108 OPTICS LETTERS / Vol. 19, No. 2 / January 15, 1994

Diffractive optical element for mode shaping of aNd:YAG laser

James R. Leger, Diana Chen, and Zhong Wang

Department of Electrical Engineering, University of Minnesota, Minneapolis, Minnesota 55455

Received July 27, 1993

A diffractive laser cavity mirror is described that can customize the amplitude and phase of a laser mode. Thedesign of this diffractive element is shown for a square, flat-topped fundamental mode. The laser cavity hasa theoretical fundamental mode loss of only 0.08% and a second-order mode loss of 48.2%, resulting in highmodal discrimination. The fabricated mirror is tested in a Nd:YAG laser system. The resulting square flat-topped mode has an rms variation of 1.5% over the two-dimensional flat-topped region and a large discriminationagainst higher-order modes.

Conventional spherical mirrors are used in virtuallyall modern laser resonators. Although the resultingGaussian mode shape is sometimes desirable, thereare applications for which a different shape may bemore useful. In addition, the separation of adjacenttransverse modes is often small, and it is desirableto maximize this discrimination while maintaininga large mode volume. Amplitude spatial filteringhas been used to produce flat-topped laser modes.'More-complex mirror shapes have been used to tailorthe modal profiles of diode-laser arrays2 -4 and CO2lasers.5 -7 In this Letter we show an extension of thelatter technique that uses diffractive optical elementsto tailor the fundamental mode of a Nd:YAG laser.In addition, careful choice of cavity length and modalfilters can provide large mode separation.

The laser cavity consists of two diffractive mode-selecting mirrors spaced by a distance 1. The designof the diffractive mirrors is chosen to establish the de-sired mode as the fundamental mode of the resonatorsystem. Let the desired mode amplitude and phasejust past the first mirror be described by the complexfunction a(x, y). This can be expressed in terms ofits angular plane wave spectrum A(u, v) as

a(x, y) = LJ r: A(u, v)exp[j2w(xu + yv)]dudv,

where u and v are spatial frequencies.The distribution at the second mirror is given by

b(x',y') = f f A(u,v)exp[j2,r(xu + yv)]

x expj jkl[1 - (Au)2 - (Av)2T]21 dudv. (2)

If we construct the second mirror with a reflectancet2(x',y') of

t2(x',y') = b(x ') '(3)b(x', y')

where the asterisk indicates the complex conjugate,the return wave is given by

b (x',t 2(x', y') = b*(x', y')

= 7 f_ A*(u, v)exp[-j27r(xu + yv)]

X exp{-jkl[1 - (Au)2 - (Av) 2 ]"2 }dudv. (4)

Propagation back to the first mirror results in

7 7 A*(u, v)exp[- j2i7r(xu + yv)]dudv = a*(x, y).

(5)

If now the first mirror reflectance is chosen to be

t,(x,y) = a(x, y)a* (X, Ay

(6)

the original distribution a(x, y) has reproduced itselfafter one round trip in the laser cavity, therebyestablishing itself as a mode of the system.

Since the reflectances of the two mirrors in Eqs. (3)and (6) are phase only, they can be fabricated asdiffractive optical elements. By making these ele-ments sufficiently large, we keep diffractive lossesto a minimum, and the loss to the fundamentalmode is very small. This phase-conjugate cavity isreminiscent of resonators based on Brillouin scatter-ing or four-wave mixing.8 Note, however, that themode-selecting mirror phases are fixed, so this low-loss imaging occurs only for the desired fundamentalmode. The diffractive mirror is designed to be lossyto higher-order modes, making it an effective filterfor single spatial mode operation.

Because diffractive mirrors can be fabricated forvirtually any phase profile, the geometric shape of themode and its amplitude and phase profile are entirelyarbitrary. We have chosen to generate a square-shaped mode with approximately uniform amplitudeand phase given by a super-Gaussian function oforder 20:

F / \ 2 0 120 20a(x, y) = exp - -~-)exp -- -'L & w oi0k ( o I

(7)

where coo is 0.6 mm. From Eq. (6) it is apparent thatthe first mirror can be replaced by a plane mirror, and

0146-9592/94/020108-03$6.00/0 © 1994 Opticil Society of America

January 15, 1994 / Vol. 19, No. 2 / OPTICS LETTERS

5

.~4

a.m2

0-3 -2 -1 0 1 2 3

Distance Across Diffractive Mirror (mm)

Fig. 1. Phase profile of the mode-selecting mirror with a16-level phase quantization.

we need fabricate only a single nonplanar element.The phase of this element is calculated by Eqs. (2)and (3) for the chosen cavity length.34 6 We thenproduce a diffractive optical element by performinga modulo-2vr operation on the phase function andquantizing the result into discrete phase levels. Thephase profile of the central portion of the diffractiveelement is shown in Fig. 1.

We performed a Fox-Li analysis9 of the lasermodes to study the effect of the mirror phasequantization, laser cavity length, and mirror aperturesizes on the mode shape and mode loss. Thetheoretical mode produced by a 16-level elementwas very close to that of the ideal 20th-ordersuper-Gaussian beam, with sharp sidewalls andan rms ripple of less than 0.67% in the flat-toppedregion. The minimum linewidth was 50 ,um, makingfabrication quite straightforward.

The planar output mirror size of 1.3 mm X 1.3 mmwas initially selected to reflect virtually the entirefundamental mode. We then optimized the cavitylength and mode-selecting mirror size by calculatingthe modal loss for the two lowest-order modes asa function of mirror separation and diffractive mir-ror size. A non-phase-quantized diffractive mirrorwas assumed. Figure 2 shows the gain required toovercome the cavity loss to the second-order mode.For each cavity length, a new mode-selecting mirrorwas calculated to produce an identical fundamentalmode as given by Eq. (7). For mode-selecting mirrorsizes of 8 mm and greater, the loss to the funda-mental mode is small and is not shown. The insetsin Fig. 2, however, show that small mirrors do tendto introduce ripples in the mode profile. This is tobe expected, since the higher harmonics from thesquare-mode diffraction are clipped by the mirror.For this reason, we have chosen to use a 16-mmmirror for the best modal shape.

The second-order mode loss for the 16-mm mirroris seen to peak at a distance of approximately z =7rTcO2 /A, or one Rayleigh range of a conventionalGaussian beam. For our experiment, too = 0.6 mm,A = 1.06 ,m, and z = 1.07 m. When we accountfor the increased index of the YAG crystal in partof the cavity (n = 1.8 over 7.6 cm), the resultantcavity length is 1.10 m. The predicted fundamentalmode from the laser cavity with these specificationsis shown in Fig. 3.

The effects of changing the output mirror sizeon the fundamental and second-order mode of theprevious laser cavity are shown in Fig. 4. The finiteoutput mirror has little effect on the fundamentalmode loss for mirror sizes greater than 1.2 mm,as expected. The loss to the second-order mode issignificant and generally increases with a reducedoutput mirror size. The shape of the fundamen-tal mode is unaffected by mirror sizes greater than-1.3 mm, with a slight degradation observed in the1.2-1.3-mm region. For our choice of 1.3 mm, thefundamental mode loss is 0.08%; the second-ordermode loss is 48.2%, requiring an additional gain of1.9 to overcome the cavity mode loss. Second-ordermodal losses as high as 59% are possible with a1.2-mm aperture (with a slight reduction in modeshape). This substantial loss difference makes itpossible to pump the laser strongly while still main-taining single-spatial-mode operation.

The mode-selecting mirror was fabricated by a four-step mask-and-etch process.10 This procedure re-sulted in a 16-level phase element with the profileshown in Fig. 1. Since the smallest features on anyof the masks were only 50 Aum in size, wet chemicaletching was used. This gave a controllable etch rateand produced an optically smooth surface.

The performance of the diffractive mode-selectingmirror was first studied outside the laser cavity.A highly expanded cw Nd:YAG laser was used to

C0

C.E0

C,

w

.0

._

._To

2.0 -

1.8

1 . 6

1.4

1 .2

1.02 . 02 0 6 0 1 00 1 40 1 80 220

Cavity Length (cm)

Fig. 2. Required second-order mode gain to over-come cavity losses as a function of cavity length andmode-selecting mirror size. The insets are modal shapesfor mirror sizes 8, 12, and 16 mm at a cavity length ofone Rayleigh range.

1.0 _

0. 8

0.6

0.4

0. 2

0.0

.1 -0.5 0 0.5 1

Distance Across Output Aperture (mm)

Fig. 3. Theoretical fundamental mode shape from themode-selecting mirror cavity. The effects of phase quan-tization, finite lithographic linewidths, and mirror aper-tures are included in the calculation.

/

8mm 12mm 16mm

I I . , , I I .

109

110 OPTICS LETTERS / Vol. 19, No. 2 / January 15, 1994

C

0

1 .,,,,,,,,,2 .... nd-ordero2dl

0.50 0.75 1.00 1.25 1.50 1.75 2.00Output Aperture Size (mm)

Fig. 4. Effect of output mirror size on the fundamentaland the second-order mode loss.

(a)

0.

0.

0.

0.

P0

20.

0.

0.

0.

. .. . .. . .. .. .. .. . .. . .. . .. . . . . .. ... ... .].7 .. . . . .. . . . .. . . . .. . . . .. . . . . . . . . .... ....... ...

6 . ... .. . .. ... . .. ... .. . .. .. . .. .. . ... . .. ... . . .

...... ... . . . . . . . . . . . . .. .. . . . .. . . . . ... .I .. . . .

t.4 .. . .. . . .. . . . . . . . . . . ......... . .. . . .

3 .. . . .. ... I . . .. . .. ... .. .. .. . .. ..I ... .. . .. . . ...... .... i. .. ... .. . .. .. . .. . .. .. .. .. .. . ..

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8Output Distance (mm)

(b)Fig. 5. Measured fundamental mode profile of theNd:YAG laser: (a) three-dimensional contour plot, (b)profile slice.

illuminate a 1.2 mm X 1.2 mm aperture with auniform plane wave. The mode-selecting mirror wasplaced 1.07 m behind this aperture, and the re-flected wave was studied after it propagated back tothe square aperture. The mirror produced a well-defined square shape. We measured the modal re-flectivity by comparing the power in this squareimage (integrated over the 1.3 mm X 1.3 mm outputmirror area) with the incident power. After com-pensation for the imperfect reflectivity of the goldcoating, the modal reflectivity was measured to be98-99%.

The performance of the mode-selecting mirror in-side a laser cavity was studied next. A laser cav-ity was constructed consisting of a planar outputmirror, the diffractive optic mode-selecting mirror,and a Nd:YAG rod pumped by a pulsed single-tubeflash lamp. The shape of the mode intensity wasmeasured by a CCD camera and frame grabber. Itwas discovered experimentally that slightly bettermode shapes were produced with output aperturesof 1.7 to 2.0 mm. The mode intensity is shownin Fig. 5 for an output aperture of 2.0 mm anda corresponding mode-selecting mirror aperture of16 mm. The rms variation across the top of Fig. 5(a)is 1.5% of the peak value. The total output energyof 50 mJ/pulse was approximately the same as theenergy of the simple Gaussian mode obtained withstandard spherical laser mirrors.

In summary, we have designed a diffractive mode-selecting mirror to produce a square flat-toppedmode with high modal separation. Other shapes(circular, multiple apertures, etc.) and profiles(tapered, phase coded, etc.) can be designed aswell. We have demonstrated the technique byusing a flash-lamp-pumped Nd:YAG laser. Theexperimentally measured mode shape in Fig. 5(b)is close to the theoretically predicted shape ofFig. 3. The fundamental mode loss from this cavitywith a non-phase-quantized mode-selecting mirrorwas predicted to be 0.08%. Experimental modereflectivity measurements of the 16-phase-levelmirror showed a loss of 1-2%. This very low lossmakes the technique suitable for both low- andhigh-gain laser systems.

We gratefully thank the Minnesota SupercomputerInstitute and the National Science Foundation (grantNSF/ECS-9109029-01) for supporting this researchand the Oregon Graduate Institute, the Naval Re-search Laboratory, and Amoco Technology Companyfor the loan of equipment.

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