Evaluation and improvement of simplified modal method for designing dielectric gratings

Fan Yang; Yanfeng Li

doi:10.1364/OE.23.031342

1. Introduction

Gratings, one of the most important optical devices, are continually finding important uses as high-contrast broadband reflectors [1, 2 ], integrated optoelectronic components [3], compressors in chirped pulse amplification [4], beam splitters [5], to name only a few. Although the simple diffraction formula gives the diffraction orders, full-vector methods are typically needed to calculate the diffraction efficiency of each order [6]. Among them, rigorous coupled wave analysis (RCWA) is widely used for the diffraction problem [7, 8 ]. In RCWA, the field is expanded by the Fourier basis, and the Maxwell equation is mapped into a matrix eigenvalue equation. However, RCWA is a numerical method which often hides the intuitive diffraction process and hence the complicated design process is not solved. In recent years, based on the original modal method [9], a physically intuitive and simpler method, i.e., the simplified modal method (SMM), has emerged [10, 11 ] and greatly simplified grating design [5, 12–14 ]. For gratings with a small period, there exist only a few propagating grating modes, two in most cases studied, and the diffraction process is attributed to the interference of these two modes, which can be understood in analogy to the well-known Mach-Zehnder interferometer [10, 11 ]. By considering the reflection of grating modes at interfaces in a fashion like those in a Fabry-Perot resonator, a more accurate multi-reflection method has been formulated to improve the accuracy [15–17 ].

However, there are two aspects which have not been fully addressed in the SMM, as far as we know. One is grating mode coupling [18], which in some cases is weak and thus can be neglected, but it can be considerable in other cases, especially for high-contrast gratings, as Karagodsky et al. have shown [1, 2 ]. Therefore, a quantitative study of mode coupling in the SMM and an improved model with mode coupling considered is needed. The other is the influence of evanescent modes, which are typically neglected in the SMM. To obtain accurate diffraction efficiencies, evanescent modes are needed in certain situations as we will show below, and so the analysis of evanescent modes is equally important.

In this paper, we provide an improved matrix formalism of the SMM for calculating the diffraction efficiency of dielectric rectangular grating. Different from a previous matrix formalism [1] where only the 0th diffraction order exists in the subwavelength gratings, the present model is formulated for an arbitrary incidence angle, and the mode numbers in both the incidence and the grating regions can be different. Also, by considering the scattering matrix at a single interface [19], the mode coupling and reflection coefficients can be obtained analytically. A former scattering matrix description is based on RCWA [20] and the transmission and reflection coefficients are not expressed analytically. With our improved method, the mode coupling in the two mode case is investigated, and it is shown that mode coupling is exactly zero at the Littrow incidence angle and non-zero for other angles and that the reflection coefficients reduce to the Fresnel’s form in this case. We also propose a parameter for measuring the boundary condition mismatch, which is effective for determining the number of evanescent modes that should be considered in the model.

2. Formulation of the method

In this section, a detailed formulation of the method is described. The grating structure is shown in Fig. 1 . For simplicity, the grating is assumed to be invariant in the y-direction and periodic in the x-direction with a rectangular profile. The blue bar denotes the dielectric material with a refractive index n ₂, the surrounding medium has a refractive index of n ₁ (air in this paper with n ₁ = 1), and the green region is the substrate with a refractive index n ₃. The incidence, grating and substrate regions are defined as regions I ( $z > 0$ ), II ( $- h < z < 0$ ), and III ( $z < - h$ ), respectively, with the grating thickness given by h. The notations Λ and f denote the period and the fill factor of the grating, respectively. In this article, we only consider the TE polarization incidence (electric field parallel to the y-direction) with the incidence angle given by θ. The formulation of TM polarization case is similar and will be discussed in a separate work.

Fig. 1 Schematic illustration of the grating structure and illumination parameters.

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The grating-air interface is first considered, as shown in Fig. 2(a) . The modal method tackles the diffraction problem with the field in the input region (I) expanded in a Fourier basis and with the field in the grating region (II) expanded in the basis of the grating modes [9]. Thus, we first express the scattered field with unknown coefficients of the modes, and then derive the scattering matrix by matching the boundary conditions. With M (M = M ₁ + M ₂ + 1) modes in region I, the electric field E_y _,I in this region is expressed as a superposition of the up-going modes and down-going modes, i.e.,

E_{y, I} (x, z) = \sum_{m = - M_{1}}^{M_{2}} u_{m}^{(I)} E_{m}^{+} (x) \exp (i k_{0} n_{p, m} z) + d_{m}^{(I)} E_{m}^{-} (x) \exp (- i k_{0} n_{p, m} z),

where

E_{m}^{+} (x) = e x p [i (k_{0} n_{1} \sin θ + 2 m π / Λ) x]

is the electric component of the m-th order up-going diffracted mode,

E_{m}^{-} (x)

is the down-going mode with

E_{m}^{-} (x) = E_{m}^{+} (x)

, and n_p _, _m is the effective index of the mode given by

n_{p, m} = {[n_{1}^{2} - {(n_{1} \sin θ + 2 π m / Λ)}^{2}]}^{1 / 2}

(for large m, the imaginary part of n_p _, _m is taken to be positive). The amplitude coefficients of the m-th up-going and down-going modes are denoted by

u_{m}^{(I)}

and

d_{m}^{(I)}

respectively. To simplify the formulation, we now introduce the matrix representation. The electric fields of the modes are represented as row vectors

E^{\pm} (x) = [E_{M_{2}}^{\pm} (x) E_{M_{2} - 1}^{\pm} (x) \dots E_{m}^{\pm} (x) \dots E_{- M_{1}}^{\pm} (x)]

, the propagation terms are represented as diagonal matrices

Φ_{\pm}^{(I)} (z)

with

{[Φ_{\pm}^{(I)} (z)]}_{m m} = \exp (\pm i k_{0} n_{p, M_{2} - m + 1} z)

, and the amplitude coefficients of the up-going and down-going modes at the surface

z = 0

are expressed as column vectors U ^(I) and D ^(I), respectively. The matrix form of Eq. (1) is then given by

Fig. 2 Scattering events at (a) grating-air interface and (b) grating-dielectric interface.The red arrows illustrate the input of the scattering matrix while the blue arrows illustrate the output of the scattering matrix.

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E_{y, I} (x, z) = E^{+} (x) Φ_{+}^{(I)} (z) U^{(I)} + E^{-} (x) Φ_{-}^{(I)} (z) D^{(I)} .

The magnetic field H_x _,I in region I can be written in a similar way, i.e.,

H_{x, I} (x, z) = H^{+} (x) Φ_{+}^{(I)} (z) U^{(I)} + H^{-} (x) Φ_{-}^{(I)} (z) D^{(I)},

where

H^{\pm} (x)

are row vectors with

{(H^{\pm})}_{m} = H_{m}^{\pm} (x)

[

H_{m}^{\pm} (x)

is the tangential magnetic component of the m-th order mode]. With the relation

H_{m}^{\pm} (x) = \mp n_{p, m} E_{m}^{\pm} (x)

obtained from Maxwell’s equations, Eq. (2b) can be further written as

H_{x, I} (x, z) = - E^{+} (x) N^{(I)} Φ_{+}^{(I)} (z) U^{(I)} + E^{-} (x) N^{(I)} Φ_{-}^{(I)} (z) D^{(I)},

where $N^{(I)}$ is a diagonal matrix with its m-th diagonal element being

n_{p, M_{2} - m + 1}

.

In region II, we use the grating modes to expand the scattered field. The effective index and the field components of the grating modes can be obtained analytically by solving the dispersion equation [11]. Similar to region I, with N grating modes considered, the electric field E_y _,II and the tangential magnetic field H_x _,II in region II are expressed as

E_{y, II} (x, z) = e^{+} (x) Φ_{+}^{(II)} (z) U^{(II)} + e^{-} (x) Φ_{-}^{(II)} (z) D^{(II)},

H_{x, II} (x, z) = h^{+} (x) Φ_{+}^{(II)} (z) U^{(II)} + h^{-} (x) Φ_{-}^{(II)} (z) D^{(II)},

where the n-th elements of the

1 \times N

row vectors

e^{\pm} (x)

and

h^{\pm} (x)

are the electric field component

e_{n}^{\pm} (x)

[we set

e_{n}^{+} (x) = e_{n}^{-} (x)

] and tangential magnetic field component

h_{n}^{\pm} (x)

of the n-th grating mode, respectively. The n-th diagonal elements of the diagonal propagation matrices

Φ_{\pm}^{(I I)} (z)

are

\exp (\pm i k_{0} n_{eff, n} z)

. U ^(II) and D ^(II) are the amplitude coefficients of the n-th grating mode at the interface

z = 0

. With

h_{n}^{\pm} (x) = \mp n_{e f f, n} e_{n}^{\pm} (x)

, Eq. (4b) is further expressed as

H_{x, II} (x, z) = - e^{+} (x) N^{(II)} Φ_{+}^{(II)} (z) U^{(II)} + e^{-} (x) N^{(II)} Φ_{-}^{(II)} (z) D^{(II)},

where

N^{(II)}

is an

N \times N

type diagonal matrix with N ^(II) _nn = n _eff, _n.

To match the boundary conditions, we need to consider the transformation between the diffracted modes in region I and the grating modes in region II. The grating modes can be linearly superposed by the diffracted modes, i.e.,

e^{+} (x) = E^{+} (x) W,

where W is an

M \times N

matrix given by

W_{m n} = \frac{1}{Λ} \int_{0}^{Λ} {[E_{M_{2} - m + 1}^{+} (x)]}^{*} e_{n}^{+} (x) d x,

provided that the modes are respectively normalized as

(1 / Λ) \int_{0}^{Λ} | E_{m}^{+} |^{2} d x = 1

and

(1 / Λ) \int_{0}^{Λ} | e_{n}^{+} |^{2} d x = 1

. Likewise, the diffracted modes can be expanded with the grating modes,i.e., E ⁺(x) = e ⁺(x)W^†, where ‘†’ stands for Hermitian conjugate. The tangential field components E_y and H_x are continuous at the interface

z = 0

, and so we obtain

E^{+} (x) (U^{(I)} + D^{(I)}) = E^{+} (x) (W U^{(I)} + W D^{(I)}),

- e^{+} (x) (W^{†} N^{(I)} U^{(I)} - W^{+} N^{(I)} D^{(I)}) = - e^{+} (x) (N^{(II)} U^{(II)} - N^{(II)} D^{(II)}),

where we have used the relation E ⁺(x) = E ⁻(x) and e ⁺(x) = e ⁻(x), and considered the transformation between the diffracted modes and the grating modes. Equation (8) then yields the matrix equation

U^{(I)} + D^{(I)} - W U^{(I)} - W D^{(I)} = 0,

W^{†} N^{(I)} U^{(I)} - W^{†} N^{(I)} D^{(I)} - N^{(II)} U^{(II)} + N^{(II)} D^{(II)} = 0.

With Eq. (9) , we finally obtain the scattering matrix S ^(I,II) linking the inputs U ^(II) and D ^(I) and the outputs U ^(I) and D ^(II) _,

[\begin{matrix} U^{(I)} \\ D^{(II)} \end{matrix}] = S^{(I,II)} \times [\begin{matrix} U^{(II)} \\ D^{(I)} \end{matrix}],

with

S^{(I,II)} = {[\begin{matrix} I_{M} & - W \\ W^{†} N^{(I)} & N^{(II)} \end{matrix}]}^{- 1} \times [\begin{matrix} W & - I_{M} \\ N^{(II)} & W^{†} N^{(I)} \end{matrix}] = [\begin{matrix} t_{u u} & r_{u d} \\ r_{d u} & t_{d d} \end{matrix}],

where I _M is an

M \times M

identity matrix, t _uu is an

M \times N

matrix representing the transmission from the grating modes to the diffracted modes, t _dd is an

N \times M

matrix representing the transmission from the diffracted modes to the grating modes, r _ud is an

M \times M

matrix representing the reflection and coupling of the diffracted modes, and r _du is an

N \times N

type matrix representing the reflection and coupling of the grating modes.

The analysis at the grating-substrate interface $z = - h$ shown in Fig. 2(b) is similar. The number of modes in region III is M, the same as in region I. Here we only give the final results,

[\begin{matrix} D^{(III)} \\ U_{s u b}^{(II)} \end{matrix}] = S^{(II,III)} \times [\begin{matrix} D_{s u b}^{(II)} \\ U^{(III)} \end{matrix}],

with

S^{(II,III)} = {[\begin{matrix} I_{M} & - W \\ W^{†} N^{(III)} & N^{(II)} \end{matrix}]}^{- 1} \times [\begin{matrix} W & - I_{M} \\ N^{(II)} & W^{†} N^{(III)} \end{matrix}] = [\begin{matrix} {t^{'}}_{d d} & {r^{'}}_{d u} \\ {r^{'}}_{u d} & {t^{'}}_{u u} \end{matrix}],

where $N^{(III)}$ is an M-th order diagonal matrix with the diagonal element being

{n^{'}}_{p, m} =

{[n_{3}^{2} - {(n_{1} \sin θ + 2 π m / Λ)}^{2}]}^{1 / 2}

(m = M ₂, M ₂−1, …, −M ₁), t '_uu is similar to t _dd, t '_dd is similar to t _uu, r '_ud is similar to r _du, and r '_du is similar to r _ud. The subscript sub represents fields in region II at the grating-substrate interface.

Now we can write a set of coupled equations to determine the unknown coefficients, that is,

D^{(II)} = t_{d d} D^{(I)} + r_{d u} Φ U_{s u b}^{(II)},

U_{s u b}^{(II)} = {r^{'}}_{u d} Φ D^{(II)},

where

Φ = Φ_{+}^{(I I)} (h)

, and use is made of the relation

U^{(I I)} = Φ U_{s u b}^{(I I)}

and

D_{s u b}^{(I I)} = Φ D^{(I I)}

. Equation (12a) implies that the down-going grating modes are contributions of transmission of the incidence field and the reflection of the up-going grating modes. Equation (12b) is written in view of the fact that the up-going grating modes are excited by the reflection of the down- going grating modes. Solving Eq. (12) , we obtain the amplitude coefficients of the grating modes

D^{(II)} = {(I - r_{d u} Φ {r^{'}}_{u d} Φ)}^{- 1} t_{d d} D^{(I)},

U_{s u b}^{(II)} = {r^{'}}_{u d} Φ {(I - r_{d u} Φ {r^{'}}_{u d} Φ)}^{- 1} t_{d d} D^{(I)} .

Finally, we can determine the transmission and reflection column vectors T and R of the gratings as (with $U^{(III)} = 0$ )

T = D^{(III)} = {t^{'}}_{d d} Φ D^{(II)},

R = U^{(I)} = r_{u d} D^{(I)} + t_{u u} Φ U_{s u b}^{(II)} .

Usually, we take the same number of modes in the grating (region II) and outside the grating (region I), namely, $M = N$ . In this case, the elements of the scattering matrix S ^(I,II) can be obtained directly

t_{u u} = 2 {(L^{(II)} W^{- 1} + W^{- 1} N^{(I)})}^{- 1} N^{(II)},

r_{u d} = {(N^{(II)} W^{- 1} + W^{- 1} N^{(I)})}^{- 1} (- N^{(II)} W^{- 1} + W^{- 1} N^{(I)}),

r_{d u} = {(N^{(I)} W + W N^{(II)})}^{- 1} (- N^{(I)} W + W N^{(II)}),

t_{d d} = 2 {(N^{(I)} W + W N^{(II)})}^{- 1} N^{(I)},

where we have used the unitarity of the matrix W (W^† = W ⁻¹). The matrix S ^(II,III) can be obtained by simply substituting N ^(III) for N ^(II) in Eq. (15) .

With the above formulation, the scattering matrix at the grating-air (or substrate) interface and a general matrix solution of the diffraction problem for TE polarization is obtained. In general, the method can takes different numbers of modes inside and outside the gratings. For the special case $M = N$ , the scattering matrices at the interfaces can be expressed more explicitly.

2. Improved simplified modal method

In this section, we discuss grating diffraction considering only two grating modes (N = 2, in decreasing order of the value of the effective index) and two diffracted modes (0th and −1st diffracted orders, M = 2, M ₁ = 1, M ₂ = 0). This is the case in which the SMM has been mostly applied.

The diffraction problem in this case can be understood in analogy to a Mach-Zehnder interferometer shown in Fig. 3 . In [11], neglecting the reflection of the grating modes at the interface, the high transmission efficiency of the low-contrast grating (the difference between n ₁ and n ₂ is small) is explained by considering the interference between the two down-going grating modes under the Littrow angle [ $θ = arc \sin (λ /2 n_{1} Λ)$ ]. In [15–17 ], adding the reflection of the grating modes with Fresnel’s coefficients and neglecting the coupling of the grating modes at the interfaces, the multi-reflection modal method is shown to be more accurate under the Littrow angle.

Fig. 3 Schematic illustration of two mode interference in SMM in the grating (left) and its analogy to a Mach-Zehnder interferometer (right).

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Here, with the method given in Section 1, we propose an improved modal method and show its advantages as compared with the previous formalisms. As illustrated in Fig. 3, we consider the reflection as well as the coupling of the grating modes at both interfaces as the main improvement in our method. The transmittance and the reflectivity of the grating are calculated using Eqs. (13) and (14) ]. It should be pointed out that for the cases considered in this work, the matrices are so small that the computation does not take any noticeable extra time.

For simplicity, we analyze the grating-air interface. The reflection and the coupling of the two grating modes are included in the scattering matrix S ^(I,II) element r _du = ρ. The diagonal elements of the matrix ρ represent the reflection and non-diagonal elements represent the coupling. With Eq. (15c), the elements of ρ are simply given by

ρ_{11} = \frac{w_{11} w_{22} (n_{eff, 1} - n_{p, 0}) (n_{eff, 2} + n_{p, - 1}) - w_{12} w_{21} (n_{eff, 1} - n_{p, - 1}) (n_{eff, 2} + n_{p, 0})}{w_{11} w_{22} (n_{eff, 1} + n_{p, 0}) (n_{eff, 2} + n_{p, - 1}) - w_{12} w_{21} (n_{eff, 1} + n_{p, - 1}) (n_{eff, 2} + n_{p, 0})},

ρ_{12} = \frac{2 w_{12} w_{22} n_{eff, 2} (n_{p, - 1} - n_{p, 0})}{w_{11} w_{22} (n_{eff, 1} + n_{p, 0}) (n_{eff, 2} + n_{p, - 1}) - w_{12} w_{21} (n_{eff, 1} + n_{p, - 1}) (n_{eff, 2} + n_{p, 0})},

ρ_{21} = \frac{2 w_{11} w_{21} n_{eff, 1} (n_{p, 0} - n_{p, - 1})}{w_{11} w_{22} (n_{eff, 1} + n_{p, 0}) (n_{eff, 2} + n_{p, - 1}) - w_{12} w_{21} (n_{eff, 1} + n_{p, - 1}) (n_{eff, 2} + n_{p, 0})},

ρ_{22} = \frac{w_{11} w_{22} (n_{eff, 1} + n_{p, 0}) (n_{eff, 2} - n_{p, - 1}) - w_{12} w_{21} (n_{eff, 1} + n_{p, - 1}) (n_{eff, 2} - n_{p, 0})}{w_{11} w_{22} (n_{eff, 1} + n_{p, 0}) (n_{eff, 2} + n_{p, - 1}) - w_{12} w_{21} (n_{eff, 1} + n_{p, - 1}) (n_{eff, 2} + n_{p, 0})},

where n _eff, _i is the effective index of the i-th grating mode, n_p _,0 is the effective index of the 0th order diffracted mode in air, and n_p _,-1 is effective index of the −1st diffraction order in air. Equation (16) can be understood as the generalized Fresnel’s reflection coefficients of the grating modes at the grating-air interface.

For Littrow mounting, there is n_p _,0 = n_p _,-1 = n ₁cosθ, and Eq. (16) reduces to

ρ_{11} = \frac{n_{eff, 1} - n_{1} \cos θ}{n_{eff, 1} + n_{1} \cos θ}, ρ_{12} = 0, ρ_{21} = 0, ρ_{22} = \frac{n_{eff, 2} - n_{1} \cos θ}{n_{eff, 2} + n_{1} \cos θ} .

Interestingly, we observe that the coupling between the two modes is exactly zero and the reflection coefficients reduce to the Fresnel’s form. Previous multi-reflection models take the reflection and transmission coefficients as the Fresnel’s form [15–17 ] by a direct analogy to the case of a flat interface. Here our exact results provide a quantitative proof of this analogy for TE polarization and Littrow angle (the transmission coefficients are given in the appendix). Furthermore, the above results allow us to conclude that the coupling of the grating modes is zero only under Littrow angle and may not be negligible for other incidence angles.

Figure 4 proves the above analysis, where a grating with Λ = 0.7λ, f = 0.5, n ₁ = 1, n ₂ = n ₃ = 1.45 is considered. The figure shows the zero order transmission efficiency as a function of the normalized groove depth h/λ, in which the RCWA results are considered accurate. In Fig. 4(a), the incidence angle is given by the Littrow angle, and we find that the results of the multi-reflection model and our improved model both have high accuracy compared with RCWA. However, for incidence angles far away from the Littrow angle, the multi-reflection model is not accurate enough in predicting the exact diffraction efficiency, while the improved model considering mode coupling still retains its high accuracy, as demonstrated in Figs. 4(b) and 4(c). Both a large (80°) and a small (20°) incidence angle have been considered in the figure, where the multi-reflection model overestimates the diffraction efficiency for the 80° case and even fails to yield the correct profile for the 20° case. The mode coupling at these angles is by no means negligible as will be shown below, so that the improved model is needed to give the more accurate results.

Fig. 4 Zeroth order transmittance versus normalized groove depth (h/λ). The grating parameters are Λ = 0.7λ, f = 0.5, n ₁ = 1, n ₂ = n ₃ = 1.45. The results are obtained with RCWA (blue circles), multi-reflection model (black dashed curves), and our improved model (red solid curves). The incidence angles are (a) 45.58° (Littrow angle), (b) 80° and (c) 20°, respectively.

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In Figs. 5(a) and 5(b) , we compare the improved model with the multi-reflection method more thoroughly [the parameters are the same as in Fig. 4 except that h is fixed at 0.9λ]. Compared with accurate RCWA results in terms of 0th order transmittance in Fig. 5(a), the improved model is shown to be valid at any incidence angle, while the multi-reflection method becomes less accurate away from the Littrow angle. In Fig. 5(b), the relative errorwith respect to RCWA results for the improved model and multi-reflection method is provided. The relative error of the improved model remains small (<1%) for all angles, while the multi-reflection method has a high relative error when deviating from the Littrow angle with the highest value being 14.8% and the mean value being 6.37%. Figures 5(c) and 5(d) shows the reflection and coupling of the grating modes. In Fig. 5(c), we plot the moduli of the non-diagonal elements of ρ under different incidence angles. The results indicate that mode coupling is zero under the Littrow angle and is considerable for large or small incidence angles, which explains the results in Figs. 5(a) and 5(b) well. From the Littrow angle to a smaller angle, there are two abrupt changes indicated by the vertical black dashed lines. The first one is caused by the change of the −1st diffracted order from being propagating to evanescent, i.e., n_p _,1 becomes an imaginary number. The second one is caused by the change of one grating mode from a propagating mode to an evanescent mode, i.e., n _eff,2 becomes an imaginary number. In Fig. 5(d), the four elements of ρ are plotted for different dielectric refractive indices (n ₃ = n ₂ and θ = 60°). Similar to the reflection at a flat interface, the reflection coefficients |ρ₁₁| and |ρ₂₂| increase as n ₂ increases. However, the increase of the coupling coefficients |ρ₁₂| and |ρ₂₁| is rather small.

Fig. 5 Comparison of accuracy of the improved model and multi-reflection method. (a) 0th order transmittance versus incidence angle, obtained with RCWA, the improved model and multi-reflection method. (b) Relative error shown in log scale between accurate RCWA results for the improved model and multi-reflection method. Reflection and coupling coefficients of grating modes at the grating-air interface. (c) Coupling coefficients ρ₂₁ and ρ₁₂ versus incidence angle θ. (d) Reflection ρ₁₁, ρ₂₂ and coupling coefficients ρ_21, ρ₁₂ versus dielectric refractive index n ₂ (n ₃ = n ₂) for a fixed incidence angle (θ = 60°). All data are for the same grating structure as in Fig. 4 except that h is fixed at 0.9λ.

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Based on the improved model, we can have a re-formulated multi-reflection model for arbitrary incidence angles. In [15–17 ], the reflection and transmission coefficients are separated into two groups corresponding to the 0th and −1st diffracted orders. Here, we provide a simpler model to consider just one group of reflection and transmission coefficients. We neglect the mode coupling and take the generalized Fresnel’s coefficients ρ_mm, ρ'_mm, t _out _,nm, t' _out _,nm, t _in, _m, r_n, [m, n = 1, 2, see Eq. (16) and Eqs. (23)-(30) in the Appendix] and then the transmission and reflection coefficients of the grating are simply given by the Fabry-Perot form

T_{n} = \frac{{t^{'}}_{out, n 1} t_{in, 1} \exp (i k_{0} n_{eff,1} h)}{1 - ρ_{11} {ρ^{'}}_{11} \exp (2 i k_{0} n_{eff,1} h)} + \frac{{t^{'}}_{out, n 2} t_{in, 2} \exp (i k_{0} n_{eff,2} h)}{1 - ρ_{22} {ρ^{'}}_{22} \exp (2 i k_{0} n_{eff,2} h)},

R_{n} = r_{n} + \frac{t_{out, n 1} t_{in, 1} {ρ^{'}}_{11} \exp (2 i k_{0} n_{eff,1} h)}{1 - ρ_{11} {ρ^{'}}_{11} \exp (2 i k_{0} n_{eff,1} h)} + \frac{t_{out, n 2} t_{in, 2} {ρ^{'}}_{22} \exp (2 i k_{0} n_{eff,2} h)}{1 - ρ_{22} {ρ^{'}}_{22} \exp (2 i k_{0} n_{eff,2} h)} .

For angles near the Littrow situation, the mode coupling is weak, and Eq. (18) provides a simpler but more approximate expression of the transmission and reflection coefficients. Figure 6 show the numerical results of the −1st order transmittance with Λ = 0.55λ, n ₂ = n ₃ = 2.5, f = 0.3. In Fig. 6(a) where the incidence angle is the Littrow angle (65.38°), the improved model and the improved multi-reflection model [Eq. (18) ] are equivalent. For θ near the Littrow angle, Eq. (18) is accurate enough to predict the diffraction efficiency as shown in Figs. 6(b) and 6(c) (θ = 70° and θ = 60° respectively). For θ far away from the Littrow angle, the results of Eq. (18) may be not accurate at the peak positions (see Fig. 6(d) with θ = 40°) but the trend of the curve is still reproduced well enough, and the improved model considering mode coupling is needed.

Fig. 6 Transmission efficiency of −1st order versus normalized groove depth. The grating structure parameters are Λ = 0.55λ, n ₂ = n ₃ = 2.5, f = 0.3. The blue circles, red curves, and black dashed curves denote the RCWA results, the improved model prediction, and the results of the improved multi-reflection model [Eq. (18) ], respectively. The incidence angles are (a) 65.38° (Littrow angle), (b) 70°, (c) 60°, and (d) 40°, respectively.

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3. Discussion of modes numbers and boundary condition mismatch

In this section, we discuss the influence of mode numbers in the SMM, the essence of which is to use a small number of modes to achieve highly accurate results in a more or less analytical manner. In most cases, two or three modes would suffice, only the propagating modes are taken into account and the evanescent modes are neglected. However, there is no parameter to quantify the degree of accuracy of the approximation using a smaller number of modes and the contribution of the evanescent modes. Here we propose a parameter as such a criterion by considering the boundary mismatch at the grating interface.

In the formulation in Section 1, we use a finite number of modes to approximately match the boundary conditions, and thus the accuracy of the method is dependent on the mismatch of the boundary conditions. Let us assume that the grating is illuminated by a normalized TE polarized incidence field $E_{y, inc} = \exp [i k_{0} n_{1} (x \sin θ - z \cos θ)] / Λ^{1 / 2}$ . When the unknown coefficients of the modes are obtained, the boundary condition of the tangential electric field is approximately satisfied,

E^{+} (x) A \approx e^{+} (x) B,

where A = [A ₁ … A_M ]^T and B = [B ₁ … B_N ]^T, corresponding to the amplitude coefficients of the diffracted modes and the grating modes, respectively. To test the validity of the above approximation, we define a function ϕ(x),

ϕ (x) = E^{+} (x) A - e^{+} (x) B = \sum_{m = 1}^{M} A_{m} E_{M_{2} - m + 1}^{+} (x) - \sum_{n = 1}^{N} B_{n} e_{n}^{+} (x) .

The mismatch of the boundary condition can be estimated by the mean square modulus of ϕ(x), i.e.,

η = \frac{1}{Λ} \int_{0}^{d} ϕ^{*} ϕ d x = \sum_{m = 1}^{M} {| A_{m} |}^{2} + \sum_{n = 1}^{M} {| B_{n} |}^{2} - \sum_{m, n = 1}^{M, N} A_{m}^{*} w_{m n} B_{n} - \sum_{m, n = 1}^{M, N} B_{n}^{*} w_{m n}^{*} A_{m},

where we have used the normalization and orthogonal relation of the modes. With the matrix representation, we finally obtain the parameter η given by

η = A^{†} A + B^{†} B - A^{†} W A - B^{†} W^{†} A .

The mismatch exists at the two interfaces $z = 0$ and $z = - h$ , and exists for two field components E_y and H_x, and so the degree of mismatch is measured by the average of those four contributions, i.e., $η = \sum_{i = 1}^{4} η_{i} / 4$ . Generally speaking, as the parameter η measures the mismatch percentage at the interface, the smaller the parameter η is, the better the matching of the boundary conditions is. In some cases, using propagating modes only is not able to match the boundary conditions well (η is large), and we need to take some evanescent modes into consideration until η becomes small enough (our numerical results show that when η>0.1, the results are not satisfactory, and when η <<0.1, the results are sufficiently accurate).

Figure 7 shows a numerical verification of the above judgment [in Figs. 7(a1)-−7(b2), we set M ₁ = M/2, M ₂ = M/2−1 when M is even, and M ₁ = M ₂ = (M−1)/2 when M is odd]. In Figs. 7(a1) and 7(a2), the grating structure parameters are Λ = 0.6λ, f = 0.5, n ₂ = n ₃ = 2.5, and θ = 10°. Comparing the improved model with the RCWA, we find that two modes do not provide an accurate diffraction efficiency, although there are only two propagating grating modes (n _eff,1 = 2.24, n _eff,2 = 1.23). Adding the evanescent mode with the smallest imaginary effective index (n _eff,3 = 0.72i), the results agree well with the RCWA results. The parameter $\bar{η}$ shown in Fig. 7(a2) is effective to predict this result. For the two mode case, the boundary condition mismatch parameter η remains large (>0.1) with an average of 0.34 when the groove depth h is varied. As for three modes, $\bar{η}$ is small enough (<<0.1) with an average of $\bar{η}$ = 0.0063. This confirms the validity of the parameter $\bar{η}$ as a criterion for choosing mode numbers. Figures 7(b1) and (b2) show another example with Λ = 0.6λ, f = 0.5, n ₂ = n ₃ = 3.2, and θ = 56.44° (Littrow angle). In this case, there still exist two propagating modes (n _eff,1 = 2.95, n _eff,2 = 2.16), but two modes or three modes (adding an evanescent mode n _eff,3 = 1.15i) are not enough to match the boundary conditions well with a large $\bar{η}$ value ( $\bar{η}$ = 0.21 and $\bar{η}$ = 0.16, respectively), thus giving the inaccurate results. Taking four modes (adding another evanescent mode n _eff,4 = 1.27i) makes the matching of the boundary conditions good enough with $\bar{η}$ drastically reduced ( $\bar{η}$ = 0.0053), and the model is in high accuracy compared with RCWA. As shown in Figs. 7(c1) and 7(c2), the improved SMM model can also be applied to the general case where M≠N. Here, we consider a multi-mode case with 9 propagating modes in the grating [Λ = 2λ, f = 0.5, n ₂ = n ₃ = 3.2, θ = 30° (second Bragg angle)]. It is shown that the results of using different mode numbers (M = 9, N = 13) are more accurate than the case of using the same mode numbers (M = N = 9). Actually, there are 13 transmission orders in region III (−7th, −6th, …, 5th transmission orders), and so using different mode numbers is naturally a better choice. The parameter $\bar{η}$ indicates that the boundary condition matching is better for this case ( $\bar{η}$ = 0.052 compared with $\bar{η}$ = 0.21 for using the same mode numbers). This example also presents the advantage of our general formulation where the mode numbers inside and outside the grating may be different.

Fig. 7 Numerical verification of the function of the parameter η. In (a1) and (a2), the grating parameters are Λ = 0.6λ, f = 0.5, n ₂ = n ₃ = 2.5, θ = 10°. In (b1) and (b2), the parameters are Λ = 0.6λ, f = 0.5, n ₂ = n ₃ = 3.2, θ = 56.44° (Littrow angle). In (c1) and (c2), the parameters are Λ = 2λ, f = 0.5, n ₂ = n ₃ = 3.2, θ = 30° (second Bragg angle).

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4. Conclusion

To sum up, we have introduced an improved matrix formalism of the SMM to solve the diffraction problem for dielectric gratings illuminated by a TE polarized incidence field. Based on a scattering matrix at each interface, the mode reflection and coupling properties can be obtained analytically. The method generally considers different mode numbers in the incidence and grating regions. Applying the method to the two mode case, we find that the coupling of the grating modes only vanishes at the Littrow incidence angle. Also, due to the symmetry of the 0th and −1st diffraction orders, the reflection coefficients of the modes reduce to the classical Fresnel’s form in this case. For a general case, we propose an improved model considering both mode reflections and couplings, which shows significantly improved accuracy compared with previous formulations and agrees well with results by RCWA for a wide range of grating parameters. For incidence angles near Littrow mounting, a simplified method neglecting mode couplings is also presented with generalized Fresnel’s formulae. Based on the matrix description, a parameter η is defined as a quantitative criterion for measuring boundary condition mismatch and then used to determine the number of evanescent modes to be used in the method. Numerical results demonstrate that when η is smaller than 0.1, a sufficient number of modes have been considered. Otherwise, more evanescent modes are needed. Therefore, we believe that this improved SMM will further facilitate the fast and accurate design of goal-oriented diffraction gratings.

Appendix

In this appendix, the rest [ρ is given in the main text, Eq. (16) ] of the transmission and reflection coefficients of the two mode case are given.

First, we give the reflection and coupling coefficients at the grating-substrate interface. Similar to Eq. (16) , these coefficients are included in the matrix r '_ud= ρ', that is,

{ρ^{'}}_{11} = \frac{w_{11} w_{22} (n_{eff, 1} - {n^{'}}_{p, 0}) (n_{eff, 2} + {n^{'}}_{p, - 1}) - w_{12} w_{21} (n_{eff, 1} - {n^{'}}_{p, - 1}) (n_{eff, 2} + {n^{'}}_{p, 0})}{w_{11} w_{22} (n_{eff, 1} + {n^{'}}_{p, 0}) (n_{eff, 2} + {n^{'}}_{p, - 1}) - w_{12} w_{21} (n_{eff, 1} + {n^{'}}_{p, - 1}) (n_{eff, 2} + {n^{'}}_{p, 0})},

{ρ^{'}}_{12} = \frac{2 w_{12} w_{22} n_{eff, 2} ({n^{'}}_{p, - 1} - {n^{'}}_{p, 0})}{w_{11} w_{22} (n_{eff, 1} + {n^{'}}_{p, 0}) (n_{eff, 2} + {n^{'}}_{p, - 1}) - w_{12} w_{21} (n_{eff, 1} + {n^{'}}_{p, - 1}) (n_{eff, 2} + {n^{'}}_{p, 0})},

{ρ^{'}}_{21} = \frac{2 w_{11} w_{21} n_{eff, 1} ({n^{'}}_{p, 0} - {n^{'}}_{p, - 1})}{w_{11} w_{22} (n_{eff, 1} + {n^{'}}_{p, 0}) (n_{eff, 2} + {n^{'}}_{p, - 1}) - w_{12} w_{21} (n_{eff, 1} + {n^{'}}_{p, - 1}) (n_{eff, 2} + {n^{'}}_{p, 0})},

{ρ^{'}}_{22} = \frac{w_{11} w_{22} (n_{eff, 1} + {n^{'}}_{p, 0}) (n_{eff, 2} - {n^{'}}_{p, - 1}) - w_{12} w_{21} (n_{eff, 1} + {n^{'}}_{p, - 1}) (n_{eff, 2} - {n^{'}}_{p, 0})}{w_{11} w_{22} (n_{eff, 1} + {n^{'}}_{p, 0}) (n_{eff, 2} + {n^{'}}_{p, - 1}) - w_{12} w_{21} (n_{eff, 1} + {n^{'}}_{p, - 1}) (n_{eff, 2} + {n^{'}}_{p, 0})} .

At Littrow incidence angle, ${n^{'}}_{p, 0} = {n^{'}}_{p, - 1} = {(n_{3}^{2} - n_{1}^{2} \sin^{2} θ)}^{1 / 2}$ , Eq. (23) reduces to

{ρ^{'}}_{11} = \frac{n_{eff, 1} - {n^{'}}_{p, 0}}{n_{eff, 1} + {n^{'}}_{p, 0}}, {ρ^{'}}_{12} = 0, {ρ^{'}}_{21} = 0, {ρ^{'}}_{22} = \frac{n_{eff, 2} - {n^{'}}_{p, 0}}{n_{eff, 2} + {n^{'}}_{p, 0}} .

Then, with Eqs. (15a), (15c) and (15d) , we can derive the transmission coefficients from the grating modes to the diffracted modes (t _out=t _uu and t ' _out=t '_dd), the reflection from the incidence to the diffracted orders [r_n=(r _ud)_n, ₁], and the transmission coefficients from the incidence to the grating modes [t _in, _n=(t _dd)_n, ₁], that is,

t_{out, 11} = 2 w_{11} (w_{11} w_{22} - w_{12} w_{21}) n_{eff, 1} (n_{eff, 2} + n_{p, - 1}) / g,

t_{out, 12} = 2 w_{12} (w_{11} w_{22} - w_{12} w_{21}) n_{eff, 2} (n_{eff, 1} + n_{p, - 1}) / g,

t_{out, 21} = 2 w_{21} (w_{11} w_{22} - w_{12} w_{21}) n_{eff, 1} (n_{eff, 2} + n_{p, 0}) / g,

t_{out, 22} = 2 w_{22} (w_{11} w_{22} - w_{12} w_{21}) n_{eff, 2} (n_{eff, 1} + n_{p, 0}) / g,

r_{1} = \frac{w_{11} w_{22}}{g} (n_{p, 0} - n_{eff, 1}) (n_{eff, 2} + n_{p, - 1}) - \frac{w_{12} w_{21}}{g} (n_{eff, 1} + n_{p, - 1}) (n_{p, 0} - n_{eff, 2}),

r_{2} = 2 w_{21} w_{22} n_{p, 0} (n_{eff, 1} - n_{eff, 2}) / g,

t_{in, 1} = 2 n_{p, 0} w_{22} (n_{eff, 2} + n_{p, - 1}) / g,

t_{in, 2} = - 2 n_{p, 0} w_{21} (n_{eff, 1} + n_{p, - 1}) / g .

where g = w ₁₁ w ₂₂(n _eff,1 + n_p _,0)(n _eff,2 + n_p _,-1)−w ₁₂ w ₂₁(n _eff,1 + n_p _,-1)(n _eff,2 + n_p _,0), and the elements of t ' _out can be obtained simply by substituting n'_p _, _m for n_p _, _m in Eq. (25) . When the incidence is at the Littrow angle, the transmission and the reflection coefficients reduce to a simpler Fresnel’s form (where the unitarity of the matrix W is also considered),

t_{out, 11} = \frac{2 w_{11} n_{eff, 1}}{n_{eff, 1} + n_{p, 0}},

t_{out, 12} = \frac{2 w_{12} n_{eff, 2}}{n_{eff, 2} + n_{p, 0}},

t_{out, 21} = \frac{2 w_{21} n_{eff, 1}}{n_{eff, 1} + n_{p, 0}},

t_{out, 22} = \frac{2 w_{22} n_{eff, 2}}{n_{eff, 2} + n_{p, 0}},

r_{1} = \frac{1}{2} [\frac{n_{p, 0} - n_{eff, 1}}{n_{p, 0} + n_{eff, 1}} + \frac{n_{p, 0} - n_{eff, 2}}{n_{p, 0} + n_{eff, 2}}],

r_{2} = \frac{w_{22}}{w_{12}} \frac{n_{eff, 1} - n_{eff, 2}}{(n_{p, 0} + n_{eff, 1}) (n_{p, 0} + n_{eff, 2})},

t_{in, 1} = \frac{1}{w_{11}} \frac{n_{p, 0}}{n_{eff, 1} + n_{p, 0}},

t_{in, 2} = \frac{1}{w_{12}} \frac{n_{p, 0}}{n_{eff, 2} + n_{p, 0}} .

Acknowledgments

F. Yang thanks Professor Haitao Liu of Nankai University, China for guidance and many helpful discussions. This work was supported in part by the National Basic Research Program of China (Grants 2014CB339800 and 2011CB808101), the National Natural Science Foundation of China (Grants 61377047, 61377041, 61322502, 61077083, and 61027013), and Program for Changjiang Scholars and Innovative Research Team in University (Grant IRT13033).

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Evaluation and improvement of simplified modal method for designing dielectric gratings

Abstract

1. Introduction

2. Formulation of the method

2. Improved simplified modal method

3. Discussion of modes numbers and boundary condition mismatch

4. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (7)

Equations (59)

Optics Express