Free and Guided Optical Beams: Proceedings of the International School of Quantum Electronicserice Slcily, Italy 20 - 27 November 2002

FREEHGUIDEHPJICAL BEAMS INTERNATIONAL SCHOOL OF QUANTUM ELECTRONICS

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EDITORS SERGIO MARIELLUCCI • MASSIMO SANIARSIERO University "Tor Vergata" Rome, Italy

University "Roma Tre" Rome, Italy

(

FREE AID GLIIDED OPT GAL BEAMS INTERNATIONAL SCHOOL OF QUANTUM ELECTRONICS Erice Sicily, Italy

20 - 27 November 2002

'World Scientific NEW J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I

• H O N G K O N G • TAIPEI •

CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: UK office:

Suite 202, 1060 Main Street, River Edge, NJ 07661 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

FREE AND GUIDED OPTICAL BEAMS Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-238-950-4

Printed in Singapore by World Scientific Printers (S) Pte Ltd

Table of Contents Preface

ix

List of Participants

xi

Invited Lectures Semiperiodic Zones, Critical Points and Numerical Calculation of Diffraction Integrals 5. Bosch and J. Ferre-Borrull

3

Partial Polarization in Arbitrary Three-Dimensional Electromagnetic Fields A.T. Friberg

15

Optical Beams as Quantum Models (abstract) F. Gori

27

Micro-Optics: Fundamentals and Recent Topics H. P. Herzig

29

Generation and Propagation of Coherent Matter Waves F.S. Cataliotti, I. Herrera, S. Bartalini, C. Fort, L. Fallani and M. Inguscio

37

Broad-Band Active Optical Waveguides based on Thin Films R.M. Montereali

51

Spatial Laser Beam Characterization and Propagation Through Abed-Type Optical Systems using the Second-Order Moments Method G. Nemes

61

Beams in Resonators (abstract) M.R. Perrone

77

On the Effects of Diffraction in the Propagation of Ultrashort, Femtosecond, Optical Pulses M.A. Porras

79

Gaussian and Bessel Beams and Pulses beyond the Paraxial Approximation (abstract) C. Sheppard

99

Partially Coherent Beams in Free Space and in Lenslike Media R. Simon

v

101

VI

Beam Polarization Modulation in Wave-Optical Engineering J. Tervo and J. Turunen

103

Spatial Coherence: Definitions and Measurements B. Eppich, G. Mann and H. Weber

109

Invited Seminars Optical Beams in Uniaxial Crystals G. Cincotti, A. Ciattoni and H. Weber

131

Light Beam Shaping: The Integration Method P. Di Lazzaro, S. Bollanti and D. Murra

143

Beam Propagation in Quadratic Media G. Leo

153

Generation of Polarization Entangled Photons by a Universal Source of Entanglement M. Barbieri, F. De Martini, G. Di Nepi and P. Mataloni

171

Polarization Instabilities in a Quasi-Isotropic CO2 Laser R. Meucci, I. Leyva and E. Allaria

185

Superluminal Localized Solutions to the Maxwell Equations for Vacuum and for Dispersive Media (with Arbitrary Frequencies and Bandwidths) M. Zamboni Racked, K.Z. Nobrega and E, Recami

193

Angular Momentum in Optical Beams J. Serna

207

Experimental Aspects in Beam Characterization (abstract) A. Sana

219

Seminars Applications of the Time-Resolved Integral-Geometric Methods for the Composite Materials Diagnostics (abstract) A. A. Aliverdiev

223

The Electrodynamics of Processes having placed in the Volume of a Monochromatic Coherent Polarized Radiation Beam (abstract) A. A. Aliverdiev, A. A. Aliverdiev and A. A. Amirov

225

VII

Naturally Generated Beams by Quasi-Optical Phenomena M. Branescu

227

Laser and Satellite Remote Sensing of the Ocean L. Fiorani

237

Time-Resolved Spectroscopy of Semiconductor Photodetectors (abstract) B. Pura, J. Strzeszewski, A. Tadeusiak and Z. Wrzesinski

245

Polymer Waveguides for Optical Modulators (abstract) B. Pura, J. Strzeszewski, A. Tadeusiak and Z. Wrzesinski

247

Acceleration of Electrons in Free Space by a TMoi Laser Beam C. Varin and M. Piche

249

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Preface

Optical beams are electromagnetic waves that remain essentially concentrated around a mean axis upon free propagation or that are guided by suitable structures. The study of such kind of electromagnetic fields was greatly stimulated by the invention of the laser, almost forty years ago. In recent years, however, the interest on optical beams has further increased, due to the availability of many types of laser sources, characterized by very different properties, as far as their polarization, coherence, spectral content, and spatial distribution are concerned. On the other hand, many of the most recent applications of laser radiation require a precise knowledge of the behavior of the beam characteristics on propagation. In November 2002, propagation of optical beams was the subject of one of the courses organized within the International School of Quantum Electronics at the Ettore Majorana Centre in Erice, Sicily. It was the 35th Course of the School and its subject was "Free and Guided Optical Beams". The purpose of the Course was to give an updated overview, as complete as possible, on topics concerning the propagation of light beams in free space, in materials, and in guiding structures, and to provide the tools needed to its study, both from a theoretical and from an applicative point of view. As an important byproduct, the school was a great opportunity for many world's acknowledged scientists to meet together and to discuss the latest developments and emerging perspectives on the study and characterization of light beams. Most of the lecturers attended all the lectures and devoted their spare hours to stimulating discussions. We would like to thank them all for their admirable contributions, in particular Prof. Raja Simon for his enthusiastic action. The Course also took advantage of a very active audience; most of the participants were active researchers in the field and contributed with discussion and seminars. Some of these seminars are also included in this book. The present book reports most of the lectures, as extended review-like papers, given during that course and provides a fundamental introduction to the topic, as well as reports on recent research results. Arguments of the lectures range, just to give some examples, from beam characterization to numerical techniques for beam diffraction, from vector theory of propagation to optical waveguides, from generation of matter beams to applications in remote sensing, electron acceleration, or micro-optics. As it is evident, the spectrum is rather wide and, as a consequence, some of the topics are not treated with a great detail. This is the case, for instance, of the propagation in anisotropic or non-linear media, or of active waveguides, which would require specific courses. The choice of the organizers, however, was to cover the subject of the course in a way as complete as possible. The main reason of such a choice was that often the boundaries between different topics are actually very fuzzy and it may happen - it is indeed very likely - that people who is used to work, for instance, on characterization of scalar light beams in free space, sometimes have to deal with propagation in different media, or with partially coherent fields, or with fields for which the scalar approximation, or even the paraxial approximation, are no longer valid. On the other hand, we believe that such a wide-spectrum choice not only did not reduce the

IX

x didactical quality of the course, and of the present book as well, but it was indeed a way to enhance it. In this book, within each Section, the Chapters give a fairly complete accounting of the Course invited lectures and seminars, with the exception of the informal panel discussions. We did not modify the original manuscripts in editing this book, except to assist in uniformity of style. We regret that some very busy invited lecturers were not able to transfer into a written record their magnificient oral contributions given during the Course. Only the abstracts are included in the book. We are grateful to Prof. Riccardo Borghi for contributing to the secretarial organisation of the Course and to our editor at World Scientific Publishing Co. Pte. Ltd., Yim Kok Kean, for professional support. This International School was held under the auspices of Prof. Antonino Zichichi, Director of the "Ettore Majorana" Centre and Foundation for Scientific Culture, Erice, Italy. Finally, we acknowledge with gratitude the generous financial support of the organizations who sponsored the Conference: the Italian Ministry of University and Scientific Research, the Sicilian Regional Parliament, the I.N.F.M. (Rome), and the Universities of Rome "Tor Vergata" and "Roma Tre".

Sergio Martellucci Professor of Physics University "Tor Vergata" Rome (Italy) Massimo Santarsiero Professor of Physics University "Roma Tre" Rome (Italy) February, 2004

DIRECTOR OF THE "ETTORE MAJORANA" FOUNDATION AND CENTRE FOR SCIENTIFIC CULTURE Antonino Zichichi E.M.F.C.S.C. Via Guarnotta 26 91016 Erice (Italy) Phone:+39 0923 869133 Fax: +39 0923 869226 e-mail: [email protected]

CO-DIRECTORS OF THE INTERNATIONAL SCHOOL OF QUANTUM ELECTRONICS Arthur N. Chester

Sergio Martellucci Engineering Faculty University of Rome "Tor Vergata" Via del Politecnico 1 00133 Rome (Italy) Phone:+39 06 7259 7144 Fax: +39 06 7259 7145 E-mail: [email protected]

Senior Vice President, Hughes Electronics and President Hughes Research Labs. (Retired) 667 Neff Road Grosse Pointe, MI 48230 (USA) Phone:+1 3138849425 Fax:+l 3138849425 E-mail: [email protected]

CO-DIRECTORS OF THE 35th I.S.Q.E. COURSE Massimo Santarsiero

Jari Turunen Department of Physics University of Joensuu P.O.Box 111, 80101 - Joensuu (Finland) Phone:+ 358 (0)13 2513209 Fax:+ 358 (0)13 251 3290 E-mail: [email protected]

Dipartimento di Fisica University Roma Tre via della Vasca Navale, 84 00146 Roma (Italy) Phone: (+39) 06 5517 7207 Fax: (+34) 06 5579 303 E-mail: [email protected]

SCIENTIFIC SECRETARIAT Riccardo Borghi Dipartimento di Elettronica Applicata Universita Roma Tre via della Vasca Navale, 84 00146 Roma (Italy)

Phone: (+39) 06 5517 7056 Fax: (+39) 06 5517 7026 E-mail: [email protected]

xi

xii INVITED LECTURES BY: Salvador Bosch Departament de Fisica Aplicada i Optica Universidad de Barcelona Marti i Franques, 1 08028 Barcelona (Spain) Phone: (+34) 93 402 1203 Fax: (+34) 93 402 11 42 E-mail: [email protected]

Ari T. Friberg Department Microelectronics & Inf. Tech. The Royal Institute of Technology Electrum 229 164 40 Kista (Sweden)

Phone:+46 8 790 4191 Fax: +46 8 789 6672 E-mail: [email protected]

Rosa Maria Montereali Advanced Physical Technologies ENEA C.R. Frascati, Via Enrico Fermi, 45 00044 FRASCATI - RM (Italy) Phone: + 39 06 9400 5296 Fax: +39 06 9400 5400 E-mail: [email protected]

George Nemes Laser Department National Institute for Laser, Plasma and Radiation Physics 76900 Bucharest (Romania) also Astigmat, San Jose, CA 95117, USA Phone: +1 408 244 3005 Fax: +1 408 244 3005 E-mail: [email protected]; [email protected]

Maria Rita Perrone Franco Gori Dipartimento di Fisica Universita Roma Tre via della Vasca Navale, 84 00146 Roma (Italy) Phone: (+39) 06 5517 7036 Fax: (+34) 06 5579 303 E-mail: [email protected]

Hans Peter Herzig Institute of Microtechnology University of Neuchatel Rue A.-L. Breguet 2 CH-2000 Neuchatel (Switzerland) Phone:+41 327183270 Fax:+41 327183201 E-mail: [email protected]

Universita' di Lecce Dipartimento di Fisica Via per Arnesano 73100 Lecce (Italy) Phone: +39 0832 3204-98 (-82, -84) Fax: +39 0832 320505 E-mail: [email protected] Miguel Angel Porras Politecnic University of Madrid Department of Applied Physics Rios Rosas, 21 28003 Madrid (Spain) Phone:+34 91 3367011 Fax: +34 91 336 6952 E-mail: [email protected]

Giancarlo Righini Massimo Inguscio Universita' di Firenze Via Sansone, 1 50019 Sesto F.no - FI (Italy) Phone: +39 055 4572035 Fax: +390554572121 E-mail: [email protected];

Head, Optoelectronics & Photonics Dept. IROE CNR Via Panciatichi 64 50127 Firenze, Italy Phone: +39 055 4235239 Fax: +39 055 412878 E-mail: [email protected]

XIII

Saiedeh Saghafi Physics Department University of Shahid Beheshti Evin Ave. - Parkway Rd. 1983963113 Tehran (Iran) Phone:+98 21 2413350 Fax:+98 212418698 E-mail: [email protected]

Colin Sheppard Physical Optics Dept., School of Physics A28, University of Sydney NSW 2006 (Australia) Phone:+61-2-9351-3941 Fax:+61-2-9351-7727 E-mail: [email protected]

Rajiah Simon The Institut of Mathematical Sciences C I T Campus Tharamani, Room No. F9, Main Building Chennai 600113 (India) Phone: 044 - 235 1856 (extn 232) Fax: 044 - 235 0586 E-mail: [email protected]

Horst Weber Optical Institute Technical University Berlin Strasse des 17 Juni 135 10623 Berlin (Germany) Phone:+49 30 31423585 Fax: +49 30 27850 E-mail: [email protected]

INVITED SEMINARS BY: Francesco Saverio Cataliotti LENS Universita di Firenze Via Nello Carrara 50019 Sesto Fiorentino (Italy) Phone: +39-055-457 2478 or 2458 Fax:+39-055-4572451 E-mail: [email protected]

Gabriella Cincotti Dipertimento di Elettronica Applicata Universita' Roma Tre via della Vasca Navale, 84 00146 Roma (Italy)

Phone: (+39) 06 5517 7399 Fax : (+39) 06 5579 078 E-mail: [email protected]

Paolo Di Lazzaro FIS ACC ENEA Via Enrico Fermi 45 00044 Frascati (Italy) Tel: +39 06 9400 5722 Fax: +39 06 9400 5334 E-mail: [email protected]

Giuseppe Leo Dipartimento Ingegneria Elettronica Universita Roma Tre Via della Vasca Navale 84 00146 Roma (Italy) Phone:+39 06 55177096 Fax: +39 06 5579078 E-mail: Ieo@uniroma3. it

Paolo Mataloni Dipartimento di Fisica Universita di Roma "La Sapienza" P.le Aldo Moro 2 00185 Roma (Italy) Phone:+39 06 49913478 Fax:+39 06 4463158 E-mail: [email protected]

XIV

ATTENDEES: Riccardo Meucci Istituto Nazionale di Ottica Applicata Largo E. Fermi 6 50125 Firenze (Italy) Phone:+39 055 23081 Fax: +39 055 2337755 E-mail: [email protected] Erasmo Recami Faculty of Engineering Bergamo State University Viale G. Marconi 5 24044 Dalmine (Italy) Phone:+39 035 461880 Fax: +39 035 562779 E-mail: [email protected] Julio Serna Galan Optics Department Complutense University Facultad CC. Fisicas 28040 Madrid (Spain) Phone:+34 91394 4554 Fax: +34 91 394 4683 E-mail: [email protected]

Alberto Sona Department of Physics University of Insubria Via Valleggio 11 22100 Como (Italy)

Phone: +39 031 2386272 - 2132810 (home) Fax: +39 02 213 5410 (home) E-mail: [email protected] Jani Tervo Department of Physics University of Joensuu P.O.Box 111 80101 Joensuu (Finland) Phone:+ 358 (0)13 251 3207 Fax:+ 358 (0)13 251 3290 E-mail: [email protected]

Praveen Aghamkar Department of Physics Guru Jambheshnar University 125-00 IHisar (India) Phone:+91 1662251057 Fax:+91 16622760240 E-mail: [email protected]

Aboutrab Aliverdiev Institute of Physics Daghestan Scientific Centre of Russian Academy of Sciences (DSC RAS) M. Yaragskogo 94 367003 Makhachkala (Russia) Phone: +7 8722 629070 Fax: +7 8722 628900 E-mail: [email protected]

Andrea Alu Dipartimento di Elettronica Applicata Universita di Roma 3 Via della Vasca Navale 84 00158 Roma (Italy) Phone:+39 06 5517 7065 Fax:+39 06 5517 7026 E-mail: [email protected]

Stefan Amarande Laser Section National Institute for Laser, Plasma and Radiation Physics 76900 Bucharest Magurele (Romania) Phone: 0040 21423 1470 Fax: 0040 21423 1470 E-mail: [email protected]

Maria Branescu Laboratory of Advanced Materials for Special Applications National Institute for R&D on Physics of Materials 76900 Bucharest (Romania) Phone:+4021-493-0195 Fax:+4021-493-0267 E-mail: [email protected]

xv Maria Victoria Collados Departamento de Fisica Aplicada Universidad de Zaragoza Pedro Cerbuna 12 50009 Zaragoza (Spain) Phone:+34 976761000 (ext: 3451) Fax:+34 97676123 3 E-mail: [email protected]

Luca Fiorani

Luigi Moretti DEIS University of Calabria Via P. Bucci Rende (Italy) Phone: +39 081 6132373 Fax.+390816132598 E-mail: [email protected]

Maria Arcangela Nigro DIMET

FIS-LAS ENEA Via Enrico Fermi 45 00044 Frascati (Italy) Tel:+39 06 9400 5861 Fax:+39 06 9400 5312 E-mail: [email protected]

University "Mediterranea" Via Graziella Reggio Calabria (Italy) Phone:+39 081 6132373 Fax:+39 0816132598 E-mail: [email protected]

Simonetta Grilli

Lara Pajewski

Istituto Nazionale di Ottica Applicata Via Campi Flegrei 34 80078 Pozzuoli (Italy) Phone:+39 081 8675040 Fax:+39 081 8042519 E-mail: [email protected]

Electronic Engeneering University "La Sapienza" Via Eudossiana 00184 Rome (Italy) Ul. Szmidtal3bl. 3m. 79 97300 Piotrkow Trybunalski

Phone: +48446486890 E-mail: lara.pajewski@uniromal .it

Montserrat Hidalgo Nunez Istituto per i Processi Chimico-Fisici Consiglio Nazionale delle Ricerche Via Moruzzi 1 56124 Pisa (Italy) Phone:+39 050 3152259 Fax:+39 050 3152230 E-mail: [email protected] Maurizio Mattarelli Dipartimento di Fisica Universita degli Studi di Trento Via Sommarive 14 38050 Trento (Italy) Phone:+39 0461881695 Fax:+39 0461881696 E-mail: mattarel(2)science.unitn.it

Fabio Paleari Scienze Chimiche Fisiche Matematiche University dell' Insubria Via Valleggio 11 Como (Italy) Phone:+39 0312386253 Fax:+39 031 2386119 E-mail: [email protected]

Massimo Palma Dipartimento di Tecnologie dell'Informazione Universita degli Studi di Milano Via Bramante 65 26013 CREMA (Italy) Phone: +39 02 503 30052 Fax:+39 02 503 30010 E-mail: [email protected]

XVI

Carmine Sapia

Charles Varin

Department of Applied Electronics University "Roma Tre" Via della Vasca Navale 84 00146 Rome (Italy) Phone:+39 06 5517 7801 Fax:+39 06 5517 7026 E-mail: sapia@uniroma3. it

Physics department, Laval University Centre for Optics, Photonics and Lasers (COPL) G1K 7P4 Quebec, Qc (Canada) Phone: (418) 656-2131 #6220 Fax:(418)656-2623 E-mail: [email protected]

Danuta Slawinska Department of Physics August Cieszkowski Agricultural University of Poznan Wojska Polskiego 38/42 60-637 Poznan (Poland) Phone: +48-61-8487496 Fax:+48-61-8487495 [email protected]

Jan Strzeszewski Faculty of Physics Warsaw University of Technology Koszykowa 75 00661 Warsaw (Poland) Phone: +48226605068 Fax:+48226282171 E-mail: [email protected]

Andrea Tadeusiak Faculty of Physics Warsaw University of Technology Koszykowa 75 00661 Warsaw (Poland) Phone: +48226605068 Fax:+48226282171 E-mail: [email protected]

INVITED LECTURES

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SEMIPERIODIC ZONES, CRITICAL POINTS AND NUMERICAL CALCULATION OF DIFFRACTION INTEGRALS SALVADOR BOSCH AND JOSEP FERRE-BORRULL Universitat de Barcelona, Departament de Fisica Aplicada i Optica, Diagonal 647, E-08028, Spain E-mail: sbp(a)fao.ub.es Basic theoretical elements enabling the numerical calculation of the diffraction integrals are introduced. The related practical procedures are also explained in some detail and several application cases corresponding to significant problems in the field of optics are addressed.

1

Overview

In this work we present a novel approach to the modeling of diffraction phenomena, with some examples of application. The development will be divided in several parts: • • • •

Statement of the problem for spherical waves diffracted by plane apertures Extension to aberrated waves Extension to non-monochromatic waves Conclusions

2

Statement of the problem for spherical waves diffracted by plane apertures

The basic underlying idea for the forthcoming numerical methods can be summarized with a paragraph taken from the Born&Wolf book, Appendix III, when speaking about the evaluation of integrals of the form \g(z) e>kf(x)dz (for large values of k), when referring to the stationary phase methods initially proposed by Lord Kelvin for their evaluation [1]: "... the only significant contributions to the integral arise from portions of the path in the vicinity of the saddle points or end points..." Similarly, when the integral is of the form fjg(.x,>') e'V(*'y)dx dy (for large values of k), the main contributions arise from the critical points of the integral. These points, according to their very nature (and usually their relative importance), are designated as being of the first, second or third kind. Critical points of the first kind are those corresponding to stationary points of the function (integrand). Critical points of the second kind are defined by the stationary points of the integrand when this is restricted to the border of the aperture. Finally, critical points of the third kind are the corners of the aperture (those points along the curve of the border with no defined tangent). 2.1

The semiperiodic zones

Let us analyze the case of a spherical (divergent or convergent) monochromatic wave incident upon a plane aperture. We can assume that the source point (S) and the calculation point (P) lie in the same half-space defined by the plane of the aperture (in the case of a convergent spherical wave) and in the two different half-spaces (in case of a divergent spherical wave). We will define the semi-periodic zones on the aperture plane

3

4 in close accordance with the function that is to be integrated [2, 3]. In the case of the Rayleigh-Sommerfeld formulation of scalar diffraction, the complex amplitude at P that is due to a distribution U(x,y) in the aperture S is exp(/Ar /> ) / .,, ( ik)cos(n,rp)dxdy, U(P)=\\U(x,y)- i (1) X

'P

'P

where S is in the plane z=Q, k=2n/h, rp is the vector from P to Q=(x,y), and n is the normal to the aperture at Q away from the volume closing P. The character of the incident wave leads to different expressions for U(x,y). Let us consider first the case of a diverging wave (see Fig. 1): U(x,y) = exp(ikrs)/rs, where rs is the vector from S to (x,y).

Figure 1. Notation for the calculation of the diffraction for a diverging spherical wave.

According to the Fresnel zone construction, adapted to the situation in which a plane aperture is present, the external border of the «th semi-periodic zone is defined by points Q of the plane, which intersect the three-dimensional ellipsoid A

rs+rp=r0+n-, n = .

__

T

(2)

where K is the wavelength and rs = SP , rs = SQ , rp = PQ. This ellipsoid is the surface in space that is generated by the ellipse with foci at S and P and major-axis length r0 + n(/l / 2), when the ellipse is rotating around the line SP. Thus the external border of the nth semi-periodic zone will always be an ellipse: the intersection of the ellipsoid (2) with the plane of the aperture. The internal border of the nth semi-periodic zone is naturally defined by the external part of the (n-l)th zone. The case of a converging wave requires a definition of the zones with some additional care, since there is a maximum valid number n given by the integer number of times that the distance r0 contains the quantity X/4. This is: -rs+rp=r0-n-, n = l,2,..nm

(3) A/4 It is important to note that, either for the diverging or the converging case, the zones completely fill the plane of the aperture so that any point in this plane belongs to one and only one zone. Restricting ourselves to the diverging case for conciseness, we note that the successive zones n =1, 2, ... will cover the plane of the aperture and will have welldefined intersections with the diffracting aperture. This allows us to express the diffraction integral as a sum of contributions due to the parts of the zones that lie within the aperture. Within this framework, each critical point of the diffraction integral will belong to one and only one zone. Since the number of critical points is finite, only several

5 zones in our construction will contain a critical point, even though some zones may contain more than one of these points. The interest of the subdivision of the diffracting aperture according to the zones is clear: the two-dimensional highly oscillatory integrand varies smoothly within a zone, so the integration of a zone is easy in terms of the required number of evaluations of the integrand. This allows the formulation of an elementary integration method consisting of the subdivision of the aperture according to its intersection with the zones, followed by a summation of the contributions of the intersections. It has been shown that this elementary method is indeed useful for rectangular and circular apertures, whereas any standard two-dimensional integration algorithm in Cartesian coordinates is absolutely impractical owing to the extremely high number of evaluations of the integrand that would be required [4]. 2.2

The critical points in a scheme for numerical methods.

Moreover, since the contribution of the critical points is crucial for the diffraction integrals, general numerical procedures for its computation may be proposed based on the fact that the semi-periodic zones containing critical points will be those that contribute most decisively to the integral. This approach was developed in detail in Ref. 4 for the particular case of the plane perpendicular incidence. Our idea now is to demonstrate the validity of this approach in the general case of a convergent or divergent wave, thus eliminating the intrinsic limitations of stationary-phase methods for cases of multiplicity in the critical points [5]. Following the same process as in part 4 of Ref. 4, we conclude that there are two key points for designing general fast and accurate numerical algorithms: 1) considering only the zones with critical points inside gives a good approximation to the integral (as in Method 1 of Ref. 4), and 2) taking into account the subsequent neighboring zones, we can reach any predefined accuracy (as in Method 2 of Ref. 4). 2.3

Examples

To illustrate the real significance of our approach, we consider a rectangular aperture (2 mm x 1 mm) lying in the XY plane, centered with respect to the origin, and a diverging spherical wave (^=632.8 run) with its origin at (0,0,-100) incident upon the aperture. In Fig. 2 we show the modulus of the diffracted wave (in arbitrary units) computed along the line joining the points (0,0,100) and (3,1.5,100). Three types of calculation have been performed: exact calculations (for each calculation point, the contributions of all the zones are added), and two approximate calculations according to the precision criteria defined for Methods 1 and 2 of Ref. 4. Only two results are visible, since method 2 is virtually exact at that graphic scale. Note that in this example the point (2,1,100) is the shadow of one corner of the rectangle, so that for the computation of its diffracted intensity, that corner is simultaneously a critical point of the first, second, and third kinds. This poses no problem to our methods, whereas it is a definite limitation for stationaryphase or geometrical theory of diffraction techniques, because of the multiplicity of the critical point, as mentioned above. It is also worth noting that the computation times required for our methods are independent of the position of the calculation point: all points are equally difficult to compute.

6 0.06-

0.055 • P 0.050-

< Q 3 0.04-

(D

"B 0.045"o. < 0.040 -

"5.

I

-

P

0)

1

V; 0.02-

0.035Method 1 0.030-

0.000.0

0.4

0.8 1.2 1.6 X coordinate (mm)

2.0

0.025 • 1.0

•

Method 2

1.1 1.2 1.3 X coordinate (mm)

1.4

Figure 2. Examples of calculations for a spherical diverging wave using different methods based on the semiperiodic zones and the critical points.

3

Extension to aberrated waves

The next natural step in the present approach is to consider cases of more practical interest, for instance diffraction in real optical systems [6]. The ideas presented in the previous section relating to the semiperiodic zones and the critical points are based on basic concepts of wave physics. These are independent on whether they are applied to a converging or diverging spherical wave or any other kind of wave. Consequently, the methods presented above can be directly adapted, provided the correct expressions for the optical path are used. 3.1

Fundamentals of the extension

Let us consider an imaging system such as the one in Figure 3. By analogy with the previous Fig. 1, the system images the point S onto its Gaussian image point S' and we wish to calculate the complex amplitude at a point P after the light has passed trough the aperture stop Z (placed at the exit side of the optical system).

Calculation Point,

0P 7

Lens thickness: 5.4 mm

S'

4.0 mm

^235.0 mm J ~ ~

xyxxyy

-xzxxzz

4>xx4>yy + xxxzz

y z y y z z

,„•.

yyzz

This result states that the degree of polarization PS, when squared, always is greater than or equal to the averaged squared correlations between the electric-field components, weighted by the corresponding intensities. The left-hand side of Eq. (8) does not depend on the orientation of the coordinate axes, but the right-hand side does.

21

The equality holds for an orientation in which the diagonal elements (component intensities) are the same. In such a situation P32 reduces to the plain average of the squared correlations. These conclusions are fundamentally in agreement with the analogous 2D results. Hence, the 3D degree of polarization PS is a measure of the correlations that, at any point in space, exist between the three orthogonal electric-field components. We stress that the 3D formalism, in general, gives for the degree of polarization different values than the usual 2D method. For example, while a fully unpolarized plane wave in 2D description has a zero degree of polarization (P2 = 0), in our 3D analysis its degree of polarization P3 must be non-zero, since the electric field is confined to oscillate in a plane perpendicular to the direction of wave propagation; in fact, the 3D degree of polarization of plane waves is restricted between 0.5 < PS < 1. Intuitive understanding of these differences is gained by considering Fig. 1. In upper row, an unpolarized planar field (<j>xx = <j)yy, no correlations) passes through a polarizer. The 2D formalism gives P-2 = 0 and P% = I for the field before and after the polarizer, respectively. In lower row, an unpolarized 3D field (4>xx = (f>yy = (j>zz, no correlations between any components) is passed through two devices that cut off the wave oscillations in orthogonal directions. For the initial field, which cannot be described using the 2D method, the 3D formalism gives a zero value for P3. On passing through the first device the field becomes partially polarized, i.e., 4>xx = 0, (j)yy = zz with no correlations between the y and z components, and consequently PS = 0.5. After the second device, the field is fully polarized (Ps — 1), since the oscillations now take place only in a single direction.

pa=o

7

Figure 1. Schematic illustration of the differences in the 2D and 3D coherence-matrix formalisms in the treatment of the polarization state of an electromagnetic field.

22

Random electromagnetic fields that are statistically homogeneous and isotropic, whether in thermal equilibrium or not, exhibit spatial correlations that are identical with those of blackbody radiation. Such fields can be represented by an ensemble of unpolarized, angularly uncorrelated, electromagnetic plane waves, uniformly distributed within the entire 4?r solid angle.25 The 3D degree of polarization is zero (as customarily assigned to blackbody fields). On restricting the size of the plane-wave acceptance cone, as often happens e.g. in imaging, the 3D degree of polarization is also altered and approaches 0.5 as the solid angle tends to zero.

4.2. Degree of Polarization in Optical Near Fields The new 3D formalism of partial polarization has been applied to electromagnetic near fields emitted by semi-infinite thermal half-space sources.26 The cross-spectral density tensors of the current distributions, acting as sources, are found on the basis of the statistical fluctuations in the (slightly) absorbing medium. The methods of the electromagnetic coherence and polarization calculations, as well as the results, are explained in more detail elsewhere.27

1.0T

0.8-

SiC, 11.36 urn Au, 620 nm

Au, 400 nm SiC, 9.1 urn

glass, 500 nm

2 z/X Figure 2. Degree of polarization PS (z) in the optical near field, at different frequencies, for thermal half-space sources consisting of glass, gold (Au), and silicon carbide (SiC).

In Fig. 2 we illustrate the behavior of the new 3D degree of polarization PS(Z), introduced in Eq. (7), as a function of distance from the boundary of the source consisting of gold (at A — 620 nm and A = 400 nm), SiC (at A — 11.36/mi and A = 9.1//m), and glass (at A = 500 nm). At wavelength A = 620 nm gold exhibits

23

a surface-plasmon resonance, and plasmon waves are known to be highly polarized in the plane spanned by their direction of propagation and the surface normal. Due to the resonance, the near field emitted by the gold source then is highly polarized. Similar behavior is observed in the near field of the SiC source at A = 11.36^m, at which wavelength SiC supports a surface-phonon resonance. However, tuning the wavelength out of the resonance, the near-field degree of polarization becomes greatly reduced, as evidenced by the curves for gold at A = 400 nm and for SiC at A = 9.1/xm. Further, we note that the near-field degree of polarization PZ(Z) for a thermal glass sample decays monotonically as the distance z from the source increases. Glass does not support surface waves, and the decay is directly related to the loss of the evanescent modes.

5. Conclusions The 3D electromagnetic coherence and polarization in this article are analyzed in the space-frequency representation, at a single frequency of the random field. The method applies to any stationary field, no assumption of quasi-monochromaticity is made. For nearly monochromatic waves one normally assumes that the states of spatial coherence and polarization do not change appreciably within the narrow spectral band. In such a case the results are directly valid at the center frequency of the quasi-monochromatic field. From a mathematical point of view the conventional 2D formulation and our new 3D formulation of partial polarization have many similarities. These have to do, for example, with the basic nature of the respective coherence matrices, the decompositions of these matrices in suitable basis sets, and the identification and meanings of the expansion coefficients. Likewise, the relations to the correlations of electric-field components, and the limits of completely polarized and unpolarized light are fundamentally similar in both treatments. So, analogies between the 2D and 3D cases take us a long way. But there are also major differences. Maybe the main (and most disturbing) difference is that, unlike in the 2D formalism, there appears to be no unambiguous way of expressing the 3D coherence matrix as a sum of two matrices, one representing completely polarized and one completely unpolarized light. Hence also the definition of the 3D degree of polarization lacks the physical foundation used in the 2D analysis. We already noted that the dimensionality (2D vs 3D) plays a crucial role in the quantitative values of the degree of polarization. That a uniform plane wave, regardless of how unpolarized in the 2D measure, should necessarily have a non-zero degree of polarization in the 3D space, is how we feel it should also be physically, but others may disagree. As regards optical near fields, the 3D degree of polarization appears to behave in a manner that is fully consistent with the underlying physical phenomena. Far from the thermal half-space source Pj, (z) approaches a non-zero value that depends on the source material. However, on the source boundary we always have Pa(0) = 0.25, regardless of the media or wavelengths involved. This result seems to be related

24

to the half-space geometry, but its true origin and meaning are as yet unresolved problems. Other matters that would benefit from further study are the measurement (e.g., with a scanning near-field optical microscope) of the 3D degree of polarization, and an extension of the theory to 3D electromagnetic coherence. Acknowledgments The author thanks T. Setala, A. Shevchenko, and M. Kaivola (Helsinki University of Technology) and J. Tervo (University of Joensuu) for useful discussions and collaboration on the subject of this article. Financial support from the Swedish Research Council (Vetenskapsradet) is gratefully acknowledged. References 1. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991). 2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995). 3. E. Wolf and D. F. V. James, Rep. Prog. Phys. 59, 771 (1996). 4. E. Collett, Polarized Light in Fiber Optics (PolaWave, Lincroft, NJ, 2003). 5. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998). 6. F. Gori, Opt. Lett. 23, 241 (1998). 7. F. Gori, M. Santarsiero, G. Piguero, R. Borghi, A. Mondello, and R. Simon, JEOS A: Pure Appl. Opt. 3, 1 (2001). 8. M. J. Beran and G. B. Parrent, Jr., Theory of Partial Coherence (SPIE, Bellingham, WA, 1974). 9. J. Pefina, Coherence of Light (D. Reidel Publ. Co., Dordrecht, Holland, 1985). 10. P. Ostlund and A. T. Friberg, Opt. Commun. 197, 1 (2001). 11. J. Tervo and J. Turunen, Opt. Commun. 209, 7 (2002). 12. R. Carminati and J.-J. Greffet, Phys. Rev. Lett. 82, 1663 (1999). 13. C. Henkel, K. Joulain, R. Carminati, and J.-J. Greffet, Opt. Commun. 186, 57 (2000). 14. J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, Nature 416, 61 (2002). 15. A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, Phys. Rev. Lett. 85, 1548 (2000). 16. L. D. Landau, E. M. Lifshits, and L. P. Pitaevskii, Statistical Physics, 3rd edition (Pergamon Press, Oxford, UK, 1980). 17. J. E. Sipe, J. Opt. Soc. Am. B 4, 481 (1987). 18. S. A. Ponomarenko and E. Wolf, Phys. Rev. E 65,,016602 (2001). 19. T. Setala, K. Blomstedt, M. Kaivola, and A. T. Friberg, Phys. Rev. E 67, 026613 (2003). 20. J. Tervo, T. Setala, and A. T. Friberg, Opt. Express 11, 1137 (2003). 21. T. Carozzi, R. Karlsson, and J. Bergman, Phys. Rev. £61, 2024 (2000). 22. T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002). 23. J. C. Samson, Geophys. J. R. Astr. Soc. 34, 403 (1973). 24. R. Barakat, Opt. Acta 30, 1171 (1983). 25. T. Setala, M. Kaivola, and A. T. Friberg, Opt. Lett. 28, 1069 (2003). 26. T. Setala, M. Kaivola, and A. T. Friberg, Phys. Rev. Lett. 88, 123902 (2002).

25

27. T, Setala, Spatial Correlations and Partial Polarization in Electromagnetic Optical Fields: Effects of Evanescent Waves, PhD thesis (Helsinki University of Technology, Espoo, 2003).

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OPTICAL BEAMS AS QUANTUM MODELS FRANCO GORI Dipartimento di Fisica, Universita Roma Tre via della Vasca Navale, 84 - 00146 Roma (Italy) E-mail: gori@uniroma3. it

Abstract The differential equation governing, within the paraxial regime, the propagation of an optical field through a medium with variable refractive index is formally identical to the time-dependent Schroedinger equation for a particle in two dimensions. In particular, the wave function of the particle plays the role of the transverse disturbance of the optical field, the potential acts as the modulation of the refractive index, and time evolution is replaced by the propagation along the longitudinal coordinate. An interesting consequence of such analogy is that well known propagation features of light in some media can afford a model for the time evolution of quantum wavefunctions of a particle within the corresponding potential. Finding the stationary states of a given potential, for instance, turns out to be equivalent to finding the propagation modes in a suitably chosen guiding structure. The simplest case is that of an optical beam propagating inside a homogeneous medium and it corresponds to a free particle on a plane evolving in time. Well, the time evolution of the particle wavefunction can be predicted, starting from its expression at the starting time, using a propagator that can be immediately derived from the Fresnel integral. Yet, it is not difficult to conceive, by suitably superposing eigenstates of the energy of the free particle, a wavefunction whose squared modulus does not change with time, in perfect analogy with an optical nondiffracting Bessel beam. A slightly less trivial case is that of the propagation inside an optical fibre having parabolic refractive index. It is known that the modes of such a fibre are HermiteGaussian modes, whose width is determined by the parameter governing the refractive index variation. The quantum analog is a two-dimensional harmonic oscillator, whose energy eigenstates are, of course, the same Hermite-Gaussian functions, and this is, actually, even better known. But the optical analogy allows us also to predict behaviors of the quantum particle that are not completely trivial. It is the case, for instance, of a fundamental Gaussian beam that is injected into the fiber off axis. If the width of the Gausian is chosen as that of the propagation modes of the fiber, the beam will propagate inside the fibre without changing its width, but its center will oscillate around the axis of the fiber. This is due to the "lens-like" effect of a fiber with parabolic refractive index. This corresponds to a coherent state of the harmonic oscillator. Due to the same effect, if we inject off-axis a Gaussian beam with a different spot size, then the beam not only oscillates around the axis, but also changes its width, passing periodically from smaller to larger values of the spot size. This corresponds to a squeezed state of the harmonic oscillator.

27

28

Further significant cases can be considered, such as those pertinent to optical waveguides, that correspond to potential wells for the quantum particle. Even in such cases, classical results concerning the propagation of the optical fields can help to solve analogous quantum problems, as far as the determination of the stationary states and of the energy eigenvalues, and the time evolution of the system is concerned. Finally, starting from the above analogy, a correspondence between partially coherent beams and mixed quantum states can be established. It stems from the equivalence between the cross-spectral density function, defined within the second-order coherence theory, and the density matrix, used to describe quantum mixed states. Results obtained in the field of the partial coherence optics, such as those pertinent to the wellknown Gaussian Schell-model sources, can be easily transposed into the quantum realm.

MICRO-OPTICS: FUNDAMENTALS AND RECENT TOPICS HANS PETER HERZIG Institute of Microtechnology, University ofNeuchatel, RueA.-L Breguet2, CH-2000 Neuchatel, Switzerland E-mail: hansveter. herzie. @ unine. ch

Micro-optics includes a family of optical components and systems that are fabricated with the aid of modern micromachining. Elements utilizing either refractive or diffractive surfaces are now found in applications ranging from laser beam shaping in laser material processing to optical interconnects in telecom applications. We introduce the micro-optics family members and present system concepts.

1

Introduction

Micro-optics includes a family of optical components that are fabricated by modern micromachining, such as optical lithography, direct laser and e-beam writing, and reactive ion-etching. In the past decade micro-optics has emerged as a powerful tool to realize various optical functions that have not previously been possible or feasible using conventional optical elements. Elements utilizing either refractive or diffractive surfaces are now found in applications ranging from laser beam shaping in laser material processing to optical interconnects in telecom applications. In general terms one of the major strengths of micro-optics compared to conventional optics lies in the fact that micro-optics allows integration of large, complex optical systems into much more compact form. Additionally, emergence of replication techniques such as injection molding, allow the lost cost mass-production of micro-optical elements. For further reading we refer to the literature.1'2'3 2 Deflection of light by micro- and nanostructures There are various possibilities to control the propagation of light. Basically we can modify the polarization, the amplitude and the phase distribution of an incident beam as function of space and time. Consequently the light propagation will change. For this chapter we concentrate on elements fabricated by lithographic techniques to generate a modulation of the surface-profile. 2.1 Refractive and diffractive micro-optics The basic difference between refractive and diffractive structures is explained in Fig. 1 with the example of a prism and the corresponding grating structure. In the case of the prism the light is deflected by varying the optical path length d(x)An, where d(x) is the thickness profile and An the refractive index difference. In air An = n - 1. An incident plane wave is deflected by the angle d. For a thin prism the angle 9 is given by (1)

29

30

where a is the prism angle (see Fig. 1). The diffractive blazed grating profile can be obtained by wrapping the depth profile d(x) to an interval between 0 and /I/An [Fig. l(b)]. The optical path difference A/An corresponds to a phase change of 2n. The blazed grating has period A and each period has the same slope as the original prism. The binary grating is then obtained by replacing the prism shape of each period by a binary profile with depth 0 and A/2An [Fig. l(c)]. The deflection angle 6 varies linearly with the wavelength A and the inverse of the grating period A:

(2) sm9 =m — A m is the diffraction order. In the ideal case, the blazed grating generates only one diffraction order (e.g. m = 1), whereas the binary grating generates several orders. Binary gratings are therefore used for applications where several diffraction orders are desired, such as fan-out elements4'5 or far-field beam-shaping elements6 generating symmetric light distributions. Note, that gratings with periods near the wavelength are strongly polarizing. They can therefore be used as birefringent elements.7

A/An (a)

(b)

(c)

t Figure 1. Optical elements: (a) prism, (b) blazed grating and (c) binary grating.

2.2

Artificial index material

If the period A of a grating is reduced then the number of propagating diffraction order diminishes. For grating periods in the order of the wavelength (A ~ A), high efficiency can be achieved also with binary gratings when they are illuminated under oblique incidence. In that case, only one diffraction order is propagating. When the grating

31

period is further reduced, no diffraction order (except the zero-order) is propagating. Such a binary gratings appears as a thin film with an artificial index. For grating periods A « A, the effective electric permittivity e for TE polarization becomes8

(3) and for TM-polarization (4) £

eff

£

1

where t is the fill factor and E^2 is the electric permittivity of the substrate and the surrounding medium (in general air). Thus it is possible to realize antireflection coatings by structuring the surface.89 Such AR coatings are attractive, because they can be replicated. It has also been proposed to realize blazed gratings with binary technology (Fig. 2).10 Initially, these elements worked only for longer wavelengths, e.g. for a CO2 laser working at 10.6 fum. Due to the improved technology, these devices become realistic for telecom wavelengths." Blazed grating

Blazed-binary grating Figure 2. Blazed diffraction grating and the corresponding blazed-binary grating. The structure size of the blazed-binary grating has to be considerably smaller than the wavelength.

2.3 Resonant grating filters

R

d H = 0.664 d 2 = 0.45 03 = 1.9163

Figure 3. Resonant filter with a grating on top of a waveguide.11

32

Periodic surface-relief structures can also be combined with waveguide structures as in the case of resonant grating filters. The basic concept is shown in Fig. 3.12 Resonant grating filters use the guided-mode resonance effects in waveguide gratings, showing sharp intensity variations of the observable propagating waves. The grating period is chosen so that no orders propagate outside the filter other than the 0 order. This resonance results from an evanescent diffracted wave that is parametrically near to a corresponding leaky mode of the waveguide. Because the propagating and evanescent diffracted waves of gratings are all coupled to the adjacent orders, a resonance in an evanescent wave can cause a redistribution of the energy in propagating waves. Resonant grating filters offer the advantage of obtaining a sharp linewidth response with a relatively small number of layers, compared to standard thin film techniques. 3

Photonic crystals

If the incident light propagates within the periodic structure and is reflected back and forward then the element becomes selective, such as a Bragg (1-D crystal). These structures, which are periodic in one-, two- or three-dimensions are known as photonic crystals. The term photonic crystal has been first introduced in 1987 by Yablonovitch13'14. The goal was to control spontaneous emission. The mechanism of operation is already familiar from solid-state physics where the wave function of the electron is modulated by the periodic potential of the crystalline lattice. This coherent scattering modifies the wave function of the electron, which in some cases results in the opening of an electronic bandgap. The presence of such a bandgap has consequences in electrical and thermodynamic properties of solids. Similarly, when periodicity is introduced in the dielectric constant of a medium, a photonic bandgap can open. When it happens, electromagnetic waves of certain frequencies are forbidden to propagate through the structured medium. This principle has been applied to suppress spontaneous emission. If light of a certain frequency is forbidden to propagate through the medium surrounding an atom, this atom will not be able to radiate photons of the corresponding energy. Thus, certain direct transitions between energy levels will be forbidden. For the existence of a complete photonic bandgap a fully 3-D periodic structure is required. However, 2-D photonic crystals are also of interests as optical waveguides or sensors. In the 2-D case the light is confined in the third dimension by total internal reflection. Similarly, we can consider a Bragg grating within a fiber as 1-D photonic crystal. There is a strong interest in 2-D crystals, because they can be realized with lithographic technology, enabling a large variety of shapes. 4

Fabrication of elements

Fabrication techniques for realizing the microstructures resulting from the design of micro-optical elements are based on a variety of high-resolution lithographic and optical processes.1'2'3 The typical procedure to fabricate elements is based on lithography. First a mask is generated by e-beam or by laser beam lithography. The desired pattern is encoded as an amplitude distribution. Uniform illumination of the mask is used to expose a photosensitive coating (e.g. photoresist) on the substrate. After the exposure of

33

the photoresist layer a development step converts the exposed photoresist into a surface profile. Finally, the mask is transformed into the substrate by dry or wet etching. Using several masks multilevel profiles can be generated to improve the efficiency. The illumination through an amplitude mask can also be replaced by the direct writing of the phase relief in photoresist by e-beam or laser beam. The relief structure can then be transferred into fused silica by dry etching or it can be converted into a metalized master relief by electroplating to emboss or cast low-cost replica. Continuous profile can also be generated by graytone lithography. There the binary mask is replaced by a graytone mask. A well-known graytone mask is the high-energy beam sensitive (HEBS) glass. Another method, suitable for manufacturing fine grating structures, is to generate the photoresist pattern by holographic recording. Depending on the material, surface-relief structures can also be realized without illuminating a light sensitive material. Wellknown examples are diamond turning and laser ablation. Various micro-optical elements are shown in Figs. 4-9.

Figure 4. Spherical lenses (diameter 3 |um).

Figure 5. Interferometrically recorded crossed grating (period A = 1 jam).

Figure 6. Grating on top of a (A = 1 urn).

Figure 7. Arbitrary surface produced by gray-tone lithography. The element is used for the collimation of high-power semiconductor lasers.15

microlens

34

Figure 8. Birefringent micro-optics

Figure 9. Photonic crystal waveguide in silicon.16

5 Micro-optical systems Another challenge in micro-optics is the efficient integration of micro-optical elements with other optical components (fibers, laser sources, and detectors). Figure 10 shows different system concepts, namely stacked micro-optics, slab micro-optics, surface mounting techniques and optical MEMS. These concepts have been applied to realize various systems, such as a collimator for high-power semiconductor lasers,15 a miniaturized analytical system (uTAS)17, and a miniaturized spectrometer18'19. The miniaturization of microsystems is limited by diffraction of the light at small apertures, by off-axis aberrations and by stray-light. Stacked micro-optics

Miniaturized optical table

Figure 10. System concepts.

Slab microoptical system

Optical MEMS

35

jIMT-Neuchatel

200 um Figure 11. View on a miniaturized chemical system containing microchannels, stops and microlenses.17

Figure 12. Scanning electron microscopy photograph of a miniaturized spectrometer based on silicon technology.18'1

Chemical microchips and miniaturised analytical systems (uTAS) are of increasing interest in life sciences. Usually, a complex arrangement of lasers, detectors, filters, optics, and high precision mechanical stages is required for illumination and optical detection. Microlens arrays offer a large potential to reduce the size and to simplify the architecture of analytical systems. Such a microsystem is shown in Fig. 11. It is the first chemical microsystem that integrates etched microchannels with microlenses and stops on both sides of the glass wafer.

36

A miniaturised Fourier Transform Spectrometer (FTS) based on optical MEMS technology is under development at IMT-Uni NE.18'19 Its design, combining silicon technology with micro-optics, is a Michelson interferometer with one scanning mirror. The motion of the mirror is carried out by an electrostatic comb drive actuator, resulting in a mirror displacement of 39 urn with a reproducibility of ±13 nm, which leads to a resolution better than 10 nm in the visible wavelength range. An example of such a system is shown in Fig. 12. References 1. H. P. Herzig, ed., Micro-Optics: Elements, Systems, and Applications (Taylor & Francis, London, 1997). 2. J. Turunen, F. Wyrowski, eds 1997 Diffractive Optics for Industrial and Commercial Applications (Akademie Verlag, Berlin, 1997). 3. S. Sinzinger, J. Jahns, Microoptics (Weinheim: Wiley-VCH, Weinheim, 1999). 4. H. Dammann, K. Gortler, "High-efficiency in-line multiple imaging by means of multiple phase holograms", Opt. Commun. 3, 312-315 (1971). 5. A. Vasara, et al., "Binary surface-relief gratings for array illumination in digital optics", Applied Optics 31, 3320-3336 (1992). 6. H. P. Herzig, P. Kipfer, "Aperture modulated diffusers (AMDs)", International Trends in Optics and Photonics ICO IV, Springer Series in Optical Sciences Vol 72, T. Asakura, ed. (Springer, Berlin, 1999) pp. 247-257. 7. C. W. Haggans, R. K. Kostuk, "Polarization transformation properties of highspatial frequency surface-relief gratings and their applications", in Ref. 1. 8. D. H. Raguin, G. M. Morris, "Antireflection structured surfaces for the infrared spectral region", Appl. Opt. 32, 1154-1167 (1993). 9. S. J. Wilson, M. C. Hutley, "The optical properties of 'moth eye' antireflection surfaces", OpticaActa 29, 993-1009 (1982). 10. H. Haidner, P. Kipfer, W. Stork, N. Streibl, "Zero-order gratings used as an artificial distributed index medium", Optik 89, 107-112 (1992). 11. Ph. Lalanne, S. Astilean, P. Chavel, E. Cambril, H. Launois, "Blazed-binary subwavelength gratings with efficiencies larger than those of conventional echelette gratings", Opt. Lett. 23, 1081-1083 (1998). 12. R. Magnusson, S. S. Wang, "New principle for optical filters", Appl. Phys. Lett. 61, 1022-1024 (1992). 13. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics", Phys. Rev. Lett. 58, 2059-2062 (1987). 14. E. Yablonovitch, "Photonic band-gap structures", J. Opt. Soc., Am. B 10, 283-295 (1993). 15. A. Schilling, H. P. Herzig, L. Stauffer, U. Vokinger, M. Rossi, "Efficient beam shaping of linear, high-power diode lasers using micro-optics", Appl. Opt. 40, 58525859 (2001). 16. I. Marki, M. Salt, I. Marki, U. Staufer, H. P. Herzig, "Practical and detailed analysis of photonic crystal waveguides", 2003 Digest of the LEOS Summer Topical Meetings, Vancouver, British Columbia, July 14-16, 2003, pp. 83-84. 17. J.-Ch. Roulet, R. Volkel, H. P. Herzig, E. Verpoorte, N. F. de Rooij, R. Dandliker, "Performance of an integrated microoptical system for fluorescence detection in microfluidic systems", Anal. Chem. 74, 3400-3407 (2002). 18. O. Manzardo, H. P. Herzig, C. Marxer, N. F. de Rooij, "Miniaturized time-scanning Fourier transform spectrometer based on silicon technology", Opt. Lett. 24, 17051707 (1999). 19. W. Noell, P.-A. Clerc, L. Dellmann, B. Guldimann, H. P. Herzig, O. Manzardo, C. Marxer, K. Weible, R. Dandliker, N. de Rooij, "Applications of SOI-based optical MEMS", IEEE Journal of Selected Topics in Quantum Electronics 8, 148-154 (2002).

GENERATION AND PROPAGATION OF COHERENT MATTER WAVES

FRANCESCO SAVERIO CATALIOTTlt*, IVAN HERRERA*, SAVERIO BARTALINI*, CHIARA FORT*, LEONARDO FALLANI*, MASSIMO INGUSCIO* f INFM - Dipartimento di Fisica, Universita di Catania, via S. Sofia 64, 1-95125 Catania, \ Laboratorio Europeo di Spettroscopia nonlineare LENS, via N. Carrara 1, 1-50019 Sesto F.no (FI), Italy Bose-Einstein condensates (BECs) realized with dilute alkali atoms can be considered as a source of coherent matter waves, what is sometimes termed atom laser. We will review the experimental methods used to create and manipulate BECs. We will also illustrate some simple atom-optical elements that can be created. We will review the propagation of coherent matter waves in periodic potentials showing how it is possible to control matter wave dispersion and create gap solitons.

1. Introduction The first experimental realization of a Bose-Einstein condensate (BEG) in a dilute gas of rubidium atoms1 has marked the birth of a completely new field of optics, that of coherent matter wave beams. Indeed it was immediately recognized that a BEG is in very many aspects the analogous of an optical laser for atoms. After the first pioneering experiment at MIT where a pulsed atom laser was created by simply switching off the confining magnetic potential2 other more refined experiments followed. The first continuous wave atom laser was realized in Munich by coupling atoms out of a magnetic trap using a radio frequency field3 while in Yale a BEG was loaded into a vertical standing wave to produce the analogous of a mode-locked laser4. Coherent matter wave beams were soon showed to produce interference fringes5 and double slit and multiple beam interferometers were soon realized6'7. On the other hand atoms, different from photons, interact even in the dilute gas limit. The nonlinearity introduced by atomic interactions in coherent matter waves propagation through vacuum was readily shown to be the same of the third order susceptibility for electromagnetic waves propagating through a nonlinear crystal. It was soon demonstrated that it was indeed possible to observe matter wave amplification8 and four wave mixing9. Indeed the XXVII course of this same school was dedicated to Bose-Einstein condensates and atom lasers10. More recently the intrinsic nonlinearity of condensates was successfully exploited in the observation of dark11 and bright solitons12. The last result could also be obtained by controlling the dispersion of matter waves using a periodic potential13.

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38

In the following we will try to review part of the development of this rapidly expanding field. The first section will illustrate the experimental method used to reach Bose-Einstein condensation in a dilute gas of alkali atoms. In the second section we will describe the tools used to manipulate coherent atoms and realize atom-optical elements. In the third and fourth section we will report on experiments on the propagation of coherent matter waves through periodic potentials showing how it is possible to engineer dispersion and eventually create gap solitons. In the conclusion we will give some outlook on the development of the field. 2. Bose—Einstein Condensates in dilute gases A dilute gas of particles can be cooled to a temperature such that the De Broglie wavelength associated to each particle becomes larger than the mean interparticle distance. At these temperatures the quantum statistics of the particles fully dominates the behavior of the gas. In a trapped atomic gas of bosons all the particles will tend to occupy the trap state with the largest population in a very similar way to photons in a laser cavity being pulled to the mode with the highest gain. In thermal equilibrium the state with the highest occupancy is the ground state of the trap. In a gas of interacting bosons however, the ground state of the system will not necessarily be the ground state of the potential holding the atoms. In a dilute ultracold atomic gas of bosons only binary collisions are allowed and the system is conveniently described by a nonlinear Schrodinger equation for the atomic density amplitude known as the Gross-Pitaevskii model. (1)

Where m is the atomic mass and V is the trapping potential. The s-wave scattering length a is positive for repulsive interactions and negative for attractive interactions. This parameter is a function that strongly depends on the atomic internal state. In particular it is possible to change the value of a by different orders of magnitude and even change its sign by tuning a magnetic field thanks to Fano-Feshback resonances14. The parameter a strongly influences the properties of the condensate; condensates with negative a are unstable but can form bright solitons. On the other hand the ground state of condensates with repulsive interactions (a > 0) significantly differs from the ground state of the confining potential. Indeed for most experimentally realized condensates the nonlinearity totally dominates dispersion i.e the interaction energy is much larger than the kinetic energy. The condensate wavefunction then takes the so called Thomas-Fermi shape which has the same simmetry of the trapping potential and, for a condensate in a harmonic trap, is an inverted parabola15. The standard experimental procedure for the creation of a Bose-Einstein condensate in a dilute atomic gas starts with laser cooling of an atomic vapor in ultrahigh vacuum conditions. This first step, performed in a magneto-optical trap16

39

takes the atoms from a phase space density of 10 20 at room temperature to 10 5 below 100 /UK. At this phase space density laser cooling stops essentially because of spontaneous light scattering from the atoms17. For this reason the following cooling step has to be performed in a non-dissipative trap created either by a magnetic field or by a very far-off resonance laser beam. In non-dissipative traps cooling is achieved by removing the high energy tail of the atomic distribution and by letting the atoms thermalize via binary collisions. Removal of atoms is realized either by reducing the trap depth in optical traps or by RF-induced transitions to untrapped Zeeman sublevels in magnetic traps. Let us concentrate on magnetic traps, the interaction energy of an atom in a magnetic field is E = —jl • B where jl is the atomic dipole moment. Of the atomic ground sublevels only low-field seeking states can be trapped in a magnetic field minimum. If the atomic motion is not very fast the atomic dipole adiabatically follows the magnetic field therefore the energy is proportional to the modulus of the field. However if the field variation is very rapid adiabaticity cannot be maintained and atoms are lost from the trap. To avoid these so called Majorana spin-flips it is necessary to avoid a zero of the magnetic field in the trap. Conventional magnetic traps are realized with a few centimeter size coils carrying more than 100 Ampere of current. In this way traps with 10-100 Hz oscillation frequencies are realized. The cooling procedure, known as evaporative cooling strongly relies on atomic collisions and can be very different for different atomic species. In particular since the most probable collisions at temperatures below the milliKelvin are spherically symmetric binary collisions, due to the Pauli principle18 this method cannot work for spin-polarized fermions as those trapped in a magnetic trap. Indeed fermions were cooled to degeneracy either using spin mixtures19 or via collisions with a bosonic atomic species20. 3. Atom-optical elements Matter wave beams can be manipulated in very much the same way as optical beams however the role of matter and electromagnetic fields is totally reversed in the field of atom-optics. Indeed condensates can be manipulated with conservative potentials created either by far-off resonance laser beams or by magnetic fields. In the first case the potential is obtained via the interaction of the induced atomic dipole with the electric field of the laser. This dipole potential is dependent on the laser intensity and detuning and for a two level atom in interaction with a far-off resonance beam can be written as

where WQ is the atomic resonance frequency, F the natural linewidth of the atomic transition, A the laser detuning from resonance and I(r) the intensity of the laser

40

beam. From eq. (2) we note that when the laser detuning is negative atoms are pulled towards the region with the highest laser intensity. On the other hand when the detuning is positive atoms are expelled from high intensity regions. This can be used to create very different kind of potentials with a single tunable laser beam. For example it is possible to create atomic waveguides either by using collimated gaussian laser beams with negative detuning or collimated hollow laser beams with positive detuning21. With a sheet of light created by rapidly moving a collimated laser beam with an acousto-optic modulator the group of IQO-Hannover has been able, by simply varying the laser beam intensity to create an atom mirror, a beam-splitter or a phase-shifter. The laser detuning for these experiments was positive. The experiment was performed by creating a condensate via evaporative cooling in a magnetic trap1 and then dropping it under the effect of gravity onto the sheet of light. When the laser intensity was very high the atoms did not acquire sufficient kinetic energy during their fall to go over the dipole potential and were totally reflected22. When the laser intensity was reduced part of the atoms could go over the dipole potential and the system realized a beam-splitter. By further reducing the laser intensity all the atoms were able to go over the potential but were retarded with respect to free fall in analogy to an optical phase-shifter23.

Light sheet

Time Figure 1. Scheme of the IQO experiment. The condensate falls under the effect of gravity and bounces off the potential generated by laser light.

This examples demonstrate that laser radiation is a very versatile tool, since it is possible to create many different potentials simply from the interference of laser beams coming from different directions as will be discussed in the next sections. On the other hand it is also possible to manipulate atoms with magnetic field gradients. Magnetic traps are, for neutral atoms, a particularly versatile class of these manipulation methods as they can be used for any atomic species with a magnetic moment and they can produce conservative potentials also for very long times. Techniques to trap and manipulate atoms with magnetic fields once integrated with surface deposition techniques, either lithographic or of other kind, realize what is termed an atom chip2'1. Atom chips are based on the possibility to create a 2-D-quadrupole

41

magnetic field close to a current carrying wire by compensating the field generated by the wire B = ^ 4 at the height ZQ with a constant magnetic field. If the magnetic moment // of the atom remains aligned with the magnetic field the resulting potential for the atoms can be approximated as

(3)

where we have assumed the wire to be along the axis x. With a current of just 0.4 A in the wire and a constant field of 35 G it is possible to create a waveguide with a confining frequency of 10 kHz for 87Rb much larger than conventional magnetic traps.

wire field

bias field

2-D-quadrupole

Figure 2. Using a bias field to compensate at a point ZQ the magnetic field generated by a current carrying wire it is possible to create a 2-D-quadrupole trap.

The wire guide represents the building brick for magnetic microtraps. Indeed by bending the wire in a U shape as illustrated in Fig. 3b) it is possible to create a 3-D quadrupole. As already pointed out this trap configuration have high atomic losses due to Majorana Spin-flips. In order to avoid these losses it is better to create a harmonic trap with a filed minimum different from zero as can be done by bending the wire in a Z shape (Fig. 3a)). The bias field can also be realized in a planar configuration using three parallel wires with opposite currents as in Fig. 3d). This configuration can then be used to create more complicated structures as in Fig. 3c) where a possible analogue of a SQUID is illustrated. These devices represent one of the most promising schemes for coherent atom optics and may be the basis for a totally new class of integrated sensors and quantum logic instruments. Indeed, thanks to component miniaturization, in atom chips it is possible to reach huge field gradients (above 1 Tesla/cm) with 1 Ampere currents, of the same order of magnitude as those used in electronic circuits and a few orders of magnitude lower than those employed in conventional apparata. Furthermore the substrates employed are compatible with the ultra-high vacuum technology needed for atom cooling. Nowadays the microstructures used experimentally are rather simple and the conductors sizes are of the order of 10 micron with a total chip size of a few square centimetres. However they are capable of substituting experimen-

42

(a)

(b)

(c)

(d)

Figure 3. Various microtrap configurations: a) Z-Trap; b) U-Trap. Both Z-Trap and U-Trap need an external bias field to achieve a 3-D-confining potential for cold atoms; c) A ring guide with two diametral potential barriers, working with cold atoms like a SQUID works with superconductors; d) A linear guide completely realized in planar geometry, without any bias field.

tal systems currently spread in about 1 cubic meter. Integrating many elements to control atoms onto a single device, an atom chip, will make atom physics experiments much more robust and simple. This may allow much more complicated tasks in atom manipulation to be performed in a way similar to how integration of electronic elements has allowed the development of new powerful electronic devices. Ideally one wants routine single atom state selective loading, preparation and manipulation. All will be achieved with minimal heat load and power consumption. Potentials with sizes smaller than the particle deBroglie wave-length will allow tight traps with large energy level spacing. The large level spacing reduces the probability of the environmental noise to induce unwanted excitations. Consequently, coherent manipulation will be more stable. Using well developed nanofabrication technology from microelectronics to build the atom optics will allow integration of many atom optical devices into complex quantum networks combining the best of two worlds: the ability to use cold atoms, a well controllable quantum system, and the immense technological capabilities of nanofabrication and microelectronics to manipulate the atoms. Experiments based on atom chips have allowed the demonstration of trapped atom interferometers25, of atomic conveyor belts26 and of atomic waveguides24. 4. Condensates in periodic potentials We now describe some experiments we have performed on BECs in periodic potentials, that well demonstrate the flexibility of optical potentials in the manipulation of condensates. In our experiment we load cold 87Rb atoms in the \F — 1 mp — — 1} state from a double magneto-optical trap system into a loffe type harmonic magnetic trap27. The trap is cylindrically symmetric with an axial frequency of w x /2?r = 9 Hz and a radial frequency of u>j_/27r = 92 Hz. We cool the atoms in this trap via

43

rf-forced evaporation until they reach a temperature below 300 nK, just above the critical temperature for condensation, which is around 150 nK dependent on the number of atoms we load in the trap. At this stage of evaporation we suddenly switch on an optical standing wave formed by retroreflecting light from a laser blue detuned of ~ 3 nm with respect to the Dl transition at A = 795 nm. The laser beam is aligned horizontally along the axis of the magnetic trap and, since the laser beam waist is much larger than the atomic cloud transverse size and does not produce any appreciable radial force, forms an array of disk shaped traps together with the magnetic potential. The potential is therefore V = im(w 2 z 2 + wiri) + sER cos2(27nr/A)

(4)

Zi

The optical potential is given in units s of the energy ER = /i2/2mA2 gained by an atom (of mass m) absorbing one lattice photon corresponding in rubidium to a temperature of ~ 170 nK. In our experiments we could vary the optical potential up to s = 15. When s » 1 the atoms are confined in an array of classically independent traps since the optical potential barriers are much higher than the thermal energy of the atoms. Indeed it is possible to "freeze" the degree of freedom associated with the motion along the axis and study condensation in this quasi 2D system28. After switching on the laser light we continue the evaporation ramp until the desired temperature is reached. This ensures that the atoms reach the equilibrium state in the combined trap. When we evaporate to well below the critical temperature, so that no thermal fraction is experimentally visible, we typically obtain ~ 200 condensates separated by a distance of A/2, each containing ~ 1000 atoms. Due to the blue detuning of the laser beam the atoms are trapped in the nodes of the standing wave reducing light scattering below ~ 1 Hz. When the height of the optical barriers is much larger than the condensates chemical potential we are justified in describing the condensate as a sum of wavefunctions localized in each potential well:

*o(r)=

Y,

exp[-(z-fcA/2) 2 /2k of the individual condensates. In Fig. 4 left A I show a typical image of the cloud taken after an expansion time texp = 29.5 ms, corresponding to a total number of atoms N ~ 20000 and to an optical potential s — 5. The image shows a clear structure with three interference peaks separated by 2h/m\texp i.e. by the distance corresponding to the reciprocal of the lattice constant. We remark that, differently from the case of two separated condensates, interference fringes appear only if the initial configuration is mutually coherent. In other words, since one single interference experiment with an array of condensates is equivalent to averaging a series of interference experiments with two condensates, an interference pattern will appear only in presence of a fixed relative phase between condensates belonging to consecutive wells. What is locking the phase difference across the array of BEG is tunnelling through the optical barriers, in a classical picture no interference peaks would arise. The width of the central peak (n = 0) of the interferogram is of the order Apx ~ h/mRxtexp where Rx ~ kMd is half of the length of the whole sample in the z-direction. The occurrence of these peaks is the analog of multiple order interference fringes in light diffraction. The relative population of the n ^ 0 peaks with respect to the central one (n = 0) obeys the simple law Pn = exp[-167T2nV/A2]

(6)

holding also in the presence of a smooth modulation of the atomic occupation number Nk in each well. Result (6) shows that, if a is much smaller than A/2 the intensity of the lateral peaks will be high, with a consequent important layered structure in the density distribution of the expanding cloud. The value of a, which characterizes the width of the condensates in each well, is determined, in first approximation, by the optical confinement. By using a numerical minimization of the energy we can determine the relative population Pn of the n = I peak as a function of the intensity factor s. This is shown in Fig. 4 right together with the experimental results. The good comparison between experiment and theory reveals that the main features of the observed interference patterns are well described by this model. 5. Expansion of a BEC in a moving optical lattice

We now discuss a different experiment where we study the expansion of a condensate inside a moving ID optical lattice29. This experiment allows us to load the

45

A) 0.012-

0.15-,

0.12-

0.008-

0.09-

0.06 0.004

0.030.00 -400

-200

0

200

distance (nm)

400

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

S

Figure 4. LEFT: A) Absorption image of the density distribution of the expanded array of condensates. B) Experimental density profile (crosses) obtained from the absorption image (A) integrated along the vertical direction. The wings of the central peak result from a small thermal component. The continuous line corresponds to the calculated density profile for the expanded array of condensates for the experimental parameters (s = 5 and texp = 29.5 ms). RIGHT: Experimental (circles) and theoretical (triangles) values of the relative population of the n = 1 peak with respect to the n = 0 central one as a function of the intensity factor s of the optical potential Vopt-

condensate with different quasi-momenta q in the periodic structure realized by the optical lattice. From the solid state physics it is well known that in the presence of an infinite periodic potential the energy spectrum of the free particle is modified and a band structure arises30. In the rest frame of the lattice the eigen-energies of the system are En(q), where q is the quasi-momentum and n the band index. According to band theory, the velocity in the n-th band is vn = h~ldEn/dq and the effective mass is m* = h2(d2En/dq2) l. The effective mass can be negative for a range of quasi-momentum and this has been recently recognized as a possibility to realize bright solitons in BEG with repulsive interactions31'13. In this experiment we first produce the condensate in a pure harmonic trap, then we switch off the magnetic harmonic potential let the BEG expand for 1 ms and we switch on a moving periodic potential. After 1 ms of expansion the density of the condensate decreases enough to neglect the non linear term in the Gross-Pitaevskii equation describing an interacting BEG. This means that, as a first approximation, we are allowed to consider the BEG as a linear probe of the periodic potential energy spectrum. The moving periodic potential is created by the interference of two counterpropagating laser beams with a slightly different frequency and blue detuned 0.5 nm from the D% resonance at 780 nm. The two beams are obtained by the same laser and are controlled by two independent acousto-optic modulators. The resulting light field is a standing wave moving in the laboratory frame with a velocity VL = AA///2 where A^ is the frequency difference between the two laser

46

beams. In our experiment, we typically vary the optical lattice velocity between 0 and 2vR where VR = hkL/m is the recoil velocity of an atom absorbing one lattice photon and corresponds, in the frame of the band theory, to the limit of the Brillouin zone. We switch on the moving optical lattice adiabatically by ramping the intensity of the two laser beams in 2 ms. This ensures we are loading the condensate in a Bloch state of well-defined energy and quasi-momentum32. We let the condensate expand in the lattice and after a total expansion time of 13 ms we take an absorption image of the cloud along the radial horizontal direction looking at the position and dimensions of the condensate inside the optical lattice. Prom the position after the expansion, we extract the velocity of the condensate inside the optical lattice. In particular we repeat the experiment for different velocities of the lattice and compare the position of the expanded condensate inside the lattice with the position of the condensate expanded without the optical lattice. Let call this difference in position along the axial direction Az, then the velocity of the BEG inside the optical lattice is given by v = Az/Ai — VL where At is the expansion time inside the lattice.

2.0 s=1.3 s = 3.8

. 13 0.5 ,***

0.0

0.0 0.5 1.0 1.5 2.0 quasimomentum (q/c^)

Figure 5. Velocity of the condensate inside the optical lattice as a function of the quasi-momentum q in units of the recoil momentum qn = Kki,' The open circles corresponds to data obtained with Vopt = 1.3.Efl and the filled circles to data obtained with Vopt = 3.8ER. The dashed and dotted lines are the correspondent curves given by the band theory.

In Fig. 5 we show the results obtained for the velocity of the condensate as a function of the quasi-momentum q for two different values of the lattice potential depth. The experimental data points are compared with the theoretical results obtained from the band theory and show a very good agreement. With an adequate sampling of the velocity we can extract the effective mass values given by dv/dq. The results for an optical potential depth of 1.3 ER are shown in Fig. 6. As we increase the lattice velocity (corresponding to increasing the quasi-momentum q) the effective mass rapidly increases and between 0.7 q and 0.8 q it first becomes infinite positive and then negative.

47

0.0 0.5 1.0 1.5 2.0 quasimomentum (q/qR) Figure 6. Effective mass of a condensate moving in an optical lattice of 1.3-Ejj as a function of the quasi-momentum. The data points correspond to the values extracted from the measured velocity and the solid line is the corresponding theoretical prediction of the band theory.

The consequence of the strong variation of the effective mass is expected to consistently modify the expansion of the condensate along the axial direction 33. As a matter of fact the effective mass enters the diffusive (kinetic) term in the Gross-Pitaevskii equation.

0.0

0.5

1.0

1.5

quasimomentum (q/qg) Figure 7. Axial and radial dimensions of the condensate after the expansion in an optical lattice Vopt = 2.9ER. The experimental points (filled and open circles) show the Thomas—Fermi radii of the cloud extracted from a 2D fit of the density distribution. The dotted lines show the dimensions of the expanded condensate in the absence of the optical lattice. The continuous and dashed lines are theoretical calculations obtained from the ID effective model.

In Fig. 7 we report the radii of the condensate measured as a function of the quasi-momentum after the expansion inside the optical lattice compared to numerical predictions based on an effective ID theoretical model33. The axial radius (filled circles in Fig. 7) decreases until the quasi-momentum reaches QR, this is first due to the increase of the effective mass (causing a slower expansion) and then by the

48

II

m FigureS. Absorption images of the expanded condensate. From left to right: a) normal expansion of the condensate without lattice; b) axial compression in a lattice of 2.9 ER and VL = O.Qvft; c) enhanced axial expansion in a lattice of 2.9 EH and VL = \.\VR.

fact the the effective mass becomes negative (causing a contraction of the axial direction during the time spent in the optical lattice). When q > qn the effective mass becomes positive again but with a value smaller then the real mass m. As a consequence the expansion becomes faster in this region of quasimomenta. In Fig. 7 we also report the measured values of the radial dimension of the BEG. A deviation from the expansion without optical lattice (dashed line) is observed also in this direction for q < qpi, even if this dimension is not directly affected by the presence of the lattice. This is consistent with the theory (continuous line) and can by explained in terms of a coupling between the axial and the radial dynamics. For q < qp. the compression along the lattice direction increases the mean-field energy and causes a faster radial expansion. Instead, when the condensate is loaded with q > qH, the axial expansion is enhanced (0 < m* < m) and the residual mean-field energy is further reduced, causing a suppression of the non-linear coupling between the axial and radial dynamics. This behaviour is evident in the absorption images reported in Fig. 8 where we show the shape of the condensate expanded without optical lattice (image a)), and with an optical lattice of 2.9 ER and respectively quasimomenta q < q& (b)) and q > qp. (c)). In the first case a contraction along the axial direction is accompanied by a faster expansion along the radial direction, while in the second case the condensate expands faster in the axial direction. 6. Conclusions In this paper we have illustrated the methods to obtain and manipulate BoseEinstein condensates, in particular we have concentrated on experiments on condensates in periodic potentials. Indeed in these experiments the role of light and matter are completely reversed with respect to optics were radiation is used to gain information about the crystal structure. In atom-optics instead crystals created by light are used to have a detailed study of phenomena typical of solid-state physics. This is just one example of the multidisciplinary approach of ultracold atom physics. At present one of the most promising applications of coherent matter wave beams is towards the realization of ultraprecise inertial sensors. These sensors will take advantage of the ability, illustrated in this work, to coherently manipulate atoms to realize atom interferometers. In this applications the intrinsic nonlinearity of the condensates can be an advantage. Indeed these sensors promise to breach the Heisenberg limit thanks to squeezing. In conclusion we can affirm that the field of

49 atom-optics is a rapidly growing and extremely promising new branch of optics. Acknowledgments We acknowledge F. Ferlaino and P. Maddaloni for contributing in the first part of our experimental work. The authors would like to thank also J. Catani and M. Zawada for their contribution on the experiment on the expansion of the condensate inside the optical lattice. We also acknowledge useful discussion and theoretical contribution of M. Modugno, P. Massignan, A. Smerzi, A. Trombettoni, C. Menotti, P. Pedri, S. Stringari, and L. P. Pitaevskii. This work has been supported by the EU under Contracts Numbers HPRI-CT 1999-00111 and HPRN-CT-2000-00125, by the MURST and by the INFM Progetto di Ricerca Avanzata "Photon Matter". References 1. See as a general reference Bose-Einstein Condensation in Atomic Gases, M. Inguscio, C. E. Wieman, S. Stringari, Eds. (IOS Press Amsterdam, Oxford, Tokio, Washington, 1999). 2. M.-O. Mewes et al Phys. Rev. Lett. 78, 582, (1997). 3. I. Bloch, T. W. Hansch, T. Esslinger Phys. Rev. Lett. 82, 3008, (1999). 4. B. P. Anderson, M. A. Kasevich, Science, 282, 1686, (1998). 5. M. R. Andrews, et al Science, 275, 637, (1997). 6. C. Fort, et al, Optics Lett., 26, 1039, (2001). 7. P. Pedri et al, Phys. Rev. Lett, 87, 220401, (2001). 8. M. Kozuma et al, Science 286, 2309, (1999), S. Inouye et al, Nature 402, 641, (1999). 9. I. Deng et al, Nature, 398, 218, (1999). 10. Bose-Einstein Condensates and Atom Lasers, S. Martellucci, A. N. Chester, A. Aspect, M. Inguscio, Eds. (Kluwer Academic/Plenum Publishers 2000). 11. S. Burger et al, Phys. Rev. Lett., 83, 5198, (1999); J. Denschlag et al, Science, 287, 97, (2000). 12. L. Khaykovic et al, Science, 296, 1290, (2002). 13. B. Eiermann et al, Phys. Rev. Lett, 91, 060402, (2003). 14. M. Inguscio, Science, 300, 1671, (2003). 15. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys., 71, 463, (1999). 16. E. L. Raab et al, Phys. Rev. Lett. 59, 2631, (1987). 17. S. Chu, Rev. Mod. Phys., 70, 685, (1998); C. N. Cohen-Tannoudji Rev. Mod. Phys., 70, 707, (1998); W. D. Phillips Rev. Mod. Phys. 70, 721 (1998); 18. B. DeMarco, et al, Phys. Rev. Lett. 82, 4208, (1999). 19. B. DeMarco, D. S. Jin Science, 285, 1703, (1999). 20. A. Truscott et al, Science, 291, 2570, (2001); F. Schreck et al, Phys. Rev. Lett, 87, 080403, (2001); Z. Hadzibabic et al, Phys. Rev. Lett, 88, 160401, (2002); G. Roati, F. Riboli, G. Modugno, M. Inguscio, Phys. Rev. Lett, 89, 150403, (2002). 21. K. Bongs et al, Phys. Rev. A Rapid Comm., 63, 031602, (2001). 22. K. Bongs et al, Phys. Rev. Lett. 83, 3577, (1999). 23. K. Bongs et al, Cr. Acad. Scie. IV-Phys. 2, 671 (2001). 24. R. Folman et al, Phys. Rev. Lett. 84, 4749, (2000). 25. W. Hansel, J. Reichel, P. Hommelhoff, and T. W. Hansch Phys. Rev. A 64, 063607 (2001).

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26. 27. 28. 29. 30. 31.

W. Hansel, J. Reichel, and T. W. Hansch Phys. Rev. Lett. 86, 608 (2001) C. Fort et al, Europhys. Lett. 49, 8, (2000). S. Burger et al, Europhys. Lett, 57, 1, (2002). L. Fallani et al, cond-mat/0303626. N. Ashcroft and N. Mermin, Solid State Physics (Saunders, Philadelphia, 1976). G. Lenz,P. Meystre, E. M. Wright, Phys. Rev. A 50, 1681 (1994); V. Konotop, M. Salerno, Phys. Rev. A 65, 021602 (2002); K. M. Hilligse, M. K. Oberthaler, K.-P. Marzlin, Phys. Rev. A 66, 063605 (2002). 32. J. H. Denschlag, et al, J. Phys. B At. Mol. Opt. Phys. 35, 3095 (2002). 33. P. Massignan and M. Modugno, Phys. Rev. A, 67, 023614, (2003).

BROAD-BAND ACTIVE OPTICAL WAVEGUIDES BASED ON THIN FILMS ROSA MARIA MONTEREALI ENEA, Advanced Physical Technologies, C.R. Frascati, Via E. Fermi 45, 00044 Frascati (Rome), Italy

1

Introduction

In recent years a renewed interest has been increasing for novel materials exhibiting active optical properties. The area of growth and characterization of thin films has seen a considerable expansion due to the need for a substantial shrink in the scale and cost of optical devices. The miniaturization of light emitting devices is a key objective of optoelectronics and new configurations allowing a higher compatibility with integrated optics technologies' are under investigation. Optical waveguides, which consist, by definition, in dielectric regions of a high refractive index surrounded by lower index dielectric materials2, are among the most used ones. They act as light "traps" within which optical rays remain confined by multiple total internal reflections at the boundaries3. They allow to transport light from one location to another, and, in the case of a luminescent medium of high refractive index, the waveguiding geometry facilitates the confinement of light generated inside it. So these configurations are particularly suitable for the realization of optically pumped miniaturized amplifiers and lasers4. For luminescent materials the optical waveguides preparation, characterization and optimization is strongly related to a careful investigation of the spectroscopic features of the active medium, its gain properties as well as to the choice of the final device configuration, amplifier and/or laser, under development. We will discuss these aspects considering two specific cases of vacuum-deposited thin films characterized by similar broad-band emissions in the visible spectral range: small molecular weight organic thin films5 and inorganic Lithium Fluoride (LiF) films containing active color centers (CCs)6. These two classes of materials show high potentialities for the development of novel active waveguide devices, due to their peculiar optical characteristics and/or to the lower cost and simpler fabrication processes. 2

Optical waveguides: basic definition

The fabrication of miniaturized optoelectronic devices requires the confinement of light in an optical waveguide. Different geometries can be chosen, like slab, strip and fiber3, the last one characterized by a circular cross-section. In contrast, in the case of thin film based technologies, planar and channel waveguides are used. These device configurations are particularly suitable for the realization of optically pumped optical amplifiers and lasers in integrated optics, allowing for the reduction of the optical losses both at the emission and exciting wavelengths4. In this way high gain can be achieved with low pump powers. The simplest dielectric waveguide consists in a planar film of refractive index nf sandwiched between a substrate and a cover material with lower refractive indices ns and nc (nf > ns > nc). All materials are assumed loss-less and isotropic. Often the cover

51

52

material is air, in which case nc=l; instead, if nc=ns, the guide is defined symmetric, as sketched in Fig.l. Typical differences between n/and «c range from 10~3 to 10"1.

n

f unguided ray

guided ray

Figure 1. Sketch of a planar dielectric waveguide. Rays making an angle less than the critical one are guided by total internal reflection.

The confined propagation of light in a dielectric film structure of thickness on the order of the wavelength A, of a traveling electromagnetic (e.m.) wave is dictated by the wave theory of confined e.m. propagation in non conducting media7. The formal approach consists to develop solutions to Maxwell's equations in the inner and outer media with the appropriate boundary conditions. Only this wave picture fully describes the propagation properties of a dielectric confining structure; however, in a simple approach, the ray-optical picture, the light guidance in a slab waveguide is that of light rays tracing a zig-zag path in the film, with total internal reflection at the film-substrate and filmcover interfaces8, as sketched in the right part of Fig. 1. It associates with each optical ray a transverse e.m. plane wave and provides an overall insight of the phenomena and order of magnitude calculation of the mode properties. The total e.m. field is the sum of these plane waves. These approaches are clearly described and compared7. Guided modes are fields that maintain the same transverse distribution and polarization at all distance along the waveguide axis. Each mode can be viewed as a standing wave in the transverse direction, traveling in the axial direction. The condition for confinement is obtained from the so-called transverse resonance condition: only a discrete set of angles, and sometimes none, allow the propagation3. An asymmetric guide cannot always support a guided mode, i.e there is a minimum cut-off frequency, below which no-propagation is possible. There is always a solution, even when the film thickness gets very small, in the case of symmetric dielectric waveguides9. The number M of TE modes (electric fields perpendicular to the plane of incidence defined by the wave normal and the perpendicular to the interfaces) supported by a symmetric dielectric waveguide is the greatest integer given by the approximate formula M = 2 dA/NA

s 1/2

where NA is the numerical aperture of the waveguides. When M<ns LiF crystal as substrate, ns<ns

Substrate, «s

Figure 6. Planar waveguide structures based on thin colored layers of LiF in crystals (left) and films (right).

This behavior should be ascribed to the polycrystalline nature of LiF films, which increases the formation efficiency of visible active CCs, and can also reduce the final densities of larger complex re-absorbing defects. Further investigation and optimization

59

of the waveguide design could allow for a selective generation of the requested defects and therefore an improvement of light emission performance. However, in the investigated cases, an external feedback or etched faces, which tend to reduce the performances, are required. An interesting case to consider is that of colored thick LiF films grown on silica on silicon substrates31 in order to form the facets by cleaving of the substrate after film deposition and waveguide preparation. In every case, to design an active optical waveguide based on LiF, crystal or film, it is important to determine the influence of the coloration conditions, i.e. dose, depth of coloration, irradiation temperature, dose-rate, on the CCs formation and on the induced refractive index modification. Moreover, in the case of polycrystalline LiF films grown on amorphous and crystalline substrates, several properties, in particular refractive index, density, structure and microstructure, and their influence on the CCs formation induced by low energy electron beam bombardment should be taken into account. The principal challenge to fabricating an optically pumped inorganic waveguide laser based on CCs in LiF films is to optimize high gain and low optical loss confining structures by the proper choice of irradiation conditions, film growth parameters and device design. Acknowledgments I would like to thank G. Baldacchini, E. Burattini, M. Montecchi, E. Nichelatti, S. Pelli, M. Piccinini, G.C. Righini for valuable discussions. The precious contribution of F. Bonfigli, S. Gagliardi, S. Gambino, T. Marolo and A. Pace is also acknowledged.

References 1.

S. Martellucci and A. N. Chester, eds., Integrated Optics: Physics and Applications, Plenum Press, New York, 1983. 2. B. E. Saleh and M. C. Teich, Fundamentals of Photonics, J. W. Goodman, ed., J. Wiley &Sons, New York, 1991. 3. F. Gori, Elementi di Ottica, Ed. Accademica, 1995, cap.9, p.410. 4. D. C. Hanna, Proceedings of the European Conference on Integrated Optics, ECIO93, Neuchatel, Switzerland, April 18-23, 1993, 3-1. 5. G. Baldacchini, S. Gagliardi, R. M. Montereali, A. Pace and R. B. Pode, Phylosophical Magazine B 82,6 (2002) 669. 6. R. M. Montereali, Point Defects in Thin Insulating Films of Lithium Fluoride for Optical Microsystems, in Handbook of Thin Film Materials, H. S. Nalwa ed., Vol.3: Ferroelectric and Dielectric Thin Films, Academic Press, 2002, Ch.7, p. 399. 7. H. Kogelnik, in Guided Wave Optoelectronics, T. Tamir, ed., Springer Verlag, 1988, ch.2. 8. P. K. Tien, Appl. Opt. 10 (1971) 2395. 9. H. Kogelnik and V. Ramaswamy, Applied Optics 13 (1974) 1857. 10. Y. Sato, Organic LED system considerations, in Electroluminescence I, Semiconductors and Semimetals, Academic Press, San Diego, 2000, Vol.64, Ch.4, p.209. 11. V. Bulovic, P. E. Burrows, and S. R. Forrest, Organic LED system considerations, in Electroluminescence I, Semiconductors and Semimetals, Academic Press, San Diego, 2000, Vol.64, Ch.5, p.255.

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12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

C. W. Tang, S. A. Van Slyke, Appl. Phys. Lett., 51 (1987) 913. V. G. Kozlov, V. Bulovic, P. E. Burrows, and S. R. Forrest, Nature 389 (1997) 362. V. Bulovic, V. G. Kozlov, V. B. Khalfm, S. R. Forrest, Science 279 (1998) 553. S. Riechel, U. Lemmer, J. Feldmann, S. Berleb, A. G. Muckl, W. Bruiting, A. Gombert, V. Wittwer, Opt. Lett. 26,9 (2001) 593. P. E. Burrows, V. Bulovic, G. Gu, V. Kozlov, S. R. Forrest, M. E. Thompson, Thin Solid Films 331 (1998) 101. B. H. Soffer and B. B. McFarland, Appl. Phys. Lett. 10 (1967) 266. H. Kogelnik and C. V. Shank, Appl. Phys. Lett. 18 (1971) 152. M. Pope and C. E. Swemberg, Electronic Processes in Organic Crystals, Oxford University Press, New York, 1982. W. B. Fowler, Physics of Color Centers, W. B. Fowler ed., Academic Press, New York and London, 1968. L. F. Mollenauer, Tunable Lasers, L. F. Mollenauer ed., Springer Verlag, Berlin, 1987. G. Baldacchini, M. Cremona, R. M. Montereali, L. C. Scavarda do Carmo, R. A. Nunes, S. Paciornik, F. Somma, V. Kalinov, Opt. Commun. 94 (1992) 139. R. M. Montereali, G. Baldacchini, L. C. Scavarda do Carmo, Thin Solid Films 201 (1991) 106. R. M. Montereali, M. Piccinini, E. Burattini, Appl. Phys. Lett.78,26 (2001) 4082. G. Baldacchini, A. Mancini, R. M. Montereali, S. Pelli, G. C. Righini, in Proc. 8th European Conference on Integrated Optics, ECIO'97, Stockolm, 1997, p.373. R. M. Montereali, A. Mancini, G. C. Righini, S. Pelli, Opt. Commun. 153 (1998) 223. L. Fornarini, S. Martelli, A. Mancini, R. M. Montereali, S. Pelli, G. C. Righini, in Proc. 9th European Conference on Integrated Optics, ECIO'99, Turin, Italy, 1999, p.343. A.Belarouci, F.Menchini, H.Rigneault, BJacquier, R.M.Montereali, F.Somma, P.Moretti M.Cathelinaud, Optical Materials 16 (2001) 63-67 M. Montecchi, E. Nichelatti, A. Mancini and R. M. Montereali, J. Appl. Phys. 86,6 (1999)3745. M. Montecchi, R. M. Montereali, E. Nichelatti, Opt. Mater. 17 (2001) 347. L. Fornarini, A. Mancini, S. Martelli, R. M. Montereali, P. Picozzi, S. Santucci, J.NonCrystalline Solids 245 (1999) 141.

SPATIAL LASER BEAM CHARACTERIZATION AND PROPAGATION THROUGH ABCD-TVPE OPTICAL SYSTEMS USING THE SECOND-ORDER MOMENTS METHOD GEORGE NEMES ASTIGMAT, 1441 Detroit Ave. Ste. 253, Concord, CA 94520, USA

E-mail:[email protected]; [email protected] The main concepts and results in laser beam characterization and their propagation through ABCD-type optical systems are reviewed by using the method of second-order moments. A better understanding of laser beams and optical systems is obtained.

1.

Introduction

Understanding beams is an important task. To simplify this task, the real optical systems and beams are replaced by mathematical models representing idealized cases. In introduction the idealized concepts and some general issues are discussed. Section 2 introduces the simplest beam, called the idealized gaussian beam (IGB). In Section 3 the beams with separable properties in the two transverse directions are discussed. The beams with transverse coupling, requiring a 4x4 matriceal treatment, are discussed in Section 4. Section 5 concludes the paper. The laboratory reference frame has x and y as the transverse horizontal and vertical axes, and z as the horizontal, longitudinal, optical axis, and also the beam axis. For matrices the notation of the software Mathematica® is used. The concepts and some of the main results of this paper were previously presented in [1-6]. 1.1. Idealized concepts A paraxial ray is represented as a 4x1 column matrix "R containing the two coordinates x(z), y(z) of its intercept at the z = constant plane, and the two paraxial [7] angles u, v, representing the angles between the ray and its projections on the yOz and xOz plane, respectively, or by its transpose ft1 (z) = (x(z) y(z) u v). For the decoupled case 7?.T is a 1x2 row matrix (x(z) u), or (y(z) v). The mathematical space (x(z), y(z), u, v), or its reduced version (x(z), u), is called the phase-space of the paraxial geometrical optics. An optical system, made of spherical and cylindrical lenses, and sections of freespaces, is represented by the real 4x4 ray transfer matrix S = {{A,B};{C,D}}. All lenses are considered thin, centered, at normal incidence to the beam axis, with no absorption and no aberrations. The 2x2 matrices A,..., D have elements with: no physical dimensions (A, D); dimension of length (B); dimension of reciprocal length (C). Such optical systems are called ABCD-type optical systems. The ray transfer property of the optical system allows one to specify the ray at the output of the system, ??„„,, from the ray at its input, Rim through the linear relation: K>ut = S Rjn. The matrix S is symplectic, meaning that S J ST = J, where J = {{0,1}; {-1,0}}, and 0 and I are the 2x2 zero and

61

62

identity matrix, respectively. For a symplectic matrix S, det S = 1 (det stands for determinant) [8]. The maximum number of independent elements of the matrix S is ten. Any real, symplectic matrix S as above, and with A,...., D submatrices with physical dimensions as above, can be constructed using only a finite number of thin lenses (or mirrors) and free spaces. A theoretical proof was given in [9], and a systematic procedure to synthesize any such matrix S was given in [4]. For the decoupled, lower dimensional case, the 2x2 symplectic matrix S has A,. . ., D scalars with det S = AD - BC = 1 . The beam is represented by sixteen second-order moments arranged in a real 4x4 beam matrix P [10]; they are defined as the averages (denoted < >): (z) = [ flffri Hw(x, y, u, v; z) dx dy du dv]/[ ffiJHw(x, y, u, v; z) dx dy du dv],

(1 )

where £, r\ are any of the x, y, u, v phase-space coordinates, and Hw(x, y, u, v; z) is the Wigner distribution function associated to the beam, interpreted as a power (or energy, for pulsed beams) density with respect to all four phase-space coordinates, at the transverse plane z = constant. The first-order moments, defining the beam axis, are zero, because the beam axis is chosen as the optical axis z. The integrals in (1) are extended formally from - oo to + oo, even for the angular variables u and v, by assuming that the Wigner function has no significant value outside the paraxial range. For beam measurements, the three spatial second-order moments are most important, because they are the only ones directly measurable. From (1), by integrating with respect to the angular variables u and v, they are obtained as (z) = [ I^)spatial H(x, y; z) dx dy] / [ p(x, y; z) dx dy]

(2)

where £ and r| is any of x, or y, and H(x, y; z) = jfHw(x, y, u, v; z) du dv is the beam irradiance distribution. The beam matrix P is constructed using (1) on the ray matrix, P = = { {<x2>,<xy>,<xu>,<xv>} ; {<xy>,,,} ; {<xu>,,,} ; {<xv>,,,}}={{W,M}; {MT,U}}. P is written in its 4x4 matrix form and also in its four 2x2 matrices form, respectively. It is a real, positive definite, symmetric matrix: P > 0; PT = P (or WT= W; UT = U; MT * M), with maximum ten independent elements. The 2x2 submatrices have the following physical dimensions: W-length squared, M-length, U-dimensionless. W contains information on the transverse beam shape and size, U - on the transverse far-field shape and size, and M - on the radii of curvature of the average wavefront of the beam, and on the orbital angular momentum carried by the beam. A physical interpretation in the gaussian-Schell-model for the beam including the coherence properties and the twisted phase parameter [1 1], is given in [6]. For the decoupled case, P is a 2x2 matrix, and its elements W, M, and U are scalars. Only quasi-monochromatic and linearly polarized beams are considered here. 1 .2. General issues related to optical systems and beams The following issues are important to consider for optical systems and beams:

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Characterization. It means the mathematical formalism used to describe the physical systems. Here we use real matrices to describe them. One drawback of the matrix P is that the same matrix P may correspond to several different physical beams, i.e., the physical beam -> beam matrix is a many-to-one correspondence. Symmetry. Definitions and terminology based on symmetry properties. For an optical system we distinguish between its structural symmetry - that of its lenses and of their relative orientation, and its overall symmetry - that of its matrix S, through the properties of its submatrices A,..., D. Similarly, for beams, which we look at in free space, we distinguish between three types of symmetry along z: that of the transverse irradiance pattern; that of the second-moments-defined spot (represented by the 2x2 submatrix W); and the symmetry of the whole beam matrix P, through its submatrices W, M, U. They all could be different. Denoting by K any of the A,..., D submatrices of S, or any of the W, M, U submatrices of P, the following definitions to designate stigmatic (ST), simple astigmatic (SA), and general astigmatic (GA) optical systems and beams are used [1,4-6]. A ST optical system or beam is that having all its 2x2 submatrices proportional to the identity 2x2 matrix: K = K I. There are maximum three independent elements for a ST matrix. An aligned SA (ASA) optical system or beam have all 2x2 submatrices K diagonal, i. e., K = Kj. There are maximum six independent elements for an ASA matrix. A rotated (by a) SA (RSA) system or beam is that one which can be transformed into an ASA one by rotating it back about the z axis with the angle (-a). Therefore, the RSA systems or beams have all their submatrices K symmetrical, and with their principal axes all rotated with respect to the laboratory axes x, y by the same angle a. We designate them Ks(a). There are maximum seven independent parameters, a mandatory, for a RSA matrix. A system or a beam with a matrix neither ST, ASA, nor RSA, is a GA one. A GA matrix can have maximum ten independent elements. The matrices of ST systems or beams have rotational symmetry, but not every rotationally symmetric matrix is a ST one. Examples are the GA matrix (for a # kn/2, k - integer) of the optical system representing the rotation Sa = {{Ra,0};{0,Ra}}, Ra = {{cos a, sin a};{- sin a, cos a}} [12], or of the system S = {{AR^BRJ; {CR^DRa}} with AD-BC = 1, called "4-parameter rotation" [4], having overall rotational symmetry (invariance). For beams a similar property holds. There are GA beams having in free space a symmetrical behavior of their secondmoment-defined spot identical to that of a ST, ASA, or RSA beam. They are called pseudo-ST (PST), pseudo-ASA (PASA), and pseudo-RSA (PRSA) beams, respectively, or in general pseudo-symmetrical (PS) beams [5], or even non-rotating GA (NRGA) beams. The submatrices M of the PS beams have an additional antisymmetrical term mJ in comparison to the submatrices M of the corresponding symmetrical beams (ST, ASA, or RSA); the submatrices W and U are the same for both types of beams. Our definition for ST, ASA, RSA, and GA beams is based on the properties of the beam matrix P, rather than on the symmetry properties of the beam spot (irradiance-defined or secondmoments-defined) in free space propagation, as used in the classical paper [13], Figure 1 [14] shows the irradiance patterns of an expanded TEM0oHe-Ne laser beam at progressively increasing z = constant transverse planes, after a passage through (from left to right): a focusing spherical lens; a focusing cylindrical lens with power on x axis; two orthogonally positioned, focusing cylindrical lenses, rotated « 10° with respect to x and y

64

axes, respectively, and separated by a free space; two focusing cylindrical lenses, the first one with power on the x axis, the second one rotated at 45° with respect to the x axis and separated from the first one by a free space. These beams show ST, ASA, RSA, and GA

Figure 1. Experimental ST, ASA, RSA, and GA beams generated from a ST TEM00He-Ne laser (see text).

symmetries, the same for all three definitions of the symmetries: of the irradiance spot in free space, of the second-moment-defined spot in free space, and of the beam matrix. Classification [2]. For optical systems, the classifications is based on their S matrix and was given above, by defining the ST, ASA, RSA, and GA types of optical systems. For beams, one classification is based on the symmetry of the beam matrix P, and another one on the beam matrix invariants at propagation (see Section 4.2). Identification [2]. To identify an optical system or a beam means to establish that its matrix belongs to a given class defined by the classification procedure. Identifying optical systems. The identification is done directly by looking at their matrix S. A ST or an ASA systems is identified simply from their definition. We proved that a symplectic matrix S is RSA if and only if it has all four submatrices symmetrical [2, 4-6]. Therefore, a system is GA if at least one of its submatrices is non-symmetrical. Identifying beams. Identifying ST and ASA beams is done directly by looking at their matrix P. A beam P which is not ST or ASA has its W and U submatrices symmetrical, but with different principal axes, Ws(aw), Us(au), aw * a0; its submatrix M = Ms (aM) + mJ is generally not symmetrical. A beam is RSA if and only if all its three submatrices are symmetrical and have the same rotation angle, i. e., MT = M (or m = 0), and a\y = <XM = av. A beam is GA only if it is not ST or ASA, and not all the conditions to be RSA are satisfied. The identification of a physical beam is done after measuring its beam matrix elements [2]. Synthesis of optical systems. The synthesis means to construct, from simpler matrices representing simpler optical systems (or building blocks), the symplectic matrix S of a desired physical optical system. The feasibility to synthesize any optical system using lenses and free spaces was analyzed theoretically in [9]. A constructive procedure to obtain any 2x2 and 4x4 matrix S representing a physical optical system was given in [4]. Beam transformations. A beam transformation means the change of the beam matrix P by passing the beam through an optical system S. A certain beam can by transformed by all optical systems S into a specific set of equivalent beams, and not in any other beam. The set of equivalent beams is defined by the specific values of two independent beam invariants at the transformation. The optical systems S do not change the value of the beam invariants and therefore cannot change the set of equivalent beams, or cannot transform beams from one set of equivalent beams into beams of other, different set of equivalent beams. The main transformation are: beam generation; beam decoupling and

65

diagonalization; beam identification; and beam measurement. The beam generation means to obtain any given beam matrix of a set of equivalent beams from the simplest beam of that set (usually a beam with a diagonal matrix). The beam decoupling means to obtain from a given GA beam the equivalent beam having the ASA or (when possible) the ST symmetry [4, 14-16]. The diagonalization means to bring the decoupled, ASA or ST beam matrix into the diagonal form, with submatrix M = 0, meaning to transform the beam such as to have a common waist, and expressing the beam matrix in its common waist plane. Beam identification means to assess a given beam to a certain class, subclass, or family of beams, based on measurements. The measurement transform means to determine all beam matrix elements by using appropriate ABCD-type systems with appropriate detector systems and software [3,14,17,18]. Beam invariants and equivalent beams. The beam matrix P is transformed by an optical system S, from input to output, as: Pout = S Pjn ST. This transformation law has two invariants, specific for the beams Pln and Pout. They are: A = del P; T = tr [(PJ)2] (tr is the trace of the matrix). We call the beams Pin and Pout related by all possible matrices S, equivalent beams [2]. Each pair of the independent invariants (A, T), or any other independent pair derived from these, defines a set of equivalent beams (or matrices) P, in the sense that all these beams can be transformed into each other by using all existing ABCD-type optical systems S. Methods for beam characterization and measurement are described in the approved ISO 11146 standard (for ST and SA beams) [19] and in its current revision which includes also GA beams [20].

2.

The elementary level: the idealized gaussian beam (IGB)

In this section we discuss the spatial parameters of the simplest beam, which we call the idealized gaussian beam (IGB). This beam is rotationally symmetric (similar to the patterns of Fig. 1, first picture), has a gaussian irradiance profile (Eq. (3)), and a perfectly coherent, quadratic phase wavefront (the coherence means spatial, or transverse coherence). It is an idealized beam, because physical beams approaching best these characteristics, as, for example, the lowest order transverse mode (TEMoo) beams of low power, stable cavity lasers (as a visible He-Ne) depart slightly from these characteristics. The theory of the IGB is found in laser textbooks [21]. The irradiance in the point x, y, of a transverse section z = constant, is given by H(x,y;z) = H(0,0;z)exp[-2(x2 + y2)/w2(z)] = H(0,0;z)exp[-2x2/w2(z)]exp[-2y2/w2(z)]

(3)

where by definition w(z) is the beam radius (half diameter) at the plane z = constant, and H(0,0;z) is the peak, on axis irradiance at the same plane. From (3) results w(z) as the radial distance from axis corresponding to the irradiance level H(x,y;z) = (1/e )H(0,0;z) = 0.135 H(0,0;z). The lateral beam boundary is conventionally defined by 2w(z) = D(z), the beam diameter. w(z) represents a hyperbola which minimum WQ = W(ZQ) is called the waist radius. The positive branch of the hyperbola has the equation:

66

w(z) = w 0 [l+(z-z 0 ) 2 /z R 2 ] 1/2

(4)

and it is symmetric with respect to the waist position. Other constants in (4) are: the coordinate (positive or negative) z0, called the waist location; and ZR, the Rayleigh range (or Rayleigh length), the longitudinal distance (positive number) from the waist plane to the transversal planes (on any side of the waist) where the beam radius is 2 I/2 larger than the waist radius. For the IGB (and only for this type of beam) the Rayleigh range is determined by the waist radius: ZR = 7iw02/X, with X - the beam wavelength. Therefore the IGB with a given wavelength is completely characterized by two spatial parameters, w0 and z0, or ZR and z0. Any ST real beam needs an additional third parameter, because ZR no more depends only on w0 for a real beam. The longitudinal range |/-z0| < ZR, symmetrically positioned on both sides of the waist having the total length of 2zR = b, is called the near-field of the beam; its length is called the confocal length and it is a measure of the collimation length of a beam. The far-field is the longitudinal range, from a plane |z| > ZFF| far away from waist, ZFF - z0| » ZR, from where the beam waist size can be neglected (is like a point) with respect to the local beam size, w0 « W(ZFF). The farfield zone starts, by convention, for (ZFF - z0| > (3-10)zR. Between the near-field and the far-field there is an intermediate-field. The beam divergence angle characterizes the farfield; it is defined either as the half-angle 6h, or the full-angle 9 = 26h, by the limit (for z -> °o): 6h = lim [w(z)/z] = WO/ZR. The beam parameter product, BPP, is defined as: BPP = w00h = TJn ~ 0.318 A,, and it is a beam invariant at beam propagation through ST optical systems. For the IGB of a certain wavelength, this is the minimum possible value of the BPP; for any other real beam with the same wavelength, its BPP (provided the waist size and the divergence angle are defined using second-order moments) is larger than for the IGB. Care should be taken when specifying the spatial beam parameters by using either full size quantities (beam diameter and full angle divergences) or half size quantities (beam radius and half-angle divergences). ISO 11146 defines the beam size by the beam diameter and the beam divergence by the full-angle divergence; however, the BPP uses the half-size quantities (as defined above). Another quantity of interest for the IGB is the radius of curvature of the wavefront at a certain z, R(z). It varies along z as: R(z) = (z-z0)[l + zR2/(z-z0)2]

(5)

From (5) it follows that: R > 0 for the diverging side of the beam, z > z0; R < 0 for the converging side of the beam, z < z0; the wavefront is flat in the waist, R(z0) = °o; in the far-field |R(zFF)| » ZFF; the minimum radius of curvature (in absolute value) is obtained in the two planes located a distance ZR from waist, |R(z0 ± ZR)| = 2zR = minimum. The ST IGB is characterized by a 2x2 beam matrix with only one spatial second-order moment, <x2>(z) = (z), one angular moment, = , and one mixed moment, <xu>(z) = (z). By calculating these moments, the following results are obtained: <x2>(z) = (z) = W(z) = (1/4) w2(z) = (1/16) D2(z) <xu>(z) = (z) = M(z) = (1/4) (z - z0) 0h2 = (1/16) (z - z0) 02

(6)

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= = u = (i/4) eh2 = (i/i6) e2 The three relations (6) link the spatial parameters of the IGB, w(z), z0, 6h, to the secondorder moments of its beam matrix. R(z) is also related to M(z) through the relation W(z)/M(z) = R(z); a beam with M = 0 (a diagonal P matrix) is represented in its waist plane. Note also that BPP = 4 (det P2x2)1/2 = Mil = invariant (the explicit 2x2 subscript is used because the BPP is different if P is a 4x4 matrix).

3. The current level: decoupled beams (2x2 matrix case) All optical systems and beams can be described by 4x4 matrices. However, because of symmetry, a ST system or beam can be described each by one 2x2 matrix. The ASA systems and beams need each two 2x2 matrices. A RSA system or beam requires each a 4x4 matrix, but this can be changed into two 2x2 matrices by an appropriate rotation of the reference axes such as the new matrices to represent an ASA system or beam. GA systems or beams require each a 4x4 matrix. This section deals with ST and ASA systems and beams, therefore with description using one or two 2x2 matrices. 1. 1. Real ST beams The real ST beams have a round spot everywhere, but differ from the IGB in one or more of the following ways: having a non-gaussian irradiance profile; being coherent, but having other-than-quadratic wavefront (aberrations); being partially coherent. In practice [19,20] a spot is considered round if its ellipticity (the ratio between the large and the small ellipse axis) is < 1.15. The spatial parameters for the real ST beams are defined directly from the beam matrix elements which are determined through measurements and calculations according to Eq. (1). From the three independent beam matrix elements, W(z), M(z), and U, the following spatial parameters are defined, similar to (6): D(z) = 4 (W(z))1/2; w(z) = 2 Do = 4(Wo) 1/2 ;wo=2(W 0 ) 1/2 0 = 4(U)1/2; 0h = 2(U)1/2

z-z 0 = M(z)/U R(z) = W(z)/M(z) BPP = 4(det PST2x2)12 = 4(W0U)12 = 4[W(z)U - M2(z)]12 = (1/4)D09 = w0eh

(7)

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M2 = (det PST2x2)1/2/(det PIGB)1/2 = (n/XXBPP) = (7i/^)w0eh = 7iD09/(4A) > 1 The parameters defined in (7) are: the beam diameter D(z) and the beam radius w(z); the waist diameter D0 and the waist radius wa; the full-angle divergence 9 and the half-angle divergence Q^\ the Rayleigh range ZR; the waist position coordinate z0; the radius of curvature of the wavefront, R(z); the beam parameter product, BPP, where PsT2x2is the 2x2 beam matrix of the real ST beam, PST2x2 (z)= {{W(z), M(z)},{M(z), U}}; the beam propagation ratio M2 [19,20]; the IGB has the same wavelength as the real ST beam. The parameter M2, called also the "beam quality parameter" or the "beam propagation factor", represents the third independent parameter necessary to characterize a real ST beam, besides the other two discussed at the IGB. For the IGB M2 = 1, and for any real ST beam M2 > 1 ; the closer to 1 is M2 for a real beam, the closer to the IGB the real beam is. With the above definitions of the spatial parameters for the real ST beam, the beam radius w(z) of the real ST beam transforms as the beam radius of an IGB, Eq. (4). Because det P$T2x2 is invariant at beam transformation through ST ABCD-type optical systems, M2 is also invariant at these transformations. Non-ABCD-type, aberrated optics increase the M2 value of the beam. Usefulness of M2. M2 allows one to compare in several ways a ST real beam with parameters w0, 0h, M2, with an IGB having the same A, and with parameters WOIGB> QMGB. because the relation w09h = M2WoiGB6hiGB holds: (a) in the near-field, assuming w0 = WQIGB, and resulting 9h = M2Q^iGB (whence also the name "times diffraction limit" for the M2 parameter of the real beam); (b) in the far-field, assuming 9h = SKIGB, (and correspondingly, the real beam can be focused through a certain lens to the same small spot as an IGB), and resulting w0 = A/^WQIGB, showing that the necessary focusing lens (if used in near-field), should have a linear aperture M" times larger for the real beam than for an IGB; (c) when w0 = A/WOIGB and 9), = MB^IGB, leading to both beams having the same ZR and the same R(z); in this case the IGB is called the "embedded gaussian beam" for the ST real beam, and it represents the lowest order mode of the laser resonator supporting also the real beam. This latter case is used to design the optical path for real beams, first by designing the optics as for the IGB, and then, to accommodate the real beams, by multiplying M times the linear apertures of all lenses in the beam path. 3.2. Real ASA beams To characterize the spatial properties of the ASA real beams similar quantities to those defined for real ST beams in Eqs. (7) are defined, by working independently in the x and y axes, i. e., by doubling the number of quantities. Also, to each of the Eqs. (4) and (5) for a real ST beam there will correspond two independent "propagation" equations (for x and for y) for a real ASA beam. The second-moments-defined spot for an ASA beam is generally an ellipse with its principal axes oriented along x and y axes (analog to the second picture of Fig. 1). Therefore, the names "beam diameter" and "beam radius" used for ST beams are replaced by the names "beam widths (in x and y)" and "beam halfwidths (in x and y)" for ASA beams, respectively. In the ASA case the optics may (or may not) contain some cylindrical lenses aligned with x and y axes, and the beams have

69

in general different and independent properties in x and y. Typical examples are beams from some diode lasers or from excimer lasers. The existence of independent, x, y sets of beam parameters for ASA beams gives rise to some new situations, discussed below. The qualitative property of intrinsic stigmatism (IS) or intrinsic astigmatism (IA). Some ASA beams have Mx2 * My2, and some other ASA beams have M2 - My2. ST beams have Mx2 = My = M2. Because Mxy2 are independent beam invariants, an ASA beam having Mx2 •£ My cannot be transformed into (or cannot be obtained from) a ST beam by using ABCD-type optics. Such a beam has IA. On the other hand, an ASA beam having Mx = My2 can be transformed into (or can be generated from) a ST beam by using ideal optics; it has IS. Beams with IA cannot be transformed into beams with IS by using M2preserving ABCD-type optics [1-6]. Classical astigmatism. This is the qualitative property of an ASA beam having the two waists located at different z, z0x ^ z0y. The distance |z0x - z0y| is called the astigmatic waist separation or the astigmatic difference. It can be made zero by using ABCD-optics. For the ASA beams with IS, somewhere between the two separated waists there is one round spot named circle of least confusion (see Fig. 1, second picture). An ASA beam with IA may, or may not have a circle of least confusion. Other ASA beams asymmetries. Other possible asymmetries for ASA beams, irrespective of their IS or IA property, are: the waist asymmetry, or different waist widths, DO* ^ Doy; and the divergence asymmetry, or different divergence angles, 9X ^ 0y. An ASA beam with IA has at least one type of asymmetry, otherwise it would be an IS beam. The near-field, intermediate-field, and far-field range for ASA beams. Because of different properties in x and y, it is possible that certain transverse locations z to belong to the near-field in x, and to the intermediate- or the far-field in y. A certain range of the beam (for example, the near-field) has a well defined meaning only when there are locations z corresponding to the same range (near-field) for both, x and y.

4. The advanced level: beams with coupling (4x4 matrix case) 4.1. Beams described by 4x4 matrices. Identifying pseudo-symmetrical beams There are beams characterized by a 4x4 matrix, either because they are SA but rotated with respect to the x, y frame, or because they are GA (Fig. 1, the third and the fourth picture). A RSA beam can still be characterized with 2x2 matrices, provided its angle a is known and the coordinate frame is no more x, y, but a frame rotated with a; in this case the beam is actually an ASA beam in the new coordinate frame. Some GA beams have a rotating irradiance pattern in free space (Fig. 1, the fourth picture) named rotating GA (RGA), and some others can have a non-rotating, symmetrical irradiance pattern in free space, the NRGA beams, or the PS beams. The beam matrix symmetry of the PS beams was discussed in Introduction. By using the current measuring procedures employing only rotationally-symmetric optics [19], the PS beams cannot be distinguished from the true ST, ASA, or RSA beams [5].

70

Figure 2 illustrates the impossibility to distinguish in free space between ST and PST beams, and also shows the distinction between the symmetry of the irradiance-defined beam spot and the second-order moments-defined beam spot for the same physical beam. In the upper side of the drawing, in perspective, a rotationally-symmetric beam in free space has the spot sizes defined using second-moments. It corresponds to three physically different beams. The coherent irradiance patterns of the three different physical beams are illustrated in the first three rows under the perspective drawing; the fourth row shows the corresponding second-moments-defined spots, which are the same for all three different beams. The representation is done at different z planes: the waist (minimum size) plane, and the planes of the first four multiples of the Rayleigh range ZR. The three different beams, along the first three rows, are beams made of: a single Hermite-Gauss mode of order 11, rotated with 30° with respect to the x, y axes (HGn;7t/6); a single Laguerre-Gauss mode LG02; and a single Laguerre-Gauss mode LGi° [1,5]. By comparing the first row (of the HG11;7t/6 beam) with the circles of the second-moments sizes (the fourth row), we see the distinction between the symmetry in free space of the irradiance pattern of the HGn;7t/6 beam (looking like a RSA beam rotated by 30°), and the symmetry in free space of the second-order moments-based beam size (the circles, having rotational symmetry), corresponding to a ST beam. The first row and the third row correspond to ST beams in second-order moments definition. The second row, having also rotational symmetry both for the irradiance pattern, and for the second-momentsdefined spots, looks ST-like, but actually its P matrix corresponds to a PST beam [5], or twisted-phase beam [11]. Such a beam can be distinguished from a ST beam only by a cylindrical lens inserted in the beam path. After a cylindrical lens followed by a free space, a ST beam becomes an ASA beam, while a PST beam becomes a RGA beam. This identification procedure of the PST beam is illustrated in Fig. 3. Here, a converging

O O

O

o

o OO

Figure 2. Three different beams looking the same in free space by using second-moments-defined spots. The first three rows show the irradiance patterns, and the last row the spot sizes of the three beams (see text).

Figure 3. Comparison of second-momentdefined beam spots for a ST beam (left) and a PST beam (right) before and after a cylindrical lens followed by a free space (see text).

71

cylindrical lens is inserted at the plane denoted IN, in the beam path of a ST beam (left) and PST beam (right), and the second-moment-defined spots of these beams are depicted at several distances (multiples -1, 0, 1, 2, 3 of the focal length f) in the free space before and after the lens. Both beams look identical in the free space before the lens (distances -f and 0), but they look different after the lens. Example of measurements of real GA beams are given in [17,18] and recommended in the new draft of ISO 1 1 146 standard [20]. 4.2. Generalized beam propagation invariants. Beam classification Generalized invariants. Based on combinations of the two independent general invariants A and T mentioned in Introduction, several pairs of independent invariants, valid for any beams, were defined [1-6] and extensively discussed in [2]. Three normalized invariants, only two of them being independent, are defined and discussed. The normalization is done by comparison to the same invariants defined for the IGB with the same wavelength as the real beam. One invariant is defined by combining the two invariants above (note that T is always negative and its absolute value is used in the definition; also, k =

a = k2(|T | - 4 A1/2) > 0

(8)

It is the quantitative measure of the intrinsic astigmatism of a beam, and we call it the normalized intrinsic astigmatism. For beams with IS a = 0, while for beams with IA, a > 0. The other two invariants are defined only with respect to A, as follows: Me{f = 4k2A1/z > 1 aM=(l/2)(Meff4-\?>a

(9) (10)

We call them the effective beam propagation ratio, and the normalized maximum intrinsic astigmatism, respectively. The invariant au is an upper limit for a: 0 1; fllA,ASA = (1/2)(M/ - M/)2 > 0; M/-l) 2 >a, A ,, ASA

(12)

M./ST = (M2)2 > 1; aST = 0; a MS T= (1/2) [(M2)2- I]2 > 0

(13)

Mtffoo = 1; floo = 0; aMoo = 0

(14)

M,ff (and through it also a^) is a measure of the overall beam localization (or spread) with respect to all four phase-space coordinates-angles, or with respect to the near-field and the far-field simultaneously; the closer to 1 has Meg its value, the closer to the "best localized" beam (for which Me/ = 1) is the current beam. Note that the "best localized" beam can have any symmetry, including GA; only when it is ST it becomes the IGB or the TEMoo beam. The invariant a is a measure of the intrinsic astigmatism of a beam, or the departure of the current beam from a ST beam chosen to have the same overall beam spread (the same Mef). By using Eqs. (12) or (13), from the value of classical invariants (M*, My2) one can determine the values of any pair of the generalized invariants (a, Mey) or (a, aM), and vice-versa. The pair (a, Meff4) is used in the new ISO 11146 draft standard to characterize the GA beams [20]. Intrinsic beam classification. The pair (a, aM) is well suited for the intrinsic beam classification. There are two disjoint classes of beams, intrinsic stigmatic (IS), a = 0, and intrinsic astigmatic (IA), a > 0, and each class has two disjoint families. The families are numbered (I) - (IV) using the four possible cases of the pair (a, aM): (I) -> (a = 0; aM = 0); (II) •* (a = 0; aM > 0); (III) -> (a > 0; aM > 0; a < aM); (IV) -> (a > 0; aM > 0; a = au\ The same four families can be rewritten by using the relation between the M^2 invariants: (I) -» Mx2 = My2 = 1; (II) -> M2= My2 > 1; (IE) -» M,,/ > My2 > 1; (IV) -» Mx2 > Myjc2 = 1. In this latter case, note that the family (I) represents all the beams equivalent to a TEMoo beam for a certain wavelength. We may call this family the 00type family of beams (where 00 stands either for the TEMoo beam, or for the (0, 0) values of the pair (a, aM)). In Fig. 1 all the experimentally generated beams belong to this family because the starting ST beam had M2 ~ 1. Another interesting case is the family (IV); all beams of this family, when transformed to most symmetrical ones, ASA, have "the best quality" (i. e., a minimum value Af*,/ =1) along only one of the transverse axis, x or y. Figure 4 represents all four families of beams in the (a, aM) parameter plane. Each dot on the diagram represents a set of equivalent beams. Only the first quadrant has meaning, including the horizontal and the vertical axes together with the origin, a > 0; aM > 0. The points on the bisector of the first quadrant have a = «M- Family (I) of beams is the origin; it contains only one set of equivalent beams. Family (II) is the vertical axis, excluding the origin; it contains a simple infinity set of equivalent beams, each set for a specific value of aM. Family (III) is the area in the second octant comprised between the vertical axis and the bisector, excluding the vertical axis, the bisector, and the origin; it contains a double infinity set of equivalent beams, each set determined by the values of the pair (a, aM). Family (IV) is the bisector of the first quadrant, excluding the origin; it contains a simple infinity set of equivalent beams, each set defined by the value of a = aM.

73

(ID

f (I) Figure 4. The (a, au) plane and the four types of disjoint families of beams

Table 1, below, provides a beam classification based on intrinsic and also on geometrical properties (i. e., the matrix symmetry) of the beams. For comparison, the old classification, based on the second-moment-defined spot symmetry in free space propagation is also included.

Table 1. Combined (intrinsic and geometrical) beam classification

Class

Family

(intrinsic)

(intrinsic)

Beam matrix symmetry (geometrical)

Type I

ST

IS

(a = 0;«M = 0)

SA

a=0

Type II

(old classification) Rotational

ASA

Aligned, orthogonal

RSA

Rotated, orthogonal

(a = 0; aM > 0)

GA

RGA

Non-orthogonal

Type III

SA

ASA

Aligned, orthogonal

GA

NRGA

Rotated, orthogonal

RSA

(a>0;aM>0;a0

Spot (2-M) symmetry

PST PSA

Type IV (a = aM > 0; aM > 0)

RGA

Rotational PASA

Aligned, orthogonal

PRSA

Rotated, orthogonal Non-orthogonal

74

4.3. Formal definition of a beam There is no formal definition of the beam concept. A paraxial light distribution and the condition that the smallest beam size is » X is most often assumed but not formalized. The goal here is to formalize the beam definition such as to be valid for light distributions of any symmetry, ST to GA (defined by a matrix of second-order moments). The formal definition contains three steps. First, criteria for a ST distribution of light to be a ST beam are formulated [7]. Second, it is shown that it is possible to construct in a unique way, from a distribution of any symmetry (ST to GA), a ST distribution in the neighborhood of the original one; we call this ST distribution the "associated average ST distribution", in short AD. The final step is to search whether the AD fulfills the criteria to be a beam, and to extend, by definition, the property of the AD of being (or not) a beam, to the original distribution. In the first step the two conditions (paraxial, and size » A.) are formalized for a ST distribution; it is assumed that the matrix of the second-order moments is measured by using existing techniques [19]. The paraxial criterion for the ST distribution is defined by imposing that its half-angle divergence, 0h, to be smaller than a certain maximum conventional paraxial angle 0M, to be determined, 0h < 0M. Because the distribution is ST, it has a waist radius w0 and a beam propagation ratio M2. The paraxial condition can be rewritten as M2 < (wo/A.)(7i0M), and by choosing 0M = (!/TI) rad « 0.32 rad «18.2°, it is written simply as M2 < (wo/X). The second condition can be stated conventionally as (wo/A.) > 10. We can relax this condition by saying that distributions with 1 < (W(A) < 10 can be considered still almost beams, or quasi-beams, provided the paraxial condition still holds. The distributions which do not obey both conditions (paraxial and w0 » X) are not considered beams (they are general distributions, or nonbeams).

10'1

1

10

102

103

w0A.

Figure 5. Areas representing beams (B), quasi-beams (Q-B) and general distributions, or non-beams (N-B). Each dot in the diagram corresponds to a ST light distribution.

75

Fig. 5 summarizes the two criteria for the first step. Each dot in the diagram corresponds to a certain ST distribution determined by A,, w0, and M2. The oblique line M2 = Wo/A, and the vertical line Wo/A, = 10 divide the plane in three areas representing beams (B), quasibeams (Q-B) and non-beams (N-B). The second step starts from the second-moments matrix of a given distribution of any symmetry (supposedly a 10 parameter GA distribution), obtained by measurements [3, 14, 17, 18, 20]; therefore the submatrices W, M, and U are known. By definition, the associated average ST distribution AD is that one having the three beam matrix elements (they are scalars because the AD distribution is ST) WAD. MAD, and UAD as the arithmetic mean of the traces of the submatrices W, M, U of the original distribution: WAD = (l/2)tr (W); MAD = (l/2)tr (M); UAD = (l/2)tr (U). This ST AD is located in the neighborhood of the original distribution, and it is unique, in the sense that every GA distribution has one, and only one ST AD. However, the same ST AD may correspond to different GA distributions, because of the rotational invariance of the trace. The third step is to assess whether the ST AD is a beam, by calculating its beam parameters WQAD and M2AD and by using the criteria summarized in Fig. 5. If the ST AD is a beam (or Q-B, or N-B), then, by definition, the original distribution is also a beam (or Q-B, or N-B).

5. Conclusion The main results in characterizing the beams and their propagation through optical systems were presented, gradually increasing the level of complexity. A relatively extended introduction defines the main concepts, formulates the objectives of the work, and give the main results in understanding the optical systems used to transform the beams, and also some general results for beams. The beam characterization and propagation is reviewed starting from the simplest beam (Section 2), advancing through the current level of interest (Section 3), and progressing to the advanced level (Section 4). Generalized beam invariants, in-depth beam classification, and a formal definition of the beam concept are the relatively new and compact results presented here.

Acknowledgments I thank the Organizing Committee, Prof. M. Santarsiero, Universita Roma Tre, and Prof. S. Martellucci, University "Tor Vergata", Rome, Italy, for the invitation and support to attend and contribute to the ISQE-35. Many thanks also to Prof. A. E. Siegman, Stanford University, USA, Prof. H. Weber, Technical University, Berlin, Germany, and Prof. J. Serna, Complutense University, Madrid, Spain, for their permanent active interest in the subject and many interesting discussions.

References 1.

G. Nemes, J. Serna, in Diode Pumped Solid State (DPSS) Lasers, ed. M. W. Dowley, TOPS 17, 200 (OSA, Washington, DC, 1998).

76

2.

G. Nemes, SPIE Proc. 4932, 624 (2003).

3.

G. Nemes, A. E. Siegman, J. Opt. Soc. Am. A 11,2257 (1994).

4.

G. Nemes, in Laser Beam Characterization, eds. H. Weber, N. Reng, J. Ludtke, and P. M. Mejias, (Festkorper-Laser-Institut Berlin GmbH, Strasse des 17. Juni 135, D-10623, Berlin, Germany, 1994, pp. 93-104).

5.

G. Nemes, J. Serna, in Laser Beam and Optics Characterization 4, eds. A. Giesen and M. Morin (EFSW, Stuttgart, Germany, 1998, pp. 29-49).

6.

G. Nemes, J. Serna, in Laser Beam and Optics Characterization 4, eds. A. Giesen and M. Morin, (IFSW, Stuttgart, Germany, 1998, pp. 92-105).

7.

Maria Branescu, G. Nemes, in Proc. 7-th International Symposium on Recent Advances in Microwave Technology, eds. C. Camacho Penalosa and B. S. Rawat, (CEDMA, Malaga, Spain, 1999, pp. 344-347).

8.

M. Kauderer, Symplectic Matrices. First Order Systems and Special Relativity (World Scientific, Singapore, 1994, pp. 10).

9.

E. C. G. Sudarshan, N. Mukunda, and R. Simon, Optica Acta 32, 855 (1985).

10.

M. J. Bastiaans, J. Opt. Soc. Am. 69, 1710 (1979).

11.

R. Simon, N. Mukunda, J. Opt. Soc. Am. A 10, 95 (1993).

12.

G. Nemes, A. G. Kostenbauder, in Laser Beam Characterization, eds. P. M. Mejias, H. Weber, R. Martinez-Herrero and A. Gonzales-Urena, (SEDO, Madrid, 1993, pp. 99-109).

13.

J. A. Arnaud, H. Kogelnik, Appl. Opt. 8, 1687 (1969).

14.

G. Nemes, in Laser Beam Characterization, eds. P. M. Mejias, H. Weber, R. Martinez-Herrero, and A. Gonzales-Urena, (SEDO, Madrid, 1993, pp. 325-358).

15.

J. Serna, G. Nemes, Opt. Lett. 18, 1774 (1993).

16.

Yu. A. Anan'ev, A. Ya. Bekshaev, Optics and Spectroscopy 76, 558 (1994).

17.

B. Eppich, C. Gao, and H. Weber, Opt. Laser Technology 30, 337 (1998).

18.

J. Serna, F. Encinas-Sanz, G. Nemes, J. Opt. Soc. Am. A, 18, 1726 (2001).

19.

ISO 11146:1999 - Optics and optical instruments - Lasers and laser-related equipment - Test methods for laser beam parameters - Beam widths, divergence angle, and beam propagation factor.

20.

Documents ISO/TC 172/SC 9/WG 1: N 277, ISO/DIS 11146-1, Lasers and laserrelated equipment - Test methods for laser beam widths, divergence angles and beam propagation ratios -Part 1: Stigmatic and simple astigmatic beams; N 279, ISO/DIS 11146-2, Part 2: General astigmatic beams; N 274, Revised manuscript for ISO/TR 11146-3, Part 3: Intrinsic and geometrical laser beam classification, propagation, and details of test methods (ISO, Geneva, 2003).

21.

E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986, Ch. 17).

BEAMS IN RESONATORS MARIA RITA PERRONE Dipartimento di Fisica, Universita' di Lecce Via per Arnesano - 73100 Lecce (Italy) E-mail: [email protected]

Abstract The evolution of the main parameters (spot size, radius of curvature, far-field divergence and/or beam quality factor) of laser radiation pulses propagating in stable and unstable cavities with variable reflectivity mirrors is investigated both theoretically and experimentally as a function of the number of round trips. It is shown that a theoretical analysis, formerly developed to study the growth of coherence of a Schell-Gauss model beam propagating through a periodic sequence of Gaussian apertures, provides a satisfactory description of the laser radiation temporal evolution. The study of transient states is very important in practice because high-gain, short-pulse lasers as excimers or copper-vapor lasers, generate pulses making only a few transits inside the resonator. In this situation, beam parameters seldom reach a steady state hence, an analysis yielding the beam parameters at each round-trip is needed for a better characterization of the laser beam quality and can be of great interest in many application fields.

The experimental techniques used to analyze the temporal evolution of the main parameters of laser radiation pulses few ten of nanoseconds long are at first presented. Then, the propagation of laser radiation pulses in plane-parallel cavities with a variable reflectivity mirror is analyzed. Mirrors with Gaussian, super-Gaussian and step-reflectivity profile are tested in order to investigate the effects of the mirror reflectivity profile on the temporal evolution of the laser radiation parameters. Gaussian reflectivity mirrors of different spot size E are also considered to investigate the effect of E on the radiation beam-quality-factor evolution. An experimental study of coherence evolution versus the Fresnel number of Gaussian cavities is also presented. To this end, plane parallel cavities of different length and with a Gaussian reflectivity mirror have been applied to a XeCl excimer laser (A=308 nm) and the results on the resonator-length effects on the coherence growth of the laser radiation are presented. It is also shown that in Gaussian cavities of equal length and effective Fresnel number, the properties of the intracavity radiation are dependent on the Gaussian mirror distance from the active medium, since the onset of laser action. Finally, experimental and numerical results on the beam quality and coherence growth of the laser radiation in unstable cavities equipped with a variable reflectivity mirror as output coupler are presented. Variable reflectivity unstable cavities are generally applied to pulsed laser to get a faster coherence growth of the oscillating radiation.

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ON THE EFFECTS OF DIFFRACTION IN THE PROPAGATION OF ULTRASHORT, FEMTOSECOND, OPTICAL PULSES MIGUEL A. PORRAS

Departamento de Fisica Aplicada. Universidad Politecnica de Madrid. Rios Rosas 21. E-28003 Madrid. Spain.

1 Introduction The problem of diffraction of ultrashort, femtosecond light pulses, is a subject of increasing interest '"8 from the advent of femtoseond laser devices in the nineties. For femtosecond pulses, having a wide spectrum of frequencies, the well-known and well-understood laws of diffraction for monochromatic or quasimonochromatic light have been shown to fail in predicting the spatiotemporal form of the diffracted pulse, and should be therefore modified. Generally speaking, there are two different but cmplementary points of view when studying diffraction of ultrashort pulses. The first one, often referred to as transient diffraction, analyzes the temporal dynamics of the diffraction pattern, that is, the effects of pulse temporal form on transversal shape. On the opposite, the second one is centered on the study of transversal effects on pulse temporal form. The second approach is the most widely used up to now. Transversal effects are included among the different phenomena, as nonlinear interactions, dispersive and absorption effects that make a pulse to change during propagation. There are numerous studies on diffraction-induced transformations of femtosecond pulses, most of them limited to the case of Gaussian transversal profile. '"3 They revealed however interesting effects such as pulse front curvature (time-delay), pulse broadening, spectral changes leading to red or blue shift of optical oscillations, and the time-derivative effect at the far field, that will be explained in the next sections. The main difficulty with the study of diffraction of femtosecond light pulses is the lack of closed-form, general enoguh, analytical expressions for the propagated field. Seemingly simple cases as the propagation of a pulse with a Gaussian profile become intractable in the ultrashort case. For this reason, most of diffraction-induced pulse transformations have been described qualitatively, from numerical simulations, ' from particular on-axis 3 or asymptotic expressions at the field, ' or from inspection of the corresponding spectra.2 It is the purpose of this lecture to show that rather elemental, well-known concepts on the propagation of three-dimensional wave packets 10 such as the pulse front surface, its group velocity, and group velocity dispersion, can still be used to characterize accurately and classify diffraction-induced ultrashort pulse transformations, and to write down simple accurate analytical expressions for them and for the propagated field of general (nonGaussian) pulsed beam sources. We find it is possilbe to express diffraction-induced transformation of pulses in the familiar language of one-dimensional pulse propagation in media with phase and gain dispersion. " To illustrate the theory, we apply these basic concepts to obtain a detailed characterization of the field of pulsed Gaussian beams and pulsed Bessel beams.

79

80

In the following, a function of time f(t) will be represented by its complex analytic signal F(t), 12 whose real part gives the real field f(t). The real and imaginary parts of F(t) are Hilbert transform pairs. lo If the frequency spectrum offft) is f(coi) , i.e.,

1 °° /(O = — f/(a>) exp(-/fi*)

2^_i

(1)

then the frequency spectrum of the analytic signal is F(cai) = 2&((0)f(a>) , where i9((u) is the Heaviside step function. The analytic signal

1 °° F(t) = — \f(a>)exp(-icot)dct)

(2)

thus has no negative frequencies and the positive ones are doubled. In this paper, the symbols A/ and Aft> will mean full width at half maximum (FWHM) of \F(t)\2 and | /(«) |2 , respectively.

2

Theory

Preliminaries We consider a spatially and temporally localized optical disturbance E(rtt), r, =(x,y), of frequency spectrum Em(r,) at a certain plane z = 0 of a suitable Cartesian coordinate system, and study how the radiated field E(r,t), r = (x,y,z), propagates in the half-space z > 0. Without loss of generality, we shall conveniently write the frequency spectrum of the initial disturbance in the form Em(rJ = pw U^r,), with Um (0) = 1. In this way pm is the frequency spectrum at the point r, = 0, a representative point of the source about which the source is approximately located, and

1 "rex.p(-iaX) , = -\p a

(3)

is the temporal waveform at this point. The function Um(r,) represents a monochromatic disturbance of frequency CO and normalized amplitude. If the spectrum EJrj) = pm U^frJ factorizes into a function of frequency and a function of position, i.e., if Um(r,) ~ U(r,) does not depend on frequency, the source is said to be spectrally pure. In this case, monochromatic disturbances of different frequencies have the same transversal profile U^r^apan from their variable amplitude pm , and all points of the source present the temporal field variation P(t) [apart from a global amplitude U^rJ]. Beyond the source (z > 0), the frequency spectrum pw Uw(r) of the radiated field E(r, f) is determined by Helmholtz equation AL^ + (CO I c) Um = 0, or by the paraxial wave equation for the slowly varying complex amplitude if/m [ Um = l]/m exp(/z 0 and phase ,(r)

=(Oo/\ gradpoo (r)l.

(16)

84

The instant of time at which the pulse peaks (\A(r,t )\ is maximum) at position r is given by 10 ' = Vr) .

(17)

This equation defines pulse front surface, which advances at the group velocity 10 v(r)=l/\gTad(P'a0(r)\.

(18)

The quantity (p'^o (r) thus measures the time of arrival of the pulse at position r. The time delay induced by diffraction, or delay with respect to a plane pulse, is given by (p' 0, and towards redder frequencies where a'^r) < 0. The local frequency of the oscillations can be calculated from

I 2 a2a,(r)dco (20) o

and approached, from Eq. (15), by

85

4corffis2

(21)

where A (O^ is the root mean square width of \pcf . The term with o, allows us to describe the transformations experienced by a pulse due to its finite transversal size. It is also possible to construct an explicit expression for the complex envelope at any point of space that carries the pulse transformations as described above. Introducing Eq. (15) into Eq. (14) and retaining up to the second derivative of the phase and the first derivative of the amplitude, we obtain A(r, t) = am0 (r)ACVD [ s^z) /21 ). The local frequency of the oscillations, calculated from Eqs. (21) and (36), is given by

(37) -'fflO '

and is represented in Fig. 5 as a function of the propagation distance for a single cycle pulse. In particular, along the beam axis, the blue shift is

Aco,

(38)

which increases from zero up to the asymptotic value (Oo[\ + 2(/JoWcub)2] in the far field, as shown in Fig. 5. As the number of oscillations in the pulse grows (4fi^ms tends to 0), frequency shifts diminish, and disappear in the many-cycle limit.

92

g;

r=0

c 0>

I

01

13-

N

,P 8

r=s (z)

Q. I

CJ

O)

o o> o

CJQ

Figure 5. Relative frequency shift of the optical oscillations in pulsed Gaussian beams as it varies with axial distance z along several caustic surfaces. On the caustic surface r - Sea(z)/2la there is no frequency shift. The values are calculated for a single-cycle Gaussian pulse of Gaussian envelope, for which 40)rms/fflb = 0.187.

Diffraction-induced spectrum chirp, and hence pulse broadening, appear to be very small for pulsed Gaussian beams. We adopt here the convention that pulse broadening is negligible when the phase variation within pulse frequency band due to its chirp is much smaller than 1 radian, i.e., when (1/2) (p"' ^(r^Ao),^ « 1. On account that At^^Ao)^,, is of the order of 1/2 for nearly dispersion limited pulses, the above inequality rewrites as

(p"tao(r)«2Atn

(39)

This criterion is equivalent to the usual condition for negligible pulse broadening z « L&sp = 2Atrms2/\k"a*,! (dispersion length) when spectrum chirp originates from propagation in a material with group velocity dispersion k"aa (second derivative of the propagation constant kao). Numerical inspection of the expression of ^")], Jo(.) is the Bessel function of zero order and first kind, and K< (Oo/c is a constant independent of frequency that determines the size of the transversal Bessel function. The above spectrally pure disturbance must be distinguished from the Bessel-X disturbance EJ,r,)= pJ^Kr), with K proportional to (O/c. Contrary to Bessel-X pulses, pulsed Bessel beams experience diffraction changes, which have been studied Ref. 22 on the basis of numerical simulations. It has been shown that the pulse P(t) broadens during propagation, while the transversal profile remains a JgfKr) function.22 This behavior can be easily understood from the diffraction-induced spectrum chirp. The propagated spectrum is the well-known monochromatic Bessel beam Ua (r) = Jd(Kr) exp/7(«/ cf - K2]1/2z]},

(42)

whose amplitude and phase are ajj) = \JdKr)\ , (T )= [; / Pixel

s = 0,16

(s m = 1025pm) 0,80,70,6-

-0,5-

io,«"0,30,20,10,00

25 SO

75 100 126 150 175 200 Z& 250 27S 306 Oft X / Pixel

Figure 6 Shear interferogram16 of a He-Ne-laser with M2 = 1.2 .

0-

0

—I

1

1

sm/wR

1

1

Figure? Degree of coherence16 |yi2| vs normalized shift S/WR,X, for M 2 = 1 . 2 .

122

Interferogram

R1

M3

Camera eg. Gated Diode Array

Figure 8 A shear-interferometer with a rotating Dove-prism to scan the transverse plane'7.

2 Round Trips

4 Round Trips

25 mm

Figure 9 Transverse coherence of a copper-vapour laser'7. Parameters are the magnification M of the unstable resonator and the number of round trips.

123

It is a representation of the field in the sx, ax-phase space. The second moments of ccx and sx are the far-field coherence angle 0coh x in x-direction and the transverse coherence length ^ g l o b x :

(30)

2 ^ Q Js x Z A (s x ,aJ ds x da, _o

(31)

* glob.x

The global degree of coherence reads:

(32)

dsx da x

Eqs. 30, 31,32 can be proved by inserting eq. 29 into eqs. 9, 12. The propagation of ZA in free space is very simple. It turns out that with increasing z the ambiguity function is sheared as shown in fig. 10. The following transformation18 of the coordinates holds:

s x -»s x -zcc x ax->ax Using this property a simple relation between the intensity J(x,z) and ZA can be deduced

= Jj(x,z)-exp[ikaxx]dx

z=-sv /a.

(33)

Finally the procedure is as follows measure J (x,z) at 10-20 different positions around the focal region calculate ZA by the Fourier transform and replace z by -s/a. from ZA, global coherence, coherence length and far-field divergence can be evaluated by another Fourier transform the cross-correlation function is obtained use the second moments of J(x,z) to calculate beam width and beam divergence Wigner function13'19

Another complete representation of the field in the x, 6X phase-space (position, divergence) is the Wigner-function hx(x,9x), which has a certain similarity with the radiance, the power per angle 9X at a position x. It is related to Fx also by a Fourier transform with respect to sx

h*Mx) = jr x (x,s x )exp[ik0 x sjds x

(34)

124

J(x,z)=rx(x,s=0)

(s x ,cc x )= Jr x (x,s x )exp[ikxa x ]dx - JJ(x, z)exp[ikxax]dx z=-s, /a,

transverse coherence ^ g i 0 b, x Figure 10

Propagation of the ambiguity function in free space-

and is related to the intensity distribution by a three-fold Fourier transformation (equivalent to the Radon-transformation13, well know from tomography). The Wignerfunction propagates in the same way as the ambiguity function and delivers beam width and beam divergence directly

_A Jx 2 h x (x,6 x )dxd9, W

R K,Xx =

Ql =•

4

'

Jh x (x,e x )dxd0 x Jh x (x,e x )dxde x

(35)

(36)

and of course all other parameters as phase, phase gradient. But numerically the construction of the Wigner-function is more difficult and more time-consuming than the ambiguity function.

125

Experimental results Experiments were performed with a mode generator. This is a diode off-axis pumped YAG-laser, producing by mechanical adjustment of the gain profile the Gauss-Hermite polynomials20. The resulting global coherence vs adjustment is shown in fig. 11. The global degree of coherence has always a maximum when a pure TEMom mode oscillates, but never reaches Kx = 1, which means that always a mode mixture oscillates. In table 2 the results obtained with a Young-interferometer are compared with the nonmterferometric measurements. The theoretical values are based on the assumption that only two modes are oscillating. Two-dimensional fields If the field cannot be separated E(x,y) ^ E0-f(x)-g(y) or if the field is not of circular symmetry, the field has two transverse dimensions. In this case a four-dimensional crosscorrelation function r i2 (r 1 ,r 2 ) results, which again includes the full information. The corresponding ambiguity- and Wigner-functions are also four-dimensional functions. But the intensity J(f, z) is only three-dimensional. The missing information is hidden in the azimuthal phase structure. Inserting a cylinder lens and rotating by p, delivers the lacking information. Now the intensity becomes four-dimensional J(r,(3,z), and can be used to evaluate the other functions and all beam parameters.

I

10 -

TEM04

TEM03.

M ., TEM02 TEM01

1,0 0.8

—

0.2

I 0.5

0.6

I 0.7

I 0.3

diode laser adjustment (mm)

Figure 11 Beam propagation factor M^ and global degree of coherence Kx vs diode laser adjustment.

126

TEMOO: TEM01 a 0 :a,

beam propagation theory exp.

transverse coherence

global degree of coherence theory

Young

Wigner

0.96-1.0

1.0

1.0:0

1.0

1.01

1.0

0.75:0.25

1.75

1.71

0.79

0.5:0.5

2.0

2.04

0.25:0.75

2.5

0:1.0

3.0

theory Young

Wigner

1.0

0.99

1.01

0.70-0.75 0.74

0.63

0.63

0.60

0.71

0.66-0.73 0.71

0.71

0.74

0.71

2.39

0.79

0.72-0.77

0.94

0.89

0.87

3.04

1.0

1.0

0.97

0.96

0.75

0.89-0.99 0.95

Table 2 Comparison of beam propagation factor M2X, global degree of coherence Kx, and global transverse coherence length 4oh,x. obtained with Young's interferometer and by non-interferometric measurements.

Notation w0, QO :

WR (z), 9R: WT: ZR = 9R/wT:

TEMOO beam radius / half divergence beam-radius, half divergence waist radius Rayleigh length beam propagation factor

A./71

This research project was supported by the Federal Ministry of Education and Research; EUREKA-CHOCLAB II (E12359; 13EU0153).

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

M. Born, E. Wolf, Principles of Optics, Cambridge University Press (1999) Th. Graf, I.E. Balmer, "Laser beam quality, Entropy and the Limits of Beam Shaping", Optics Comm. 131, 77, (1996) E. Tervonen, J. Turunen, A.T. Friberg, "Transverse laser-mode structure determination from spatial coherence measurements", Appl. Phys. B. 49, 409-414 (1989) M.J. Bastiaans, "Application of the Wigner distribution function to partially coherent light", JOSA , A3, 1227, (1986) B. Eppich, "Definition, meaning and measurement of coherence parameters", Proc. SPIE Vol. 4270, 71, (2001) F. Gori, Opt. Comm. 34, 301 (1980) M.J. Bastiaans, Optica Acta 28, 1215 (1981) H. Laabs, B. Eppich, H. Weber, "Modal decomposition of partially coherence beams using the ambiguity function", JOSA 19A, 497 (2002) E. Wolf, "New theory of partial coherence in the space-frequency domain",

127

10. 11. 12. 13. 14. 15.

16. 17. 18. 19. 20. 21. 22.

JOSA 72, 343 (1982) R.Simon, K. Sundar, N.Mukunda, "Twisted Gaussian Schell-model beams" JOSA, A10,2008, (1993) R.Simon, E.C.G. Sudarshan, N. Mukunda, Phys. Rev. A29, 3273, (1984) R.Simon, N.Mukunda, E.C.G. Sudarshan, Opt.Comm.65, 322, (1988) A. Siegman, Proc. SPIE 1224, 2, (1990) C.M. Warnsky, B.L: Anderson, C.A. Klein, "Determining spatial modes of lasers by coherence measurements", Appl. Optics 39, 6109 (2000) B. Eppich, Dissertation Technical University Berlin, D83, (1998), "Die Charakterisierung von Strahlungsfeldern mil der Wigner-Verteilung und deren Messung" E. Waetzmann "Interferenzmethode zur Untersuchung der Abbildungsfehler optischer Systeme", Ann. Phys. 39, 1043 (1912) M.V.R.K. Mruthy, "A compact lateral shearing interferometer based on the Michelson interferometer", Appl. Optics 9, 1146 (1970) S.M. Jackisch, "Shearing Interferometrie zur Laserstrahl-charakterisierung", Diploma Thesis, Optical Institute, Technical University Berlin, (1999) D.W. Coutts, "A versatile angular shearing interferometer for measurement of spatial coherence", Techn. Digest, Conf. Lasers Electro-Optics-Europe 1998, paper CWI4, pp.217 A. Papoulis, "Ambiguity function in Fourier optics", JOSA 64, 779 (1974) MJ. Bastiaans, "Wigner distribution function and its application to first order optics", JOSA A69, 1710, (1979) H. Laabs, B. Ozygus, "Excitation of Hermite Gaussian modes in end-pumped solid-state laser via off axis pumping", Optics & Laser Technology, 28, 213 (1996)

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INVITED SEMINARS

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OPTICAL BEAMS IN UNIAXIAL CRYSTALS GABRIELLA CINCOTTI Electronics Department University Roma Tre via della Vasca Navale 84,1-00146 Rome, Italy, and National Institute for the Physics of the Matter (INFM) Unity Roma 3, Rome, Italy email: s.cincotti(a),uniroma3.it Url: www.ele.uniromaS.it/fotonica ALESSANDRO CIATTONI Physics Department University Roma Tre, 1-00146 Rome, Italy, and National Institute for the Physics of the Matter (INFM) Unity Roma 3, Rome, Italy HORST WEBER Institute for Optics Technical University of Berlin, D-10623 Berlin (Charlottenburg)

1 Introduction Many different methods have been proposed in literature to evaluate the electromagnetic field inside anisotropic media; some of them are very general and comprehensive, but the mathematics involved are rather cumbersome. On the other hand, the simplest approach of a single plane wave or ray propagating in an anisotropic medium, which can be found in many textbooks, does not take the diffraction effects into account, so it cannot accurately describe the propagation of beams with transverse widths comparable with the wavelength. Moreover, the study of light propagation in anisotropic media requires a vectorial approach, as a complete description of different polarization states and the coupling between the field components is mandatory. The propagation of a paraxial beam along the optical axis of an unbounded uniaxially anisotropic crystal is described by means of a vectorial angular spectrum representation of the electromagnetic field, to evidence the combined effects of anisotropy and diffraction. Within this approach, it is possible to solve the boundary problem in the most general case, i.e. to evaluate the analytical expression of the propagated beams, when the boundary field distributions are assigned. In addition, these results allow us to identify some relevant feature of propagation in uniaxial crystals. The optical field inside the crystal is the superposition of an ordinary and extraordinary part, which satisfy two parabolic decoupled equations. Therefore, the ordinary and the extraordinary beams propagate independently and superimposed along the optical axis. Due to the intrinsic polarization states of the ordinary and extraordinary plane waves, the ordinary component is a solenoidal field, whereas the extraordinary one is irrotational. If the Cartesian components of the boundary field are cylindrically symmetric, the analytical expression of the propagated field presents a very simple dependence on the azimuthal angle. This result clearly shows the way in which the anisotropy changes the state of polarization and how the boundary cylindrical symmetry is lost. In addition, in this case a very simple correspondence between the paraxial propagation formulas inside a uniaxial crystal and in an isotropic medium can be found, so that it is possible to evaluate closed-form expressions for Hermite-Gauss beams, Laguerre-Gauss beams and Bessel-Gauss beams, which are well-known solutions of the paraxial equation in vacuum.

131

132

The effect of the anisotropy on the polarization state of an optical beam can be also described by means of the propagation equations for the Cartesian components of the electric field; in fact, the x- and ^-components of the electric field satisfy a system of coupled 'paraxial-like' equations, that highlights the energy coupling between them. The energy exchange between the x- and y-components undergoes to a saturation mechanism that is due to both diffraction and coupling between the components. For Hermite-Gauss beams of any order, linearly polarized along the x-axis at the entrance facet of the crystal, it is possible to furnish a full characterization of their diffraction properties, evaluating the spot-size evolution formula and the corresponding angular divergence: due to anisotropy, the beam parameters evaluated along the x- and y-axes have different expressions. The analytical results are confirmed by a set of experimental measurements at the output of a calcite crystal, for a fundamental Hermite-Gauss and for the field diffracted by circular apertures. 2

Fundamental Hermite-Gauss mode

Let us consider a uniaxially anisotropic crystal whose optical axis is the z-axis of a suitable reference frame, and with n0 and ne being the ordinary and extraordinary refractive indexes, respectively. Without any loss of generality, we assume that entrance facet of the crystal coincides with the z=0 plane, and that the input fundamental HG mode is linearly polarized along the x-axis, i.e.

— I E x (x,y,0)=Eexp

x (1)

E y (x,y,0)=0, where E is a constant, and w0 the spot size at the waist plane. The field propagated through the crystal has the following expression [6,7] 2 exp(ik 0 n 0 z) = —-

(2) \ =—

2 exp(ik 0 n 0 z)

l^ttr

;

1

-+

(Q e (z)

' i

x2+y

where k0=2n/K is the wavenumber in vacuum, and (3)

are the ordinary and extraordinary propagation parameters, which are related to the complex propagation parameters of Gaussian beams propagating through isotropic media with index of refractive indices n0 and ne2/n0, respectively.

133

Figure 1 shows the intensities \EX\2 and \Ey\~ evaluated at the planes 80=11, 26, 45, and 78, where 80=z/zRo and zRo =7iw02n0/h is the Rayleigh distance in an isotropic medium, with refractive index «„. It is evident that the ^-component of the electric field loses its boundary cylindrical symmetry, due to the anisotropy; besides, the anisotropy gives origin to ay-component of the electric field. Ex and Ey experience both diffraction and mutual coupling, the latter being responsible for the change of polari/ation state; to investigate their energy exchange, we consider the optical power carried by Ex and Ey IE

/

IE

/

IE/

IE/

-20

80 =z/zRo=ll

0 „/*„

20

S0 =z/zRo=26

80 =z/zRo=78

S0 =z/zRo=45

Figure 1: Intensities of the x- and y-components of the electric field originated in a calcite crystal by a fundamental HG mode linearly polarized at the crystal entrance facet: analytical (upper rows) and experimental results (lower rows) for four propagation distances.

P (j) (z)=jdxdy(|E j (x,y,z)|

= x,y.

Substituting Eqs. (2) into Eq. (4), after a simple algebra, we get

(4)

134

3 +-

(5)

where P0=¥t*)(Q) is the input power, and 8e=z/zRe with z/ee =mv02ne2/foi0 being the Rayleigh distance in an isotropic medium, with refractive index ne2/n0

Figure 2: Evolution of the optical power and spot-sizes associated to the x- and y-components of the electric field inside a calcite crystal: numerical (lines) and experimental results (marks).

Figure 2 shows the evolution of P6^ as a function of S0: we observe that the energy associated to the y-component of the electric field monotonically increases and undergoes a saturation: the asymptotic power exchange between Ex and Ey amounts to a quarter of the total power for every linearly polarized circularly symmetric input beam [8]. To investigate the diffraction properties of Ex and Ey, we evaluate the evolution of their spot-sizes, along the x- andy-axes

w

=4

Jdxdyi7 2 E j (x.y > z)| :

(6)

135

getting as a result

wn

-=—+9

- + 2-

I o —O

-In

1 1 1

.

-

1

°

1

1

!

1 2 JJ

! /I 1 7

.

e

1

o 2

JJ

8.80-\

9

6 (7)

1+ 4 +3

w,. wn

- + 21n 1 +

wn

1+ -

Figure 2 reports the spot-sizes of the Cartesian components of the electric field, along with the spot size of a Gaussian beam propagating in an isotropic medium with refractive index n0, that is plotted for reference. From the expressions of Eqs. (7), we can evaluate the angular divergences of the beams and the waist spot-size of ^distribution (x) _

w

x

(X) z

( )_

A

!

(x) ,

(8) 0(y)=0(y)=lim y

~»

w

(y)

z

(z)_ ;ra 0 w 0 > /2\n*

w x (y) (0) = w y (y) (0) = lim w x (y) (z) = V3w0. Of course, all the beam parameters coincide with those of a Gaussian beam propagating in an isotropic medium if we put n0—ne\ let us also observe that the waist plane of the y-

136

component coincides with the z=0 plane and that the corresponding spot size is equal to V 3w0 , even though it is Ey-0 at the entrance facet of the crystal. 2.1 Higher-order Hermite-Gauss modes For a HG10 mode, linearly polarized at the entrance facet of the crystal, the electric field distribution is [6]

IE/ HS,0

IE/ HG0,

("I

ff

HG,,

IE/ HO,,

= z/zRo =26 Figure 3: Intensities of the x- and y-components of the electric field of higher-order HG modes inside a calcite crystal, numerically evaluated for two propagation distances.

137

Qe2l (9)

for a HG0i mode we have

X +y

Q»(z)

x +y"

~ow

(10)

and for a HG. i mode it is

"Q.W

138

E y (x,y,z) = Ew4elkon°:

" x'+y*

-3Q* (z)(x 6 -5x 4 y 2 -5x 2 y 4 + y6)-3Q^ (z)(x 4 -6x 22y2

-3Q200(>

6

y 4 )]

;(z)(x 4 -6x 2 y 2 + y4)]}.

-5x 4 y 2 5x 2 /

30

40

(11)

50

sn=z/zDn

Figure 4: Evolution of the optical power associated to the y-component of the electric field of higher-order laser modes inside a calcite crystal.

Figure 3 shows the intensity distributions of the Cartesian components of the electric field corresponding to the HG10, HG0i and HGu laser modes, propagated for two distances z inside a calcite crystal; from an inspection of this figure, it is evident how the anisotropy modifies the beam profiles. Substituting Eqs. (9)-(ll) into Eq. (4), we obtain the expressions of the energies pertaining to the Cartesian components of the laser modes, that are plotted in Fig. 4: it is worth noting that laser modes of different order present different saturation values; moreover, the energy exchange does not ever exceed the half of the input power [8]. Tables I and II summarize the main results of the present investigation: the angular divergences, evaluated along the .x-and >>-axes, of Ex and Ey are reported, along with the waist spot-size of the y-component and the saturation values of P*y).

139

Table I: x-component of the electric field

mode

3 00

wx(0)

Wy(0)

wn

wn

ex

wn

Px(oo)/P(0)

*- (3)

HG oi

+

« wn

(3)

• 129n

3/4

f

5/8

(5)

w

13/16

11/16

Beam parameters of the x component of the electric field: wx(0)Wy(0): waist spot size along the x and y axes, respectively, 0x, 0 y : angular divergence along the x and y axes, respectively, Px(oo)/P(0): normalized energy.

140 Table II: y-component of the electric field

mode

wx(0)

Wy(0)

HGoo

V3w

V3w

Mx

M

Py(«>)/P(0)

1/4

1/4

HGoi

V3w0

1/4

3/8

HG

V3w

3/16

3/16

5/16

Beam parameters of the y component of the electric field: Wx(0)wy(0): waist spot size along the x and y axes, respectively, MX, My: M factor along the x and y axes, respectively, Py(°o)/P(0): normalized energy.

141

3 Experiments To confirm the analytical results for the fundamental HG beam, we used the laboratory setup illustrated in Fig. 5. From the right-hand side to the left-hand side there are: an Argon-Kripton laser tuned at the A=0.514 jum line on the fundamental Gaussian mode, with 1 mm spot size, / mrad divergence and linear polarization; a neutral density filter to attenuate the beam; an additional polarizer, as the linear polarization is strongly requested; a lens that focuses the beam on the entrance facet of the calcite crystal: the input spot size wg depends on its focal length; a calcite crystal with refractive indices n0=L658 and ne=1.486, dimensions 10x10x20 cm, and the optical axis coincident with the longest axis, along which the beam is made to propagate; a rotating polarizer that can be oriented to transmit the components of the electric field along the x-axis, or y-axis; a lens that, together with the objective mounted on the camera, forms a magnified image of the exit facet of the crystal on the CCD detector (3.3x4.4 mm, 576x768 pixels) of a Sony TV camera; a the polarizer that attenuates the intensity on the CCD detector. A computer is connected to the camera to record and process the images, that are reported in Fig. 1. Starting from the intensity distributions of Ex and Ey acquired with the camera, we measured the power dynamics and the beam variances, that are shown in Fig. 2. From an inspection of these figures, it is evident that the experimental results confirm the theoretical approach.

Figure 5: Experimental setup

4 Conclusions We have investigated the propagation of laser modes along the optical axis of a uniaxially anisotropic crystal: starting from the analytical expressions of the Cartesian components, we have characterized their diffraction properties. All the beam parameters, as the spot sizes and the angular divergences have been computed. The combined effect of anisotropy and diffraction gives rise to a change of the input polarization state and an interaction between Ex and Ey: we have determined the energy exchange dynamics and their saturation values. The analytical results have been contrasted with experimental measurements, showing a good agreement.

142

References 1 2

3 4 5 6 7 8

9 10 11

12 13 14

15

16 17

A. Ciattoni, B. Crosignani, and P. Di Porto, "Paraxial vector theory of propagation in uniaxially anisotropic media, " J. Opt. Soc. Am. A, 18, 1656-1661 (2001). A. Ciattoni, G. Cincotti, and C. Palma, "Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals, " Opt. Commun., 195, 55-61 (2001). G. Cincotti, A. Ciattoni, and C. Palma, "Hermite-Gauss beams in uniaxially anisotropic crystals," IEEE J. of Quantum Electron., 37, 12, 1517-1524. (2001). A. Ciattoni, G. Cincotti, C. Palma, "Optical beams in uniaxially anisotropic crystals," Sixth International Workshop on Laser Beam and Optics Characterization, Munich (Germany), (2001). A. Ciattoni, G. Cincotti, and C. Palma, "Propagation ofcylindrically symmetric fields in uniaxial crystals " J. Opt. Soc. Am. A, 19, 792-796 (2002). A. Ciattoni, G. Cincotti, and C. Palma, "Nonparaxial description of reflection and transmission at the interface between an isotropic medium and a uniaxial crystal, " J. Opt. Soc. Am., 19, 7, 1422-1431 (2002). G. Cincotti, A. Ciattoni, and C. Palma, "Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals, "' J. Opt. Soc. Am. A, 19, 9, 1680-1688 (2002) A. Ciattoni, G. Cincotti, C. Palma, and H. Weber, "Energy exchange between the Cartesian components ofaparaxial beam in a uniaxial crystal," J. Opt. Soc. Am., 19,9,1894-1900(2002). G. Cincotti, A. Ciattoni, and C. Palma, "Propagation-invariant beams in anisotropic crystals," J. Mod. Opt., 49, 13, 2267-2272 (2002). D. Provenziani, A. Ciattoni, G. Cincotti, C. Palma, "Diffraction by elliptic and circular apertures in uniaxially anisotropic crystals: theory and experiments, " J. Opt. A Pure Appl. Opt., 4,424-432 (2002). D. Provenziani, A. Ciattoni, G. Cincotti, C. Palma, F. Ravaccia, C. Sapia, "Stokes parameters of a Gaussian beam in a calcite crystal, " Opt. Expr., 10, 15, 699-7062 (2002). G. Cincotti, A. Ciattoni, D. Provenziani, C. Palma, and H. Weber, "Characterization of laser beams in uniaxial crystals," 7th International Workshop on Laser Beam and Optics Characterization (LBOC7), Boulder Co. (USA), (2002). A. Ciattoni, G. Cincotti, D. Provenziani, C. Palma, "Paraxialpropagation along the optical axis of a uniaxial medium," Phys. Rev. E, 66, 036614-1-11, (2002). A. Ciattoni, G. Cincotti, C. Palma, "Paraxial beams in uniaxially anisotropic crystals, " 19th Congress of the International Commission for Optics, Firenze (Italy), (2002). A. Ciattoni, G. Cincotti, C. Palma, "Circularly polarized beams and vortex generation in uniaxial media," Journal of the Optical Society of America A, 20, 1, 163-171 (2003). A. Ciattoni, G. Cincotti, C. Palma, "Angular momentum dynamics of a paraxial beam in a uniaxial crystal," Physical Review E, 67, 036618-1-109 (2003). G. Cincotti, A. Ciattoni, C. Sapia, "Radially and azimuthally polarized vortices in uniaxial crystals, " Optics Communications 220, 1-3, 33 - 40 (2003)

LIGHT BEAM SHAPING: THE INTEGRATION METHOD PAOLO DI LAZZARO, SARAH BOLLANTI, DANIELE MURRA ENEA, C.P. 65, 00044 Frascati (Italy) E-mail: dilazzaro@frascati. enea. it http://www.frascati. enea. it/fis/lac/excimer/index-exc. html

1 Introduction The general light beam shaping problem consists of an optical system which operates upon an input light beam to modify its spatial or temporal shape, producing an output beam having the desired space distribution or time evolution. Let us limit our attention to low-loss shaping techniques acting on the beam space distribution. They may be divided in two categories: field mapping and beam integration. Field mapping techniques include optical systems operating on both phase and amplitude of the input beam to produce an appropriate intensity distribution on the output plane1. Beam integration methods mix fractions of the original beam to smooth out intensity spikes. Usually, field-mapping techniques are successfully applied to honest beams, having well defined modes. On the contrary, beam integration is convenient when the beam is partially coherent and its intensity distribution is irregular. As the authors have a fatal attraction to bad laser beams, only beam integration methods will be discussed in the following.

2 Motivation A large number of laser applications need a strict control of the beam profile. Many of them require beam coverage uniformly extended across target areas, and it is necessary using high-quality homogenized profiles, i.e. top-hat spatial energy distributions with steep edges and high uniformity in the plateau region. Typical examples include laserbased material processing (e.g., surface cleaning, ablation, marking, drilling, metal hardening), thin film transistors for flat panel displays, chip-making, medical applications (e.g., corneal reshaping, cosmetic surgery, UV-curing dermatology diseases). Unfortunately, none of the high-power laser systems available in the market emit beams with flat-top, uniform intensity profiles. As a consequence, different beam integration techniques for homogenizing laser radiation are employed. 3 Design elements Figure 1 illustrates the principle of a typical homogenizer along one transverse axis. An array of cylindrical lenses (the "divider") breaks the incident beam into secondary beamlets. A "condenser" lens provides the overlap of the beamlets on the focal plane of the homogenizer. The addition of different portions of the initial (non-uniform) beam smoothes the intensity profile fluctuations, finally creating a flat-top beam. The averaging

143

144

effect is proportional to the number of the secondary beamlets. That is, the larger the number of the divider lenses, the more uniform the beam in the focal plane of Fig. 1. Focal plane

Figure 1. 2-D scheme of a conventional homogeniser. The beam travels from left to right.

A simple exercise shows that the size D of the homogenized beam in the focal plane of the optical system of Fig. 1 is given by:

(D

J D

where S = size of a single divider lens; fD = focal length of each divider lens; fc = focal length of the condenser. Note that D is independent of the distance between divider and condenser. To optimize the homogeniser performance, we have to take into account interference and diffraction effects that can modulate the homogenized intensity shape, thus frustrating the averaging effect of the homogenizer. Let us discuss these issues. Interference: if beamlets exiting by adjacent lenses of the divider are mutually coherent, the output intensity will be dominated by speckles (high-contrast interference patterns). The period P of this interference pattern, in the simplest case of rectangular lenses (1-D case) is P ~ Kxfc/S and its impact is minimized decreasing the value of the ratio fc/S. However, this cannot be directly achieved by increasing 5, as this would obviously lead to a worse averaging effect of the homogenizer. Diffraction: diffraction effects arise from the boundaries of the divider lenses, and create an additional pattern having the same period P above and a modulation depth in the homogenized intensity shape. The modulation depth is minimized by a sufficiently large (> 100) equivalent Fresnel number1 = S2 /(4Xx/D) = SxD/(4Kxfc),

(2)

where A. is the laser wavelength. Again, it is not convenient increasing P by augmenting S, to preserve the essential averaging effect of the homogenizer. When reducing to practice the above considerations, the size S of the divider lenses should be small enough to guarantee an effective averaging of the input beam intensity profile fluctuations, but larger than the transverse coherence length of the input beam to avoid speckles caused by interference. Moreover, the choice of S affects also the equivalent Fresnel number, see Eq. (2). As a consequence, the designer makes a tradeoff between S and P and this usually yields a homogenizer able to integrate only the average fluctuations of the input intensity. Then, relatively poor results are expected in the case of light beams having local intensity fluctuations stronger that the average fluctuations (e.g., beams with asymmetric non-uniformity profiles emitted by discharge-pumped lasers, like

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excimer lasers, copper vapor, TEA CO2 lasers, and beams emitted by diode laser arrays). This is one of the two main inconveniences of conventional homogenizers. The second one is evident from Eq. (1), which shows that, once fixed the geometry and lens power, the energy density on the focal plane can be changed only by changing the energy of the input beam. Sometimes it is not convenient changing the output energy of a laser, either because this may reduce the efficiency and stability of the laser emission, either because this may be hardly done continuously. This means those irradiation processes requiring different energy density values and/or different spot sizes need distinct homogenizer systems. A possibility to overcome the above limitations is given by a recently proposed homogenizer with zoom and asymmetric divider arrays2. This patent pending ENEA homogenizer on the one hand can continuously modulate the spot size (and then the energy density) of the homogenized beam along one or both axes in a fixed target plane; on the other hand, it is able to make homogeneous any beam, including beams having strong local intensity spikes.

4 Advanced design Figure 2 illustrates the principle of the ENEA homogenizer. The two main differences with respect to the scheme of Fig. 1 are, respectively, 1) the asymmetric divider (the more homogeneous part of the beam impinges on the larger size lenses of the divider array, and vice versa) and 2) the zoom lens added after the condenser lens. In spite of their different size, all the secondary beamlets have the same dimension on the focal plane, provided that a constant ratio Si/fa is maintained, where S{ is the size of the i-th lens of the divider and /Di its focal length. The zoom element can be a spherical lens or a couple of cylindrical lenses (according to the condenser lens). In the last case, it is possible to independently change the spot size along the two orthogonal axes.

Intensity profile

Focal plane

Figure 2. 2-D scheme of the ENEA homogeniser. When the beam has an asymmetric profile, the divider array is made by different lenses, such that the smaller lenses intercept the less homogeneous part of the beam, and that each lens size is much larger than the local coherence length of the corresponding portion of the beam.

The size D of the homogenized beam in the focal plane of the optical system of Fig. 2 is given by:

146

fc*fz (3) fz+fc~dcz where fz = focal length of the zoom lens; dcz = distance between condenser and zoom. The following equation gives the total length of the homogenizer, from the divider to the focal plane: _,_, , z = d, DC+d cz+-r- ----—,,

(4)

where d^c is the distance between the divider and the condenser. From Eq. (3) it is clear that changing the distance dcz we change the size of the homogenized beam in the focal plane and, from Eq. (4), that a corresponding proper adjustment ofdDC allows keeping fixed the position of the focal plane of the homogenizer (i.e., of the target plane) with respect to the divider and to the laser system. In short, the ENEA beam integrator allows to make homogeneous beams having asymmetric intensity spikes, and to vary the spot size of the homogenized beams (which can be either a square or a rectangle with a fixed or a variable aspect ratio) without changing any optical element and keeping constant the optical path length from the laser to the target plane.

5

Software

In order to fully exploit this homogenizer technology, ENEA developed a proprietary software, useful to design the best optical system to achieve the wished output beam performance (once known the input beam characteristics), and to know how the output beam shape and size change when changing position of each optical element. This software is written in Visual C++ and runs on MS Windows 95/98. It is user friendly, thanks to a properly designed user interface. In particular, this software can: *J* give the optimum choice of the optics parameters once the boundary conditions are known (input laser beam characteristics, mechanical constraints, desired beam shape and desired output beam size); •J» simulate the homogenizer performance by inserting the input beam profile data and by carrying out the ray-tracing of the light beam passing through the optics. The optical elements can be simulated according to the paraxial optics (focal length) or by physical parameters (curvature radius, thickness, refraction index); * show the profile of the beam on any transverse plane along the homogenizer path, including the focal plane; *!* calculate meaningful beam parameters, like the baseline size, the plateau size, the average and peak intensity, the steepness, the plateau fluctuations root mean square (RMS), the threshold uniformity; * give the position of the divider, condenser and zoom elements when a particular final size is wanted, leaving the distance laser - target unaltered.

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6

Experiment

Up to date (November 2002) the ENEA homogenize! has been successfully applied to a number of lasers, namely excimers, HeNe, diode. Let us present the results achieved with the most challenging attempt, made by homogenizing the large-size, high-energy beam (10 cm x 5 cm, 8 J) emitted by the XeCl (A, = 308 nm) excimer laser-facility Hercules4'5. Hercules Near Field - Horizontal

Hercules Near Field - Vertical 7000 6000 H 5000 o 4000 Q 3000 y 2CKH) 1000

0

Figure 3. Typical cross-sections of the near-field profile of the XeCl laser Hercules, 0.5 m behind the laser housing.

Figure 3 shows the typical near field profiles of the laser beam emitted by Hercules. The horizontal profile (along the discharge electric field) is asymmetric because of typical discharge non-uniformity (streamers6) near the grounded electrode. Here and in the following, the beam profiles are measured by using the UV-sensitive CCD camera Andor model DV438-BV. This back-illuminated camera7 has an active sensor area of (1.73 x 2.59) cm2, and the image is digitized in 770 x 1152 pixels. To avoid saturation of pixels, we attenuated the laser beam down to an energy density of 5 x 10"8 J/cm2. The attenuation factor was maintained constant during all the measurement runs. In the case of laser cross sections larger than the CCD sensor size (like in Fig. 3), the overall image was obtained adding images captured in consecutive shots, by rastering the CCD with the help of a step-motor-driven slide (Physik Instrumente, model M521-DD) moved along the cross section profile. We have designed and tested a homogenizer made by two crossed arrays of 4+4 cylindrical lenses as divider, by a condenser along the horizontal direction and by a condenser and a zoom along the vertical one. The transmission factor @ K = 308 nm of this homogenizer is 80%. The subaperture sizes and the focal lengths are: 5H= 2.5 cm, /HD= 22cm;/HC= 120 cm, Sv = 1.25 cm, fVD= 50cm, fvc= 44 cm,fvz.= 21 cm. Here the subscripts H and V mean horizontal and vertical, respectively. By substituting these values into Equation (3) we obtain a fixed 14-cm spot along the horizontal direction, while, along the vertical one, the homogenized beam size is variable from 0.6 cm up to about 7 cm, with a maximum zoom factor larger than 11. Equation (4) gives an homogenizer length dDC + dcz « 1.7 m, while the software provides the suitable distances between dividers, condensers and zoom to keep fixed L (see Eq. (4)) as well as the distance between the laser housing and the focal plane of the homogeniser. Equation (2) gives the equivalent Fresnel numbers along the horizontal (0H = 2200) and vertical (Pv « 4200) axes. Thanks to the short A., both p values are large enough to make negligible the intensity modulation depth due to diffraction effects. Interference effects can be neglected as well, because the period of the pattern (P » 15 ^m) is very small and the associated

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modulation depth is close to zero, being SH and Sy respectively 25 times and 10 times larger than the corresponding transverse coherence lengths of Hercules.

7

Results

Figures 4 and 5 show the vertical cross-sections of the homogenized beams with the smallest and the largest measured size, respectively. The horizontal cross-section of the homogenized beams is shown in Fig. 6: the horizontal length of the beam spots is the same for all the vertical zoom factors. 6 mm-wide Homogenised Spot

66 nun-wide Homogenised Spot

30000 25000 a 20000

s

U 15000

y 10000 5000 0-

)

5 1 Bottom I

10

15

pftp]

nun

Figure 4 Typical cross-section of the Hercules laser beam in the focal plane of the homogeniser along the vertical axis, zoomed from 50mm down to 6mm.

Figure 5 Typical cross-section of the Hercules laser beam in the focal plane of the homogeniser along the vertical axis, zoomed from 50mm up to 66mm.

Homogenised Profile-Horizontal Fig. 6 Typical cross-section of the Hercules laser beam in the focal plane of the homogeniser along the horizontal axis, resized from 100mm to 140mm.

H.V side

A comparison of Figs. 4, 5 and 6 with Fig. 3 shows that the homogenizer made the beam shape very smooth and with steeper edges. A substantial reduction of the high-spatialfrequency intensity fluctuations on the plateau is evident comparing Fig.3 and Fig. 6. The apparent noise increment in the plateau of Fig. 5 with respect to Fig. 4 is a device-related effect, due to the different laser energy density impinging on the CCD in these two cases (note the different vertical scales), which yields a worse signal-to-noise ratio in Fig. 5.

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8

Characterization

There are two important questions one can rise: how "good" are the homogenized beam profiles shown in Figures 4 to 6? How it can be made quantitative the improvement in terms of beam flatness and steepness between the input raw beam (Fig. 3) and the homogenized beam (Figs. 4, 5 and 6)? In order to characterize the quality of homogenized laser beams, the ISO standard 13694 proposes two parameters, respectively named "edge steepness" and "plateau uniformity"8' 9 . The edge steepness is defined as '10%, 90%

(5)

= (A10% - A90%) / A 10%

where AK>% (A90%) is the effective irradiation area at 10% (90%) of the maximum energy density Hmax. The plateau uniformity is defined as

(6)

= AH/Hn

where AH is the full width at half maximum of the peak in the energy density histogram curve N(Hi). The energy density histogram curve is the number of data points plotted as a function of the corresponding energy density H;. As an example, Figure 7 shows the energy density histogram curve of the cross section profile of Fig. 4. The width AH of the high-energy-density peak approaches zero for ideally flat distributions.

Figure 7. Histogram of the homogenised laser beam cross section shown in Fig. 4. Vertical: number of pixels. Horizontal: energy density (% of

40

Hjnax)-

AH is the full width at half maximum of the peak just below the maximum energy density Hmax.

I

0.4 0.6 Energy density (% of Hmax)

0.8

Specializing Equations (5) and (6) to our experimental data we can see, for example, that along the horizontal axis the steepness decreased from 47% (raw beam) to 11.3% (homogenized beam), and the plateau uniformity decreased from 13.5%, to 5.5% (see Fig. 6). Concerning the vertical axis, the results are summarized in the Figures 8 and 9.

150

Figure 8. Comparison of the steepness (see Eq. (5)) along the vertical axis between the raw beam (without) and the homogenized beam zoomed in to 6 mm, 7 mm and 13 mm, zoomed out to 66 mm and reshaped to 50 mm.

Vertical axis: steepness

without

6mm

7mm

13mm

50mm

66mm

V e r t i c a l axis: p l a t e a u u n i f o r m i t y

without

6mm

7mm

13mm

50mm

Figure 9. Comparison of the plateau uniformity (see Eq. (6)) along the vertical axis between the raw beam (without) and the homogenized beam zoomed in to 6 mm, 7 mm and 13 mm, zoomed out to 66 mm and reshaped to 50 mm. 66mm

Some trends can be identified from these data. For example, the edge steepness improves when the beam is zoomed to larger spots (see Fig. 8). This effect is due to a not perfect quality of the zoom lens, which is aberrated when working with large numerical apertures. The plateau uniformity of the homogenized beams clearly improves with respect to the one of the input beam, decreasing by a factor between 1.5 and 3 (see Fig. 9). This oscillation of the improvement factor does not exhibit a clear dependence on the zoom factor. It is probably related to a not perfect alignment, which was made difficult by the very strong attenuation factor of the Hercules laser beam to avoid saturation of the CCD camera (see Sect. Experimental). 9 Applications The laser sources equipped to date with the ENEA homogenizer were successfully used by different customers to a number of laser material processing applications. They include the recrystallization of amorphous silicon films, cleaning semifinished aluminum plates, the selective removal of paints and graffiti without damaging different substrates (including glass, weather-strip rubber, marble, and varnished metal). Some illustrative images of irradiated samples are reported in Fig. 10.

151

Figure 10: Examples of material processing done by laser beams reshaped by the ENEA homogenizer. a) SEM image of recrystallized silicon film; b) photos of graffiti stripped without damaging the glass (top) and white marble (bottom) substrates; c) SEM images of a semifinished aluminum plate before (top) and after (bottom) laser irradiation. The homogeneous irradiating beam removed the dirty layers, without affecting the original surface structure: note the sub-micrometer scratches and digs on the cleaned sample do not exhibit any material ablation or melting effects.

10 Ut breviter dicam Beam shaping is the process of redistributing the irradiance and phase of a beam of optical radiation. A correct beam shaping is an essential requisite to optimize a large number of laser-based materials processing. We have discussed in some detail the beam integration method, especially suitable to reshape partially coherent beams having an irregular intensity distribution. In particular, we have presented the design elements, the experimental performance and the main characterization parameters of a new beam integrator homogenizer technology (patent pending3), able to homogenize and reshape bad light beams, including those having asymmetric-spike profiles. In particular, the new homogenizer is useful in the following cases: 1) Uniform irradiation processes where the optimum energy density and/or spot size of the irradiation (illumination) process are not known in advance. Operators can vary the output spot size and/or the energy density of the homogenized beam until the optimal working point is reached - all without making the expensive and time-consuming equipment adjustments (e.g., change of optical elements, change of the homogenizer length, change of distance laser-target) necessary with existing beam-handling systems. 2) Homogenization of light beams having local intensity spikes stronger than the average intensity fluctuations, like, e.g., beams with asymmetric non-homogeneity profiles (discharge-pumped lasers, diode laser arrays). The homogenizer prototype presented in this note has been designed using a proprietary software and experimented on the laser beam emitted by the XeCl (k = 308 nm) excimer laser facility Hercules: it was tested up to a zoom factor 11 along the vertical direction. The cross-sections of the homogenized beams were characterized in terms of edge steepness and plateau uniformity.

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Thanks to the licensing agreement between ENEA and Info & Tech S.p.A.10, the ENEA's homogenize! technology is commercially available. At the moment, our transfocal homogenizers have been successfully used in several laser applications, including recrystallization of amorphous silicon films, selective paints and graffiti stripping from several substrates (glass, rubber weather-strip, marble and painted metal), and cleaning of semifinished aluminum plates.

References 1

F. M. Dickey and S. C. Holswade, "Laser beam shaping: theory and techniques" (M. Dekker, Inc. New York Basel, 2000). 2 P. Di Lazzaro, S. Bollanti, D. Murra, G. Felici: "A novel light beam homogeniser" Proc. Int. Conf. on Gas Flow and Chemical Lasers, High Power Lasers (Wrocklaw, August 2002), SPIE, to be published. 3 ENEA patent pending n° IT RM 000229 (UIBM, April 28, 2000); n° 09/727.268 (US PTO, November 30, 2000); n° 00830807.4-2210 (EPO, 6 December 2000). 4 T. Letardi, S. Bollanti, P. Di Lazzaro, F. Flora, N. Lisi, C.E. Zheng: "Some design limitations for large-aperture high-energy per pulse excimer lasers", II Nuovo Cimento D 14, 495 (1992). 5 P. Di Lazzaro: "Hercules, an XeCl laser facility for high-intensity irradiation experiments" 2nd GR-I Int. Conf. on New Lasers, Technologies and Applications, Proc. SPIE vol. 3423, pp. 35 - 43, A. Carabelas, P. Di Lazzaro, A. Torre, G. Baldacchini Eds. (1998). 6 S. Bollanti, P. Di Lazzaro, F. Flora, T. Letardi, N. Lisi, C.E. Zheng: "Space- and time-resolved discharge evolution of a large volume X-ray triggered XeCl laser system", Appl. Phys. B 55, 84 (1992). 7 Andor Technology Limited, 9 Millennium Way, BT12 7AL Belfast Northern Ireland. Web site: http://www.andor-tech.com 8 K. Mann, J. Ohlenbusch, V. Westphal: "Characterization of excimer laser beam parameters" Third Int. Workshop on LBOC, Proc SPIE vol. 2870 pp. 367 - 377, M. Morin, A. Giesen Eds. (1996). 9 ISO/DIS 13694: "Test method for laser beam parameters: Power (energy) density distribution", May 1999. 10 Info&Tech S.p.A. Via Teognide 24, 00124 Roma (Italy). E-mail: [email protected]

BEAM PROPAGATION IN QUADRATIC MEDIA GIUSEPPE LEO NOOEL, INFM & Dept. of Electronic Engineering, University "Roma Tre" Via delta Vasca Navale 84, 00146 Roma, Italy E-mail: [email protected] This lecture is intended to provide a general introduction to quadratic nonlinear optics, along with a view on a few selected research topics of present interest. In Section 1 I will recall the quadratic effects arising from non-resonant light-matter interactions, in the absence of absorption and carriers. Specifically, 1 will illustrate the concept of phase matching and overview the related techniques, with emphasis on guided-wave interactions. In the last decade, novel parametric sources have been conceived and fabricated in both bulk and integrated geometries, and new phenomena and applications have been demonstrated in the domain of soliton propagation. In Section 2, on the side of novel integrated optical sources, I will focus on the recent progress towards a monolithic semiconductor OPO. In Section 3, in the framework of the phase effects associated to the cascading of %(2), I will deal with quadratic spatial optical solitons, i.e. free-propagating optical-field configurations where diffraction is balanced by nonlinear self-focusing.

1.

Introduction

As compared to other nonlinear sciences, nonlinear optics (NLO) is relatively young, since it only started after the demonstration of the laser in 1960. The first observed NLO effect, second harmonic generation (SHG) [1], was a quadratic one. Quadratic nonlinearities stem from the material-dependent valence-electron anharmonic oscillator and occur in non-centrosymmetric molecules or crystals [2]. Far from resonances, the related interactions rely upon virtual transitions, and are intrinsically ultra-fast since they do not depend on material lifetimes (like electron recombination). In the case of a spatially and temporally local medium response, the electric polarization associated to an incident field E takes on the general form

P(r.f) = £ 0 x (l) (r.f )• E(r,Q +£ oX (z) (r.f ) ; E(iy)2 + £0x'3) (r.f ).:E(r,r)3 JIG

p(3)

where EQ is the vacuum permittivity, and j£(1) the linear susceptibility tensor; %(l) is a rank(i+1) tensor defining the ith-order susceptibility, with its structure and components determined by the type of nonlinearity and crystal symmetry. The linear term P(L) is adequate to describe the response to weak natural light, while further terms, involving increasing powers of E, come into play for higher (typically laser) powers. In general the electric field consists in the superposition of n (quasi)monochromatic waves E(r,f) = £E,.(r,0 1=1 where c.c. stands for complex conjugate, and the complex amplitude E.(r,o>.) is the Fourier component with frequency (Oj and wavevector £,=, and second harmonic (SH) output at . The plane-wave evolution of FF and SH fields along the z direction in the ^ medium is derived considering the total electric field: E(Z, 0 = (l/2^E(z,0)) exp[i(fflf - *.z)]+E(z,2ffl)x

exp[i(2fl» - k2(a z)]}+ c.c. .

The relevant EjE^ terms in Eqs. 3, 4 induce the polarization terms P®(z,co)=2e0^jkdijk(

-to;2aa,-€o)E'J(z,to)Et(z,2to)exp[-i(kIa

Pt(2) (z,2co)= e0 ^jt dljk< - 2w; to, co)EJ (z,co)Ek(z, co) exp[- 2»Bz]

-ka)z]

155 CO,=03

generally experiences a larger phase velocity than the field at 2a> (Ak^O), and this periodically brings out of phase the coherent contributions to the SHG at different stages of propagation. When the dephasing between such contributions reaches the value of n, power starts to flow back from SH to FF field, giving rise to the oscillatory evolution of Figure 2b. It is common to express this behaviour in terms of the coherence length Lc = 7T/AA: — -^(B/4(n2(U — M m ), corresponding to the distance after which FF and SH field get to phase opposition (half a cycle in the oscillatory evolution of Figure 2b). In Type-I SHG, wave-vector conservation implies that nlo=n2a>. The most common way to fulfill this condition is to use two different FF and SH polarizations in a birefringent nonlinear medium, by exploiting the angular dependence of the extraordinary index. An alternative SHG phase-matching scheme requires two orthogonally polarized pump fields, with Ak=k3(2a>)-kl(a>)-k2((o), and is known as Type-II SHG. In this scheme the condition to fulfill is nj=(ni+n2)/2. Besides angle tuning, either type of birefringent phase matching (BPM) can be achieved through the extra degree of freedom provided by the temperature dependence of material dispersion curves or, if the SHG wavelength is not a tight constraint, by the use of a tunable pump source. Among the most relevant rf2> devices there are optical parametric amplifiers (OPA), optical parametric generators (OPG), and optical parametric oscillators (OPO) [4]. This class of devices were invented in the early days of nonlinear optics [5]. Until the mid-80's, however, they were only considered as research setups, as dye lasers were largely preferred as coherent tunable sources. In the last fifteen years, mostly due to progress in nonlinear materials, research on OPA's and OPO's was greatly boosted. Today large segments of the related technology are at a maturing stage, for the most diverse industrial, militar, environmental and research applications, with new commercial products being frequently launched on the market. Furthermore, parametric devices are used in fundamental research, as valuable sources of squeezed light and twin photons, and to investigate quantum systems. OPA is associated to a DFG process (see Figure 1) where: the intense, higher-frequency input at a>i acts as the "pump" (p), the lower-frequency input at a>2 is the "signal" (s), and the DFG output at a>3 is named the "idler" (i). Besides the DFG at (o^Wp-cOs, the input signal is amplified through the parametric interaction. At PM, for weak parametric gain g, neglecting losses, the signal output power is readily found as Ps(z,)Kps(o)(l + g 2 L 2 ) [2]. In OPG an incident pump photon splits into two lower-energy, idler and signal photons, fulfilling energy conservation. The output frequencies are selected by momentum conservation (PM condition), with possible angle-, temperature-, or pump-wavelength tuning. This spontaneous "decay" of the pump photon, formerly studied in 1961 [6] and also known as Parametric Fluorescence (PF), is a quantum phenomenon associated to zero-point fluctuations in signal-idler inputs. These fluctuations, interacting with the pump field via the %(2), locally induce a nonlinear polarization which radiates in all the possible directions, and at all the possible frequencies between 0 and cq, . However, output radiation only grows for those frequencies and directions which, being close to the PM condition, experience constructive interference of the nonlinearly-radiating dipoles. The OPO is an OPA within a cavity, and it can operate in either cw or pulsed regime. In the most common configurations, the cavity mirrors are transparent at (q,, and highly reflecting at en, and/or

157

or (Of . In the presence of a pump beam, quantum noise triggers PF which, upon repeated reflections on the cavity mirrors, is parametrically amplified at subsequent passes through the nonlinear crystal. An analogy is apparent between the OPO and the laser, with the roles of spontaneous (stimulated) emission played by PF (OPA). Relative to the laser, however, the OPO exhibits a few major differences: two coherent beams are emitted at different wavelengths, which can be tuned by varying crystal angle, temperature, or pump frequency; parametric amplification only occurs as pump, signal and idler co-propagate, implying that the oscillation threshold occurs when single-pass gain g equals the roundtrip losses (as in a ring-cavity laser). Efficient frequency generation requires high intensities and long interaction lengths. For a pump laser beam with a given power, the former requirement implies tight focusing (at the price of large diffraction and small effective interaction length), whereas the latter implies loose focusing (long Rayleigh range). These conflicting requirements are simultaneously satisfied in optical waveguides, where light is confined in a highrefractive-index channel surrounded by a low-refractive-index medium [7]. In such waveguides Maxwell equations result in an eigenvalue problem, with two discrete sets of solutions (fim „} representing the propagation constants along z of two orthogonal sets of guided modes. In each set of modes there is a prevailing polarization component of the electric field, along either the horizontal transverse axis (y : TEmn) or the vertical transverse axis (x : TMmn). The transverse distributions of the guided modes are stationary waves described byfmn(x,y), with m (n) zeroes along x (y). Through the same procedure leading to Eq. 7, guided-wave SHG efficiency is found to be 77 =J]_^(0)L2 sine2 (Aj8L/2), with ^ =7]0/Pffl(o)L2 =C2/Aeff [W'cm2], and the effective area

* = \L K (*> For Type-I SHG, the mismatch Afts^mn(2a))-^m'n{o}) can be expressed in terms of effective indices N=fi/k0 as A/3=ko((a)[Nmn(2(o)-Nm'n'(a])], and the corresponding PM condition is Nmn(2co)=Nm'n'((o). Since N is higher for stronger confinement, and usually waveguides are much more confining in the vertical direction than in the horizontal direction, in general its approximate calculation can be carried out referring to the corresponding planar waveguide. The corresponding one-dimensional (ID) problem is easier to handle: eigenvalues and eigenvectors are identified by one index m, the guided modes have pure transverse-electric (TE) or transverse-magnetic (TM) polarization, and their profiles are as in Figure 3, with m zero crossings along x axis. As the thickness-towavelength ratio h/A. increases, guided modes get more confined, and higher-order modes become guided. This is an effect of modal dispersion, which is also exemplified in Figure 3, for a GaAs waveguide. In principle, PM in waveguides can be pursued with the same techniques that prove useful for bulk media. However, once a waveguide is fabricated, BPM is constrained by the substrate cut and the channel orientation. This often results in waveguides cut for noncritical PM (0=90°), with the consequent need of a temperature control. With respect to bulk, moreover, waveguides offer an additional PM option, which stems from their multimode dispersion. Since e.g. the PM condition for Type-I SHG in (planar) waveguides

158

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0.1

3.0

0.3

0.5

0.7

0.9

1.1

1.3

1.5

GaAs thickness h (urn)

(a)

(b)

Figure 3 TE modal profiles (a) and dispersion (b) in an air / (1 .5 \un) GaAs / Alo.3Gao.7As planar waveguide. 3.5

-n,(2co)

3.3 (D

S

3.2 3 1

-

3.0 n,(2w)

2.9 ns(TE2(2

0

x(2)

0

x 0

x(2)

0

A=2LC

QPM (-

A=2LC

x(2) -X' ' 2

X'2'

-x'2>

X'2'

-x2 laser pump beam propagating normally to the plates [19]. In the mid-infrared, the related scattering losses at each GaAs-air interface could be reduced only several years later, with the diffusion-bonding technique [20,21]. Alternative QPM implementations consist in GaAs epitaxial regrowth over a template substrate with periodically reversed orientations [22]. To date, the templates have been prepared with two main techniques: 1) a "flip-chip" procedure introduced by Yoo et al. [17,18] and 2) a selective GaAs/Ge/GaAs heteroepitaxy [23]. 2) The first method of this class consists in creating an artificial %(2) through asymmetric quantum wells [24,25]. In AlGaAs this asymmetry makes d is =10 pm/V [26], and allows (-1/+1) QPM schemes, through sequences of x(2> domains with alternating sign [27]. (0/1) QPM was also implemented in AlGaAs waveguides through asymmetric-quantum-well disordering, i.e. periodically destroying the asymmetry-induced nonlinearity along propagation [28]. The second method consists in (0/l)-modulating the d14 bulk coefficient along a GaAs/AlAs superlattice waveguide, through selective superlattice disordering along propagation [29]. 3) Reminiscent of band-gap engineering [30], today semiconductor technology grants a true refractive-index-engineering approach to integrated-optics design. Refractive-index engineering is possible in a multilayer stack, according to the textbook notion of form birefringence (FB) [31]. FB was formerly proposed for BPM SHG in a GaAs/AlAs stack, with a CO2 pump [32]. Due to small index contrast nQaAs-nAiAs and high GaAs dispersion, shorter pump wavelengths could not be used until AlAs oxidation technology emerged in the early 90's, providing

161

the low-index AlOx [33]. Besides triggering significant breakthroughs in optoelectronic technology [34,35], AlOx has recently allowed form-birefringent phase matching (FBPM) in AlGaAs multilayer waveguides [36]. The first two approaches share a high fabrication complexity and the related generation efficiencies are disappointing. This is due either to high scattering losses in the near infrared (1), or to poor %(2) modulation (2). We therefore detail briefly the AlOx-based method: the related structure is fabricated through the double etching of a planar waveguide (with AlGaAs claddings and GaAs/ALAs multilayer core), which is epitaxially grown on a (100) GaAs wafer. As can be appreciated in Figure 6, the first etching defines a ridge for mode confinement. The second etching defines a mesa for the lateral oxidation of AlAs layers (see inset). After the oxidation, carried out at 400°-500°C in a water vapour atmosphere, the waveguide exhibits a strongly birefringent GaAs/AlOx core [37]. With AlAs "" ..

guided mode oxidation —+

ridge

AlGaAs / ••*— oxidation

Figure 6 Double-etched AlGaAs multi-layer structure for optical guidance and oxidation.

this type of waveguide, both DFG and SHG have been reported [38,39]. Recently, mainly due to the growing importance of environmental issues, an increasing interest has arisen towards infrared tunable compact sources operating at roomtemperature. Whereas traditional lead-salt diodes are being rapidly superseded by quantum cascade lasers [30] and antimonide lasers [40], waveguide OPO's are an additional competitor, owing to their wide tunability, integrability, and cw operation. To date, due to the unique combination of mature waveguide technology, very low losses and good nonlinearity, all the operating integrated OPO's have been fabricated in LiNbO3 [41,42], but of course their semiconductor counterparts are greatly appealing. Within this framework, parametric fluorescence in GaAs waveguides has recently attracted a great deal of interest, since it constitutes the basic element towards an integrated OPO. Specifically, PF has been demonstrated in the above AlOx-AlGaAs waveguides, with a cw Ti:Sa source and a »3 mm long waveguide [43]. As the pump wavelength was varied around 1.06 urn, the total PF power in the 1.9 to 2.5 um spectral window varied as in Figure 7a, with the output power depending linearly on the pump power (TJPF = 4> = TT from the Bell state |_). This can be demonstrated by referring to Fig. 2. Starting from the condition d — R and neglecting for simplicity the BBO thickness, a displacement Ad — OOl = AAl ^ 0 determines different optical paths of the UV beam, 2(OAf), and of the photon pair generated toward the left in Fig.2 and reflected by mirror M, 2(OJ5/ + BlC}. The factor 2 which compares in the second optical path accounts for the two photon emission over two symmetric directions. We can calculate the phase difference: 4,

Xp

A

A

function of the distances OAl, OB/ and BlC. Since OA = OB = Of At = OiBl — R, we have

OAl = R + Ad.

(4)

By approximating all = at ~ a and applying the Carnot theorem to the triangle OO/B/, we find the following expression for OB/: OB/ = y(Ad) 2 + R2 + 2R&dcos(a).

(5)

Finally, we have BlC — OllC + OltBl, where O//C and OnBl are obtained from a careful analysis of Fig. 2: 0,,C

where

=

cos (all)

=

cos (a)

'

^'

175

«0

«8

Figure 2. Scheme representing the optical path difference within the single arm interferometer.

Figure 3. Maxima and minima interference fringe distribution vs. the BBO displacement.

The expression of OllBi is again obtained by applying the Carnot theorem to the triangle OB/O/I. By approximating a —> 0, we have (OllBi) = -(OOn) cos(a) + (OBi)

(8)

By the above equations it's easy to find that the transition |$+) —> |^>-} corresponds to a a value of the displacement |Ad| = 60^m, in very good agreement with the experimental results. Note that a value A.d ^ 0 implies a lateral displacement OC of the reflected SPDC beams (Fig. 2). Because of the intrinsic cylindrical symmetry, OC may be viewed as the radius of an annular-shape region which grows with Ad on the BBO plane. This geometrical effect makes distinguishable the two emission cones, it introduces a spatial decoherence which becomes relevant as far as OC becomes comparable with the diameter of the active pumped region of the crystal (~ ISO^m). In our experimental conditions, OC ~ ^>Ad, we have observed that any coherent superposition on the state vanishes for |Ad| ^ 600/^m. This is confirmed by the results of Fig. 3, where the maxima and minima of coincidences are reported as a function of the BBO displacement Ad.

176

4 4-1

Experimental results Violation of Bell inequalities

The Bell state |$_) over the whole Entanglement- ring has been adopted to test the violation of a Bell inequality by the standard coincidence technique 17, by the following angle orientations of the A (1) and B (2) sites l?-analyzers:{^i = Q,0{ — 45°} and {6-2 = 22.5°, 0'2 — 67.5°}, together with the respective orthogonal angles: WiS^i"1 f and 102 i^-i" (• By these values, the standard Bell-inequality parameter could be evaluated 15:

92) + p(0'1,92)\

(9)

where

and C(6i,02) is the coincidence rate measured at sites A and B. The measured value S — 2.5564 ± .0026, obtained by integrating the data over 180s, corresponds to a violation as large as 213 standard deviations respect to the limit value S = 2 implied by local realistic theories 12>18. Fig. 4 shows the "^-correlation obtained by varying the angle 6\ in the range (45° — 135°), having kept fixed the angle 0? — 45°. The interference pattern demonstrates the high degree of polarization entanglement of the source. The measured visibility of the coincidence rate, V > 94%, gives a further strong indication of the entangled nature of the state over the entire cone of emission, while the single count rates don't show any periodical fringe behaviour as expected. The experimental data are compared in Fig. 4 with the dotted line which correspondes to the limit boundary between the quantum and the "classical" regimes and with the theoretical (continuous) curve expressing the ideal interferometric pattern with maximum visibility: V = 1. We have characterized the robustness and the brightness of the source by measuring coincidences for different values of the of the radius r of the iris diaphragms (I.D.) (Fig. 1). This corresponds to select different portions of the entanglement ring, with area A = 2DS arcsin(^). The experimental results of Fig. 5 demonstrate that a coincidence rate of more than 4000 sec"1 are measured over the entire Entanglement ring with a still relevant value of visibility. By taking into account the UV pump power (P ~ lOOmW) and the overall efficiency of the apparatus (optic transmissions + detector efficiencies), we can evaluate that more than 2 • 105s~1 entangled photon pairs are generated. 4-2

Generation and characterization of mixed states

Because of the peculiar spatial superposition property of the output state, the present apparatus appears to be an ideal source of any bi-partite, two-qubit entangled state, either pure or mixed. In particular of the Werner state: pw = £>|*-) (*-| + ^j2! consisting of a mixture of a pure singlet state |\&_) = 2~5 {\HV) - \VH)} with probability p (0 < p < 1) and of a fully mixed-state

177

A-©|t

(Dcg)

Figure 4. Measurement of the polarization entanglement.

The selected state is

• -

*

0.6-