Direct inclusion of the proximityeffect in the calculation of kinoforms

J6rgen Bengtsson

Within the direct inclusion method the proximity effect is already taken care of when one calculates thekinoform relief, whereas conventionally the proximity effect is compensated for after one has calculatedthe relief. In particular, when proximity effects are considerable, that is, for small structures, the directinclusion method is shown to give significantly better results than the conventional two-step method.

Provided that the proximity effect is correctly modeled, it is shown that for an 8 x 8 array illuminatornearly perfect uniformity can be achieved even for a kinoform with very small structures.

Key words: Direct inclusion method, proximity, virtual amplitudes, spatial filtering, kinoform.

1. Introduction

Because of the relief etched onto one of its surfaces, akinoform changes the phase distribution over thecross section of a traversing laser beam. This phasedistribution is designed to generate the desired inten-sity distribution in the diffraction plane.

However, when one is manufacturing a kinoform,the processes involved, for example, electron-beamlithography and chemical development, invariablysmear out the relief given as input to the process.This is called the proximity effect, and it may degradethe diffraction pattern.

A method for including this effect in the calculationalgorithm is presented here. Input to this algorithmis the diffraction pattern to be created (e.g., an 8 x 8spot array) and a model for the proximity effect.Output is a relief to be used as input in the manufac-turing system (e.g., an electron-beam lithograph).

It is important to model the smearing-out effectproperly. The effect of a specific manufacturingmethod, direct-writing, electron-beam lithography,and chemical development, can be described as aconvolution of the input relief with a double Gaussianpoint-spread function (PSF)':

1 r fr21 [r2PSF(r) = 2- exP -(-)] + j expt-()jp (1)

The author is with the Department of Microwave Technology,Chalmers University of Technology, S-41296 Gbteborg, Sweden.

Received 15 July 1993; revised manuscript received 13 December1993.

0003-6935/94/224993-04$06.00/0.© 1994 Optical Society of America.

where r is the lateral distance from the primaryelectron beam. The parameters a and can bethought of as arising primarily from the forward andbackward scattering of electrons in the electron-beamlithograph. They also include the effects of develop-ment. Thus they are best thought of as fit param-eters to our model of the proximity effect as aconvolution. Accordingly they have to be deter-mined experimentally, which was done earlier." 2

That the proximity effect can be modeled as aconvolution has been established by my group in thecase of kinoforms manufactured by electron-beamlithography (with 25- and 50-kV acceleration voltage)in SAL 110 resist (Shipley) and then developed. Anumber of patterns (with feature sizes of 4-10 plm)and sets of electron doses (as high as 400 ,uC/cm2)have been used. It also turns out that the param-eters a and a depend on the mean electron dose (orequivalently the mean relief depth) and accelerationvoltage used in the exposure. Specifically, in Ref. 2 aand are determined as functions of the meanelectron dose for an acceleration voltage of 50 kV.

2. Method

A. Direct Inclusion Algorithm

The cornerstones of this algorithm are a modifiedGerchberg-Saxton method3 and spatial filtering.It functions as follows (see Fig. 1). (Throughout, Axdenotes a phase distribution and Ax the correspondingreal relief.)

(A) In the first iteration step the desired diffractionpattern amplitude distribution (with a random phase

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Pattern with virtual amplitudesIn first cycle, use desired amplitudes Compare to the

AAAA ideal patternIVVV\*7F-=7I

pT-fE

Kinoform A

FT(PSF)

Diffraction patternfrom kinoform relief

tWA(§) 4D AD

Relief (c swing too high Convolve (c with PSF-(D

Relief (C swing acceptable

FThf }Fig. 1. Direct-inclusion-method cycle. (The letters in circles refer to corresponding sections in the text.)

distribution) is inverse Fourier transformed. [Inlater iteration steps virtual amplitudes are usedinstead of the desired ones, see (E).] A new phasedistribution PA (i.e., a kinoform relief) is obtained butmay not be possible to manufacture with good accu-racy.

(B) What relief B will give rise to the kinoformrelief MA after it has been smeared out by the proxim-ity effect? Because the proximity effect can be de-scribed by a convolution, the Fourier transform (FT)of this relief is simply

FT{4B) = FTPASF}' (2)

and the relief we seek is thus readily obtained.RB differs from A. Its valleys are deeper and its

mountains higher. B is in a sense a compensatedkinoform relief, which the proximity effects of themanufacturing will smear out to A. However, wecannot permit the input relief swing to be too high.The higher the swing, the deeper into the resist weare, and the smearing-out effects are known to be-come more and more pronounced the deeper into theresist we are. 3 Besides we have assumed a space-invariant convolution. The parameters of the PSF,a and 1, are taken to be depth independent within theresist interval that we are using. Hence the relief weseek, c, should have a swing as low as possible butstill be as close to A as possible after having beensmeared out.

(C) The relief tc can be found by a spatial filtering:

FT{Jc1 = G x FT{(B1, (3)

where the bandwidth of the low-pass filter G is chosenso that the swing in tc is not greater than is permitted.This is done in an iterative process within each outerloop, where the bandwidth decreases stepwise.

The low-pass filtering can be done abruptly, cuttingoff all spatial frequencies above some radius from thecenter. Here a softer filter is used in which allfrequencies below a certain radius f (which is de-creased in each iteration) are completely transmitted,and frequency components above that radius aremultiplied by an exponential factor with a 1/e dis-tance of 13 pixels from f, (for details of matrix sizesetc., see Section 3). We thus obtain a relief (c thatfairly well resembles relief MA after convolution andhas an acceptably high swing.

(D) The diffraction pattern of a kinoform that ismanufactured with c as input and convolved withthe PSF is calculated as the Fourier transform of thephase distribution of the convolved relief AD as usual.If the result is the best so far, (c is saved as a potentialcandidate for the input to the manufacturing system.

(E) We now enter a new iteration step and utilizethe result from the former using the Gerchberg-Saxton method with weights with memory. Thismethod is equivalent to the one described by Farnwith the inner loop reduced to just one run.4 Virtualamplitudes are simply used in step (A) instead of thedesired ones; after calculating the amplitudes in step(D), check how well they match the desired ones.If, for example, an amplitude value is too small, thevirtual amplitude of that spot is increased a littlecompared with its value in the preceding loop.(Thus that value must be saved.)

4994 APPLIED OPTICS / Vol. 33, No. 22 / 1 August 1994

i 4 ~ P A possible input relief c

^A\JVJ'a \ 4+

0low pass\ filtering

a

(a) (b)Fig. 2. (a) Input relief from the direct-inclusion method for an 8 x 8 spot array. (b) Input relief from the two-step method for an 8 x 8spot array.

The formula used here for calculating a new virtualamplitude is

new virtual amplitude = old virtual amplitude

[ desired intensity 0.35

[intensity from (D)j

(4)

The exact form of the formula is not critical; the fairlylow exponential 0.35 is used to ensure stability.

The input relief c that gives the best diffractionpattern is chosen as input relief to the manufacturingsystem. Figure 2(a) shows an example of one periodof an 8 x 8 array illuminator input relief obtainedafter 2000 iteration steps.

Two-Step MethodFor completeness, a two-step method is also pre-sented here briefly.

Conventionally compensation for the proximityeffect is done in two steps. First, one calculates akinoform relief with no proximity effect taken into

Uniformity (%) L E Efficiency (%)

0 0No Two-step Directproximity method inclusion

methodFig. 3. Comparison between the two-step and direct-inclusionmethods.

account. This calculation is done, for example, withthe algorithm suggested by Farn.4 In step two theeffects of proximity are compensated for; by itera-tively changing the relief shape somewhat, i.e., add-ing relief height where the smearing out tends tolower it and vice versa, we obtain a relief that moreand more resembles the relief from step one after ithad been smeared out. As described above, the reliefswing must not be too high, and a characteristicfeature of reliefs calculated with this two-step methodis that they are truncated at their deepest and highestlocations, which can be seen from Fig. 2(b). Theresults for the two-step method that are presentedhere were all obtained with this method.

Another way to perform step two is to go throughsteps (B)-(D) in the direct inclusion algorithm oncewith the relief from step one as PA. The drawbackwith all two-step methods, however, is that the relieffrom step one is fixed when one enters step two, and itmay not be possible to manufacture it with goodaccuracy whatever compensation method is used.

Results and ConclusionsThe direct inclusion method was tested for an 8 x 8even intensity spot array with ot = 0.22 jlm and L =12.0 jim in a 128 x 128 matrix with a pixel width of0.18 Am. These conditions determine the maximumallowable swing since these values are valid for amean relief depth of 1.2 jim (corresponding to amaximum allowable swing of the input relief Cc of 2.4jm).2 In the Fourier plane the pattern is centeredaround zero spatial frequency, and the spots in thearray are 1 pixel apart, which corresponds to anangular separation between two neighboring spots of1.60 (X = 633 nm).

As quality measures I used efficiency and unifor-mity, defined as

* . Imaxspot-I1 *spot

uniformity - spot + spotmax ' mins

(5)

where I is intensity. Ideally the uniformity shouldbe zero (illogically enough).

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The uniformity is a quality measure that is usefulfor patterns with many equally bright spots because itis sensitive to any kind of relief imperfection; forexample, errors in the experimental determination ofa and 13 will immediately give rise to a much worseuniformity.

The direct-inclusion-method calculations werefound to give a uniformity of better than 1% with anefficiency of 84% for the input relief after convolu-tion, which should be compared with the results withthe two-step method for the same 8 x 8 spot array.Note, however, that these results depend on the reliefobtained in step one. Depending on the startingconditions for the Farn method,4 for example, thismethod gives very different step one reliefs, whichyield virtually identical diffraction patterns. Afterstep two the uniformity was typically found to be30-40% for the input relief after convolution.

If no proximity effects were present, the Farnmethod4 would give a relief (which is also the relieffrom step one in the two-step method) with virtuallyperfect uniformity and an efficiency of 93%. Theresults are shown in Fig. 3.

In practice there are always uncertainties in a- and13 and these parameters may also fluctuate from timeto time. Therefore the effects of small changes in ahave been studied. A simulation shows that if a is20% (50%) greater in reality than the value used forcalculating the kinoform with the direct inclusionmethod, the uniformity will be 15% (36%) with the

same development time. However, much of thisworsening in uniformity comes from the relief beingsomewhat shallower, and in practice this can beadjusted by developing for a little longer time. If thedevelopment time is increased by 2% (4%), the unifor-mity improves by 8% and 21%, respectively.

It has been shown that the direct inclusion methodenables the calculation of array illuminators withnearly perfect uniformity even when proximity ef-fects are considerable. In this case the proximityeffect manifests itself mainly as a reduction in effi-ciency. Most likely the method functions well forother diffraction patterns and where unequal spotintensities are desired. Whether near-perfect unifor-mity is reached in practice depends eventually on howwell the proximity model chosen reflects reality.

References1. G. Owen, "Methods for proximity correction in electron lithog-

raphy," J. Vac. Sci. Technol. B 8, 1889-1892 (1990).2. F. Nikolajeff, M. Ekberg, M. Larsson, J. Bengtsson, and S. Hard,

"Shape distortion of diffractive optical elements, directly writ-ten with electron beam lithography," IEE Conf. Publ. 379,60-61 (1993).

3. R. W. Gerchberg and W. 0. Saxton, "A practical algorithm forthe determination of phase from image and diffraction planepictures," Optik 35, 237-246 (1972).

4. M. W. Farn, "New iterative algorithm for the design of phase-only gratings," in Computer and Optically Generated Holo-graphic Optics, I. Cindrich and S. H. Lee, eds., Proc. Soc.Photo-Opt. Instrum. Eng. 1555, 34-42 (1991).

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