1756 OPTICS LETTERS / Vol. 24, No. 23 / December 1, 1999

Optimized phase quantization for diffractive elementsby use of a bias phase

Karsten Balluder and Mohammad R. Taghizadeh

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

Received July 22, 1999

Many applications of diffractive phase elements involve the calculation of a continuous phase profile that issubsequently quantized for fabrication. The quantization process maps the continuous range of phase valuesto a limited number of discrete steps. We report our observation of the inf luence of this quantization processon the performance of mode-selecting diffractive elements and show that the quantization process producessignif icantly better results by use of an optimized bias phase. In principle this process can be employed to agreater or lesser extent in any quantization process. 1999 Optical Society of America

OCIS codes: 050.0050, 050.1970, 140.3300, 140.4780, 230.1950.

The f lexibility offered by diffractive optical elements(DOE’s) has resulted in their use in a wide rangeof applications. We examine the consequences of thequantization process on the performance of intracavitymode-selecting elements (MSE’s).

Our interest in the quantization process comes fromour research on intracavity mode selection,1 – 4 for whichwe studied the properties of a setup that combines aMSE and a conventional lens. Some of the techniquesfor calculating DOE’s first calculate a continuous phaseprofile, which is then quantized to a number of discretephase values for fabrication. This quantization intro-duces an error to the phase profile. The signif icanceof this error depends on the purpose of the DOE. Inthe case of MSE’s, it can be the dominant source of er-ror in the beam profile that is produced.

The MSE profile is similar to the parabolic profile ofa lens. Removing this lens component from the MSEprofile and adding a separate conventional lens gener-ally promise improved performance. In this setup, thequantized MSE will only carry the difference betweenthe lens profile and the required profile. The combi-nation of the continuous lens with a quantized MSEproduces a better approximation to the required phaseprofile than a MSE alone, as is demonstrated in Fig. 1.

Variations in the focal length of the lens led to largevariations in performance of the total setup, althoughthe resultant phase profile for the combination beforequantization remained the same. These variationscould not be explained by the different focal length,and they did not occur if the MSE was left unquantizedor was quantized to a large number of levels. In fact,the lower the number of quantization levels in thefinal MSE, the more pronounced the effect. No directrelation between the element’s total deviation from thedesired profile and the performance of the elementexists: That is, measurement of a large total phasedeviation between the ideal profile and the quantizedprofile does not imply poor performance. Assessingthe quality of a quantization and the performance ofthe setup by comparing the phase structure of theMSE and the lens with that of the ideal profile istherefore not possible.5,6 One can easily understand

0146-9592/99/231756-03$15.00/0

that this is so by regarding the deviations that areintroduced as constituting an additional DOE that isbeing added to the ideal phase profile. To evaluatethe performance of this additional DOE, one has tocarry out a full analysis of the element’s performance,a computationally expensive procedure.

In the design of DOE’s, only phase values in therange 0 . . . 2p are of interest. The most commonlyused method for quantization maps this range ontoN evenly spaced discrete values fi � i�2p�N�, i [�0, . . . , N 2 1�, ranging from f0 � 0 to fN21 �2p�N 2 1��N , as illustrated in Fig. 2. All the phasevalues are then mapped to the nearest discrete level fi.If Q represents the quantization mapping function,mapping the continuous phase range to the discretesteps as described above, it will produce a certainfi � Q�f�. One can also add a bias phase to theoriginal phase value before quantizing it, which has nophysical significance because our relative phase values

Fig. 1. Scan through the required phase profile (solidcurve) for forming a f lat-topped output beam from aFabry–Perot laser resonator. The ideal continuous profilecannot be fabricated directly; it needs to be quantized first.Dotted curve, the quantized eight-level MSE that can beused to approximate this profile. Dashed curve, the phaseprofile of a combination of a MSE and a lens that is betterapproximation to the required profile.

1999 Optical Society of America

December 1, 1999 / Vol. 24, No. 23 / OPTICS LETTERS 1757

Fig. 2. Commonly used quantization scheme, simply map-ping the total range 0 . . . 2p onto N evenly spaced discretevalues.

matter. (In cases when a constant phase shift wouldmake a difference, it can be compensated for by thepositioning of the element in the final experimentalsetup.) The addition of a bias phase fB fi 0 to thephase profile before it is quantized changes the quan-tization mapping function to fj � Q�f 1 fB�, withi fi j or i � j , depending on f. Thus the applicationof the bias phase will lead to a different quantizedstructure for the same continuous profile. This modi-fied quantization by the addition of a bias phase isillustrated in Fig. 3. The addition of the bias phasebefore the quantization is equivalent to subtracting thebias phase from the discrete steps, thus being equiva-lent to a rotation of the discrete phase levels throughthe phase range. Although this does not modify theideal profile in any way nor the spacing or numberof quantization steps, we found that the performanceof the quantized structures varies strongly with thebias phase.

Finding the optimal value for the bias phase re-quires a full analysis of the final setup. To reducecomputation time, we used an algorithm consistingof two stages. The first stage, operating at reducedresolution, quickly selects the best few values of thebias phase, which will be analyzed in a full simula-tion in the second stage. The simulations involve cal-culations scaling between R log2 R in one dimensionup to R 3 R in two dimensions, with R being a mea-sure of the resolution, e.g., the number of pixels cal-culated. Reducing the resolution by a factor of 2 or4 greatly speeds up the calculation. The first stagevaries the value of the bias phase through a large num-ber of values in the range 0 . . . f1, where f1 � 2p�N.Although this approximation cannot choose the cor-rect best value, it can reliably select the few best-performing values. Only this selection of promisingcandidates for the bias phase is used in the secondstage to perform a full resolution simulation of theexperimental setup, which reliably chooses the opti-mal value.

We applied this technique to a MSE design in a 1-mNd:YAG laser oscillator (the design of which is de-scribed in Ref. 7). The MSE is placed at one end ofthe resonator together with a lens (or a curved mirror)to produce a super-Gaussian, f lat-topped output beamprofile at the output coupler. We investigated combi-

nations of MSE’s with lenses of various focal lengths.In this comparison the ideal phase profile remains thesame; only the quantized one changes. As a measureof performance, the total squared deviation from thedesired output beam profile of the modified resonatorwas used. The resultant simulated performance var-ied strongly with the focal length of the lens (Fig. 4,dotted curve), something that was not observed for theunquantized MSE. We then modified the design al-gorithm to use a bias phase during quantization andto optimize its value for optimal performance as de-scribed above. Figure 5 shows beam profiles obtainedwith two different values for the bias phase in anotherwise unmodified simulation setup. The two biasphases are for the best and the worst performance ob-tained and demonstrate the large inf luence that thebias phase can have on the elements’ performance. Asthe normal, nonoptimized, quantization is equivalentto an arbitrarily chosen bias phase of fB � 0, its per-formance will usually lie somewhere between these twoextremes. (The ideal beam profile as it would have

Fig. 3. The effect of the additional bias phase on thequantization process is equivalent to a rotation of thediscrete phase levels relative to the original phase value.

Fig. 4. Improvement in beam quality of a MSE combinedwith a lens. Variations in the focal length f of thelens lead to different phase profiles for the MSE. Smallchanges in f produce large variations in the beam quality.This variation can be reduced by as much as 1 orderof magnitude by use of an optimized bias phase in thequantization.

1758 OPTICS LETTERS / Vol. 24, No. 23 / December 1, 1999

Fig. 5. Beam profiles obtained with a particularly badchoice of bias phase, leading to a MSE design thatproduces a large deviation from the desired beam profile,and a bias phase chosen by our optimization algorithm,compared with the ideal beam profile as produced by theunquantized MSE.

been produced by an unquantized MSE is shown forcomparison.)

The improvement obtained for different setups com-pared with the normal quantization is also shownin Fig. 4 (solid curve). The introduction of the biasphase generally improves the performance of the ele-ment, the extreme cases by almost an order of mag-nitude � f � 0.95 m�. Figure 4 also shows two setupsfor which the introduction of a bias phase slightly de-creased performance. This decrease in performance isdue to the imperfection of the first stage of our opti-mization algorithm, which as a result of the approxi-mation of using a lower resolution falsely discards thebetter value fB � 0 as a possible candidate for the ex-act analysis. The number of possible bias phases forexamination in both stages of the algorithm can be cho-sen freely. Keeping a larger number of them for theexact analysis can improve the final result at the ex-pense of computational speed. If performance is of su-perior importance, the first stage can be dropped and afull simulation of a large number of possible values forthe bias phase can be made, thus guaranteeing that aperfect value will be obtained.

Previously suggested optimization techniques forMSE’s use algorithms such as simulated annealing

or direct binary search8 to optimize them by f lippingindividual pixels between the fixed quantization levelsand evaluating the resultant change. Although thesealgorithms produce some improvement in the elements’performance, they are computationally expensive andoften take many days to run, compared with the fewhours taken by the algorithm described here. The twomethods complement each other in that they optimizethe DOE in different ways, and one can obtain bestresults by first optimizing the bias phase and thenrunning the simulated annealing algorithm on thisquantized structure. Overall, this approach not onlyproduces a better element but also is still faster thanusing simulated annealing alone.

We have shown that the errors introduced duringquantization of a mode-selecting diffractive phase ele-ment can be significantly reduced by choice of an op-timal bias phase for the quantization process. Usingthis method for the quantization of a MSE, we were ableto reduce the error in the generated beam profile byas much as 1 order of magnitude. Although we havedemonstrated this technique for MSE design, it can beapplied to any diffractive element design that involvesa quantization step.

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

References

1. I. M. Barton, P. Blair, A. J. Waddie, K. Balluder, M. R.Taghizadeh, H. McInnes, and T. H. Bett, Proc. SPIE3492, 437 (1999).

2. K. Balluder, I. M. Barton, P. Blair, M. R. Taghizadeh,H. McInnes, and T. H. Bett, in European Conferenceon Lasers and Electro-Optics (CLEO/Europe) (OpticalSociety of America, Washington, D.C., 1998), p. 363.

3. K. Balluder and M. R. Taghizadeh, in Postgraduate Re-search in Electronics and Photonics—PREP99 Proceed-ings (Institute of Physics, London, 1999), p. 17.

4. K. Balluder and M. R. Taghizadeh, in Conference onLasers and Electro-Optics (CLEO/US) (Optical Societyof America, Washington, D. C., 1999), p. 482.

5. J. W. Goodman and A. M. Silvestri, IBM J. Res. Dev. 14,478 (1970).

6. Y. J. Guo and S. K. Barton, IEEE Trans. AntennasPropag. 142, 115 (1995).

7. K. Balluder, M. R. Taghizadeh, H. R. McInnes, and T.Bett, Appl. Opt. 38, 5768 (1999).

8. I. M. Barton and M. R. Taghizadeh, Opt. Lett. 23, 198(1998).