Abstract

First a rigorous formulation is presented for the scattering of a uniform plane wave by an infinite dielectric grating waveguide, under the most general condition of oblique incidence. The results are then applied to the analysis of the guidance of waves by the dielectric grating waveguide. By using a simple coordinate rotation, the TE and TM Floquet mode functions determined previously for the special case of principal-plane incidence are combined to treat the general case of oblique incidence. In terms of the coupling between the known Floquet mode functions of both polarizations, the general case of oblique incidence is formulated in an exact fashion, as a three-dimensional boundary-value problem, so that the hybrid nature of waves supported by the structure can be investigated rigorously.

© 1989 Optical Society of America

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References

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  1. C. Elachi, “Waves in active and passive periodic structures: a review,” Proc. IEEE 64, 1666–1698 (1976).
    [Crossref]
  2. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [Crossref]
  3. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [Crossref]
  4. K. Handa, S. T. Peng, T. Tamir, “Improved perturbation analysis of dielectric gratings,” Appl. Phys. 5, 325–328 (1975).
    [Crossref]
  5. S. T. Peng, T. Tamir, “TM-mode perturbation analysis of dielectric gratings,” Appl. Phys. 7, 35–38 (1975).
    [Crossref]
  6. W. H. Lee, W. Streifer, “Radiation loss calculations for corrugated dielectric waveguides,”J. Opt. Soc. Am. 68, 1701–1707 (1978).
    [Crossref]
  7. W. H. Lee, W. Streifer, “Radiation loss calculations for corrugated dielectric waveguides. II. TM polarization,”J. Opt. Soc. Am. 69, 1671–1676 (1979).
    [Crossref]
  8. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chaps. 3 and 4.
  9. R. Ulrich, R. Zengerle, “Optical Bloch waves in periodic planar waveguides,” in Digest of Topical Meeting on Integrated and Guided-Wave Optics (Optical Society of America, Washington, D.C., 1980),paper TUB1.
  10. J. Van Roey, P. E. Lagasse, “Coupled wave analysis of obliquely incident waves in thin film gratings,” Appl. Opt. 20, 423–429 (1981).
    [Crossref] [PubMed]
  11. G. I. Stegeman, D. Sarid, J. J. Burke, D. G. Hall, “Scattering of guided waves by surface periodic gratings for arbitrary angles of incidence: perturbation field theory and implications to normal-mode analysis,”J. Opt. Soc. Am. 71, 1497–1507 (1981).
    [Crossref]
  12. L. A. Weller-Brophy, D. G. Hall, “Analysis of waveguide gratings: a comparison of the results of Rouard’s method and coupled-mode theory,” J. Opt. Soc. Am. A 4, 60–65 (1987).
    [Crossref]
  13. L. A. Weller-Brophy, D. G. Hall, “Measured TM–TM couplings in waveguide gratings,” J. Opt. Soc. Am. A 4(13), P6 (1987).
  14. S. T. Peng, “Oblique guidance of surface waves on corrugated dielectric layers,” in Proceedings of 1980 International URSI Symposium on Electromagnetic Waves (International Union of Radio Scientists, Brussels, 1980), pp. 341/B1–341/B4.
  15. A. A. Oliner, S. T. Peng, “New physical effects on periodically grooved open dielectric waveguides,” Radio Sci. 19, 1251–1255 (1984).
    [Crossref]
  16. S. T. Peng, M. J. Shiau, “Scattering of plane waves by corrugated dielectric layers,” in Proceedings of 1983 International URSI Symposium on Electromagnetic Waves (International Union of Radio Scientists, Brussels, 1983) pp. 175–178.
  17. M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), Sec. 1.6.
  18. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chaps. 1 and 2.

1987 (2)

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

1984 (1)

A. A. Oliner, S. T. Peng, “New physical effects on periodically grooved open dielectric waveguides,” Radio Sci. 19, 1251–1255 (1984).
[Crossref]

1981 (2)

1979 (1)

1978 (1)

1976 (1)

C. Elachi, “Waves in active and passive periodic structures: a review,” Proc. IEEE 64, 1666–1698 (1976).
[Crossref]

1975 (3)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[Crossref]

K. Handa, S. T. Peng, T. Tamir, “Improved perturbation analysis of dielectric gratings,” Appl. Phys. 5, 325–328 (1975).
[Crossref]

S. T. Peng, T. Tamir, “TM-mode perturbation analysis of dielectric gratings,” Appl. Phys. 7, 35–38 (1975).
[Crossref]

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), Sec. 1.6.

Burke, J. J.

Elachi, C.

C. Elachi, “Waves in active and passive periodic structures: a review,” Proc. IEEE 64, 1666–1698 (1976).
[Crossref]

Felsen, L. B.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chaps. 1 and 2.

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

Hall, D. G.

Handa, K.

K. Handa, S. T. Peng, T. Tamir, “Improved perturbation analysis of dielectric gratings,” Appl. Phys. 5, 325–328 (1975).
[Crossref]

Lagasse, P. E.

Lee, W. H.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chaps. 3 and 4.

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chaps. 1 and 2.

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

Oliner, A. A.

A. A. Oliner, S. T. Peng, “New physical effects on periodically grooved open dielectric waveguides,” Radio Sci. 19, 1251–1255 (1984).
[Crossref]

Peng, S. T.

A. A. Oliner, S. T. Peng, “New physical effects on periodically grooved open dielectric waveguides,” Radio Sci. 19, 1251–1255 (1984).
[Crossref]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[Crossref]

K. Handa, S. T. Peng, T. Tamir, “Improved perturbation analysis of dielectric gratings,” Appl. Phys. 5, 325–328 (1975).
[Crossref]

S. T. Peng, T. Tamir, “TM-mode perturbation analysis of dielectric gratings,” Appl. Phys. 7, 35–38 (1975).
[Crossref]

S. T. Peng, M. J. Shiau, “Scattering of plane waves by corrugated dielectric layers,” in Proceedings of 1983 International URSI Symposium on Electromagnetic Waves (International Union of Radio Scientists, Brussels, 1983) pp. 175–178.

S. T. Peng, “Oblique guidance of surface waves on corrugated dielectric layers,” in Proceedings of 1980 International URSI Symposium on Electromagnetic Waves (International Union of Radio Scientists, Brussels, 1980), pp. 341/B1–341/B4.

Sarid, D.

Shiau, M. J.

S. T. Peng, M. J. Shiau, “Scattering of plane waves by corrugated dielectric layers,” in Proceedings of 1983 International URSI Symposium on Electromagnetic Waves (International Union of Radio Scientists, Brussels, 1983) pp. 175–178.

Stegeman, G. I.

Streifer, W.

Tamir, T.

S. T. Peng, T. Tamir, “TM-mode perturbation analysis of dielectric gratings,” Appl. Phys. 7, 35–38 (1975).
[Crossref]

K. Handa, S. T. Peng, T. Tamir, “Improved perturbation analysis of dielectric gratings,” Appl. Phys. 5, 325–328 (1975).
[Crossref]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[Crossref]

Ulrich, R.

R. Ulrich, R. Zengerle, “Optical Bloch waves in periodic planar waveguides,” in Digest of Topical Meeting on Integrated and Guided-Wave Optics (Optical Society of America, Washington, D.C., 1980),paper TUB1.

Van Roey, J.

Weller-Brophy, L. A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), Sec. 1.6.

Zengerle, R.

R. Ulrich, R. Zengerle, “Optical Bloch waves in periodic planar waveguides,” in Digest of Topical Meeting on Integrated and Guided-Wave Optics (Optical Society of America, Washington, D.C., 1980),paper TUB1.

Appl. Opt. (1)

Appl. Phys. (2)

K. Handa, S. T. Peng, T. Tamir, “Improved perturbation analysis of dielectric gratings,” Appl. Phys. 5, 325–328 (1975).
[Crossref]

S. T. Peng, T. Tamir, “TM-mode perturbation analysis of dielectric gratings,” Appl. Phys. 7, 35–38 (1975).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[Crossref]

J. Opt. Soc. Am. (3)

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

Proc. IEEE (2)

C. Elachi, “Waves in active and passive periodic structures: a review,” Proc. IEEE 64, 1666–1698 (1976).
[Crossref]

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

Radio Sci. (1)

A. A. Oliner, S. T. Peng, “New physical effects on periodically grooved open dielectric waveguides,” Radio Sci. 19, 1251–1255 (1984).
[Crossref]

Other (6)

S. T. Peng, M. J. Shiau, “Scattering of plane waves by corrugated dielectric layers,” in Proceedings of 1983 International URSI Symposium on Electromagnetic Waves (International Union of Radio Scientists, Brussels, 1983) pp. 175–178.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), Sec. 1.6.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chaps. 1 and 2.

S. T. Peng, “Oblique guidance of surface waves on corrugated dielectric layers,” in Proceedings of 1980 International URSI Symposium on Electromagnetic Waves (International Union of Radio Scientists, Brussels, 1980), pp. 341/B1–341/B4.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chaps. 3 and 4.

R. Ulrich, R. Zengerle, “Optical Bloch waves in periodic planar waveguides,” in Digest of Topical Meeting on Integrated and Guided-Wave Optics (Optical Society of America, Washington, D.C., 1980),paper TUB1.

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Figures (5)

Fig. 1
Fig. 1

Configuration of dielectric grating waveguide: (a) plane-wave incidence in an arbitrary direction, (b) side view of the grating waveguide.

Fig. 2
Fig. 2

Periodic array of dielectric layers.

Fig. 3
Fig. 3

Relative orientations of eigen and structure coordinate systems.

Fig. 4
Fig. 4

Incidence of a surface wave onto a grating waveguide: (a) top view, (b) side view.

Fig. 5
Fig. 5

Ray picture of waveguiding process.

Tables (4)

Tables Icon

Table 1 Characteristic Solutions in the Periodic Medium

Tables Icon

Table 2 Fourier Representations for the Field Components

Tables Icon

Table 3 Field Components in the Structure Coordinate System

Tables Icon

Table 4 Fourier Components of the Tangential Fields in the Structure Coordinate System

Equations (160)

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k x = k sin θ cos ϕ ,
k y = k sin θ sin ϕ ,
k z = k cos θ ,
k x n = k x + 2 n π d             for n = , - 2 , - 1 , 0 , 1 , 2 , ,
k z n = ( k 2 - k x n 2 - k y 2 ) 1 / 2             for n , - 2 , - 1 , 0 , 1 , 2 , ,
k 2 = k 0 2 ,
H t n ( t g ) = - Y ˜ n ( up ) · [ z 0 × E t n ( t g ) ] ,
H t n ( 0 ) = Y ˜ n ( dn ) · [ z 0 × E t n ( 0 ) ] ,
Y ˜ n ( up ) = A ˜ n Y ˜ n ( a ) A ˜ n T ,
Y ˜ n ( dn ) = A ˜ n Y ˜ n ( i ) A ˜ n T ,
A ˜ n = 1 k t n [ k x n - k y k y k x n ] ,
Y ˜ n ( a ) = [ Y a n 0 0 Y a n ] ,
Y ˜ n ( i ) = [ Y i n 0 0 Y i n ] .
Y i n = Y f n Y s n + j Y f n tan k z n ( f ) t f Y f n + j Y s n tan k z n ( f ) t f ,
Y r n = { k z n ( r ) ω μ for TE modes ω 0 r k z n ( r ) for TM modes ,
V i ( x ) = E i + exp ( - j κ i x ) + E i - exp ( j κ i x ) ,
I i ( x ) = Y i [ E i + exp ( - j κ i x ) - E i - exp ( j κ i x ) ] ,
k i = ( k 0 2 i - k t 2 ) 1 / 2 ,
Y i = { κ i ω μ for TE modes ω i κ i for TM modes ,
E 1 + = ( 1 / 2 ) [ V 1 ( 0 ) + Z 1 I 1 ( 0 ) ] ,
E 1 - = ( 1 / 2 ) [ V 1 ( 0 ) - Z 1 I 1 ( 0 ) ] ,
[ V 1 ( x ) I 1 ( x ) ] = T ˜ 1 ( x ) [ V 1 ( 0 ) I 1 ( 0 ) ] ,
T ˜ 1 ( x ) = [ cos κ 1 x - j Z 1 sin κ 1 x - j Y 1 sin κ 1 x cos κ 1 x ] .
E 2 + = ( 1 / 2 ) [ V 2 ( d 1 ) + Z 2 I 2 ( d 1 ) ] exp ( j κ 2 d 1 ) ,
E 2 - = ( 1 / 2 ) [ V 2 ( d 1 ) - Z 2 I 2 ( d 1 ) ] exp ( - j κ 2 d 1 ) .
[ V 2 ( x ) I 2 ( x ) ] = T ˜ 2 ( x - d 1 ) [ V 2 ( d 1 ) I 2 ( d 1 ) ] ,
T ˜ 2 ( x - d 1 ) = [ cos κ 2 ( x - d 1 ) - j Z 2 sin κ 2 ( x - d 1 ) - j Y 2 sin κ 2 ( x - d 1 ) cos κ 2 ( x - d 1 ) ] .
[ V 1 ( d 1 ) I 1 ( d 1 ) ] = T ˜ 1 ( d 1 ) [ V 1 ( 0 ) I 1 ( 0 ) ] ,
[ V 2 ( d ) I 2 ( d ) ] = T ˜ 2 ( d 2 ) [ V 2 ( d 1 ) I 2 ( d 1 ) ] ,
[ V 1 ( d 1 ) I 1 ( d 1 ) ] = [ V 2 ( d 1 ) I 2 ( d 1 ) ] .
[ V 2 ( d ) I 2 ( d ) ] = T ˜ [ V 1 ( 0 ) I 1 ( 0 ) ] ,
T ˜ = T ˜ 2 ( d 2 ) T ˜ 1 ( d 1 ) .
T ˜ = [ T 11 T 12 T 21 T 22 ] ,
T 11 = cos κ 1 d 1 cos κ 2 d 2 - Z 2 Y 1 sin κ 1 d 1 sin κ 2 d 2 ,
T 12 = - j ( Z 1 sin κ 1 d 1 cos κ 2 d 2 + Z 2 sin κ 2 d 2 cos κ 1 d 1 ) ,
T 21 = - j ( Y 1 sin κ 1 d 1 cos κ 2 d 2 + Y 2 sin κ 2 d 2 cos κ 1 d 1 ) ,
T 22 = cos κ 1 d 1 cos κ 2 d 2 - Z 1 Y 2 sin κ 1 d 1 sin κ 2 d 2 .
T ˜ f = λ f .
λ 2 - ( Tr T ˜ ) λ + det ( T ˜ ) = 0 ,
λ 1 = exp ( - j κ d ) ,
λ 2 = exp ( j κ d ) ,
cos κ d = cos κ 1 d 1 cos κ 2 d 2 - 1 2 ( Y 1 Y 2 + Y 2 Y 1 ) sin κ 1 d 1 sin κ 2 d 2 ,
k t = k t v o ,
r t = u u o + v v o ,
[ y z ] = 1 k t [ k z k y - k y k z ] [ u v ] ,
ψ = sin - 1 ( k y / k t ) .
u = y cos ψ - z sin ψ ,
v = y sin ψ + z cos ψ .
u o = y ^ o cos ψ - z ^ o sin ψ ,
v o = y ^ o sin ψ + z ^ o cos ψ .
u y = v z = k z / k t ,
u z = v y = k y / k t .
k z m = [ ( k t m ) 2 - k y 2 ] 1 / 2             for TE modes ,
k z m = [ ( k t m ) 2 - k y 2 ] 1 / 2             for TM modes .
E x n ( g ) ( z ) = m V m n [ c m exp ( - j k z m z ) + d m exp ( j k z m z ) ] ,
- E y n ( g ) ( z ) = m V m n u y m [ c m exp ( - i k z m z ) + d m exp ( j k z m z ) ] + m G m n v y m × [ c m exp ( - j k z m z ) + d m exp ( j k z m z ) ] ,
H x n ( g ) ( z ) = m I m n [ c m exp ( - j k z m z ) - d m exp ( j k z m z ) ] ,
H y n ( g ) ( z ) = - m G m n v y m [ c m exp ( - j k z m z ) - d m exp ( j k z m z ) ] + m I m n u y m × [ c m exp ( - j k z m z ) - d m exp ( j k z m z ) ] .
E x = V ˜ [ exp ( - j K ˜ z ) c + exp ( j K ˜ z ) d ] ,
- E y = V ˜ [ exp ( - j K ˜ z ) c + exp ( j K ˜ z ) d ] + G ˜ [ exp ( - j K ˜ z ) c + exp ( j K ˜ z ) d ] ,
H x = I ˜ [ exp ( - j K ˜ z ) c - exp ( j K ˜ z ) d ] ,
H y = - G ˜ [ exp ( - j K ˜ z ) c - exp ( j K ˜ z ) d ] + I ˜ [ exp ( - j K ˜ z ) c - exp ( j K ˜ z ) d ] .
V ˜ = ( V m n u y m ) ,
V ˜ = ( V m n ) ,
I ˜ = ( I m n ) ,
I ˜ = ( I m n v y m ) ,
G ˜ = ( G m n v y m ) ,
G ˜ = ( G m n v y m ) .
E t ( z ) = P ˜ [ exp ( - j K ˜ z ) c + exp ( j K ˜ z ) d ] ,
H t ( z ) = Q ˜ [ exp ( - j K ˜ z ) c - exp ( j K ˜ z ) d ] .
E t ( z ) = [ - E y ( z ) E x ( z ) ] ,
H t ( z ) = [ H x ( z ) H y ( z ) ] ,
c = [ c c ] ,
d = [ d d ] ,
P ˜ = [ V ˜ G ˜ 0 ˜ V ˜ ] ,
Q ˜ = [ I ˜ 0 ˜ - G ˜ I ˜ ] ,
K ˜ = [ K ˜ 0 0 K ˜ ]
H t ( 0 ) = - Y ˜ ( up ) E t ( 0 ) ,
H t ( t g ) = Y ˜ ( dn ) E t ( t g ) ,
Y ˜ ( up ) = A ˜ Y ˜ ( a ) A ˜ T ,
Y ˜ ( dn ) = A ˜ Y ˜ ( i ) A ˜ T ,
A ˜ = [ A ˜ 11 A ˜ 12 A ˜ 21 A ˜ 22 ] ,
Y ˜ ( a ) = [ Y ˜ a 0 ˜ 0 ˜ Y ˜ a ] ,
Y ˜ ( i ) = [ Y ˜ i 0 ˜ 0 ˜ Y ˜ i ] .
d = exp ( - j K ˜ t g ) Γ ˜ out exp ( - j K ˜ t g ) c ,
Γ ˜ out = [ Q ˜ + Y ˜ ( dn ) P ˜ ] - 1 [ Q ˜ - Y ˜ ( dn ) P ˜ ] .
H t ( 0 ) = Y ˜ in E t ( 0 ) ,
Y ˜ in = Q ˜ [ 1 ˜ - exp ( - j K ˜ t g ) Γ ˜ out exp ( - j K ˜ t g ) ] × [ 1 ˜ + exp ( - j K ˜ t g ) Γ ˜ out exp ( - j K ˜ t g ) ] - 1 P ˜ - 1 .
E t ( t g ) = T ˜ E t ( 0 ) ,
T ˜ = P ˜ ( 1 ˜ + Γ ˜ out ) [ 1 ˜ + exp ( - j K ˜ t g ) Γ ˜ out exp ( - j K ˜ t g ) ] - 1 P ˜ - 1 .
E t ( 0 ) = A ˜ ( a + b ) ,
H ˜ t ( 0 ) = A ˜ Y ˜ ( a ) ( a - b ) ,
b = Γ ˜ a ,
Γ ˜ = [ A ˜ Y ˜ ( a ) + Y ˜ i n A ˜ ] - 1 [ A ˜ Y ˜ ( a ) - Y ˜ i n A ˜ ] .
H t ( 0 ) = Y ˜ ( up ) E t ( 0 ) ,
H t ( 0 ) = - Y ˜ ( dn ) E t ( 0 ) ,
[ Y ( up ) + Y ( dn ) ] E t ( 0 ) = 0.
det [ Y ( up ) + Y ( dn ) ] = 0 ,
det ( A ˜ Y ˜ ( a ) + Y ˜ in A ˜ ) = 0.
× E ( r ) = - j ω μ H ( r ) ,
× H ( r ) = j ω E ( r ) .
E ( r ) = E o exp ( - j k · r ) ,
H ( r ) = H o exp ( - j k · r ) ,
k × E o = ω μ H o ,
k × H o = - ω E o ,
k × ( k × E o ) = ω 2 μ E o ,
k 2 = ω 2 μ ,
v = v t + z ^ o v z ,
k t × E t = ω μ H z z ^ o ,
k t × z ^ o E z + k z z ^ o × E t = ω μ H t ,
k t × H t = - ω E z z ^ o ,
k t × z ^ o H z + k z z ^ o × H t = - ω E t .
z ^ o × E t = Z ˜ · H t ,
Z ˜ = ω μ k z k 2 ( k z 2 1 ˜ 2 + k t k t ) ,
H t = Y ˜ · ( z ^ o × E t ) ,
Y ˜ = ω k z k 2 ( k 2 1 ˜ 2 - k t k t ) ,
Z ˜ = Y ˜ - 1 .
Z = ω μ k z ,
Z = k z ω ,
a ^ = k t / k t ,
a ^ = z ^ o × ( k t / k t ) ,
H t = H o a ^ ,
E t = - E o a ^ ,
H z = k t ω μ E o ,
E z = 0
E t = E o a ^
H t = H o a ^ ,
E z = k z ω H o ,
H z = 0 ,
E o = Z H o ,
E o = Z H o .
z ^ o × E t = E o a ^ + E o a ^ ,
H t = H o a ^ + H o " a ^ .
Y = 1 Z = k z ω μ ,
Y = 1 Z = ω k z ,
H o = Y E o ,
H o = Y E o .
H t = Y ˜ · ( z ^ o × E t ) ,
Y ˜ = Y a ^ a ^ + Y a ^ a ^ .
a ^ = 1 k t [ k x k y ] ,
a ^ = 1 k t [ - k y k x ] .
[ - E y E x ] = A ˜ [ E o E o ] ,
[ H x H y ] = A ˜ [ H o H o ] ,
A ˜ = 1 k t [ k x - k y k y k x ] .
[ - E y E x ] = Y ˜ [ H x H y ] ,
Y ˜ = A ˜ Y ˜ c A ˜ T ,
Y ˜ c = [ Y 0 0 Y ] .
k z = ± ( k 2 - k t 2 ) 1 / 2 ,
E t = exp ( - j k t x ¯ ) V ( z ) a ^ ,
H t = exp ( - j k t x ¯ ) I ( z ) a ^ ,
E t = exp ( - j k t x ¯ ) V ( z ) a ^ ,
H t = exp ( - j k t x ¯ ) I ( z ) a ^ .
V ( z ) = E + exp ( - j k z z ) + E - exp ( j k Z z ) ,
I ( z ) = H + exp ( - j k z z ) + H - exp ( j k z z ) ,
H + = Y E + ,
H - = - Y E - ,
d d z V ( z ) = - j k z Z I ( z ) ,
d d z I ( z ) = - j k z Y V ( z ) ,
E t ( x ¯ , y ¯ , z ) = exp ( - j k t x ¯ ) [ V ( z ) a ^ + V ( z ) a ^ ] ,
H t ( x ¯ , y ¯ , z ) = exp ( - j k t x ¯ ) [ I ( z ) a ^ + I ( z ) a ^ ] .

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