Beam-shaping longitudinal range of a binary diffractiveoptical element

Renaud de Saint Denis, Nicolas Passilly, Mathieu Laroche, Tayeb Mohammed-Brahim,and Kamel Aït-Ameur

An experimental and theoretical investigation of laser beam shaping using a simple binary diffractiveoptic is presented. Beam tailoring has been characterized by the experimental determination of tworelevant parameters: beam propagation factor M 2 and the beam-shaping longitudinal range, whichrepresents the propagating distance for which the tailored beam remains nearly unchanged. © 2006Optical Society of America

OCIS codes: 050.1380, 140.3300.

1. Introduction

In many experiments involving lasers, as well as inmilitary, industrial, and medical applications, it issometimes useful or even necessary to take advan-tage of the use of laser beams with a specific intensityprofile distinct from the usual Gaussian shape. Apossible solution is to convert a standard Gaussianlaser beam into a given profile by using a transparentdiffractive optical element (DOE). Laser beam shap-ing is a topic that has been intensively developed fora long time. For instance, a Gaussian beam can betransformed into a uniform beam by using a binarydiffractive optic1–3 or a distributed phase plate.4,5

Note that flat-top profiles can also be obtained byusing refractive optics.6–8 Laser beams with uniformintensity profiles are useful for material processingand nonlinear frequency conversion. Other profilessuch as the doughnut profile are also desirable fortrapping the small particles9 that can be obtainedfrom the conversion of a Gaussian beam.10,11

Usually, DOEs are designed so that the laser beamtailored gives rise to the desired pattern in the focalplane of a converging lens. However, it is importantto note that because of diffraction the beam profilewill deteriorate along the propagation direction. Con-sequently, two important characteristics of the beamconversion may be considered:

(i) The longitudinal range on which the beam pro-file remains nearly unchanged on both sides of thefocal plane. Here this particular distance will be calledthe beam-shaping longitudinal range (BSLR).

(ii) The beam propagation factor M2.

The two parameters above are relatively unremarkedin the literature dealing with laser beam shaping. Weconsider them for the case of a simple DOE when theincident beam is Gaussian. The latter is a binarydiffractive optical element, referred to in this paperas a � plate, which introduces a � phase shift in thecentral part of the incident beam. The transmission(amplitude ratio) of the � plate of radius �0 is given by

���� ���1 for � � �0

�1 for � � �0. (1)

The normally incident field on the � plate is thatof a collimated Gaussian beam with axial symmetrywhose unit-amplitude distribution is described by

Ein��� � exp���2

w2�, (2)

where W is the width of the incident beam. An im-portant parameter that governs the BSLR and M2

values is the ratio � � �0�W.

R. de Saint Denis, N. Passilly, M. Laroche, and K. Aït-Ameur(kamel.aitameur@ensicaen.fr) are with the Centre Interdisciplinaire deRecherche Ions, Lasers, Equipe Lasers, Instrumentation Optique etApplications, Unité Mixte de Recherche 6637, Ecole Nationale Su-perieure d’lngenieurs de Caen and Centre de Recherche, 6 BoulevardMaréchal Juin, F-14050 Caen, France. T. Mohammed-Brahim iswith the Institut d’Electronique et de Télécommunications deRennes, Unite Mixté de Recherche 6164, Bâtiment 11B, Campusde Beaulieu, F-35042 Rennes, France.

Received 13 March 2006; revised 21 June 2006; accepted 23 June2006; posted 26 June 2006 (Doc. ID 68875).

0003-6935/06/318136-06$15.00/0© 2006 Optical Society of America

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The motivation for considering a � plate, as theDOE under study through the BSLR and M2 of thebeam tailored, is based on the remarkable properties ofthe � phase step discontinuity relative to its simplicity.Indeed, a circular � plate can be used to convert aGaussian beam into a super-Gaussian of the order of6, doughnut or ring-shaped profile, depending on thevalue of parameter �.2,3 Similar 1D beam shaping hasbeen achieved by using a � phase slit,12 while a Gauss-ian beam has been transformed into a Hermite–Gaussian TEM01 beam that keeps its profile from thenear to the far field, by using a � phase step.13 Anotherability of the � phase slit is the transformation of anelliptic Gaussian beam, in the near field, into a circularquasi-Gaussian beam in the far field.14 The � platecan also be used inside a laser cavity, for instance,to improve the transverse mode discrimination,15–17

or to speed up the dynamics of a self-Q-switchedCr:LiSrAlF6 (Cr:LiSAF) laser so that its beam bright-ness is improved in a ratio of approximately 2.5 with-out increasing the laser threshold.18

2. Experimental Setup

The setup used for reshaping a Gaussian beam usinga � plate is shown in Fig. 1. An SP-980-M CCD cam-era and a Spiricon LBA-400PC beam-imaging systemhave been used to record diffracted beam patterns onboth sides of the focal plane of an imaging lens of focallength f � 0.75 m. The latter, placed quite close afterthe � plate, has been used to restrict the longitudinalextent of the far-field region. The distance z betweenthe (� plate � lens) and the CCD planes will be ex-pressed in units of Rayleigh distance zR of the focusedbeam given by

zR �f2

W2, (3)

where � 532 nm is the wavelength of the incidentbeam. We have also measured the beam-propagationfactor of the diffracted beam emerging from the �plate by using a beam scanner (Model M2-200,SPIRICON) equipped with a Cohu Model 4800 CCDcamera.

The DOE has been fabricated by the microelectron-ics group of the Institut d’Electronique et de Télécom-munications de Rennes. It consists of a series of 12 �plates of diameters ranging from 0.1 to 3.6 mm inorder to vary parameter � for a constant incidentbeam width W. The 12 � plates were fabricated byfirst coating a 70 mm � 70 mm Corning 1737 glassplate by a thin film of indium tin oxide (ITO) havinga thickness that produces the desired � phase shift at � 532 nm. This ITO coating was obtained by mag-netron sputtering at 13.56 MHz with a bench 822(Material Research Corporation, Orangeburg, NewYork.). The second step consists in patterning the 12plots by standard UV photolithography and chemicaletching by using the Merck solution 11 881-9028.

Relative to the incident beam at � 532 nm, weused two kinds of source. The first one is a passivelyQ-switched Yb-double-clad fiber able to generate

pulses having a duration of 2.7 ns and a peak powerof approximately 9 kW at a repetition rate of 4 kHz.19

In a second step the laser beam at � 1064 nm isfrequency doubled by a type II KTP nonlinear crystal.The beam propagation factor of the green laser beamwas measured to be M2 � 1.16.

The second source used in the experiments wasbased on a cw doubled Nd:YVO4 (Beijing Viasho Tech-nology Company) with an output power of 30 mW butunfortunately a low beam propagation factor that wemeasured at M2 � 3.5. Consequently, the beam prop-agation factor was improved by injecting the beaminto a single-mode fiber (Thorlabs #460HP) deliver-ing an approximately 3 mW beam with M2 � 1.25.We obtained similar results with the two sources.

3. Results and Discussion

The intensity patterns in the focal plane of the imag-ing lens, resulting from the diffraction of the incidentGaussian circular beam on the � plate, are computedby using the Fresnel–Kirchhoff formula3 and areshown in Fig. 2. These plots demonstrate that beamshaping is sensitive with parameter � since, as itdecreased, the intensity pattern changed from aflat-top profile to a doughnut profile and then to aprofile that resembled the symmetrical Laguerre–Gauss TEM10 mode.

Fig. 1. (Color online) Experimental setup for measuring, on bothsides of the lens focal plane, the longitudinal evolution of intensitypatterns of the beam tailored by the � plate.

Fig. 2. Theoretical plots showing that the � plate is able to trans-form a Gaussian beam into a flat-top profile, a doughnut profile, ora TEM10-shaped profile, depending on the value of parameter �. ris the radial coordinate in the diffraction plane. Intensity patternshave been calculated in the focal plane of the imaging lens.

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Now the aim is to make a comparison between thedifferent kinds of intensity profiles shown in Fig. 2through their BSLR and M2 factors. The first intensitypattern in Fig. 2 that we consider is the flat-top profileobtained for � � 1.25, which can be fitted by one superGaussian of the order of 6. Figure 3 shows the exper-imental evolution of the beam profile on both sides ofthe focal plane from z � f to z � f � 4zR. It is seen thatthe beam profile does not remain flat for a distancehigher than zR from the focal plane. To evaluate theBSLR, the longitudinal scale has to be spread asshown in Fig. 4. One notices that the BSLR cor-responding to the super-Gaussian profile is onlyapproximately �zR�5. To be more concrete on thepractical consequence of such a property, let us con-sider the case where a solid state laser is longitudi-nally pumped by a flat-top beam instead of aGaussian beam. It is clear that the thickness of theactive medium has to be smaller than zR�2.5 to keepup radially uniform pumping. If one considers, forinstance, a pumping wavelength of � 810 nm (forneodymium-doped material) and a collimated Gauss-ian pump beam of width W � 2 mm incident on afocal length f � 50 mm lens, one finds a Rayleighrange zR � f2��W2� of approximately 3.2 mm. Ifone sets a � plate in front of this lens so that� � 1.25, it is guaranteed that the laser mediumwould be pumped almost uniformly only if its thick-ness is smaller than approximately 1.2 mm.

Other types of interesting intensity profile are thedoughnut and the ring obtained in the focal planewhen � is approximately 1.1 and 0.9, respectively, asshown in Fig. 2. Such beams with an annular inten-sity profile are particularly well adapted for the trap-ping of low-index microscopic particles20,21 or the

guiding of atoms or particles.22 The BSLR value is animportant quantity that limits the volume of trap-ping. Its value has been found to be of the order of�zR�3. In fact this distance should correspond to avery small one since the microscope objective, typi-cally used in trapping applications, is characterizedby a magnification of the order of 60 to 100.

The last beam that we now consider has a profilethat resembles the symmetrical Laguerre–GaussTEM10 mode and is obtained in the focal plane for� � 0.5 as shown in Fig. 2. The longitudinal evolutionof the beam shaping on both sides of focal planez � f is displayed in Fig. 5 over a range of �4zR. It isseen that the BSLR is approximately 2zR before thefocal plane and 3zR after. This value is remarkablyhigh since it is approximately 12 to 20 times theBSLR achieved for the doughnut or flat-top profiles.

Fig. 3. Experimental evolution of the tailored beam profile, for� � 1.25, from z � f to z � f � 4zR. Note that, in plane z � f, theintensity distribution is a flat-top profile that can be fitted by asuper Gaussian of the order of 6.

Fig. 4. Evolution of the tailored beam profile for � � 1.25 fromz � f to z � f � 0.3zR.

Fig. 5. Evolution of the tailored beam profile for � � 0.5 fromz � f to z � f � 4zR. This value of � is well adapted for theconversion of a TEM00 into a TEM10 beam.

8138 APPLIED OPTICS � Vol. 45, No. 31 � 1 November 2006

The reason for such behavior is that the reshapedbeam for � � 0.5 is very close to a true TEM10, i.e., aneigenmode for which the shape does not change asit propagates. The intensity profile of a symmetricTEM10 Laguerre–Gauss mode is given by

I10 � �1 � 2�2

W2�2

exp��2�2

W2�, (4)

which shows that the position of the null intensityring is such that ��W � 0.707. Intuitively, but incor-rectly, one could believe that the best value of param-eter � that allows the transformation of a TEM00 intoa TEM10 is � � 0.707. On the plots in Fig. 6 thatdisplay, for � � 0.707, the longitudinal evolution ofthe beam shaping, it is seen that the transformationTEM00 → TEM10 is never achieved. We have foundthat the best value of parameter � allowing the trans-formation TEM00 → TEM10, with the highest possiblevalue for the BSLR, is � � 0.5 (Fig. 5).

The second beam-conversion parameter that has tobe discussed is the beam-propagation factor M2 of thebeam emerging from the � plate. Generally, in beam-shaping literature, authors report information aboutthe calculation and fabrication of DOEs that are usedto achieve a particular beam transformation; how-ever, the question of the degradation of beam qualitythat arises from beam diffraction is usually not con-sidered. Most of the laser applications require notonly a high power level P, but also a good beam-propagation factor M2. The ability of a laser beamto produce high intensities, after being focused, incombination with a large Rayleigh range can be ex-pressed by its brightness B defined as23

B �2P

2�M2�2. (5)

Consequently, it is important to determine thechanges in the beam-propagation factor M2 of the

Gaussian beam diffracted through a � plate. The nu-merical evaluation of the M2 factor is based on the fielddecomposition in terms of Laguerre–Gauss functionsthat constitute an orthonormalized set of eigenfunc-tions.24 The result is shown in Fig. 7, which displaysthe evolution of the M2 factor as a function of param-eter �. The solid curve represents the theoretical val-ues of the M2 factor, while the squares represent theexperimental measurements. The theoretical and ex-perimental results are in a good agreement. The mainpractical information that can be deduced from Fig. 7is that the conversion of a Gaussian beam into aflat-top beam using a � plate for � � 1.25 leads to abrightness reduction in a ratio of approximately 4.

Another question of interest concerns the ex-pected shift of the lens focal plane. Indeed, thiseffect has already been established in the case of ahard-apertured converging lens.25,26 If one takes

Fig. 6. Evolution of the tailored beam profile for � � 0.7 fromz � f to z � f � 4zR.

Fig. 7. (Color online) Evolution of M 2, the beam-propagation fac-tor of the beam emerging from the � plate, as a function of param-eter �: solid curve, theoretical; squares, experimental.

Fig. 8. (Color online) Variations of the on-axis intensity Id�0, z� asa function of the normalized propagation distance.

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into account the similarities between the diffractedfield characteristics in the case of a hard aperture orin the case of a � plate,3 one can expect a longitu-dinal shift of the on-axis intensity maximum asparameter � varies. The results shown in Fig. 8display the evolution of the on-axis intensity Id�0, z�as a function of the normalized propagation distancez�zR. The following points should be recognized fromexamination of Fig. 8. In the case where � 0.5, it isobserved that the axial intensity is maximum in thefocal plane, while for � � 0.5 the axial intensity hastwo maxima located on both sides of the focal plane.Consequently, the couple (� plate � converging lens)can be considered as a bifocal device. Note that thefirst maximum, occurring before the geometric focus,is higher than the second one. Figure 9 shows thatbetween the bifocal points the beam is ring shaped for� � 0.9, with a BSLR of approximately �0.5zR. The3D plot in Fig. 9 shows very clearly the relative im-portance of the different intensity maxima. Anotherfactor that characterizes a DOE is its diffraction ef-ficiency. Typically, the intensity pattern in the focalplane of the lens is made up of a central useful pat-

tern surrounded by ripples. The diffraction efficiency� for the � plate is calculated from the ratio of thepower in the useful pattern to the total power. Fig-ure 10 shows that � varies with parameter �, and hasa value of approximately 90% for the flat-top profile�� � 1.25�.

4. Conclusions

Considerable attention has been devoted for manyyears to the design of diffractive optics for the conver-sion of a Gaussian beam into another pattern. Wehave considered a � plate, which is probably one ofthe simplest diffractive optical elements, but is notwithout interest. In particular, the same device canbe used to transform a Gaussian beam into a super-Gaussian profile, a doughnut profile, or a TEM10-shaped profile. We have considered, for each of thesebeam transformations, a parameter called a beam-shaping longitudinal range, which expresses the lon-gitudinal range on which the beam profile remainsnearly unchanged. The second factor of interest thathas retained our attention is the beam-propagationfactor M2 of the tailored beam. These two parametersare generally not considered in beam-shaping stud-ies, and we hope that our experimental and theoret-ical results, obtained with the use of a � plate, willconvince everyone of the utility of their use in thecharacterization of any beam-shaping device.

This work was carried out within the framework ofthe Contrat de Plan Etat-Région “Pôle ITIC,” and theauthors acknowledge the support of the DélégationRégionale à la Recherche et à la Technologie BasseNormandie, the Conseil Régional Basse Normandie,and the Fond Européen de Developpement Régional.

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