Reshaping collimated laser beams with Gaussian profileto uniform profiles

Chang-Yuan Han, Yukihiro Ishii, and Kazumi Murata

A set of holographic filters was developed to convert the Gaussian intensity distribution of a collimated laserbeam into a uniform one. The design and the fabricating method of the holographic filters are presentedand experimental results are shown.

1. Introduction

In many laser applications the conversion of theGaussian distribution into a uniform one is useful.Many papers have recently been written on this subject.In laser fusion and laser projection printing it is desir-able to obtain a uniform intensity at the focal plane ofthe lens. Lacombat et al.1 described a method in whicha Gaussian laser beam is split into four beams which arethen recombined with proper lateral displacements sothat the total beam irradiance becomes approximatelyuniform. Lee2 suggested a method using a phase filter.For laser radars, Veldkamp et al. 3 '4 described a methodof obtaining a uniform far-field intensity in which dif-fraction gratings are used.

In optical data processing and optical gauging, werequire a collimated uniform laser beam that maintainsthe phase of the incident wave front. For this purpose,a beam expander is generally used which consists of amicroscope objective of high magnification, a pinhole,and a collimating lens. h5 described a beam expanderin which a high-magnification microscope objective isnot needed, but the collimating lens is made of neutraldensity glass to produce the Gaussian absorption profileso that the output beam becomes uniform. These beamexpanders have a low energy efficiency. Rhodes andShealy6 designed an afocal system having high energyefficiency to obtain a collimated uniform laser beamfrom the collimated Gaussian laser beam. This systemconsists of two aspherical lenses, and their fabricationis rather difficult. Shafer7 designed an afocal systemwhich consists of four spherical lenses that were easily

The authors are with Hokkaido University, Department of AppliedPhysics, Sapporo, Hokkaido 060, Japan.

Received 18 June 1983.0003-6935/83/223644-04$01.00/0.© 1983 Optical Society of America.

fabricated. However, since the profiling of the beamis accomplished by balancing the spherical aberrationsof the four spherical lenses, the uniform output is ob-tained only between the l/e points.

In this paper we show that, because a computer-generated hologram (CGH)8 can easily generate a re-quired wave front, the aspherical lenses in Rhodes'safocal system may be replaced by CGHs. Moreover, thediffraction efficiency of volume phase holograms canbe high, so that high diffraction efficiency could be ob-tained. Therefore, to realize a collimated uniform laserbeam from an incident collimated Gaussian laser beam,we suggest a new afocal system using a set of CGH phasefilters. We design these holographic filters to gain theuniform output between the 1/e2 points, which is widerthan that between the lie points. This paper will il-lustrate the theory and design of the holographic filtersand the method of fabrication and will describe theexperimental setup and the intensity-distributionmeasurement. Finally, the holographic filters aretested in an interferometer.

II. Theory and Design of Holographic Filters

Referring to Fig. 1, let a collimated Gaussian laserbeam be incident on the first holographic plane P1 .The first hologram deflects the incident rays so that theintensity distribution is converted to a uniform onewhen the laser beam reaches the second holographicplane P 2 . The second hologram deflects the rays so thatthe uniform distribution leaves plane P 2 as a collimatedbeam.

Let the distance of the incident ray to the center ofthe beam in plane P1 be r. Then the incident Gaussianintensity distribution i (r) can be represented as

i(r) = e-r 2/2a

2, ()

where the central intensity is normalized to unity andae is a constant. If the intensity of the incident colli-mated beam drops to 1/e2 at a circle of radius ro,

3644 APPLIED OPTICS / Vol. 22, No. 22 / 15 November 1983

TCO Ir

z

P2

Fig. 1. Afocal system for converting the beam intensity.

i(r) = e -2r2lr 2 (2)

The energy E(r) collected in a circle of radius r is

E(r) = 27r fr i(r)rdr = 7rrO(1- e-2 r2

/r). (3)so 2 3

The collected total energy of the laser beam is E(o) =irr2/2. The normalized energy ratio is

E(ro)= 1-1/e 2 = 0.865. (4)

If the beam inside a circle of radius r on plane P1 reachesa uniform intensity o in a circle of radius R on plane P2 ,equating both light energies yields

2ir f i(r)rdr= rR2. (5)

Substituting Eq. (2) into Eq. (5), we obtain the rela-tionship between the radius r and R for converting theGaussian intensity distribution on P1 into the uniformintensity on P2:

R = rO2 (1 - e-2r2/r2)] 1/2

[2r = ino1 - 2 2/r2 1I

If radius r of the incident beam and radius Ro of theoutput beam are taken to be equal, i.e., r = R0 , from Eq.(6) the uniform intensity is = 1/2 (1 - 1/e2 ) = 0.432.Let the distance between two planes P1 and P2 be Z; theobliquity of the ray which emerges from a point at r onplane P1 and reaches a point at R on plane P2 is (R -r)/Z. Since this obliquity is equal to the derivative ofthe wave front W1 (r) which leaves from plane P 1 , wehave the following differential equation:

dWl(r) R - rdr -Z W()0

and CGH2, from the calculated wave fronts W1(r) andW2(R), respectively. When the CGH1 is set at plane Pand a plane wave enters, the desired wave front W1(r)will be reconstructed. The reconstructed wave frontWi(r) propagates to plane P2 and generates the wavefront W2(R). With CGH2 set at plane P2, a plane wavefront will be reconstructed. The collimated Gaussianlaser beam which enters plane P1 is converted into anewly collimated uniform laser beam with holographicfilters.

Ill. Fabrication of Hologaphic Filters

By solving the differential Eqs. (8) and (9) with thehelp of Eq. (6) or (7) by a Runge-Kutta integrationroutine, we can extract the wave fronts W1(r) andW2(R). Here, the radii of incident and output beamsare taken as 4 mm. The filter separation Z is 500 mm,and the radiation wavelength is 632.8 nm. The nu-merical results indicate that the maximum amounts ofthe optical path difference become W = W2 = 5.7 atr = R = 4 mm. With oblique reference plane wavefronts added to the wave fronts W1(r) and W2 (R), thecontour lines with 1 - X intervals are computed anddrawn by a computer-controlled plotter. The obliquityof the reference plane wave front determines the num-ber N of contour lines of the CGH. We have taken N= 96. After plotting, these contour lines are photo-graphically reduced to the required-size high-resolutionplates. The synthesized CGH1 and CGH2 are shownin Figs. 2(a) and (b), respectively. To test these CGHs,the optical setup shown in Fig. 3 is required, namely, aninterferometer using the +1st-order diffracted wavesfrom the CGHs. In Fig. 3, 0 is a microscope objective,PH a pinhole, and C a collimating lens. The grating

(6)

(7)

(a)

(8)

In a similar way, wave front W2(R) reaches plane P2,

dW2(R) R-rdR Z

W2(0) = 0.

(b)

(9)

Solving these differential equations, the wave frontsW1(r) and W2(R) can be determined, and we can syn-thesize the two computer-generated holograms, CGH, Fig. 2. Computer-generated holograms: (a) CGH 1 and (b) CGH 2 -

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y

M He-e LASER

0 C Li L2 L3 , ' 14 : 14 -M2

PH Grating Di CGH D2 F

Fig. 3. Optical setup for testing the CGHs.

(a) (b)

Fig. 4. Interferograms (a) and (b) with dlst-order diffracted wavesfrom CGH1 and CGH 2 , respectively.

Ar LASER

Fig. 5. Optical setup for recording the DCG hologram.

MM He-Ne LASER

-I =.= - I -

Fig. 7. Gaussian intensity input profile (upper) and the uniformoutput profile (lower)./~ ~ ~ ~ ~~~~, , ,F

Fig. 6. Experimental arrangement with an interferometer fordemonstrating the feasibility of the method.

(a) (b)

Fig. 8. Interferograms for testing plane mirrors using (a) a collimatedGaussian beam and (b) a collimated uniform beam.

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frequency is 12 lines/mm and L1 , L2 , L3 , and L4 are theFourier transform lenses. Diaphragms D1 and D2 onlyallow the ±1st-order diffracted waves, correspondingto the reconstructed wave of the CGH and its conju-gated wave, to pass. These conjugated wave frontsinterfere and the fringes are recorded in plane F. Fig-ures 4(a) and (b) show the interference fringes of the±1st-order diffracted waves from CGH1 and CGH2,respectively. The total number of fringes indicate thatthe maximum optical path difference between theconjugate wave fronts is 11.4X, which shows twice theamount of the computed results, i.e., W1 = W2 = 5.7Xat r = R = 4 mm. Given the pattern of these interfer-ence fringes, we know how the wave front W1 (r) isconverted into the wave front W2(R) by the propagationdistance Z between P1 and P2 -

To increase the carrier frequencies of the holograms,we optically generate new holograms with reconstruc-tion waves from the CGHs using the setup shown in Fig.5. The diffracted wave front from the CGH is imagedat the optically holographic recording plane H with aone-to-one telescope. The diaphragm D is positionedat the frequency plane so that only the first-order dif-fracted wave from the CGH is passed to the recordingplane H. The optical holograms H, and H2 are inter-ferometrically made in sequence with these diffractedwaves from CGH1 and CGH2 and a plane wave with 100offset angle. These optical holograms H1 and H2 areused as filters in later experiments. Recording materialfor these holograms is dichromated gelatin (DCG).9Because the DCG is grainless, the diffraction efficiencycan increase while the coherent noise decreases. In thisexperiment transmission-type holograms are used andthe diffraction efficiency is as high as 30%. The CGHsare designed and generated at the 632.8-nm wavelengthof a He-Ne laser. The CGHs are then illuminated atthe 514.5-nm wavelength of an Ar laser, and the dif-fracted wave is used as the object wave for high effi-ciency recording in DCG. The reillumination of thehologram at the 632.8-nm wavelength produces no re-construction phase error in the conversion of the in-tensity distribution.

IV. Experiments

A feasibility demonstration of our method is shownin Fig. 6. A He-Ne laser is used as the light source. Amicroscope objective 0 is used with 4X magnificationand a 0.1 N.A., a 150-,um diam pinhole PH, and a100-mm focal length collimating lens C. The distancebetween the He-Ne laser source and the microscopeobjective is adjusted so that the collimated Gaussianbeam, which emerges from lens C, projects the /e2 in-tensity on a circle of ro = 4-mm radius. The intensitydistribution along the diameter of the beam is measured

by a scanning photomultiplier with a 0.5-mm circularaperture and the results are recorded on an X - Y re-corder. The upper curve in Fig. 7 shows the Gaussianintensity input distribution. Next, we place holo-graphic filters H1 and H2 at planes P1 and P2 , respec-tively, and remeasure the intensity distribution of thecollimated output beam behind plane P2. The lowercurve in Fig. 7 shows that measurement. The deviationfrom uniformity is less than ±5%.

Finally we compared the effect of the collimatedGaussian beam with that of the collimated uniformbeam on an interferometer. We show two experimentalresults. We took off the holographic filters H1 and H2shown in Fig. 6 and arranged the interferometer (whichconsists of the half-mirror HM, plane mirrors M 3 , M 4 )and a camera F behind the collimating lens C. We thenrecorded the interference fringe formed by the colli-mated Gaussian beam, and that result is shown in Fig.8(a). Then, we replaced the holographic filters H1 andH2 at planes P1 and P2, respectively, as shown in Fig.6. The new interference fringe formed by the uniformcollimated beam is shown in Fig. 8(b). Comparing theresults shown in Fig. 8(a) with those in Fig. 8(b), it isobvious that this setup of holographic filters is usefulin interferometric testing.

V. Conclusion

We have designed a set of holographic filters thatconverts collimated Gaussian laser beams into colli-mated uniform laser beams and have experimentallyverified their use. These holographic filters have thesame capability as aspheric lenses, and they are easierto fabricate. The remaining problem is to improve theholographic recording techniques for making opticalholograms to increase diffraction efficiency and reducecoherent noise.

The holographic filters might be used in other opticalmeasurements or processing, in which a Gaussian laserbeam is used as the light source.

References1. M. Lacombat, G. M. Dubroeucq, J. Massin, and M. Brevignon,

Solid State Technol., 23, Aug. 115 (1980).2. W-H. Lee, Opt. Commun. 36, 469 (1981).3. W. B. Veldkamp and C. J. Kastner, Appl. Opt. 21, 345 (1982).4. W. B. Veldkamp, Appl. Opt. 21, 3209 (1982).5. C. S. Ih, Appl. Opt. 11, 694 (1972).6. P. W. Rhodes and D. L. Shealy, Appl. Opt. 19, 3545 (1980).7. D. Shafer, Opt. Laser Technol. 14, 159 (1982).8. W-H Lee, "Computer-Generated Holograms: Techniques and

Applications," in Prog. Opt. 16,121 E. Wolf Ed. (North-Holland,Amsterdam, 1978).

9. D. Meyerhofer, "Dichromated Gelatin," in Holographic RecordingMaterials, H. M. Smith, Ed. (Springer, Berlin, 1980).

C.-Y. Han is on leave from the Changchun Instituteof Optics and Fine Mechanics, Academia Sinica, Changchun, China.

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