1. Introduction

As diffuse scattering constrains the optical performance of gratings that are used, e.g., in high performance spectrometers [1–5], the occurrence of stray light is highly undesired. Basically, straylight originates in any disturbance of the ideal periodicity of the grating structure. In case of 1D-gratings, e.g. mechanically ruled gratings, holographic gratings or lithographically manufactured lamellar gratings, there are various types of shape deviations and fabrication inaccuracies that may cause qualitatively different stray light scenarios.

For example, periodic large scale deviations of the grating geometry from its ideal shape cause distinct stray light artefacts, which are known as grating ghosts [6–10]. Further, the surface roughness of the substrate or of additional optical coatings produces a diffuse stray light background, e.g. for high reflective gratings [11, 12]. The problem of surface roughness of coatings and its optical scattering are already investigated profoundly [13–16]. Additionally, especially in binary gratings, line edge roughness (LER) of the grating lines also provokes a continuous stray light background. The stochastic nature of LER, its effects onto the diffraction efficiencies and even onto the stray light spectrum is examined in various papers, especially with respect to scatterometry measurements [17–25]. E.g., in [22, 24] the finite element method was used for calculating the influence of line position roughness (LPR), line width roughness (LWR) onto the diffraction efficiencies of special reflection gratings. Bergner et al. [19] applied a special effective medium approach for modelling the LER and thereby the effects onto scatterometry measurements. Investigations on the distribution of the diffusely scattered light were firstly performed by Germer et al. [17], who rigorously calculated the Stokes parameters and the intensities of stray light orders in a one-dimensional model. Unfortunately, the applied approach doesn’t carry information about the two-dimensionality of the LER-problem. The field stitching approach [18, 23] overcomes this restriction and allows for calculating the 2D scattered light distribution of LER disturbed gratings. Within this method, the near field of slightly different elementary cells is stitched together and then propagated to a common far field. Though, this method still might suffer from its stitching approach neither does it give a physical understanding of the observed diffuse stray light pattern. Kato et al. [20] presented an analytically derived one-dimensional model of the intensity distribution of the scattered light. They found sine- and cosine-terms in the stray light spectra of LPR- and LWR-disturbed gratings that explain the simulation results of Germer et al. [17].

But still, the correct prediction of the diffuse scattering and the angle resolved scattering (ARS), respectively, due to LER is a great challenge and there is a need for models that allow a correct prediction of LER-induced straylight both in high performance spectrometry and scatterometry measurements [24–27]. On the other hand, a physical insight into LER-scattering and an understanding of the mechanisms responsible for the stray light generation is highly desirable.

In this paper, we introduce a one-dimensional method, which allows the correct prediction of the ARS within an angular range surrounding the dispersion direction of arbitrary 1D-gratings with 2D-stochastical disturbances of the grating geometry. The method bases on standard rigorous simulation tools, e.g. the rigorous coupled wave analysis (RCWA), which is mostly used for simulating diffraction efficiencies of periodic structures [28]. It further relies on the property of the ARS (for small disturbances) that it can be mathematically described by a product of optic factors depending on the undisturbed structure and the power spectral density describing the stochastic disturbance. For verification, several 1D- and 2D-simulations on the example of LER-disturbed lamellar gratings will be performed.

In a second part, we present a scalar analysis of light scattering on lamellar large-period gratings due to LER with the purpose of a physical understanding of the complex mechanisms. For simplification, the analytic model bases on Fourier optics applying thin element approximation (TEA). It is shown that this model allows to calculate even wide angle scattering of TE-polarized light. The relation to the widely accepted Rayleigh-Rice-theory for rough surfaces and the influence of different parameters (roughness, grating geometry, illumination) onto the straylight spectrum is discussed.

Therefore, the paper is structured as follows. In Sec. 2, we summarize the basic characteristics of LER and give an overview about the parameters that are necessary to describe roughness. In Sec. 3, the basic principles of the applied one- and two-dimensional simulation methods are explained and their relation is examined. Here, LER serves as an example of the disturbance, even though the method is valid for any kind of stochastic irregularity. However, a physical insight into the scattering effects due to LER is given in Sec. 4. Based on the derived TEA-model, the effect of the roughness parameters, the illumination and the grating geometry onto the distribution of the scattered light is examined and its validity range is explored.

2. Line edge roughness (LER)

As the scattering due to LER is of high interest in optics, this phenomenon will be investigated in this paper, even though the introduced algorithm is valid for every kind of stochastic disturbance. Other stochastic disturbances are generally treated similarly. The term line edge roughness (LER) describes the deviation of a grating line edge from its ideal position, which can be mathematically expressed by the function X(y) [Fig. 2(a)]. LER is a widely analyzed topic and its basic characteristics are already investigated profoundly [29–32] and are well understood. Basically, LER is a stochastic process and must thus be described by both probability distribution functions and correlation functions and the corresponding parameters. These parameters are known as the standard deviation σ, the correlation length ξ and the roughness exponent α [29, 30], and are found e.g. in the analytic expression for the autocorrelation function, $ACF(\mathrm{\Delta }y)={\sigma }^2{e}^{-{(\mathrm{\Delta }y/\xi )}^{2\alpha }}$. Furthermore, in optics it is usually necessary to analyze the power spectral density (PSD) of a certain roughness as it provides the link to the mathematical description of scattered power distribution. The PSD can be evaluated by the Fourier transform of the ACF according to $PSD(f)={\displaystyle {\int }_{-\infty }^{\infty }ACF(y)e^{-2\pi ify}dy}$, with f denoting the spatial frequency. Figure 1 shows the power spectrum for different parameters and an example of their corresponding rough lines. Unfortunately, there exists no general analytical model to describe the PSD of a stochastic process determined by σ, ξ and α. Though, there are models, which give an approximate description of the PSD. The so called Palasantzas-PSD is given by [29]

 

Fig. 1 Power spectral density for different values of the roughness parameters: standard deviation σ (a) correlation length ξ (b) and roughness exponent α (c) and typical rough line edges according to each set of parameters.

Download Full Size | PDF

 

Fig. 2 (a) Illustration of the underlying principle of the applied one- and two-dimensional model that was used for simulating the scattered light distribution of LER-disturbed gratings. Left: 2D-model applies full LER-characteristics within an area P_x × P_y and calculates stray light orders (SO) in the full half space (not illustrated). Right: 1D-model deduced from the profile of the 2D-grating-structure calculates SOs in dispersion plane. (b) 1D- and 2D-simulation results (average of 32 single simualtions) for a monolithic grating in fused silica (n_i = 1.457) with p = 667 nm, d = 1640 nm, b = 430 nm, θ_i = 22° and λ = 633 nm and a roughness characterized by σ = 3 nm, ξ = 50 nm, α = 0.5. The 99%-CI (colored shading and error bars) represents the error. A factor S harmonizes the different simulation results (blue solid curve, cf. Eq. (7)).

Download Full Size | PDF

For the purpose of theoretically investigating the effects of LER onto the scattering spectrum, there is a need for algorithms that numerically generate random rough edges. In this article the moving average method [33] is applied. There, the discretized rough edge $\overline{X}=(X_0,\dots ,X_{\mathrm{L}})$ originates in a vector $\overline{\epsilon }=({\epsilon }_0,\dots ,{\epsilon }_{\mathrm{L}})$ of Gaussian white noise with standard deviation σ_ϵ via $X_n={\displaystyle {\sum }_{l=0}^L{\delta }_l{ϵ}_{n-l}}$. The vector $\overline{\delta }=({\delta }_0,\dots ,{\delta }_{\mathrm{L}})$ is determined by the ACF of $\overline{X}$, which is on the one hand given by the parameters σ, ξ and α. On the other hand, the ACF can be calculated by the expectation value 〈X_nX_n+τ〉. This yields a conditional equation for $\overline{\delta }$, which can be numerically evaluated.

3. Simulation of angle resolved scattering (ARS)

3.1. One-dimensional ARS-simulation

For simulating the light propagation and eventually the intensity distribution in the half-space of a transmission grating with period p, we used the rigorous coupled wave analysis (RCWA) [28]. For simulating the intensity distribution of the scattered light, we still use RCWA by introducing a super-lattice with period P = N · p. As illustrated in Fig. 2(a), such a compound of many single periods p possesses additional diffraction orders modeling the quasi-continuous scattering background. In the following, the additional diffraction orders will be referred to as stray light orders (SO, red arrows in Fig. 2(a)), whereas the ordinary diffraction orders of the undisturbed grating still will be referred to as main diffraction orders (DO, yellow arrows in Fig. 2(a)).

In the case of an ideal undisturbed grating the diffuse stray light orders have an intensity of zero and thus, give the same result as for N = 1. In the case of LER, there are statistical disturbances in each grating period and the SOs carry a certain amount of energy. The diffraction efficiency η_m in each stray light order m is then calculated using the RCWA. According to the definition of angle resolved scattering (ARS) [12] the scattered light distribution can be evaluated by

3.2. Two-dimensional ARS-simulation

The simulation of 2-dimensional scattering can be performed using standard optics modelling methods like RCWA, too. The simulation principle is exactly the same as in the one-dimensional case, but now also the y-direction of the grating is taken into account. Hence, we have to consider a domain-period P_x = N_xp in x-direction and a domain-period P_y = N_yΔy in y-direction, with P_y being the length of the rough edges and Δy the pixel size or sampling distance, respectively. The rough line edges were artificially generated by means of the moving average method as described above and thus, all roughness parameters σ, ξ and α are taken into account. The corresponding structure of dimension P_x × P_y generates diffraction orders both in dispersion (x-direction in Fig. 2(a)) and cross-dispersion direction (y-direction in Fig. 2(a)), of which we can distinguish between the main diffraction orders (DOs) and diffuse stray light orders (SOs). Similar to the 1D-approach every stray light order (m_x, m_y) is associated with a certain solid angle $\mathrm{\Delta }{\Omega }_{m_{\mathrm{x}}{m}_{\mathrm{y}}}$. Accordingly, the angle resolved scattering is calculated by

As we consider stochastic problems it is necesarry to calculate the mean value of several single simulations (“ensembles”) in order to achieve a accurate result. In Fig. 2(b) N_e = 32 ensembles were taken into account and additionally the 99%-confidence interval ($\text{CI}\propto {\sigma }_{\mathrm{e}}/\sqrt{N_{\mathrm{e}}}$, with σ_e being the standard deviation of one ensemble) was calculated (depicted as shading for the 1D-simulation and error bars for the 2D-simulation). The parameter N was found to affect the total error but not the mean value as a growing N only results in a reduced σ_e. In summary the parameters N_e and N should be large in order to reduce the total error.

3.3. Relation between 1D- and 2D-ARS-simulation

Following the simulation results shown in Fig. 2(b), the one-dimensional simulation is able to compute a qualitatively correct scattering behavior with a sufficient number of data points within a fast computation time. But the simulation principle neither considers the complete roughness descriptors nor gives the right values of the 2D-ARS that are necessary in order to make it comparable to stray light measurements. On the other hand, the two-dimensional simulation considers the full roughness characteristics and is able to compute the ARS in the correct way. Unfortunately, the 2D-ARS requires a huge numerical effort, which makes it unfeasible for every-day-application, especially for large periods. In order to achieve a deeper understanding of the relation between 1D- and 2D-scattering the general structure of the ARS will be examined. Generally, for sufficiently small disturbances the amplitude of the scattered electric field s_scat can be expressed as a product of the random variable that describes the stochastic shape deviation (e.g., ΔX_left/right for LER in lamellar gratings), the amplitude of the incident field s_i and an optic factor F_left/right depending on the undisturbed structure (first order expansion) [34, 35]

This formula can be deduced from integrating Eq. (5) in the special case of classical mount, meaning no scattering in off-dispersion direction. As mentioned above, the optic factors depend on the geometry of the undisturbed grating but also on the incident and scattered electromagnetic waves propagating inside the grating structure. Hence, considering the relevant case of scattering in an angular range around the dispersion plane, meaning δk_y = |k_t,y − k_i,y| ≪ 2π/λ, the 1D- and 2D-optic factors are approximatively equal |F_{LER,left/right}|² ≅ |F_{1D,left/right}|². Applying this identity to Eq. (5) and (6), the complex 2D-scattering phenomena in the approximation range around the dispersion plane can be finally expressed in terms of 1D-scattering according to

It must be pointed out that this relation can be applied for arbitrary polarization state $\overline{P}$ and gives exact results within the dispersion plane. It is usually even valid in a large angular range surrounding the dispersion direction. But close to Wood’s anomalies [37] of gratings the optic factors may exhibit sharp resonances, which reduce the range of validity to smaller off-plane angles.

As a proof, an adaptation of the one-dimensional simulation result to the two-dimensional situation via Eq. (7) is shown in Fig. 2(b). An almost perfect agreement between ARS_1D and ARS_2D is obtained. The significant difference of the curves is the computational effort.

As a further verification, Fig. 3 shows conical ARS-simulations of the same grating as investigated in Fig. 2. The 2D-simulation allows to calculate the scattering into the full transmission half space [Fig. 3(a)], whereas the 1D-simulation only gives exact results along the dispersion direction of the grating (classical mount, see Fig. 3(b)). Nevertheless, by applying the scaling factor S in Eq. (7) to Δk_y ≠ 0, it is possible to evaluate scattering parallel to the dispersion direction, too. As can be seen in Fig. 3(c), the 1D- and 2D-ARS-curves still show a very good agreement even far off m_y = 0. There, for m_y = 3 the diffraction angles in cross-dispersion direction amounts to θ_y ≈ 30°. However, the higher θ_y the higher the deviation between 1D-and 2D-simulation becomes (as also can be seen in Fig. 3(b–c) for small θ_x), i.e., the difference between 1D- and 2D-simulation in Fig. 3(c) for small θ_x is not due to the different sizes of the unit cells (N_x = 12 vs. N = 100 in Fig. 3) but only due to θ_y ≠ 0.

 

Fig. 3 Comparison of the numerically calculated scattering spectra within the 1D- and 2D-approach (red line and black dots, respectively) for a contemporary spectrometer grating (p = 667 nm, b = 430 nm, d = 1640 nm, λ = 720 nm, θ_i = 22°, $\overline{P}=\text{TE}$, n_i = 1.457, n_t = 1, σ = 4 nm, ξ = 500 nm, α = 0.5). (a) Complete 2D-simulation for a domain P_x × P_y = 12p × 5 µm with the 0^th and −1^st DO at (m_x, m_y) = (0, 0) and (m_x, m_y) = (−12,0), respectively. (b) Comparison with 1D-simulation (N = 100) for straylight along the dispersion direction. (c) Comparison with 1D-simulation (N = 100) for conical scattering parallel to the dispersion direction with θ_y ≈ 30°.

Download Full Size | PDF

The 1D-simulation method can hence be used in order to investigate scattering spectra of various 1D-grating structures (where the range of validity depends on the resonance behavior of the structure). In this way, even low-frequency lamellar gratings become within reach of a rigorous examination. In Fig. 4 the stray light spectra of very shallow (d = 30 nm) and deep (d = 1500 nm) monolithic gratings are compared. It is found that deep gratings possess a conspicuously higher stray light level with local scattering maxima and minima throughout the angular half space [Fig. 4(c), cf. Fig. 3(b)]. The shallow gratings instead show a monotonically decreasing ARS for increasing scattering angle [Fig. 4(a–b)]. Independent of the grating period or incident angle they seem to show the very same qualitative characteristics. As we will see in the next chapter, the case of shallow gratings can be understood in the frame of Fourier optics applying thin element approximation (TEA). This also allows a comprehensive understanding of the underlying physical effects, e.g., of the roughness parameters, the grating geometry, the optical constants and the illumination onto the scattering spectrum. Additionally, the relation between the well-known Rayleigh-Rice theory of rough interfaces and LER is revealed. Finally we want to mention, that for a single stochastic grating disturbance different from LER the treatment is fully equivalent. In the presence of several stochastic processes various scaling factors S have to be determined for the different PSDs and cross-correlations.

 

Fig. 4 Comparison of the numerically calculated scattering spectra (mean of 32 simulations) within the 1D- and 2D-approach (red line and black dots, respectively) for shallow and deep monolithic gratings (with different periods). The gray circles belong to a single simulation. The LER of the grating lines was set to be σ = 3 nm, ξ = 50 nm and α = 0.5 and the illuminating wave is characterized by λ = 633 nm, $\overline{P}=\text{TE}$.

Download Full Size | PDF

4. Scalar diffraction analysis

A simple analytic description of the diffuse scattering in monolithic lamellar binary gratings, which is caused by statistical disturbances of the grating lines, is desirable in order to understand the basic mechanisms. The analytic derivation will be performed within the framework of TEA on the example of TE-polarized light. Within this approximation, next to the phase shift φ_b/g = k_b/g,zd caused by the grating bars and grooves, respectively, the Fresnel transmission coefficient t at the interface is taken into account [38,39]. Therefore, the transfer function of an arbitrary binary phase grating is given by

It is found that this formula describes the wide angle scattering of TE-polarized light correctly, if the Fresnel coefficient is a function of the incidence angle and the phase shift is defined according to φ_b/g = k_b/g,z(θ_x)d, with k_b/g,z being the wave vector components of the scattered fields in z-direction. Hence, the phase shift is not assumed to be constant, instead it is a function of the scattering angle θ_x and, thus, different for each outgoing direction, i.e. $k_{\mathrm{b}/\mathrm{g},\mathrm{z}}({\theta }_{\mathrm{x}})=\sqrt{{(n_{\text{b/g}}{k}_0)}^2-{(n_{\mathrm{t}}{k}_0\sin {\theta }_{\mathrm{x}})}^2}$ (see Appendix). The Rayleigh-Rice-scattering of TE-polarized light on rough interfaces also suggests the use of t(θ_i) and Δ_φ(θ_x) = |φ_b − φ_g|, as will be shown in the following section.

4.1. Relation to Rayleigh-Rice-theory of rough surfaces

In Eq. (9) we can now see the basic behavior of diffuse scattering of shallow large-period gratings caused by LER. A comparison to well-established models that describe the scattering of rough surfaces, e.g., the widely accepted Rayleigh-Rice-model [40], reveals several similarities that might be considered as fundamental principles for scattering effects. Usually, the Rayleigh-Rice-model is used for describing diffuse reflection of rough mirrors. Nevertheless, for TE-polarized light the scattering into the transmission half space within Rayleigh-Rice-theory is given by

Here, the transmission coefficient t is replaced by its corresponding reflection coefficient r = t − 1 at a single surface. The similarities to LER-induced scattered light become more evident if Eq. (9) is reformulated slightly. Using the definition of the Fresnel coefficients, Eq. (9) can be expressed in terms of the amplitude reflection coefficients r. Further, in the limit of a small grating depth d ≪ λ a Taylor expansion leads to

It must be pointed out that the approximative transition to the Rayleigh-Rice-like ARS-formula Eq. (11) was only possible by means of a θ_i-dependent Fresnel-coefficient t(θ_i) and a phase shift Δφ(θ_x) only depending on the scattering angle.

As the analytical model bases on Fourier optics, its validity range is restricted to large grating periods and small depths. In the following we will present various one-dimensional rigorous simulations with the purpose to explore the range of validity in more detail and, further, to show the influence of the different parameters. The investigations were performed based on a shallow low frequency gating (p = 10 µm, b = 6.45 µm, d = 100 nm, θ_i = 0°, λ = 1 µm, $\overline{P}=\text{TE}$, n_i = 1.457, n_t = 1, σ = 10 nm, ξ = 200 nm, α = 0.5, N = 20), unless not stated otherwise.

4.2. Influence of LER-parameters σ, ξ and α

The 1D-simulation results for different roughness parameters are shown in Fig. 5. Furthermore, the figure contains the corresponding analytical model according to Eq. (9), which is drawn as a thick line and excellently fits the numerically evaluated curves. According to this model, the influence of the roughness parameters is simply determined by the corresponding PSD. In particular, its value in dispersion direction is determined by P₀ = PSD (0) = 2σ²ξ. Hence, both parameters σ and ξ directly affect the intensity of the scattered light, whereas σ has a much stronger effect. The roughness exponent α has no influence onto the ARS in dispersion direction. At first view, it seems to be surprising as a higher ξ and α usually smooth the rough lines [15][Fig. 1(b–c)] and a stray light reduction could be expected. But on the other hand, it is clear that for very small correlation length the stray light has to vanish due to averaging of an overall statistical independent, uncorrelated process. For increasing correlation length, there is a transition to the 1D-limit of the scattering problem (see right picture in Fig. 2(a)) where less averaging and thus increasing stray light occurs.

 

Fig. 5 Influence of the roughness parameters onto the stray light spectra of a thin monolithic low frequency grating (parameter set as given in Sec. 4.1). The thick line represents the TEA-model whereas the thin line is the 1D-RCWA result.

Download Full Size | PDF

4.3. Influence of the grating geometry and the refractive indices

Figure 6 shows the comparison of various 1D-simulations and TEA-model results of gratings with different refractive indices n_i and n_t and a varying thickness d. In Eq. (9) the ARS is dependent on the grating thickness only via the phase difference Δφ between the bar and the groove of the cosine interference term. Thus, the ARS equals 0 for d = 0. For small grating depth the ARS increases quadratically in d according to Eq. (11) and as discussed above, for larger d the cosine-function reaches a first maximum at a critical thickness d_c = λ/(2Δn), as marked by the blue line in the bottom graphs in Fig. 6. The simple TEA-formula predicts a saturation of the scattering just below d_c and even a decrease of the ARS for just above d_c.

 

Fig. 6 Above: Influence of the grating thickness d onto the straylight spectrum of a thin monolithic low frequency grating (parameter set as given in Sec. 4.1) for different refractive indices n_i and n_t. The thick line represents the TEA-model whereas the thin line corresponds to the 1D-rigorous simulation. Bottom: Interference term as a function of the grating depth with the colored marker indicating the depths of the corresponding stray light simulation. The blue marker corresponds to d_c.

Download Full Size | PDF

Although the TEA-approximation looses its validity for large grating depth, this behavior can be partially reproduced by the ARS simulations. On the one hand, the graphs show that there is a perfect agreement between simulation and analytic model for a small grating depth d. On the other hand, the deviation of the TEA-model from the simulation data increases with growing d, especially for high scattering angles θ_x. In this case of larger ray angles the bar-groove-interaction, which is not included in the TEA, becomes stronger. But even for thicknesses well above d_c the TEA-ARS-values for small scattering angles coincide almost perfectly with numerical simulation. For example, the straylight reduction for a fused silica grating with an even 2 µm thickness at θ_x = 0° is well described by the TEA-model [Fig. 6(b)]. Additionally, the small angle TEA-ARS-values hold true also for varying refractive indices. Most impressively, a decrease of the ARS over the whole angular spectrum is observed for low n_t and high n_i = 5 [Fig. 6(a)].

Figure 6(c) shows the opposite situation with small n_i and large n_t. There, TEA and RCWA agree excellently for d = 10 nm. Even the discontinuity at θ_x = 47°, which is determined by the angle of total internal reflection θ_TIR = asin(n_i/n_t), is reproduced by the TEA-model. A slightly higher grating thickness d = 100 nm, which is still much less than the critical thickness d_c = 921 nm, already leads to a perceivable deviation of the TEA-model from the RCWA-data for diffraction angles θ_x > θ_TIR. Though, there is still a perfect agreement for θ_x < θ_TIR. For even higher d the deviation quickly increases and for d = d_c there is a coincidence between model and data only for θ_x ≈ 0° ± 10°.

In summary, an agreement between TEA-model and RCWA-data is achieved best for high n_i and low n_t. A growing n_t instead, seems to reduce the validity range of the TEA-model (regarding the grating thickness). The reason for this behavior originates from the fact that the potential obliqueness of light rays and thus, the validity of TEA is governed by those indices. The maximum transverse wave number k₀n_t of propagating scattered light in transmission is determined by the index n_t. Therefore, smaller n_t values means less transverse effects inside the grating structure which are not covered by TEA. If additionally, a large bar index n_i exists, the light inside the bar is enforced to propagate forward, which is beneficial for TEA validity.

The influence of the grating period p and the duty cycle FF onto the scattering spectrum is shown in Fig. 7. According to the proportionality ARS ∝ 1/p predicted by the TEA-model, the RCWA-data show an increasing stray light level for shrinking p. Further, the stray light distribution and the qualitative curve progression, respectively, is basically not affected by p. Nevertheless, for very low periods we find a weak deviation between TEA and RCWA. E.g., for p = 2 µm (red curve in Fig. 7(a)) the RCWA-data starts to deviate from the TEA-model. Even smaller periods show the same qualitative behavior but a slightly reduced stray light level than predicted by the TEA. Though, the difference is very small. If d is chosen even smaller than 250 nm, TEA and RCWA do not show a difference for the investigated periods. In consistency with the TEA-model, the duty cycle is found to marginally affect the stray light spectrum of shallow low-frequency gratings. Figure 7(b) shows that only utmost high and low duty cycles (FF = 0.05 and FF = 0.95) slightly influence the stray light distribution.

 

Fig. 7 Influence of the grating period p (a), the duty cycle FF (b) and the angle of incidence θ_i (c) onto the stray light spectrum of a thin monolithic low frequency grating (parameter set as given in Sec. 4.1). The thick line represents the TEA-model whereas the thin line corresponds to 1D-RCWA Data.

Download Full Size | PDF

4.4. Influence of the angle of incidence

Usually, the angle of incidence is a critical parameter in stray light modelling [41]. Therefore, the effect of θ_i ≠ 0° onto the stray light spectrum is of high interest. For the examined low-frequency grating, a comparison of the TEA-model with numerical simulations for different θ_i is shown in Fig. 7(c). It turns out, that there is only a minor deviation of the TEA-model from the simulated scattering spectra also for very high θ_i > 80°. It should be mentioned that there is a correlation to the grating depth, too, and the agreement between TEA and RCWA further improves if d decreases. As Eq. (9) is mainly dependent on the cosine of θ_i, the angle of incidence only affects the stray light level but not the stray light distribution.

5. Conclusion

In this article we present a fast and accurate 1D-method for simulating the scattered light distribution of arbitrary 1D-gratings with two-dimensional stochastic disturbances. The method bases on the separation of the ARS into a product of the PSD describing the stochastic disturbance and additional factors depending on the undisturbed structure. On the example of lamellar gratings and line edge roughness (LER) as disturbing effect, this method is proofed by comparison to full 2D-simulations that consider the whole 2D-characteristics of the disturbance, i.e. the standard deviation σ, correlation length ξ and roughness exponent α. The 1D-approach allows a very fast computation of the angle resolved scattering compared to the full 2D-approach, i.e. the numerical effort is reduced by several orders of magnitude. Thus, even low frequency gratings that are not possible to investigate within the 2D-approach become within reach of straylight simulation.

Inherently, the 1D-approach gives exact results in the range of the dispersion plane of the grating, nevertheless, also a conical range parallel to the dispersion direction can be addressed as long as the grating structure doesn’t show strong resonances. On the example of a contemporary spectrometer grating it is shown, that a conical range up to diffraction angles in cross-dispersion direction of θ_y ≈ 30° show good agreement to 2D-simulations.

Further, the scattering of TE-polarized light on lamellar shallow low-frequency gratings caused by LER is analytically investigated in the framework of Fourier optics (TEA). By applying the 1D-simulation method the correctness of the TEA-model over the whole angular half space is proven. It is shown that the derived model possesses a high analogy to the Rayleigh-Rice-formula for scattering of rough surfaces. The analytics eventually allow a physical understanding of scatter generation in binary gratings and reveal the influence of relevant parameters onto the scattered light distribution. In fact, it is found that next to the roughness parameters also the grating geometry affects the stray light level, e.g., the scattering increases quadratically in the grating depth and inversely with the grating period.

6. Appendix

In Sec. 4 we found that wide-angle scattering (of TE-polarized light) due to LER can be analytically described by Eq. (9) under the assumption of a Fresnel coefficient t(θ_i) only dependent on the angle of incidence and a phase shift Δφ(θ_x) = |φ_b(θ_x) − φ_g(θ_x)| depending on the scattering angle (b … grating bar, g … grating groove). Further, for small d, a transformation to a Rayleigh-Rice-like ARS-formula in Sec. 4.1 was only possible by assuming these dependencies. In the following, these assumptions will be further verified.

This assumption should also be valid for the ordinary diffraction orders of a perfect periodic grating without disturbances. As the RCWA provides the Rayleigh coefficients RC_m of every diffraction order, it is possible to calculate the dependency of the phase shift Δφ(θ_x) on the diffraction angle θ_x. This was done for a grating with period p = 60 µm, n_i = 1.457, n_t = 1, FF = 0.645, λ = 1 µm, θ_i = 0° and different d. The result is depicted in Fig. 8(a) compared to the analytic curves. For a very thin grating with d = 10 nm we find an almost excellent agreement between the RCWA-result and the analytic curves. For higher d the RCWA results increasingly deviate from the phase shift used for the TEA-model. However, the mean curve progression of the RCWA-data still excellently fits the analytic model.

 

Fig. 8 Calculated phase shift (a) and Fresnel coefficient (and relative amplitude, respectively) (b) of every single diffraction order of a fused silica grating with p = 60 µm, FF = 0.645 and different grating depths compared to the phase function and Fresnel coefficient used for the TEA-model.

Download Full Size | PDF

Moreover, the Rayleigh coefficients RC_m also allow for determining the relative amplitude of each diffraction order m (which can be interpreted as the Fresnel coefficients t_m). Based on the Fourier coefficients of the undisturbed binary phase grating given by $F{C}_m=t_m\left(e^{i{\phi }_{\mathrm{b}}}-e^{i{\phi }_{\mathrm{g}}}\right)$ FF · sinc(mπFF) a comparison to the Rayleigh coefficients, i.e. RC_m = FC_m, allows for analyzing t_m. The numerically determined parameter t_m is shown in Fig. 8(b) in comparison to the constant Fresnel coefficient t(θ_i = 0°) = 1.186, which is used within the TEA-model. Especially thin gratings show a very good agreement with the constant t(θ_i). For d = 10 nm only slight deviations between RCWA and TEA occur and the mean value of t_m always equals t. However, with growing thickness the calculated coefficients more and more deviate from the theoretical value t. For d = 400 nm, the relative amplitude of single diffraction orders already differs strongly from the expected value. Nevertheless, this deviation still seems to vary around an averaged value t = 1.186. Thus, the numerically determined phase shift Δφ_m and amplitude factors t_m affirm the assumption of a θ_x-dependent phase and θ_i-dependent Fresnel coefficient that was applied for the TEA-model in order to describe wide-angle scattering of TE-polarized light.

Funding

German Ministry of Education and Research (BMBF) under contract 03Z1H534.

References

1. D. Lobb and I. Bhatti, “Applications of immersed diffraction gratings in earth observation from space,” in ICSO 2010, 10565 (SPIE, 2017), p. 105651M.

2. S. Kraft, U. Del Bello, B. Harnisch, M. Bouvet, M. Drusch, and J.-L. Bézy, “Fluorescence imaging spectrometer concepts for the earth explorer mission candidate flex,” in ICSO 2012, 10564 (SPIE, 2017), p. 105641W.

3. G. Kopp, P. Pilewskie, C. Belting, Z. Castleman, G. Drake, J. Espejo, K. Heuerman, B. Lamprecht, P. Smith, and B. Vermeer, “Radiometric absolute accuracy improvements for imaging spectrometry with hysics,” in Geoscience and Remote Sensing Symposium (IGARSS), 2013 IEEE International, (IEEE, 2013), pp. 3518–3521.

4. B. Guldimann, A. Deep, and R. Vink, “Overview on grating developments at esa,” CEAS Space J. 7, 433–451 (2015). [CrossRef]  

5. M. Kroneberger and S. Fray, “Scattering from reflective diffraction gratings: the challenges of measurement and simulation,” Adv. Opt. Technol. 6, 379–386 (2017).

6. C. A. Palmer and E. G. Loewen, Diffraction Grating Handbook (Newport Corporation, 2005).

7. B. Schnabel and E.-B. Kley, “On the influence of the e-beam writer address grid on the optical quality of high-frequency gratings,” Microelectron. Eng. 57, 327–333 (2001). [CrossRef]  

8. M. Heusinger, T. Flügel-Paul, and U.-D. Zeitner, “Large-scale segmentation errors in optical gratings and their unique effect onto optical scattering spectra,” Appl. Phys. B 122, 222 (2016). [CrossRef]  

9. M. Heusinger, M. Banasch, T. Flügel-Paul, and U. D. Zeitner, “Investigation and optimization of rowland ghosts in high efficiency spectrometer gratings fabricated by e-beam lithography,” in Advanced Fabrication Technologies for Micro/Nano Optics and Photonics IX, 9759 (SPIE, 2016), p. 97590A.

10. M. Heusinger, M. Banasch, and U. D. Zeitner, “Rowland ghost suppression in high efficiency spectrometer gratings fabricated by e-beam lithography,” Opt. Express 25, 6182–6191 (2017). [CrossRef]   [PubMed]  

11. M. D. Perry, R. D. Boyd, J. A. Britten, D. Decker, B. W. Shore, C. Shannon, and E. Shults, “High-efficiency multilayer dielectric diffraction gratings,” Opt. Lett. 20, 940–942 (1995). [CrossRef]   [PubMed]  

12. S. Schröder, D. Unglaub, M. Trost, X. Cheng, J. Zhang, and A. Duparré, “Spectral angle resolved scattering of thin film coatings,” Appl. Opt. 53, A35–A41 (2014). [CrossRef]   [PubMed]  

13. J. E. Harvey, N. Choi, S. Schroeder, and A. Duparré, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2012). [CrossRef]  

14. T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004). [CrossRef]  

15. Y. Zhao, G.-C. Wang, and T.-M. Lu, Characterization of Amorphous and Crystalline Rough Surface – Principles and Applications (Elsevier, 2000).

16. S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, and J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Express 19, 9820–9835 (2011). [CrossRef]   [PubMed]  

17. T. A. Germer, “Modeling the effect of line profile variation on optical critical dimension metrology,” in Advanced Lithography, (SPIE, 2007), pp. 65180Z. [CrossRef]  

18. T. Schuster, S. Rafler, V. F. Paz, K. Frenner, and W. Osten, “Fieldstitching with kirchhoff-boundaries as a model based description for line edge roughness (ler) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009). [CrossRef]  

19. B. C. Bergner, T. A. Germer, and T. J. Suleski, “Effective medium approximations for modeling optical reflectance from gratings with rough edges,” JOSA A 27, 1083–1090 (2010). [CrossRef]   [PubMed]  

20. A. Kato and F. Scholze, “Effect of line roughness on the diffraction intensities in angular resolved scatterometry,” Appl. Opt. 49, 6102–6110 (2010). [CrossRef]  

21. E. Gogolides, V. Constantoudis, and G. Kokkoris, “Towards an integrated line edge roughness understanding: metrology, characterization, and plasma etching transfer,” in Advanced Etch Technology for Nanopatterning II, 8685 (SPIE, 2013), p. 868505.

22. H. Gross, M.-A. Henn, S. Heidenreich, A. Rathsfeld, and M. Bär, “Modeling of line roughness and its impact on the diffraction intensities and the reconstructed critical dimensions in scatterometry,” Appl. Opt. 51, 7384–7394 (2012). [CrossRef]   [PubMed]  

23. J. Bischoff and K. Hehl, “Scatterometry modeling for gratings with roughness and irregularities,” in Metrology, Inspection, and Process Control for Microlithography XXX, 9778 (SPIE, 2016), p. 977804.

24. H. Gross, S. Heidenreich, and M. Bär, “Impact of different stochastic line edge roughness patterns on measurements in scatterometry-a simulation study,” Measurement 98, 339–346 (2017). [CrossRef]  

25. M. H. Madsen and P.-E. Hansen, “Scatterometry: A fast and robust measurements of nano-textured surfaces,” Surf. Topogr. Metrol. Prop. 4, 023003 (2016). [CrossRef]  

26. B. Harnisch, A. Deep, R. Vink, and C. Coatantiec, “Grating scattering brdf and imaging performances: A test survey performed in the frame of the flex mission,” in ICSO 2012, 10564 (SPIE, 2017), p. 105642P.

27. U. D. Zeitner, M. Oliva, F. Fuchs, D. Michaelis, T. Benkenstein, T. Harzendorf, and E.-B. Kley, “High performance diffraction gratings made by e-beam lithography,” Appl. Phys. A 109, 789–796 (2012). [CrossRef]  

28. M. Moharam, E. B. Grann, D. A. Pommet, and T. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” JOSA A 12, 1068–1076 (1995). [CrossRef]  

29. C. A. Mack, “Analytical expression for impact of linewidth roughness on critical dimension uniformity,” J. Micro/Nanolithography, MEMS MOEMS 13, 020501 (2014). [CrossRef]  

30. V. Constantoudis, G. Patsis, A. Tserepi, and E. Gogolides, “Quantification of line-edge roughness of photoresists. ii. scaling and fractal analysis and the best roughness descriptors,” J. Vac. Sci. & Technol. B: Microelectron. Nanometer Struct. Process. Meas. Phenom. 21, 1019–1026 (2003). [CrossRef]  

31. T. Verduin, P. Kruit, and C. W. Hagen, “Determination of line edge roughness in low-dose top-down scanning electron microscopy images,” J. Micro/Nanolithography, MEMS MOEMS 13, 033009 (2014). [CrossRef]  

32. A. R. Pawloski, A. Acheta, S. Bell, B. La Fontaine, T. Wallow, and H. J. Levinson, “The transfer of photoresist ler through etch,” in Advances in Resist Technology and Processing XXIII, 6153 (SPIE, 2006), p. 615318.

33. T. H. Naylor, J. L. Balintfy, D. S. Burdick, and K. Chu, “Computer simulation techniques,” Tech. rep., Wiley (1966).

34. M. Sharpe and D. Irish, “Straylight in diffraction grating monochromators,” Opt. acta 25, 861–893 (1978). [CrossRef]  

35. E. Marx, T. A. Germer, T. V. Vorburger, and B. C. Park, “Angular distribution of light scattered from a sinusoidal grating,” Appl. Opt. 39, 4473–4485 (2000). [CrossRef]  

36. H. Rigneault, F. Lemarchand, and A. Sentenac, “Dipole radiation into grating structures,” JOSA A 17, 1048–1058 (2000). [CrossRef]   [PubMed]  

37. A. Hessel and A. Oliner, “A new theory of wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275–1297 (1965). [CrossRef]  

38. D. A. Pommet, M. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” JOSA A 11, 1827–1834 (1994). [CrossRef]  

39. E. N. Glytsis, “Two-dimensionally-periodic diffractive optical elements: limitations of scalar analysis,” JOSA A 19, 702–715 (2002). [CrossRef]   [PubMed]  

40. J. C. Stover, Optical Scattering: Measurement and Analysis (SPIE Optical Engineering, 1995). [CrossRef]  

41. A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” JOSA A 28, 1121–1138 (2011). [CrossRef]   [PubMed]