reason to believe that the MLS error signal at many airports may approach an essentially flat power spectral density function. Multipath bursts, when they exist, may be of short duration. If this is the case, then at the very least complementary rate filter designs, when required, will be easier to implement in MLS-equipped aircraft. MLS control laws, as well as the tracking error and control activity criteria, are currently under development not only for the autocoupled mode, but also for flight director and manual-coupled flight. These development programs are directed at approach and landing operations in the terminal area. Critical issues are the following questions: What are the operational benefits of a curved approach? Given that the autopilot error is less than 30 ft at the Category I1 window in the presence of steady wind and wind turbulance, is an autopilot error larger than 100 to 150 ft acceptable in the terminal area? Similar control law design considerations are relevant to the enroute application for the "cruise mode." One control law currently in use for aircraft speeds of 500 knots at 30,000 ft is A4c = (0.0025 deg/ft)Ay + (1.6deg/deg)A+,. Using 5 = and the equation in Fig. 34, the guidance bandwidth for this control law is 0.04 rad/s. In this case, the position deviation Ay can be derived from VOR/DME or DME/DME. The dynamic damping is derived from the track angle deviation A+T as determined from an INS. The roll anomaly for the above cruise mode control law will be well within 2". In summary, navaid sensors (not only MLS) should satisfy the following three criteria or inertial aiding may be required:

l/fi

(1) PFE, when combined with either the flight technical error or the autopilot error, should permit the aircraft tracking accuracy to be achieved. (2) Filter bandwidth should not introduce more than 6"lag or more than 1 dB of gain at the guidance loop bandwidth. (3) The CMN level must not include more than 2" rms unwanted roll activity for autocoupled flight. The MLS guidance signals satisfy the first two criteria. Only the third criterion must be analyzed when developing MLS control laws for the terminal area and the approach/landing phases of flight. If criterion 3 cannot be satisfied, then a control law implementation having one of the forms illustrated in Fig. 38 may be necessary. 8. Interaction of DMEIP Signal with AFCS

In the ICAO SARPs, the DME/P accuracy components are described in terms of PFE and CMN. The purpose of this section is to provide the technical basis for specifying the range measurement in these terms rather than the more conventional ones of bias and noise.

104


(a) CONVENTIONAL

COMPLEMENTARY

2 K%zc drp 9

(b) INERTIALLY DERIVED RATE COMPLEMENTARY FILTER

-

RADIO NAVAID

,

AY

b COMPLEMENTARV

Ayc

FILTER

AV INERTIAL.

INERTIAL DATA h

i

(GI

COMPLEMENTARY FILTER

FIG.38. Alternate control law implementations.

Depending upon the aircraft’sorientation with respect to the MLS ground facility, azimuth, elevation, and DME signals can be inputs to any of the two autopilot channels when properly converted to linear path deviations. The precision DME accuracy standard must ensure that the DME/P has approximately the same operational accuracy as the angle MLS in specific portions of the approach path. For example, two-dimensional (constant altitude) curved and segmented approaches may require that the DME/P


105

accuracy be equivalent to azimuth guidance cross-course deviations. On the other hand, for most straight-in approaches, range data will not be required in either the lateral or vertical channels of the AFCS. As mentioned earlier, CMN has a negative influenceupon pilot acceptance factors. Resolution of this difficultyrequires the CMN to be at least limited by the system design to a maximum of 0.1" everywhere in the MLS coverage sector while utilizing an airborne output data filter bandwidth which limits the phase delay to 5" or 6" (see Section IV,D). It is against this angular error requirement that the DME/P CMN accuracy standard (when it is applicable) is derived. Since the range information is already linear, it is specified to be equivalent to the angular error of 0.1" at all points within the coverage volume. Future tests are required to determine whether 0.1" is an adequate CMN upper limit. The situation is different for the PFE component. At the MLS entrance gate, the PFE can be large because the aircraft separation standard is large ( rfi 2 nmi); however, as the aircraft proceeds toward runway threshold, the guidance error must tighten up so that the aircraft can comfortably fly through the 100-ft decision height window. Thus, unlike CMN, the PFE can degrade with increasing distance from runway threshold by a 10:1 factor. Figure 39 graphically illustrates the differences between PFE, intended course, and autopilot error. It should be emphasized again that for DME/P, CMN is only a consideration when the application requires vertical or lateral guidance using the DME/P, e.g., curved approaches. It is not critical for example in determining the 100-ft decision height. MLS INDICATED POSITION

/ /

INTENDED COURSE

/

,

'
900 kHz 8 p s after 60 dB pulse None None k 500 ft

None >700 pps 80 dB; >900 kHz 8 p s after 60 dB pulse None None f 500 ft

IA Mode: -86 dB W/m2 FA Mode: - 75 dB W/m2 Same as DME/N < 1200 pps

80 dB; >900 kHz 8 p s after 60 dB pulse -75 dB at f 2 p s f 1 p s (1 dB degrad.) f 500 ft

- 89 dB W/m2 7 nmi coverage limit -75 dB W/m2 ”’

(40)

and is applicable when the noise spectrum is flat from dc to the sampling frequency. The 2a range noise level is thus reduced, by filtering, to l/g of its initial value. Large noise reduction factors correspond to narrow-bandwidth filters. For a single-pole filter, the noise bandwidth is n/2 x (3 dB bandwidth). For example, a 10 rad/s single-pole filter utilizing a 40 Hz sample rate has a noise reduction factor of 2.8 and a phase delay of 4” at 1.5 rad/second. In addition to data smoothing with adequate transient responses, the data processing algorithm should perform two other functions that will enhance the quality of the range data: prediction of missing data samples and limiting of spurious measurement values (e.g., those due to garble) which would otherwise deviate significantly from their predicted values. When the system efficiency is less than loo%, replies will not be received for all interrogations. To “coast” through periods of missing data, the data filter algorithm replaces the missing samples with predictions based on past position and velocity data. A filter with velocity memory is the a-j? filter (Kelly and LaBerge, 1980; Benedict and Bordner, 1962).An a-j? filter which satisfies CTOL FA mode applications has a = 0.125 and j? = 0.0078. The noise


23 1

reduction factor for this filter is 3.3 and it will satisfy the “not-to-exceed aircraft flight control phase delay requirement of 6” (i.e., 4.6” at 1.5 rad/s). Assuming that the output of the data filter is close to the true range, significant deviations (greater than 300 ns) from the predicted range are then physically unrealistic in terms of aircraft response and can therefore be removed. This process is called “outlier rejection.” If the rejected sample is replaced by a predetermined value, the entire algorithm is called a “slew-rate limiter.” The outlier window width and the slew-rate limit value are determined by the error budget associated with the user application and the data filter noise reduction factor. Acceptable values are a 300 ns outlier limit with a 40 ns slew-rate limit. These parameters, together with the above specified a-8 filter coefficients, permit the DME/P to accurately track all CTOL maneuvers. Figure 91 summarizes the a-B filter algorithm which incorporates outlier rejection and slew-rate limiting.

f. Interrogation Rate Jitter. It is common practice in DME to “jitter” the time between interrogations to preclude the occurrence of synchronous interference between interrogating aircraft. This jittering process has the additional effect of reducing the aliased spectrum peaks of the range error frequency response such that the sampled frequency components outside the filter pass band are uncorrelated. This means that sinusoidal errors, such as those that can be generated by multipath having nonzero scalloping frequency,appear noiselike to the filter and are therefore attenuated according to the filter noise reduction factor given in Eq. (40). A jitter value of f50% of the intersample time is adequate to remove the aliased peaks, as illustrated in Fig. 92. E. System Field Tests

The feasibility of the DME/P concept was confirmed by a comprehensive FAA field test program. Figure 93 is an example flight error trace recorded in June, 1980, as part of this test program. It illustrates many of the accuracy concepts discussed earlier. The ground transponder was a modified Cardion conventional 100 W terminal area DME. The airborne interrogator was a modified version of a Bendix enroute DME. This equipment constituted the range element of the FAA’s basic wide MLS which was contracted to Bendix. The ground station was installed on Runway 22 at Wallops Station Flight Center, Chincoteague, Virginia, December, 1979. A laser ground tracker was used within 5 nmi of the runway threshold, and a beacon tracker was used at ranges greater than 5 nmi. The airborne antenna used on the FAA’s Convair CV-880 was a standard L-band omni type.

232

i f (data sample available) then

Fk = uk - Yk else

4

(missing data) then

if

YK

Fk = 0 end i f

A i'k I

--- - -- _+_a % ? _

__

&--

-

OUTLIER REJECTION AND SLEW R A T E LIMITER

---

--_

41

yk

Uk

~

RAN GEMEASUR EMENT

Ek

=

DIFFERENCE BETWEEN MEASURED A N D PREDICTED RANGE AUGMENT ED BY OUT LlER iSLEW R AT EiMlSSlNG D A T A ALGORITHM

Vk

=

VELOC IT Y ESTIMATE

Xk

=

POSITION ESTIMATE

Yk

=

PREDICTED POSITION

a,C

=

F ILT ER COEFFICIENTS

T

-

INTER-SAMPLE TIME

la1

=

OUT LIER REJECTION L I M I T

Ibl

~

SLEW R AT E L I M I T

DESIGN R ELAT ION S FOR OPT IMAL F ILT ER G I VEN

az

(3-

a

12-ai

3dB BAN D WIDT H IRADIANSISECI NOISE BAND WIDT H (RADIANSISECI

~

F3dB

JTa

~

3

FNB

-

'i

v a m n c e I" VeloClty 0"tP"t variance in Raw Position InDut

FgdB

PERFORMANCE MEASURES 'x

=

Variance ~n Parition Output Variance in ~ a Position w ~nput

z0 m

Dxz

~

( Unit Ramp)

=

zo m

-

Lpos,tmn Ramp

RBS~OI~IC]

Di2

=

IVeloCity o f Unit Rampl~ V e l o c ~ tRamp y Respanse~

(2-a) (i-aI2 i a2iz-al+zP i i - a i ~ap re-p-zai T~ a0 i4-0-2ar FIG.91. Block diagram of a-B filter with outlier rejection, slew-rate limiting, and missing data capability. The optimal filter provides the maximum noise reduction while providing the best transient tracking capability to a unit ramp input.


-.-.-. ---------................ -

--- _----.

----I-

233

.O .1

.2

.3 .4

.5

FIG.92. Frequency response of an a-fi filter for uniformly distributed interrogation jitter that is Ox,lo%, 20%, 30%, 40%, and 50% of the nominal intersample time of 25 ms. The aliased peaks are reduced as the jitter percentage is increased. 40 Hz sample rate, a = 0.125, = 0.0078.

A detailed description of these field tests, which verified the validity of the precision mode concept, is found in Edwards (1981). F. Conclusion This article has described the internationally standardized radio aid to navigation known as “Distance Measuring Equipment” and its evolving role in support of air navigation in the world today. Based on the beacon concept developed in the 1940s, DME was first standardized as an international system by ICAO in the early 1950s.In conjunction with VOR and TACAN, DME has since become one of the primary navigation aids in the U.S. The most recent revision of the DME standard provides for two forms of the DME system. The DME/N serves the needs of the enroute navigation, while the newly adopted DME/P standard provides the high data rates and accuracy necessary for approach/landing operations in conjunction with MLS. VOR and TACAN, together with DME, have defined the basic air route structure in the U.S. with aircraft flying station to station along inter-


234

3 DEG CENTERLINE APPROACH RUN 1 JUNE 5,1980 OUTPUT DATA (FROM FILTER) 150

N 2 2 2 FT (MEASURED)

I; 100

U

a-

ga

50

o

.

'

-50

w -100

-150

-*O0

STANDARD 1

t

STANDARD 1 PFE ERROR LIMITS

E 0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

N M FROM AZ PHASE CENTER

FIG.93. Example flight error trace from the Wallops Island precision DME field tests.

connecting radials. To put the DME system in its proper perspective, this article has described the enroute navaids, VOR and TACAN, and the landing aids, ILS and MLS, showing the role of the DME as an integral component of these systems. With the availability of economical airborne computing facilities, aircraft need no longer fly along VOR radials. Instead, with an RNAV computer, arbitrary flight paths are possible. In an RNAV environment, the navigation sensor data, be it DME, VOR, or TACAN, are used to compute the aircraft's position. The superior position fix performance of a multi-DMEbased RNAV clearly emerges from the alternative position fixing techniques based on VOR and TACAN. The basic concepts of aircraft navigation, guidance, and control were introduced by defining how the information from the navigation sensors is utilized to execute these functions.In doing so, the characteristics of the sensor


235

output data that affect the performance of an aircraft in both manual and autocoupled flight were defined. This article has described the conventional DME system-DME/Ndetailing its primary components and its evolution. The most important aspects of the DME/N signal in space, accuracy, coverage, the channel plan, and service capacity, were discussed. The standardization of DME/N has been achieved through the definition of a handful of system parameters, particularly the pulse detection and calibration methods. It has been shown that, through a combination of equipment characteristics, namely frequency rejection, decoder rejection, and channel assignment rules, an adequate number of facilities can be accommodated without mutual interference to serve the enroute, terminal, and approach/landing navigation needs of an expanding and dynamic aviation industry. Finally, this article addressed the newest member of the DME family, the DME/P. DME/P, which provides slant range accuracy on the order of 100 ft at runway threshold, is to be an integral part of the MLS. Only with DME/P can the full capabilities of MLS (i.e., full 3D guidance within the airport environment) become reality. The newly defined ICAO standard represents the successful integration of the precision DME function with that of DME/N. Its signal format provides for full interoperability between DME/N and the initial approach mode of DME/P, eliminating the need for dual equipment on the ground and in the air. In addition, the DME/P waveform satisfies the strict spectral requirements imposed on ground facilities and therefore minimizes frequency compatibility issues. The discussions have emphasized those factors which make DME/P a viable system, even in an environment characterized by strong multipath, 50% missing data, and an ever-increasing number of undesired pulses. Careful selection of the pulse shape (cos/cos2),the IF bandwidth (4 MHz), and the pulse detection technique (delay-attenuate-compare), ensures immunity to multipath in nearly all foreseeable runway environments. Use of even simple data filter algorithms, incorporating such features as outlier rejection and slew-rate limiting, will minimize the effects of missing data and accuracy degradations due to the presence of undesired pulses. Finally, the prevalence of high-speed digital technology has rendered obsolete the analog and mechanical timing/tracking techniques of the past, ensuring the availability of economicalground and airborne equipment exhibitinga 10-foldimprovement in instrumentation error. When combined, these factors yield a precision DME system that is compatible with its DME/N counterpart and that provides the accuracy required to support complex operations in the terminal area, including automatic landings. In summary, the DME system is one whose capability and utility will well serve the aviation community into the next century.

236


LISTOF ACRONYMS AiC AFCS AGC ARINC ASR ATA ATARS ATC ATCRBS AWOP

Aircraft Aircraft flight control system Automatic gain control Aeronautical Radio, Inc. Airport surveillance radar Air Transport Association Automatic Traffic Advisory and Resolution Service Air traffic control Air traffic control radar beacon system All Weather Operations Panel

CMN

Control motion noise

Diu DAC DME DOD DOT

Desired-to-undesired signal ratio Delay-attenuate-compare Distance measuring equipment Department of Defense Department of Transportation

EUROCAE

European Organization for Civil Aviation Electronics

FA mode FAA FAR

Final approach mode Federal Aviation Administration Federal Air Regulation

GPS

Global positioning system

IA mode ICAO IF IFR ILS INS

Initial approach mode International Civil Aviation Organization Intermediate frequency Instrument flight rules Instrument landing system Inertial navigation system

Mi0 MLS Mode S

Multipath-to-direct signal ratio Microwave landing system A digital data link system (formerly DABS)

NAS NOTAMs

National airspace system Notices to Airmen

PAF PFE

Peak amplitude find Path following error

rf RNAV RSS RTCA

Radio frequency Area navigation Root-sum-square Radio Technical Commission for Aeronautics

SARPs SNR SSR

Standards and recommended practices Signal-to-noise ratio Secondary surveillance radar Standard service volume

ssv

DISTANCE MEASURING EQUIPMENT IN AVIATION TACAN TCAS

Tactical air navigation system Traffic alert and collision avoidance system

VFR VOR VORTAC

Visual flight rules Very high frequency omnirange transmitters A TACAN colocated with a VOR station

237

ACKNOWLEDGMENTS The authors wish to dedicate this chapter to Jonathan and Anne Kelly and Woodrow and Christopher Cusick. We wish to thank C. LaBerge, W. Reed, and A. Sinsky of the Allied Corporation-Bendix Communications Division for their helpful technical comments and G. Jensen and K. Jensen for their much-needed editorial comments. Finally, we would like to thank M. Morgan and M. ODaniell for their efforts preparing the manuscript and Dick Neubauer for the excellent graphics.

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Office of Telecommunications (1978). “EMC Analysis of JTIDS in the 960-1215 MHz Band,” Vols. 1-7. US. Department of Commerce/Office of Telecommunications, Washington, D.C. Pierce, J. A. (1948). Electronic aids to navigation. Adu. Electron. 1,425. Porter, W. A., ODay, J., Scott, R., and Volq R. (1967). “Modern Navigation Systems,” Engineering Summer Course. University of Michigan, Ann Arbor. Quinn, G. (1983). Status of area navigation. J. Inst. Nauig. 30. Radio Technical Commission for Aeronautics (RTCA) (1946). “Report of SC-21 Meetings.” RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1948a).“Air Traffic Control,” RTCA DO12, Spec. Comm. 31. RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1948b). “Final Report of SC-40,” Doc. 121/48/CO 24. RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1963). “Standard Performance Criteria for Autopilot/Coupler Equipment,” Doc. DO-1 18 (SE-79). RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1978). “Minimum Performance Standards Airborne DME Equipment Operating Within the Radio Frequency Range of 960-1215 MHz,” Doc. DO-151A. RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1982). “Minimum Operational Performance Standards for Airborne Area Navigation Equipment Using VOR/DME Reference Facility Inputs,” RTCA DO-180. RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1984). “Minimum Operational Performance Standards for Airborne Area Navigation Equipment using Multi-Sensor Inputs,” RTCA DO-187, RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1985a). “Minimum Operational Performance Standards for Airborne MLS Area Navigation Equipment,” RTCA Pap. 532-85/SC151-79,5th Draft. RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1985b). “Minimum Performance Standards Airborne DME Equipment Operationwithin the Radio-Frequency Range of 960-1215 MHz,” Doc. DO-189. RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1985~).“Minimum Aviation System Performance for the Global Position System,” Spec. Comm. SC-159. RTCA, Washington, D.C. Ragazzini, J., and Fanklin, G. (1958).“Sampled Data Control Systems.” McGraw-Hill, New York. Redlien, H., and Kelly, R. (1981). Microwave landing systems: The new international standard. Adv. Electron. Electron Phys. 57,311. Roberts, A. (1947). “Radar Beacons,” MIT-Radiat. Lab. Ser., Vol. 3. McGraw-Hill, New York. Rochester, S. I. (1976). “Take-off at Mid-Century: FAA Policy 1953-1961.” Federal Aviation Admin., Washington, D.C. Roscoe, S. (1973). Pilotage error and residual attention: The evaluation of a performance control system in airborne area navigation. J. Inst. Nauig. 20, No. 3. Ruhnow, W., and Goemaat, J. (1982).VOR/DME automated station selection algorithm. J . Inst. Nauig. 29, No. 4. Sandretto, P. C. (1956). Development of TACAN at federal telecommunications laboratories. Electr. Commun. 33(1). Sandretto, P. C. (1958). “Electronic Aviation Engineering.” International Telephone and Telegraph, New York. Sandretto, P. C. (1959). Principles of electronic navigation systems. IRE Trans. Aeronaut. Nauig. Electron. ANEX(4). Schmidt, S . (1966).Application of state space methods to navigation problems. Adv. Control Syst. 3.


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Schneider, A. (1959). Vector principles of inertial navigation. IRE Trans. Aerosp. Nauig. Electron. ANE-6(3). Silver, S . (1949). “Microwave Antenna Theory and Design,” MIT-Radiat. Lab. Ser., Vol. 12. McGraw-Hill, New York. Steiner, F. (1960).Wide-base doppler VHF direction finder. IRE Trans. Aerosp. Nauig. Electron. ANE-1. Tanner, W. (1977). Inertially aided scanning DME for area navigation. J . Inst. Naoig. 24, No. 3. Tyler, J., Brandewei, D., Heine, W., and Adams, R. (1975). Area navigation systems: Present performance and future requirements. J . Inst. Nauig. 22, No. 2. Wax, M. (1981). Position location from sensors with position uncertainty. IEEE Trans. Aerosp. Electron. Syst. AES19, No. 5. Williams, C., Bowden, B., and Harris, K. (1973). Pioneer award for contribution in field of secondary radar. IEEE Trans. Aerosp. Electron. Syst. AES-9, No. 5. Winick, A. B., and Hollm, E. (1964). “The Precision VOR System,” IEEE International Convention, New York. Zimmerman, W. (1969). Optimum integration of aircraft navigation systems. IEEE Trans. Aerospace Electron Syst. AFS-5(5).

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ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS, VOL. 68

Einstein-Podolsky-Rosen Paradox and Bell’s Inequalities W. DE BAERE Seminarie voor Wiskundige Natuurkunde Rijksuniversiteit Gent B-9000 Gent, Belgium

1. INTRODUCTION

Quantum mechanics (QM) is a statistical theory that makes predictions about the outcome of measurements on microphysical systems. The peculiarity of this physical theory is, according to the standard, or Copenhagen, intepretation of QM, that its statistical predictions are claimed to be inherent and cannot be explained further by completing our knowledge by some supplementary physical parameters. This inherent statistical feature of QM is so strange as compared with all known (classical) theories before the advent of QM, that resistance to accept such a counterintuitive world view has always been present in the scientific community, from the early days of QM. On the contrary, classical theories give us a rather comfortable feeling just because our most intuitive notions of physical systems (e.g., having definite properties of their own, being localized and separable from the rest of the world) are somehow inherent in this classical kind of theory. Moreover, the dogmatic claim of the proponents of the Copenhagen School, that at the atomic level physical events are no further analyzable in more detail than is allowed by QM (e.g., the impossibility of a space-time picture or analysis of individual events) has resulted in a critical attitude by a large number of scientists,among whom are de Broglie, Schrodinger, Einstein, and Bohm. In the words of Stapp (1982123, the origin of the uneasy feeling of those people with the present state and orthodox interpretation of quantum theory (QT) may be understood as follows: “Some writers claim to be comfortable with the idea that there is in Nature, at its most basic level, an irreducible element of chance. I, however, find unthinkable the idea that between two possibilities there can be a choice having no basis whatsoever. Chance is an idea useful for dealing with a world partly known to us. But it has no rational place among the ultimate constituents of Nature.” 245 Copynght @ 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.

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The most important critical quantitative analysis of QM appeared in 1935 when Einstein, Podolsky, and Rosen (EPR) (Einstein et al., 1935) made a thorough analysis of the orthodox interpretation of QT, based on a Gedankenexperiment with correlated systems. Starting from a realistic conception of nature, according to which physical systems may have independent intrinsic properties (whether observed or not), EPA came to conclusions which contradicted QM (the EPR paradox), from which they inferred that QM was not a complete theory. This conclusion was at the origin of the famous Bohr-Einstein debate over the interpretation of QT, the end of which is now, 50 years later, apparently not yet in sight, in spite of claims to the contrary. An important step forward in the clarification of the EPR problem came from a development by Bell (1964), who used the ideas of EPR to set up a simple inequality, the Bell inequality (BI), which has to be satisfied if EPR are correct. There are several reasons for which this BI is interesting. First of all, it is violated by QT in appropriate circumstances. Furthermore, the original EPR issue about the completeness(a rather vague concept which is not easy to define)of QT has shifted to the question whether or not QT is compatible with locality and separability. Finally Clauser et al. (1969) showed that this issue of basic importance may be decided on the empirical level. Hence, as a result of these developments initiated by Bell, the original EPR Gedankenexperiment has become since then within the reach of experimental realization. The result of most of the existing correlation experiments agree very well with QT and violate the BI, from which it is concluded that on the quantum level nonlocality is now a basic property. However, it appears that, due to various criticisms, this conclusion is not yet final. It is the purpose of this article to present a critical review of the problems related to the EPR argumentation and to the Bell inequalities. As a complement to this review we advise the reader to consult other books or reviews on these subjects, e.g., Hooker (1972),Belinfante(1973),Scheibe(1973), Jammer (1974),Clauser and Shimony (1978), Selleri and Tarozzi (1981),Selleri (1983b),and DEspagnat (1983, 1984). In the following we will use the terms “EPR paradox,” “EPR problem,” and “EPR argumentation” interchangeably,without bothering about which is preferable. 11. THEEINSTEIN-PODOLSKY-ROS~ ARGUMENTATION A . Original Version

EPR made their analysis on a pair of correlated physical systems for which they showed that, according to some set of particular definitions and

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hypotheses, a physical system can have both position and momentum, in contradiction to the QM postulates, according to which noncommuting observables cannot be determined or known simultaneously. The main definitions in EPR are those of “complete theory” and of “element of physical reality.” As a necessary requirement for a complete theory they adopt the following definition:

(Dl) Complete theory: “Every element of the physical reality must have a counterpart in the physical theory.” As to the concept of “element of physical reality” which is contained in (Dl), EPR use the following definition, which they find reasonable enough for their purpose. (D2) Element of physical reality: “If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.” To illustrate the meaning of these definitions, consider a quantummechanical system S in one dimension x, described by a wave function $(x). If $(x) is an eigenfunction of an operator A with eigenvalue a

444 = M x )

(1)

then measurement of A will give with certainty the value a. Moreover, from the preparation of the state of the ensemble of systems S, this result is known even without measuring A, i.e., without disturbing S. According to (D2)there exists in this case an element of physical reality corresponding to the physical quantity A. Using (Dl) this element of physical reality must have a counterpart in the underlying theory, QT. This must then be the case for every other physical observable quantity C with operator C for which Eq. (1) is valid C W ) = c$(x)

(2)

i.e., [A, C] = 0. Hence, according to (Dl) the only allowed elements of physical reality within orthodox QT are those corresponding to a complete set of commuting observables if the ensemble of systems S is described by an eigenfunction of one of them. If, on the contrary, the operator C corresponding to the physical quantity C does not commute with A: [ A , C ] # 0, then one has W(X)

# clCl(4

(3)

i.e., the result of measuring C on this state is not known without disturbing S. Certainty about the value of C, among the different possible values, will in this case only be obtained by actually measuring C, i.e., by disturbing S. Hence, as long as C is not measured actually, with C there does not correspond an

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element of physical reality of S , according to (D2). Up to now all this is consistent with orthodox QM. However, at this point EPR remark that, in order to criticize the completeness of QT, it is sufficient to find at least one particular situation for which all the above is not valid. This means that, if it can be shown that A and C are elements of physical reality of S, even if [ A , C ] # 0, then the alleged completeness of QT would no longer be tenable. Now, to show that this actually happens in some particular cases is the intention of the EPR reasoning. In particular by their argumentation they try to show that Heisenberg’s uncertainty principle may be violated. To this end EPR consider two correlated systems S , and S, which are supposed to interact only in a finite time interval (0, T ) .Suppose that for t > T the wave function $(x1,x2)of the combined system ( S , , S , ) is known completely. At the basis of the EPR argumentation, which leads to their contradictory conclusions, are furthermore the following three hypotheses:

(H 1) Einstein locality (separability), or the validity of the relativity principle: in any reference frame physical influences between nonseparated physical systems S , and S2(ie., between interacting systems) should propagate with a maximal velocity c, the velocity of light. If S , and S2 are separated, then no mutual influence at all should exist. In both cases, the idea of action at a distance or influences propagating with infinite velocity is considered as an absurdity and empirically incorrect. As a consequence, a measurement on one of two separated systems, say S , , cannot influence at all the outcome of a measurement made on the second system S , , if the two measurements are separated in space. Einstein, in his proper and direct way of expressing his intuitive insights in such fundamental matters, expressed his view on locality as follows (Einstein, 1949):“But on one supposition we should, in my opinion, absolutely hold fast: the real factual situation of the system S, is independent of what is done with the system S , , which is spatially separated from the former.” (H2) Universal validity of Q M : In particular it is supposed that the process of state vector reduction applies when a certain measurement is made, or when information about a system is gained. (H3) Counterfactual dejiniteness (counterfactuality)(Stapp, 1971) is valid: This is essentially the assumption that, if on a certain system S , an observable A is measured with the result a, it does make sense or one is allowed to speculate about what would have been the result if, instead of A , another (eventually noncompatible) observable B were measured on precisely the same system S, in exactly the same individual state [which is not contained in the quantum formalism, but may be contained in some future more fundamental

EINSTEIN-PODOLSKY-ROSEN PARADOX

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hidden variable theory (HVT)]. Hence the above hypothesis amounts to assuming that physically meaningful conclusions may result from combining actual measurement results with hypothetical ones. Suppose then that at t > T we measure the physical quantity A on S , , with eigenvalues a , , a , , . . . , and corresponding eigenfunctions u , ( x , ) , u,(x,), . . . : = akuk(xl)

(4)

According to standard QM rules we have to write II/(x,,x2) in terms of the wave function U k ( x 1 ) $(xl, x2)

=

uk(x1)$k(x2) k

(5)

with $ k ( x z ) the expansion coefficients. If the result of the measurement is ak, then the prescription of the reduction of the wave function says that I&,,x 2 ) reduces to one single term, namely uk(xl)II/k(xz).In QM this must be interpreted to mean that, after measurement of A on S , , S , must be described quantum mechanically by uk(x1) and S, by II/k(x2). Again, up to this point everything is quantum mechanically correct. I think that at this point it is important to note that, in order to understand the origin of the EPR paradox, the foregoing conclusion is correct only if the measurement of A on S , has actually been carried out. If not, the wave function $(xl, x,) does not reduce to one single term; in particular S, is not described further by $ k ( x Z ) . Now, EPR would propose to measure, instead of A, another quantity B with eigenvalues b , , b,, . . ., and corresponding eigenfunctions u 1(x 1 u 2 (x 11, . * * BUk(X1) = bkUk(X1) (6) 19

Instead of Eq. (5) one should now write $(xl,xZ)

=~uk(x1)(Pk(x2) k

(7)

This means that if B is measured actually (say at t = t o ) with the result bk,then $ ( x l , x 2 )reduces again to one single term, namely uk(xI)(Pk(xz),and after this measurement any further predictions for measurements on S , should be derived from its wave function (Pk(X2). In the foregoing we have stressed the condition that the measurements were actually carried out. It is important at this point to note that the EPR reasoning violates what is allowed by standard QT: Indeed, if A and B are incompatible, then either their values cannot be measured or known simultaneously, or A and B are measured simultaneously but on different systems, prepared analogously by different sources. Now it is precisely this violation of QM which EPR allow themselves in their argumentation, namely, as they state it: “If, instead of this, we had chosen another quantity, say B,

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having the eigenvalues bl, b,, ...,and eigenfunctions ul(xl), uz(xl), u3(x1),.. ., we should have obtained,. . ..” When discussing the Bell inequalities in Section 111, we will see, however, that this counterfactual reasoning, “If, instead of this,. .,we had chosen . ..we should have obtained ...” lies also at the origin of the disagreement between QM predictions and the BIs. Furthermore we will see (Sections I11 and IV) that the BIs are violated empirically in favor of QM. We will also make plausible there that it is the breakdown of counterfactuality (H3), rather than of locality (Hl), that is at the origin of the disagreement between BI and QT. Hence, at this point it already seems reasonable to suspect that it is incorrect and unjustified to speculate about the outcome of an already-carried-out experiment if some parameters of the total experimental arrangement were changed. In fact, on closer examination, it appears that it is reasonable to assume that, once an experiment is carried out, a subsequent experiment can never be done again under exactly the same conditions as those which were responsible for the specific outcome in the actual experiment. Therefore, it is to be expected that the result of any argumentation, like the EPR, which violates at some stage some QM principle, will lead to conclusions which are contradicted by QM (see below). Before continuing the original EPR reasoning to its end, let us remark that, even before choosing the alternative to measuring B, there already exists a conflict with the initial assumption, namely that for t > T the wave function is represented by the pure state, Eq. (5). Let us consider therefore the consequences of the hypothesis of locality (H1)and of the universal validity of QT (H2) and show that these hypotheses lead to contradictory statements. Indeed, for simplicity suppose for a moment that the sum on the right-hand side of Eq. (5) contained only a finite number of terms, and let us introduce time

Then the measurement of A = uk at c = t o > T reduces J/(x1,xZrt) to uk(x1, tO)+k(XZ, to). Now, because of the absence of any mutual influence between S1 and S, from t = T on, it may be concluded that S2 is also described by +&, t) for all t satisfying T < t c t o . This means that a statistical ensemble of couples (Sl,S,) for t > T may be decomposed into a series of subensembles described by ul(xl,t)+l(xZ,t),uZ(xl,t)+z(xZ,c),...; i.e., the original ensemble should be described by a mixture of states uk(x1, t)+k(xz, t), which is in contradiction to the original assumption of the ensemble being described by the pure state, Eq. (5). There is also an observable difference between the two possible descriptions (Furry, 1936a; Selleri, 1980; Selleri and Tarozzi, 1981). In the literature, the last description is by means of so-called


251

“state vectors of the first type” (corresponding to proper mixtures), while the first way is by means of “state vectors of the second type” [i.e., Eq. ( 5 ) ] (corresponding to improper mixtures) (DEspagnat, 1965). QM observables which have different values in the two schemes are sometimes called “sensitive observables” (Capasso et al., 1973; Baracca et al., 1976; Bergia and Cannata, 1982; Cufaro Petroni, 1977a,b; Fortunato et al., 1977). For later reference we shall call the above argumentation, which uses only hypotheses (Hl) and (H2), the modified or simplified EPR argumentation. From the assumed legality of measuring B instead of A on S,, EPR conclude: “We see therefore that, as a consequence of two different measurements performed upon the first system, the second system may be left in states with two different wave functions. On the other hand, since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system. This is, of course, merely a statement of what is meant by the absence of an interaction between the two systems. Thus, it is possible to assign two diflerent wave functions (in our example, $k and rp,) to the same reality (the second system after the interaction with the first).”This conclusion contradicts the QM formalism and constitutes what is known as the EPR paradox. Note again that here the spirit of QM is violated because it is supposed that the wave function represents the system itself. Inasmuch at the time of EPR much attention had been paid to whether or not physical systems (viewed as “particles”) followed individually a welldetermined path (a world line) in space-time, in the remaining part of their paper EPR are concerned with showing by an explicit example of the above conclusion that one and the same physical reality may be described by eigenfunctionsof the two noncommuting operators P and Q: [P, Q ] # 0. To this end EPR consider as the wave function describing the couple (S,, S,)

1

m

vwX19

xz) =

expCi(x1 - X Z + XO)P/51 dP

(9)

-m

with xo being a constant. For observable A they choose the momentum PI = h/i a / a x , of S , with eigenvalues p and eigenfunctions u,(x 1) = exp(ipx 1 lh)

(10)

Because of the continuity of p , Eq. (5) takes the form

1

m

$(XI,XZ)

=

up(x1)$p(x2)dp

(1 1)

-m

where the wave function for S, $Jx2)

= exPC - i(x, - xo)p/hl

(12)

W.DE BAERE

252

is an eigenfunction of P2 = h/i a/ax, with eigenvalue -p. Applying the above general results of the EPR analysis we have to conclude the following. Suppose the measurement of Pl on s, gives the result p. Then $(x1,x2) in Eq. (11) reduces to ~ , ( x , ) ~ ~ ( x ,This ) . allows us to conclude that P2 = - p without disturbing in any way S, (S, and S, are separated by assumption). Hence P, = --p is an element of reality for S,. For the observable B EPR take the coordinate Q , of S,: B = Q1,with eigenvalues x and eigenfunctions ux(xl) = 6(x, - x). $(x,,x,) may now be written

with (Px(x2)=

rm

expCi(x - x2

= hd(x - x,

+ xg)

+ XO)P/hl

dP (14)

which is an eigenfunction of Q , with eigenvalue x2 = x + xo. Applying again the EPR criteria we have: If the measurement of Q1 on S, has the result Q1 = xl, then $(xl, x,) reduces to u,(xl)~,(x2).This allows EPR to conclude that x2 = x xo without in any way disturbing (again by assumption) S,. Hence, x2 = x xo is also an element of reality for S,. Now [P2,Q2] = h/i # 0, and from this particular example EPR conclude that the same physical reality, S,, may be represented by two wave functions which may be eigenfunctions of two noncompatible observables P2 and Q 2 , in contradiction with what is allowed by orthodox QT. Hence they conclude that QT is not complete, because in its formalism either position or momentum is contained, whereas according to EPR both should be contained in it as elements of physical reality. But remember the crucial role played by the hypothesis of counterfactuality (H3) in the EPR argumentation. The particular situation above considered by EPR (measurement of P, or Q1 on one member of two coupled systems) is up to now only a kind of Gedankenexperiment. However, recently Bartell (1980b) has shown how the EPR proposal can be transformed into a real experiment. It has been shown that inequalities of the Bell type (Section 111) can be constructed, on the basis of which the validity of the EPR hypotheses (Hl) and (H3) may be verified. However, we consider it as not justified, as Bartell argues, that violation of the proposed inequalities should be considered as evidence for the existence of nonlocal influences. Indeed, as we will show in Section V, locality or the relativity principle may be saved by assuming a breakdown of counterfactuality (H3) as a more plausible origin of an eventual violation.

+

+

253


B. E P R Paradox in the Bohm Version Instead of measuring two noncompatible observables A = P and B = Q which have a continuous spectrum of eigenfunctions, Bohm (1952a)proposed to consider two systems S , , S , having the property of possessing spin 3. Instead of the original EPR Gedankenexperiment, this proposal has afterwards turned out to be realizable experimentally. In this case $ ( x , , x , ) has to be replaced by x ( x l , x z ) $ ( a l , a z ) , with $(u1,a2)being the spin part of the wave function and the Pauli matrices, a,,a, representing the spins of S, ,S, . Suppose the couple (S, ,S,) is prepared somehow in the singlet spin state: $(u1,a2)=I+s) = 100). For simplicity it may be supposed that we have a spin-0 system S(0)which decays according to S(0) + Sl(# S,(i). If this process ends at T, then for t > T, S1 and S2 may be supposed to be separated or at least no longer influencingeach other. From elementary QM it follows that, if angular momentum is conserved in the decay process, 100) may be written

+

I$J = 100)

= (1/Jz)(lu:Y)Iub3

-b .I">":).!I

(15)

in terms of basis vectors II&) for S , and IU:~) for S2 in which S , , S 2 have spin projections & 1 (in appropriate units) along an arbitrary unit vector a. Let us now illustrate the EPR argumentation on this conceptually simple example of correlated systems with only dichotomic variables. Take as a physical observable A of s,:A = a, z = q Z= an operator for the measurement of the spin projection m,,(z, t) along the unit vector z . The eigenvalues are f 1, and the respective eigenvectors, lu$!), Iui!!). An alternative option is to take as an observable B for S,: B = a, y = olY= an operator for the measurement of the spin projection ms,(y, t) along the unit vector y. The eigenvalues are again k l , and the respective eigenfunctions, It&)>, IuY2). EPR option ( I ) : Measure A = a, z, with the result m,,(z, t ) by sending S , through a Stern-Gerlach device SG, ( z ) with its analyzer direction (i.e., its magnetic field) along z . According to Eq. (15) we should write for this case

-

-

-

loo) = ( l / J z ) ( ~ u : ~ ) ~ u : ? )- ~ u ~ ! ! ) ~ u : ~ ) ) (16) From Eq. (16) it is seen that m,,(z, t) = + 1 and m,,(z, t) = - 1 will be found

with equal probability = 3. Suppose that at t = to the result obtained is ms.(z, t o ) = + 1. According to QT, 100) reduces to l u ~ ! ) l u : ? ) .It follows then that for t 2 to measurement of msz(z,t 2 to) will give - 1 with certainty without disturbing S, . According to the EPR criterion, this means that with msz(z, t 2 t o ) = - 1 there corresponds an element of physical reality and this must have a counterpart in a complete theory.

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W. DE BAERE

EPR option (2): Applying the counterfactuality hypothesis (H3), one can argue that one could equally well have measured the quantity B = a, * y with the result ms,(y, to), by replacing SG,(z) by SG,( y). Instead of Eq. (16) we must now start from

loo) = (l/fi)(luy2)lup)

- luy?)lu$3)

(17)

Again msl(y, to) = + 1 and ms,(y, t o ) = - 1 will be found with equal probability 3. Suppose again that at t = to we would have found msl(y, to) = - 1. This measurement may now be considered as a state preparation of S2 for t 2 to: In fact 100) is reduced by the measurement process to the single term 1ur?)1u1‘+‘). As a result of this process, the result ms,(y, t 2 t o ) = + 1 is then predicted with certainty, again without disturbing S2. Hence with this value for ms,(y, t 2 to) there must also correspond an element of physical reality and this must again have a counterpart in a complete theory. Now this is certainly not the case in orthodox QT because of the noncompatibility of oZty and 02*: [02,, 02,] # 0. Hence the EPR conclusion that QT is incomplete. As already mentioned, this incompletness conclusion may be avoided by insisting that for this to be true, ms2(z,t) and ms,(y, t) should be available at the same time. This way out of the problem is anticipated by EPR; indeed they offer it themselves at the end of their analysis: “One could object to this conclusion on the grounds that our criterion of reality is not sufficiently restrictive. Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but not both simultaneously, of the quantities P and Q can be predicted, they are not simultaneouslyreal. This makes the reality of Q and P depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this.” However, their remark in the last sentence of this quotation can be circumvented rather easily by extending their definition of an element of reality; indeed each measured value should be considered as representing an element of reality not only of the object system but of the combined system, object plus measuring apparatus. However, the simplified EPR argumentation in the Bohm version is again much more difficult to reject. To see this, let us apply the EPR reasoning on each of the couples ( S , , S 2 ) (Selleri and Tarozzi, 1981): (1) Measure mS,(a,tO)= + 1 at an instant t = to when S,,S2 are separated (located) so far that it may be supposed that they no longer influence each other (principle of separabilty). (2) From the standard QM formalism (assumed to be universally valid

255


by EPR), it follows then that at this moment the state 100) reduces to lu:!+))lu:?), which describes henceforth (Sl,S,) for t 2 t o . In particular, as a result of measuring ms,(a, to), the state describing S, is now luf!) for t 2 t o . (3) Because of the supposed separability and absence of any influence, this must also have been the case for t < to (at least for those times where separability holds); the state vector of S , must have been lu:?) and mS2(a,t < t o ) = - 1. (4) From msI(a, t) + ms2(a, t) = 0 (conservation of the component of the total spin along an arbitrary direction) it follows further that mS1(a,t < to>= -1 and that the state vector of S , is It&!) for t < t o (notice that this is already in conflict with the usual QM formalism, according to which it is only after ms2(a, to) = 1 is measured by SGl(a), thus for t 2 to that the state vector of S , is 1.)~': (5) Making the same reasoning for each couple ( S , , S,), one arrives at the contradictory conclusion that, the original ensemble was not described by the pure state, Eq. (15), but by a mixture of states Iu::')lu:?) and I u ! , ! ! ) l u : ~ ) with equal probability (see also Table I).

+

TABLE I SG,(a) IN PLACE, SG2(a)NOT IN PLACE: EPR PARAWX Time

SI

s2

State vector of (Sl,S2):100)

t < to

msl(a,t < t o ) = ? m,,(a, t < t o ) = -ms,(a, t

to

< to)=?

ms,(a,to)is measured: S, moves through

No measurement on S,, but SGI(a)(physical, local interaction) m,,(a, t o ) = -m,,(a, t o ) predicted Reduction of (00): if t L to State vector becomes Iu:?!) by nonlocal ms,(a,to)= + 1:100> reduces to influence Iu~!)tu~?!)for (s,,s,) Inferences: m,,(a,t = to) = - I (separability,locality) m,,(a, t < to) = + 1 c- m,,(a,t < to) = - 1 (conservation of angular momentum) state vector = 1)~':; state vector = Iu;!!) f =

1

I

I

(Sl,S2jdescribed by mixture of Iub:' )I ub? ) and 1 ub? )I ub:' ) = observably different from (00)

EPR Paradox

256

W. DE BAERE

In order to show the contradiction in a quantitative way, let us introduce the triplet state

I$,)

= 110) = (1/&)(lu:Y)lu:9

+ lu:?)lu:Y))

(18)

Then one may write lu:Y)lu:3

= (l/JZ)(lW

+ 110))

( 194

lu:?)iu:Y)

= (l/Jz)(lOo) - 110))

(19b)

such that the original ensemble of couples ( S , , S , ) should now be a superposition of singlet and triplet spin states, instead of a pure singlet state. In fact there are observable differences between the two kinds of descriptions (Selleri and Tarozzi, 1981). In the pure state one has

( o o ) J ~ ~ o =o )0

(20)

Obviously there is something that has to be wrong in the above reasoning. Noting that no contradiction arises if one does not go outside the strict QM framework [Bohr's resolution of the EPR paradox (Bohr, 1935a,b)]; Selleri and Tarozzi (1981) arrive at the conclusion that it is step (3) in the EPR reasoning, namely the locality hypothesis (Hl), which is incompatible with QM, and which is responsible for the above contradiction.

C . Critical Analysis of the Original EPR Argumentation When the EPR paper appeared in 1935, it was already clear that the quantum formalism was very powerful and successful in the treatment of physical problems at the atomic level. As a consequence, it had become the daily working tool of a large majority of physicists who did not bother about fundamentals. Yet the original EPR argument was considered a serious challenge of the completeness of QT, especially by those interested in the soundness of its basis. In this section we will review the main reactions immediately after the EPR argumentation was published. We will see that most of the early reactions emphasized the difference between the definition of state of a physical system in EPR, which is that of classical mechanics (CM), and that adopted in QM. In CM the state is supposed to be a representation of the system itself, while in QM the state represents, at least in the Copenhagen interpretation (Stapp,


257

1972),merely our knowledge about the system. Also attention is drawn to the fact that measurement of noncompatible observables requires mutually exclusive experimental arrangements. This may be seen as an indirect criticism of EPRs implicit hypothesis of counterfactuality (H3). Kemble (1935) was one of the first to defend the orthodox viewpoint against EPR. He argues that, if it were true, as EPR contend, that “it is possible to assign two different wave functions ... to the same reality ...,” the QM description would be erroneous. According to Kemble’s view, the EPR problem may be avoided by sticking to “. . . the interpretation of quantum mechanics as a statistical mechanics of assemblages of like systems.” Hence in the above EPR contention, one of the wave functions describes the future statistics of one subassemblage, while the other describes the future of another, different, subassemblage. And which wave function is to describe the subassemblage depends on the choice of what will be measured on one of two correlated systems in the original assemblage. Kemble concludes that there is no reason to doubt the completeness of QM on the grounds advanced by EPR. Once it is accepted that the wave function does not describe the intrinsic state of a system itself, but represents only the regularities between future observations on a system, the original EPR problem on the completeness of QM disappears. According to Wolfe (1936), the origin of the EPR troubles lies in an unjustified extrapolation of ideas on “physical reality,” which are certainly satisfactory and reliable in classical physics, to the domain of QM. EPRs own specific example of a system having both position and momentum illustrates this point. In CM both may indeed be determined and known simultaneously, and states in CM accordingly contain this information. However, in QM only one of the two may be known, and this is again reflected in the quantum state. According to Wolfe, the quantum state only represents our knowledge about a system. Hence Wolfe concludes that “Viewed in this light, the case discussed by Einstein, Podolsky, and Rosen simply dissolves. Two systems have interacted and then separated. Nothing that we do to the first system after this affects the “state” of the second. But ..ieasurements on the first system affect our knowledge of the second and therefore affect the wave function which describes that knowledge. Different measurements on the first system give us different information about the second and therefore different wave functions and different predictions as to the results of measurement on the second system.” The reply of Bohr (1935a,b) goes along similar lines. After explaining in detail his general viewpoint, called “complementarity,” Bohr shows that the QM formalism is mathematically coherent and that the limitations on the measurement of canonically conjugate observables are already contained in the formalism. According to Bohr, the origin of the EPR contradiction is

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W. DE BAERE

“. .. an essential inadequacy of the customary viewpoint of natural philosophy for a rational account of physical phenomena of the type with which we are concerned in quantum mechanics.” Bohr continues to explain in detail that the experimental arrangements and procedures for measuring noncompatible observables, in particular x and p , are essentially different and mutually exclusive. Choosing one arrangement means the renunciation of knowledge of the variable measured by the other. Otherwise stated: Measurement of one variable means an uncontrollable reaction on the system which definitely prohibits the measurement of the other variable. Hence Bohr concludes that only if two or more observables can be known or measured simultaneously,can they have simultaneous “reality” status. This was also the point made by Ruark (1935). According to Bohr then, it is therefore clear that the original EPR argumentation is invalid for noncommuting observables A and B. Also EPR were apparently aware of the above criticism because of their final remarks towards the end of their paper (Section 11,B): “... one would not arrive at our conclusions if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted.” They rejected this solution to their problem on the ground that “This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this.” However, the last viewpoint is not correct because it must admitted that any measurement result is somehow characteristic for elements of reality of both the instrument and of the physical system, and not only of the system itself. Hence, according to Stapp (1971), it appears that EPR violates “. . . Bohr’s dictum that the whole experimental arrangement must be taken into account: The microscopic observed system must be viewed in the context of the actual macroscopic situation to which it refers.” For a recent discussion of the debate between EPR and Bohr, see de Muynck (1985). An interesting discussion of the EPR criticism against QM has been given by Furry (1936a,b). According to Furry (1936a) the essence of Bohr’s reply to EPR is that “one must be careful not to suppose that a system is an independent seat of “real” attributes simply because it has ceased to interact dynamically with other systems.”However, one should be warned that such an extreme positivistic standpoint may also be criticized. Indeed, it may well be assumed that both the apparatus and the system itself have their own “real” and independent (in the sense of not being influenced instantaneously by the rest of the world) attributes of which any measurement result is but a representation. It will be shown in Section V,B that such a viewpoint is perfectly consistent with the Bell inequalities and the hypothesis of Einstein locality (H l), provided the hypothesis of counterfactuality (H3) is abandoned.

EINSTEIN- PODOLSKY-ROSEN PARADOX

259

Furry then studies in a quantitative way the observable differences between the above assumption of EPR that the QM wave function describes the “real” attributes of the system itself and the orthodox QM standpoint that applies the usual QM formalism and denies that physical systems possess independent real properties. Consider then two correlated systems S1,S 2 which are described by a QM state vector l$(xl,x2)). According to von Neumann (1955; see also Baracca et al., (1974), l $ ( x l , x2)) may in general be written

Moreover in Eq. (22) all &’ are different, and also all P k . The state vector Eq. (15) or (16),corresponding to the Bohm version of the EPR paradox with correlated spin-) systems in the singlet state, may be considered as an illustration of Eq. (22), for which moreover the expansion on the right-hand side is not unique. From the one-to-one correspondence between & and p k in Eq. (22) it follows that if L = & is measured on S1,then R = Pk is predicted on S2 and vice versa. Furry then gives a definite form to the above-mentioned opposite viewpoints of systems possessing real attributes and of standard QM in the following way: Method A assumes that, as a result of the interaction which causes the correlation between S , and S 2 , S, made a transition to one of the states I q A k ( x , ) and ) S2 made a transition to the corresponding state 15Pk(x2)).Also it is assumed that the probability for transition to JCP,,(X,)> is wk. Note that the assumptions of Method A are precisely the conclusions of the modified EPR argumentation (Sections II,A and I1,B) according to which a mixture of states 1 q A k ( x l )1)t p k ( x 2 ) )should describe an ensemble of correlated systems (S,, S2), instead of a pure state as in Eq. (8) or (22),which was the assumption from the outset. Method B, on the contrary, uses only standard QM calculations based on the pure state, Eq. (22). Furry considers then four types of questions which may be answered by using either Method A or Method B. The particular type of question for which the two methods give different answers is the following.Suppose observable M is measured on S1with eigenvaluesp and eigenstates1 &(x1)) and observable S is measured on S , with eigenvalues c and eigenstates 147,(x2)). The question

260

W. DE BAERE

is: If M = p on S , is measured, what is the probability that S = (r will be measured on Sz? According to Method A the probability is

and according to Method B

IT

~(rl,(xz)l 5pk(XZ)>

1 k

wkl

(25)

It is seen that the difference between Eqs. (24) and (25) consists in the appearance of interference terms in Eq. (25) which are not present in Eq. (24). The origin of these terms is usually ascribed to the influence of the measuring apparatus on the internal conditions within the system, whose future behavior has therefore been influenced. We will see in Section III,F in a direct way that this must happen with each system that passes some apparatus. Moreover there does not exist a way by which Method A can be made consistent with Method B. Hence, Furry claims to have shown mathematically the inconsistency between standard QM and the EPR viewpoint that physical systems may have objective properties which are independent of observation. However, we will show below that this is not entirely correct. It is only correct if Furry’s assumptions of his Method A are accepted. The link with the original EPR argumentation is obtained by consideringa particular example for which the expansion of Eq. (22) is not unique. Furry then applies his Method A to each of the expansions, which amounts to assuming that in each case the system S1 or Sz made a transition to a definite state for which a certain observable has some well-defined value. As in the case of EPR, contradiction with QM occurs when both observables are noncompatible. To illustrate these points clearly, Furry analyzes a thought experiment of the EPR type and shows that the final conclusions are in conflict with Heisenberg’s uncertainty relations. Furry (1936b) concludes that “. . . there can be no doubt that quantum mechanics requires us to regard the realistic attitude as in principle inadequate.” In defense of the realistic attitude of EPR we should remember, however, what has been said at the start of the discussion of Furry’s contribution. Also it may be questioned whether Furry’s Method A is identical to that which Einstein had precisely in mind. Indeed, the eigenstates l q A k ( x l ) ) of the observable L, which Furry ascribes to subensembles of systems S 1 , are QM


26 1

states. This means that, for the subensemble described by l q A k ( x l ) (selected ) by retaining only those S1 for which measurement of L is A&, each subsequent measurement of L will again give the result 1,.However, if another observable M is subsequently measured instead of L, then only the probabilities ~ ( $ , , ( x , ) ~ q A k ( x l for ) ) ~ obtaining z the value p are known. Now the idea of Einstein was probably that, once the state of S, is known completely, it should be possible to predict with certainty all possible measurement results at one instant. Perhaps EPR’s only weak point was to assume that all these possible measurement results also correspond to actual results; i.e., they assumed implicitly the validity of counterfactuality (H3). We will see in Section V that, on account of the empirical violation of the BIs either Einstein locality (Hl) or counterfactuality(H3) has to be abandoned. In the latter case this means that, among all possible measurement results, there is only one that has physical sense, namely the one that is actually obtained in a real measurement. It is seen that the difficulty with considerations on states of individual quantum systems is that there does not exist any formal scheme within which such problems can be discussed. Tentatively this may be done along the following lines. Suppose we represent the individual state of each quantum system S at each instant t by a set of (field) functions { d ( x , t ) } ( i = number of field components).In the spirit of Bohr, we shall consider the individual state of the apparatus, say a Stern-Gerlach device SG(a) with analyzer direction a, as equally important for the determination of the measurement result, e.g., the spin projection ms(a, t) along a at time t. This means that SG(a) too should be represented by a similar set {tl,&a)(x, t)}. Measurement of the spin projection ms(a, t) of S along a should then formally be represented by the mapping

In this formal scheme the “elements of reality” corresponding to the particular value rn,(a, t) should therefore be represented by { a i ( x ,t ) } for S and by {cL$~(,,)(x, t ) } for SG(a); i.e., rn,(a, t ) should not be considered as an attribute of S alone, but of S and SG(a) together. Hence, within such a scheme, a reconciliation between the realistic standpoint of EPR and the orthodox or Copenhagen standpoint should be possible (see also de Muynck, 1985). D . First Attempt at an Experimental Discrimination between EPR and Standard QM

We have seen above that the simplified EPR argumentation leads to the conclusion that an ensemble of correlated systems S,, Sz should be described by a mixture of product states, instead of a pure state as required by QM. The

262

W. DE BAERE

quantitative analysis of Furry (1936a)(Section I1,C)has shown very clearly the difference between the predictions based on both kinds of description. Subsequently, Bohm and Aharonov (1957) were the first to suggest that, eventually, the original situation of correlated systems, described in standard Q M by means of a pure state, could evolve spontaneously to a situation which should be described by means of a mixture of product states of the individual systems when the separation becomes sufficiently large. In the special case of correlated photons it is assumed that their polarizations are opposite, and, in order to retain the QM rotational symmetry, that the polarization directions are uniformly distributed in an ensemble of coupled systems. In order to discriminate between the two alternatives Bohm and Aharonov discuss an experiment (Wu and Shaknov, 1950)in which correlated photons y1 and y2 originate from the annihilation of an electron and a positron, each of which is supposed to be at rest. Under these conditions the state of the couple (yl, y 2 ) has total spin and parity J p = 0-(Kasday, 1971)and according to QT one can write either = ( l / f i ) ( I R l ) l R Z ) - IL1)ILz))

(27)

or = (1/JZXlX,>lY2) - IYl)lX2))

(28) for the pure state vector of the coupled photons. In Eq. (27) the notation is such that IR1)(R2)(ILl)lL2)) represents photons moving in opposite directions (e.g., along z) which are right (left) circularly polarized. Similarly, in Eq. (28), IXl)l Y,) represents two photons which are linearly polarized in, respectively, the x and the y direction. Hence, according to standard QM, measurement of one kind of polarization on y1 immediately gives information on the corresponding polarization of y 2 , because of the correlation in Eq. (27) or (28). Now, because such ideal polarization measurements on individual annihilation photons do not yet exist, this kind of one-to-one correlation between individual photons is not verifiable. Therefore, to distinguish between the two alternatives, one has to proceed in an indirect way, namely by means of the process of Compton scattering, which also depends on the photon polarization. Suppose then that y1 and y 2 , with momenta pyl and py2= -py,, are Compton scattered at points A and B with final momenta and ptz (Fig. 1). In Bohm and Aharonov (1957)two different situations are considered, namely for the scattering planes (pyl,pi,) and (pyz,pi2)being perpendicular or parallel. In each case coincidence counts of photons are registrated for which the scattering angle 8 is the same for both photons. The ratio of the number of coincidence counts in both cases is called R. It is this experimentally determined ratio R, for an ideal angle of 8 = 82", which is compared with the l~ylyz)

PI,


263

Y FIG.1. Compton scattering of correlated annihilation photons.

predictions following from three assumptions:

(Al) Standard QT is universally correct, i.e., for large as well as for small separations between y,l y2 ; (A2) Standard QT applies only for small separations, e.g., when the wave packets overlap. For those separations for which this is not the case, it is assumed that the photons are already in QM states which are circularly but oppositely polarized about their direction of motion; (A3) The same as (A2), but now it is assumed that each photon y1 is already in a state of linear polarization in some arbitrary direction, and the other, y2, is in a state of perpendicular polarization. All directions are equally probable. The theoretical predictions as calculated by Bohm and Aharonov (1957) for the different assumptions, taking into account the characteristics of the experimentalconfiguration (e.g., finite experimental solid angle), are as follows RA1 = 2.00,

R,, = 1.00,

RA3 = 1.5

(29)

These values are to be compared with the experimental result R = 2.04 f.0.08 of Wu and Shaknov (1950). From this it is evident that in this case standard QM gives the correct result. More recent, similar experiments on annihilation photons (Bertolini et al., 1955; Langhoff, 1960; Section IV,A) give further evidence for the correctness of the QM description of correlated and widely separated photons; hence the above hypothesis of Bohm and Aharonov Lie., assumption (A2) or (A3)] is not correct. The relevance of the experiment of Wu and Shaknov with respect to the EPR paradox has been criticized by Peres and Singer (1960). They argued that photons, being massless, have their spin always along their direction of propagation, and that the spin components orthogonal to this direction do not have physical meaning because these quantities are not gauge invariant. However, Bohm and Aharonov (1960) subsequently showed that these arguments were invalid because they were based on an incorrect interpretation of photon polarization in QT.

264

W. DE BAERE

E. Attempts to Resolve the EPR Paradox Margenau (1936) apparently was the first who tried to resolve the EPR paradoxical conclusions by emphasizing that the assumed validity of the QM process of state-vector reduction may be the origin of the troubles. Following EPR,Margenau remarks that, if this process is correct, then “. . . the state of system 1, which, by hypothesis, is isolated from system 2, depends on the type of measurement performed on system 2. This, if true, is a most awkward physical situation, aside from any monstrous philosophical consequences it may have.” However, note again that from this statement it is seen that Margenau too accepts implicitly the validity of counterfactuality (H3). Margenau then argues with respect to the EPR problem that “. . .if it be denied that in general a measurement produces an eigenstate, their conclusion fails, and the dilemma disappears.” However, we have already shown above that the EPR contradictions are most striking when (H3) is not used at all. In this respect, we will discuss below (Section II,F) some interesting recent proposals for a direct experimental verification of the correctness of the process of statevector reduction in the case of correlated systems, or of the validity of locality or the relativity principle. According to Breitenberger (1965) there can be no question of a real paradox, neither in the original EPR situation nor in the later Bohm version, because the contradictory conclusions result from statements which are unverifiable (i.e., counterfactual) and therefore are devoid of physical sense. It is also noted that the EPR reasoning is based implicitly on the assumption of the knowledge of a precise value of a conserved physical quantity, the importance of which was first emphasized by Bohr (1935b). Moldauer (1974)re-examines the EPR argumentation from the standpoint that physical theories, in order to be verifiable, should deal with reproducible phenomena. It is argued that under this requirement the EPR conclusion of incompleteness of QT does not apply, although it is admitted that nonreproducible events, such as the decay of a radioactive atom, are objective observations which endow physical reality to such events. EPR claim that these events too should be described in a complete theory. According to Moldauer such inherently nonreproducible events (e.g., no ensemble of radioactive atoms can be prepared which decay at precisely the same instant) ought not to be the object of theoretical investigation. Zweifel (1974) uses Wigner’s theory of measurement to resolve the EPR paradox. This theory is based on the idea of the existence of some kind of interaction between the system under investigation and the mind of the observer. It is argued that S , and S , are interacting with each other, because both interact with the mind of some observer via the Wigner potential. The observed correlation between S , and S, is then viewed as a direct consequence

EINSTEIN- PODOLSKY -ROSEN PARADOX

265

of their interaction with the mind of the respective observers. It is argued that both observers must exchange information in order to be sure that their measurements on S , and S2 concern two systems which have a common origin, and this exchange should be responsible for the correlation. The physics of the EPR paradox has also recently been re-examined by Kellett (1977). It is argued that the EPR argumentation against the completeness of Q T is unsatisfactory on two grounds: (1) The EPR argument is invalid as it stands because it is based on a Gedankenexperiment which is physically unrealizable [however, see Bartell, 1980b, Section II,A and Section 11,F); (2) The basic EPR assumptions are equivalent to assuming that the QM description of physical systems is independent of any observation or measurement. It is argued that the present experimental evidence for the violation of the BI (Section IV) rules out these basic assumptions, hence make invalid the EPR argumentation itself. Kellett admits, however, that the concept of electron “in itself‘’ is not necessarily meaningless and that individual electrons may well exist. The difficulty is that up to now no theory for such individual systems exists and that questions pertaining to such systems are not relevant to the state vector II/ of QM. The completeness claim of Q M is considered as being the claim that all information that is necessary to any future observation is already contained in I). Hence, I) is a summary of all possible outcomes of future measurements. Nevertheless, in spite of admitting the existence of individual systems, according to Kellett “. . . it is clearly the EPR definition of physical reality that is at fault.” An approach to the EPR paradox that sounds rather strange consists in assuming that influences are not only propagating into the future, but equally well into the past (retroactivity). Adherents of such a view are, e.g., Costa de Beauregard (1976, 1977, 1979), Rietdijk (1978, 1981, 1985), and Sutherland (1983). However, it seems very difficult to grasp the physical significance, if any, of the idea that events or processes that already happened in the past could have been influenced by events that happen just now. The only significance the present author may recognize in such a picture is within a completely deterministic evolution of the world: What happens in the future depends on what happened in the past, and in this way both the future and the past are related. Yet most of us will prefer the picture that all events which will happen later on are influenced by events that happened earlier, and not the other way around. Of course, in such a scheme, knowledge of the present would also provide knowledge of the past, however without making it necessary that influences are propagated backwards in time toward the past. Also Sutherland (1985) has recently shown that such a model is implausible,

266

W. DE BAERE

because a new paradox arises from the requirement that a nonlocal influence of a measurement on S,, exerted on the measurement on S,, should be independent of whether S, is inside or outside the forward light cone of S, . Another approach to the EPR problem is that of Destouches (1980), in terms of De Broglie's theory of the double solution. It is shown that within this framework a more general and more satisfactory theory than QT may be set up, satisfying the following set of conditions: (1) the results of QM are recovered, (2) the Bell inequality is violated, (3) the EPR paradox disappears and (4) there do not exist retroactive influences in the sense of Costa de Beauregard (1976,1977,1979), of Rietdijk (1978,1981,1985), or of Sutherland (1983). Other attempts to resolve the EPR paradox are in terms of the density matrix formalism of Jauch (1968) and of Cantrell and Scully (1978). In both cases it is claimed that within this formalism a satisfactory answer to the EPR problem may be given. However, Whitaker and Singh (1982) remark that in this way one is implicitly using the ensemble interpretation of QT, inside which no paradoxical conclusions appear (see also Ballentine, 1970,1972).It is stated that the EPR paradox requires a resolution only within the Copenhagen interpretation (CI) (Stapp, 1972) of QM, because this maintains that it gives a complete (probabilistic) description of individual systems. The EPR paradox has been discussed in different QM interpretations in Whitaker and Singh (1982). Muckenheim (1982) presents the following solution to the Bohm version of the EPR problem. It is based on the allowance of negative probability distributions. It is assumed, in the EPR spirit, that each of the correlated spin4 systems S,, S, has a definite spin direction on its own, e.g., s and -s. Hence for each system it is accepted that spin components along different directions have simultaneous reality, even if these correspond to noncompatible observables, although it is admitted that these cannot be observed simultaneously. Call w,(a,s) the probability for obtaining the result s* = ki along a, with w,(a,s) + w-(a,s) = 1. It is assumed further that w+(a,s)w_(a,s) = (const)a s, such that w,(a,s) = 3 k a s. With s2 = 2, it follows from a s = ( a / 2 ) c o s 8 that the probabilities w+ may become negative. This model allows Miickenheim to reproduce all QM results. For a single system it is found that

-

-

-

(s(a,s))

=

-

s.

p(s)[w+(a,s)s+

+ w-(a,s)s-]dR

=0

(30)

with p(s) = (47c-',a s = &/2)cos0, dC2 = sinOd8dz and s+ = -l while +, for correlated systems the result for the correlation function is P(a,b) = (s(a,s)s(b,s)) = -$a

-b

(304


267

in accordance with standard QM. However, no explanation is given of the physical significance of the negative probabilities. In an interesting paper de Muynck (1985) investigates the relation between the EPR paradox and the problem of nonlocality in QM. In this paper it is discussed, along the lines of Fine (1982a,b) (Section V,C) whether the EPR pr6blem and the violation of the BIs by recent experiments has anything to do with a fundamental nonlocality or inseparability at the quantum level. In this respect it is interesting that de Muynck makes a clear distinction between unobserved, objective EPR reality and observed reality that is described by QT. Because of the absence of any experimentally verifiable consequence on the basis of this EPR reality, the EPR analysis itself is incomplete and on a metaphysical level. Remember also that similar points were stressed when pointing to the necessity of introducing counterfactuality as well in the EPR reasoning (Section II,A,B) as in the Bell reasoning (Section 111,C). In fact it cannot be denied that on the quantum level the result of what would have happened if, instead of an actual experiment, one had carried out another one, with other macroscopic apparatus parameters, is completely unverifiable. de Muynck repeatedly stresses not to forget the central lesson of QM (and of Bohr) that it is impossible to gain knowledge without taking into account the measuring arrangement. However, by remarking that the existence of an objective reality is not excluded at all by QM, de Muynck tries to show that “. .. the positions of Bohr and Einstein with respect to reality are less irreconcilable than is often taken for granted.” Hence, in this article no extreme standpoints are taken and it is a virtue that a reconciliation between EPR and Bohr is strived for. As de Muynck remarks: “. . . Bohr’s completeness claim,. . . restricts the possibility of defining properties of a system . . .,”and: “Einstein, indeed, may be right in pointing at the possibility that we might obtain knowledge about a system exceeding the quantum mechanical knowledge.” Hence, “. .. the Einstein-Bohr controversy is not a matter of principle. It is just about the domain of application of quantum mechanics.” Other proposals for a resolution of the EPR paradox and of the problem of locality in relation with the Bell inequalities will be discussed in Section II1,G. A generalization of the EPR paradox by Selleri (1982) and the resulting new Bell-type inequalities will be mentioned in Section III,D,6.

F. Recent Proposals for Testing the Validity of Einstein Locality In Section II,D we have seen that the agreement of the experimental results of Wu and Shaknov (1950) with Q M (and also with the results of more recent similar experiments) was used by Bohm and Aharonov (1957) to rule out the

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W. DE BAERE

hypothesis that a pure state vector evolves spontaneously into a mixture of product states (known as the Bohm-Aharonov hypothesis).Such an evolution is indeed expected if Einstein locality (Hl) is supposed to be valid. Hence the conclusion that (Hl) must be wrong. In Section IV,A we will see that further evidence supporting this conclusion comes from atomic cascade experiments suggested by the Bell inequality, such as the very convincing recent experiment of Aspect et a!. (1982). However, a drawback of this kind of experiment is that the validity of the hypothesis of counterfactuality (H3) has to be assumed in order to justify the combination of results obtained in at least two different experiments. If one neverthelesscontinues to stick to nonlocality,then it appears that the EPR paradox is avoided by introducing another paradox, namely the violation of the relativity principle. Now, eliminating paradoxes by introducing new ones is certainly not a recommended method to solve physical problems. Moreover Fine (1982a,b) and others have recently shown (Section V,B,l) that locality is irrelevant for the derivation of the BI. Therefore, there is some ground to suspect that the resolution of the EPR paradox, which consists in the rejection of the existence of separated entities and a finite speed of propagation of influences,is not the correct one. Bearing in mind that the process of state-vector reduction plays a crucial role in the EPR paradox, we will discuss below two recent proposals of Gedankenexperiments for verifying (non)locality. Although it was not the purpose of EPR to criticize the QM formalism, it will appear that the EPR reasoning may be used to criticize the CI of Q M (knowledge of the result of an experiment is sufficient to determine the subsequent state vector)as it is applied to the case of two correlated systems S , and S,. Remember that Margenau’s resolution of EPR (Margenau, 1936) consisted precisely in the rejection of state-vector reduction. In fact in the following we shall assume unlimited validity of the relativity principle (i.e., no action at a distance) and ask what may be wrong in the use of the CI of the quantum formalism to two correlated systems S , and S , . We will try to make plausible that it is indeed the reduction of the state vector for the system ( S , , S,) which may be subjected to criticism.

I . Poppers’s New E P R Experimenl As a first example in which state-vector reduction in the case of correlated systems may be seriously in trouble, we mention Popper’s recent EPR experiment. Recently Popper (1985) proposed a new version of the classic EPR Gedankenexperiment. As in the original EPR paper, he is mainly concerned with showing that physical systems may have both position and momentum at


*L

BL

I

TY

I

269

*R

FIG.2. Setup for Popper's new EPR experiment.

the same time, and hence follow a (classical) trajectory. However, we will use this proposal to criticize state-vector reduction in the case when only one measuring device is in place. Popper's new proposal is essentially as follows (Fig. 2): 0 is the origin of correlated systems ( S , , S,) which move in opposite directions: S, moves to the right towards a screen A, which has a slit of extension Ay, while S, moves to the left. Because of their common origin, S, and S, may be represented by a state vector which reflects their correlation I$S1S$O))

=

s

14Y(1))l -d,YfZ) = - Y 9

x (d,ycl'; -d,y(,) = - ~ ' " ~ $ ~ , s , ( t o ) )dy'"

(3 1)

It follows that if at t = t o S , is observed at (d,y(')), then at this moment the state of Eq. (31) reduces to Id,y('))l -d,y'" = -y(')). This means that at t = to the position of S , will be (- d , - y'") if this were actually measured. In particular, those systems S , passing through the slit Ay will be diffracted (to be observed at 4)as a result of the Heisenberg relation Apy x h/Ay; with the knowledge of the position of S, with an accuracy Ay corresponds an uncertainty Apy of its corresponding momentum component. Now, because of the correlation, the same must happen with S,; indeed, the resulting statevector reduction provides us with the knowledge of the position of s, with an accuracy of the same order Ay. Hence there must be, according to standard QM, an uncertainty Apy of the momentum, which means that S, will be diffracted too (to be observed at screen B,), even when the screen A, with a slit Ay is not in place. If this were actually the case, this would be a direct confirmation of the validity of state-vector reduction in the case of correlated and widely separated physical systems, and as a consequence also a strong argument for the existence of action at a distance. Thus the physical interaction of S, with the slit in A, would be responsible for both the diffraction of S , and, by a nonlocal influence, for a diffraction of S, at the imagined screen A,. On account of separability and locality, however, diffraction of S, is not expected

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W. DE BAERE

to occur (at least as long as A, is not in place);to all observations of diffraction on 4 should correspond observations on B, in a small region around the x axis. In a quantitative analysis of Popper's proposal, taking into account the initial (inevitable) uncertainties at the source, Bedford and Selleri (1985) have shown that the above experiment is possible in principle under certain conditions. With annihilation photons, for example, the uncertainty at the source is such that an observed effect on S , cannot be related to the observed diffraction of S , . In the case of photon emissions in opposite directions in the center of mass, however, with sufficiently large mass of the emitting source, a relation between both diffractions should be possible. Indeed, these authors have shown that the angular deviation between the momenta of S, and S2 in the laboratory system is inversely proportional to the mass of the source, hence in principle may be made arbitrarily small. 2. Double-Slit Experiment with Correlated Systems

As a second example of a situation in which troubles with state-vector reduction in the CI of QM may be expected, we discuss a double-slit experiment connected with an EPR-type situation (Fig. 3) (De Baere, 1985). 0 is the source of two correlated systems S,, S,. S, moves towards a screen A , which contains two slits s1 and s2 . S, moves toward a screen A , where its position can be observed. To the right of A, we have a third screen 4 where S, may be observed, in the case S , has passed A,. Suppose further that all relevant parameters (distances OAR,OA,, slsz, momentum of S,) are such that on 4 an interference pattern is observed in the case that 0 is the source of systems S, only. The question we are interested in then is the following: If S, is observed at the screen A, at about the moment that S , reaches A,, will the subsequent interference pattern at 4 subsist or will it disappear (as demanded

"i FIG.3. Double-slit arrangement with correlated systems.


27 1

by the CI)? Again, the argumentation of Bedford and Selleri may be used here to conclude that, with each impact of S , on A,, there must correspond a unique position of impact of S , on A,. To answer the above question, let us look at the QM description of this situation. The correlated couple ( S , , S,) is again described by the state vector of Eq. (31). Suppose we observe the impact of S, at A, at t = t o . At this moment the state vector I$s,s,(to)) reduces to Id, y(l) = -y(”)1 -d, y‘”). If it happens that y(,) = y:) = -s then we may say that observation of S , on A, has indirectly determined the slit s, through which S, will move. More precisely, if a counter were placed behind the slit sl,S1 would have been observed by it. In any case (counter behind s, in place or not), by the observation of S, on A, the CI assures that at t = to the state vector for S, is Id,y(’) = s). Having determined in this way the slit s, through which S , will pass, standard QM predicts that the corresponding part of the incident S, beam will give rise to a diffraction pattern at &. The same can be said for observations of S, at A, for which y(’) = yji’ = +s. For those cases, statevector reduction steers the corresponding s, systems in the state vector Id, y“’ = - s) (which corresponds in the usual terminology to those systems S , which “pass” slit s,), which again will give rise to a diffraction pattern at 4 . For all other observations of S , at A,, the original state vector will reduce to a state Id, y ( ’ ) # fs), which will correspond to absorption (or at least an impact) of the corresponding systems S , on the left side of screen A,. In any case, these systems will not pass either through slit s, or s,, and hence cannot contribute to the final pattern observed on &.All this can be visualized by imagining that photographic plates are mounted on the appropriate sides of screens A,, A,, and &. Therefore, one may expect that, under the validity of the QM process of state-vector reduction (as used by EPR and in the CI), the pattern observed at 4 will be a superposition of two diffraction patterns, each corresponding to observations of systems S, at positions s; and s; . On the contrary, under the conditions of Einstein locality and separability, a measurement on S, will not disturb or influence the conditions within S, at the moment it interacts with the screen A,. It follows that the interference pattern at 4 will subsist in this case, independently of what is done with S,. If this turns out to be the case experimentally, then this would be evidence for our conjecture that in the case of correlated systems S , , S,, observation on S , does not steer S, in some QM state, unless an appropriate measurement apparatus is present. At the same time this would constitute a resolution of the EPR paradox (see also Table 11). However, we want to remark here that for single systems the so-called noresult Gedanken experiment of Renninger (1960) may be used to gain information about the system (and its wave function) without subjectingit to a measurement apparatus.

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W. DE BAERE

TABLE I1 SG,(a) I N PLACE,SG,(a) IN PLACE:EPR PARADOX Time t

< to

t

=

to

t 2 to

S,

S,

State vector of (Sl,S2):100) m,,(a, t < t o ) = ? m,,(a, t < to) = -m,,(a, t < t o ) = ? m,,(a, t o ) is measured: S, moves through SG,(a) (physical, local interaction) Reduction of 100): if m,,(a, to) = + 1: 100): reduces to 1)~’: for S , -+ State vector for ( S , , s , ) : lu!,:’)Iu~?!)

ms,(a, t o ) is measured: S, moves through SG,(a) (physical, local interaction)

Inferences ms,(a,t 2 to) = 1 No inference about S, for t < to

+

m,,(a, t o ) = - 1: 100) reduces to for S ,

Iub!!)

ms,(a, t 2 t o ) = - 1 No inference about S, for t < t o 3 No EPR Paradox

Considering the double-slit experiment with a source of single quantum systems, then according to Feynman et al. (1965) this is “. . . a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by explaining how it works.” Perhaps the proposal made in this section with a source of correlated quantum systems may help to get some insight in this mystery by allowing for the possibility, via coincidence measurements on A, and 4 , to explain or to follow how the pattern on 4 is built up. Let us note in this respect that recently Wootters and Zurek (1979) studied in a quantitative way, in the context of information theory, Bohr’s complementarity concept and Einstein’s version of the double-slit experiment (with a source of single photons), in which both the path and the interference pattern of the photons are attempted to be observed, it was found that the more information one has about the path, the less one has about the interference pattern. Subsequently Bartell (1980a) made a proposal of simpler realizable systems than that of Wootters and Zurek for observing intermediate waveparticle behavior. It is based on Wheeler’s reformulation of the Einstein version of the double-slit experiment (Wheeler, 1978). G . Conclusion

We have seen that the EPR argumentation [using (Hl), (H2) and (H3)], which resulted in the claim of incompleteness of QM, has been criticized


’

273

rather convincingly, mainly on the grounds of the impossibility in principle of knowing simultaneously noncompatible observables. We have made plausible that this amounts to the invalidity of the hypothesis of counterfactuality (H3), i.e., that one is not allowed to combine in an argumentation actual measurement results and hypothetical ones (supposed to be carried out on the same systems in exactly the same individual states). A disagreement on the observational level exists between QM and what we called the simplified argumentation [using only (Hl) and (H2)]. The discussion above has shown that the problem of QM nonlocality and nonseparability deserves further critical analysis, both on the theoretical and the experimental level. According to a new approach to the EPR paradox (Section II,F), it may be interesting to question the validity of the CI of the QM formalism, especially in its application to correlated but widely separated physical systems.

111. THEBELLINEQUALITIES

A . Introduction

The status of the EPR paradox, after the early criticism, remained unchanged for about 25 years. Several reasons may be given for this. First of all, EPR were not able to elevate their objections to QM from the purely epistemological level to the empirical level. This was done for them by Bell (1964). Moreover, QM predicted with great success and accuracy all results in atomic and molecular physics, and for every proposed new and realizable experiment a definite and unambiguous answer could be given. Hence, there was no practical need at all to supplement this successful formalism by means of hypothetical or hidden variables (HVs). Also, attempts to derive QM from classical ideas (Fenyes, 1952; Weizel, 1953a,b) were in general not very convincing because they did not lead to clear, verifiable differences with QM. As a result of all this, opinion had settled that Bohr’s (1935a,b) reply to EPR was convincing and final. Another important reason for this was the existence of the von Neumann theorem. Indeed, already in 1932 von Neumann (1955)had proved mathematically, starting from a number of plausible assumptions, that no deterministic theory for the individual system, based on the idea of dispersion-freevariables, is able to reproduce all statistical predictions of QT. However, since Bohm (1952b,c) proposed a concrete counterexample of von Neumann’s theorem, and Bell (1966) and Bohm and Bub (1966b) showed that one of the underlying assumptions, namely the linearity hypothesis, was not necessarily valid in all HV theories, the interest in the fundamentals of QM

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W. DE BAERE

was again awakened. Also, at about the same time, Bell (1964)made his famous analysis of the EPR paradox which consisted in translating the EPR ideas in precise mathematical form and deriving a simple inequality, the Bell inequality. The importance of this BI stems from the fact that it made very clear the conflict between QM and the EPR point of view, and that Clauser et al. (1969)showed that by generalizing the BI the whole issue could be brought to the experimental level. B. Von Neumann’s Theorem

The essentials of von Neumann’s proof of the impossibility of dispersionfree states are as follows (von Neumann, 1955; Albertson, 1961; Bell, 1966; Bohm and Bub, 1966a; Capasso et al., 1970; Jammer, 1974, pp. 265-278). The starting points are a number of hypotheses: (vN1) To each observable R there corresponds in a one-to-one way a Hermitian operator R in Hilbert space. It is assumed that R is hypermaximal, i.e., that its eigenvalue problem is solvable; (vN2) With the observable f ( R )there corresponds the operator f ( R ) ; (vN3) If to the observables R, S , . . . (not necessarily simultaneously measurable) there correspond Hermitian operators R, S, . . ., then to the observable R + S + ... there corresponds the operator R + S + ” ’ ; (vN4) Hypothesis of linearity: If R, S , . . . are arbitrary observables and a, b, .. . real numbers, then the following relation holds between expectation values (uR

+ bS + ...) = u ( R ) + b ( S ) + ...

(32)

with

(R)

(33)

= CWn($nIRI$n) n

if the ensemble is described by the pure states I$,)

with probabilities w,.

The dispersion A R of an observable R in an ensemble may be defined by (AR)’ = ( ( R - ( R ) ) ’ )

=

( R 2 ) - (R)’

(34) A dispersion-free ensemble is then defined by the condition AR = 0 for all R, i.e. (35) A further definition introduced by von Neumann was that of the homogeneous or pure ensemble. Any partition of such an ensemble results in a series of subensembles, which, by definition, have the same statistical properties as the original one. < R 2 ) = ’,

VR.


275

On the basis of the above postulates and definitions, von Neumann was able to prove his theorem on the impossibility of reconstructing QM from any kind of theory which starts from dispersion-free ensembles, usually identified with a HVT. In the words of von Neumann “we need not go any further into the mechanism of the ‘hidden parameters,’ since we now know that the established results of quantum mechanics can never be re-derived with their help” (von Neumann, 1955, p. 324). The proof proceeds by showing that the assumption of dispersion-free ensembles leads to inconsistencies. In the particular case of spin 3,this proof runs as follows. Consider a dispersion-free ensemble of spin-3 systems corresponding to a definite HV 1.For such an ensemble, each observable has a well-defined value when measured. Consider in particular the observables R = ox, S = oy,and T = R + S = ox + ay,with a,, oy,and a, the Pauli matrices. Then the results of measuring R, S, and T on each element of the ensemble may be represented by r(A), s(A), t(A). Hence

(R)

= r(4,

( S ) = s(4,

( T ) = t(4

(36)

and, according to (vN4), one should have (R

or

+S) =(T)=(R) +(S) t(2) = r ( 4 + s(1)

(37)

Now, because of the fact that eigenvalues of R, S, and T are, respectively, 1, f 1, and Eq. (37) cannot be true, from which follows von Neumann’s impossibility theorem. Challenged by these assertions, Bohm constructed in 1952 (Bohm, 1952a,b) a concrete counterexample of von Neumann’s theorem. In Bohm’s model, which is a reinterpretation of QM, the notion of quantum potential is introduced, and position and momentum are treated as hidden variables. Subsequently, Bell (1966) and Bohm and Bub (1966a) criticized the general validity of von Neumann’s hypotheses, in particular the hypothesis of linearity (vN4). Bohm and Bub (1966a) came to the conclusion that only a limited class of HV theories was excluded by von Neumann’s theorem [in fact, only those satisfying the von Neumann assumptions (vNl)-(vN4)]. Moreover, Bell (1966)explicitly constructed a physically reasonable HV model for spin-4 systems, which did not satisfy (vN4). Bell showed (Selleri, 1983b, pp. 49-52) that, although (vN4) is correct for standard QM states, it is unjustified to require it to be valid for all conceivable HV models. Other impossibility proofs for HV theories were set up by Gleason (1957), by Jauch and Piron (1963), and by Kochen and Specker (1967). All these are apparently more general than von Neumann’s proof and are not based upon the criticized hypothesis (vN4).The proof of Jauch and Piron has in turn beer,

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W. DE BAERE

refuted by Bohm and Bub (1966b). The general problem with proofs of this kind is not their mathematical correctness but the physical relevance of their basic assumptions. For this reason it may be stated that no such proof will ever be able to rule out a priori all possible HV models. For more details we refer to the already mentioned papers, Clauser and Shimony (1978)and Jammer (1974, pp. 296-302).

C . The Original Bell Inequality 1 . Derivation and Discussion

In 1964 Bell re-examined the Bohm version of the EPR argumentation. Bell investigated the properties of any local deterministic HV theory which satisfies the requirements of EPRs realistic conception of reality. The main result is the famous Bell inequality, which has to be satisfied by any such theory. The fact that the BI is violated by QM constitutes Bell's theorem: No local deterministic HVT can reproduce all results of QT. To prove this, a couple of correlated spin-+systemsS,($), &(+) in the singlet state is considered, and it is assumed that the internal state of each system may be represented, because of their correlation, by some common set of timedependent parameters (HVs),collectively denoted by A(t). Suppose that s, and S2 move in opposite directions toward Stern-Gerlach devices SG,(a) and SG2(b)which measure the spin component m,, along unit vector a and of mS2 along unit vector b. Bell then remarks that, under the validity of Einstein locality (separability) (H l), the value of m,, is independent of band the value of ms2is independent of a. Furthermore, these values are assumed to depend on the HVs A(t), representing the individual state of S, and S 2 . This suggests the existence of functions msl(a,A(t)), ms2@,A(t)) which completely determine the spin projections (whether measured or not). Hence the basic assumption is that at any time t , S , and S, have correlated properties which are definite and locally determined (in the above sense),in agreement with the EPR idea of elements of physical reality. To simplify notation we shall write A(a,A) = m,,(a, A)

=

k 1,

B(b,A) = m,,@,A)

=

k1

(38)

and adopt the convention that the spin projections are k 1 instead of i-3 (in units of h). The central quantity in Bell's reasoning is the correlation function A(a, A)B(b,A ) p ( l )dA

(39)

277


in which A represents the whole HV space, and p(A) a normalized distribution function of the HVs il (40) Now, the QM prediction for the correlation function P(a, b) is (Peres, 1978)

. -

P(a, b) = (OOJa, aa, b100)

= -a

-b

(41) In Eq. (41), 100) is the singlet state vector of Eq. (15) and a, a,@, b are the operators for the spin projections along a and b. In particular, one has P(b,b) = - 1. Hence, if the HV expression in Eq. (39) reproduces this result, then one should have that B(b,A) = - A @ , A) and Eq. (39) becomes P(a,b) = -

I

A(u,il)A(b,A)p(A)dA

.

.

(42)

Bell then goes on to reason along the lines of EPR and considers the hypothetical situation that instead of analyzer direction b one had chosen another direction c. Now, it is seen immediately that at this point,just as in the EPR argumentation, the validity of counterfactuality (H3) is needed to allow such reasoning. As remarked while discussing the EPR argumentation, QM forbids, however, in principle such a reasoning because the spin projections B(b,A) and B(c,A) are incompatible quantities, and hence cannot be known simultaneously because of the practical impossibility of measuring them together in one single measurement. We believe that this is a central lesson from the successful QT, that one necessarily has to incorporate in any future theory. Simultaneous measurements of incompatible observables was studied long ago, e.g., by She and Heffner (1966), Park and Margenau (1968), and recently by de Muynck et al. (1979) and by Busch (1985).In de Muynck et al. it has been shown that a simultaneous measurement of such observables causes an unavoidable mutual disturbance of both results; i.e., the value of B(b,A) will depend on whether it has been measured alone or together with B(c,A) (after having given an operational meaning to such combined measurements). Otherwise stated, the joint probability distributions will not reproduce the marginal distributions. We will see in Section V,B that this is precisely a point in the derivation of the BI which is severely criticized. Yet, to make physical sense of the above counterfactual hypot!iesis (Lochak, 1976; De Baere, 1984a,b), it may be assumed that in similar correlation experiments (which may be carried out either simultaneously at different places, or subsequently at the same place) the HVs are distributed according to the same HV distribution p(A). Under this assumption one may

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W. DE BAERE

write then, according to Eq. (42)

h lA

A(a, A)A(c,A)&)

P(a, c ) = -

dA

(43)

Subtracting Eq. (43) from Eq. (42) one obtains P(a, b) - P(a, c) = -

= --JA

Because A(a,A)

=

CA(a,4A(b,4- 4, A)A(c,4 l P ( 4

&a,

(44)

& 1, B(b,A) = f 1, it follows that

I

IP(a, b) - P(a, c)l I

or

W ( b ,4L-1 - A@, AMc, AllP(A)dJ.

[1 - A(b, A)A(c,A)]p(A)dA

(45)

+ P(b,c)

(46)

IP(a,b) - P(a,c)l I 1

This is the original form of Bell's inequality, which expresses in very simple mathematical terms the consequences of Einstein locality (H 1) and counterfactuality (H3). Equation (46) is the inequality that each local, deterministic HVT in which counterfactuality is valid has to satisfy. It is easily seen that Eq. (46)is violated by the QM prediction in Eq. (41).To this end it suffices to take the following configuration of spin analyzer directions (Fig. 4)

O,, = O,,

= O,,

f 2 = 60"

(47)

for which P(a,b) = P(b,c) = -P(a,c) = -3

(48) Inserting Eq. (48) into Eq. (46), one gets the contradiction 1 5 3 . From this result follows Bell's theorem: No local deterministic HVT is able to reproduce

FIG.4. Configurationof orientations for which the BI is violated by QM.


219

all predictions of QT. We will see in Section IV that most correlation experiments verify QM, and hence violate the BI, from which it is concluded almost generally that the locality hypothesis (H 1) is invalid. In other words, it is believed that any acceptable HVT which does not violate QT a priori should be nonlocal. However, we have seen that Bell implicitly assumes the validity of counterfactuality (H3) as self-evident. From the above conclusion of nonlocality, it may be argued then that perhaps the importance of counterfactuality (H3) is being overlooked. It may be stated, therefore, that locality may be saved if counterfactuality is abandoned. 2. Wigner‘s Version of Bell’s Theorem

Wigner (1970) considers two correlated spin-$ systems S,($), S,($) in the singlet state and assumes that the HVs A determine the spin components in any number of directions. Wigner’s argument uses only three directions a, b, and c. For a given 1, and under the condition of locality, the HVT gives the predictions A(a,A) = a, = & 1, A(b,A) = b, = k 1, and A(c, A) = c, = k 1 for the spin components of S , and predictions B(a,A) = a, = & 1, B(b,A) = b, = L- 1, and B(c,A) = c, = & 1 for S,. Let us denote then by p(a,, bl,cl;az,b,,c,) the probability for having the configuration (al,b,, c,; a,, b,, c,) of spin components for S , and S, . In the singlet state one must have that a, = -a,, b, = -b,, c, = -cl. Again QM forbids from first principles that the above characterization be verified experimentally. Indeed, Heisenberg’s uncertainty principle forbids the simultaneous measurement, hence knowledge, of spin components such as a,, b,, c, for S, or of a2,b2,czfor S,. Refusing to accept this basic truth will again necessarily lead to contradictory conclusions, as we will see. Also, and for the same reason, only one couple of values such as, e.g., a,, a, or b,,c, will represent values that correspond to actual results, while all the other combinations such as b , , a , , etc. are then hypothetical. Hence, as in all other derivations of Bell-type inequalities, the validity of counterfactuality is already built in implicitly. To see that the above HV scheme contradicts QM, let us derive probabilities from p(a,, b,, c,; a,, b,, c,) which are experimentally verifiable and, hence, for which QM makes specific predictions. Consider then the probability p ( a , + ,b 2 + )for the results a, = 1, b, = 1

+

+

p ( a , + , b , + )= ~ ~ ( ~ , + , ~ , ~ ~ = ~ -l C J~ ~ z - ~ ~ z + ~ ~ c1

and thus one should have from Eq. (50)

which is not satisfied if c bisects oab. Wigner’s argumentation has been criticized by Bub (1973), who shows that the assumption of a probability distribution for the values of incompatible observables leads to inconsistencies in the case of single spin-4 systems. Freedman and Wigner (1973) reply that Bub’s criticism is based on a misinterpretation of Bell’s locality postulate.

D. Generalized Bell Inequalitiesfor Dichotomic Variables It has been remarked that a weak point in the derivation of the original BI is the requirement that the condition P(a,a) = - 1 has to be satisfied exactly. Now, no experimental setup is perfect; so that this condition will never be realized in practice. The solution to this problem led to the first derivation of the generalized Bell inequality (GBI) by Clauser et al. (1969). It has been argued, furthermore, that neither the restriction to a deterministic scheme nor the use of the HV concept is necessary for the derivation of generalized forms of the BI. It appears that even the concept of locality may be defined in different ways with respect to Bell’s inequalities (Eberhard, 1978; Rastall, 1981; Bastide, 1984; Stapp, 1985). As a result, a large number of generalized Bell inequalities (GBI) has been constructed since the appearance of the original BI, the most well known of which we will review below.


28 1

I . Generalization by Clauser et al. Clauser et al. (1969) were the first to derive a GBI and to show how it may be transformed so as to allow direct experimental verification. Consider again the correlation function P(a, b) defined by Eq. (39). With the notation and conventions of Section II1,C one may write the following inequality for IP(a, b) - P(a, b’)l IP(a,b) - P(a, b’)J I

= =

=

I I

IA(u, l)B(b,A) - A(a, l)B(b‘,l ) l p ( l )d l I A k , WW,4lC1 - B(b,W ( b ’ ,m(4 dl C1 - B(b,

1-

WW, 4 l p ( 4 dl

B(b,l)B(b’,l ) p ( A )d l

(53)

If the criticized condition A(b,A) = - B ( b , l ) were valid, Eq. (53) would lead immediately to the original BI, Eq. (46). However, suppose that we now have

P(u’,b) = 1 - 6,

0I 6 I 1

(54)

for some directions a‘ and b, and such that 6 differs from zero for a’ = -b. Writing A = A+ u A - , with A+ = (A1 A(a’,A) = &B(b,A))

(55)

it may be shown that

With this result the right-hand side of Eq. (53) may be written B(b,WV’,M 4 d l =

P(a’, b’) - 2

J1,-

A@’, /Z)B(b’,&(A)

dl

IA(a’,A)B(b’,I)(p(A)dA = P(u‘,b’) - 6 = P(u’,b‘)

+ P(u’,b) - 1

(57)

282

‘i= W. DE BAERE

+

450

2

-a‘

FIG.5. Analyzer directions for which the GBI is violated by QM.

From Eqs. (53)and (57) it follows that IP(u, b) - P(u, b’)l

+ P ( d , b) + P(u’,b’) I2

(58)

This is the GBI, also called the Clauser-Horne-Shimony-Holt or CHSH inequality, which again is violated by QM; e.g., for %,b = %,,./3 = Oarb= %arb, = 45” (Fig. 5) one has the contradiction 2& I2. The GBI of Eq. (58) may be transformed to an inequality which contains only directly available experimental quantities for correlated optical photons yl, yz, originating from an atomic cascade. In this case the Stern-Gerlach apparatuses SG,(a), SGz(b) are replaced by linear polarization filters Pl(a), Pz(b). Normally A(a) = + 1 would correspond to the detection of y , which passed P,(a), and A(a) = - 1 would denote nondetection. However, because of problems connected with small photoelectric efficiencies for optical photons, another convention must be introduced: A(a) = - 1 means emergence from P,(a) and A(a) = - 1 nonemergence. With this convention, P(a, b) is an emergence correlation function, and in order to relate it to experimental data one has to introduce the supplementary assumption that the probability for coincidence detection of y,, y, is independent of a, b. If, furthermore, the following experimental counting rates are introduced: R(a,b) = rate of coincidence detection of y1,y2 with Pl(a),P,(b) in place; R, = R,(a) = rate of detection of y , with P,(b) removed, assumed independent of a; R, = R,(b) = rate of detection of yz with P,(a) removed, assumed independent of b; then Eq. (58) may be transformed to IR(u,b) - R(u,b’)l + R(u’,b) + R(u’,b’) - R , - R , I 0

(59)

For R(a,b) = R(a - b) and relative polarizer orientations la - bl = la - b‘1/3 = la’ - bl = la’

-

b’l

= ~p

(60)

283


Eq. (59) simplifies to 3R(q) - R ( 3 q ) - R , - R , I 0

(61)

Defining

Ncp) = C3R(cp)- R(3cp) - R ,

-

(62)

R,l/Ro

with Ro being the rate of coincidence detection of yl, y z with Pl(a), P,(b) removed, Freedman and Clauser (1972)arrived at a lower bound - 1 for A(cp). Hence, instead of Eq. (61) one may write finally -

1 I [3R(cp) - R(3cp) - R1 - R , ] / R o I 0

(63)

Now, it may be shown that if the QM prediction for the correlation function is of the form r + scosncp, then there is maximal violation of the upper limit of Eq. (63) for ncp = n/4 and of the lower limit for ncp = 3n/4. In the case of correlated cascade photons one has n = 2, and these angles become, respectively, cp = n/8 and 3n/8. Inserting these two values for cp into Eq. (63),and noting that R(9n/8) = R(n/8),one has

- 1 5 [3R(n/8)- R(3n/8) - R1

0

(64)

[ 3 R ( 3 ~ / 8-) R(n/8) - R1 - R , ] / R o I 0

(65)

- R,]/Ro I

and -1I

By subtracting Eq. (65)from Eq. (64),both inequalities may be combined to one single inequality

a

(R(22.5")/R0- R(67.5")/R0(- I 0

(66)

which no longer depends on R , , R , . Actually it is Eq. (66) which has been tested in most cascade-photon correlation experiments. 2. The Proof by Bell

In his own derivation of the GBI, Bell (1972) starts from the idea that the respective Stern-Gerlach apparati SG, (a)and SG,(b) carry their own hidden information, which may be represented formally by HVs A,, & with domains A,, I \ b and normalized distributions pu(Aa),p b ( & ) satisfying JAa

and n

pa(Au)

d l a = 1,

la

E Aa

(67)

284

W. DE BAERE

The measurement results are now A(u,A,&) = f 1,O and B(b,A, &) = the value 0 assigned to A or B if S1 or S2 is not detected. For the correlation function P(a,b) we now have

+

- 1,0,

with

44 4= B(b,A) = and

1. 1.,

N u , A,&)pa(Aa)

(704

B(b,A, i b ) p b ( & ) d&

(70b)

I&&i)I I 1, lB(b,i)l I 1 (71) If for SGl(a),SG2(b)alternative settings a’ and b’ are considered, then one may write P(a, b) - P(a, b’) =

=

I

[A(a,i)B(b,A) - A(a,A)B(b’,i ) l p ( l )dA

b

A(a,A)B(b,A)[ 1 f A@’,i)B(b’,i ) ] p ( i )d i

A(a,i)B(b’,i)[l & A(a’,i)B(b,A ) ] p ( A ) d l (72) -

or, because of Eq. (71) IP(a,b) - P(a, b’)l I

I

k A(a’, A)B(b‘,i)]p(i) d A

+

c1

[l

jAf

A(a’,i)B(b,41p(A)dA

< 2 f [P(a’,b’) + P(a’,b)]

(73)

which finally may be written

+

IP(a,b) - P(~,b’)l IP(a’,b’)

+ P(~’,b)lI 2

(74) This form of the GBI is similar to that first derived by Clauser et al. (1969),

EINSTEIN- PODOLSKY-ROSEN PARADOX

285

Eq. (58). It is seen that, as a result of the conditions in Eq. (71), the GBI of Eq. (74) has to be satisfied by any stochastic HVT for which locality and counterfactuality are assumed to be valid. 3. The Proof by Stapp In his version of Bell’s theorem, Stapp (1971,1977) considers an ensemble of N correlated spin4 systems S,(i), S2($) moving toward Stern-Gerlach devices SG,(a), SG,(b). We will review this version in some detail because it is claimed and believed (DEspagnat, 1984) that the concept of HV is not needed. However, we will see that this is not correct, because the HVs I may be considered as a formal representation of some elements in Stapp’s argumentation. It is assumed that each device has two alternative settings: a, a’ for SG,, b, b’ for SG,. Call Aj(a,b) the prediction of a hypothetical, more complete theory for the measurement of the spin component of the jth system S , along a, and likewise for Bj(a,b). If Aj(a,b) and Bj(a,b) are actually measured, then Stapp supposes that the theory predicts definite values for the numbers Aj(a,b’), Bj(a,b’),Aj(a’,b), Bj(a’,b), Aj(a‘,b‘),and Bj(a’,b‘) which correspond to alternative settings (a, b’), (a‘,b) and (a’, b’) of the Stern-Gerlach devices. Moreover, it is assumed that these numbers would have been the measurement results if, instead of the couple of actual settings (a,b), one of the three alternative possibilities were chosen. This assumption is the essence of what Stapp calls the hypothesis of counterfactual definiteness, and he further assumes that QT also makes correct predictions for these hypothetical measurement results. Now, if Aj(a,b) and Bj(a,b) are measured, then according to QT Bj(a,b’) can never be measured simultaneously with Bj(a,b) (unless b and b’ are parallel). Hence, if the values Bj(a,b’),j = 1,. . .,N are to be used to define a correlation function whose value is determined in a real experiment, then the hypothesis of counterfactuality is equivalent to the following assumption. Suppose that in one actual experiment with settings (a,b) we have obtained N couples of results Aj(a,b),Bj(a,b), j = 1 , . .. ,N and from another actual measurement with settings (a,b’) we have obtained the series of results Aj(a,b’),Bj(a,b’). Counterfactuality then assumes that between the [ ( N / 2 ) ! I 2 permutations which make the series Aj(a,b‘), j = 1 , . .. ,N coincide with the series Aj(a,b), j = 1,. ..,N there is at least one permutation such that all internal conditions (which are supposed to be responsible for each actual result) within S, and S, coincide exactly in both series. It is readily seen that these “internal conditions” may be identified with the former concept of HV 1. Hence, although HVs do not appear explicitly in Stapp’s reasoning, they are implicitly contained in it. Note that only when the

286

W. DE BAERE

above condition is satisfied does it have sense physically to consider what would have been the result if instead of an actual apparatus setting, one had chosen another possible one. Suppose then further that such a reasoning is allowed. Consider then the series of results Aj(u,b),

j = 1,. ..,N

Bj(u,b),

Aj(u,b’),

Bj(u,b’),

Aj(U’,b),

Bj(U’,b)

A ~ ( u b’), ’,

j = 1,. . ., N

.. ,N = 1,. .. ,N

j = 1,.

B~(u’, b’),

j

(754 (75W (754 (754

If the first series are actual results, then the other three are hypothetical, but, on account of the assumed validity of counterfactuality, may be assumed to be equivalent to actual results in some subsequent experiment. Under this condition, then, Stapp introduces locality by the following requirements: j = 1,. . .,N

Aj(u,b) = Aj(u,b’) = Aj(u), A ~ ( u b) ’ , = A ~ ( u b’) ’ , = A~(u’),

Bj(u,b) = Bj(u’,b) = Bj(b), B~(u, b’) = B~(u’, b’) = Bj(b’),

. . ,N = 1,. . . ,N j = 1,. . .,N j = 1,.

j

(764 (76b) (764 (764

The relevant correlation functions are then defined by (for N sufficiently large) 1

N

1

P(u, b) = N j = 1 Aj(u)Bj(b) 1

(774

c N

P(u’,b) = - Aj(U’)Bj(b) N j=1 1

P(u, b’) = -

1N Aj(a)Bj(b’)

Nj=l

1

1N

P(a’,b’) = Aj(U’)Bj(b’) N j=1

(774

Stapp shows then that Eqs. (77a-d) are not compatible with the Q M prediction of Eq. (41) for all conceivable settings a, a’, b, b’. Indeed, for O,, = O”, eaZb= 135”, Oob, = 0”,and Oarb,= 45” one arrives at the contradiction f i I 1. Stapp concludes that the locality conditions of Eqs. (76a-d) have to be false. Recently Stapp (1985) has been defending his counterfactuality hypothesis further. According to Stapp there are two different concepts of locality, both


287

due to Einstein. The first is the one that is used in Einstein’s relativity theory, according to which no physical signal can travel faster than light. The second is the idea that events separated in space may in no way disturb each other, and this should be incompatible with QM. However, both concepts must necessarily have something to do with the propagation of influences, and it is not clear why both propagation speeds should be different, at least not in a unified world view. In Section V we will see, on the contrary, that many arguments point to the validity of locality in both the above senses. 4 . The Proof by Selleri

Selleri (1972) has given a very simple proof of the GBI by using the result [Eq. (53)] of Clauser et al. (1969) IP(a, b) - P(a, b’)(= 1 -

I I

B(b,A)B(b’,A)p(A)d I

(78)

Noting that on both sides of Eq. (78) the sign may be changed and that the right-hand side does not depend on a, one may also write IP(a’,b)

+ P ( d ,b’)l = 1 +

B(b,A)B(b’,A)p(A)d A

(79)

Adding Eqs. (78) and (79), one immediately gets the GBI of Eq. (74). 5 . The Proof b y Clauser and Horne

An alternative to the proof by Bell (1972) of the GBI which is valid for deterministic, as well as for stochastic HV theories, is given by Clauser and Horne (1974). Assume analyzer orientations a and b, and suppose measurements are made on N couples (Sl,S2).Call N , ( a ) and N,(b) the number of counts at detectors D,and D, behind the respective analyzers. Call N,,(a, b) the number of coincidence counts. If N is large enough, then one may write for the probabilities of these counts

p12(a,b) = N,,(a, b ) / N

(804

The internal conditions within the correlated systems S , , S , are again represented formally by means of HVs I with a normalized distribution p ( l ) . Clauser and Horne then assume that knowledge of these HVs and of the

288

W. DE BAERE

analyzer directions determines only the probabilities pl(a, A), pz(b,A), and p12(a,b, A) for counts to be registrated at Dl, at D,,and at both D1 and D,. Now, here locality is introduced by assuming that pl(a, A), pz(b,A) do not depend on b and a and that

4= P l h 4P,(b, 4

(81) Equation (8 1) is known as the Clauser-Horne factorability condition. Also Plz(a9

Pl(4 = P2(W =

Pl&,b) =

I I I

Pl(44 P ( 4d l

(82a)

A M 4d l

(82b)

Plz(a,b,A)P(A)dA

(824

P&

To prove the GBI, Clauser and Horne consider two alternative orientations a, a’ for the first analyzer and two alternatives b, b’ for the second one. From the inequality - X Y Ix y - xy’

+ x ’ y + x’y’ - X’Y - yx I

0

(83)

which is valid for real numbers x , x’,y , y‘, X , Y satisfying 0 I x , x’ 5 X , 0 5 y , y’ IY, it is concluded that - 1 5 p,,(a,b,A) - Pl,(4b‘,A) -P1(U’,4

+ Pl2(af,b,4 + P1z(a’,b’,4

- pz(b,A) 5 0

(84)

Multiplying Eq. (84) by p ( l ) and integrating over A, one gets -

1 5 P l 2 ( 4 b) - P l 2 ( 4 b’) + P1&’,

4 + P l Z b ’ , b’) - P 1 ( 4

-P

2 W

I0

In the above inequality the probabilities may be replaced by the observed counting rates &(a‘), R2(b),R(a,b), etc. defined in Clauser et al. (1969)(Section III,D,l), to obtain R(u,b) - R(u,b’) + R(u’,b) + R(u’,b’) &(a’)

+ R2@)

I1

(86)

which is essentially the same as Eq. (59). As already remarked when discussing the derivation by Clauser et al., to compare Eq. (86) with atomic cascade experiments necessitates, as long as no


289

perfect polarizers exist, the introduction of a supplementary assumption. This is the so-called “no-enhancement’’ assumption, according to which the probability for photon detection with a polarizer in place, e.g., pl(u,A), is not larger than the corresponding probability pl(A) with the polarizer removed

0I p1(a,A) I p1(A) s 1

(874

0 I pz(b, A) I p#)

(87b)

s1

for all I, a, and b. To save locality, the validity of Eqs. (87a,b) in the existing cascade experiments has recently been seriously criticized (Section V,C). The general validity of the factorability condition [Eq. (Sl)] may be criticized (Selleri and Tarozzi, 1980; Selleri, 1982) by making the following remarks. In the above scheme, the correlation function P(a, b) may be written as p(a?b) =

h

PAa+ A)PZ@+? 4- PI@+ 9

-Pk-

9

A)PZ(b+

9

4+ P l k -

9

9

4Pz(b-

A)P,(b-

7

2

4 M A )&

f

with pl(a*, A) the probabilities for A(a,A) = f 1. Now, suppose that A is just a short-hand notation for the set of HVs A’,A“, . . .: A = (A’,A”,. . .). Then one may write

s

P(a, b) = ql(a,A‘, I“,. . .)q,(b, 2,il”,. . .) x p ( X , A”,. . .) dA’ d l ” . . .

(89)

and, integrating Eq. (89) formally over A’, the new expression should also satisfy the factorability condition, i.e.,

s

P(a, b) = rl(a,I”,.. .)rz(b,A”, . . .)p(A“, ...) dI“ dA”’ . . .

(90)

and one should have that rl(u, A”,. . .)r,(b,A“,. . .) =

jql(..r,A!!,. ..)qz(b,Z,i”, .. .)p(A’, A”,. .. ) d X

(91)

290

W. DE BAERE

Dividing Eq. (91) by its a derivative, one obtains r

with

r\(a,I’’,

. . .) = dr,(a,I”,. . .)/da

q\(a, I’,A“, . . .) = dql(u, I’,I”,.. .)/da

(93)

It is seen that the left-hand side of Eq. (92) is independent of b, while its righthand side depends on b. This cannot be true in general; hence the same applies to the factorability condition [Eq. (81)] on which Eqs. (88) and (90) are based. 6 . The Proof by Eberhard and Peres

Like Stapp, Eberhard (1977) and Peres (1978) claim to give a proof of the GBI which starts only from the principle of locality and does not need to introduce the HV concept nor determinism. The validity of counterfactuality is considered as self-evident or at least a “rather natural way to thinking.” To give an outline of the simple proof, we shall use the same notation as in Section III,D,3 and start from Eqs. (77) for the correlation functions and from Eqs. (76), which express the requirement of locality. Eberhard’s derivation is based on the remark that for the j t h event one has Xj

=

Aj(a)Bj(b)+ Aj(~’)Bj(b) + Aj(a)Bj(b’)- Aj(~’)Bj(b’) I 2

(94)

because Aj(a’)Bj(b’) is the product of the other three terms, and Aj = f 1 and Bj = 5 1. It follows from Eq. (94) that 1 -C

N j

Xj = P(u, b) + P ( d , b) + P(a, b’) - P ( d , b’) I2

(95)

which is the generalized Bell inequality. From the violation of Eq. (95) by QT, Eberhard concludes that Eqs. (76) are not correct, hence requiring nonlocality, because he unconditionally assumes counterfactuality to be correct. Eberhard also gives an alternative formulation of Bell’s theorem: It amounts to saying that, for certain relative orientations of four analyzer directions a, a’, b and b, it is impossible to find four sets of actual spin measurement results {Aj(a,b),Bj(a,b ) } , {Aj(u’,b), Bj(a’,b)}, {Aj(a,b’),Bj(a,b ’ ) } , and { Aj(a’,b’), Bj(a’,b‘)}for which Eq. (76) and hence the GBI Eq. (95)are valid.

EINSTEIN-PODOLSKY- ROSEN PARADOX

29 1

However, according to us, in this reformulation of Bell‘s theorem the invalidity of Eq. (95) does not imply the breakdown of locality but rather the breakdown of counterfactuality. A proof of the GBI along similar lines has been given by Peres (1978). Remembering that the derivation of the GBI is the result of combining one series of actual data with three series of hypothetical ones, Peres distinguishes two possible attitudes with respect to the violation of the GBI Eq. (95) by QM: “One is to say that it is illegitimate to speculate about unperformed experiments. In brief, ‘Thou shalt not think.’ Physics is then free from many epistemological difficulties. For instance, it is not possible to formulate the EPR paradox.” At this point Peres refers to the EPR statement “If ... we had chosen another quantity . .. we should have obtained .. .,” which is according to him “The key point in the EPR argument.” The second attitude is: “Alternatively, for those who cannot refrain from thinking, we can abandon the assumption that the results of measurements by A are independent of what is being done by B.” Here A and B stand for SG,(a) and SG,(b) in our notation. 7 . Other Bell-Type Inequalities

We have seen that the hypothesis of locality (Hl) and of counterfactuality (H3) leads in various ways either to the original BI [Eq. (46)] or to the GBI [Eq. (58)l.Although the theoretical and experimental disagreement between these inequalities and QT (Section IV) is already sufficient to discard one of these underlying hypotheses, various people have extended the BI further by constructing what we shall call Bell-type inequalities. In most of these generalizations Eq. (46) or (58) is somehow contained. Pearle (1970) considers n possible analyzer orientations a,, a,,. . . ,an for one measuring device and n orientations b,, b,, . ..,bn for a second one. In terms of a HV A with a normalized distribution p(A), and dichotomic measurement results A(a,,A), B ( b j ,A) = 5 1, a correlation function P(a,, b j ) is defined in the usual way as P ( a i ,b j ) =

Starting from the identity

A(a,,A)B(bj,A)p(A)dA

(96)

292

W. DE BAERE

one easily derives the inequality

+ P(a,,b,) + ... + P(a,,b,)

P(a,,b,)

I 2n - 2

+ P(a,,b,)

(98a)

or n

1 CP(ai,bi) + P(ai+ ,, bill I 2n - 2 + P(a,, b,)

(98b)

i= 1

For equal angles between adjacent vectors in the sequence a,, b,, a,,. Eq. (98b) simplifies to

(2n - I)P(a,b ) I (2n - 2)

+ P(a,, b,)

. .,b, (984

in which a, b represent any two adjacent vectors. Applying the approach of Wigner (1970),DEspagnat (1975) considers an ensemble, for which it is somehow possible to predict the number n(rl,r2,. . ., r m ) of systems having dichotomic values r , , r , , . . .,rm for observables R,, R,, . . .,R,. A correlation function P(ri,r j ) is defined as the mean value of the product rirj. D’Espagnat is then able to derive the general inequality

On the basis of locality and counterfactuality Herbert and Karush (1978) derived the following set of Bell-type inequalities

-(m

-

1) I mP(8) - P(m8) - (m - 1) I 0

-n I nP(8) - P(n8) - n

+ P(0) I 0

(100)

with m representing any odd integer and n any even integer. Another method to generate Bell-type inequalities (Selleri, 1978)is to start from the inequality CaA(a.1)

+ /3B(b,A)+ yA(c,A)]’

2

1

(101) with a,& y = k 1, A(a,A) = f 1, etc., and B(a,A) = -A(u, A). It may be shown that Eq. (101) generates inequalities such as

+ P(a,c) + P(b,c) I 1

(102a)

P(u, b ) - P(u,c) - P(b,c) I 1

(102b)

-P(a,b)

+ P(a,c) - P(b,c) I 1

( 102c)

-P(a,b)

-

P(a,c) + P(b,c) I 1

(102d)

P(a,b)

Garuccio (1978) generalizes this method by considering the inequality [aA(a, A)

+ /?A@, A) + yA(c, A)]’

2 min( k a

k /3 k y)’

(103)

293


in which a, fl, and y are positive, but otherwise arbitrary, and a 2 With these conditions min(_+a_+

a

_+

Y )= ~ (-c(

B 2 y > 0.

+ + y)‘

( 104)

and the following Bell-type inequality may be deduced P(u, b )

+ xP(u,C ) + yP(b,C) I 1 + x - y

(105)

with a = y//? and y = y/a. Other equivalent inequalities may be derived, such that instead of Eqs. (102a-d) one may write

P(u,b)

+ xP(u,C ) + yP(b, c) I 1 + x

y

(106a)

+ x -y -P(u, b ) + xP(u,c)- y f ( b , c ) I 1 + x - y P(u,b ) - xP(u,C ) + yP(b,C ) I 1 + x - y

(106b)

-

P(u,b ) - xP(u,c)- y f ( b , c ) I 1

-

(106c) (106d)

All these inequalities are again violated by QM for certain values of u, b, c. From Eqs. (106a-d) follows an inequality with four correlation functions: From Eqs. (106a) and (106b) we have P(u, b )

+ IxP(u,c)+ yP(b,c))I 1 + x - y

(107a)

and from Eqs. (106c) and (106d), replacing x by x’, y by y’, and c by c’

+

- P ( u , ~ ) Ix’P(a,c’)- y’P(b,c’)l I1

+ X ’ - y’

(107b)

Adding Eqs. (107a) and (107b), one obtains

+ yP(b,c)[ + Ix’P(u,c’) - y ’ f (b,c’)l 2 + x + x’ - y - y’

IxP(a,C )

I

(108)

in which 1 2 x 2 y > 0, 1 2 x‘ 2 y’ > 0. From numerical calculations for variable parameters x, y, x’, y’ it follows that maximal violation of Eq. (108) occurs for x = y = x’ = y’ = 1; i.e., the original GBI is the strongest inequality with respect to the QM singlet state. A systematic derivation of all Bell-type inequalities for dichotomic measurement results has been given by Garuccio and Selleri (1980). They use the probabilistic formulation of Clauser and Horne (1974), in which the correlation function is given by Eq. (88): %(a, 4q2(b,4

P ( 4d i

(109)

294

W. DE BAERE

with

and - 1 Iq,(u,A.) < 1, - 1 < q , ( b , l ) I 1. Suppose we have analyzer directions a,, a,,. . .,a,,, for S1 and analyzer directions b, ,b,, . ..,b, for S,. The intention of Garuccio and Selleri is then to derive inequalities, for all possible linear combinations of correlation functions, of the type

1cijP(ai,b j ) I M

(111)

i,j

in which cij are real coefficients. The result of their study is that appropriate upper bounds M are given by

in which ti = & 1, yli = f 1. Another method to construct all generalizations of the BI has been proposed by Froissart (1981).The method appears to be very general because it covers not only an arbitrary number of orientations of each measuring device, but also an arbitrary number of correlated systems. By introducing a “quality factor” for two-systems inequalities, Froissart shows that the original Bell inequality has the best quality factor. This is consistent with the conclusion of Garuccio (1978). Other Bell-type inequalities have been derived by Selleri (1982) as a result of a generalization of the current versions of the EPR paradox. The original EPR definition (D2) of an “element of physical reality” has been criticized by Selleri because of its insistence on predictability “with certainty.” Selleri argues that, on account of the existence of the Wigner-Araki-Yanase theorem (Wigner, 1952; Araki and Yanase, 1960; Yanase, 1961), it is in principle necessary to have imperfect measurements, if Q T is to be correct. Therefore, it is not justified to base an argumentation on the possibility in principle to predict some physical observable with certainty. Selleri proposes some new, generalized criteria for elements of reality and separability, which are based on the possibility of predicting probabilities pi for measurement results ui of an observable U on objects belonging to a subensemble E’ of an ensemble E, without disturbing the objects. In this case Selleri speaks of a physical property l l ( U ; p , , p , , . . .) of the subensemble E’. Furthermore, it is said that, if another subensemble E“ of E is considered, E‘ and E” are separated, if the physical property l l ( U ; p , , p , , . . .) of E‘ does not depend on what property ll( V; q l , q 2 , .. .) is measured on E“.


295

On the basis of these generalized definitions, Selleri then derives the following Bell-type inequalities IP(a,b) - P(a,b’)l

+ IP(a’,b) + P(a’,b’)l + 81P(a,a)l I 10

P(a, b) = - (P(a,a)la.b

(113) ( 1 14)

Combining Eqs. (1 13) and (1 14), and choosing all analyzer orientations such that maximal violation of the resulting inequality occurs, one arrives at the inequality

2 4

+ 8 I10/IP(a,a)l

(115)

which is violated if IP(a,a)l > 0.923. E. Generalization of the Bell Inequality to Arbitrary Spin

Until now we have considered only generalizations of Bell’s inequality which were concerned with two-state systems or with two-dimensional subspaces of many-state systems for which only dichotomic measurement results are possible. In recent years, however, it has been shown that the conflict between QM and all the above-mentioned Bell-type inequalities is not restricted to spin-$ systems, but may be extended to systems with arbitrary spin s. The first consideration of correlated systems on which nondichotomic quantities A(a, A), A(a’,A), B(b, A), and B(b’,A) may be measured came from Baracca et al. (1976). It was shown that, if values M, and M, exist such that IA(441I MI,

t’@’,4l 4 M ,

IB(b,A)I I M , ,

lB(b’,A)l IM ,

(116)

then the following generalization of the GBI [Eq. (74)] exists: IP(a, b ) - P(a, b’)l

+ /P(U’,b’) + P ( d , b)l I 2M2

(117)

with M 2 = M1M2

However, inequality (1 17) is apparently not a very strong one for the following reason. If A(a,A), B(b, A), . .. are the results of measuring the observables j a, j b,. . . for the spin projections of systems S,, S , along a, b,. . ., then Baracca et al. (1976) show that in the singlet state one has

- -

P(a,b)=(001j-aj-b100) = - * j ( j + 1)a.b

(1 19)

and that this QM result agrees with Eq. (1 17) for all possible orientations a, a’, b, and b’ for j > 5.

296

W. DE BAERE

Yet it appears that observables exist, without having a direct physical meaning such as spin projections, for which the QM predictions violate Eq. (117). Interesting spin-s Bell-type inequalities which are more successful with respect to their violation by nondichotomic observables that have a clear physical meaning were constructed by Mermin (1980). The basic inequality from which Mermin starts is slms,(a)

+ ms,(b)I 2 -ms,(a)ms,(c) - ms,(b)ms,(c)

(119a)

in which mS,(a),m,,(b), and m,,(c) denote spin components of S , along arbitrary directions a, b, and c, s the spin value, and

S1)* als, ms,(4> = ms,(a)ls, ms,(a)>

.

( 120)

etc., with Sf1) a the operator corresponding to the spin projection of S , along a, and Is, ms,(a)) the corresponding state vector. Assuming perfect correlation between S1, S2,for which mS,(a)= - ms2(a),and taking expectation values of both sides of Eq. (119), one has

-

s(OO11(S(’). a - S(’) b)llOO)

-

2 (OO1S(’) as(’) clOO)

+ (001S(’)- b S 2 )- C

l W

(121)

in which (Edmonds, 1957) 1

+S

1

l o o ) = ~ m m s , = - s

Is,ms,(a))ls,ms2(a) = -ms,(a)>(- l)s-msJe) ( 122)

Mermin shows then that for the above expectation values the following results hold true:

-

(001S(” * as(’) b100) = -$(s

+ l)a - b

(1 23a)

and

with direction y perpendicular to the plane (a, b), and the spin components m and m‘ both taken along a. If it is assumed that a, b, and c are coplanar and a and b make an angle 8 + n/2 with c (Fig. 6) then Mermin’s GBI for spin s takes on the form

This inequality has to be satisfied for all 8 if locality is supposed to be compatible with QM. However, it may be shown that Eq. (124) is violated by 8

297

EINSTEIN-PODOLSKY -ROSEN PARADOX *

C

FIG.6. Relative orientiations a, b, and c for which Mermin’s spin-s version of the GBI applies.

values in the range

0 < sin0 < sine,

=

112s

(125)

If x = sin 8, then explicit expressions of Eq. (124) and corresponding values 8, for low spin values are as follows: spin 3:x 2 2 x,

0,

spin 1: 2x2 - x 4 2 x,

e, = 38.170

spin 3: 5x2 - 6x4 + 3x6 2 $x,

0,

= 24.08”

spin 2: 20x2 - 42x4 + 48x6 - 20x8 2 5x,

0,

=

= 90”

17.58”

In the classical limit s + co, the above contradiction disappears and ms,(a), etc., become simply components of a classical angular momentum along a. It may be shown further that in this limit the inequality (119) reduces to s sin 0 2 $ 9 sin 8 and hence is always satisfied. Further work on spin-s Bell-type inequalities has been pursued by Bergia and Cannata (1982), Mermin and Schwarz (1982), and Garg and Mermin (1982, 1985).

It may be observed that all possible BIs and GBIs can be classified according to how they are derived in a deterministic local HVT [in which the results A(a,A),B(b,A) are fixed] or a probabilistic one [in which only probability functions pl(a,, A ) , p 2 ( b i , A) exist]. The relation between both kinds of approaches has been discussed by Garuccio and Selleri (1978) and by Garuccio and Rapisarda (1981). F. Direct Proof of QM Nonlocality

Because of the crucial importance of the issue of locality, it is interesting to note that recently Rietdijk and Selleri (1985) claimed to have proven “that the locality hypothesis is in any case untenable if the predictions of quantum mechanics are all correct.” We shall give below a physical outline of their argumentation.

unpolarized beam

A

I.$>'

No transmission 7

2

To illustrate how the process of state-vector reduction, applied to an ensemble of correlated systems S,, S,, strongly suggests the existence of nonlocal influences, we shall consider the following four steps: Step I. (Fig. 7) An unpolarized beam of spin-$ systems S,($) moves in the + z direction towards a polarizer Pl(+x) with its spin analyzer in the + x direction. Suppose that only systems polarized along this direction are transmitted; i.e., the spin state of the transmitted beam is described by the state vector luili ). A second polarizer P,( - x) will subsequently stop all transmitted systems. Step 2. (Fig. 8) Insert a third polarizer P3(a)between Pl(x)and P,(-x). Now we have luilj)

= cos(8/2)lu!y)

luil?) = sin(8/2)lu;y)

- sin(8/2)lubl))

( 126a)

+ COS(~/~)~U:~!)

(126b)

which means that a fraction c0s2(8/2) of the beam transmitted by Pl(x)will pass P3(a).From Iu!?) = cos(8/2)lui11)

+ sin(8/2)lui1!)

Iub'?) = -sin(8/2)1ui1i)

+ COS(~/~)~U~~!)

(127a) (127b)

it follows that P,( - x) will transmit a further part sin2(8/2); i.e., P,(a) is responsible for the fact that from the beam transmitted through Pl(x)a part cos2(8/2)sin2(8/2) will now pass Pl(-x), instead of zero in Step 1. Hence, it may be concluded that the passage of S, through some apparatus (e.g., a

FIG.8. Unpolarized beam moving toward three polarizers.


299

?”

FIG.9. Two correlated systems moving toward two polarizers with opposite analyzer directions.

polarizer) changes by a local interaction the internal physical conditions within S , . Step 3. (Fig. 9) Consider now a source which produces a beam of spin-* systems S,($) and S,($) in the singlet state IOO), such that S , moves toward a polarizer P,(a), and S2 toward a polarizer in the opposite direction, P2( - a) (Fig. 3). With each S , that passes P,(a), there corresponds an S2 that passes Pz( -a). From the state vector of Eq. (15), it follows that the beams that are transmitted through P,(a) and P2(- a) are described by the state vector Iu;J)luh’l). From Step 2 it follows that the internal conditions within each S2 that has passed P2(- a) are changed by a local interaction between S , and

PZ(- 4. Step 4. (Fig. 10) The same situation as in Step 3, but now P2(- a) has been removed. Quantum mechanically we have the same description as in Step 3: The statistical properties of the couples ( S , , S,) for which S , passed P,(a) must again be calculated by means of the reduced state vector Iub?)lub”l). Now we have that S , has changed locally by passing through P,(a), and apparently this process has steered the correlated system S2 such as to become a member of an ensemble described by a well-defined state vector, namely, Iuh?). The difference with Step 3 is that there a local physical interaction was responsible for this steering, while here this is lacking. Rietdijk and Selleri argue that the only way to avoid the conclusion that these systems S, were already described by lub2!) before S , passed through P,(a) is to assume that

FIG. 10. Two correlated systems; only one polarizer is present.

300

W. DE BAERE

precisely at this moment the internal conditions within S, are changed nonlocally (i.e., by action at a distance), so as to become described by luL2!). In this way state-vector reduction in the case of correlated systems strongly suggests a nonlocal behavior at the quantum level. G . Recent Developments

An understanding of the origin of the BIs or of the nonlocality of QT in terms of Einstein’s own unified field theory has been attempted by Sachs (1980) and by Bohm and Hiley (1981). It is well known that Einstein’s attempts to construct a fundamental theory of matter are based on an extension of his general relativity and on a representation of matter and radiation by a continuous field, obeying deterministic nonlinear field equations. It was hoped that the probabilistic and alleged nondeterministic predictions of QT could be derived somehow from such a deterministic and classical scheme. It is argued by Sachs that in his own approach Einstein finally rejected the concept of “Einstein locality,” in the sense of the existence of physical systems which could be supposed to be described independently of anything else in the world, in particular of measuring devices. Hence, according to Einstein, the idea of separability of physical systems should not be considered as a fundamental one (Einstein, 1949). Sachs then goes on to discuss the conflict between QT and the BI within his own version of the above deterministic scheme, in which a maximum velocity c (the velocity of light) for the propagation of physical influences is built in, and which reproduces QT in the linear limit. A similar attempt at understanding the Q M nonlocality along the same lines comes from Bohm and Hiley (1981). After recalling that Einstein rejected the “. . . fundamental and irreducible feature of the quantum theory, i.e., nonlocality,” and describing Einstein’s views on locality, these authors propose a way how nonlocality could follow from Einstein’s approach. Remembering that in Wheeler and Feynman’s absorber theory of radiation (Wheeler and Feynman, 1949) a photon is emitted when there is somewhere matter to absorb it, Bohm and Hiley suggest a similar mechanism in the case of correlated photons. If the analyzer directions are a and b, then it may well happen that these circumstances help to determine the way the photons leave the source. In this way they are led to assume that the distribution of HVs will no longer be represented by p(A), but instead by p(a, b, A). And this would lead to an HV framework in which the usual BIs can no longer be derived. Cramer (1980) constructs a QM version of the Wheeler-Feynman absorber theory (Wheeler and Feynman, 1949) as an explanation for the nonlocality which follows from the theoretical and experimental violation of the BI. It is shown that within this approach there can be a nonlocal


30 1

communication between measurement devices separated in space, so as to produce a violation of the BIs. In recent years an interesting connection has been revealed between the existence of a joint probability distribution for the values of some observables and the validity of the BI. As in Clauser and Horne (1974) (Section III,D,5) let us introduce the experimental probability distributions plz(a, b), p , ,(a, b’), plz(a’,b), plz(a’, 0 p l ( 4 pda‘), p,(b), and p,(b’) for spin projections or polarizations of systems separated in space S,, S, along directions a, a’ and b,b. Then Fine (1982a,b)has proved that in any theory (with or without HVs, deterministic or stochastic, local or nonlocal) whose framework provides a joint probability distribution p(a,a‘, b, b’) for the values a *, a;, b +, b+ ( = f1) of the respective dichotomic observables, such that the above experimental distributions are obtained as marginals of p(a,a’, b, b’), the GBI [Eq. (85)] should hold: -1

Plz(a,4 - Pl2@,6’)

+ Plz(a’,b) + P,&’,b’)

-P l ( 4

-PAN I 0 (128)

It may be shown that by appropriate interchanging of the parameters in Eq. (128) seven similar equivalent inequalities exist. Writing for the previously defined correlation functions P(a, b) P(a,b) = P I Z ( Q + , ~ + ) Plz(a+,b-) - P I Z ( ~ - , ~ + +) ~ i z ( a + , b + )(129) and similarly for the other correlation functions P(a, b‘), P(a‘, b), and P(a’,b’), one may arrive (de Muynck, 1986)at the GBI [Eq. (58)l in the form given by Clauser et al. (1969) and by Bell (1972). These results are very interesting because it appears that for the derivation of the BIs the locality hypothesis does not play any role. In this respect it may be worthwhile to mention that Edwards (1975) and Edwards and Ballentine (1976) have constructed nonlocal HV models which satisfy the BI. And in contrast with this Scalera (1983) and Caser (1984b)(see below) have developed local models which violate the BI. These examples seem to support the thesis of the irrelevance of the locality condition with respect to the BIs. As a first illustration of such models with anomalous properties, we note that Scalera (1983) constructed a kind of local classical helix model for two correlated photon beams moving in opposite directions. Each beam is assumed to be represented by a continuous helix-shaped ribbon. If a photon passes some polarizer with its analyzer direction along a, it means that the associated ribbon passes undistorted, but somehow the directions f a are marked on it. If a photomultiplier is to receive and to count such a photon, it is assumed that the energy contained within two marked directions is integrated. A correlation between the two ribbons produced at the source originates by

302

W. DE BAERE

their being marked in the same direction. Scalera then shows that the BIs may be violated by this model. Another local model that violates the BIs has been constructed by Caser (1984b).The peculiarity of this model is that it starts from the assumption that the state of a measuring apparatus depends on some or all previous measurements made by means of it. It is argued that QM is compatible with such a memory effect. By showing quantitatively that the model satisfies all physical requirements of locality, however without being identical to Bell’s definition, it appears that the BIs can be violated. That the violation of the BIs is not a peculiar property of correlated quantum systems has been shown by Aerts (1982a, 1984).If one considers two macroscopic systems which are correlated because of the fact that they are not completely separated (e.g., there are some connecting tubes or wires), then results of measurements made on them will also be correlated, and the GBI may be shown to be violated. Hence, according to Aerts, the problem is to separate physical systems completely, whether quantum mechanical or classical. If it were possible really to produce such separate systems, then the GBI would be verified and QM should turn out to be wrong in describing such systems. Hence, Aerts (1982b) concludes that the present QM is unable to describe separated systems. For this reason, a more general scheme than QM is proposed that encompasses both CM and QM (Aerts, 1983). To explain the observed violation of the GBI, hence the alleged nonlocality, one may attempt to set up concrete physical models. This has been done by Caser (1982), who remembers that, according to Mach’s principle, the inertial properties of matter are determined by the distribution of far-away matter in the form of stars, etc. In the same way, Caser argues, it may be that, although the magnetic field H of a Stern-Gerlach analyzing device is different from zero only in the region of the device, the vector potential A (H = V x A) is nonvanishing in a much larger region, which may include the source of correlated spin4 systems. Now from the existence of the Aharonov-Bohm effect (Aharonov and Bohm, 1959), it follows that the behavior of a quantum system depends on the local values of A. Hence, it may be, according to Caser, that the values of A at the region of the source influence the distribution of HVs in such a way that one has to write p(a,b, A) instead of p(A) (a,b being the orientations of the fields H in both SternGerlach devices). It follows that Bell’s theorem is no longer valid under these conditions, and it is shown explicitly how such a model can reproduce the results of QM. In a subsequent work Caser (1984a) shows that in an anisotropic space, in which two measuring devices act in a different way on physical systems, a local HVT may reproduce the QM results for spin correlation experiments. Moreover it is shown that, unlike the argumentation of Clauser et al. (1969),

303


no supplementary assumption about the detection probabilities is needed when the HVs lie in the plane of the analyzer directions. A further illustration of the disagreement between Q M predictions and the BI when the relevant observables do not all commute pairwise, has been given by Home and Sengupta (1984). They show that the violation of the BIs is not only characteristic for systems having correlated components separated in space. It is argued that instead one may equally well consider an ensemble of single systems and measure the compatible pairs (A(a),B(b)), @(a), B ( b ) ) , (A(a’),B(b)),and (A(a’),B(b’))on the individual members of the ensemble, of course under the condition that only one pair is measured at a time. Adopting a specific labeling scheme which transforms nondichotomic results to dichotomic ones, they are able to derive Bell-type inequalities of the form - P(a, C) + P(b, C ) P(b, a) I 1 (130) or IP(a,b) + P(a, b’) + P ( d , b) - P ( d ,b’)l < 2 (131)

+

with the usual definition [Eq. (39) or (77)] of the correlation functions. They consider then an ensemble of atoms (e.g., alkali atoms), which have a single valence electron in the 2p1,, state. In such a state the total angular momentum is J = +,the orbital angular momentum is L = 1, and the spin of the valence electron is s = +. If m,, rnL, and m, denote the components of the respective angular momenta along some arbitrary direction a, then one may write for m, = ++

1.1

= +,m, = $) =

( ~ / J I )=(l,m, ~ ~= 1)ls I L= + , m , - +> -IL

=

l , m L = O)ls = +,m, = ++))

(132)

Now Home and Sengupta take as observables and their operators

-

.

A(b):L b,

.

B(b):a b,

A(a):L a,

B(a):a a,

-

-

A(c): L c,

-

B(c):a c

(133)

and calculate P(a, c), P(b,c), and P(b,a) P(a,c) = (+,+1L . a a - c l + , + ) = $c0s2(e,/2) - $sin2(8,/2) -

3

(134a)

P(b,c) = (+,+lL*ba-c1+,3) =

+os2(el

-

e,/2) - $sin2(6,/2)

- -

-

5

(134b)

P(b,a) = (f,+IL b a a[+,+) = $cos2

o1 - 71

(1 34c)

304

W. DE BAERE

In Eqs. (133) and (134), a, b and c are coplanar unit vectors and 8, = Oab, 8, = Oat. The left-hand side of Eq. (130) becomes

-5 + 4

~ 0 ~ 2 +cos2(ez/2) 0 ~

+ $cosz(e, - 02/2)

(135)

which for, e.g., 0, = 7c/6 and 0, = 2n/3 becomes 4, such that the BI [Eq. (130)] is violated. Franson (1985) remarks that the recent experiment of Aspect et al. (1982) (Section IV,A) does not exclude local theories in which a measurement result may be known only some time after the measurement event has occurred. Under these circumstancesit should be possible that information between two measurement devices could be exchanged with subluminal velocities. In a recent paper Summers and Werner (1985) show that the violation of the BIs is also a general property of a free relativistic quantum field theory. Use is made of the Reeh-Schlieder theorem (Reeh and Schlieder, 1961; Streater and Wightman, 1964), according to which any local detector has a nonzero vacuum rate. It is shown that the vacuum fluctuations in free-field theories are such that the BIs are maximally violated. It is concluded that all physical and philosophical consequences with respect to the violation of the BI apply also in free quantum field theories. Some proposals for a resolution of the (non)localityproblem have already been discussed in Section II,E,F. Other recent investigationson that issue with respect to the BI will be reviewed in Section V, where the significance of the BIs and their violation will be considered from a critical point of view. A number of papers which may be of help in the study and clarification of the problems raised by EPR and the BIs are those of Berthelot (1980), Bub (1969), Corleo et al. (1975), Flato et al. (1975), Gutkowski and Masotto (1974), Gutkowski et al. (1979), Liddy (1983,1984), Rastall (1983,1985), and Stapp ( 1979).

Iv. EXPERIMENTAL VERIFICATION OF BELL’SINEQUALITIES In this section we will give a brief survey of the results of existing experiments which were intended to verify the GBI [Eq. (86) or (66)] of Clauser et al. (1969) (Section III,D,l) and at the end we will mention some new proposals. For more details on the experiments themselves and for a thorough discussion of the experimentalproblems connected with verification of the GBI [Eq. (86) or (66)], we refer to the review papers of Paty (1977), Pipkin (1978), Clauser and Shimony (1978), or to the original papers themselves.

305


In their paper Clauser et al. (1969) also discussed the inefficiency of the then-available data on polarization correlation of photons (Wu and Shaknov, 1950; Kocher and Commins, 1967) with respect to an experimental discrimination between QM and HVT via their GBI [Eq. (58) or (63)]. They proposed a concrete new experiment which would allow the verification of Eq. (63). The idea of Clauser et al. was to optimize the experiment of Kocher and Commins (1967),i.e., to choose optimal relative polarizer orientations of 22.5" and 67.5O, to increase polarizer efficiencies, and also to make observations with one polarizer and then the other removed. Under these conditions one will get sufficiently reliable data to be used in the GBI or CHSH inequality (66) lR(22.5")- R(67.5")]/RoI I

(136)

They actually made two proposals: One was for the J = 0 -+ J = 1 + J = 0 electric-dipole cascade, while the other was for the J = 1 J = 1 J = 0 cascade. In terms of the relative angle cp between the two polarizer orientations and of the half-angle 8 of the cone in which the photons are detected, the QM predictions for the polarization correlations are for both cases --+

R(cp)IRo = &f

--+

+E m f i +4

+ $(&;I

- E:)(Efi

R i / R o = ( ~ f l+ ~ 1 ) / 2 ,

- Efi)F1(8)COS243

i

=

1,2

(137)

(138)

In Eqs. (137) and (138) and E\ (i = 1,2) are the efficiencies of the ith polarizer for light polarized parallel and perpendicular to the polarizer axis and, further, for the cascade J = 0 + J = 1 J =0 -+

+

Fl(8) = 2G:(8)[G2(8) iG3(8)]-1 G,(8) = - cos 8 + sin2 8 - 3cos38)

(139) ( 140a)

G2(@ = 8 - +(sin28 + 2) cos 8

( 140b)

G3(8)= 4 - cos 8 - ~

( 140c)

a(+

C O 0S ~

For the second cascade J = 1 --+ J = 1 -+ J = 0, the relations in Eq. (140) remain valid, but in Eq. (137) -Fl(8) should be replaced by

+

F2(8) = 2G:(8)[2G2(8)G3(8) iG:(8)]-'

(141)

Assuming that E: and E: are vanishingly small and takings1 = ~ f =i it may be seen that Eq. (137) may be violated in both cascades for experimental parameters satisfying the relation

J Z e ( 8 ) + 1 > 2 / ~ ,, ,

j = 1,2

(142)

306

W. DE BAERE

A , Cascade Photon Experiments 1 . Experiment of Kocher and Commins (1967) This was the first correlation experiment with cascade photons. The source was the 6 IS, excited state of Ca. Via a two-step cascade J = 0 + J = l+J=O (4 lP1) 4,= 4227 (4 lS0) (143) (6 IS,)

4

4

towards the ground state, two photons y1 and y2 are emitted. Because the total angular momentum of y1 and y2 is zero, the QM transition probability is proportional to (el .e2)', with e, the polarization vector of photon y i . The results were in agreement with this prediction: For perpendicular polarizer settings no coincidences above background were measured, while there was a high coincidence rate for parallel polarizer settings. However, the experiment was not appropriate for a verification of the GBI [Eq. (136)] because the efficiencyof the polarization was not high enough and only the angles cp = 0" and 90" were considered. 2. Experiment of Freedman and Clauser (1972) In this experiment correlated photons y1 and y2 were obtained in the following J = 0 + J = 1 + J = 0 cascade in calcium (4p2 IS,)

.

=

5513 A

* (4P4S Pl)

I,,,

= 4227

A

,(4s

IS,)

The polarizer efficiencies had the following values

~ f =i 0.97 f 0.01, ~ f =i 0.96 f 0.01,

E:

= 0.038

E:

= 0.037

f 0.04 0.004

(144)

The half-angle 6, within which the photons were detected, was 8 = 30". In different runs of the experiment the results R(22.5")/R0= 0.400 f 0.007 and R(67.5")/R0 = 0.100 f 0.003 were determined. Denoting by dexp,and, ,a the experimentally determined and theoretically predicted values for the left-hand side of Eq. (136), then it is found that aexpt= 0.300 k 0.008, which violates the BI of Eq. (136), but is in agreement with the Q M prediction , ,a = 0.301 0.007. This was the first clear evidence for the validity of Bell's theorem which excludes all local HV theories.

+

3. Experiment of Holt and Pipkin (1973)

J

Here the source was the spin-0 isotope Ig8Hg of which the J 1 + J = 0 cascade was used to produce photons y1 and y 2

=

= 1+

307

EINSTEIN - PODOLSKY -ROSEN PARADOX

The polarizer efficiencies were as follows: = 0.910

0.001,

E:

=

loo> = (l/~)Clu:)lu:)Iu3)Iu4) -(1/$)(lu: > b 2 ) + lu’>lu: ))(1/$) x (lu:)lu:)

+ Iu3)Iu4+))+

lu’)lu2)lu:>lu4+>1

(164)

320

W. DE BAERE

The results for the relevant correlation functions are P(U, b) = (0()1U1 * aU2 * bJOO) = 3 C O S

(165a)

~ ( u , b ’ ) =

=

vko

=0

1 r2 3

1 --pa

via

I2

>

via

=

-3pbgi>

(134)

as well as involving Eqs. (39), (41), (63), (82), (94), (95), and (1 18), we rewrite Eq. ( 1 15) as the expansion

3 59

ELECTRON MIRRORS AND CATHODE LENSES

-".>I"

+ m{[A9gb + k(2qpa 4a ~b

gb - iu4:}

There is a number of correlations between the quantities entering the last expression. On comparing Eqs. (120) and (123) it is clear that € 7 3 2 = sbA5 (136) As shown below, the following relations hold

[A3g:

+

k(2+ ;)]

- [A4&

A7gb---2 R

~

4

Phsb

+ f($ +

-a - 0 1

F)]

=1

(137) (138)

The left-hand side of Eq. (137), involving Eqs. (121) and (122), may be represented as follows:

360

E. M. YAKUSHEV AND L. M. SEKUNOVA

On differentiating the equations

- 4 ~ ”- ‘4Q 2 p’ = -4d2 - $Qzg’

241; - 4‘11 =

(141)

241; - 4 ’ 1 3

(142)

we obtain

+ 4‘ql - 4”11 = -+Q;p’ 241; + 6’1;- 4”13 = -$Q;g’ 241;

(143) (144)

taking into account that

( 4 ~ ’+ ’ $QzP’)’ = $Qh2 (4s’’ + bQzgZ)’= $Qh2

(145) (146)

because p(z) and g(z) satisfy Eq. (74). Multiplying Eq. (143)by q 3 ,and Eq. (144) by ql ,subtracting the first from the second, and dividing the obtained equation by 2&, we find

- 1;13)I’

(147) Using the resulting equation, we carry out the integration in Eq. (140) directly proving Eq. (137), if we take into account Eqs. (106) and (134). To verify Eq. (138), substitute A7 and q4 in the forms in Eqs. (125) and (46) into its left-hand side. The twofold integral entering the resulting expression is carried out by parts, taking into account Eqs. (145) and (106) C f i ( ~ i &

= (Q;/~&)(P’v~ - g21i)

( 148)

This expression is rewritten realizing that p satisfies Eq. (74)

On carrying out the latter integration by parts, we find, after eliminating the uncertainty occurring at the point Zk


36 1

The right-hand side of Eq. (138), determined by Eq. (130), may be reduced to the same form, if q1 is substituted into Eq. (138) in the form of Eq. (43), and double integration is carried out by parts. Equation (139) is verified in the same way. Taking into account Eqs. (136)-( 139), we reduce the expansions of Eq. (135) to the form

where 1 5 1

=-

JZ

=

+ 4Q;p2p’ + 32p”(W2 + aQ2p2)]dz

2’pCQ4~3

1 9 -CQ4p2g 84kPaga I z k f i

+ 4 Q W g + 32~”(4p’g’+ aQ2pg)l dz

ok J , = 16pbz~ ~ ~ & ( B ” p4B’p’

+

J

(152) (153)

+ 8Bp”)dz

ok 6 -

(157)

362

E. M. YAKUSHEV A N D L. M. SEKUNOVA

The expansions in Eqs. (135) and (151) differ by the circumstance that there is no double integrationin the latter. We have eliminated them by the same way used for verifying Eqs. (137)-( 139). Using Eqs. (84), (85), (101), (104), (105), and (109), we rewrite the expansion in Eq. (151) as follows

Equation (162) determines the aberration of an electron mirror in an arbitrary plane, located beyond the field. If this plane is the plane of the Gaussian image, z = z b , then using Eqs. (108)-(112), we may simplify the expressions for abberrations

Equation (163) determines the image aberration in an electron mirror when a limiting diaphragm is absent. If an incident ray is determined by the coordinates of an object point and by the coordinates rD and R, of the point of its intersection with the plane of the diaphragm z = z D ,which also located beyond the field, then the following relation is valid uD=

D e i R ~=

r,

+ uh(zD - z,) = r, + ubZDa

(175)

364


Using it, we write expression for the image aberrations in a mirror with the limiting diaphragm

where

With E = 0 Eq. (176) gives geometrical aberrations of a mirror in a form quite analogous to the case of geometrical aberrations of electron axially symmetric lenses. This allows one to use for electron mirrors the classification of aberrations adopted for electron lenses. According to this classification, G D is the coefficient of spherical aberration, FD is the coefficient of isotropic coma, while f , is the coefficient of anisotropic coma, C, is the coefficient characterizing astigmatism, c, is anisotropic astigmatism, D D is the image curvature coefficient, and ED and e, are the coefficients of isotropic and anisotropic distorsions, respectively. The coefficients at c/4a give one the possibility to determine the mirror chromatic aberrations. K I D represents the coefficient of chromatic aperture aberration, the coefficient K 2 , characterizes the change of the image scale, and the coefficient k, denotes the change of the image turning angle. The coefficients f,, c,, e,, and k , differ from zero only in the presence of a magnetic field.


365

D . Time-o f-Fligh t Focusing

Investigation of time-of-flight focusing of nonstationary fluxes of charged particles is one of the basic problems when calculating and designing a large number of electron-optical and ion-optical devices, in particular, time-offlight mass spectrometers. Now we study this problem theoretically, in the general case, i.e., without imposing any restriction, besides their rotational symmetry. The time of flight of a particle from the initial plane z = z, to an arbitrary plane z = const after reflection, according to Eqs. (11) and (12), equals

When deriving this equality, a change of the sign of has been taken into account in the particle reversal point at i= zk, and the notation q, = ~(2,) has been adopted. The planes z = z, and z = const are assumed to be located beyond the mirror field. In this case Eq. (188) takes a simpler form t = to

+ (d/k&)(?,

+ 'I)

(189)

is the time of flight of a particle along the z axis with E = 0 between the planes considered, z = z, and z = const, before and after reflection, 4a denotes the potential of the field-free space. The second term of the sum in Eq. (189) determines time-of-flight aberration. We shall find the structure of this aberration, taking account the quantities of first order in E and of second order in I , substituting q in the form of Eq. (41)into Eq. (189) and using Eqs. (81) and (82). t

-

to

=(

+

~ / 4 a ) ~ t e (a/k&){aa*Cqt(za)

+ V~(Z)I

+ (ab* + a*b)Cqz(z,) - qz(z)l + ia(a*b - ab*)C~4(z,)+ G'4(Z)I + bb*C%(Z,) + %(Z)1) (191) where Dra(Zu,Z) = Dre = ( D / ~ ) & C V J Z J

+ V,(Z)I

(192)

is the time-of-flight energy dispersion of the first order. Let us show that the particle time-of-flight focusing in energy may be achieved with a mirror and deduce the equation of focusing. As follows from

366


Eq. (42), the function q,(z) in a field-free space may be represented as being linearly dependent on z 110 = (1/24cz)(z - zT)

(193)

Note that the quantity zT does not depend on the presence of the magnetic field in a mirror or on the choice of the point z, beyond the field and is determined only by the electric potential distribution along the z axis. Taking into account Eq. (193), Eq. (192) takes the form

The first-order condition of time-of-flight focusing will be understood as the independence of the time of flight of particles on their spread in initial energy, i.e.,

Then it follows that 2,

+

ZB

= 22,

(197)

This expression means that time-of-flight focusing is achieved if the plane z = zar from which particles are emitted, and the plane z = zB, where they are registered, are located symmetrically with respect to the plane z = z T , determined, according to Eq. (194), by the electric field distribution over the principal optical axis of a mirror. The plane z = zT will be called the principal plane of time-of-flight focusing. In mirrors, time-of-flight focusing of particles with a small spread in energy is realized based on the fact that, when reflecting, the higher the particle energy, the longer the path it covers in the reflecting field. Assuming that the principal plane z = zT is located before a mirror, in the field-free space we rewrite Eq. (190) as follows:

where


367

is the time interval between the instants of intersection of the plane z = zT by a particle before and after its reflection. This interval will be called the timefocusing interval. Its dependence on the particle mass determines the amount of a mirror time-of-flight mass dispersion dTo

D,, = mam

1 2

=-T O

It is a characteristic feature of the principal plane z = zT of time-of-flight focusing that the time-focusing interval To does not depend, to a first approximation, on the initial energy of the particles. It means that particles with equal specific charges, elm, and with a small spread in energy traversing the principal plane simultaneously before reflection, will also traverse the plane simultaneously after reflection. The time-focusing interval Todetermines the effectivedrift distance L

It is worth noting that both L and zT depend only on the mirror electric field. If the position of the principal plane of time focusing is known, the position of two mutually conjugated, in the sense of time-of-flight focusing, planes, z = z, and z = zB is determined by Eq. (197). The latter will be called the equation of time images, an object/image understood as the structure of substantially nonstationary flux (of a short current pulse) in the plane of the object/image. The mirror time magnification, M,, determined as a ratio of the current pulse duration (At)Bin the image plane, z = z B ,to the initial duration of the same pulse, (At),, in the object plane, z = z,, is always constant, due to the stationary nature of the electric and magnetic fields forming a mirror, and equals

It means that to a first approximation a mirror adequately transmits the time structure of the nonstationary charged particle flux from the object plane to the image plane. As it is seen from Eqs. (197)and (198),the object and the image are shifted, relative each other, in time by To,independent of the values of z, and zg. Now we shall find the position of the mutually conjugated planes, z = z1 and z = z 2 , for which both space and time-of-flight focusing are realized. Setting into the equations of an image, Eqs. (111) and (197), z, = z, = zl, zb = zB = z2, and solving these equations with respect to z l and z 2 , we obtain 21,2

= zT

kJ(zT

- zB)(zT

- zC)

(203)

368


When this equality is satisfied, the correct point-by-point space and time image of an object is formed in the same plane, i.e., space-time-of-flight focusing of the nonstationary flux of charged particles is realized. For an electrostatic mirror having a specific distribution of the field along its axis, only a pair of the mutually conjugated planes satisfying the conditions of space-time-of-flight focusing, Eq. (203),may be found. When a magnetic field is present, one may find a set of such pairs due to the fact that the quantities zB and zc depend now both on electric and magnetic fields. Note that, as for the possibility of realizing space-time-of-flight focusing of a charged particle nonstationary flux, a mirror is a unique element in the class of the axially symmetric fields considered. Now let us consider the second-order time-of-flight aberrations, tracing the motion of two particles with initial energies E = 0 which simultaneously traverse the plane z = z , located before a mirror. Let one of them move along the z axis and, after reflection in a mirror, reach the plane z = const in a time interval t o . Then the time of flight up to this plane for the other particle remote from the z axis, due to the presence of aberrations determined by Eq. (19l), will differ somewhat from to t

= to

+ (At)!”

(204)

where

+ + + $(ab* + a*b)pbgb(z - z,) + 2ia(a*b - ab*)q, + bb*gbZ[zB- +(z, + + R I , ] }

(At)!2)= ( ~ ~ / k f i ) { ~ ~ *-p&z, b ~ [ Z~) c R I , ]

Z)

(205)

and

When deriving these expressions, it was taken into account that the initial and final planes, z = z , and z = const, are located beyond the field of a mirror. In this case, according to Eqs. (101) and (104), the functions p ( z ) and g ( z ) are approximated by the straight lines p = pb(z - zc) and g = gb(z - zB), and Eqs. (43), (44), and (45) for the quantities q l ,q,, q3 are simplified; the quantity q4, determined by Eq. (46), becomes constant. Represent the structure of geometrical time-of-flight aberrations as the dependences on a radial deviation of particles ra and on a slope of their trajectories r: and r,& given in the initial plane z = z,. For this purpose, we use the expressions in Eqs. (84),(85), (94),and (95) determining the parameters


369

+ ir,$k. Then, combining in a proper way the quantities of second order, we find

a, b and the relation uh = rk

(At):2) = T,(ri2 + r:$i2)

where T -

-b, ff

-k R f i

-

zJ21,

+ T2r,rh + T,r:$i + T,r;

(208)

+ (za - z,)~Z,

The aberration associated with the coefficient TI describes the deformation of the time structure of the charged particle's nonstationary flux emitting from the point located in the z axis and in the plane z = z,. The aberration of this sort may be called the time-of-flight spherical aberration. The coefficient T4 determines the time deformation of a structure of the flux entering the mirror field parallel to the z axis. An aberration of this sort depends only on the particle coordinates in an object plane, and so it may be called the timeof-flight distortion. The coefficients T2,T3 determine mixed time-of-flight aberrations, and in pure electrostatic fields T3 = 0. If the first-order time-of-flight focusing in an energy E is available as well as the initial spread in energy which is relatively great, it is of interest to calculate time-of-flight aberrations of second order and above. In order to calculate these aberrations, it is necessary to consider the particle motion along the z axis and to define, with a required accuracy, the quantity q = q ( ( ) as the dependence on the initial energy E. Then for the space free of field the total time-of-flight chromatic aberration may be written as follows

It is possible to calculate chromatic aberrations with any specified accuracy, as the equations of motion of axial particles can be integrated strictly at arbitrary values of E. For axial particles (at r = 0) Eq. (15), with the

370


use of Eqs. (12) and (17), reduces to an extremely simple form

1:'s

allowing solution by the method of separation of variables

=j

:

+

dz V

k

d

m

where the quantity q k satisfies the equality

4(zk + qk) +

(216) Equation (215) combined with Eq. (216) determines the required dependence 9 = V(Z,&). In order to make Eq. (215) more convenient for carrying out the analysis of time-of-flight aberrations, it is necessary to fulfill certain transformations in it. We rewrite it using Eq. (216) as follows: =

The expression under the radical on the right-hand side of the equation will be rewritten in the form

4(z+ q k ) - 4 ( z k + q k ) = 4(z)[1+ pL(z,q k ) ] where p is a small quantity. Expanding the functions 4,4(z + v k ) , + ( z k

+

(2 18) q k ) in

a power series in q k , we obtain

It is seen from this expression that the function p is continuous and differentiable everywhere including the singularity z = zk. The value of p = pk at the singularity will be found after eliminating the uncertainty of the 0/0 type

In the space free of field the quantity p becomes constant, pa = &Ida.Recall that the object and its image are located beyond the field of a mirror. This provides a possibility of solving Eq. (217) with respect to 9. On carrying out this operation and involving Eqs. (218), we find


371

This equation, combined with Eqs. (213), (216), and (219), determines the overall time-of-flight aberration (At), with any specified accuracy with respect to the spread in energy in the nonstationary flux of particles. So, restricting ourselves by the values of second order, on the basis of Eq. (221), one may obtain (At), = (&/4Pt, + Wi2’

(222)

where D,, is the energy dispersion determined by Eq. (195), and (At)L2’ = (e2/&)TV

(223)

is the time-of-flight chromatic aberration of second order, and

(224) with PI

=

(4‘ - 4;M

p2

=

(4“ - 4;1/4

(225)

The second-order total time-of-flight aberration of an electron mirror is composed of geometrical aberrations [Eq. (208)] of four kinds and of chromatic aberration [Eq. (2231 of one kind (At)‘,’ = (At)!,)

+ (At)!”

(226)

In the case for which space-time-of-flight focusing is realized, the coefficients of geometrical and chromatic time-of-flight aberrations in the image plane take the form

372


IV. MIRROR ELECTRON MICROSCOPE

An electron mirror possesses a remarkable feature-it creates an image of its reflecting electrode. For the first time this effect was noticed by Hottenroth (1936, 1937). Investigating experimentally electron-optical properties of electrostatic mirrors, he noticed that, with a solid reflecting electrode, tiny roughness of the electrode surface is observed in a mirror image. The mirror ability to create a magnified image of the surface of its electrode reflector allows one to use it as an original electron microscope like a shadow lightoptical microscope. The scheme of a mirror electron microscope (MEM) is shown in Fig. 5. The main part of the microscope is the mirror M (in the simplest case: aflat sample, a cathode and a diaphragm with an aperture, and an anode). An electron beam emerges from the source S and, retarded in the field of the mirror objective, reflects at a very short distance from the sample surface K. When near a sample a slow electron beam is modulated by microfields produced by micrononhomogeneities of the sample surface, which may be of electric, magnetic, or geometrical character. After changing the direction of its motion, the beam again traverses the objective field, which now acts upon it as accelerating one.

FIG.5 . Scheme of the simplest mirror microscope: (a) of the direct construction (the incident and the reflected beam axes coincide), (b) with separation of the incident and reflected beams by virtue of the magnetic field B.


373

On the screen P the reflected beam creates a brightness picture which represents the magnified image of the nonhomogeneities of the sample surface. It is the advantage of the device that in it a sample does not undergo bombardment by an electron beam. As a result, the sample is not destroyed, its subcharging is reduced to minimum, there is no sample heating, and no secondary emission of its surface or other undesirable phenomena. MEM is extremely sensitive to the micrononhomogeneitiesof the surface under study, because in the region occupied by fields associated with these micrononhomogeneities which are very close to the sample surface, the velocities of particles belonging to the beam incident on the sample are extremely small, and even insignificant disturbances of the field in the near-cathode region lead to noticeable redistribution of the velocities in its components. In the neighborhood of the reversal point the character of the particle motion is such that its trajectory is parallel to the surface under consideration, and it still raises the device sensitivity because of an increase in the time of interaction between the beam and microfields. If the potential difference between the sample and the cathode of the source of illuminating electrons (bias voltage) varies, it is possible to study the surface microfields of the sample at various distances from its surface. Owing to the merits mentioned above, the application field of MEMs is rather wide (Luk’yanov et al., 1973). An image in a mirror electron microscope operating in a shadow regime is formed in a different way than in a microscope of a transmission type. In a mirror microscope an image represents a brightness picture formed by a reflected beam in which the density of electron current is redistributed as a result of its interaction with micrononhomogeneitiesof the cathode surface. Conformity between the object micrononhomogeneities and the received image is of a rather complex character. The contrast theory explaining this conformity is constructed based on the calculation of the current density of the reflected beam of electrons moving in the field of a mirror objective disturbed by the nonhomogeneities of the cathode surface in any cross section. As the region of action of micrononhomogeneitiesin the direction perpendicular to the cathode surface is small compared with the objective field extent, one may agree with Wiskott (1956) that the field disturbance caused by them, is concentrated in a narrow near-cathode region, and that the entire additional pulse is imparted to the electron at its reversal point. The problem of contrast calculation then involves two tasks. The first is to define the value of an additional pulse which is imparted to the electron in the disturbed field (a microfield) near the cathode. Here, in the absence of micrononhomogeneities on the cathode surface, the field may be considered as homogeneous. The field disturbance is determined by a concrete character of micrononhomogeneities of various types on the cathode surface. Calculations are based on determination of electron motion in such a field. The study by Wiskott (1956)

3 74


underlying the theory of a mirror microscope was the pioneering work. Later much effort was devoted to this problem. The most comprehensive papers to be mentioned are those by Barnett and Nixon (1967-1968) and Sedov (1970). The second task is to calculate the electron trajectory in the basic field of a mirror objective (in a macrofield). In doing so it is necessary to take into account that in the reversal point of an electron, due to a disturbance, the initial conditions for its motion in the reversed direction are changed. When interpreting the image contrast, the calculation of trajectories in a macrofield is no less important than the calculation of disturbances in a microfield. It is known (Sedov et al., 1971) that disappearance of the contrast and even its inversion are possible for certain geometries of a mirror objective and for appropriate parameters of illuminating electron beams. However, the complete studies of trajectories in the macrofield of a mirror objective are not enough. Therefore, the image deformation associated with geometrical aberrations of electron mirrors have not been studied, and theoretical investigations in a paraxial approximation have been carried out only in application to the simplest designs of the MEM objectives (e.g., Luk’yanov et aZ., 1968; Artamonov and Komolov, 1970; Schwartze, 1965-1966). Here the geometrical optics of MEMs operating in the shadow regime is considered taking into account the mirror’s geometrical and chromatic aberrations of third order in r and of first order in rE without imposing any restriction on its design but the rotational symmetry of the field forming a mirror. In doing so it is taken into account that, equally with electrostatic fields, magnetic fields can take part in forming a shadow image, and the cathode surface (a mirror reflecting electrode), in general, has a form of the surface of an axially symmetric body. Electron-optical properties of a mirror, including its aberrations, were dealt with, as a whole, in Section 111. However, a mirror operating in the MEM regime has a peculiarity that it creates on a screen an image of one of its electrodes-a cathode, i.e., the object located in the neighborhood of the reversal points of electron trajectories. Therefore, the formulas obtained in Section 111, to deduce which it is assumed that both the object and the image are located beyond the mirror field, cannot be applied for calculation of mirror microscope aberrations. However, such calculations may be made using the equations deduced in Section I1 that determine the trajectories of charged particles in the field of an electron mirror taking into account the terms proportional to the initial energy spread of the particles and the terms of third order with respect to a particle’s deviation from the axis of field symmetry. An electron mirror is considered to be the entire optics of a mirror microscope, except for the system of separation (when it is available) of the direct and the reversed beams, as well as the emission system specifying the illuminating beam of electrons. Then the source of illuminating electrons is understood as


375

the electron beam crossover formed by the emission system. The position of an electron source and its size are assumed to be the position of the crossover and its size that are formed by the emission system in the absence of a mirror field. Such definition of a source gives one the possibility of separating investigations of the mirror electron-optical properties and those of the emission system. We shall assume that there is a field-free region between the mirror and the illuminating system, and it is there that the microscope screen is located. Introduce the cylindrical coordinate system r, $, z , whose origin is set into the point of intersection of the z axis and the cathode surface, while the positive direction of the z axis is chosen along the direction from the cathode to the screen. The potential on the cathode surface is assumed to be constant and equal to zero. Under these conditions the equalities hold o=-l,

z,=o

which are to be introduced when using the formulas deduced in Section 11. The energy of illuminating electrons will be characterized by the quantity cp + 5, 5 now being associated not only with the initial energy spread E of the illuminating particle beam, but also with the bias voltage 6 between the cathode reflector and the cathode of the emission system specifying the flux of illuminating electrons

(=&+S

(233)

To investigate the mirror electron-optical properties in the MEM regime we shall use both Eq. (72), determining the explicit dependences of I and $ on z in a region far from the cathode surface, and the parametric dependences of the trajectory [Eqs. (69) and (70)], which are also valid in the near-cathode region. The constants a and b entering these equations are to be expressed in terms of the coordinates of the point on the cathode surface and the coordinates of the point of an electron source. In doing so, it is necessary to keep in mind the specificity of the MEM operating in the shadow regime, the essence of which is that when illuminating monoenergetic electrons by a homocentric beam, the information at a certain point on the cathode surface is transmitted by a unique trajectory (a reflecting trajectory) which passes by this point at the smallest distance (in the limit, it is a tangent of the cathode surface at this point). Every reflecting trajectory is associated with the appropriate point on the cathode surface by definite conditions: First, it intersects the normal to the cathode surface drawn through this point, and, second, in the place of their intersection the reflecting trajectory and the normal are mutually perpendicular (Fig. 6). If the source of illuminating electrons has a finite size and emits electrons with an energy spread, then every point belonging to the

376

E. M. YAKUSHEV AND L.M. SEKUNOVA

FIG.6. Scheme determination of the connection between a point on the cathode and the reflecting trajectory. ANB, cross section of the cathode surface by a meridional plane; A'N'B', projection of the reflecting trajectory section on this plane; NN', normal to the cathode surface.

cathode surface is associated not with one reflecting trajectory, but with a set of trajectories (a reflecting beam of electrons), each of which satisfies the specified conditions. A . Determination of Arbitrary Constants

In order to establish the connection between the constants a and b referring to the reflecting trajectory and the coordinates of the point on the cathode corresponding to this trajectory, it is necessary to consider the trajectories traversing within close proximity of the cathode surface. Then, on account of aberrations, one should keep the terms up to and including third order in r. Let the cathode surface be an equipotential; then its equation near the axis may be approximated by the paraboloid of revolution = (4i/44;)uku?

(234)

where U f = rkei$*;rk,$k are the coordinates of the point on the cathode surface. The equation of the normal to this surface will be

NOWwe shall find the point of intersection of this normal and the trajectory. Using Eq. (62),we write re'*

=

[u(()+ x ( ( ) + i ~ ( ( ) ~ ( ~ ) ] e ' @ ' ~ '

(236)

Setting Eqs. (235) and (236) equal, and taking into account that z = [ + '1, we

377


obtain

c

The value = incorresponding to the intersection point will be found from the condition of perpendicularity of the normal and the reflecting trajectory, which, in the required approximation, may be written as follows:

(4;:/44;)[uku*’(c)

+ u:v’(c)15=5,

(238)

=

Then, using the equality U = ap(c) + b g ( c ) and g(c) = q ( c ) m , as well as the initial conditions of Eq. (35) for p ( c ) and q(c), we obtain = $4;r)(Ukb*

From this equality it is clear that and (239) give

+ Uzb)

(239)

enis the fourth-order value. Equations (237)

When deducing these expressions, due to the smallness of equations are adopted: p(in) = 1, ei8(‘n)= 1

g(cn) =

+ Wn),

a? 4’(cn)

O(5,)

=

= 4;?

- kBk&$5/24;,

en, the following

q ( c n ) = q(O) = q k ,

42 = x(i,)

=0

Equation (240)connects the constants a, b of the reflecting trajectory and the coordinates of the corresponding point on the cathode surface. The relation between these constants and the coordinates rs, Rs,zs of the source point may be determined on the base of Eq. (72).Extrapolating the rectilinear section of the trajectory direct branch from a certain plane z = z, located beyond the mirror field to the source plane z = zs, we obtain us = IseQ - aps

+ bgs + x s - ( a ~ h+ bgb)qs + i(aps + bgs1k.a

(241)

where ps

= (zs - zdpb, Ka

g s = (zs = K(Za),

-

Zs)gb,

x s = x(za)

v ~ s=

~ ( 2 , )+ q’(za)(zs - za)

+ ~ ’ ( z , ) ( z-s z,)

the points z = zB,z = zc determining, as before, the positions of the vertex and the center of curvature of an electron mirror on the z axis. On solving the set of equations, Eqs. (240) and (241) with respect to a, b with accuracy up to values

378


of third order, we obtain

- x s - 4aPS

+ BSS)~,l

(243)

Here a, p are values of first order in r (244)

a = uk

B = (l/gs)(us

- Psuk)

(245) As can be seen from the expressions in Eqs. (242) and (243), to a first approximation one may set a = a, b = /?.This approximation is enough to calculate the values I],x,K to an order not higher than first by Eqs. (41), (47), and (63). Thus, Eqs. (69) and (70) [or (72)] combined with Eqs. (242) and (243), completely determine the charged particle trajectories in the field of a mirror objective under specified conditions of the object point and of the point of an electron source. B. Electron-Optical Properties of MEM in the First Approximation

Now we shall investigate the properties of a mirror microscope in the first approximation. In it, the equation of a reflecting trajectory in the region far from the cathode surface, according to Eqs. (72) and (242)-(245), will be T @(@I

=

ukP(z) f [ g ( z ) / g s l ( u s

- Psuk)

(246)

Recall that the functions p(z),g(z) describe the trajectories lying in the meridional plane of the rotating coordinate system and determining the cardinal elements. In the field-free region these functions are approximated by straight lines and, according to Eqs. (101), (104), (105), (108), and (109), are as follows:

dZ)= ( z - zF + f)pb

(247)

g ( z ) = ( z - zF - f)gb

(248)


379

where f is the focal distance, zF is the position on the z axis of the focal plane z = zF.Just as in light optics, the position of the mirror curvature center, z c , and the position of its principal plane, z B , are given by the relations zc = ZF - f

(249)

+f

(250) In the screen plane, z = z b , which is the plane of a shadow image, Eq. (246), written for the reversed branch of the trajectory, gives z g = ZF

where b - r be i[#b+@(Zb)l P b = ( z b - ZF + f ) P b , g b = ( z b - Z F - f ) g b . Equation (251) means that it is the correct image of the cathode surface that is formed in the screen plane for the illuminating electron point source. Indeed, if such a source of electrons is found on the mirror axis (r, = 0),then Eq. (251) becomes

Hence, it follows that there is a linear conformity between the points of the cathode and the microscope screen. The microscope magnification is constant in all directions and equals = rb/rk = P b

+ (Ps/gs)gb

(253)

From the expression in Eq. (208) it also follows that, in the presence of a magnetic field, the image plane is rotated with respect to the object plane around the z axis by the angle (254) The infinity sign in the upper limit of the integral means that the latter is calculated up to a certain arbitrary value z = z , belonging to the region free of field. Now we consider the case when the point source of illuminating electrons is off the z axis (us # 0). In this case, from Eq. (251) it follows that the central point of the cathode (rk = 0,z = 0)corresponds to the point (q,,u b o ) of the plane of the microscope screen ObO

= -(gb/gs)%

(255)

The above conclusions on the adequacy of a shadow image, its rotation with respect to the object plane, and the microscope magnification also remain valid when the point source of illuminating electrons is displaced. However,

380


establishing the point-to-point conformity between the object and the image, it is necessary to take into account the displacement of the image central point from the z axis caused by the displacement of the point source and determined by Eq. (255). Equation (253)gives one the possibility to determine the magnification of the mirror microscope in any position of the planes of the screen, z = zb, and the source, z = z,. It is common for these planes to coincide (z, = zb); the magnification of a microscope is M = 2(z, - zF f ) p b . If the source is at infinity (the illuminating beam is parallel), then M = 2(zb - zF)ph.In the case when the source is in the center of curvature of a mirror, M = (zb - z F - f)&. When the positions of a point source and the principal plane of a mirror are brought close together (z, + zB), the microscope field of view degenerates to a point, while its magnification tends to infinity. In this case an electron probe is formed which illuminates the only point of the cathode surface uk = u,/p,.

+

C . Image Forming Now we consider the reflecting trajectories in the presence of small disturbances near the surface of the mirror cathode. These disturbances may be caused both by (electric or magnetic) microfields on the cathode surface and by micrononhomogeneities of the surface relief. However, whatever the character of disturbances, it is common that after their action redistribution of electron velocities in its components takes place, and, as a result, the intersection of the proper reflecting trajectory and the device screen is changed. If one disregards the details of the disturbing process, the mathematical result may be characterized by some changes of the constants u and P in Eq. (246) for the reversed branch of the reflecting trajectory, writing the equation for this branch as follows:

Taking into account that the disturbance is concentrated in the neighborhood of the reversal point of the reflecting trajectory, we find from the condition of continuity of a trajectory in this point that Au = 0. In order to express the value Ab in terms of the particle velocity change caused by the disturbance, we find the dependence of /3 on the particle velocity in the reversal point of the reflecting trajectory. Taking into account values of third order, this dependence is determined by Eqs. (69) and (70), in which it is necessary to set [ = T k m ) . If we restrict ourselves by values of first order, this dependence may also be derived from Eq. (246), if we set i = T k m) On. differentiating Eq. (246) with respect to time in the neighborhood of the


38 1

reversal point, we find

Analogously, from the expression for the reversed branch of the reflecting trajectory [Eq. (256)], we find the value fl + Afl and, comparing it with Eq. (257), obtain Afl

= -L e i * k ( A i k

k4L

+ irkA$k)

where Aik and i k A$k are, respectively, the radial and azimuthal electron velocity changes undergone in the perturbation process. Then, caused by this disturbance, the displacement Aub of the point the reflecting trajectory traverses the microscope screen is determined by the equality 2 . At+, = y e r S (Aik k

k4k

+ irkA$k)(Zb - zF

-

f)gb

(259)

Note that the electron velocity, as a result of a disturbance, changes only its direction and, therefore (ik

+ At,)’ + r i ( $ k + A&)’ + (Aik)’

= i,’

+ li$i

(260)

where A i k is the electton velocity component normal to the cathode surface received by an electron in the perturbation process. If the disturbance is so small that the quantity [(Aik)’ ri(All/k)2 (Aik)2] is negligible compared with (if ri$:), then, instead of the expression in Eq. (260), we may write the equality

+

+

ik

Aik

+

+

rf$k

=0

(261)

implying the perpendicularity condition of the velocity change .vector Av = Aik + irkAIjk and the velocity component vector vll = i, + irk$k parallel to the cathode surface in the neighborhood of the reversal point. With this condition Eq. (259) takes the form

where y = arctan( - i k / r k $ k )

The quantities

ik

(264)

and rk$k entering the last equation may be found with the

382


help of the trajectory equation [Eq. (69)]. Differentiating it with respect to time and taking into account that = f k& as well as Eqs. (242)-(245), we obtain in the neighborhood of the reversal point

i

rssin($s - $k

-

)]

0,) - 7kBk gsrk 2d)k

(265)

Then Eq. (264) takes the form

Thus, Eq. (262) combined with Eqs. (263) and (266) gives one the possibility to determine the deviation of the disturbed trajectory relative to the undisturbed one both in the magnitude and in the direction in the screen plane. Note that it is a very common case for the source of illuminating electrons to be on the axis (rs = 0), and the magnetic field to be absent (Bk = 0) in the near-cathode region, for the velocity of the reflecting beam in the place of reversal to have only a radial component (rk& = o), and for the velocity change to come about in the azimuthal direction ( A i , = 0). It is just this case for which the value Au is calculated for some concrete forms of nonhomogeneities of the sample surface in a number of papers (e.g., Sedov et al., 1968a,b; Sedov, 1970). It is evident that these calculations are also suitable for the case when the source of illuminating electrons is far from the axis; however, now Av denotes the change of the velocity component parallel to the cathode surface and not that of the radial velocity. In any case, if the velocity change Av, caused by nonhomogeneities of various types, is known, then Eq. (262) combined with the expression for magnification [Eq. (253)], gives one the possibility to calculate the contrast of a shadow image in any mirror. Here, besides electrostatic fields, a magnetic field may also take part in forming an image; in the general case, the cathode surface may be curved, and the source of illuminating electrons may be located outside the symmetry axis of the system. As follows from Eq. (262), the deviation of the disturbed trajectory from the undisturbed one, Arb, in the plane of the microscope screen equals

Arb = K Av/kJ& where

(267)


383

The last quantity has a sense of the microscope sensivity to relatively small changes of the electron velocity caused by the disturbing nonhomogeneities. This increase of sensivity is always desirable. However, one should keep in mind that such an increase involves deterioration of the microscope resolution, because in this case identification of the micro-objects located close to each other becomes difficult. In reality, if two identical objects are at a distance Ark < Arb/M, where M is the microscope magnification, then their shadow images are superimposed. When the resolution is high, the case seems to be optimal when Arb/M

2:

2

(269)

where I is the limiting resolution of the microscope.

D . Impact Parameter and the Circle of Secondary Emission The effect of disturbance is determined, to a great extent, by the distance h (see Fig. 6) between the cathode surface and the reflecting trajectory in the neighborhood of the analyzed point. Taking into account third-order values, this distance may be calculated on the base of Eqs. (12) and (234).In doing so one should, in general, substitute the value 5 = [, into Eq. (12) corresponding to the intersection of the reflecting trajectory and the normal to the cathode surface. However, according to Eq. (239),5, is of fourth order; therefore, in the adequate approximation one may write

where

While deducing the last relations, besides Eqs. (12) and (234), Eqs. (41)(46),(232),and (242)-(245), as well as the conditions in Eqs. (35),(36),and (37), imposed on the functions p , q near the cathode surface, are taken into account. For a point source of monoenergetic electrons ( E = 0,d # 0) Eq. (270) determines the surface which represents the set of the reversal points of reflecting trajectories. In the adopted approximation this surface is a paraboloid of revolution with the axis parallel to the z axis and intersecting the cathode surface at the point uk = u, = r,ei*". As seen from Eq. (270), the quantity h, in the general case, is not a constant for all points of the cathode surface. The only exception is when there is no magnetic field near the surface of the cathode (Bk = 0), and the source of illuminating electrons is located in

384


the plane z = z c , passing through the center of curvature of a mirror. If the negative bias 6 is given to the cathode, so that ( = E 6 < 0, then electrons never reach cathode surface.The minimal distance at which they can approach the surface equals ISl/4;. With positive bias (6 2 0) some part of the electrons reach the cathode, forming the circle of absorption. The coordinates of the circle center coincide with r, and II/. and are determined by Eq. (271),while its radius, as follows from Eq. (270),equals

+

s++k2Bzg: E

= 4gs(44;’p:

YI2

Note that the center position of the circle of absorption does not depend on the bias voltage 6.

E. Aberrations of a Shadow Image When deducing the formulas describing the first-order electron-optical properties of a mirror microscope operating in the shadow regime, the source of illuminatingelectrons was supposed to be pointlike, and the influenceof the terms of third order in r and of first order in rE on forming a shadow image was assumed to be small. The account of these effects determines aberrations of a shadow image. We shall consider, first, aberrations associated with a finite-sizedsource of illuminating electrons. Suppose that in the plane z = z, the source represents a circular spot of a radius p , , and its every point may be considered as a selfdependent point source of illumination. If the quantity us is given a sense of the coordinates of the spot center, the coordinates of any spot point upmay be written in the form up = us + peinp,and 0 5 0,5 2n, p 5 p , . In order to take into account the influence of finite-sized source on the microscope resolution, one should substitute up instead of us into Eq. (246). Then for the reversed branch of a trajectory we obtain 9 U + -us gs

uk

- -(pg,

gs

+ p,g) =

-

9 -peiRp

(273)

ss

From the equality obtained it is seen that after reversal the cross section of the reflecting beam by the plane z = const is a circle. Thus, single-valued conformity between the points of an object and its shadow image is broken in this case. In the screen plane the radius of the spot of diffusion equal to represents the aberration determining the smearing of the shadow image caused by the finite-sized source of illuminating electrons. The aberration

385


limits the microscope’s resolving power up to the value

For a specified aperture angle of the electron reflecting beam yp = psgb/gs,the dependence of microscope resolution on the positions of the source and the screen is determined by the equality

Now we turn to a consideration of geometrical aberrations of third order in rand of chromatic aberrations of the order of rE. Using Eqs. (72),and (242)(245),we represent the coordinates of the point of intersection of the reflecting trajectory reversed branch and the plane of a microscope screen, z = zb, located beyond a field, in the form rbeiRb=

u b

+ AM

(277)

where 0, gives the first-order coordinates of the point determined by Eq. (251), and AMis the total aberration in the plane of a shadow image

Bk

- i-a(aP*

8

-

+

+ Bsb)a - x s - 4aPS + /?s,)..l

- %Pb 9s

- bgb)wb + x b

+ i(aPb

-

Pgb)lcb

Using Eqs. (41), (47), and (63), in which one should set a = a,b Eqs. (82), (232), and (233), one may reduce Eq. (278) to the form

= /?,and

AM= A u ( ~ + ’ AM

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