April 1, 2001 / Vol. 26, No. 7 / OPTICS LETTERS 417

Optimized quantization for diffractive phase elements by useof uneven phase levels

Karsten Ballüder and Mohammad R. Taghizadeh

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

Received October 29, 2000

Many applications of diffractive phase elements involve the calculation of a continuous phase profile, which issubsequently quantized for fabrication. The quantization process maps the continuous range of phase valuesto a limited number of discrete steps. We present a new scheme with unevenly spaced levels for the designof diffractive elements and apply it to the design of intracavity mode-selecting elements. We show that thismodified quantization can produce significantly better results than are possible with a regular or even thebias-phase-optimized quantization scheme that we reported here earlier. In principle this process can beemployed to a greater or lesser extent in any quantization process, allowing the fabrication of diffractiveelements with much improved performance. © 2001 Optical Society of America

OCIS codes: 050.1950, 050.2770, 720.3740, 230.1950, 140.3300.

The f lexibility offered by diffractive optical elements(DOE’s) has resulted in their use in a wide range ofapplications. In this Letter we examine the conse-quences of quantization on the elements’ performancefor the example of intracavity mode-selecting elements(MSE’s). MSE’s are used to customize the beam shapeof the fundamental mode of a laser resonator.1 – 4 Ourinterest in the quantization process comes from our re-search on intracavity mode selection,5 – 10 for which westudied the properties of a setup combining a MSEand a conventional lens. However, the quantizationscheme discussed here can be applied to improve theperformance of a wide range of DOE designs.

Some of the techniques for calculating DOE’s f irstcalculate a continuous phase profile, which is thenquantized to a number of discrete phase values forfabrication. This quantization introduces an errorinto the phase profile. The significance of this errordepends on the purpose of the DOE. In the case ofMSE’s this error can be the dominant source of errorin the beam profile produced.

We previously discussed this problem and showedthat the use of an optimized quantization scheme witha bias phase can improve the MSE performance sig-nificantly.11 – 14 Here we extend our earlier discussionby reviewing the assumption that quantization levelsshould be equidistant and showing that we can sig-nificantly improve the elements’ performance by usingunevenly spaced phase levels instead. To make fabri-cation of these elements possible, the phase levels arecharacterized in terms of the individual etch depthsused in binary optics fabrication. As in this scheme,the conventional fabrication technology that uses pho-tolithography and binary masks can be used with nochanges, except that the etch depths are different fromthose in an evenly spaced quantization scheme.

For quantizing a phase profile one usually assumesthat all phase values are equally common in the phaseprofile.15 This assumption is in many cases a grosssimplification. For many phase profiles certain phasevalues are more common than others or are more im-portant to the overall performance of the diffractive

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element. A good example of this are mode-selectingelements. The central section of the MSE with itsslowly varying phase profile receives the highest beamintensity, whereas the outer regions of the MSE con-tribute only little to the element’s performance, as thebeam intensity there is negligible. In fact it is the de-viation from the normal lens profile at the center ofthe MSE that makes the MSE a mode-selecting ele-ment rather than a simple imaging lens. The obviousconsequence of these considerations is that an optimumquantization scheme should aim to preserve the mostimportant phase values as accurately as possible, evenat the expense of phase values, which are less impor-tant. This goal has been partially achieved throughuse of the optimized bias phase,11 which moves the re-alizable phase values so they better fit the problem.

The next logical step is not only to use a bias phaseto move the quantization levels but also to adjust theirspacing, as illustrated in Fig. 1(c). The quantizationlevels are no longer evenly spaced but accumulateabout certain phase values. How the realizable phasevalues can be distributed depends on the fabricationmethod of the diffractive element. We are interestedin the case of photolithographic fabrication withbinary masks. For this kind of fabrication the phaselevels are determined by the individual etch depths.Normally these are chosen such that each etch depth

Fig. 1. (c) Irregularly spaced quantization scheme com-pared with (a) the standard 8-level quantization and(b) the regular quantization with bias phase.

© 2001 Optical Society of America

418 OPTICS LETTERS / Vol. 26, No. 7 / April 1, 2001

is twice the next-lower etch depth, with the f irst beinghalf of the total profile height. For a 16-level elementspanning a total phase range of 2p, this processrequires four etches, with etch depths of p, p�2, p�4,and p�8 (Fig. 2). The total dynamic range of suchan element is then 2p�1 2 1�16�. As 2p � 0, thisvalue spans the whole 2p range divided into 16 evenlyspaced levels, as shown in the upper half of Fig. 2.To produce an irregular profile with N masks or N2

phase levels, one has N 1 1 parameters, one each forthe etch levels plus one for the bias phase. Thesemasks can be freely chosen to best suit the particularproblem. A particular example of such a quantizationis shown in the bottom half of Fig. 2. Notice that thisirregular quantization has a lower dynamic range,leaving a relatively large step between the highestlevel and 2p. This is not necessarily the case forirregular quantizations, but this example happens tospan a smaller range of phase levels than the 16-levelregular quantization. Whereas such an irregularquantization scheme is likely to be of little use fora diffractive element for which all phase levels areequally common, it can be applied to improve the per-formance of some. In particular, the mode-selectingelements studied are promising candidates.

Searching the whole parameter space for the optimalvalues for the etch depths and the bias phase is hardlypracticable. This is particularly true for the MSE’s,for which the evaluation of a single parameter set re-quires many simulated round trips through the cavity.Optimizing a complete set of parameters would takeseveral months. Fortunately, a good combination ofparameters can be chosen directly fairly easily.

The MSE’s reproduce a given beam shape at the out-put coupler. Their phase profiles are almost identicalto that of a lens whose focal length is half of the lengthof the cavity. Only the central section of a MSE devi-ates from the lens profile. At the same time most ofthe beam energy is concentrated in the central partsof the MSE. If one produces a quantized element thatreproduces the slow variation of the central MSE pro-file particularly well at the expense of the outer areas,improved performance of the element should result.

To achieve a good representation of the central el-ement requires a number of closely spaced quantiza-tion levels. For this purpose one can reduce in sizethe smallest few etch depths to reduce the spacing be-tween them. The larger etch depths should be left atthe normal sizes such that the element still spans thewhole phase space. For a 16-level element in whichtwo phase levels are left unchanged, this process can bethought of as producing two superimposed four-level el-ements: one a normally quantized element with phaselevels 3�2p, p�2, p�2, and 0 and a second element thathas a finer spacing, for example, p�6, 3�12p, p�12,and 0. The result would be an accumulation of finelyspaced �DF � p�12� levels about the four major steps.

The approach described above was used for the de-sign of a 16-level MSE. The first two phase stepswere held fixed to guarantee a relatively wide rangeof phase levels. The bias phase was directly chosenso the closely spaced levels match the slowly vary-ing profile at the center of the MSE. The remain-

ing two etch depths were then optimized by an auto-mated brute-force approach that tried various valuesfor the third phase step and kept the fourth at exactlyhalf that height. The resultant profile is shown inFig. 3. For comparison, a normal 16-level MSE andthe unquantized ideal profile are shown too. It canbe seen that the irregularly quantized element repro-duces the f ine features in the MSE profile much better.The quantization found to produce the optimal MSE isidentical to the example from Fig. 2. Although the dy-namic range of this element is smaller than in the regu-lar 16-level quantization scheme, its performance isimproved because it is better at reproducing the centralfeatures of the MSE. To verify the performance of theMSE and to choose the best combination of etch depthswe performed a full simulation of the resonator, us-ing Fox–Li analysis16 and measuring the total squaredeviation from the desired beam shape. The resultproved that the irregularly quantized MSE can indeedoutperform the regularly spaced one (Fig. 4). The to-tal squared deviation from the desired beam profilecould be reduced by 84.3% from that of the regularquantization.

Fig. 2. Difference between two 16-level quantizationschemes. Top, the standard equidistant quantizationin which each level is twice the size of the next-smallerone. Bottom, a particular irregularly sized quantization.Right, the respective resultant phase levels.

Fig. 3. Central sections of three MSE profiles: the ideal,continuous phase profile, a 16-level DOE quantized with aregular quantization with bias phase �FB � p�, and a DOEquantized to 16 irregularly spaced levels with bias phase.The irregularly quantized MSE reproduces the small vari-ations in the central profile section much better than doesthe standard quantization.

April 1, 2001 / Vol. 26, No. 7 / OPTICS LETTERS 419

Fig. 4. Resultant beam shapes from the two quantizedMSE profiles in Fig. 3. Both profiles use the same biasphase value and 16 phase levels. As it uses an irregularspacing for the phase levels, the MSE exhibits a signifi-cantly improved performance. The total squared devia-tion from the desired prof ile has been reduced by 84.3%.

It is important to understand that this method canbe applied only in situations in which phase levels arenot randomly distributed and for which some kind ofinformed judgement about the required phase levelscan be made. The time required to optimize all fiveparameters fully (bias phase plus four etch depths for a16-level element) by running the full simulation wouldexceed the lifetime of most researchers by approxi-mately an order of magnitude. Assuming that oneround-trip simulation takes �1 h to calculate (whichwe found to be a typical value for the four-processorworkstation used) and that the values should be opti-mized to within 2% of 2p (i.e., by trying 50 values forthe smallest step and the bias phase), we estimate thetime required to be approximately 50 3 25 3 12 3 6 3

50 � 45 3 106 h, or �500 years.We have shown that the errors introduced during

quantization of a mode-selecting diffractive phase el-ement can be significantly reduced by choice not onlyof an optimal bias phase but also of a specific set ofirregularly spaced levels for the quantization process.By using this method for the quantization of a MSEone can signif icantly reduce the error in the gener-ated beam profile. Whereas this technique has been

demonstrated on a MSE design, it can in principle beequally applied to any element design that involvesa quantization step. Even for 16-level element exam-ined here this technique produced a 84.3% reduction inthe deviation from the desired beam profile.

K. Ballüder’s e-mail address is karsten@phy.hw.ac.uk.

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