198 OPTICS LETTERS / Vol. 23, No. 3 / February 1, 1998

Application of optimization algorithms to the design ofdiffractive optical elements for custom laser resonators

Ian M. Barton and Mohammad R. Taghizadeh

Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, UK

Received September 2, 1997

We report what we believe to be the first applications of numerical optimization algorithms to the designof diffractive elements that customize the fundamental mode profile of a laser system. Standard designtechniques treat these elements as specific phase-conjugation devices, which leads to performance loss whenthey are quantized to permit fabrication. Numerical optimization can account for quantization of the elementto increase the effective performance. Also, it is shown that allowing a slight increase in the intrinsic loss ofthe cavity can substantially increase the fidelity of the fundamental mode of the customized cavity. The gooddiscrimination qualities of the mode-selection elements are shown to be unaffected by this process. 1998Optical Society of America

OCIS codes: 050.1970, 050.2230, 140.3410, 140.3530, 140.3570.

Customized mirrors have been demonstrated thatalter the fundamental mode of a laser resonator to adesired profile. One example of this is the so-calledgraded-phase mirrors of Belanger and co-workers,1,2

which are fabricated by diamond turning in metals.Although these mirrors are commonly used in CO2lasers, recently a similar device was used for opera-tion in a solid-state laser system.3 Diffractive opticalelements were also successfully applied to this task,notably by Leger and co-workers.4,5 These so-calledmode-selecting elements (MSE’s) usually operatewithin a Fabry–Perot laser cavity by replacing one, orboth, of the mirrors. MSE’s have been applied mainlyin solid-state laser systems, e.g., Nd:YAG. Theadvantages of these customized cavities include su-perior discrimination against higher-order modes,6

permitting single-mode operation, and the inherentlyuseful properties of the fundamental mode profiles,e.g., f lat-top profiles, that they can generate.

Usually one can operate MSE’s as specific phase-conjugation devices to select the desired mode profile.In this case MSE’s are designed to phase conjugatethe desired mode profile once it has been propagatedhalfway around the cavity, thus re-forming the profileon return. A single MSE that replaces one mirror ina Fabry–Perot cavity will generate an arbitrary realmode profile at the opposite mirror. In previous de-signs of this type of element the numerical solutionof the phase-conjugation problem was considered ex-clusively. However, quantization of these elements topermit fabrication reduces their actual performance.In this Letter we apply nonlinear optimization algo-rithms to the design of MSE’s to compensate for thesequantization effects. Furthermore, the optimizationprocess is set up to increase the fidelity of the modethat is formed. Such a design process is also rele-vant for alternative mode-selection devices, such as thegraded-phase mirrors described in Refs. 1 and 2.

Some amount of control over the quality of these cus-tomized cavities was achieved by optimization of thephysical parameters of the cavity.5 These parameters

0146-9592/98/030198-03$10.00/0

include the cavity dimensions, including apertures atthe end mirrors, and the size and the shape of thedesired mode. Minor improvements can be made tothe fundamental mode loss and the discrimination ofa custom laser resonator by alteration of these param-eters. However, this control is limited and requiresthat some amount of freedom be available in the cavityconfiguration.

The Fox–Li analysis7 was used to evaluate the ef-fect of the MSE on the eigenmodes of the resultingcavity. The performance of the laser resonator wasmeasured in terms of the intrinsic loss L and the fi-delity of the fundamental eigenmode. The discrimi-nation of the fundamental mode compared with thatof the next-lowest-loss mode was also evaluated. Thesecond-order mode was calculated by use of the Fox–Lianalysis, with all components that are not orthogonalto the calculated fundamental mode profile systemati-cally removed at each pass. The fidelity was mea-sured in this case by the signal-to-noise ratio (SNR),which is defined as

SNR ­

R`

2` a2jU0sx, ydj2dxdyR`

2`fUmsx, yd 2 aU0sx, ydg2dxdy. (1)

U0sx, yd is the desired wave front, and Umsx, yd is thefundamental mode of the system, as calculated by theFox–Li analysis. a is defined as

a ­

R`

2` Umsx, ydU0sx, yddxdyR`

2` jU0sx, ydj2dxdy. (2)

Optimization is initially performed for the one-dimensional case. The target mode was initiallychosen to be a super-Gaussian profile, which is approx-imately f lat-topped in appearance. In one dimension,the super-Gaussian is defined as

U sxd ­ expf20.5sxyw0dng . (3)The parameters of the super-Gaussian in this case areorder n ­ 14 and width w0 ­ 0.9 mm. The laser that

1998 Optical Society of America

February 1, 1998 / Vol. 23, No. 3 / OPTICS LETTERS 199

was considered is a Nd:YAG system sl ­ 1.064 mmdthat is 1 m in length in which the input mirror isreplaced with a ref lecting MSE. The element wasconsidered to be apodized at a width of 9 w0, andan aperture was also placed at the output mirror ofwidth 1.1 w0. Sixteen-level quantization of the MSEwas assumed. For the quantized phase-conjugatingMSE, the fundamental mode was found to havea one-dimensional loss of 1.0% and the SNR was870. The loss of the next-lowest-loss mode wasmeasured to be 38%. We applied the simulatedannealing (SA) optimization algorithm8 to optimizethe SNR of the fundamental mode. In the opti-mization the initial profile was taken to be thequantized numerical solution. The algorithm then

Table 1. Performance of the Optimized MSE’s inTerms of Intrinsic Loss sLd and Fidelity of the

Customized Mode Prof ile (Measured by the SNR)for Different Values of LMAX

LMAXs%d SNR L s%d

2.0 3518 1.982.5 14,507 2.493.0 74,450 3.003.5 125,268 3.434.0 218,216 3.97

Fig. 1. Fundamental mode profiles for the initial and theoptimized MSE, LMAX ­ 3%.

Fig. 2. Fundamental mode profiles of the initial and theoptimized MSE (two-dimensional case), LMAX ­ 3%.

optimized the MSE structure purely in terms of in-creasing the f idelity of the fundamental mode. Im-provement of the fidelity comes at the expense ofincreased intrinsic loss of the cavity. The optimizationprocess also considers this, setting a maximum accept-able value for the intrinsic cavity loss. The optimiza-tion process will not accept any solution that violatesthis maximum.

Table 1 shows the performance of the optimizedMSE’s for a number of values for the maximumallowable loss, LMAX. The fidelities of the funda-mental modes formed by the optimized MSE’s aresubstantially better than those of the original phase-conjugate version. If a small amount of additionalintrinsic loss is allowed in the cavity, then the fidelityof the fundamental mode can be increased by ordersof magnitude. The discrimination is virtually un-changed in all cases. The predicted profiles of theinitial and the optimized fundamental modes for thecase of LMAX ­ 3.0% are shown in Fig. 1.

The above optimization process is now expanded tothe two-dimensional case. The parameters of the ex-amined laser cavity remain as before, except that thedesired mode profile is a circular super-Gaussian func-tion. The output aperture is now circular, but theelement and its associated aperture remain square.Owing to the substantial increase in computational ef-fort required for the two-dimensional Fox–Li analy-sis, the SA algorithm is not feasible. Instead a directbinary search algorithm is applied.9 A direct binarysearch requires many fewer steps than SA because im-provement begins immediately, although the final per-formance that can be achieved with a direct binarysearch is usually not so good as with SA.

The standard quantized MSE for the two-dimensional case has an intrinsic loss of 2.0%with a SNR of 11,350. The next-lowest-loss mode hasan intrinsic (two-dimensional) loss of 24%. As in theone-dimensional case, increasing the f idelity of thefundamental mode comes at the expense of increaseddiffractive losses within the resonator. If the intrinsicloss of the fundamental mode is allowed to increaseto 3.0%, then the SNR increases to 89,272. Again,the discrimination of the system is unaffected. Theoptimized fundamental mode profile is detailed inFig. 2.

In summary, we have demonstrated that numericaloptimization algorithms can be applied successfully tothe design of MSE’s. The resulting elements generatethe desired modes very accurately, at the expense ofslightly increased loss compared with that of the more-common specific phase-conjugation elements. We arecurrently fabricating a number of these elements sothat the theoretical improvements can be verifiedexperimentally.

References

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(1997).

200 OPTICS LETTERS / Vol. 23, No. 3 / February 1, 1998

4. J. R. Leger, D. Chen, and Z. Wang, Opt. Lett. 19, 108(1993).

5. J. R. Leger, D. Chen, and G. Mowry, Appl. Opt. 34, 2498(1995).

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