Abstract

We present a method to increase the second-harmonic generation (SHG) by an antiferromagnetic (AF) film for a fixed input power. This film is put between two different dielectrics, and an obliquely incident electromagnetic wave is used to generate the second-harmonic (SH) waves. This method can be considered as an extension to the previous work by Lim [J. Opt. Soc. Am. B 19, 1401 (2002)] in which a single AF film and normally incident light were used. We find that the SH outputs depend sensitively on the incident angle and dielectric constants of the dielectrics. The numerical calculations based on the three examples containing an AF MnF2 film show that the SH outputs are raised to a few hundred times that of the single AF film.

© 2008 Optical Society of America

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  1. N. S. Almeida and D. L. Mills, “Nonlinear infrared response of antiferromagnets,” Phys. Rev. B 36, 2015-2023 (1987).
    [CrossRef]
  2. L. Kahn, N. S. Almeida, and D. L. Mills, “Nonlinear optical response of antiferromagnetic superlattices: multi-stability and soliton trains,” Phys. Rev. B 37, 8072-8081 (1988).
    [CrossRef]
  3. Z. Wu and Q. Wang, “Propagation properties of magnetic surface waves on the interface between two nonlinear antiferromagnets,” Acta Phys. Sin. 50, 1178-1183 (2001).
  4. Q. Wang, Y. F. Wang, and J. S. Bao, “Nonlinear TE surface waves on an antiferromagnet,” Acta Phys. Sin. 46, 0145-0151 (1997).
  5. A. D. Boardman, S. A. Nikitov, and N. A. Waby, “Existence of spin-wave solitons in an antiferromagnetic film,” Phys. Rev. B 48, 13602-13606 (1993).
    [CrossRef]
  6. S. C. Lim, J. Osman, and D. R. Tilley, “Calculation of nonlinear magnetic susceptibility tensors for an uniaxial antiferromagnet,” J. Phys. D: Appl. Phys. 33, 2899-2910 (2000).
    [CrossRef]
  7. M. Fiebig, D. Frohlich, B. B. Krichevtsov, and R. V. Pisarev, “Second harmonic generation and magnetic-dipole-electric-dipole interference in antiferromagnetic Cr2O3,” Phys. Rev. Lett. 73, 2127-2130 (1994).
    [CrossRef]
  8. K. Abraha and D. R. Tilley, “Theory of far infrared properties of magnetic surfaces, films and superlattices,” Surf. Sci. Rep. 24, 125-222 (1996).
    [CrossRef]
  9. X. Z. Wang and H. Li, “Nonlinear polaritons in antiferromagnetic/ nonmagnetic superlattices,” Phys. Rev. B 72, 054403 (2005).
    [CrossRef]
  10. J. Bai, S. Zhou, F. L. Liu, and X. Z. Wang, “Nonlinear infrared transmission through and reflection off antiferromagnetic films,” J. Phys.: Condens. Matter 19, 046217 (2007).
    [CrossRef]
  11. S. C. Lim, “Magnetic second harmonic generation of an antiferromagnetic film,” J. Opt. Soc. Am. B 19, 1401-1410 (2002).
    [CrossRef]
  12. S. C. Lim, “Second harmonic generation of magnetic and dielectric multilayers,” J. Phys.: Condens. Matter 18, 4329-4343 (2006).
    [CrossRef]
  13. M. Fiebig, D. Frohlich, T. Lottermoser, R. V. Pisarev, and H. J. Weber, “Second harmonic generation in the centrosymmetric antiferromagnet NiO,” Phys. Rev. Lett. 87, 137202 (2001).
    [CrossRef]
  14. M. Fiebig, V. V. Pavlov, and R. V. Pisarev, “Second-harmonic generation as a tool for studying electronic and magnetic structures of crystals,” J. Opt. Soc. Am. B 22, 96-118 (2005).
    [CrossRef]
  15. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984), pp. 86-107.
  16. N. S. Almeida and D. L. Mills, “Dynamical response of antiferromagnets in an oblique magnetic field: application to surface magnons,” Phys. Rev. B 37, 3400-3408 (1988).
    [CrossRef]

2007 (1)

J. Bai, S. Zhou, F. L. Liu, and X. Z. Wang, “Nonlinear infrared transmission through and reflection off antiferromagnetic films,” J. Phys.: Condens. Matter 19, 046217 (2007).
[CrossRef]

2006 (1)

S. C. Lim, “Second harmonic generation of magnetic and dielectric multilayers,” J. Phys.: Condens. Matter 18, 4329-4343 (2006).
[CrossRef]

2005 (2)

2002 (1)

2001 (2)

M. Fiebig, D. Frohlich, T. Lottermoser, R. V. Pisarev, and H. J. Weber, “Second harmonic generation in the centrosymmetric antiferromagnet NiO,” Phys. Rev. Lett. 87, 137202 (2001).
[CrossRef]

Z. Wu and Q. Wang, “Propagation properties of magnetic surface waves on the interface between two nonlinear antiferromagnets,” Acta Phys. Sin. 50, 1178-1183 (2001).

2000 (1)

S. C. Lim, J. Osman, and D. R. Tilley, “Calculation of nonlinear magnetic susceptibility tensors for an uniaxial antiferromagnet,” J. Phys. D: Appl. Phys. 33, 2899-2910 (2000).
[CrossRef]

1997 (1)

Q. Wang, Y. F. Wang, and J. S. Bao, “Nonlinear TE surface waves on an antiferromagnet,” Acta Phys. Sin. 46, 0145-0151 (1997).

1996 (1)

K. Abraha and D. R. Tilley, “Theory of far infrared properties of magnetic surfaces, films and superlattices,” Surf. Sci. Rep. 24, 125-222 (1996).
[CrossRef]

1994 (1)

M. Fiebig, D. Frohlich, B. B. Krichevtsov, and R. V. Pisarev, “Second harmonic generation and magnetic-dipole-electric-dipole interference in antiferromagnetic Cr2O3,” Phys. Rev. Lett. 73, 2127-2130 (1994).
[CrossRef]

1993 (1)

A. D. Boardman, S. A. Nikitov, and N. A. Waby, “Existence of spin-wave solitons in an antiferromagnetic film,” Phys. Rev. B 48, 13602-13606 (1993).
[CrossRef]

1988 (2)

L. Kahn, N. S. Almeida, and D. L. Mills, “Nonlinear optical response of antiferromagnetic superlattices: multi-stability and soliton trains,” Phys. Rev. B 37, 8072-8081 (1988).
[CrossRef]

N. S. Almeida and D. L. Mills, “Dynamical response of antiferromagnets in an oblique magnetic field: application to surface magnons,” Phys. Rev. B 37, 3400-3408 (1988).
[CrossRef]

1987 (1)

N. S. Almeida and D. L. Mills, “Nonlinear infrared response of antiferromagnets,” Phys. Rev. B 36, 2015-2023 (1987).
[CrossRef]

1984 (1)

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984), pp. 86-107.

Abraha, K.

K. Abraha and D. R. Tilley, “Theory of far infrared properties of magnetic surfaces, films and superlattices,” Surf. Sci. Rep. 24, 125-222 (1996).
[CrossRef]

Almeida, N. S.

N. S. Almeida and D. L. Mills, “Dynamical response of antiferromagnets in an oblique magnetic field: application to surface magnons,” Phys. Rev. B 37, 3400-3408 (1988).
[CrossRef]

L. Kahn, N. S. Almeida, and D. L. Mills, “Nonlinear optical response of antiferromagnetic superlattices: multi-stability and soliton trains,” Phys. Rev. B 37, 8072-8081 (1988).
[CrossRef]

N. S. Almeida and D. L. Mills, “Nonlinear infrared response of antiferromagnets,” Phys. Rev. B 36, 2015-2023 (1987).
[CrossRef]

Bai, J.

J. Bai, S. Zhou, F. L. Liu, and X. Z. Wang, “Nonlinear infrared transmission through and reflection off antiferromagnetic films,” J. Phys.: Condens. Matter 19, 046217 (2007).
[CrossRef]

Bao, J. S.

Q. Wang, Y. F. Wang, and J. S. Bao, “Nonlinear TE surface waves on an antiferromagnet,” Acta Phys. Sin. 46, 0145-0151 (1997).

Boardman, A. D.

A. D. Boardman, S. A. Nikitov, and N. A. Waby, “Existence of spin-wave solitons in an antiferromagnetic film,” Phys. Rev. B 48, 13602-13606 (1993).
[CrossRef]

Fiebig, M.

M. Fiebig, V. V. Pavlov, and R. V. Pisarev, “Second-harmonic generation as a tool for studying electronic and magnetic structures of crystals,” J. Opt. Soc. Am. B 22, 96-118 (2005).
[CrossRef]

M. Fiebig, D. Frohlich, T. Lottermoser, R. V. Pisarev, and H. J. Weber, “Second harmonic generation in the centrosymmetric antiferromagnet NiO,” Phys. Rev. Lett. 87, 137202 (2001).
[CrossRef]

M. Fiebig, D. Frohlich, B. B. Krichevtsov, and R. V. Pisarev, “Second harmonic generation and magnetic-dipole-electric-dipole interference in antiferromagnetic Cr2O3,” Phys. Rev. Lett. 73, 2127-2130 (1994).
[CrossRef]

Frohlich, D.

M. Fiebig, D. Frohlich, T. Lottermoser, R. V. Pisarev, and H. J. Weber, “Second harmonic generation in the centrosymmetric antiferromagnet NiO,” Phys. Rev. Lett. 87, 137202 (2001).
[CrossRef]

M. Fiebig, D. Frohlich, B. B. Krichevtsov, and R. V. Pisarev, “Second harmonic generation and magnetic-dipole-electric-dipole interference in antiferromagnetic Cr2O3,” Phys. Rev. Lett. 73, 2127-2130 (1994).
[CrossRef]

Kahn, L.

L. Kahn, N. S. Almeida, and D. L. Mills, “Nonlinear optical response of antiferromagnetic superlattices: multi-stability and soliton trains,” Phys. Rev. B 37, 8072-8081 (1988).
[CrossRef]

Krichevtsov, B. B.

M. Fiebig, D. Frohlich, B. B. Krichevtsov, and R. V. Pisarev, “Second harmonic generation and magnetic-dipole-electric-dipole interference in antiferromagnetic Cr2O3,” Phys. Rev. Lett. 73, 2127-2130 (1994).
[CrossRef]

Li, H.

X. Z. Wang and H. Li, “Nonlinear polaritons in antiferromagnetic/ nonmagnetic superlattices,” Phys. Rev. B 72, 054403 (2005).
[CrossRef]

Lim, S. C.

S. C. Lim, “Second harmonic generation of magnetic and dielectric multilayers,” J. Phys.: Condens. Matter 18, 4329-4343 (2006).
[CrossRef]

S. C. Lim, “Magnetic second harmonic generation of an antiferromagnetic film,” J. Opt. Soc. Am. B 19, 1401-1410 (2002).
[CrossRef]

S. C. Lim, J. Osman, and D. R. Tilley, “Calculation of nonlinear magnetic susceptibility tensors for an uniaxial antiferromagnet,” J. Phys. D: Appl. Phys. 33, 2899-2910 (2000).
[CrossRef]

Liu, F. L.

J. Bai, S. Zhou, F. L. Liu, and X. Z. Wang, “Nonlinear infrared transmission through and reflection off antiferromagnetic films,” J. Phys.: Condens. Matter 19, 046217 (2007).
[CrossRef]

Lottermoser, T.

M. Fiebig, D. Frohlich, T. Lottermoser, R. V. Pisarev, and H. J. Weber, “Second harmonic generation in the centrosymmetric antiferromagnet NiO,” Phys. Rev. Lett. 87, 137202 (2001).
[CrossRef]

Mills, D. L.

L. Kahn, N. S. Almeida, and D. L. Mills, “Nonlinear optical response of antiferromagnetic superlattices: multi-stability and soliton trains,” Phys. Rev. B 37, 8072-8081 (1988).
[CrossRef]

N. S. Almeida and D. L. Mills, “Dynamical response of antiferromagnets in an oblique magnetic field: application to surface magnons,” Phys. Rev. B 37, 3400-3408 (1988).
[CrossRef]

N. S. Almeida and D. L. Mills, “Nonlinear infrared response of antiferromagnets,” Phys. Rev. B 36, 2015-2023 (1987).
[CrossRef]

Nikitov, S. A.

A. D. Boardman, S. A. Nikitov, and N. A. Waby, “Existence of spin-wave solitons in an antiferromagnetic film,” Phys. Rev. B 48, 13602-13606 (1993).
[CrossRef]

Osman, J.

S. C. Lim, J. Osman, and D. R. Tilley, “Calculation of nonlinear magnetic susceptibility tensors for an uniaxial antiferromagnet,” J. Phys. D: Appl. Phys. 33, 2899-2910 (2000).
[CrossRef]

Pavlov, V. V.

Pisarev, R. V.

M. Fiebig, V. V. Pavlov, and R. V. Pisarev, “Second-harmonic generation as a tool for studying electronic and magnetic structures of crystals,” J. Opt. Soc. Am. B 22, 96-118 (2005).
[CrossRef]

M. Fiebig, D. Frohlich, T. Lottermoser, R. V. Pisarev, and H. J. Weber, “Second harmonic generation in the centrosymmetric antiferromagnet NiO,” Phys. Rev. Lett. 87, 137202 (2001).
[CrossRef]

M. Fiebig, D. Frohlich, B. B. Krichevtsov, and R. V. Pisarev, “Second harmonic generation and magnetic-dipole-electric-dipole interference in antiferromagnetic Cr2O3,” Phys. Rev. Lett. 73, 2127-2130 (1994).
[CrossRef]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984), pp. 86-107.

Tilley, D. R.

S. C. Lim, J. Osman, and D. R. Tilley, “Calculation of nonlinear magnetic susceptibility tensors for an uniaxial antiferromagnet,” J. Phys. D: Appl. Phys. 33, 2899-2910 (2000).
[CrossRef]

K. Abraha and D. R. Tilley, “Theory of far infrared properties of magnetic surfaces, films and superlattices,” Surf. Sci. Rep. 24, 125-222 (1996).
[CrossRef]

Waby, N. A.

A. D. Boardman, S. A. Nikitov, and N. A. Waby, “Existence of spin-wave solitons in an antiferromagnetic film,” Phys. Rev. B 48, 13602-13606 (1993).
[CrossRef]

Wang, Q.

Z. Wu and Q. Wang, “Propagation properties of magnetic surface waves on the interface between two nonlinear antiferromagnets,” Acta Phys. Sin. 50, 1178-1183 (2001).

Q. Wang, Y. F. Wang, and J. S. Bao, “Nonlinear TE surface waves on an antiferromagnet,” Acta Phys. Sin. 46, 0145-0151 (1997).

Wang, X. Z.

J. Bai, S. Zhou, F. L. Liu, and X. Z. Wang, “Nonlinear infrared transmission through and reflection off antiferromagnetic films,” J. Phys.: Condens. Matter 19, 046217 (2007).
[CrossRef]

X. Z. Wang and H. Li, “Nonlinear polaritons in antiferromagnetic/ nonmagnetic superlattices,” Phys. Rev. B 72, 054403 (2005).
[CrossRef]

Wang, Y. F.

Q. Wang, Y. F. Wang, and J. S. Bao, “Nonlinear TE surface waves on an antiferromagnet,” Acta Phys. Sin. 46, 0145-0151 (1997).

Weber, H. J.

M. Fiebig, D. Frohlich, T. Lottermoser, R. V. Pisarev, and H. J. Weber, “Second harmonic generation in the centrosymmetric antiferromagnet NiO,” Phys. Rev. Lett. 87, 137202 (2001).
[CrossRef]

Wu, Z.

Z. Wu and Q. Wang, “Propagation properties of magnetic surface waves on the interface between two nonlinear antiferromagnets,” Acta Phys. Sin. 50, 1178-1183 (2001).

Zhou, S.

J. Bai, S. Zhou, F. L. Liu, and X. Z. Wang, “Nonlinear infrared transmission through and reflection off antiferromagnetic films,” J. Phys.: Condens. Matter 19, 046217 (2007).
[CrossRef]

Acta Phys. Sin. (2)

Z. Wu and Q. Wang, “Propagation properties of magnetic surface waves on the interface between two nonlinear antiferromagnets,” Acta Phys. Sin. 50, 1178-1183 (2001).

Q. Wang, Y. F. Wang, and J. S. Bao, “Nonlinear TE surface waves on an antiferromagnet,” Acta Phys. Sin. 46, 0145-0151 (1997).

J. Opt. Soc. Am. B (2)

J. Phys. D: Appl. Phys. (1)

S. C. Lim, J. Osman, and D. R. Tilley, “Calculation of nonlinear magnetic susceptibility tensors for an uniaxial antiferromagnet,” J. Phys. D: Appl. Phys. 33, 2899-2910 (2000).
[CrossRef]

J. Phys.: Condens. Matter (2)

J. Bai, S. Zhou, F. L. Liu, and X. Z. Wang, “Nonlinear infrared transmission through and reflection off antiferromagnetic films,” J. Phys.: Condens. Matter 19, 046217 (2007).
[CrossRef]

S. C. Lim, “Second harmonic generation of magnetic and dielectric multilayers,” J. Phys.: Condens. Matter 18, 4329-4343 (2006).
[CrossRef]

Phys. Rev. B (5)

A. D. Boardman, S. A. Nikitov, and N. A. Waby, “Existence of spin-wave solitons in an antiferromagnetic film,” Phys. Rev. B 48, 13602-13606 (1993).
[CrossRef]

N. S. Almeida and D. L. Mills, “Nonlinear infrared response of antiferromagnets,” Phys. Rev. B 36, 2015-2023 (1987).
[CrossRef]

L. Kahn, N. S. Almeida, and D. L. Mills, “Nonlinear optical response of antiferromagnetic superlattices: multi-stability and soliton trains,” Phys. Rev. B 37, 8072-8081 (1988).
[CrossRef]

X. Z. Wang and H. Li, “Nonlinear polaritons in antiferromagnetic/ nonmagnetic superlattices,” Phys. Rev. B 72, 054403 (2005).
[CrossRef]

N. S. Almeida and D. L. Mills, “Dynamical response of antiferromagnets in an oblique magnetic field: application to surface magnons,” Phys. Rev. B 37, 3400-3408 (1988).
[CrossRef]

Phys. Rev. Lett. (2)

M. Fiebig, D. Frohlich, B. B. Krichevtsov, and R. V. Pisarev, “Second harmonic generation and magnetic-dipole-electric-dipole interference in antiferromagnetic Cr2O3,” Phys. Rev. Lett. 73, 2127-2130 (1994).
[CrossRef]

M. Fiebig, D. Frohlich, T. Lottermoser, R. V. Pisarev, and H. J. Weber, “Second harmonic generation in the centrosymmetric antiferromagnet NiO,” Phys. Rev. Lett. 87, 137202 (2001).
[CrossRef]

Surf. Sci. Rep. (1)

K. Abraha and D. R. Tilley, “Theory of far infrared properties of magnetic surfaces, films and superlattices,” Surf. Sci. Rep. 24, 125-222 (1996).
[CrossRef]

Other (1)

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984), pp. 86-107.

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Figures (5)

Fig. 1
Fig. 1

Geometry and coordinate system. An AF film with the anisotropy axis normal to the incident plane is put between two different dielectrics. The incident plane is the x y plane. I, R, and T represent the incident, reflection, and transmission waves with the fundamental frequency, respectively. R s and T s indicate the output SH waves.

Fig. 2
Fig. 2

SH outputs of a single AF film ( Mn F 2 film), S R , and S T versus the incident angle and frequency.

Fig. 3
Fig. 3

SH outputs of Si O 2 Mn F 2 /air, S R , and S T versus the incident angle and frequency.

Fig. 4
Fig. 4

SH outputs of Zn F 2 Mn F 2 /air, S R , and S T versus the incident angle and frequency.

Fig. 5
Fig. 5

SH outputs of Si O 2 Mn F 2 / air, S R , and S T versus the film thickness for ω = 9.84 cm 1 and θ = 41.3 .

Equations (30)

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M l t = γ M l × H eff l + τ M 0 M l × M l t ,
H eff l = e ̂ z ( H 0 + H a M l z M 0 ) H e M l M 0 + H ( ω ) ,
χ x x ( 1 ) = χ y y ( 1 ) = ω a ω m { 1 [ ω r 2 ( ω ω 0 ) 2 ] + 1 [ ω r 2 ( ω + ω 0 ) 2 ] } ,
χ x y ( 1 ) = χ y x ( 1 ) = i ω a ω m { 1 [ ω r 2 ( ω ω 0 ) 2 ] 1 [ ω r 2 ( ω + ω 0 ) 2 ] } ,
m z ( 2 ) ( ω s ) = χ x x ( 2 ) ( ω s ) ( H x H x + H y H y ) ,
χ x x ( 2 ) ( 2 ω ) = χ y y ( 2 ) ( 2 ω ) = i ω m [ χ x x ( 1 ) ω 0 ( ω r 2 + ω 2 ω 0 2 ) χ x y ( 1 ) ω ( ω r 2 ω 2 + ω 0 2 ) ] M 0 [ ω r 2 ( ω ω 0 ) 2 ] [ ω r 2 ( ω + ω 0 ) 2 ] .
E = e ̂ z [ A + exp ( i k y y ) + A exp ( i k y y ) ] exp ( i k x x i ω t ) ,
E a = e ̂ z [ E 0 exp ( i k 0 y y ) + R 0 exp ( i k 0 y y ) ] exp ( i k 0 x x i ω t ) ,
E b = e ̂ z T 0 exp ( i k 0 y y ) exp ( i k 0 x x i ω t ) .
H = exp ( i k x x i ω t ) ω μ 0 μ v { e ̂ x k y [ ( 1 + δ ) A + exp ( i k y y ) + ( 1 δ ) A exp ( i k y y ) ] + e ̂ y k x [ ( δ 1 ) A + exp ( i k y y ) ( δ + 1 ) A exp ( i k y y ) ] } ,
H a = exp ( i k 0 x x i ω t ) ω μ 0 { e ̂ x k 0 y [ E 0 exp ( i k 0 y y ) R 0 exp ( i k 0 y y ) ] e ̂ y k 0 x [ E 0 exp ( i k 0 y y ) + R 0 exp ( i k 0 y y ) ] } ,
H b = T 0 ω μ 0 ( e ̂ x k 0 y e ̂ y k 0 x ) exp ( i k 0 y y + i k 0 x x i ω t ) .
f ± = [ Δ ( 1 δ ) ± 1 ] exp ( i k y d ) cos k y d ( Δ + Δ ) + i δ ( Δ Δ ) sin k y d i [ 1 + Δ Δ ( 1 δ 2 ) ] sin k y d ,
( 2 x 2 + 2 y 2 ) H s z ( ω s ) ε ( ω s c ) 2 H s z ( ω s ) = ε ( ω s c ) 2 m z ( 2 ) ( ω s ) .
H s z ( ω s ) = [ A s exp ( i k s y y ) + B s exp ( i k s y y ) + a exp ( 2 i k y y ) + b exp ( 2 i k y y ) + c ] exp ( 2 i k x x i ω s t ) ,
a = E 0 2 ε 0 χ z x x ( 2 ) ( ω s ) f + 2 μ 0 ( μ ν 1 ) ( ω μ ν c ) 2 [ k y 2 ( 1 + δ ) 2 + k x 2 ( 1 δ ) 2 ] ,
b = E 0 2 ε 0 χ z x x ( 2 ) ( ω s ) f 2 μ 0 ( μ ν 1 ) ( ω μ ν c ) 2 [ k y 2 ( 1 δ ) 2 + k x 2 ( 1 + δ ) 2 ] ,
c = E 0 2 2 ε 0 ε ( ω s c ) 2 χ z x x ( 2 ) ( ω s ) f + f μ 0 [ 4 k x 2 ε ( ω s c ) 2 ] ( ω μ ν c ) 2 [ k y 2 ( 1 + δ ) ( 1 δ ) + k x 2 ( δ 1 ) ( 1 + δ ) ] ,
E s a = R s exp [ i ( 2 k x x k s y ω s t ) ] ω s ε 0 ε 1 [ k s e ̂ x + k s x e ̂ y ] ,
E s = exp [ i ( 2 k x x ω s t ) ] ω s ε 0 ε { e ̂ x [ k s y ( A s exp ( i k s y y ) B s exp ( i k s y y ) ) + 2 k y a exp ( 2 i k y y ) 2 k y b exp ( 2 i k y y ) ] 2 k x e ̂ y [ a exp ( 2 i k y y ) + b exp ( 2 i k y y ) + c ] } ,
E s b = T s ω s ε 0 ε 2 ( k s e ̂ x + k s x e ̂ y ) exp [ i ( 2 k x x + k s y ω s t ) ] .
θ s = θ ,
θ s = arcsin ( ε 1 ε 2 sin θ ) .
R s = A s + B s + a + b + c ,
R s = ε 1 ε k s [ k s y ( A s + B s ) + 2 k y ( a + b ) ] ,
T s exp ( i k s d ) = A s exp ( i k s y d ) + B s exp ( i k s y d ) + a exp ( 2 i k y d ) + b exp ( 2 i k y d ) + c ,
T s exp ( i k s d ) = ε 2 ε k s { k s y [ A s exp ( i k s y d ) B s exp ( i k s y d ) ] + 2 k y [ a exp ( 2 i k y d ) b exp ( 2 i k y d ) ] } .
R s = 1 S { [ ( Δ 2 Δ 0 ) cos k s y d + i ( Δ 2 Δ 0 1 ) sin k s y d + exp ( 2 i k y d ) ( Δ 2 + Δ 0 ) ] a + [ ( Δ 2 + Δ 0 ) cos k s y d i ( Δ 2 Δ 0 + 1 ) sin k s y d ( Δ 2 + Δ 0 ) exp ( 2 i k y d ) ] b + [ Δ 2 ( cos k s d 1 ) i sin k s d ] c } ,
T s = exp ( i k s d ) S { [ ( Δ 1 + Δ 0 ) ( e 2 i k y d cos k s y d 1 ) i ( 1 + Δ 0 Δ 1 ) exp ( 2 i k y d ) sin k s y d ] a + [ ( Δ 1 Δ 0 ) ( exp ( 2 i k y d ) cos k s y d 1 ) + i ( Δ 0 Δ 1 1 ) exp ( 2 i k y d ) sin k s y d ] b + [ Δ 1 ( cos k s y d 1 ) i sin k s y d ] c } ,
S = [ ( Δ 2 + Δ 1 ) cos k s y d i ( 1 + Δ 2 Δ 1 ) sin k s y d ] ,

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