Abstract

An extension of the beam propagation method to anisotropic media is presented and the formalism is employed to study several integrated optics devices.

© 1982 Optical Society of America

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References

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  1. M. D. Feit, J. A. Fleck, Appl. Opt. 17, 3990 (1978).
    [CrossRef] [PubMed]
  2. J. Van Roey, J. van der Donk, P. E. Lagasse, J. Opt. Soc. Am. 71, 803 (1981).
    [CrossRef]
  3. D. Yevick, L. Thylén: “Analysis of Gratings by the Beam Propagation Method,” to be published in J. Opt. Soc. Am.
  4. D. Yevick, L. Thylén, “A Numerical Analysis of Grating Structures,” to be published in Proc. ASI, Erice, Italy, Aug.1981.
  5. J. McKenna, F. K. Reinhart, J. Appl. Phys. 47, 2069 (1976).
    [CrossRef]
  6. R. C. Alferness, in Technical Digest, Second International Conference IOOC (Koninklijk Instituut van Ingenieurs, Amsterdam, 1979), paper 19.6.
  7. M. Minakata, S. Saito, M. Shibata, S. Miyazawa, J. Appl. Phys. 49, 4677 (1978).
    [CrossRef]
  8. J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
    [CrossRef]
  9. P. Danielson, D. Yevick, to be published.

1981 (1)

1978 (2)

M. Minakata, S. Saito, M. Shibata, S. Miyazawa, J. Appl. Phys. 49, 4677 (1978).
[CrossRef]

M. D. Feit, J. A. Fleck, Appl. Opt. 17, 3990 (1978).
[CrossRef] [PubMed]

1976 (2)

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

J. McKenna, F. K. Reinhart, J. Appl. Phys. 47, 2069 (1976).
[CrossRef]

Alferness, R. C.

R. C. Alferness, in Technical Digest, Second International Conference IOOC (Koninklijk Instituut van Ingenieurs, Amsterdam, 1979), paper 19.6.

Danielson, P.

P. Danielson, D. Yevick, to be published.

Feit, M. D.

M. D. Feit, J. A. Fleck, Appl. Opt. 17, 3990 (1978).
[CrossRef] [PubMed]

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Fleck, J. A.

M. D. Feit, J. A. Fleck, Appl. Opt. 17, 3990 (1978).
[CrossRef] [PubMed]

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Lagasse, P. E.

McKenna, J.

J. McKenna, F. K. Reinhart, J. Appl. Phys. 47, 2069 (1976).
[CrossRef]

Minakata, M.

M. Minakata, S. Saito, M. Shibata, S. Miyazawa, J. Appl. Phys. 49, 4677 (1978).
[CrossRef]

Miyazawa, S.

M. Minakata, S. Saito, M. Shibata, S. Miyazawa, J. Appl. Phys. 49, 4677 (1978).
[CrossRef]

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Reinhart, F. K.

J. McKenna, F. K. Reinhart, J. Appl. Phys. 47, 2069 (1976).
[CrossRef]

Saito, S.

M. Minakata, S. Saito, M. Shibata, S. Miyazawa, J. Appl. Phys. 49, 4677 (1978).
[CrossRef]

Shibata, M.

M. Minakata, S. Saito, M. Shibata, S. Miyazawa, J. Appl. Phys. 49, 4677 (1978).
[CrossRef]

Thylén, L.

D. Yevick, L. Thylén: “Analysis of Gratings by the Beam Propagation Method,” to be published in J. Opt. Soc. Am.

D. Yevick, L. Thylén, “A Numerical Analysis of Grating Structures,” to be published in Proc. ASI, Erice, Italy, Aug.1981.

van der Donk, J.

Van Roey, J.

Yevick, D.

D. Yevick, L. Thylén: “Analysis of Gratings by the Beam Propagation Method,” to be published in J. Opt. Soc. Am.

D. Yevick, L. Thylén, “A Numerical Analysis of Grating Structures,” to be published in Proc. ASI, Erice, Italy, Aug.1981.

P. Danielson, D. Yevick, to be published.

Appl. Opt. (1)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

J. Appl. Phys. (2)

J. McKenna, F. K. Reinhart, J. Appl. Phys. 47, 2069 (1976).
[CrossRef]

M. Minakata, S. Saito, M. Shibata, S. Miyazawa, J. Appl. Phys. 49, 4677 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (4)

D. Yevick, L. Thylén: “Analysis of Gratings by the Beam Propagation Method,” to be published in J. Opt. Soc. Am.

D. Yevick, L. Thylén, “A Numerical Analysis of Grating Structures,” to be published in Proc. ASI, Erice, Italy, Aug.1981.

R. C. Alferness, in Technical Digest, Second International Conference IOOC (Koninklijk Instituut van Ingenieurs, Amsterdam, 1979), paper 19.6.

P. Danielson, D. Yevick, to be published.

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Figures (3)

Fig. 1
Fig. 1

Schematic of a double heterostructure GaAs pn-junction waveguide electrooptic modulator showing the crystal orientation. By applying the indicated RF electric field E, assumed homogeneous for the sake of simplicity, the isotropic waveguide medium is rendered anisotropic, with principal axes along x′,y′,z. n1 denotes the waveguiding layer (GaAs) with refractive index n1, while n denotes cladding layers (GaAlAs) of refractive index n0, where n1 > n0.

Fig. 2
Fig. 2

Calculated optical field amplitudes Ex (TE) and Ey (TM) for an electrooptic semiconductor pn-junction modulator of the type described in Ref. 5. Complete power transfer from the TE mode to the TM mode is obtained after 200 μm. The fields are graphed at 25-μm intervals in the z direction from 25 to 225 μm.

Fig. 3
Fig. 3

Evolution of the optical field amplitudes Ex (TE) and Ey (TM) along an electrooptic device of the type described in Ref. 6 with a large difference between βTE and βTM. (A) shows the optical field at 1.25-μm intervals from 1.25 to 13.75 μm when ɛxy is a constant. For a phase matching variation of ɛxy we obtain (B) which shows the optical fields at 20-μm intervals from 20 to 220 μm and a complete power conversion. Note the similarity between Figs. 2 and 3(B). The coupling length in (B) corresponds to a peak applied electric field of ~15 V/μm.

Equations (21)

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2 E + n 2 k 0 2 E = 0 ,
E ( z = Δ z ) = exp [ - i ( 2 + n 2 k 0 2 ) 1 / 2 Δ z ] E ( z = 0 ) ,
( 2 + n 2 k 0 2 ) 1 / 2 2 / [ ( 2 + k 2 ) 1 / 2 + k ] + k + k 0 δ n ( x , y )
E ( z = Δ z ) = P · Q · P · E ( z = 0 ) ,
P = exp { - i ( Δ z 2 ) 2 / [ ( 2 + k 2 ) 1 / 2 + k ] } , Q = exp [ - i Δ z k 0 δ n ( x , y ) ] .
( I 2 / z 2 - M 2 ) E ¯ = 0 ,
I = identity matrix , M 2 = - ( 2 + k 0 2 ɛ x x k 0 2 ɛ x y k 0 2 ɛ y x 2 + k 0 2 ɛ y y ) .
ɛ x x = ɛ x x 0 + δ ɛ x x ( x , y ) ,
ɛ y y = ɛ y y 0 + δ ɛ y y ( x , y )
δ ɛ x x ( x , y ) = 2 ɛ x x 0 δ n ( x , y ) ,
δ e y y ( x , y ) = 2 ɛ y y 0 δ n ( x , y ) ,
M = M 0 + M Δ + M κ ,
M 0 = ( - i ) ( 2 / [ ( 2 + k x x 2 ) 1 / 2 + k x x ] + k x x 0 0 2 / [ ( 2 + k y y 2 ) 1 / 2 + k y y ] + k y y ) , M Δ = ( - i ) k 0 δ n I , M κ = ( - i k 0 ɛ x y ) / ( n x x + n y y )             ( 0 1 1 0 ) , k x x = n x x k 0 , k y y = n y y k 0 .
exp [ ( - i q ) ( 0 1 1 0 ) ] = ( cos q - i sin q - i sin q cos q ) ,
[ E x ( x , y , Δ z ) E y ( x , y , Δ z ) ] = [ P x x 0 0 P y y ] [ Q 0 0 Q ] [ cos κ - i sin κ - i sin κ cos κ ] [ P x x 0 0 P y y ] [ E x ( x , y , o ) E y ( x , y , o ) ] .
P x x = exp ( - i ( Δ z / 2 ) { 2 / [ ( 2 + k x x 2 ) 1 / 2 + k x x ] + k x x } ) ;
n ( x ) = n sub + Δ n / cos h 2 ( 2 x / h )
M = exp { - i Δ z ( δ n x x ɛ x y / ( n x x + n y y ) ɛ y x / ( n x x + n y y ) δ n y y ) } exp ( A B B D )
M = exp { ( 0 0 0 ( D - A ) / 2 ) } exp { ( A B B A ) } exp { ( 0 0 0 ( D - A ) / 2 ) } .
( δ n x x ɛ x y / ( n x x + n y y ) ɛ y x / ( n x x + n y y ) δ n y y )
1 Δ z z - Δ z / 2 z + Δ z / 2 δ n d z ,

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