S

Fa

b

a

ARA

2111KACP2

1

tfirdt(o

gtrig[oswrpi

S

0d

Optik 124 (2013) 818– 823

Contents lists available at SciVerse ScienceDirect

Optik

j o ur nal homepage: www.elsev ier .de / i j leo

ingle closed fringe pattern phase demodulation in alignment of nanolithography

eng Xua,b,∗, Song Hua, Yong Yanga, Jinlong Lia,b, Lanlan Lia,b

Institute Optics & Electronics, Chinese Academy of Sciences, Chengdu 610209, ChinaGraduate University of Chinese Academy of Sciences, Beijing 100039, China

r t i c l e i n f o

rticle history:eceived 18 September 2011ccepted 20 January 2012

20.114000.5070

a b s t r a c t

The single closed fringe pattern that occurs in two superposed grating marks applied in the previouslydesigned moiré alignment scheme based on dual-grating for lithography is processed and analyzed usinga frequency domain method based on two-dimensional (2-D) analytic wavelet transform (AWT) and 2-D wavelet ridge algorithm. The sign ambiguities, which always occur in the process of single closedfringe pattern analysis, are removed through the discontinuities of the angle in the 2-D wavelet ridge.Theoretical analysis regarding application of 2-D AWT and 2-D wavelet ridge to the interference fringein alignment is performed. Verification of this process is carried out through numerical simulation and

00.300800.2650eywords:lignmentlosed fringehase demodulation

experiment. Results indicate that the background and noise in the fringes can be filtered effectivelythrough our method, and the phase information can be obtained successfully.

© 2012 Elsevier GmbH. All rights reserved.

-D wavelet ridge

. Introduction

With the rapid developments and wide applications of nano-echnology, more and more attention has been paid to this frontiereld from scientists and engineers. On one hand, nano-devices fab-ication, as the basis of nanotechnology, has placed more and moreemands on the resolution of nanolithography. On the other hand,he continuously shrunk feature size of highly integrated circuitsIC) and related elements also accelerates the improvement of res-lution in lithography.

Optical alignment, one of the critical technologies of the litho-raphic system, has always been an important factor in improvinghe resolution of lithography. Traditionally, alignment is mainlyealized through three categories of approaches: the geometricmaging method that directly compares two crosses or bars likeeometric marks located on both wafer and mask via a screen1,2], the intensity detection that detects the critical intensity valuef diffracted beams [3,4], and the beat signal detection that mea-ures the phase of a beat signal generated by two diffracted beamsith close frequencies [5,6]. Further, a scheme based on manual

ecognition of moiré fringes was applied in the X-ray and nanoim-rint lithography [7,8]. However, the accuracy of the geometric

maging method is always limited, and the intensity- or phase-

∗ Corresponding author at: Institute Optics & Electronics, Chinese Academy ofciences, Chengdu 610209, China.

E-mail address: casxufeng@gmail.com (F. Xu).

030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved.oi:10.1016/j.ijleo.2012.01.032

based approaches cannot avoid the problem of high complexityand beam fluctuation caused by wafer process, e.g. the resist layer.The moiré fringe scheme cannot avoid inducing systematic errors inthe process of pattern recognition. Recently, Zhou et al. proposed ananalogous moiré alignment scheme based on dual-grating, whichcombined the inherent high detectivity of interferometric mea-surement with the convenience of fringe pattern processing [9–12].Compared with other alignment techniques, higher detectivityand better robustness can be realized because alignment offset isembedded into spatial phase of the fringe pattern. And the processof fringe pattern analysis and spatial phase extraction is regardedas a necessity and critical factor to guarantee the efficiency andaccuracy of the presented scheme. In practice, the optical beamfluctuation, wave distortion and noises induced by the dusts orspots on wafer and mask in the process of lithography may smearthe fringe pattern or degrade the image quality. Therefore, imagingprocessing, especially the fringe pattern analysis and phase extrac-tion, is mainly studied and discussed in this paper.

Traditionally, phase shifting technique (PST) [13,14], regular-ized phase tracking (RPT) [15], Fourier transform (FT) [16], windowFourier transform (WFT) [17–20] and 1-Dimensional (1-D) wavelettransform (WT) [21–24] are several typical fringe pattern analysismethods. PST extracts the phase information through several fringepatterns related to the original fringe with a series of pre-set phase

shifts. Though the corresponding algorithms are simple and fast,the fact that phase-shift image is difficult to obtain and the phaseshift process is prone to introducing error limits its wide applica-tions. RPT regards the fringe pattern as the smooth area with certain

k 124

ppsahdatwwdfwmissmamtatvpftc2

2

2

slattrsdlfmtf

fcpsTaTmspb

f

wtr

F. Xu et al. / Opti

hase distribution, and establishes the cost function related to thehase and frequency at each pixel. According the cost function, themooth phase area can be obtained through estimating the phasend frequency with iterative approximation, but the algorithm is ofigh complexity. FT processes the fringe pattern in whole frequencyomain with high efficiency, but the local phase information islmost lost easily when filtering the noises. WFT extracts the phasehrough moving window function and processing each windowith FT. It obtains good accuracy and wide range of applicationsith the common advantage of PST and FT. However, the fixed win-ow size makes it not suitable for the fringe pattern with variablerequency. 1-D WT uses the specific wavelet basis to do convolutionith the fringe patterns, and extracts the phase through the maxi-um value of wavelet coefficients which called “wavelet ridge”. It

s strongly adaptable to the variable frequency, but it is also sen-itive to noise. Therefore, we present a method with regard to theingle closed fringe pattern obtained in the previously designedoiré alignment scheme based on dual-grating, which may also be

pplicable in related fields, such as optical measurement. By thisethod, 2-D analytic wavelet transform (AWT) is used to perform

he fringe pattern analyzing and extract the phase, then the signmbiguities is removed through the discontinuities of the angle inhe 2-D wavelet ridge. In this paper, the background of the pre-iously designed alignment scheme and the principle of fringesattern processing in that scheme are firstly introduced. Next, theeasibility of this method for the extraction of the phase is verifiedhrough numerical simulation and experiments. Finally, some dis-ussions about comparing the proposed method with the 2-D FT,-D WFT and 1-D WT methods are presented.

. Theory

.1. The interference fringe in alignment

Fig. 1 shows the framework of the wafer-mask alignmentcheme based on dual-grating designed previously for proximityithography. Two gratings with slightly different periods on wafernd mask respectively are used as the alignment marks. Diffractionakes place at the incident surface of two gratings illuminated byhe collimated laser beam from splitter. Certain pairs of symmet-ic diffracted orders from wafer and mask encounter at the maskurface and interfere to generate the field with constructive andestructive intensity. Then the diffracted orders are collected by the

ens and projected onto the CCD, the interference field on mask sur-ace can be recorded. As a result, any relative displacement between

ask and wafer introduces center phase shift of these fringe pat-erns, which can be directly obtained by phase demodulation forurther alignment.

Fig. 2 shows the marks of two gratings and the correspondingringe patterns. Each of the marks shown in Fig. 2(a) and (b) is ofircular gratings with micro-level periods T1 and T2. When super-osition of two marks occurs in the alignment process, two fringeshown in Fig. 2(c) and (d) are recorded by CCD through the lens.he phase of the fringe patterns varied with different frequenciesccording to the relative displacement between wafer and mask.he fringe pattern shown in Fig. 2(c) indicated that the wafer andask are misaligned by a certain offset �x, and the fringe pattern

hown in Fig. 2(d) indicated that the wafer and mask are com-letely aligned. Generally, the intensity of the closed fringes cane expressed as

(x, y) = a(x, y) + b(x, y) cos[�(x, y)] (1)

here f(x, y), a(x, y) and b(x, y) denote the recorded intensity,he background intensity, and the amplitude modulation of fringesespectively. The coordinates (x, y) denote pixels of the fringe pat-

(2013) 818– 823 819

tern and ˚(x, y) denote the phase of it. In the alignment schemebased on dual-grating, the phase is determined by the geometricaldistribution of the two gratings on the wafer and mask respectively[10]. Each phase of fringes in Fig. 2(c) and (d) can be expressed asfollow

�1(x, y) = 2�(f1√

(x − �x)2 + y2 − f2√x2 + y2) (2)

�2(x, y) = 2�(f1√x2 + y2 − f2

√x2 + y2) (3)

where ϕ1(x, y) and ϕ2(x, y) are the phases corresponding to mis-aligned and aligned state, and f1 = 1/T1 and f2 = 1/T2 are the spatialfrequencies of two gratings respectively, �x is the relative displace-ment between two grating marks along x axis. The relative offsetbetween two marks corresponds to center shift in spatial phase ofthe fringe pattern after solving the Eqs. (2) and (3), so the relativeoffset can be easily acquired if the phase of the fringe is obtained.Specifically, phase variation �� along the x axis is directly relatedto the misalignment offset �x by the formula:

�x = ��(T1 × T2)(2�(T1 + T2))

.

2.2. Phase extraction using 2-D AWT

Generally, the 2-D WT is defined as

Wf (a, b, s, �) =∫ +∞

−∞

∫ +∞

−∞f (x, y) a,b,s,�(x, y)dxdy (4)

a,b,s,�(x, y) = 1s

(x − a

s,y − b

s, �

)(5)

where f(x, y) is the 2-D input signal, and a,b,s,�(x, y) is the waveletseries generated by translating along x and y directions by a andb respectively, dilation by s, and rotating by angle � of the motherwavelet (x, y). The amplitude and the phase can be expressed as

A(a, b, s, �) =√

{Re[Wf (a, b, s, �)]}2 + {Im[Wf (a, b, s, �)]}2 (6)

�(a, b, s, �) = arc tan

{Im[Wf (a, b, s, �)]Re[Wf (a, b, s, �)]

}(7)

From Fig. 2(c), we can find that the relative displacementbetween wafer and mask connects with the variety of the fringewith different frequencies, and each particular frequency shouldbe analyzed for phase extraction. The different frequencies canbe analyzed through the multi-scale wavelet analysis which isknown as the microscope in signal analysis. Generally, separatingthe phase and amplitude of the fringe absolutely is very importantfor phase extraction. The complex analytic wavelet usually con-structed through the frequency modulation of a real symmetricwindow function can well perform the separation. Gabor wavelethas the properties of good localization in the space and frequencydomains, thus it can be approximately regarded as analytic waveletrealizing by modulating the frequency of the Gaussian windowfunction [25]. So it is suitable for phase extraction from the fringein alignment. The 2-D AWT is applied to fringes for suppressing thenoises as far as possible in the course of phase extraction in thispaper, and the 2-D expansion of Gabor analytic wavelet is regardedas the mother wavelet in the form of

G(x, y) = 14√�

√2��

exp

(− (2�/�)2(x2 + y2)

2

)exp[j2�(x cos �

+y sin �)], � = �√

2/(ln 2) (8)

where � is the normalized parameter. The essence of AWT is thesimilarity description between the input signal and wavelet series.

820 F. Xu et al. / Optik 124 (2013) 818– 823

Fig. 1. The framework of wafer-mask alignment in proximity lithography.

Fig. 2. The two group grating marks and the corresponding fringe (a): grating maker 1, (b): grating mark 2, (c): the fringe distribution of misalignment, (d): the fringedistribution of alignment.

F d thet

WtcvsbEv

[

r

ns

ig. 3. The fringe of misalignment, the phase of the fringe, the fringe with noise anhe fringe with noise, (d): the phase map after transform.

hen the phase and the rotation angle of wavelet series are closeo phase and direction of the input signal, the wavelet transformoefficient attains the maximum value which is called ‘ridge’. Thealues of s* and �* that maximize the similarity are taken as the localcale and angle at pixel (a, b). Local scale and angle for all pixels cane estimated by rotating and dilating the wavelet basis. Shown inq. (9), s and � take the value of s* and �* respectively when thesealues maximize the amplitude spectrum A (a, b, s*, �*).

s∗(a, b), �∗(a, b)] = arg max (A(a, b, s∗, �∗)) (9)

(a, b) =∣∣Wf (a, b, s∗(a, b), �∗(a, b))

∣∣ (10)

The ridge values determined by Eq. (10) involve the phaseseeded to be extracted. However, the extracted phase is wrapped,o further phase unwrapping is needed.

phase after transform (a): fringe of misalignment, (b): the phase of the fringe, (c):

3. Numerical simulation

Numerical simulation is performed through adopting two marksformed by the circular micro-gratings with the period of 4 �mand 4.4 �m, as shown in Fig. 2(a) and (b). To confirm the phaseextraction process, two gratings on mask and wafer respectivelyare misaligned by certain predetermined offset, and the generatedfringe pattern is shown in Fig. 3(a). The size of the image is selectedas 256 pixels × 256 pixels. The wrapped phase map of the generatedfringe is shown in Fig. 3(b). It is known that the phase offset of thecenter part, which is related with the misalignment displacementbetween wafer and mask, causes the fringe to vary with different

frequencies.

Inevitably, the fluctuation in the optical path and wafer pro-cess always induce noises in the course of image acquisition,so uniform Gaussian white noises with mean of 0 and standard

F. Xu et al. / Optik 124 (2013) 818– 823 821

Fig. 4. The angle distribution in the 2-D wavelet ridge after 2-D AWT, the discontinuity bor(a): the angle distribution in the 2-D wavelet ridge after 2-D AWT, (b): discontinuity bovertical direction, (d): the phase without sign ambiguities.

Ft

dptaidfiritiri

Fab

ig. 5. The fringe of misalignment in experiment and the phase after processing (a):he fringe of misalignment, (b): the phase distribution.

eviation of 1 are intentionally added into the fringe pattern in therocess of numerical simulation, as shown in Fig. 3(c). Meanwhile,he amplification lens with 20× is used. The 2-D AWT method ispplied to the fringe pattern with noises and the phase involvedn the 2-D wavelet ridges is extracted. Fig. 3(d) shows the phaseistribution after extraction. We can see that the noises are wellltered with few sign ambiguities. The sign ambiguities are notandomly distributed. Their border is the discontinuity of the anglen the 2-D wavelet ridge. So the sign ambiguities can be removed

hrough multiplying the phase within each part of the discontinu-ty by −1. Fig. 4(a) shows the angle distribution in the 2-D waveletidge. Fig. 4(b) and (c) shows the discontinuity borders of the anglen the horizontal and vertical directions respectively. The border

ig. 6. The angle distribution in the 2-D wavelet ridge after 2-D AWT, the discontinuitymbiguities in experiment (a): the angle distribution in the 2-D wavelet ridge after 2-D Aorder of angle in the vertical direction, (d): the phase without sign ambiguities.

ders of angle in the horizontal and vertical direction, phase without sign ambiguitiesrder of angle in the horizontal direction, (c): discontinuity border of angle in the

of the sign ambiguities and the symmetry line of fringes can beeasily localized through disposing the angle in the 2-D waveletridge. And the sign ambiguities can be removed along directionof the symmetry line by the early proposed phase unwrappingmethod. Fig. 4(d) shows the phase distribution without the signambiguities.

4. Experiment

According to the framework shown in Fig. 1, the experimenthas also been performed to verify the feasibility of phase extrac-tion process. The recorded CCD fringe pattern is shown in Fig. 5(a)and the size is also selected as 256 pixel × 256 pixel. We can see thatthe fringes with dull background are influenced by noises inducedby the fluctuation in the optical path and wafer process. Fig. 5(b)shows the phase distribution after using the proposed method. Theangle distribution in the 2-D wavelet ridge, the discontinuity bor-ders of the angle along the horizontal and vertical directions areshown in Fig. 6(a)–(c), respectively. Fig. 6(d) shows the phase dis-tribution after removing the sign ambiguities. We can see that thenoise is well filtered and the phase is accurately extracted throughthe proposed 2-D AWT method.

5. Discussion

The numerical and experiment results show that our methodcan successfully extract the phase from single closed fringe pat-tern. To make a comparison, the fringes in Fig. 3(c) are alsoanalyzed using 2-D FT, 2-D WFT and 1-D WT method respectively.

borders of angle in the horizontal and vertical direction, the phase without signWT, (b): discontinuity border of angle in the horizontal direction, (c): discontinuity

822 F. Xu et al. / Optik 124 (2013) 818– 823

Fig. 7. Phase distribution after 2-D FT, 2-D WFT and 1-D WT in simulation (a): phase distribution after 2-D FT, (b): phase distribution after 2-D WFT, (c): phase distributionafter 1-D WT.

F se disa

Tr2icbewperr

6

tncut

ig. 8. Phase distribution after 2-D FT, 2-D WFT and 1-D WT in experiment (a): phafter1-D WT.

he corresponding phase distributions are shown in Fig. 7(a)–(c),espectively. Compared with the phase distribution extracted by-D AWT shown in Fig. 4(d), the phase extracted by 2-D FT varies

ntensively because the detailed spectrums disappeared in theourse of filtering, and the phase extracted by 2-D WFT is difficult toe unwrapped due to its illegible contour. Furthermore, the phasextracted by 1-D WT is also difficult to be unwrapped because itas influenced by the noises somewhere. Fig. 8(a)–(c) shows thehase distributions extracted from the fringe pattern acquired inxperiment shown in Fig. 5(a) by 2-D FT, 2-D WFT and 1-D WTespectively. We can find that the same disadvantages exist in theesult of experiments.

. Conclusion

With regard to the single closed fringe pattern that occurs inhe superposition of two grating marks in the alignment process of

anolithography, we propose a method of the fringe pattern pro-essing to extract the phase distribution. The method is performedsing 2-D AWT and 2-D wavelet ridge algorithm, and sign ambigui-ies are removed through the discontinuities of the angle in the 2-D

tribution after 2-D FT, (b): phase distribution after 2-D WFT, (c): phase distribution

wavelet ridge. The framework of dual-grating alignment scheme isfirstly introduced, and then theoretical analysis of fringe patternprocessing using the method is presented. Finally, numerical simu-lation and experiment are performed to verify the feasibility of ourmethod, and the comparisons between our method and the methodof 2-D FT, 2-D WFT and 1-D WT are made. The results indicate thatour method can extract the phase successfully with well noise sup-pressing ability, and may be applicable in related fields touching tointerference measurement.

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (nos. 60976077, 60906049, 61076100) and theNational High Technology Research and Development Program ofChina (no 2009AA03Z341).

References

[1] J.W. Whang, N.C. Gallagher, Synthetic approach to designing optical alignmentsystems, Appl. Opt. 27 (1988) 3534–3541.

k 124

[

[

[

[

[

[

[

[

[

[

[

[

[

[

F. Xu et al. / Opti

[2] T. Miyatake, M. Hirose, T. Shoki, R. Ohkubo, K. Yamazaki, Nanometer scattered-light alignment system using SiC x-ray masks with low optical transparency, J.Vac. Sci. Technol., B 16 (1998) 3471–3475.

[3] D.C. Flanders, H.I. Smith, S. Austin, A new interferometric alignment technique,Appl. Phys. Lett. 31 (1977) 428–430.

[4] B. Fay, J. Trotel, A. Frichet, Optical alignment system for submicron x-ray lithog-raphy, J. Vac. Sci. Technol. 16 (1979) 1954–1958.

[5] J. Itoh, T. Kanayama, N. Atoda, K. Hoh, An alignment system for synchrotronradiation x-ray lithography, J. Vac. Sci. Technol., B 6 (1988) 409–412.

[6] U.A.M. Suzuki, An optical-heterodyne alignment technique for quarter-micronx-ray lithography, J. Vac. Sci. Technol., B 7 (1989) 1971–1976.

[7] A. Moel, E.E. Moon, R.D. Frankel, H.I. Smith, Novel on-axis interferometricalignment method with sub-10 nm precision, J. Vac. Sci. Technol., B 11 (1993)2191–2194.

[8] N.H. Li, W. Wu, S.Y. Chou, Sub-20-nm alignment in nanoimprint lithographyusing Moire fringe, Nano Lett. 6 (2006) 2626–2629.

[9] S. Zhou, Y. Fu, X. Tang, S. Hu, W. Chen, Y. Yang, Fourier-based analysis of moiréfringe patterns of superposed gratings in alignment of nanolithography, Opt.Express 16 (2008) 7869–7880.

10] W. Chen, W. Yan, S. Hu, Y. Yang, S. Zhou, Extended dual-grating alignmentmethod for optical projection lithography, Appl. Opt. 49 (2010) 708–713.

11] Y. Wei, Y. Yong, W. Chen, S. Hu, S. Zhou, Moiré based focusing and levelingscheme for optical projection lithography, Appl. Opt. 49 (2010) G1–G5.

12] S. Zhou, Y. Yang, L. Zhao, S. Hu, Tilt-modulated spatial phase imaging method for

wafer-mask leveling in proximity lithography, Opt. Lett. 35 (2010) 3132–3134.

13] P.D. Groot, Phase-shift calibration errors in interferometers with sphericalFizeau cavities, Appl. Opt. 34 (1995) 2856–2863.

14] J. Schwider, O. Fallkenstorfer, H. Schreiber, A. Zoeller, New compensating four-phase algorithm for phase-shift interometry”, Opt. Eng. 32 (1993) 1883–1885.

[

[

(2013) 818– 823 823

15] M. Servin, J.L. Marroquin, F.J. Cuevas, Demodulation of a single interferogramby use of a two-dimensional regularized phase-tracking technique”, Appl. Opt.36 (1997) 4540–4548.

16] M. Taketa, H. Ina, S. Kobayashi, Fourier-transform method of fringe-patternanalysis for computer-based topography and interferometry, J. Opt. Soc. Am.72 (1982) 156–160.

17] Q. Kemao, Windowed Fourier transform for fringe pattern analysis, Appl. Opt.43 (2004) 2695–2702.

18] Q. Kemao, S.H. Soon, Two-dimensional windowed Fourier transform fornoise reduction in fringe pattern analysis, Opt. Eng. 44 (2005), 0705601:1–9.

19] Q. Kemao, Two-dimensional windowed Fourier transform for fringe patternanalysis: principle, applications, and implementations, Opt. Lasers Eng. 45(2007) 304–307.

20] Q. Kemao, A simple phase unwrapping approach based on filtering by win-dowed Fourier transform: a note on the threshold selection, Opt. LasersTechnol. 40 (2008) 1091–1098.

21] J. Zhong, J. Weng, Phase retrieval of optical fringe patterns from the ridge of awavelet transforms, Opt. Lett. 30 (2006) 2560–2562.

22] H. Liu, A.N. Cartwright, C. Basaran, Moire interferogram phase extraction: aridge detection algorithm for continuous wavelet transforms, Appl. Opt. 43(2004) 850–857.

23] M. Liebling, T.F. Bernhard, A.H. Bachmann, Continuous wavelet transformsridge extraction for spectral interferometry imaging, Proc. SPIE 5690 (2005)

397–402.

24] L.R. Watkins, S.M. Tan, T.H. Barnes, Determination of interferometer phasedistributions by use of wavelets, Opt. Lett. 24 (1999) 905–907.

25] S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed., Academic Press, SanDiego, 1999, 87–88.