In this paper, a model for electromagnetic scattering of line structures is established based on high frequency approximation approach - ray tracing. This electromagnetic ray tracing (ERT) model gives the advantage of identifying each physical field that contributes to the total solution of the scattering phenomenon. Besides the geometrical optics field, different diffracted fields associated with the line structures are also discussed and formulated. A step by step addition of each electromagnetic field is given to elucidate the causes of a disturbance in the amplitude profile. The accuracy of the ERT model is also discussed by comparing with the reference finite difference time domain (FDTD) solution, which shows a promising result for a single polysilicon line structure with width of as narrow as 0.4 wavelength.
© 2008 Optical Society of America
The study of electromagnetic scattering for line structures has great importance in real application. The line structures are usually formed in a periodic manner, which is known as gratings. They can be found in optical elements such as diffraction gratings. In photolithography process for integrated circuits fabrication, the alignment marks and overlay targets that are formed on the wafer are also consists of line structures [1, 2]. From the perspective of electromagnetic modeling for line structures, there are several available rigorous methods. For example, the finite difference time domain (FDTD)  method, which is a popular modeling technique for the study of electromagnetic scattering. The basis in FDTD method is relied on the discrete grid space and time-stepping algorithm to find the electromagnetic fields everywhere in the computational domain. Another method, which is dedicated more for diffraction gratings is rigorous coupled wave analysis (RCWA) [4, 5]. In RCWA, the line structures in the gratings are treated as stratified layers and the solution is obtained by solving a set of simultaneous equations.
A common feature shares by most of the rigorous methods is that there is a lack of understanding on how the total electromagnetic solution is constructed based on the existence physical field. There are only two identified fields, namely the incident and scattered fields. There are no clues about each individual physical mechanism that contribute to the scattered field. The geometrical optics field such as reflected and transmitted fields is indistinguishable from the scattered field. Similarly, one cannot predict the scattered field based on the physical structure of the obstructer since the methods rely heavily on numerical analyses. In view of that, the high frequency approximation approach with electromagnetic ray tracing is preferable in this concern.
The ray tracing approach provides an indepth understanding of physical phenomenon such as geometrical optics field for reflection and transmission that associate with the structure. The discontinuities caused by geometrical optics field at different shadow boundaries are compensated by the diffracted field . Analogously to the reflected and transmitted fields, the diffracted field is characterized by a coefficient, which is a function dependent on the properties of the obstructers such as its geometry and refractive index. For the case of wedge structure, a uniform diffraction coefficient  was developed to account for the shortcoming in the earlier version. Besides that, the Maliuzhinets diffraction coefficient  has also been developed for both lossless and lossy wedges using impedance boundary conditions. Higher order in the asymptotic expansion has been studied . However, an impedance wedge only forms part of a line structure. A full electromagnetic model for the line structures formed on substrate with incorporation of more than one impedance wedge, and its overall accuracy has yet been addressed extensively.
In this paper, an electromagnetic ray tracing (ERT) model is established to study the scattering phenomenon of line structures formed on substrate. This approach clarifies and enhances understanding of every single electromagnetic field that contributes to the final solution at an observation point above the line structures. It takes into account the existence of both geometrical optics and diffracted fields. The details of the ERT model are discussed in Section 2, which follows by the numerical results presented in Section 3. All the formulations are devoted for transverse electric (TE) polarization, where the electric field has its only component in the z direction. A time harmonic of exp(jwt) is assumed and suppressed throughout.
2. Modeling approach
Figure 1 shows the ERT modeling approach for a line structure formed on substrate. The single line structure is modeled as an impedance double wedge, which is a combination of two impedance wedges. The study of electromagnetic field for an impedance wedge is first presented in this section. Subsequently, the impedance wedge is extended to include the underneath substrate. This establishes the basic component in the ERT model, which is applied analogously to form a full single line structure. The same approach is applied when two-line structures are studied. An effective refractive index is also used for the impedance wedge.
2.1 Single impedance wedge
An impedance double wedge for the single line structure is decomposed into two impedance wedges as shown in Fig. 2. The total solution for the impedance double wedge can be obtained by combining all the fields that are contributed from the two impedance wedges. This section aims to study the electromagnetic scattering of a single impedance wedge by identifying all the associated geometrical optics and diffracted fields. The electromagnetic fields that propagate into the internal part of the impedance wedge are not considered. Only the external fields are included in the analysis. On the other hand, the normalized surface impedance, used for the wedge is obtained from the effective refractive index, n e which is discussed in Section 2.3.
For an isolated impedance wedge with interior angle of (2- p̄)π, which is illuminated by an incident plane wave to its top surface, ST the surrounding zones can be separated into three regions. These regions are constrained by the incident shadow boundary (ISB) and reflection shadow boundary (RSB) as shown in Fig. 3. A summary of the field that contributes to each region is given below.
In region I, the total field is a combination of the incident, reflected and diffracted fields. The reflected field is excluded from region II as it is beyond the reflection shadow boundary. In region III, the diffracted field is the only field that exists. It can be seen that diffracted field exist in every region to ensure continuity of the total field across different shadow boundaries. The geometrical optics field gives abrupt changes of the total field across the shadow boundaries.
2.1.1 Incident field
For an uniform incident plane wave at an angle of ϕo, the field at point P(ρ, ϕ) in Fig. 3 is represented by
with all the angles, ϕ are measured from the top surface, ST and the origin is located at the edge of the wedge. The wavenumber in the vacuum and distance from the origin to the field point are denoted by k and ρ, respectively.
2.1.2 Reflected field
In region I, the incident field that reaches the top surface, ST as shown in Fig. 3 will be reflected and the amplitude will be attenuated. At a distance of ρ from the origin as illustrated in Fig. 3, the field at observation point P(ρ, ϕ) caused by the reflection from the top surface, ST of the impedance wedge is represented as
A particular observation point corresponds to a unique reflection point on the impedance wedge surface. The location of this reflection point is obtained using direct geometrical derivation. The reflection coefficient for TE polarization is given by
and the effective refractive index, n e is given in Section 2.3.
2.1.3 Diffracted field
When the incident field illuminates the impedance wedge as shown in Fig. 3, the diffracted field in any of the regions is expressed as
where Ui(QE) is the phase reference point to the origin. When the phase reference point and the origin are coincided at the edge of the wedge, the expression simplifies to
The spreading factor, ρ -1/2 suggests that the diffracted field is emanating in the cylindrical wave form from the edge, which is acting as a line source. The Maliuzhinets diffraction coefficient for transverse electric polarization, DTE (ϕ, ϕo) is expressed as
where and represent the reflection coefficients for surfaces ST and SV, respectively. The notation, Ψ(∙) is an auxiliary function expressed in terms of Maliuzhinets functions  whereas F(∙) denotes the transition function . Since the diffracted field discussed here is obtained without any alterations to its ray path, hereafter, they are referred as direct diffracted field.
2.2 Line structures on substrate
In Section 2, the edge of a line structure without underneath substrate is modeled as a single impedance wedge. All the associated electromagnetic fields are discussed. However, the electromagnetic solution becomes more complex when the line structures are formed on a layer of substrate. In addition to the geometrical optics and direct diffracted fields discussed in Section 2, it requires a detailed analysis of the electromagnetic waves that induced in conjunction with the existence of the substrate. For an observation point above the single impedance wedge with inclusion of the underneath substrate, the additional electromagnetic fields are reflected and diffracted-reflected fields. They are discussed in this section. Similarly, the total solution of a line structure formed on substrate is obtained by combining all the electromagnetic fields contributed from each impedance wedge with underneath substrate. For ease of illustration, the half line structures are shown in the figures presented in this section, instead of using the impedance wedge model.
2.2.1 Reflected field from substrate
Referring to Fig. 4, the direct reflected field from the substrate, n 2 at surface SB contributes to the total field at point B1. At this point, the reflected field caused by reflection from the point Qr is represented using
where ρQr is the distance measured from the origin to the point Qr, and ϕQr is the angle measured from the top surface, ST. The reflection coefficient at surface, SB for TE polarization, is similar to Eq. (3), except that the used refractive index is different. Here, the exact refractive index of the substrate, n 2 is used. The term, exp(-jkρr) is used to correct the phase shift due to the movement of origin from the edge of the wedge to the point Qr as illustrated in Fig. 4.
2.2.2 Diffracted-reflected field
In Fig. 5, the total field at point BII (xBII, yBII) involves reflection of the diffracted field from the substrate, n 2 at surface SB. Here, this particular electromagnetic field is called as diffracted-reflected field. The diffracted-reflected rays are treated to emanate from an image source at point O’. These reflected rays are characterized by a spreading factor of (ρi+ρr)-1/2, where ρi is the distance from the wedge origin to the point of reflection at the substrate surface, SB and ρr is the distance from the point of reflection, Qr (xr1, yr1) to the point of observation, BII (xBII, yBII). Existence of these rays ensures continuity across Regions I & II and Regions II & III. If it is neglected, the amplitude of the total field can show a significant disruption across different boundaries. Study of the geometrical optics field in different regions is similar to Section 2.1. With this, the diffracted-reflected field at observation point, BII (xBII, yBII) is expressed as
The diffraction coefficient DTE (ϕ, ϕo) is obtained from Eq. (5). On the other hand, the reflected point, Qr (xr1, yr1) for a particular diffracted ray is required to determine the change of the phase and its attenuated amplitude. By solving the following quadratic equation,
the location of the reflected point, xr1 for an observation point at BII (xBII, yBII), is obtained. Similarly, the reflection coefficient for the surface, SB, is obtained from Eq. (3) but with the use of exact refractive index, n 2 for the substrate.
When an isolated single line structure is considered, the diffracted-reflected field in Fig. 5 is not restricted by any adjacent structures. However, this is not the case for two-line structures. In this work, when two-line structures are considered, the diffracted-reflected field emanating from the edge at point B as illustrated in Fig. 6 is truncated at the cut-off point CII. This implies that for observation level at y=h1, the diffracted-reflected field is considered from x=-ls/2 to x=CII. The selection of cut-off point CII is based on the point C’II, which is located exactly below the origin. By doing so, it assumes that the weightage of the diffracted-reflected field to the total field at point CII is relatively small. If it is non-negligible, it should show a clear discontinuity at the cut-off point, CII in the amplitude profile. The discontinuity is supposed to be compensated by the secondary diffracted field emanating from the edge at point C. This secondary diffracted field is induced by the reflected-diffracted field that follows the ray path B-C’II-C. On the other hand, the same approach is applied analogously to determine the cut-off point for the diffracted-reflected field emanating from the edge at point C of another single line structure.
2.3 Effective refractive index
An effective refractive index (n e) is extracted for the line structures (n 1) and the underneath substrate (n 2). By applying the effective refractive index, only the reflected field from the top surface, ST of the line structure is considered. This reflected field corresponds to Eq. (2). It implies that the formulation for the incident field that transmitted into the line structure through surface ST, subsequently reflected from the boundary between n 1 and n 2, and transmitted back into the vacuum through surface, ST is not required. Associated with this is the use of effective refractive index for the impedance wedge when considering different diffracted fields. The normalized surface impedance of the wedge, is given by the inverse of the effective refractive index. The exact refractive index is also used when the effective refractive index is inapplicable, particularly for any reflected fields from the substrate surface. A summary of the field type and its associated refractive index used in the ERT model is given in Table 1.
The effective refractive index is obtained from the amplitude of the film stack reflection coefficient, Rf using 
where ηo is the vacuum tilted admittance. The optical admittance of the multilayer, Y which is given by the ratio, C/B is deduced from the film stack characteristic matrix,
where δ 1 and η 1,2 are the phase thickness and the tilted admittance for the respective layer.
3. Results and discussion
The numerical results presented here are devoted for the electromagnetic scattering of single and two-line structures formed on substrate. The line structures are illuminated by plane wave with unit amplitude. Figure 7 shows the geometry of the single line structure with width of l. For a practical application in photolithography, the refractive indexes for the line structure and underneath substrate are referred to the polysilicon and silicon, respectively. For illumination wavelength, λ at 633 nm, the refractive index of the former is 3.93–0.0717 j and the later is 3.88–0.01933 j. Both are lossy materials. The line structures have height, h2 of 5000 Å (0.5 µm). Furthermore, the line structures are constructed based on right-angled wedges, which implies that the wedge interior angle, (2-p̄)π in Fig. 3 has p̄ value of 1.5. All the presented results are based on incident angle, ϕo at 90° and observation level, y of 0.8 λ (k∙h1=5). The first part of this section focuses on the results for individual electromagnetic field obtained using the ERT model. Subsequently, accuracy of the ERT model for various conditions is studied. The CPU computational time and memory used in the ERT model are also discussed.
3.1 Individual electromagnetic field for single line structure
Figures 8(a) and 8(b) show each individual field that contributes to the electromagnetic scattering of an isolated single polysilicon line structure formed on silicon substrate. The selection of line width, x=2 µm is arbitrary and mainly for elaboration purpose. In Fig. 8(a), all the geometrical optics fields that contribute to the total solution are presented. This includes the (i) incident field which has unit amplitude, (ii) reflected field from the top surface, ST of the line structure and (iii) reflected field from the substrate surface, SB. The reflected field from the top surface, ST is exist in the region of |x|<1 µm whereas the reflected field from the substrate surface, SB is exist in the region of |x|>1 µm. The amplitude of both reflected fields show a clear step jump at the shadow boundaries, i.e. x=±1 µm. These step profiles are expected to be compensated by the direct diffracted and diffracted-reflected fields. The (i) direct diffracted field shown in Fig. 8(b) exists across all the x distance. Its amplitude decays exponentially along the increasing direction of |x|>1 µm. In the region of |x|<1 µm, its amplitude gives a “ripple” trend. This can be explained by considering the direct diffracted field is constructed based on the concurrent diffraction from two different edges at x=±1 µm and interaction of these fields can occur. On the other hand, the (ii) diffracted-reflected field shown in Fig. 8(b) only exist in the region |x|>1 µm, with the amplitude decay exponentially from the increasing |x|>1 µm distance.
A step by step combination of all the geometrical optics and diffracted field is given here. In Fig. 9(a), the amplitude of the total geometrical optics field is shown. It is the combination of all the geometrical optics fields given in Fig. 8(a). The total geometrical optics field shows a step jump from 0.85 to 1.5 at the shadow boundaries, x=±1 µm. By adding in the direct diffracted field, the trend of constant amplitude is no longer exist. The amplitude is decaying smoothly in the direction of increasing |x| distance from x=±1 µm, as shown in Fig. 9(b). However, there is still a discontinuity at the shadow boundaries, x=±1 µm, which implies that some fields are still missing. The remedy to this discontinuity is by adding in the diffracted-reflected field. The final and total amplitude is given in Fig. 9(c). It is observed that the amplitude across the shadow boundaries at x=±1 µm is smooth. This concludes that all the prominent fields have been taken into consideration.
Following the numerical results presented so far, it is shown that the total solution in the ERT model was obtained by constructing all the geometrical optics and diffracted fields. This is one of the key advantages of ERT model for line structure, which allows the causes to an electromagnetic disturbance to be explained. For example, there is a disturbance of the amplitude at point X in Fig. 9(c). Throughout all the step by step constructions of the solution, it is identified that the disturbance is caused by the existence of diffracted-reflected field. On the other, rigorous electromagnetic solution such as FDTD does not provide this explanation offered by the ERT model. In FDTD approach, only the final of the total electromagnetic solution is obtained. Thus, the users have no clues about how is the geometry of the obstructer or its refractive index could have impact on the final solution. The ERT model for line structures developed in this work provides the answer to this enquiry.
3.2 Accuracy of the ERT model
The aim of this section is to verify accuracy of the ERT model. The amplitude and phase of the total electromagnetic field obtained from the ERT model is compared with the reference FDTD solution . If they are matched with small errors, this indicates that the decomposition of the total field into their individual electromagnetic field, particularly all the diffracted fields is valid. The accuracy study here includes the electromagnetic scattering for a single and two-line structures, and line structures with variation in refractive index. The used parameters are Δ|Ez|, ΔPhase, amplitude correlation coefficient (ACC) and phase correlation coefficient (PCC). The notations, Δ|Ez| and ΔPhase (in radian unit) denote the difference of the amplitude and phase results, which are defined as,
On the other hand, the ACC and PCC results give an indication of the degree of matching of ERT model results to FDTD solution. The closer the ACC and PCC values to one, the better the matching of the ERT model with the reference FDTD solution. A perfect match gives value of one for both ACC and PCC. For a particular point, i at distance x, the ACCi and PCCi are defined as,
and the presented ACC and PCC results for a particular line structure are the average values of all the points within the domain of x distance.
3.2.1 Single line structure
The amplitude and phase of the total electric field for an isolated single polysilicon line structure with width, l of 8 µm (12.6 λ) and 1 µm (1.6 λ) formed on silicon substrate are given in Figs. 10(a) and 10(b), respectively. The selected line width is based on two extreme conditions, where one is near to the wavelength and the other one is in the multiple order of the wavelength. In general, it is shown that the results obtained from the ERT model gives a good match with the reference FDTD solution. At line width of 1 µm (1.6 λ), the maximum amplitude difference Δ|Ez| is -0.067 and when the line width is increased to 8 µm (12.6 λ), the maximum amplitude difference, Δ|Ez| is improved to -0.045. The results of ACC presented in Fig. 11 are also consistent with this finding for line width dependent error. The ACC improves from 0.978 to 0.988 when the line polysilicon line width is increased from 1 µm to 8 µm (from right to left in the figure). On the other hand, the phase results show relatively significant discrepancy near to their respective shadow boundaries, x=±l/2 µm for line width given in Figs. 10(a) and 10(b). The maximum phase difference, ΔPhase is 0.14 radian found at line width of 1 µm (1.6 λ) in Fig. 10(b). As a result of this, the PCC results given in Fig. 11 are relatively low compared to the ACC results. The PCC values are also improved when the line width is increased from 1 µm to 8 µm (from right to left in the figure).
The ACC and PCC results for a single polysilicon line structure on silicon substrate with subwavelength line width ranging from 0.5 µm (0.8 λ) to 0.125 µm (0.2 λ) are also included in Fig. 11. Associated with this, Fig. 12(a) presents the amplitude and phase results of the 0.25 µm (0.4 λ) single line structure obtained using the ERT model and its comparison with the reference FDTD solution. It is shown that a good match between the two solutions is still established, particularly for the amplitude results. The phase results obtained from the ERT model shows discrepancy with the reference FDTD solution only in the vicinity of x=± 0.125 µm (indicated by the dotted lines). When the line width is further reduced to 0.125 µm (0.2 λ), the ACC and PCC results drop to 0.961 and 0.926, respectively as given in Fig. 11. Figure 12(b) illustrates that the poor PCC results are due to the prominent discrepancy of the phase results in the vicinity of x=± 0.0625 µm (indicated by the dotted lines). Furthermore, the mismatch of the phase results is also significant in the regions indicated by the arrows X in Fig. 12(b). This concludes that the line width of 0.125 µm (0.2 λ) gives relatively poor accuracy. In this case, the line width of 0.25 µm (0.4 λ) is considered as the limit of the ERT model.
3.2.2 Two-line structures
The ray tracing model is extended to electromagnetic scattering of two-line structures. Previously in a single line structure, the diffracted field is emanating from two edges of the same double wedge. When an additional line structure is added, interaction of the diffracted field emanating from the adjacent double wedge occurs. Their impact to the accuracy of the ERT model solution is examined in this section. The amplitude and phase results presented in Figs. 13(a) and 13(b) are based on the combination of polysilicon-silicon (line structure-substrate) with variation of 8 µm (12.6 λ) and 1 µm (1.6 λ) in the spacing between two identical lines. Each single line has a width of 2 µm (3.2 λ). The results obtained from the ERT model show a good match with the reference FDTD solution. It is observed that the prominent discrepancy is always located near to the shadow boundaries, which are denoted by the dashed lines in the figures. When the spacing between the line structures is reduced to 1 µm as in the case in Fig. 13(b), it gives the greatest difference in amplitude, Δ|Ez| and phase, ΔPhase with 0.166 and 0.2 radian, respectively. The results are improved when the spacing in between the lines is increased to 8 µm (12.6 λ) as in Fig. 13(a). The absolute difference in amplitude and phase is bounded within 0.1 and 0.1 radian, respectively. In general, the ACC and PCC results are also increased when the spacing of the two lines structures are increased from 1 µm (1.6 λ) to 8 µm (12.6 λ), as presented in Fig. 14 (from right to left). This finding concludes that accuracy of the ERT model is improved when there is less interaction of the diffracted field from the adjacent structures.
When the spacing in between the 2 µm line structures is reduced to a subwavelength width, 0.5 µm (0.8 λ), the ACC and PCC results are further reduced to 0.943 and 0.927, respectively as shown in Fig. 14. Associated with this, the amplitude and phase results obtained from the ERT model give a relatively poor match with the reference FDTD solution as presented in Fig. 15. There are significant mismatch of both the amplitude and phase results, particularly in the x distance of ± 0.25 µm. The maximum amplitude difference, Δ|Ez| and phase difference, ΔPhase are 0.172 and -0.399 radian, respectively. It is concluded that the ERT model gives relatively poor accuracy for the two-line structures with subwavelength spacing of 0.5 µm (0.8 λ). In this case, the smallest allowable spacing between the two identical polysilicon line structures is identified as 1 µm (1.6 λ).
3.2.3 Variation in refractive index
In this section, the accuracy of the ERT model for line structures with variation in the refractive index is studied. This is to ensure that the ERT model is applicable to a wide range of materials. Lossy materials with complex refractive index are considered. Both the line structures and underneath substrate have identical refractive index, i.e. n 1 = n 2. The ACC and PCC results for single and two lines structures across different real refractive indexes are shown in Fig. 16. The imaginary part of the refractive index is taken as -0.1j. All the line structures and spacing have a width of 1 µm (1.6 λ). The ACC results for single line structure show a relatively consistent value across the real refractive index. This is observed through the linear trendline which are almost flat at ACC of 0.98. Its companion PCC results are also bounded in the range of 0.98 to 0.96. In contrast, the ACC results for two-line structures exhibit a relatively steep increase in value when the real part of the refractive index decreases (from right to left in the figure). The same finding is also applied to the PCC results.
3.3 CPU computational time and memory used
In the FDTD method, the electromagnetic field is computed everywhere in the specified domain. The domain must be sufficiently large to include the entire geometry of the structures. This is the usual case in the FDTD method where the electromagnetic field at points which are not of interest is also generated. Furthermore, a converged FDTD solution is also required. As a result of these properties, the extensive consumption of the CPU computational time and memory is unavoidable. In this paper, the amplitude and phase results of the reference FDTD solution presented in Sections 3.2.1 and 3.2.2 for each condition of line structures required CPU computational time of more than one hour using a notebook PC with Intel Core 2 Duo Processor T5450 (1.66 GHz). The memory used exceeded 100 MB. In contrast to this, the ERT model demonstrates the advantage of relatively low CPU computational time and memory used. Figure 17(a) shows the CPU computational time of the amplitude results presented in Sections 3.2.1 and 3.2.2 for the ERT model using the same notebook PC. The CPU computational time taken for all the single and two-line structures with width or space of at least 1 µm (1.6 λ) is well below 120 s. Associated with this is the relatively low consumption of memory, which is below 10 MB as presented in Fig. 17(b). Such a relatively low CPU computational time and memory used is due to the flexibility of the ERT model that computes only the required electromagnetic field. The solution is based on direct computation using the explicit expressions derived for each electromagnetic field, be it the geometrical optics or the diffracted fields, as discussed in Sections 2.1 and 2.2.
An electromagnetic scattering model for line structure was established based on ray tracing approach. This electromagnetic ray tracing (ERT) model is able to identify different electromagnetic fields that contribute to the total solution. This includes the geometrical optics field for incidence, reflection and transmission phenomenon. Associated with this is the existence of different diffracted fields. By applying this model, the particular electromagnetic field that causes a disturbance in the amplitude profile for single line structure was identified and elucidated. The accuracy of the ERT model was verified through the comparison with the reference FDTD solution. For a single 1.6 λ (width) polysilicon line structure on silicon substrate, the ERT model was able to demonstrate ACC of 0.978 with the maximum amplitude difference, Δ|E| of as low as -0.067. In additions to that, the amplitude and phase results for a single line structure with subwavelength line width of 0.4 λ, obtained from the ERT model was able to show a good match with the reference FDTD solution. When the ERT model was applied for two-line structures, the smallest allowable spacing between the two identical polysilicon line structures was 1.6 λ. A linear trend of improving ACC and PCC results with reducing real part of the refractive index was also observed for two-line structures with lossy material.
The authors would like to acknowledge Lap Chan and Ng Chee Mang from Chartered for their support on this work.
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