Zone-boundary optimization for direct laser writing of continuous-relief diffractive optical elements

Zone-boundary optimization for direct laserwriting of continuous-relief diffractive optical elements

Victor P. Korolkov, Ruslan K. Nasyrov, and Ruslan V. Shimansky

Enhancing the diffraction efficiency of continuous-relief diffractive optical elements fabricated by directlaser writing is discussed. A new method of zone-boundary optimization is proposed to correct exposuredata only in narrow areas along the boundaries of diffractive zones. The optimization decreases the lossof diffraction efficiency related to convolution of a desired phase profile with a writing-beam intensitydistribution. A simplified stepped transition function that describes optimized exposure data near zoneboundaries can be made universal for a wide range of zone periods. The approach permits a similarincrease in the diffraction efficiency as an individual-pixel optimization but with fewer computationefforts. Computer simulations demonstrated that the zone-boundary optimization for a 6 �m periodgrating increases the efficiency by 7% and 14.5% for 0.6 �m and 1.65 �m writing-spot diameters,respectively. The diffraction efficiency of as much as 65%–90% for 4–10 �m zone periods was obtainedexperimentally with this method. © 2006 Optical Society of America

OCIS codes: 050.1970, 230.4000, 090.1760.

1. Introduction

Continuous-relief diffractive optical elements(DOEs) are used in many fields of optics and pho-tonics for light-intensity redistribution and wave-front transformation. The diffraction efficiency ofthese elements depends strongly on the fabricationtechnology. Direct laser writing (DLW) on photore-sist allows one to form a continuous relief in asingle-exposure step by scanning a focused beamacross a photoresist-coated substrate and then fol-lowing the liquid development. In comparison withsingle-diamond turning, DLW is not limited to thefabrication of elements with circular symmetry andone-dimensional DOEs. Besides that, DLW ensuresfast fabrication of large-area DOEs with the same,or even better, accuracy as that of electron-beam(e-beam) writers. In comparison with the gray-scalephotolithographic process,1 the DLW on photoresistis much better suited for limited production andprototyping, owing to cost effectiveness and absenceof distortion introduced by image transfer. It is also

important for high-efficiency and high-precision dif-fractive elements.2

One of the many advantages of DLW on photoresistis its quite linear dependence of profile depth on ex-posure dose. It simplifies modeling because relief for-mation can be well described as convolution ofexposure data P�x� with a light-intensity distributionI�x� in the writing spot3 (Fig. 1). The convolution leadsto smoothing of the exposure-energy distribution E�x�and correspondingly the profile shape after develop-ment. Such a distortion of the profile takes place to agreater or lesser extent for all fabrication techniques.These backward slopes (or dead areas) of microreliefreduce the diffraction efficiency. Compensation forthis effect is a challenge for optimization of e-beam orlaser direct writing. Several methods based onexposure-data optimization4–6 were offered to correcta profile convoluted with the writing-beam intensitydistribution. The model of the laser-writing processused in these optimization schemes usually presumesthe above-mentioned linear dependence of profiledepth on exposure. Of course, the real behavior of thephotoresist during light illumination and develop-ment is much more complicated. Its modeling re-quires that the bleaching of the photoresist in depthdimension and the nonlinear dissolution kineticprocess7–9 be taken into account. However, the linearmodel is well suited for profile depths of up to severalmicrometers, assuming that the absorption gradientin depth dimension is negligible. This assumption isespecially justified when the writing-laser wave-

The authors are with the Institute of Automation and Electrom-etry, Siberian Branch of the Russian Academy of Science, ProspektKoptyuga 1, Novosibirsk 630090, Russia. V. P. Korolkov’s e-mailaddress is [email protected].

Received 31 March 2005; revised 12 August 2005; accepted 16August 2005.

0003-6935/06/010053-00$15.00/0© 2006 Optical Society of America

1 January 2006 � Vol. 45, No. 1 � APPLIED OPTICS 53

length is at the long-wavelength end of the photore-sist absorption spectrum (442–458 nm). Below, themain optimization approaches for direct laser writingare briefly reviewed.

A. Optimization of Interscan Distance

Writing with the proper interscan distance2 can in-crease the diffraction efficiency. Excessively smallvalues lead to sufficient overlapping of the severalneighboring exposed fields and consequently broadenthe backward slope. Interscan distances that are toolarge result in profile waviness because of valleysbetween adjacent exposure pixels. Both effects arenegative, but it is possible to select an optimal inter-scan distance (see Section 3). This method can in-crease the diffraction efficiency by 6% for a 5 �mgrating period and a 1 �m spot size (1�e intensitylevel) or by 3% for a 10 �m period.2

B. Working in Higher Diffraction Orders

Design in a higher diffraction order increases zoneperiods and also requires that the profile depth beincreased proportionally to the diffraction order num-ber. It is proposed that the backward slope dependsweakly on the depth of the blazed grating profile atthe same writing spot. As a result, the relative areaoccupied by the backward slopes in a DOE decreasesand the diffraction efficiency increases. The applica-tion of DOEs operating in higher diffraction orders islimited by the inherently high sensitivity of the pro-file shape due to fabrication errors. Second and thirdorders are optimal for the acceptable profile toler-ances and ensure a rather high diffraction efficiency.5

C. Linear Scaling of Exposure Data

The diffraction efficiency for small periods can beenhanced by linear scaling of exposure data5 for aconvoluted profile. According to this method, blazeangles of diffractive zones are made to exceed thedesigned blaze angles under the condition that thecentral points of zones are fixed. A computer simula-tion demonstrated the rise of the first-order efficiencyfrom 63.7% to 66.3% when a blaze angle for a 10 �mgrating period was increased by 15% at a 1.4 �mwriting-spot radius (at an e�2 intensity level). How-ever, the application of this method tends to be morecomplicated when zone periods vary over a widerange. In this case it is necessary to change thelinear-scaling coefficients with respect to the zoneperiod.

D. Individual-Pixel Optimization

The most significant increase in the diffraction effi-ciency was obtained by the individual-pixel optimiza-tion method (IPO) based on the intensity-levelvariation for every pixel. Optimization of given expo-sure data can be carried out with different meritfunctions (MFs): maximum diffraction efficiency,minimum deviation from the desired profile, and acombination of the two. Different optimization algo-rithms (difference algorithm,4,5 sequential quadraticprogramming,5 and simulated annealing6) demon-strated the possibility of considerably increasing thediffraction efficiency. For example, the diffraction ef-ficiency of a 10 �m linear grating can be theoreticallyincreased from 63.4% to 80.8% at a 1.4 �m spot ra-dius. This effect can be achieved with a light-intensity modulation range, which is three timeswider than that required for nonoptimized writing.The center of the profile was at the point correspond-ing to the middle of the intensity-modulation range.Such ineffective use of the modulation dynamic rangeproportionally increases the profile roughness owingto intensity noises and worsens the accuracy of mul-tilevel modulation owing to a limited quantity of in-tensity levels. Besides that, all authors acknowledgethat the application of IPO to complex diffractivestructures requires so much computation time that itbecomes practically unacceptable.

2. Zone-Boundary Optimization

Analysis of the above optimization results obtainedwith the IPO method allowed us to make the follow-ing suppositions:

Y The correction of exposure data takes placemainly near boundaries of diffractive zones;

Y The shape of the corrected data can be simpli-fied by joining adjacent pixels with similar intensityinto one step;

Y The shape of backward slopes and correction ofexposure data should depend weakly on the zone pe-riod, because the tilt of a backward slope is muchsteeper than the blaze angle of the diffractive zones.

To verify these suppositions, we implemented the

Fig. 1. Convolution of exposure data P�x� with a light-intensitydistribution I�x� in the writing spot: E�x�, exposure-energy distri-bution.

54 APPLIED OPTICS � Vol. 45, No. 1 � 1 January 2006

IPO method based on the simulated annealing algo-rithm.6 The diffractive efficiency for the MF was de-fined as � � |FT1���|2, where FT1��� is the firstFourier component of the phase function of the con-voluted profile. To compare these results with theknown ones,5 we used the following parameters of thewriting process: 1.4 �m writing-spot radius [at an e�2

level or 1.65 �m diameter at a 1�2 full width at half-maximum (FWHM) intensity level] with a Gaussianintensity distribution, a 0.4 �m interscan distance,and zone periods of 6, 10, and 20 �m. The theoreticaldiffraction efficiency for a nonoptimized convolutedblazed grating (hereafter denoted as None) is shownin the first row of Table 1. These values were calcu-lated for the relative exposure-intensity range of 0–1.The IPO was made for two variants of intensityrange: from �1 to 2 (IPO1) and from 0 to 1.5 (IPO2).Practical implementation of negative intensitiesdown to �1 can be made by shifting the linear expo-sure data to a relative modulation range of 1–2, andthus optimizable exposure data should be varied in arelative range of 0–3. Such freedom of exposure mod-ulation makes for a sufficient gain in diffractive effi-ciency. However, from our point of view, the relativemodulation range of 0–1.5 is more important for prac-tical implementation for the reasons mentioned inSection 1, although it yields a diffractive-efficiencygain that is 2%–5% (IPO2 in Table 1) less than thatof the IPO1 method. The fundamental problem forIPO1 and IPO2 variants of IPO is high sensitivity to

fluctuations in the interscan distance because theconvolution of abruptly varying exposure data [Fig.2(a)] at an incorrect interscan distance results in ahigh profile roughness and a loss of efficiency.5 Mul-tiple launchings of the optimization cycle show thatthe change in exposure intensity near the boundariesof the diffractive zones has a reproducible characterand that in the middle of the zones it varies stochas-tically. To decrease the variations and single out atypical shape of corrected exposure data, we used acombined MF. It was defined as MF � w1� � w2�P,where �P is the rms deviation between optimizedexposure data and linear data and w1 and w2 are theweights, which were usually w1 � 1 and w2 � 0.01.The geometric limitation allows us to obtainsmoother and more stable exposure data [Fig. 2(b)]without a significant diffractive-efficiency decrease(IPO3 in Table 1) in comparison with the IPO2method. The shape of the exposure data was wellreproduced in an area of �1 of a writing-spot diam-eter for different interscan distances, zone periods,and writing-spot diameters. Thus a key for simplify-ing the optimization process can entail fixing nonop-timized data in the middle part of diffractive zonesand varying the exposure intensity for pixels near theboundaries in certain limits. To reduce a number ofoptimization parameters, one can combine the pixelsinto steps. A group of steps forms the stepped tran-sition function (STF), which is a subject of optimiza-tion.

Table 1. Theoretical Diffraction Efficiency (%) Calculated by Different Optimization Methodsa

Algorithm Optimization FeaturesIntensity

Range

Diffraction Efficiency for DifferentZone Periods

6 �m 10 �m 20 �m

None – 0–1 43.5 63.6 80.8IPO1 MF � max��� �1–2 69.6 79.7 89.2IPO2 MF � max��� 0–1.5 64.6 75.4 87IPO3 Combined MF 0–1.5 61.4 74 85ZBO1 MF � max���, 0–1.5 58 72.5 85

STF: three stepsZBO2 MF � max���, �1–2 61.2 76.2 87.6

STF: 4 steps, centrosymmetricalZBO3 MF � max���, �1–2 68.4 78.6 89

STF: six steps

aFor a 1.65 �m writing-spot diameter (FWHM).

Fig. 2. Bar plots of optimized exposure data: (a) for IPO2, (b) for IPO3, (c) with a stepped transition function. Chain curves, Gaussianintensity distributions for a 1.65 �m writing spot; dashed curves, ideal profiles in relative units; solid curves, convoluted profiles.

1 January 2006 � Vol. 45, No. 1 � APPLIED OPTICS 55

Such optimization with the writing parameters(intensity-modulation range of 0–1.5) used above andwith the diffraction efficiency used as the MF dem-onstrated that the three steps [Fig. 2(c)] for theseconditions are enough. At that point, steps 1 and 2are on the side of maximum exposure, and step 3 is onthe side of minimum exposure. It is obvious thatwhen the exposure-intensity range is symmetricallyextended and the nonoptimized exposure values arein the central part of the range, an optimal STFshould have the same number of steps from each sideof the boundary. Furthermore, the STF should have acenter of symmetry relative to the intersection pointof the zone boundary with a medial line of the expo-sure range. In general, when there are N steps in theSTF, then there are 2N variables for optimization:intensity Ii for each step and its width Wi. At thatpoint, intensities Ii are restricted over some range,but limitations for width Wi are not required. Wename this method zone-boundary optimization(ZBO). Practical identity of diffractive efficiencies forthe ZBO and IPO methods can be attained under thefollowing conditions: the exposure intensity for everypixel in the STF is varied during optimization over agiven modulation range and the total width of theSTF with the center-on-zone boundaries is approxi-mately 2–3 spot diameters. Such a combination ofZBO and IPO can reduce calculations compared withthe pure IPO method. The reduction coefficient can beestimated as the total DOE area divided by the areaof all optimized pixels.

The essential factor that simplifies a practical im-plementation can be the universal shape of the STFfor different periods of diffractive zones. To definethis shape, we applied ZBO to gratings with a periodin the range of 6–20 �m with a 0.4 �m increment.Optimization for the STF with three steps for differ-ent grating periods gave constant values: W1� 0.4 �m, W2 � 1.2 �m, and W3 � 1.6 �m. Figure 3shows the dependence of normalized intensities I1and I2 for steps 1 and 2 on the grating period. Thevalues I1 and I2 were limited by the range of 0–1.5 at

optimization. The intensity of the third step W3 wasset to 0 for all periods. The value I2 can be consideredequal to 1.45 for the whole considered range. One cansee from Fig. 3 that the value I1 is also quite stable forzone periods of more than 8 �m. It is possible to splitthe range of periods into two subranges of less than8 �m and more than 8 �m and to set I1 � 0 and I1� 0.2. Thus two sets of STF parameters can be im-plemented for a wide range of periods. Calculation ofthe diffraction efficiency for exposure data with suchuniversal STF demonstrated a loss in efficiency ofless than 1% in comparison with individually opti-mized STF. This trick makes the practical implemen-tation of ZBO more preferable in comparison withIPO, which requires an optimization for the wholediffractive element if zone periods change continu-ously in wide limits.

3. Experimental Verification of theZone-Boundary-Optimization Method

The results described above are of interest for the-oretical reasoning of the ZBO method and for com-parison with other methods, but modern laserwriters use an essentially smaller writing spot. Weverified ZBO by means of a circular laser-writingsystem developed at the Institute of Automationand Electrometry (IAE) and called the CLWS-300IAE. Laser-writing systems that are based onsimilar circular scanning are used fairly often indiffractive optics fabrication.2,10–13 For a better un-derstanding of the application of ZBO to such sys-tems, we give a short summary of the mainoperating principles using the example of CLWS-300IAE. Scanning a focused beam of an Ar� laser(458 nm wavelength) is performed in the followingway: a substrate coated with photoresist is rotated ata constant angular speed, while a focused objective istranslated by linear carriage along a radial direction.The objective (N.A. of 0.65) forms a 0.6 �m (FWHM)spot on the substrate surface. The exposed rings arespaced evenly, with the interscan distance rangingfrom 0.25 to 0.5 �m between adjacent tracks. Twoacousto-optic modulators ensure a wide dynamicrange of laser beam power.14 The first modulator cor-rects an exposure dose while changing the linear spotvelocity, and the second one modulates the intensityaccording to the phase function of the DOE. The pe-culiarity of circular-writing systems is that every cir-cular track of the writing beam has an individuallength. Therefore it is more convenient to use thevector data format in an angular direction instead ofpixel data representations. The vector data formatand high linear spot velocity (up to several meters persecond) make it natural to use a continuous-exposuremode in an angular direction. Most of the x–y writersfor diffractive optics fabrication have a lower linearspot velocity and use pixel data representation andpulse-intensity modulation (pulse per pixel) for bothscanning directions. Pulse modulation is also used inCLWS-300IAE, near the rotation center, for addi-tional extension of the dynamic range of exposurecontrol.1

Fig. 3. STF parameters obtained at launchings of the ZBOmethod with different zone periods: spot size, 1.65 �m (FWHM);interscan distance, 0.4 �m; intensity-modulation range, 0–1.5.

56 APPLIED OPTICS � Vol. 45, No. 1 � 1 January 2006

Our fabrication technique is quite close to the typ-ical process of DLW on photoresist.15–17 The differ-ence is in the uniform preexposure18 achieved by abroad UV beam. It permits operation in the practi-cally linear part of a characteristic curve of photore-sist (depth as a function of exposure intensity) andhelps to avoid a consideration of strongly nonlineareffects near the inflection of a characteristic curve(notch behavior).9 The preexposure also permits amore effective use of the beam intensity-modulationrange and considerably reduces the profile roughnessdue to intensity fluctuations2 and errors of interscandistance.

The first step for implementing optimization ismeasurement of the writing-beam intensity distribu-tion. We measured it by using a one-dimensionalknife-edge scan technique. Such an approach wascorrect because the intensity distribution was used tosimulate one-dimensional writing. The measurednormalized intensity distribution depicted in Fig.4(a) is close to the Gaussian distribution with a0.6 �m diameter (FWHM), but it has a pedestal be-cause of the ring around the main peak. Furthersimulations were conducted with the smoothed mea-sured distribution.

The next parameter that defines the backwardslope of the convoluted profile is the interscan dis-tance. Figure 4(b) depicts the diffraction efficiency asa function of the interscan distance for nonoptimizedwriting (None) and both types of the optimizationmethods (IPO and ZBO). Relative intensity was con-strained over a range of 0 to 1.3 for both optimizationtechniques. All three curves have a maximum at aninterscan distance of �0.5 �m. This value was se-lected for the optimization process and experimentaltesting. The application of the ZBO method showedthat the iterative process rarely gives a nonzerowidth of step 1 [Fig. 2(c)] in the STF. It allows us toreduce the quantity of optimized variables of the STFto three: width W2 and intensity I2 for step 2, andwidth W3 of step 3 with I3 � 0. Figure 5 shows thetheoretical diffraction efficiency for a 3–15 �m rangeof periods and for an exposure-intensity range of0–1.3. The efficiency for ZBO in comparison with non-optimized writing (None) changes from 4% for a

10 �m grating period to as much as 7% for a 6 �mgrating period. The gain from the optimization with asmall writing spot is noticeably less than it is for thesame period and for a larger writing spot (ZBO1 inTable 1). The evident reason is that the writing-spotsize is nearly three times smaller, and therefore thebackward slope has less influence on the diffractionefficiency. For testing our simulations, we fabricatedlinear gratings with different periods (3, 5, 7, 10, and15 �m). In the experiment the diffraction efficiencywas defined as the ratio of measured intensity in thefirst diffraction order to a total light intensity behindthe grating. The results of the measurement of theefficiency for nonoptimized (Measured None) and op-timized (Measured ZBO) gratings are depicted in Fig.5. Measured efficiencies that are lower than simula-tion results are explained by the depth fabricationerror and a simplified simulation model.

For practical implementation it is important tocheck how the STF obtained for a single period valuewill suit a wide range of periods. We optimized theSTF for a 7 �m period at a 0.25 �m interscan dis-tance and applied it to grating periods ranging from3 to 15 �m. The difference between the diffractionefficiencies for this case and the individually opti-

Fig. 4. (a) Measured intensity distribution in a writing spot and a Gaussian distribution. (b) Theoretical diffraction efficiency as a functionof interscan distance. Grating period, 10 �m; writing-spot diameter, 0.6 �m.

Fig. 5. Theoretical and measured diffraction efficiencies as func-tions of the grating period for a 0.5 �m interscan distance. Writing-spot diameter, 0.6 �m.

1 January 2006 � Vol. 45, No. 1 � APPLIED OPTICS 57

mized STF (Fig. 6) did not exceed 1%. It is necessaryto notice that the application of ZBO for a 0.25 �minterscan distance requires five variables for the STFbecause the contribution of step 1 of the STF into thediffractive efficiency becomes more sufficient whenthe interscan distance is decreased to a spot radius orless. This situation is similar to that of ZBO with a1.65 �m spot diameter and a 0.4 �m interscan dis-tance, discussed in Section 2.

4. Algorithms of Realization

Practical realization of the ZBO method is most sim-ple when boundaries of diffractive zones are arrangedalong scanning trajectories of the writing beam, be-cause it becomes the one-dimensional case consideredin Section 3. This situation appears for x–y writers ifone-dimensional structures such as linear gratingsand cylindrical diffractive lenses are oriented alongan x or y axis. For circular-writing systems, a wideclass of axially symmetric DOEs can be optimized asone-dimensional structures. In this case the inter-scan distance coincides with a constant radial step oflinear carriage displacement.

A. One-Dimensional Realization

The following steps summarize the application algo-rithm of ZBO for the one-dimensional case regardlessof the type of writing system:

1. Calculate zone-boundary positions and zone pe-riods.

2. Define parameters of the process model: inten-sity distribution, interscan distance, and character-istic curve of photoresist.

3. Define optimization parameters: quantity of al-lowed steps in STF, width of optimized area, permis-sible modulation range.

4. Optimize STF with these parameters for anumber of periods to define dependences of Wi and Ii

on zone period.5. Define the period subranges with a slightly

changing STF.6. Calculate average Wi and Ii parameters for each

subrange.

7. Calculate exposure data according to the phasefunction and characteristic curve of photoresist. Instrips with width Wi along zone boundaries, the ex-posure data are substituted with Ii accordingly to asubrange of the current period.

Normally, laser writers have typical interscan dis-tances and writing-spot sizes. Therefore it is possibleto make ZBO for a number of typical situations inadvance and then afterward to select only the properparameters of STF from this database for each newDOE.

B. Simplified Two-Dimensional Realization forAnalytical-Type Diffractive Optical Elements

Two-dimensional realization of ZBO is more compli-cated. We restrict ourselves to the case of analytical-type diffractive elements19 for which the zone-boundary positions and the zone periods can beanalytically calculated. In this case the problem canbe simplified to avoid solving the two-dimensionaloptimization task. The approach is based on the fol-lowing presumptions and constraints:

Y The interscan distance in both scanning direc-tions is rather small to ensure a low influence ofpixelization on zone boundaries;

Y The STF optimized by ZBO for a one-dimensional case can be applied for a two-dimensional case in the direction perpendicular tothe zone boundaries;

Y One STF can be applied for a whole range ofzone periods and interscan distances.

The first constraint is necessary to avoid a consider-ation of pixelization on zone boundaries, because ofthe large interscan distance. If this constraint is ful-filled, the second presumption is quite evident, be-cause the profile shape becomes practicallyindependent of the scanning directions. Therefore wecan suppose that one of the scanning directions islocally perpendicular to the zone boundary and thatwe can use a one-dimensional STF. The third pre-sumption supposes availability of a STF that is uni-versal for a large range of periods. It is based onresults obtained in Sections 2 and 4. Independence onthe interscan distance is required even if the inter-scan distance is the same for both scanning direc-tions, because of a larger diagonal distance betweenpixels. The possibility of finding a universal STF fora large range of interscan distances, including thecase of a combination of pulse and continuous expo-sure modes (typical for circular laser-writing sys-tems), is considered in Section 5.

It follows from the considered presumptions andconstraints that one can define correction strips (thequantity of strips corresponds to the quantity of stepsin the STF) along every zone boundary that is inde-pendent of the zone period and can substitute non-optimized exposure data in the strips by intensityvalues of the STF (Fig. 7). The width of the strip in

Fig. 6. Theoretical diffraction efficiency as a function of the grat-ing period for a 0.25 �m interscan distance.

58 APPLIED OPTICS � Vol. 45, No. 1 � 1 January 2006

the direction perpendicular to the zone boundaryshould be equal to the width Wi of the correspondingstep in a one-dimensional STF. Therefore the lengthLi of a segment of exposure data replaced by an in-tensity level in an ith step of STF can be calculated asLi � Wi�sin��, where � is the angle between thewriting-beam trajectory and the current zone bound-ary (Fig. 7). If the laser writer uses the pixel dataformat, this length Li is defined with an accuracy upto the pixel size; but for the vector data format, Li canbe calculated accurately. Hence the two-dimensionalsimplified algorithm of ZBO can be realized as fol-lows:

1. Calculate zone-boundary positions and zone pe-riods.

2. Define parameters of the process model: inten-sity distribution, interscan distance, permissiblemodulation range, and characteristic curve of pho-toresist.

3. Define optimization parameters: quantity ofsteps in the STF, constraint on the shape of the STF,quantity of iterations, etc.

4. Use the one-dimensional ZBO to search for theSTF, which can be effectively applied to a given rangeof periods and interscan distances.

5. Calculate the zone-boundary positions by theanalytical phase function.

6. Calculate the corrected segments Li for the in-tersection of zone boundaries with every track of thewriting spot.

7. Calculate the exposure data according to thephase function and characteristic curve of the pho-toresist. Substitute the exposure data along the zoneboundaries in segments Li by optimized values Ii.

The method can be quite easily applied to both real-time exposure-data processing and processing of pre-viously calculated data. The latter is useful in asituation in which the described algorithm cannot bebuilt directly in the writing software.

The described one-dimensional and simplified two-dimensional algorithms were applied for the fabrica-tion of different elements, such as axial-symmetricdiffractive lenses and DOEs with a spiral structure.Figure 8(a) shows a photograph of the diffractive lens(minimum zone period, 3.6 �m; diameter, 20 mm)fabricated by a reactive-ion-etching (RIE) process onthe quartz substrate. The surface relief of the diffrac-tive zones with a period of �6.5 �m for the lens isdepicted in Fig. 8(b). Because the RIE process canincrease profile roughness and change the backwardslope, we measured the diffraction efficiency on pho-toresist samples for comparison with simulation re-sults. Figure 9 depicts the diffraction efficiency (for a

Fig. 7. Geometric representation of a simplified two-dimensionalrealization of ZBO: B, boundary of the diffractive zone; x, writing-beam trajectory. Other definitions are given in the text.

Fig. 8. (a) Diffractive lens with an optimized profile. (b) Atomic force micrograph of diffractive zones with a 6.5 �m period.

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532 nm wavelength) as a function of the zone periodmeasured on several DOEs fabricated on photoresist.The top and bottom curves show the theoretical dif-fraction efficiencies for the optimized and nonopti-mized gratings, respectively. A decrease in themeasured efficiency for small periods in relation tothe simulation of the optimized profile can be ex-plained by the paraxial approximation for thediffraction-efficiency calculation and an additionalsmoothing effect of the liquid development. It is alsoconfirmed by the width of the backward slope (de-fined as the distance between the minimum and themaximum of the profile depth near the zone bound-ary). According to simulation results, the backward-slope width should be near 1.1 �m, but atomic forcemicrographs [Fig. 8(b)] showed a value of 1.4 �m. Thesmoothing effects of the liquid development and thefollowing RIE process can be included in the model bymeasuring the fabricated profile shape near the zoneboundaries. Differentiation of the profile can give adistribution function, which characterizes the com-bined action of the laser exposure and postprocessing.This can be used instead of the measured writing-spot intensity distribution at convolution with thedesired profile. However, if a small writing spot isused, an accurate characterization of the backwardslope for the calculation of this equivalent intensity

distribution requires use of a scanning electron mi-croscope and specially prepared samples.

5. Stepped Transition Function for Combination ofContinuous and Pulse Exposure Modes

The combination of pulse (or stepped) and continuousexposure modes is typical for circular laser-writingsystems for reasons discussed in Section 3. The resultis that the backward slopes are different for differentscanning directions. One approach is to equalizethem by reducing the interscan distance in the scandirection with pulse exposure. According to Fig. 4(b),the diffraction efficiency of a nonoptimized profile be-comes practically constant if the interscan distance isless than the spot radius (0.1–0.3 �m range). Thismeans that the backward slope does not change itsshape. This trend occurs for a decrease in the inter-scan distance to values equivalent to continuous ex-posure. As a result, one can use the same STF in bothscanning directions when the interscan distance isequal to or less than the spot radius (FWHM). A smallinterscan distance also permits the reduction of pix-elization noises. A disadvantage of interscan-distancereduction is the broadening of the backward slope inthe direction of stepped scanning.

There is another way to use the same STF in bothscanning directions: Determine the shape of the func-tion that operates well over a wide range of interscandistances. To check this possibility, we applied theZBO method to the exposure process with a 1.65 �mwriting spot and with interscan distances of 0.1 and0.8 �m. Exposure with a 0.1 �m interscan distance isin fact equivalent to the continuous exposure for thespot size used. The relative intensity-modulationrange was 0–1.5. We applied the ZBO method to de-termine the STF for a period of 10 �m and then usedit for simulation of the convoluted grating profile withperiods of 6 and 20 �m. The first row in Table 2 givesthe STF parameters and the diffraction efficienciesfor optimization without a constraint for Wi. Otherrows are obtained under the condition that Wi shouldbe divisible by 0.8 �m. This condition allows us tocorrectly apply the obtained STF to the exposure sim-ulation with a larger interscan distance. The first,second, and fourth rows show the optimization re-

Fig. 9. Measured and simulated diffraction efficiencies as func-tions of the grating period for ZBO.

Table 2. Theoretical Diffraction Efficiency Calculated for Grating Profiles Simulated for Different STFa

Interscan Distance STF ParametersDiffraction Efficiency forDifferent Zone Periods

for Exposure for Optimization W1 I1 W2 I2 W3 6 �m 10 �m 20 �m

0.1 0.1 0.4 0.18 1.1 1.5 1.6 55.3 71.6 84.3(42) (61.6) (79.3)

0.1 0.1 0.8 0.75 0.8 1.5 1.6 53.5 70.6 83.70.8 0.1 0.8 0.75 0.8 1.5 1.6 60.7 74.7 85.70.8 0.8 0.8 0.78 0.8 1.5 1.6 60 74.85 86

(48.3) (65.7) (81)0.1 0.8 0.8 0.78 0.8 1.5 1.6 52.8 70.6 83.9

aWidth Wi of steps in the STF and interscan distances are given in micrometers; Ii, in relative units. Theoretical values of the diffractionefficiency for nonoptimized exposure are given in parentheses.

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sults for cases in which the interscan distance wasthe same for the simulation of the exposure processduring ZBO and for the following simulation of thegrating profile without optimization. The third rowwas calculated with the STF optimized for an 0.1 �minterscan distance but was used for the simulation ofexposure with a 0.8 �m interscan distance. The fifthrow was calculated for the reverse situation. The totalefficiency of the whole DOE can be estimated by av-eraging the diffraction efficiencies in both directions.Such a calculation for second–third and fourth–fifthrow pairs shows approximately the same total effi-ciency because of the similar STF. Therefore, for theconsidered case, the fastest way to determine theuniversal STF is to optimize for the maximum inter-scan distance. However, a larger intensity-modulation range makes the selection more critical.

6. Conclusion

It has been shown that ZBO of exposure data at directwriting can considerably enhance the diffraction ef-ficiency of continuous-relief DOEs. The effect is ob-tained by optimizing only 3–6 variables instead ofIPO for the whole exposure data. ZBO partially com-pensates for smoothing of a desired surface profilebecause of convolution with a writing-beam intensitydistribution. According to the computer simulation,ZBO with a writing-spot diameter of 0.6 �m and arelative intensity range of 0–1.3 yields an increase inthe diffraction efficiency from 4% to 14% by changingthe diffractive-zone period from 3 to 10 �m, respec-tively. Measurements of fabricated DOEs demon-strated a diffraction efficiency of 65%–90% for theperiod range of 4–10 �m. A real increase in the effi-ciency for a 3–4 �m period range is not reliable. Fur-ther growth in the diffraction efficiency for smallperiods can be obtained with the ZBO method byusing a more accurate approximation for thediffraction-efficiency calculation and by including aliquid-development step in the model. The last addi-tion can be made on the basis of an equivalent inten-sity distribution that takes the measured widening ofthe backward slope into account. Besides that, ex-tending the intensity-modulation range can furtherconsiderably increase the efficiency with a high accu-racy of beam positioning and beam power control.

The main problem for IPO is the application of thiseffective method to real DOEs with a large range ofperiods and a two-dimensional field of phase assign-ment. The ZBO method can be quite easily applied tothe case of analytical-type DOEs. The implementa-tion algorithm can be reduced to a one-dimensionalcase for fabrication of DOEs in which the boundariesof diffractive zones are arranged along the scanningtrajectories of a writing beam. The optimization taskfor analytical-type DOEs with a variable phase inboth scanning directions can be replaced by a geo-metrical calculation of correction strips alongdiffractive-zone boundaries. Nonoptimized exposuredata in these strips are replaced by a STF taken fromone-dimensional ZBO. The proposed approach has

been implemented for a circular laser-writing system,but there are no principal limitations to its applica-tion to x–y laser, e-beam, and ion-beam writers.

The authors thank Anatoly Malyshev from the In-stitute of Automation and Electrometry of the Sibe-rian Branch of the Russian Academy of Science (SBRAS) for photoresist processing and preparation oftest samples and to Dmitry Scheglov from the Insti-tute of Semiconductor Physics SB RAS for atomicforce microscope measurements.

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