Program system for synthesis and digital reconstruction of holograms-projectors:synthesis parameters effect on image reconstruction quality
S. N. Koreshev,a� O. V. Nikanorov, Yu. A. Ivanov, and I. A. Kozulin
St. Petersburg State University of Information Technologies, Mechanics, and Optics, St. Petersburg�Submitted September 22, 2009�Opticheski� Zhurnal 77, 42–48 �January 2010�
A promising variant of the solution to the problem ofshort-wavelength photolithography is one using reflectiverelief-phase holograms-projectors.1 Its practical implementa-tion does not require optical media which are transparent inthe working range of the spectrum. In addition, it is wellknown that when a holographic projection is used, the size ofthe simultaneously exposed region of the photoresist ismostly limited by the power of the source of radiation thanby the aberrations of the projection system, for which ahologram-projector is used in this case. The latter can bemade by analog physical recording as well as by digitalsynthesis.2 The use of digital synthesis is most promising formaking holograms-projectors for use in the extremely short-wavelength ultraviolet and x-ray regions of the spectrum.This is mentioned in, specifically, Ref. 2, where the results ofcomputer simulation of projection holographic lithographyshowing that it can be used in the x-ray range of the spec-trum for fabricating structures with characteristic size0.06 �m. The calculations described in this work were per-formed for radiation wavelength 5 nm at a distance of0.2 mm from the hologram to the plate with photoresist. Inthe calculations the hologram itself was assumed to be trans-mitting and made from a 0.1 mm thick carbon plate. Theresults of synthesis, imaging on a carrier, and investigationsof the imaging properties of a reflective synthesized holo-gram intended for use in radiation with wavelength 13.5 nmare presented in Ref. 3. Unfortunately, the speckle structureof the image reconstructed by means of this hologram doesnot permit using it in the photolithographic process, but thevery fact that it was made and reconstructed shows that inprinciple that synthesized holograms can profitably used inthe extremely short-wavelength region of the spectrum.
The main difficulties of implementation based on the useof synthesized holograms-projectors of holographic photoli-thography are due to the hologram structure. If the represen-tation of the hologram on the carrier presupposes the use ofalready existing laser images which are used for otherpurposes,4 then the process of synthesizing holograms-
33 J. Opt. Technol. 77 �1�, January 2010 1070-9762/2010/0
generators requires a narrowly specialized program systemcapable of solving the problems of synthesis and digital re-construction of holograms. The need for including in thesystem tools not only for synthesizing but also for recon-structing holograms is explained by, in the first place, thestrong dependence of the optimal, from the standpoint ofimage quality, parameters for synthesis of holograms-projectors on the structure of the photomask and, corre-spondingly, the need to adjust them for each specific object.In the second place, physical experiments on the synthesis ofholograms and investigation of their imaging properties arerelatively expensive and complex, especially in the short-wavelength region of the spectrum.
The literature available to the present authors did nothave any information about such systems, observed againstthe potential promise of synthesized holograms-projectors inproblems of short-wavelength holographic projection photo-lithography. This made it necessary to propose and performwork on developing the program system, described in thepresent article, for synthesis and digital reconstruction ofholograms-projectors.
The system consists of two basic blocks: a block forsynthesizing holograms-projectors and a block that performsdigital reconstruction of synthesized holograms. The crux ofthe complex reduces to calculating for each point of the ho-logram the complex amplitude of the field formed by allpoints of the initial object.3 In the course of developing thealgorithm it was assumed that the virtual object—photomask—is a binary two-dimensional transparency oper-ating in transmission, placed parallel to the plane of synthe-sis of the hologram at a distance h from it, and illuminatedby a coherent radiation beam incident in a direction normalto the surface of the photomask. In this case the phase incre-ment �u,v,m,n of the radiation passing through the point of theobject with the coordinates m, n and striking the hologram ata point with the coordinates u, v in the path from the objectto the hologram is calculated from the following expression:
If it is assumed that the coordinates of the points of theobject and hologram u, v, n, m can assume only integralvalues, then the expression �3� below, describing the com-plex amplitude g�u ,v� of the electromagnetic field at an ar-bitrary point on the surface of the hologram, can be derived:
g�u,v� = �m=0
M
�n=0
N
t�n,m��sin��u,v,n,m� − i cos��u,v,n,m�� , �3�
where t�n ,m� is the amplitude transmission coefficient of thephotomask.
The reference wave in the present algorithm is a planewave incident at angle � on the synthesis plane of the holo-gram. Therefore the distribution of the phases in it on thesurface of the hologram �op can be written as
�op =2�x sin �
�. �4�
Hence, adding the complex amplitudes of the reference waveand the radiation which has passed through the transparencywe obtain an array of values of the complex amplitudes ofthe holographic field in the synthesis plane of the hologram.Squaring the modulus of each of its elements gives an arrayof values of the intensity of this field which is required forrepresenting the structure of the hologram-projector on thecorresponding carrier. Thus if the synthesized hologram is tobe represented in the form of an amplitude diffraction struc-ture, then the intensity values indicated above must be rep-
FIG. 1. Interface o
34 J. Opt. Technol. 77 �1�, January 2010
resented in the form of the values, proportional to them, ofthe amplitude transmission coefficient of the hologram.When the synthesized hologram is represented in the form ofa relief-phase structure the computed values of the intensitymust be proportional to the carrier thickness variations overthe aperture of the hologram-projector. Because modernmethods of representing holograms on carriers cannot givean exact correspondence for the hologram parameters beingmodeled, i.e. the transmission coefficient or thickness of thehologram, calculated by the spatial variations of the intensityof the holographic field, operations for binarizing theholograms-projectors must be inserted into their synthesisalgorithms.
There are two ways for reconstructing the synthesizedholograms-projectors in the present program system. One isbased on the same algorithm as the one used for synthesis,i.e. it is based on the Huygens-Fresnel principle, and theother is based on the Fresnel transformation. We note thatboth methods of reconstruction presume the reconstructionof a real image of an object, and at the present time they areimplemented for amplitude and for relief-phase reflection ho-lograms. The possibility of using the two methods of digitalreconstruction of one and the same holograms-projectorswhich is incorporated in the system makes it possible todecrease the strictness of the constraints imposed on theobject-hologram distance and also guarantees that errors ofthe same type will not occur in the synthesis and reconstruc-tion algorithms based on the Huygens-Fresnel principle.
The program system described above has a simple intui-tive interface which makes it possible to set easily the basicparameters of the hologram synthesis and reconstruction�Fig. 1�. These include the regions of coordinate space occu-pied by the object and the synthesized hologram, the distance
program system.
34Koreshev et al.
from the hologram to the object, the working wavelength, theobject and hologram discretization periods, the binarizationthreshold of the hologram, the angle of incidence of the ref-erence and reconstruction waves, the depth of the randommodulation of the phase in the simulation of diffuse illumi-nation of the photomask, and the method chosen to recon-struct the hologram.
The serviceability of the program system was checked inthe course of synthesis and reconstruction of holograms of abinary amplitude transparency, consisting of a collection ofstripes with width 4, 8, and 12 �m. The work was performedusing the main synthesis and reconstruction parameters cho-sen in accordance with the recommendations of Ref. 4 for alaser image generator, i.e. for the angle of incidence of thereference wave 10° and hologram-object distance 0.8 mm,which for a 0.3�0.3 mm hologram aperture, wavelength0.488 �m, object size 0.1�0.1 mm, and hologram and ob-ject discretization periods 1 and 4 �m corresponded to thediffraction limitation on the size of the reconstructed image2.6–4 �m for different points on the surface of the object.
The digital reconstruction of the synthesized hologramswas performed by two methods without binarizing the holo-grams. One method was based on the Fresnel transformationand the other was based on coherent summation of the am-plitudes of the radiation arriving at each point of the objectfrom all other points of the hologram, i.e. on the Huygens-Fresnel principle. The results of the reconstruction are pre-sented in halftone form in Figs. 2 and 3. We note that forconvenience of representation they are inverted, i.e. theirlight sections are represented by dark sections and vice versa.The striping of the light sections of the reconstructed image,as observed in the figures, is probably due to the noise in thediscretization of the hologram. This noise is coherent withrespect to the main image. The intensity modulation due to itin the light sections of the image is much stronger than themodulation occurring in the dark sections.
Figures 4 and 5 display these same images but havingundergone threshold transformation, simulating the responseof photoresist to illumination by actinic radiation,5 which candone using Photoshop. We note that the interval of thresholdlevels for which there is no loss of quality of the recon-structed image lay in the case of the images reconstructed by
FIG. 2. Image reconstructed with the Fresnel transformation.
35 J. Opt. Technol. 77 �1�, January 2010
means of the Fresnel transformation in the range 245–246gradations of grey and was much smaller than the interval ofthreshold levels that is admissible for the image presented inFig. 5, which lies in the range 225–251 gradations of grey.The threshold level interval that gives the best correspon-dence between the reconstructed image which has been sub-jected to threshold processing and the initial image was cho-sen by the authors as a quality criterion for the reconstructedimage. The desirability of introducing precisely such a qual-
FIG. 3. Image reconstructed on the basis of the Huygens-Fresnel principle.
FIG. 4. Image reconstructed using the Fresnel transformation, after thethreshold transformation.
FIG. 5. Image reconstructed on the basis of the Huygens-Fresnel principle,after the threshold transformation.
35Koreshev et al.
ity criterion for reconstructed images can be explained by thefact that the quality loss due to the discreteness inherent insynthesized holograms for images reconstructed using themis expressed, first and foremost, as the appearance of spuri-ous additional images—noise, which can be superposed onand distort the structure of the main image.4 The intensityvariations of a reconstructed image which are admissible inholographic photolithography are largely determined by thethreshold properties of the photoresist. As a rule, photoresistsare considered to be threshold materials whose solubility inthe developer changes at a certain critical value of the expo-sure. For higher or lower values, as compared with the criti-cal value, their solubility in the developer is practically in-dependent of the energy obtained from the exposingradiation. At the same time, for real photoresists near thecritical value there is an interval of exposures where the rateof dissolution of the photoresist in the developer depends onthe exposure. The greater this “threshold” exposure interval,the worse the photoresist works as a discriminant of the de-tails of the image in terms of intensity. Thus the selectedquality criterion for a reconstructed image essentially deter-mines the requirements for the characteristic curve of thephotoresist that is suitable for working with the reconstructedimage. The larger the interval of the admissible thresholdlevels for suppression of the discretization noise, the largerthe “threshold” interval of exposures admissible to the pho-toresist employed is. Comparing the images presented in theFigs. 2–5 shows that for the parameters chosen for synthesisand reconstruction of the holograms the reconstructionmethod based on the Huygens-Fresnel principle gives betterresults than the method based on the Fresnel transformation.As an illustration of the effect of the distance between thesynthesis plane of the hologram and the object on the effec-tiveness of any algorithm used to reconstruct the hologram,images subjected to threshold transformation are displayed inFig. 6 and 7. They were obtained by the methods of recon-struction used with a halftone hologram synthesized 4 mmfrom the object. Comparing the structures of these imagesshows that when reconstructing holograms obtained with adistance of 4 mm between the object and the synthesis planeboth algorithms tested in the work give completely identical
FIG. 6. Image reconstructed with object-hologram distance 4 mm using theFresnel transformation with the threshold transformation.
36 J. Opt. Technol. 77 �1�, January 2010
results. Even the interval of threshold levels, correspondingto the best image quality, was identical, lying in the range236–247 gradations. Therefore for relatively short distancesbetween the hologram synthesis plane and the object, notexceeding 10 hologram sizes, it is best to use the algorithmbased on the Huygens-Fresnel principle for reconstruction.The distortions of the structure of the reconstructed imagewhich are clearly observed in Figs. 6 and 7—the first two“corners” collapsed and the gap between them disappeared—are due to the inadequate resolving power of the hologram,which is limited by its aperture size. The results obtainedshow that the program system and the possibility of using itto synthesize and reconstruct holograms of two-dimensionalbinary objects works.
The possibility of using the present program system tooptimize the synthesis parameters for holograms-projectorswas checked in an investigation of the dependence of thequality of the reconstructed image on the size of the holo-gram, the angle of incidence of the reference wave on thesynthesis plane of the hologram, and the ratio of the periodschosen for discretization of the object and the hologram. Thisinvestigation was conducted with the same parameters of theobject and the hologram and for the same wavelength of thereconstructing radiation as in the previous experiments. Theprocedure consisted of using the expressions presented inRef. 4 to calculate the optimal parameters for synthesizingholograms-projectors, evaluating the quality of the image re-constructed using the Fresnel transformation in virtual spacewith the aid of holograms synthesized with theoretically op-timized synthesis parameters as well as with different param-eters. Once again, here, the quality criterion for the recon-structed image was the grey gradations interval, which couldbe chosen as the threshold in threshold processing of thereconstructed image and would give maximum correspon-dence to the object. The above-indicated working wave-length, the dimensions of the object, and the periods of dis-cretization of the object and the hologram gave the followingvalues for the optimal parameters of the synthesizedhologram-projector: distance between the synthesized holo-gram and the object—1.5 mm, angle of incidence of refer-ence plane wave on the plane of synthesis—10.6°, and size
FIG. 7. Image reconstructed with object-hologram distance 4 mm on thebasis of the Huygens-Fresnel principle with the threshold transformation.
36Koreshev et al.
of the hologram 0.471�0.471 mm. The quality of the imagereconstructed with the help of the hologram-projector syn-thesized with the parameters indicated above is illustrated inFigs. 8 and 9, which display the reconstructed image beforeand after its threshold processing, respectively. The intervalof grey gradations which could be used as a threshold forthreshold processing of the image presented in Fig. 8 wasequal to 47 gradations. The experimental results on thechange of the parameters used for the synthesis of thehologram-projector showed the following. The decrease, justas the increase, of the linear aperture of the synthesized ho-logram as compared with the aperture calculated in accor-dance with Ref. 4 decreases the interval of grey gradations.Thus decreasing the hologram size to 0.4�0.4 mm resultedin a decrease of the admissible range of threshold levels to16 gradations. Further size reduction of the hologram also
FIG. 8. Reconstructed image before threshold processing.
FIG. 9. Reconstructed image after threshold processing.
37 J. Opt. Technol. 77 �1�, January 2010
resulted in loss of resolution in the image reconstructed withits help. Increasing the aperture of the hologram to 0.6�0.6 mm likewise resulted in a decrease of the admissiblethreshold range to 13 gradations. In all probability, this canbe explained by an increase of the fraction of the hologramarea on which the spatial frequencies of the hologram struc-ture exceed the values determined by the sampling theorem.6
The character of the dependence of the quality, found in thecourse of these investigations, of the reconstructed image onthe angle of incidence of the reference wave shows that thegreatest accessible range of grey gradations which can beused as a threshold in processing reconstructed images, equalto approximately 50, corresponds to the range of angles ofincidence of the reference wave 10–14°. We note that for thehologram discretization parameters used in the present workand for the working wavelength the angle of incidence of thereference wave 15° corresponds for the center of thehologram-projector to the limiting spatial carrying frequency,determined in accordance with the sampling theorem.6
The experimental data obtained and their good agree-ment with the results of the theoretical analysis presented inRef. 4 support the validity of the recommendations madethere and attest to the serviceability of our program systemand the possibility and promise of using it to optimize theparameters for synthesizing holograms-projectors.
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