Abstract

The optical transfer matrix technique for the analysis of optical harmonic generation and mixing in laminar structures is extended to include the case of anisotropic media. This approach is applicable in the limit of low conversion efficiencies, where attenuation and phase shifts of the pump beams due to nonlinear interactions can be neglected. In this approximation one first solves the linear problem for the pump waves in the structure, using optical transfer matrices to take all reflections and linear attenuation into account. Then the nonlinear polarizations driven by these waves are found, and finally expressions for the waves at the harmonic and combination frequencies generated by these polarizations are derived, again using transfer matrices to account exactly for all reflections and linear attenuation. In the anisotropic case 4 × 4 transfer and propagation matrices are required, and all polarization components are treated simultaneously. The results are given in a computationally convenient form.

© 1991 Optical Society of America

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References

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  1. D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: analysis using optical transfer matrix techniques,” J. Opt. Soc. Am. B 6, 910 (1989).
    [CrossRef]
  2. S. Teitler and B. W. Henvis, “Refraction in stratified, anisotropic media,” J. Opt. Soc. Am. 60, 830 (1970).
    [CrossRef]
  3. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 502 (1972).
    [CrossRef]
  4. P. Allia, C. Oldano, and L. Trossi, “Polarization transfer matrix for the transmission of light through liquid-crystal slabs,” J. Opt. Soc. Am. B 5, 2452 (1988).
    [CrossRef]
  5. S. Wang, N. Shah, and J. D. Crow, “Studies of the use of gyrotropic and anisotropic materials for mode conversion in thin-film optical-waveguide applications,” J. Appl. Phys. 43, 1861 (1972).
    [CrossRef]
  6. S. Yamamoto, Y. Koyamada, and T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film waveguides of integrated optics,” J. Appl. Phys. 43, 5090 (1972).
    [CrossRef]
  7. M. S. Kharusi, “Uniaxial and biaxial anisotropy in thin-film optical waveguides,” J. Opt. Soc. Am. 64, 27 (1974).
    [CrossRef]
  8. M. Wabia and K. Gniadek, “Field excited by sources in anisotropic thin-film waveguides,” Acta Phys. Pol. A 54, 493 (1978).
  9. E. Weinert-Raczka, “Nonlinear generation of the guided mode in an anisotropic thin-film optical waveguide,” Opt. Appl. 12, 195 (1982).
  10. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984), Chap. 11.
  11. R. S. Burington, Handbook of Mathematical Tables and Formulas, 5th ed. (McGraw-Hill, New York, 1973), pp. 12–14.

1989 (1)

1988 (1)

1982 (1)

E. Weinert-Raczka, “Nonlinear generation of the guided mode in an anisotropic thin-film optical waveguide,” Opt. Appl. 12, 195 (1982).

1978 (1)

M. Wabia and K. Gniadek, “Field excited by sources in anisotropic thin-film waveguides,” Acta Phys. Pol. A 54, 493 (1978).

1974 (1)

1972 (3)

D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 502 (1972).
[CrossRef]

S. Wang, N. Shah, and J. D. Crow, “Studies of the use of gyrotropic and anisotropic materials for mode conversion in thin-film optical-waveguide applications,” J. Appl. Phys. 43, 1861 (1972).
[CrossRef]

S. Yamamoto, Y. Koyamada, and T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film waveguides of integrated optics,” J. Appl. Phys. 43, 5090 (1972).
[CrossRef]

1970 (1)

Allia, P.

Berreman, D. W.

Bethune, D. S.

Burington, R. S.

R. S. Burington, Handbook of Mathematical Tables and Formulas, 5th ed. (McGraw-Hill, New York, 1973), pp. 12–14.

Crow, J. D.

S. Wang, N. Shah, and J. D. Crow, “Studies of the use of gyrotropic and anisotropic materials for mode conversion in thin-film optical-waveguide applications,” J. Appl. Phys. 43, 1861 (1972).
[CrossRef]

Gniadek, K.

M. Wabia and K. Gniadek, “Field excited by sources in anisotropic thin-film waveguides,” Acta Phys. Pol. A 54, 493 (1978).

Henvis, B. W.

Kharusi, M. S.

Koyamada, Y.

S. Yamamoto, Y. Koyamada, and T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film waveguides of integrated optics,” J. Appl. Phys. 43, 5090 (1972).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984), Chap. 11.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984), Chap. 11.

Makimoto, T.

S. Yamamoto, Y. Koyamada, and T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film waveguides of integrated optics,” J. Appl. Phys. 43, 5090 (1972).
[CrossRef]

Oldano, C.

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984), Chap. 11.

Shah, N.

S. Wang, N. Shah, and J. D. Crow, “Studies of the use of gyrotropic and anisotropic materials for mode conversion in thin-film optical-waveguide applications,” J. Appl. Phys. 43, 1861 (1972).
[CrossRef]

Teitler, S.

Trossi, L.

Wabia, M.

M. Wabia and K. Gniadek, “Field excited by sources in anisotropic thin-film waveguides,” Acta Phys. Pol. A 54, 493 (1978).

Wang, S.

S. Wang, N. Shah, and J. D. Crow, “Studies of the use of gyrotropic and anisotropic materials for mode conversion in thin-film optical-waveguide applications,” J. Appl. Phys. 43, 1861 (1972).
[CrossRef]

Weinert-Raczka, E.

E. Weinert-Raczka, “Nonlinear generation of the guided mode in an anisotropic thin-film optical waveguide,” Opt. Appl. 12, 195 (1982).

Yamamoto, S.

S. Yamamoto, Y. Koyamada, and T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film waveguides of integrated optics,” J. Appl. Phys. 43, 5090 (1972).
[CrossRef]

Acta Phys. Pol. A (1)

M. Wabia and K. Gniadek, “Field excited by sources in anisotropic thin-film waveguides,” Acta Phys. Pol. A 54, 493 (1978).

J. Appl. Phys. (2)

S. Wang, N. Shah, and J. D. Crow, “Studies of the use of gyrotropic and anisotropic materials for mode conversion in thin-film optical-waveguide applications,” J. Appl. Phys. 43, 1861 (1972).
[CrossRef]

S. Yamamoto, Y. Koyamada, and T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film waveguides of integrated optics,” J. Appl. Phys. 43, 5090 (1972).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Appl. (1)

E. Weinert-Raczka, “Nonlinear generation of the guided mode in an anisotropic thin-film optical waveguide,” Opt. Appl. 12, 195 (1982).

Other (2)

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984), Chap. 11.

R. S. Burington, Handbook of Mathematical Tables and Formulas, 5th ed. (McGraw-Hill, New York, 1973), pp. 12–14.

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Figures (1)

Fig. 1
Fig. 1

Schematic of multilayer nonlinear medium showing the coordinate axes used and, for an assumed negative uniaxial medium with its ĉ axis in the xz plane, the free-wave index vectors n ( m ) and the eigenvectors û(m) allowed for a given value of κ. Also shown for illustration is a typical source wave index vector n s and the associated orthogonal basis vectors s(q) used in deriving the components of the bound-wave amplitude vector Es from the nonlinear polarization in the layer.

Equations (44)

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[ n 2 I - n n - ɛ ] E = 0.
E i = m = 1 4 E i ( m ) u ^ i ( m ) exp [ i κ k 0 x + i N i ( m ) k 0 ( z - z i ) - i ω t ] .
E i ( z i + d i ) = Φ i E i ( z i ) .
Φ i , m m ϕ i ( m ) exp [ i N i ( m ) k 0 d i ] .
Φ i E i = M i j E j .
[ E x B y E y - B x ] z = z j = Π j E j ,
[ Π j , 1 m , Π j , 2 m , Π j , 3 m , Π j , 4 m ] = [ u j x ( m ) , ( N j ( m ) u j x ( m ) - κ u j z ( m ) ) , u j y ( m ) , N j ( m ) u j y ( m ) ] .
Π i Φ i E i = Π j E j .
M i j = Π i - 1 Π j ,
T M f ( f - 1 ) Φ f - 1 M ( f - 1 ) ( f - 2 ) Φ f - 2 M 21 .
[ t ( 1 , 1 ) 0 t ( 3 , 1 ) 0 ] = T [ 1 r ( 2 , 1 ) 0 r ( 4 , 1 ) ] ,             [ t ( 1 , 3 ) 0 t ( 3 , 3 ) 0 ] = T [ 0 r ( 2 , 3 ) 1 r ( 4 , 3 ) ] .
r ( 2 , 1 ) = ( T 24 T 41 - T 21 T 44 ) / D , r ( 4 , 1 ) = ( T 21 T 42 - T 22 T 41 ) / D , r ( 2 , 3 ) = ( T 24 T 43 - T 23 T 44 ) / D , r ( 4 , 3 ) = ( T 23 T 42 - T 22 T 43 ) / D ,
E j = L j 1 [ E 1 ( 1 ) r ( 2 , 1 ) E 1 ( 1 ) + r ( 2 , 3 ) E 1 ( 3 ) E 1 ( 3 ) r ( 4 , 1 ) E 1 ( 1 ) + r ( 4 , 3 ) E 1 ( 3 ) ] ,
[ n s 2 I - n s n s - ɛ ] E s = D NL ,
[ n s 2 I - n s n s - ɛ ] s ^ = λ s ^ .
E s = q = 1 3 s ^ ( q ) ( s ^ ( q ) · D NL ) λ ( q ) = q = 1 3 E s ( q ) s ^ ( q ) exp [ i κ s k s 0 x + i N s k s 0 ( z - z j ) - i ω s t ] .
[ Π s , 1 q , Π s , 2 q , Π s , 3 q , Π s , 4 q ] = [ s x ( q ) , ( N s ( q ) s x ( q ) - κ s s z ( q ) ) , s y ( q ) , N s ( q ) s y ( q ) ] .
Π i Φ i E i = Π i E j + Π s E s
Π j Φ j E j + Π s Φ s E s = Π k E k .
E k = M k j Φ j ( M j i Φ i E i + S j ) ,
S j ( Φ ¯ j M j s Φ s - M j s ) E s .
S j = Δ Φ j M j s E s .
Δ ϕ j , m m = ( ϕ ¯ j ( m ) ϕ s - 1 ) = [ exp ( i Δ N j ( m ) k s 0 d j ) - 1 ] ,
R j f [ E f ( 1 ) 0 E f ( 3 ) 0 ] - L j 1 [ 0 E 1 ( 2 ) 0 E 1 ( 4 ) ] = S j ,
T ¯ [ E f ( 1 ) 0 E f ( 3 ) 0 ] - [ 0 E 1 ( 2 ) 0 E 1 ( 4 ) ] = S 1 j ,
[ E f ( 1 ) E 1 ( 2 ) E f ( 3 ) E 1 ( 4 ) ] = 1 D [ T ¯ 33 0 - T ¯ 13 0 T ¯ 33 T ¯ 21 - T ¯ 23 T ¯ 31 - D T ¯ 11 T ¯ 23 - T ¯ 21 T ¯ 13 0 - T ¯ 31 0 T ¯ 11 0 T ¯ 33 T ¯ 41 - T ¯ 43 T 31 0 T ¯ 11 T ¯ 43 - T ¯ 41 T ¯ 13 - D ] S 1 j
Φ 1 = L 11 = R f f I ,             S 1 M 1 s E 1 s , S f - M f s E f s ,
[ ɛ s P - ( ɛ o + λ ) I - Δ ɛ c ^ c ^ ] s = 0.
n ^ s n ^ s / n s ,             p ^ o n ^ s × c ^ / c , p ^ e p ^ o × n ^ s = P c ^ / c ,
λ o = ɛ s - ɛ o , λ e ± = 1 2 { ɛ s - 2 ɛ o - Δ ɛ ± [ ( ɛ s - Δ ɛ ) 2 + 4 ɛ s Δ ɛ c 2 ] 1 / 2 } ;
( b a ) ± = - Δ ɛ c c λ e ± + ɛ o + Δ ɛ c 2 ,
u ^ o ± = p ^ o ,             N o ± = ± ( ɛ o - κ 2 ) 1 / 2 .
u ^ e ± = a p ^ e + b n ^             with             ( b a ) ± = - δ ɛ c c 1 + δ ɛ c 2 ,
1 n 2 = c 2 ɛ e + c 2 ɛ o .
N e ± = 1 1 + δ ɛ c z 2 { - κ δ ɛ c x c z ± [ ( ɛ e - κ 2 ) ( 1 + δ ɛ c z 2 ) - κ 2 δ ɛ c x 2 ] 1 / 2 } .
p ^ o = [ - N c y , ( N c x - κ c z ) , κ c y ] / n c , p ^ e = [ ( c x - κ c / n ) , c y , ( c z - N c / n ) ] / c ,
u ^ ( 1 ) = ( N e ^ x - κ e ^ z ) / n ,             u ^ ( 2 ) = ( N e ^ x + κ e ^ z ) / n , u ^ ( 3 ) = u ^ ( 4 ) = e ^ y ,
Π = [ N / n N / n 0 0 n - n 0 0 0 0 1 1 0 0 N N ] , Π ¯ = 1 2 [ n / N 1 / n 0 0 n / N - 1 / n 0 0 0 0 1 1 / N 0 0 1 - 1 / N ] .
s ^ ( 1 ) = ( N s e ^ x - κ s e ^ z ) / n s ,             s ^ ( 2 ) = e ^ y ,             s ^ ( 3 ) = n s / n s
Π s = [ N s / n s 0 κ s / n s n s 0 0 0 1 0 0 N s 0 ] .
ɛ = i = 1 3 ɛ i c ^ i c ^ i .
n = 0 4 a n N n = 0 ,             n = 0 3 b n λ n = 0 ,
a 0 = [ κ 2 ( κ 2 ɛ x x - ɛ x x ɛ y y - ɛ x x ɛ z z + ɛ x y 2 + ɛ z x 2 ) + ( ɛ x x ɛ y y ɛ z z + 2 ɛ x y ɛ y z ɛ z x - ɛ x x ɛ y z 2 - ɛ y y ɛ z x 2 - ɛ z z ɛ x y 2 ) ] , a 1 = 2 κ ( κ 2 ɛ z x + ɛ x y ɛ y z - ɛ y y ɛ z x ) , a 2 = [ κ 2 ( ɛ x x + ɛ z z ) - ɛ x x ɛ z z - ɛ y y ɛ z z + ɛ y z 2 + ɛ z x 2 ] , a 3 = 2 κ ɛ z x ,             a 4 = ɛ z z
b 0 = [ N s 2 ( n s 2 ɛ z z - n s 2 ɛ x x + ɛ x x ɛ y y - ɛ y y ɛ z z + ɛ y z 2 - ɛ x y 2 ) + 2 κ N s ( N s 2 ɛ z x + ɛ x y ɛ y z - ɛ y y ɛ z x ) + n s 2 ( n s 2 ɛ x x - ɛ x x ɛ y y - ɛ x x ɛ z z + ɛ x y 2 + ɛ z x 2 ) + ( ɛ x x ɛ y y ɛ z z - ɛ x x ɛ y z 2 - ɛ y y ɛ z x 2 - ɛ z z ɛ x y 2 + 2 ɛ x y ɛ y z ɛ z x ) ] , b 1 = [ N s 2 ( ɛ x x - ɛ z z ) - 2 N s κ ɛ z x + ( n s 2 - ɛ x x ) ( n s 2 - ɛ x x - ɛ y y - ɛ z z ) + ɛ y y ɛ z z - ɛ x x 2 - ɛ x y 2 - ɛ y z 2 - ɛ z x 2 ] , b 2 = ( ɛ x x + ɛ y y + ɛ z z - 2 n s 2 ) ,             b 3 = 1.

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