Accurate characterization of nanoimprinted resist patterns using Mueller matrix ellipsometry

In order to control nanoimprint lithography processes to achieve good fidelity, accurate characterization of structural parameters of nanoimprinted resist patterns is highly desirable. Among the possible techniques, optical scatterometry is relatively ideal due to its high throughput, low cost, and minimal sample damage. Compared with conventional optical scatterometry, which is usually based on reflectometry and ellipsometry and obtains at most two ellipsometric angles, Mueller matrix ellipsometry (MME) based scatterometry can provide up to 16 quantities of a 4 × 4 Mueller matrix in each measurement and can thereby acquire much more useful information about the sample. In addition, MME has different measurement accuracy in different measurement configurations. It is expected that much more accurate characterization of nanoimprinted resist patterns can be achieved by choosing appropriate measurement configurations and fully using the rich information hidden in the measured Mueller matrices. Accordingly, nanoimprinted resist patterns were characterized using an in-house developed Mueller matrix ellipsometer in this work. We have experimentally demonstrated that not only more accurate quantification of line width, line height, sidewall angle, and residual layer thickness of nanoimprinted resist patterns can be achieved, but also the residual layer thickness variation over the illumination spot can be directly determined, when performing MME measurements in the optimal configuration and meanwhile incorporating depolarization effects into the optical model. The comparison of MME-extracted imprinted resist profiles has also indicated excellent imprint pattern fidelity. ©2014 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (120.2130) Ellipsometry and polarimetry; (290.3200) Inverse scattering; (050.1950) Diffraction gratings. References and links 1. S. Y. Chou, P. R. Krauss, and P. J. Renstrom, “Imprint lithography with 25-nm resolution,” Science 272(5258), 85–87 (1996). 2. S. Y. Chou, P. R. Krauss, W. Zhang, L. Guo, and L. Zhuang, “Sub-10 nm imprint lithography and applications,” J. Vac. Sci. Technol. B 15(6), 2897–2904 (1997). 3. H.-J. Lee, C. L. Soles, H. W. Ro, R. L. Jones, E. K. Lin, W. Wu, and D. R. Hines, “Nanoimprint pattern transfer quality from specular x-ray reflectivity,” Appl. Phys. Lett. 87(26), 263111 (2005). 4. R. L. Jones, T. Hu, C. L. Soles, E. K. Lin, R. M. Reano, S. W. Pang, and D. M. Casa, “Real-time shape evolution of nanoimprinted polymer structures during thermal annealing,” Nano Lett. 6(8), 1723–1728 (2006). 5. D. Fuard, C. Perret, V. Farys, C. Gourgon, and P. Schiavone, “Measurement of residual thickness using scatterometry,” J. Vac. Sci. Technol. B 23(6), 3069–3074 (2005). 6. R. M. Al-Assaad, S. Regonda, L. Tao, S. W. Pang, and W. Hu, “Characterizing nanoimprint profile shape and polymer flow behavior using visible light angular scatterometry,” J. Vac. Sci. Technol. B 25(6), 2396–2401 (2007). 7. H. J. Patrick, T. A. Germer, Y. Ding, H. W. Ro, L. J. Richter, and C. L. Soles, “Scatterometry for in situ measurement of pattern reflow in nanoimprinted polymers,” Appl. Phys. Lett. 93(23), 233105 (2008). #210183 $15.00 USD Received 14 Apr 2014; revised 31 May 2014; accepted 4 Jun 2014; published 12 Jun 2014 (C) 2014 OSA 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.015165 | OPTICS EXPRESS 15165 8. C. J. Raymond, “Scatterometry for semiconductor metrology,” in Handbook of Silicon Semiconductor Metrology, A. C. Diebold, ed. (CRC, 2001), Chap. 18, pp. 477–514. 9. X. G. Chen, S. Y. Liu, C. W. Zhang, and H. Jiang, “Improved measurement accuracy in optical scatterometry using correction-based library search,” Appl. Opt. 52(27), 6726–6734 (2013). 10. R. Silver, T. A. Germer, R. Attota, B. M. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits of optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007). 11. B. Bunday, T. A. Germer, V. Vartanian, A. Cordes, A. Cepler, and C. Settens, “Gaps analysis for CD metrology beyond the 22 nm node,” Proc. SPIE 8681, 86813B (2013). 12. T. Novikova, A. De Martino, S. Ben Hatit, and B. Drévillon, “Application of Mueller polarimetry in conical diffraction for critical dimension measurements in microelectronics,” Appl. Opt. 45(16), 3688–3697 (2006). 13. M. Foldyna, A. De Martino, E. Garcia-Caurel, R. Ossikovski, C. Licitra, F. Bertin, K. Postava, and B. Drévillon, “Critical dimension of bioperiodic gratings determined by spectral ellipsometry and Mueller matrix polarimetry,” Eur. Phys. J. Appl. Phys. 42(3), 351–359 (2008). 14. X. G. Chen, S. Y. Liu, C. W. Zhang, and H. Jiang, “Measurement configuration optimization for accurate grating reconstruction by Mueller matrix polarimetry,” J. Micro/Nanolith. MEMS MOEMS 12(3), 033013 (2013). 15. Z. Q. Dong, S. Y. Liu, X. G. Chen, and C. W. Zhang, “Determination of an optimal measurement configuration in optical scatterometry using global sensitivity analysis,” Thin Solid Films 562, 16–23 (2014), doi:10.1016/j.tsf.2014.03.051. 16. T. Novikova, A. De Martino, P. Bulkin, Q. Nguyen, B. Drévillon, V. Popov, and A. Chumakov, “Metrology of replicated diffractive optics with Mueller polarimetry in conical diffraction,” Opt. Express 15(5), 2033–2046 (2007). 17. T. Novikova, P. Bulkin, V. Popov, B. Haj Ibrahim, and A. De Martino, “V. Popov, B. Haj Ibrahim, and A. De Martino, “Mueller polarimetry as a tool for detecting asymmetry in diffraction grating profiles,” J. Vac. Sci. Technol. B 29(5), 051804 (2011). 18. J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9(4), 041305 (2010). 19. X. G. Chen, C. W. Zhang, and S. Y. Liu, “Depolarization effects from nanoimprinted grating structures as measured by Mueller matrix polarimetry,” Appl. Phys. Lett. 103(15), 151605 (2013). 20. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995). 21. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14(10), 2758–2767 (1997). 22. S. Y. Liu, Y. Ma, X. G. Chen, and C. W. Zhang, “Estimation of the convergence order of rigorous coupled-wave analysis for binary gratings in optical critical dimension metrology,” Opt. Eng. 51(8), 081504 (2012). 23. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977). 24. J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.) 33(2), 185–189 (1986). 25. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957). 26. T. A. Germer and H. J. Patrick, “Effect of bandwidth and numerical aperture in optical scatterometry,” Proc. SPIE 7638, 76381F (2010). 27. H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007). 28. R. W. Collins and J. Koh, “Dual rotating-compensator multichannel ellipsometer: instrument design for real-time Mueller matrix spectroscopy of surfaces and films,” J. Opt. Soc. Am. A 16(8), 1997–2006 (1999). 29. S. Y. Liu, X. G. Chen, and C. W. Zhang, “Mueller matrix polarimetry: A powerful tool for nanostructure metrology,” ECS Trans. 60(1), 237–242 (2014). 30. C. H. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multiangle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998). 31. A. R. Forouhi and I. Bloomer, “Optical properties of crystalline semiconductors and dielectrics,” Phys. Rev. B Condens. Matter 38(3), 1865–1874 (1988). 32. X. G. Chen, S. Y. Liu, H. G. Gu, and C. W. Zhang, “Formulation of error propagation and estimation in grating reconstruction by a dual-rotating compensator Mueller matrix polarimeter,” Thin Solid Films (2014), doi:10.1016/j.tsf.2014.01.049.


Introduction
Nanoimprint lithography (NIL) [1,2], in which features on a prepatterned mold are transferred directly into a polymer material, represents a promising technique with the potential for high resolution and throughput as well as low cost.In order to control NIL processes to achieve good fidelity, accurate characterization of structural parameters of nanoimprinted resist patterns is highly desirable.These parameters usually include not only the critical dimension, sidewall angle and feature height, but also the residual layer thickness.Although both scanning electron microscopy (SEM) and atomic force microscopy (AFM) can provide high precision data, they are in general time-consuming, expensive, complex to operate, and problematic in realizing in-line integrated measurement.Recently, optics-based metrology techniques have been introduced to measure imprinted nanostructures, such as specular x-ray reflectivity (SXR) [3], critical dimension small-angle x-ray scattering (CD-SAXS) [4], and optical scatterometry [5][6][7].Among these techniques, optical scatterometry is relatively ideal due to its high throughput, low cost, and minimal sample damage.
The optical scatterometry, which involves an inverse diffraction problem solving process with the objective of finding a structural profile whose theoretical signature can best match the measured one [8,9], is traditionally based on reflectometry and ellipsometry.With the ever-decreasing in features and critical dimensions, conventional optical scatterometry is quickly reaching its limit and requires improvements for future process nodes [10,11].Compared with conventional optical scatterometry, which at most obtains two ellipsometric angles, Mueller matrix ellipsometry (MME, sometimes also referred to as Mueller matrix polarimetry) based scatterometry can provide up to 16 quantities of a 4 × 4 Mueller matrix in each measurement.Consequently, MME can acquire much more useful information about the sample.Meanwhile, MME can obtain different measurement configurations by changing three measurement conditions, i.e., the wavelength, the incidence angle and the azimuthal angle.Great discrepancies in the final measurement accuracy in different measurement configurations have been reported in the literatures [12][13][14][15].It is therefore expected that much more accurate characterization of nanoimprinted resist patterns can be achieved by choosing appropriate measurement configurations and exploring the rich information hidden in the measured Mueller matrices.The present study takes this as a starting point and characterizes nanoimprinted resist patterns using an in-house developed Mueller matrix ellipsometer.
Several researchers have applied MME to characterize nanostructures in NIL processes.Novikova et al. implemented MME in different azimuthal angles to characterize onedimensional diffraction gratings used in the fabrication of molding tool for the nanoimprint process [16,17].It was shown that the Mueller matrices measured in proper conical diffraction configurations may help decouple some of the fitting parameters.Li et al. investigated the possibility of measuring grating asymmetry with MME for a patterned hard disk sample prepared by nanoimprint technique [18].They reported that certain Mueller matrix elements were proportional to both the direction and amplitude of profile asymmetry, which provides a direct indication to the sidewall tilting.In our recent work, noticeable depolarization effects were observed from the measured Mueller matrices of nanoimprinted resist patterns [19].We found that improved accuracy can be achieved for the line width, line height, sidewall angle, and residual layer thickness measurement after taking depolarization effects into account.However, to the best of our knowledge, there is no reported study yet to emphasize that much more accurate characterization of nanoimprinted resist patterns can be achieved by performing MME measurements in the optimal configuration and meanwhile fully using the rich information contained in the measured Mueller matrices.The present study is also a continuous advance of our recent work [19] where nanoimprinted resist patterns were only characterized in the planar diffraction configuration, which is not necessarily the optimal measurement configuration.

Theory
The optical responses of periodic structures can be calculated by rigorous coupled-wave analysis (RCWA) [20][21][22].In RCWA, both the permittivity function and electromagnetic fields are expanded into Fourier series.Afterwards, the tangential field components are matched at boundaries between different layers, and thereby the boundary-value problem is reduced to an algebraic eigenvalue problem.Consequently, the overall reflection coefficients can be calculated by solving the eigenvalue problem.According to the reflection coefficients, the 2 × 2 Jones matrix J associated with the zeroth order reflected light of the sample, which connects the incoming Jones vector with the reflected Jones vector, can be formulated by where E s,p refers to the electric field component perpendicular and parallel to the plane of incidence, respectively.If the sample is non-depolarizing, the 4 × 4 Mueller matrix M can be calculated from the Jones matrix J by [23] ( where the symbol denotes the Kronecker product, J* is the complex conjugate of J, and the matrix A is given by (3) In practice, the Mueller matrix M is usually normalized to the (1, 1)th element M 11 , with the normalized Mueller matrix elements .As depicted in Fig. 1, when the azimuthal angle φ = 0, this configuration is called planar diffraction.In this configuration, any one-dimensional (1D) structure exhibits mirror symmetry with respect to the plane of incidence, and thus its Jones matrix is diagonal, and the upper right and bottom left 2 × 2 submatrices of its Mueller matrix vanish [see Eq. ( 2)].Moreover, all nonzero elements of both matrices can be expressed in terms of the classical ellipsometric angles Ψ and Δ, which are commonly used to characterize isotropic samples.In contrast, both Jones and Mueller matrices are generally fully dense in conical diffraction configurations (φ ≠ 0).In this case, new anisotropic information becomes available.Previous studies have revealed that the newly added information may help decorrelate fitting parameters and make the solution of the inverse diffraction problem more robust [16,17].In our previous work, a measurement configuration optimization method was proposed to find an optimal combination of incidence and azimuthal angles, with which more accurate measurement can be achieved.According to the proposed method, the optimal measurement configuration can be achieved by [14] (4) where ( )  is the Moore-Penrose pseudo-inverse of a weighted Jacobian matrix  p J whose elements are proportional to partial derivatives of the diffraction signature with respect to elements in the structural parameter vector p.  a J is also a weighted Jacobian matrix but with elements proportional to partial derivatives of the diffraction signature with respect to elements in the configuration vector a, which consists of the incidence and azimuthal angles, i.e., a = [θ, φ] T .The matrix +   p a J J is referred to as the configuration error propagating matrix, and +   p a J J describes the maximum gain factor in the propagation of the configuration error Δa.In the optimization process, the values of +   p a J J are firstly scanned in the given parameter domain Ω for the maximum.Then all of the maxima of +   p a J J are scanned in the ranges of incidence and azimuthal angles (Θ and Φ) for the minimum.The combination of incidence and azimuthal angles corresponding to this minimum is the final optimal measurement configuration.
On the other hand, when the sample has a depolarization effect, totally polarized light used as a probe in ellipsometry is transformed into partially polarized light, and the Jones matrix formalism cannot be used to describe the optical response of this sample any more, while the Mueller matrix formalism is still applicable.In this case, the associated Mueller matrix is a depolarizing one.The depolarizing effect of a depolarizing Mueller matrix can be characterized by the depolarization index DI that is defined by [24 where M T is the transpose of M, and Tr( ) ⋅ represents the matrix trace.DI = 0 and DI = 1 correspond to the totally depolarizing and non-depolarizing Mueller matrices, respectively.Many factors may induced depolarization effects [19], such as finite spectral bandwidth of the monochromator, finite numerical aperture (NA) of the focusing lens in the measurement system, thickness nonuniformity in a thin film formed on a substrate, and large surface or edge roughness of a sample.The depolarization phenomena caused by the above factors essentially occur by the generation of different polarizations upon light reflection and are generally referred to as quasi-depolarization.In the analysis of a depolarizing sample, we need to use an optical model that incorporates depolarization effects.The optical modeling principle is based on the optical equivalence of the polarization states [25], which states that a depolarizing system is optically equivalent to a system composed of a parallel combination of several non-depolarizing systems.It can be further deduced that a depolarizing Mueller matrix can be written as the sum of various non-depolarizing Mueller matrices, i.e., where D M and ND M represent the depolarizing and non-depolarizing Mueller matrices, respectively.The variable x denotes the factors that induce depolarization.( ) x ρ is a weighting function, which can be specifically the spectral bandwidth function, numerical aperture, or the thickness distribution function etc.The calculation of Eq. ( 6) requires discretization of the range of x and calculation of the non-depolarizing Mueller matrices at all specified discrete points followed by weighted averaging [19,26].
A weighted least-squares regression analysis (Levenberg-Marquardt algorithm) is performed, during which the model parameters are varied until calculated and experimental data match as much close as possible [27].This is done by minimizing a weighted mean square error function where

Experimental setup
A dual rotating-compensator configuration is adopted to measure the sample Mueller matrices.As schematically shown in Fig. 2, the system layout of the dual rotating compensator Mueller matrix ellipsometer in order of light propagation is PC r1 (ω 1 )SC r2 (ω 2 )A, where P and A stand for the polarizer and analyzer, C r1 and C r2 refer to the 1st and 2nd rotating compensators, and S stands for the sample.The 1st and 2nd compensators rotate synchronously at ω 1 = 5ω and ω 2 = 3ω, where ω is the fundamental mechanical frequency.The emerging Stokes vector S out of the light beam can be expressed as the following Mueller matrix product [28,29]  (8)   where M i (i = P, A, C1, C2, S) is the Mueller matrix associated with each optical element.R(α) is the Mueller rotation transformation matrix for rotation by the angle α (α = P, A, C 1 , C 2 ) that describes the corresponding orientation angle of each optical element.δ 1 and δ 2 are the phase retardances of the 1st and 2nd rotating compensators.By performing Hadamand analysis [27], the sample Mueller matrix elements can be extracted from harmonic coefficients of the irradiance at the detector, which is proportional to the first element of the emerging Stokes vector S out .Based on the above measurement principle, we developed a Mueller matrix ellipsometer prototype, as depicted in Fig. 2. The spectral range is from 200 to 1000 nm.The beam diameter can be changed from the nominal values of ~3 mm to a value less than 200 μm with the focusing lens.The two arms of the ellipsometer and the sample stage can be rotated to change the incidence and azimuthal angles in experiments.

Results and discussion
As illustrated in Fig. 3, the imprinted grating line profile is characterized totally by five structural parameters p 1 ~p5 .In the experiments, we mainly care about the first four structural parameters p 1 ~p4 , while the fifth structural parameter p 5 is only used to adjust the final fitting performance.We attempt to find an optimal measurement configuration for MME, with which more accurate measurement of p 1 ~p4 can be achieved.To do this, when calculating the norm of the configuration error propagating matrix +   p a J J , the matrix  p J is only comprised of partial derivatives of the Mueller matrix elements weighted to their estimated standard deviations with respect to the first four structural parameters p 1 ~p4 .In the optimization, the 2  norm of the configuration error propagating matrix +   p a J J was in a parameter domain with p 1 varied from 340 to 360 nm, p 2 from 460 to 480 nm, p 3 from 86 to 88°, p 4 from 35 to 55 nm, and p 5 from 50 to 80 nm.In the calculations, p 1 , p 2 , p 4 and p 5 were in units of micrometer, while p 3 , θ and φ were in units of radian.Since there is not a uniform unit for the vector p, we denote its unit as PU in the rest of this paper for convenience.Since the procedure of measurement configuration optimization described in Eq. ( 4) is very time-consuming, the spectral range was varied from 200 to 800 nm with increments of 5 nm and we fixed the incidence angle at 65° and only varied the azimuthal angle in a range from 0 to 90° with increments of 5° to reduce the calculation time.Figure 4 presents the maximal values of +   p a J J calculated in different measurement configurations.According to Eq. ( 4), we know that the minimum of all the maximal values of +   p a J J corresponds to the final optimal measurement configuration.As observed from Fig. 4, the value of +   p a J J calculated at φ = 15° is smaller than those calculated in other azimuthal angles.It is therefore expected that the structural parameters p 1 ~p4 of the imprinted gratings extracted at φ = 15° will be much more accurate than other azimuthal angles.In order to examine the above prediction of the optimal measurement configuration, three nanoimprinted grating samples with imprint pressures of 10, 15 and 20 bar, respectively, were measured by the Mueller matrix ellipsometer at the incidence angle of 65° and azimuthal angles varied from 0 to 90° with increments of 5°.The systematic errors in the structural parameters are defined as the differences between the structural parameters of the imprinted grating samples extracted by the Levenberg-Marquardt algorithm and the results measured by XSEM.The 2  norm of the systematic errors in the extracted parameters p 1 ~p4 (apart from p 5 ) was then calculated for each measurement configuration and shown in Fig. 5.An examination of Fig. 5 shows that the variation of the norms of the systematic errors in the extracted parameters is not in rigorous agreement with that of the norms of the configuration error propagating matrix +   p a J J as illustrated in Fig. 4, which was also observed in our previous work [14].However, we do have a qualitative agreement.For example, the greater the azimuthal angles are, the greater the values of +   p a J J are, and then the greater the norms of the systematic errors in the extracted are.Importantly, as shown in Fig. 5, the optimal measurement configurations for the three imprinted grating samples, where the norms of the systematic errors in the extracted parameters achieve minimal values, are in accordance with the theoretical prediction given in Fig. 4. It therefore suggests that more accurate measurement of nanoimprinted grating samples can indeed be achieved by adopting +   p a J J as an objective function to optimize the measurement configuration for MME.Table 1 presents the comparison of fitting parameters obtained from MME measurements in the optimal configuration and XSEM measurements.The uncertainties appended to the MME-extracted parameter values all had a 95% confidence level and the uncertainty in the ith parameter was estimated by 1.96 , where C ii is the ith diagonal element of the fitting parameter covariance matrix [32].About 5 XSEM images were taken around the illumination spot where MME measurement were previously performed, and each XSEM image provided a set of XSEM values for parameters p 1 ~p4 .As shown in Table 1, the averaged XSEM values over the 5 XSEM images are used as a reference for accuracy evaluation.The fit1 in Table 1 illustrates the MME-extracted parameters without considering any depolarization effect, which exhibits reasonable agreement with the results measured by XSEM.The corresponding fitting errors 2 r χ of fit1 between the measured and calculated best- fit Mueller matrix spectra are presented in Table 2. Clearly, the fit1 for the three imprinted grating samples does not have 2 r χ close to 1, which is possibly attributed to the following reasons.First, the estimated standard deviations of the experimental Mueller matrix elements may not reflect all of the errors in the measurement, since it was found in our experiments that the estimates of errors in the fitting parameters underestimated the true errors.Second, there are depolarization effects in the measurement, which are not taken into account in the interpretation of the measured data.As an example, Fig. 6 depicts the depolarization index spectrum DI that corresponds to the measured Mueller matrix spectrum of the imprinted grating sample with an imprint pressure of 20 bar.As can be observed from Fig. 6, the depolarization indices are close to 1 over most of the spectrum except for the range from ~300 to 460 nm and show significant dips to ~0.87 near 310 nm and to ~0.85 near 350 nm.It is also noted that the depolarization indices slightly exceed 1 in the spectral range from ~220 to 290 nm, which may be attributed to the experimental errors.Obviously, the imprinted grating sample exhibits noticeable depolarization effects that should be included in the interpretation of the measured data.Considering that the ellipsometer-measured data for each recorded wavelength always contains contributions from a span of wavelengths and incident directions, we first investigate the depolarization effects induced by the finite spectral bandwidth and NA of the Mueller matrix ellipsometer.In the experiments, the spectral bandwidth and NA of the instrument were pre-determined through a measurement on a nominally 1000 nm SiO 2 thick thermal film on the Si substrate to isolate the effects of finite bandwidth and NA from sample-specific artifacts.The measurement yielded the spectral bandwidth σ λ = 1.0 nm and NA = 0.065.A rectangular bandwidth function was chosen in the measurement of bandwidth.During the grating reconstruction process, we fixed the bandwidth and NA and let just the structural parameters p 1 ~p5 (grating pitch was fixed at the nominal value of 800 nm) vary.The depolarization index spectrum DI 1 shown in Fig. 6 corresponds to the calculated best-fit Mueller matrix spectrum when only considering the depolarization effects induced by finite spectral bandwidth and NA.According to Fig. 6, we observe that the measured and calculated depolarization indices show good agreement within the spectral range from ~460 to 800 nm, but exhibit poor performance in the spectral range from ~300 to 460 nm.We then further took the depolarization effect induced by the residual layer thickness nonuniformity over the area of the probe light into account.According to the incidence angle of 65° and the beam diameter of about 200 μm with the focusing lens, we know that the area of the probe light is an ellipse with a major diameter of about 473 μm and a minor diameter of about 200 μm.During the regression analysis, we still fixed the spectral bandwidth and NA but let the structural parameters p 1 ~p5 as well as the standard deviation σ t of the residual layer thickness vary.A Gaussian thickness distribution function was chosen in our calculation.The depolarization index spectrum DI 2 shown in Fig. 6 corresponds to the calculated best-fit Mueller matrix spectrum when further considering the depolarization effect induced by the residual layer thickness nonuniformity.According to Fig. 6, we observe that the match between the measured and calculated depolarization index spectra is significantly improved, especially in the spectral range from ~300 to 460 nm.It thus suggests that the residual layer thickness nonuniformity is an important factor that induces depolarization effects in characterization of nanoimprinted gratings.Fig. 7. Fitting result of the measured and calculated best-fit Mueller matrix spectra of the nanoimprinted grating sample with an imprint pressure of 20 bar.The fit1 and fit2 corresponds to their counterparts in Tables 1 and 2, which are the calculated best-fit Mueller matrix spectra achieved when ignoring depolarization effects and when considering the depolarization effects induced by finite spectral bandwidth, NA, and residual layer thickness nonuniformity, respectively.Data analysis was also performed in the optimal measurement configuration.
The MME-extracted parameters for the three imprinted grating samples when considering the depolarization effects induced by finite spectral bandwidth, NA and residual layer thickness nonuniformity are illustrated by fit2 in Table 1.As observed from Table 1, the MME-extracted parameters associated with fit2 are much closer to the results measured by XSEM than fit1.It is also noted that the extracted structural parameters of the imprinted gratings are in good agreement with the Si imprinting mold, which has a top line width of 350 nm, a line height of 472 nm, and a sidewall angle of 88° [14].The corresponding fitting errors 2 r χ of fit2 are given in Table 2.As shown in Table 2, although the fit2 for three nanoimprinted grating samples still does not have 2 r χ close to 1, it is significantly improved in comparison with fit1, where the depolarization effects are not included in the interpretation of the measured data.As an example, Fig. 7 shows the fitting result of the measured and calculated best-fit Mueller matrix spectra of the imprinted grating sample with an imprint pressure of 20 bar.The fit1 and fit2 as illustrated in Fig. 7 corresponds to the best-fit Mueller matrix spectra achieved when ignoring depolarization effects and when considering the depolarization effects induced by finite spectral bandwidth, NA, and residual layer thickness nonuniformity, respectively.Compared with fit1, the improvement in fit2 is not so distinct, but can still be easily observed from Fig. 7.According to the fitting results presented in Tables 1 and 2, we can finally conclude that not only further improvement in the measurement accuracy and fitting performance can be achieved, but also the residual layer thickness variation over the illumination spot can be directly determined by incorporating depolarization effects into the interpretation of the measured data collected in the optimal configuration.

Conclusions
In this work, nanoimprinted resist patterns were characterized using an in-house developed Mueller matrix ellipsometer.Prior to the experiments, the optimal measurement configuration for the investigated nanoimprinted resist patterns was firstly achieved by minimizing the norm of a configuration error propagating matrix.The optimal configuration was found to be with the azimuthal angle φ = 15° (the incidence angle was fixed at θ = 65°), which was not the planar diffraction configuration.More accurate characterization of line width, line height, sidewall angle, and residual layer thickness of the nanoimprinted resist patterns has been achieved by performing MME measurements in the optimal configuration.In addition, noticeable depolarization effects were observed from the measured data in the optimal configuration, which were found to be induced by the finite spectral bandwidth and numerical aperture of the instrument, as well as the residual layer thickness variation of the nanoimprinted resist patterns.It was demonstrated that not only further improvement in the measurement accuracy and fitting performance can be achieved, but also the residual layer thickness variation over the illumination spot can be directly determined by incorporating depolarization effects into the interpretation of measured data.The comparison of MMEextracted imprinted resist profiles has also indicated an excellent fidelity of the nanoimprint pattern transfer process.

Fig. 1 .
Fig. 1.Representation of polarized light incidence for a one-dimensional grating structure.
k indicates the k-th spectral point from the total number N, and indices i and j show all the Mueller matrix elements except m 11 .meas , ij k m denotes the measured Mueller matrix elements, and calc , ( , ) ij km p a denotes the calculated Mueller matrix elements associated with the structural parameter vector p and the configuration vector a. standard deviation associated with m ij,k , and K is the dimension of the vector p, which also indicates the total number of fitting parameters.

Fig. 3 .
Fig. 3. XSEM micrograph and geometric model of the nanoimprinted grating sample with an imprint pressure of 20 bar.A set of STU220 resist films with thickness of ~240 nm (developed by Obducat AB Co.) were spin-coated onto (100)-orientation single crystal Si wafers.Soft UV nanoimprint lithography was then performed on an Eitre3 Nano Imprinter using a Si mold.The nanoimprint process was operated at a temperature of 70 °C with 1 min UV exposure time and 20 min imprint time.The imprint pressure was varied from 10 bar (1 bar = 10 5 Pa) to 20 bar to make imprinted resist patterns have different residual layer thicknesses.The Si mold fabricated by e-beam lithography followed by dry etching has gratings with a pitch of 800 nm, a top line width of 350 nm, a line height of 472 nm, and a sidewall angle of 88°[14].The crosssectional SEM (XSEM) (Nova NanoSEM450, FEI Co.) image of one of the nanoimprinted gratings with an imprint pressure of 20 bar is shown in Fig.3.During the regression analysis, optical constants of the Si substrate are fixed at values taken from[30].Optical properties of the STU220 resist are modeled using a two-term Forouhi-Bloomer model[31], whose parameters were determined from a STU220 resist film deposited on the Si substrate and taken as A 1 = 0.004447, A 2 = 0.03051, B 1 = 8.8611 eV, B 2 = 12.0043 eV, C 1 = 19.6703eV 2 , C 2 = 36.3258eV2 , n(∞) = 1.4842, and E g = 3.3724 eV in our calculations.An accurate geometric model for the grating line profile is essential to achieve good agreement between the experimentally measured and theoretically calculated Mueller matrix spectra.As shown in Fig.3, we apply a single-layer symmetrical trapezoidal model with a total of five structural parameters p 1 ~p5 to characterize the imprinted grating line profile, where p 1 , p 2 , p 3 , p 4 , and p 5 represent top line width, line height, sidewall angle, residual layer thickness, and radius of the bottom round corner, respectively.For RCWA calculations, the number of retained orders in the truncated Fourier series is 12, and the modulated portion of the grating is divided into 25 slices.

Fig. 4 .
Fig. 4. The maximal norms of the configuration error propagating matrix p a J J +   calculated in different measurement configurations.The incidence angle was fixed at 65°.

Fig. 5 .
Fig. 5.The norms of systematic errors in the structural parameters of different nanoimprinted grating samples extracted in different measurement configurations without considering any depolarization effect.The corresponding imprint pressures are 10, 15 and 20 bar, respectively.The incidence angle was fixed at 65°.

Fig. 6 .
Fig.6.Depolarization index spectra of the measured and calculated best-fit Mueller matrix spectra of the nanoimprinted grating sample with an imprint pressure of 20 bar.The depolarization index spectrum DI corresponds to the measured Mueller matrix spectrum.The depolarization index spectrum DI1 corresponds to the calculated best-fit Mueller matrix spectrum when only considering the depolarization effects induced by finite spectral bandwidth and NA.The depolarization index spectrum DI2 corresponds to the calculated best-fit Mueller matrix spectrum when considering the depolarization effects induced by finite spectral bandwidth, NA, and residual layer thickness nonuniformity.Data analysis was performed in the optimal measurement configuration.