Abstract

Recently, the simplified modal method (SMM) has proved to be very successful to facilitate grating design by reducing the diffraction problem to the interference (and reflection at interfaces) of a very small number of grating modes In this work, an intuitive and fully-analytical matrix formalism is developed to evaluate and improve the SMM. The present method focuses on the coupling between the grating modes and the influence of evanescent modes, which have not been touched on in detail in previous formulations of the SMM. In particular, we show that when there are only two grating modes, their coupling is exactly zero only for Littrow mounting and the reflection coefficients also reduce to the familiar Fresnel’s form as is commonly used by previous formulations. For other incidence angles, mode coupling can be significant, and our model shows greatly improved accuracy over the common SMM when compared with numerical results. A new parameter measuring the boundary condition mismatch and reflecting the accuracy of the method is proposed, which can serve as a criterion for choosing the number of evanescent modes in the model. The improved model will be of great value for grating designs.

© 2015 Optical Society of America

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References

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2014 (1)

2013 (1)

2012 (3)

2010 (1)

2008 (4)

2007 (2)

2006 (1)

2005 (2)

1996 (2)

1995 (1)

1981 (1)

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. Andrewartha, “The dielectric lamellar diffraction grating,” J. Mod. Opt. 28(3), 413–428 (1981).

Adams, J. L.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. Andrewartha, “The dielectric lamellar diffraction grating,” J. Mod. Opt. 28(3), 413–428 (1981).

Andrewartha, J.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. Andrewartha, “The dielectric lamellar diffraction grating,” J. Mod. Opt. 28(3), 413–428 (1981).

Botten, I. C.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. Andrewartha, “The dielectric lamellar diffraction grating,” J. Mod. Opt. 28(3), 413–428 (1981).

Cao, H.

Chang-Hasnain, C. J.

Chavel, P.

Clausnitzer, T.

Craig, M. S.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. Andrewartha, “The dielectric lamellar diffraction grating,” J. Mod. Opt. 28(3), 413–428 (1981).

Feng, J.

Gamet, E.

Gaylord, T. K.

Grann, E. B.

Hugonin, J. P.

Jin, S.

Jing, X.

Kämpfe, T.

Karagodsky, V.

Kley, E. B.

Kley, E.-B.

Lalanne, P.

Li, L.

Liang, P.

Lv, P.

McPhedran, R. C.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. Andrewartha, “The dielectric lamellar diffraction grating,” J. Mod. Opt. 28(3), 413–428 (1981).

Moharam, M. G.

Parriaux, O.

Peschel, U.

Pommet, D. A.

Sedgwick, F. G.

Sun, W.

Tian, Y.

Tishchenko, A.

Tishchenko, A. V.

Tünnermann, A.

Wang, B.

Wu, J.

Yang, W.

C. J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4(3), 379–440 (2012).
[Crossref]

Zhang, J.

Zheng, J.

Zhou, C.

Adv. Opt. Photonics (1)

C. J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4(3), 379–440 (2012).
[Crossref]

Appl. Opt. (5)

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. Andrewartha, “The dielectric lamellar diffraction grating,” J. Mod. Opt. 28(3), 413–428 (1981).

J. Opt. Soc. Am. A (4)

Opt. Express (5)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

A. V. Tishchenko, “Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method,” Opt. Quantum Electron. 37(1–3), 309–330 (2005).
[Crossref]

Other (1)

E. Popov, Gratings: Theory and Numeric Applications, 2nd ed. (Institut Fresnel, AMU, 2014).

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Figures (7)

Fig. 1
Fig. 1 Schematic illustration of the grating structure and illumination parameters.
Fig. 2
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.
Fig. 3
Fig. 3 Schematic illustration of two mode interference in SMM in the grating (left) and its analogy to a Mach-Zehnder interferometer (right).
Fig. 4
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.
Fig. 5
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 ρ21, 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.
Fig. 6
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.
Fig. 7
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).

Equations (59)

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E y , I ( x , z ) = 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 ) ,
E y , I ( x , z ) = E + ( x ) Φ + ( I ) ( z ) U ( I ) + E ( x ) Φ ( I ) ( z ) D ( I ) .
H x , I ( x , z ) = H + ( x ) Φ + ( I ) ( z ) U ( I ) + H ( x ) Φ ( I ) ( z ) D ( I ) ,
H x , I ( x , z ) = E + ( x ) N ( I ) Φ + ( I ) ( z ) U ( I ) + E ( x ) N ( I ) Φ ( I ) ( z ) D ( I ) ,
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 ) ,
H x , II ( x , z ) = e + ( x ) N ( II ) Φ + ( II ) ( z ) U ( II ) + e ( x ) N ( II ) Φ ( II ) ( z ) D ( II ) ,
e + ( x ) = E + ( x ) W ,
W m n = 1 Λ 0 Λ [ E M 2 m + 1 + ( x ) ] * e n + ( x ) d x ,
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 ) ) ,
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.
[ U ( I ) D ( II ) ] = S ( I,II ) × [ U ( II ) D ( I ) ] ,
S ( I,II ) = [ I M W W N ( I ) N ( II ) ] 1 × [ W I M N ( II ) W N ( I ) ] = [ t u u r u d r d u t d d ] ,
[ D ( III ) U s u b ( II ) ] = S ( II,III ) × [ D s u b ( II ) U ( III ) ] ,
S ( II,III ) = [ I M W W N ( III ) N ( II ) ] 1 × [ W I M N ( II ) W N ( III ) ] = [ t d d r d u r u d t u u ] ,
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 ) ,
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 ) .
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 ) .
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 ) ,
ρ 11 = 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 = 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 = 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 = 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 11 = n eff , 1 n 1 cos θ n eff , 1 + n 1 cos θ , ρ 12 = 0 , ρ 21 = 0 , ρ 22 = n eff , 2 n 1 cos θ n eff , 2 + n 1 cos θ .
T n = 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 ) + 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 + 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 ) + 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 ) .
E + ( x ) A e + ( x ) B ,
ϕ ( x ) = E + ( x ) A e + ( x ) B = m = 1 M A m E M 2 m + 1 + ( x ) n = 1 N B n e n + ( x ) .
η = 1 Λ 0 d ϕ * ϕ d x = m = 1 M | A m | 2 + n = 1 M | B n | 2 m , n = 1 M , N A m * w m n B n m , n = 1 M , N B n * w m n * A m ,
η = A A + B B A W A B W A .
ρ 11 = 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 = 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 = 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 = 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 11 = n eff , 1 n p , 0 n eff , 1 + n p , 0 , ρ 12 = 0 , ρ 21 = 0 , ρ 22 = n eff , 2 n p , 0 n eff , 2 + n p , 0 .
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 = w 11 w 22 g ( n p , 0 n eff , 1 ) ( n eff , 2 + n p , 1 ) 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 .
t out , 11 = 2 w 11 n eff , 1 n eff , 1 + n p , 0 ,
t out , 12 = 2 w 12 n eff , 2 n eff , 2 + n p , 0 ,
t out , 21 = 2 w 21 n eff , 1 n eff , 1 + n p , 0 ,
t out , 22 = 2 w 22 n eff , 2 n eff , 2 + n p , 0 ,
r 1 = 1 2 [ n p , 0 n eff , 1 n p , 0 + n eff , 1 + n p , 0 n eff , 2 n p , 0 + n eff , 2 ] ,
r 2 = w 22 w 12 n eff , 1 n eff , 2 ( n p , 0 + n eff , 1 ) ( n p , 0 + n eff , 2 ) ,
t in , 1 = 1 w 11 n p , 0 n eff , 1 + n p , 0 ,
t in , 2 = 1 w 12 n p , 0 n eff , 2 + n p , 0 .

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