Open Access
Add to BibSonomyAbstract
The feasibility of metrological characterization of the one-dimensional (1D) holographic gratings, used in the nanoimprint molding tool fabrication step, by spectroscopic Mueller polarimetry in conical diffraction is investigated. The studied samples correspond to two different steps of the replicated diffraction grating fabrication process. We characterized master gratings that consist of patterned resist layer on chromium-covered glass substrate and complementary (replica) gratings made of nickel. The profiles of the gratings obtained by fitting the experimental spectra of Mueller matrix coefficients taken at different azimuthal angles were confirmed by atomic force microscopy (AFM) measurements. The calculated profiles of corresponding master and replica gratings are found to be complementary. We conclude that the Mueller polarimetry, as a fast and non-contact optical characterization technique, can provide the basis for the metrology of the molding tool fabrication step in the nanoimprint technique.
©2007 Optical Society of America
1. Introduction
The micro- and nanostructures replicated from the mold into a plastic-covered metallic foil are widely used in the fabrication of optical security devices, such as hologram labels and foils for the identity documents, brand protection, etc. The hologram labels produce a unique visual effect, which cannot be reproduced by means of traditional printing techniques. Additionally, hidden images can be built into the holograms, visible only during illumination by specially formed laser beam. The most common technique to produce such structures is hot embossing. In this replication technology a nickel shim is typically used to mold the sheets of formable material in order to obtain diffractive optical structures.
The replication tool itself is a result of nickel electroplating onto the original surface of master diffraction gratings defined by the chosen optical design. These original gratings can be made in different materials, e.g. photoresist or silicon, and by different techniques, e.g. electron beam lithography and etching or diamond turning [1]. The surface relief of the original structure is quite complicated and the fabrication of master gratings can be very expensive. Thus, the metrology of this step of replication process is a key factor to achieve low cost per replica. Optical techniques have already proved to be a good choice for the control of ever-decreasing critical dimensions (CDs) in microelectronics, as they are fast and non-destructive compared with direct scanning electron microscopy (SEM) and AFM measurements. The characterization of the low-cost replicated diffraction gratings by near-field techniques is somewhat laborious, as the line roughness of such samples, unlike that of microelectronics structures, could vary significantly over the area (see Fig. 1).
It is worth to mention that only the average values of dimensional parameters yielded by optical techniques are relevant, as they actually determine diffraction efficiencies. We used a novel spectroscopic Mueller polarimeter [2] in conical configuration for the characterization of 1D diffraction gratings used in the fabrication of molding tool for the nanoimprint process, and showed that proposed optical technique is adequate even with such CDs fluctuations.
We studied two sets of the samples: the original master gratings and their nickel replicas. The former ones were made with electron beam lithography and wet etching in a polymethyl methacrylate (PMMA) layer, spin-coated on the chromium-covered glass substrate. Spectroscopic Mueller matrix coefficients of the special metrological test patterns, representing 1D sub-micron diffraction gratings, were acquired at fixed angle of incidence of 55° and at different azimuthal angles. Those spectra were fitted with a code based on rigorous coupled wave analysis (RCWA) algorithm with trapezoidal models of the grating profile.
The principle of the technique and experiment are described in the second part of the paper. In the third part we present the results of the fits of the measured data with trapezoidal model of grating profile for both master and replica gratings and discuss them. We analyze the stability and consistency of the model parameter values obtained by fitting polarimetric data in conical diffraction geometry. The results of simulations were compared with the results of AFM measurements of the samples. We conclude that this new optical technique can be a good choice for the metrology of nanoimprint process.
Fig. 1 SEM image of 1D diffraction grating test-structure (resist on Cr substrate) with the nominal period of 800 nm and resist linewidth of 400 nm. Dark lines correspond to the resist grating ridges; bright regions show the exposed chromium in the bottom of the groove. Significant variation of the resist linewidth (297 nm – 396 nm) is observed over the area. The dot line at the bottom shows 5 μm.
2. Principle of the technique and experiment.
2.1 Mueller polarimetry in conical diffraction.
Well-established optical techniques, such as spectroscopic ellipsometry, normal incidence reflectometry, and reflectance-difference spectroscopy [3–5] are already used for metrological characterization of patterned surfaces for many industrial applications. It was shown by us previously [6, 7] that Mueller polarimetry in conical diffraction could be particularly interesting for this purpose, as it can significantly improve the accuracy and robustness of the reconstruction of the grating shape.
If the plane of light incidence is not perpendicular to the direction of the grooves of 1D diffraction grating, the diffracted light orders are not within the same plane anymore. This general configuration is usually called conical mounting. For both incident and reflected light beams we note the components of the electric field, which are perpendicular and parallel to the incidence plane Es and Ep respectively. Polarimetric properties of non-depolarizing samples can be described by complex Jones matrix J:
On the other hand, the polarimetric properties of any sample, even a depolarizing one, are given by a real 4 × 4 Mueller matrix M, which relates the Stokes vectors defining the polarization states of the incident and reflected light beams.
If the sample is non-depolarizing, the Mueller and Jones matrices are related by:
While azimuthal angle of light incidence plane varies between 0° and 90° the Mueller matrix of the patterned sample is fully dense. We show below that all elements of Mueller matrix provide non-redundant information, which is an essential advantage for metrological applications. As mentioned above optical characterization techniques are non-direct and model-dependent, the problem of the model parameter variances and decorrelations is of key importance. We showed previously [7, 8], that with complete spectral Mueller matrix of the sample measured at optimal conical configuration it is possible to decouple some of model parameters and decrease the parameter variance.
2.2 Experimental set-up, samples and simulation procedure.
Throughout this investigation we used a commercially available spectroscopic Mueller polarimeter (MM16 produced by Jobin-Yvon/Horiba). This instrument features identical polarizer and analyzer heads based on liquid crystal devices. With a minimal beam spot diameter of about 200 μm, spectral range of 420–850 nm and 2048-pixel CCD it performs the measurement of full spectrum Mueller matrix coefficients in about 2 seconds. Polarimeter calibration is performed with the robust eigenvalue method without the need for modeling of the optical elements [9]. Measurements of Mueller matrix spectra allow accurate, simple and easy characterization of anisotropic and depolarizing materials, quantification of important parameters such as retardance, polarization dependence loss, depolarization degree necessary to accurately characterize substrates, coatings and diffraction gratings. The polarimeter is also equipped with azimuthal stage for the measurements of gratings in conical mounting.
As it was already mentioned, two types of the samples were studied: 1) 1D master diffraction gratings etched in a polymer (PMMA) resist on the Cr-coated glass substrate and 2) their nickel replicas, both of them fabricated by CompOptics [10].
The whole original structure of security holograms consists of many 1D gratings with different periods, filling ratios and orientations etched in a PMMA layer after patterning by e-beam lithography according to the chosen design [11]. One-dimensional 2 × 2 mm test gratings are formed on the sides of the main hologram. Two such test-structures, with nominal periods and linewidths of 1200 nm and 700 nm (S1200/700) and of 1000 nm and 400 nm (S1000/400) respectively were chosen for the analysis. We also studied the corresponding nickel replicas S1200/500 and S1000/600 with the same periods, but with the nominal values of the linewidth and spacing reversed with respect to the original master gratings.
The spectroscopic polarimetric measurements were acquired in the wavelength range from 450 nm to 850 nm at the fixed angle of incidence of 55° and at different azimuthal angles between 75° and 105°. The PMMA resist dispersion law and the thickness of the resist layer were obtained from the fit of ellipsometric spectrum measured on the homogeneous resist layer left on Cr outside of the patterned areas. This value of the resist layer thickness was taken as a nominal value of the gratings ridge height. For chromium we directly took the values of the metal dielectric function measured after removal of the resist from the chromium-covered substrate by a microwave oxygen plasma, as roughness of the film was found to be negligible. Those measurements on non-patterned structures were performed with spectroscopic phase-modulated ellipsometer UVISEL (also by Jobin-Yvon/Horiba).
Experimental spectral Mueller matrix coefficients for all four samples were fitted by Levenberg-Marquardt algorithm with symmetric trapezoidal model (Fig. 2).
The values of the height H, top CD L and slope projection A were adjusted during the fit procedure in order to minimize the mean square deviation of the measured and calculated Mueller matrix spectral coefficients of the samples. Direct calculations of the spectral Mueller matrix coefficients of the periodic structures were performed with rigorous coupled-wave analysis (RCWA) [12] combined with S-matrix propagation algorithm [13].
Fig. 2. Symmetric trapezoidal model of the grating ridge. H is the height of the ridge, L is the top width and A is the slope projection.
The modeling of the light diffraction on the metallic gratings in TM polarization with RCWA algorithm demands the use of the proper rule for Fourier factorization of a product of two discontinuous periodic functions [14, 15]. In this case the number of retained orders N in the truncated Fourier series is usually higher compared to the case of TE polarization and dielectric gratings in order to achieve the convergence to the real solution. In conical configuration both polarizations of the light are present and the value of parameter N could be crucial for the precision of calculations. On the other hand, the increase of the number of retained Fourier orders leads to significant increase of the calculation time. We used the value N=15 for master gratings simulations and N=25 for the modeling of the Ni replica gratings. The calculated profiles of the gratings were compared with the results of AFM measurements.
3. Results and discussion
3.1 Master gratings: simulations and measurements.
The accuracy of grating period definition was verified with the diffraction of laser beam (λ = 632.8 nm) in Littrow mounting. The formula sinθk = k(λ/2d) provides the relation between wavelength λ, grating period d and the angle of reflection of the k-th diffracted order θk in this configuration. From this experiment we got the value of d = 1200.6 nm for the sample S1200/700 and d = 1000.3 nm for S1000/400. As expected from electron beam lithography the accuracy of grating period definition is very good and we kept the gratings period constant during the fitting procedure.
The measured and calculated spectra of Mueller matrix coefficients for the master grating S1200/700 are shown in Fig. 3(a) for different azimuthal angles. The coefficients were normalized by M11, which represents total reflectivity of the sample. The pronounced variation of all spectral Mueller matrix coefficients with azimuthal angle φ shows clearly that we obtain very different sets of experimental data while doing measurements in conical diffraction.
The calculated optimal values of parameters H, L and A are almost constant versus azimuthal angle for the sample S1200/700 (see Fig. 4 (a)). The calculated values of the height H and bottom critical dimension L+2A do not vary significantly with azimuthal angle for the sample S1000/400, while there is local drop of the optimal top width L and local increase of the slope projection A in the vicinity of azimuthal angle φ = 90° (Fig. 4(b)). It was shown previously [7] that in classical planar diffraction (φ = 0°) there is a strong correlation between these two parameters of the trapezoidal model. Like at φ = 0°, the spectral Mueller matrix of the symmetric 1D diffraction grating at φ = 90° also contains two zero submatrices. Mainly because of that reduced set of the experimental data the linear combination of the parameters L and A only can be determined from the polarimetric measurements of 1D gratings in this configuration.
The values of parameters obtained from AFM scans are also presented on the same graphs. For both samples the results of the fit are in a good agreement with the data obtained from the AFM analysis. The fitted values of the height H are equal to the measured height of the grating ridge for the sample S1200/700 as it is shown in Fig. 4(a).
Fig. 3. Measured (solid lines) and calculated (open circles) spectra of normalized Mueller matrix coefficients for the master sample S1200/700 at different azimuthal angles φ.
The mean calculated value of the grating height H for the sample S1000/400 is equal to the low limit of the AFM measured values (Fig. 4(b)). The optimal fitted values of the bottom width of the ridge for the sample S1200/700 are within the range of AFM measured values.
Fig. 4. Results of the simulations (solid lines) and AFM measurements (dash-dotted lines) for the samples S1200/700 (a) and S1000/400 (b). Dashed lines correspond to the nominal values of the grating parameters.
Calculated values of the bottom CD for the sample S1000/400 are about 10 percent less than the low limit of AFM measured values. The convolution between AFM tip and grating ridge can affect the accuracy of lateral measurements while moving from the bottom of the groove to the ridge top. As the slope is steeper in latter case, the AFM measurements overestimate the CDs of the sample S1000/400. The AFM images of both master samples are shown in Fig. 5 (a, b).
The mean optimal value of the height H is only 2 percent less than nominal value of 200 nm for the sample S1200/700, while this value is about 40 percent less than nominal one for the sample S1000/400. From both fit and AFM measurements the bottom width of the ridge is larger and the top width is smaller than corresponding nominal values for both samples. It means that the area of chromium-covered substrate in the grating grooves where resist was removed is less then one can expect from nominal values. As a consequence, the shape of the ridge is not rectangular, but trapezoidal. This is quite normal for electron beam lithography exposure of positive resist, like PMMA. The large area of security holograms (several cm2) requires exposure time to be minimized, unless the cost becomes prohibitive, so the dose is minimized too. Step of nickel electroplating necessitates full removal of the exposed resist from the groove and thus considerable development time. Another factor with the same consequences is a Gaussian shape of the electron density distribution inside the electron beam. It affects stronger the features with smaller CDs, exactly what we see in our case. The top of the grating ridge is rounded according to AFM measurements (Fig. 5(a, b)); but even with simple symmetric trapezoidal model for the fit of experimental data we still can obtain average CD values, which are constant versus azimuthal angle and provide good agreement between experimental and simulated spectra.
Three sets of Mueller matrix spectral coefficients of the sample S1200/700 at the azimuthal angle φ = 75° are plotted in Fig. 6(a). The same data for the sample S1000/400 at the azimuthal angle φ = 105° are shown in Fig. 6(b). Experimental spectra of Mueller matrix coefficients are shown by solid curves. The calculated Mueller matrix coefficients of the nominal rectangular and optimal trapezoidal profiles are plotted with solid and open circles, respectively. These superimposed profiles are depicted at the bottom of Fig. 6(a) and 6(b). The symmetry (anti-symmetry) of the corresponding transposed Mueller matrix coefficients supports the hypothesis of the symmetry of the grating profile [6,16].
All spectra of nominal rectangular profile look just shifted in the wavelength space with respect to the measured ones for the sample S1200/700. In contrary, the spectra of the nominal rectangular profile are very different from the measured data for the sample S1000/400. For the former sample the optimal values of L and A are different from nominal values, but optimal and nominal values of the height H are very close. For the latter sample all three optimal values of H, L and A are quite far from the nominal ones. It means that proposed optical technique of 1D diffraction gratings characterization can directly provide a qualitative estimation of the deviation of the master grating profile from the nominal one.
3.2 Replica gratings: simulations and measurements.
The replica gratings are the result of electroplating of master gratings with nickel. Once the master grating fabrication step is completed, a thin layer of silver (about 10 nm) is deposited onto its surface by thermal evaporation, in order to provide a conductive film on the resist ridges and facilitate electroplating. After depositing several hundred micron-thick layer of Ni, the master and replica are separated mechanically and the silver layer is etched away chemically. As nickel fills completely all grooves, the replica is nearly complementary structure to the original master structure.
Fig. 6. Normalized spectral Mueller matrix coefficients of the master grating samples S1200/700 at azimuthal angle φ = 75° (a) and S1000/400 at azimuthal angle φ = 105° (b). Experimental data are shown by solid line, the spectra of optimal trapezoidal profile are plotted with open circles. The curves with solid circles represent the spectra of nominal rectangular profile. Both profiles are depicted in the bottom of the figure.
Two replica samples S1200/500 and S1000/600 (corresponding to master samples S1200/700 and S1000/400) were measured with the spectroscopic polarimeter at fixed angle of incidence 55° and at different azimuthal angles varying between 75° and 105°. Experimental data were fitted with the same symmetric trapezoidal model as for master gratings. Calculated profiles of the replica gratings were compared with the results of its AFM measurements and with profiles, calculated from the spectra of corresponding master gratings.
The measured and calculated spectra of Mueller matrix coefficients for nickel replica S1000/600 are plotted in Fig. 7. Some off-diagonal spectral Mueller matrix coefficients are much less sensitive to the variation of azimuthal angle φ compared to the spectra for the master grating S1200/700 (see Fig 3). Nevertheless we will show that Mueller matrix spectral coefficients measured in conical configuration contain non-redundant information, which can be used for the metrological characterization of replica gratings.
Fig. 7. Measured (solid lines) and calculated (open circles) spectra of normalized Mueller matrix coefficients for nickel replica S1000/600 at different azimuthal angles φ.
For replica samples S1200/500 and S1000/600 the results of the fit are presented in Fig.8(a, b). The optimal values of the height H are very stable with respect to azimuthal angle and consistent with the values obtained from AFM measurements for both samples. The calculated bottom CD values are somewhat smaller than those extracted from the AFM measurements for both samples. The optimal value of bottom CD is almost constant versus azimuthal angle for replica grating S1200/500, while the optimal value of this parameter varies noticeably with azimuthal angle for the sample S1000/600.
Fig. 8. Results of the simulations (solid lines) and AFM measurements (dash-dotted lines) for the nickel gratings S1200/500 (a) and S1000/600 (b).
The calculated values of the top width L are within the interval of the values extracted from AFM measurements for the sample S1000/600, but the calculated values of this parameter are underestimated compared to AFM measurements for the sample S1200/500 at all azimuthal angles. As follows from AFM measurements of both replica samples the shape of the grating ridge is well described by trapezoidal model (Fig. 9(a, b)).
Nevertheless, for both nickel replica samples the variation of optimal parameters L and A with azimuthal angle is more pronounced compared to the master gratings (see Fig 8. (a, b)).
Mueller matrix spectral coefficients of the sample S1200/500 at azimuthal angle φ = 85° are plotted in Fig. 10(a), the same data for the sample S1000/600 but at azimuthal angle φ = 95° are shown in Fig. 10(b). Experimental data are depicted by solid curves, the spectra, corresponding to nominal rectangular and to optimal trapezoidal profiles are plotted with solid and open circles, respectively. Superimposed profiles are presented at the bottom of Fig. 10(a) and 10(b).
Like for the master gratings the symmetry (anti-symmetry) property of the measured Mueller matrices of both replica samples proves the symmetry of the corresponding grating profiles. Measured and fitted values of the elements of upper right and bottom left 2 × 2 submatrices of Mueller matrix of both replica gratings are closer to zero compared to the corresponding values of Mueller matrix elements of the master gratings at all azimuthal angles.
Fig. 10. Normalized spectral Mueller matrix coefficients of the nickel replica gratings S1200/500 at azimuthal angle φ = 85° (a) and S1000/600 at azimuthal angle φ = 95° (b). Experimental data are shown by solid line, the spectra of optimal trapezoidal profile are plotted with open circles. The curves with solid circles represent the spectra of nominal rectangular profile. Both profiles are depicted in the bottom of the figure.
The SEM images of replica gratings S1200/500 and S1000/600 are presented in Fig. 11. Despite the pronounced roughness of the lines and given the complex shape of the grating ridge profile, the results of the fit of experimental polarimetric data in conical diffraction show that we are able to obtain quite accurately the average values of replica grating CDs.
Fig. 11. SEM images of the nickel replica samples S1200/500 (a) and S1000/600 (b) The white bar at the bottom shows 1 μm.
The profiles of master grating S1200/700 and complementary replica grating S1200/500 extracted from AFM measurements are shown in Fig. 12(a, b). The equivalent profiles for the samples S1000/600 and S1000/400 can be found in Fig. 13(a, b). The mean values of the fitted parameters H, L and A averaged over azimuthal angle φ were calculated for all samples.
Fig. 12. Profiles of the master grating S1200/700 (a) and replica grating S1200/500 (b) from AFM measurements. The calculated optimal profiles of the same gratings obtained from polarimetric measurements, meshed (c).
The meshed profiles of the gratings with corresponding averaged optimal parameters and with period of 1200 nm and 1000 nm are drawn in Fig 12(c) and Fig. 13(c) respectively. The averaged slope of the nickel gratings does not reproduce exactly the averaged slope of master grating; it is steeper for both replica samples. There is a remaining gap of 10 to 20 nm between the meshed profiles of master gratings and replicas. We attribute this fact to the presence of silver layer during electroplating, since these values fall within expected thickness range of evaporated silver film. Silver was removed by chemical etching after mechanically separating one grating from another. Thus the calculated profiles of both master and replica gratings are found to be complementary, taking into account the specifics of Ni electroplating process during the replica grating fabrication step.
Fig. 13. Profiles of the master grating S1000/400 (a) and replica grating S1000/600 (b) from AFM measurements. The calculated optimal profiles of the same gratings obtained from the polarimetric measurements, meshed (c)
One also needs to compare the values of diffraction efficiencies calculated for profiles obtained with the fit against the measured efficiencies. That was done for a He-Ne laser wavelength of 632.8 nm at near-normal incidence for both TE and TM polarizations.
Fig. 14. Measured and calculated diffraction efficiencies for zeroth and first orders in TE and TM polarization at near-normal incidence at 632.8 nm for S1000/400 (a) and S1000/600 (b).
For all samples the absolute efficiency values were normalized to the reflection outside of test boxes, from the resist-covered chrome layer and clean nickel shim surface, respectively.
The results are shown in Fig. 14. Despite some discrepancy the accuracy of prediction is quite remarkable. Many parameters come here to play – inaccuracy of determination of refractive indices of resist and chromium, precision of polarized laser beam orientation, absolute measurements of reflected energy, as well as complexity of the gratings profile. We believe that it is a strong demonstration of the success for our replicated diffraction optics characterization approach.
4. Conclusion
The set of measured data provided by Mueller polarimetry in suitably chosen geometry (typically conical diffraction) is clearly larger than the one or two quantities yielded by the classical optical techniques like reflectometry or spectroscopic ellipsometry. The optimal values of the diffraction gratings parameters calculated from polarimetric measurement data are in a very good agreement with the results of AFM measurements for both master and replica gratings. The calculated profiles of the gratings for both samples are found to be complementary.
While AFM measurements contain information on local topology of the sample surface, which can vary somewhat within the test pattern, the Mueller polarimetry provides the averaged values of the features size, which essentially define the visual quality of the resulting hologram. Thus, Mueller polarimetry, as a fast and non-contact optical characterization technique, can be used for the verification of master gratings before molding tool fabrication step and for the on-line analysis of the final product in the process of fabrication of replicated diffractive optics elements.
References and links
1. M. Gale, “Replicated Diffractive Optics and Micro-Optics,”, Optics and Photonic News, 24–29, August (2003). [CrossRef]
2. A. De Martino, Y.K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals”, Opt. Lett. 28, 616–618 (2003). [CrossRef] [PubMed]
3. B. K. Minhas, S. A. Coulombe, S. S. H. Naqvi, and John R. McNeil, “Ellipsometric scatterometry for the metrology of Sub-0.1-μm-linewidth structures”, Appl. Opt. 37, 5112–5115 (1998). [CrossRef]
4. Y. -S. Ku, S. -C. Wang, D. -M. Shyu, and N. Smith, “Scatterometry-based metrology with feature region signatures matching”, Opt. Express 14, 8482–8491 (2006). [CrossRef] [PubMed]
5. H.-T. Huang, W. Kong, and F. L. Jr Terry, “Normal-incidence spectroscopic ellipsometry for critical dimension monitoring”, Appl. Phys. Lett. 78, 3983 – 3985 (2001). [CrossRef]
6. T. Novikova, A. De Martino, R. Ossikovski, and B. Drévillon, “Metrological applications of Mueller polarimetry in conical diffraction for overlay characterization in microelectronics”, Eur. Phys. J. Appl. Phys. 31, 63–69 (2005). [CrossRef]
7. 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, 3688–3697 (2006) [CrossRef] [PubMed]
8. A. De Martino, T. Novikova, Ch. Arnold, S. BenHatit, and B. Drévillon, “Decorrelation of fitting parameters by Mueller polarimetry in conical diffraction” in Metrology, Inspection, and Process Control for Microlithography XX,Chas N. Archie, ed., Proc. SPIE 6152, 530–541, (2006).
9. A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, “General methods for optimized design and calibration of Mueller polarimeters“, Thin Solid Films , 455–456, 112–119, (2004). [CrossRef]
10. CompOptics Ltd., http://www.compoptics.ru/.
11. S. Yu Serezhnikov. “Preparation, treatment and visualization of data for the fabrication of holograms using electron beam system ZBA-21”, Numerical Methods and Programming , 3, 110–115 (2002) (in Russian).
12. 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, 1068–1076 (1995). [CrossRef]
13. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings”, J. Opt. Soc. Am. A 13, 1024–1035 (1996). [CrossRef]
14. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization” J. Opt. Soc. Am. A 13, 779–784 (1996) [CrossRef]
15. L. Li. Use of Fourier series in the analysis of discontinuous periodic structures. J. Opt. Soc. Am. A13, 1870–1876 (1996). [CrossRef]
16. L. Li. Symmetries of cross-polarization diffraction coefficients of gratings, J. Opt. Soc. Am. A17, 881–887 (2000). [CrossRef]
