Abstract

A wave-vector zigzag analysis is developed to investigate the complicated interference effects among the four extraordinary waves in a biaxial slab with general principal-axes orientation; the four corresponding zigzag Poynting vectors have distinct walk-off directions. Expressions for transmission and reflection coefficients, analogous to Airy’s summation for isotropic slabs, are determined for an arbitrarily oriented linearly polarized monochromatic plane wave at oblique incidence. A KTP slab is analyzed with the derived method, and the results are compared with those obtained with the 4 × 4 matrix method. Some common applications of this theory include analysis of multilayer structures and waveguides.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. S. Weis, T. K. Gaylord, “Fabry–Perot/Solč filter with distributed Bragg reflectors: a narrow-band electro-optically tunable spectral filter,” J. Opt. Soc. Am. A 5, 1565–1570 (1988).
    [CrossRef]
  2. R. S. Weis, T. K. Gaylord, “Electromagnetic transmission and reflection characteristics of anisotropic multilayered structures,” J. Opt. Soc. Am. A. 4, 1720–1739 (1987).
    [CrossRef]
  3. X. Wang, J. Yao, “Transmitted and tuning characteristics of birefringent filters,” Appl. Opt. 31, 4505–4508 (1992).
    [CrossRef] [PubMed]
  4. See, for example, L. Thylén, “Integrated optics in LiNbO3: recent developments in devices in telecommunications,” J. Lightwave Technol. 6, 847–861 (1988).
    [CrossRef]
  5. F. Flory, D. Endelema, E. Pelletier, I. Hodgkinson, “Anisotropy in thin films: modeling and measurement of guided and nonguided optical properties: application to TiO2films,” Appl. Opt. 32, 5649–5659 (1993).
    [CrossRef] [PubMed]
  6. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991). See, for example, P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  7. R. S. Weis, T. K. Gaylord, “Magnetooptic multilayered memory structure with a birefringent superstate: a rigorous analysis,” Appl. Opt. 28, 1926–1930 (1989).
    [CrossRef] [PubMed]
  8. Z. M. Li, B. T. Sullivan, R. R. Parsons, “Use of the 4 × 4 matrix method in the optics of multilayer magnetooptic recording media,” Appl. Opt. 27, 1334–1338 (1988).
    [CrossRef] [PubMed]
  9. J. Staromlynska, “A double-element broad-band liquid crystal tunable filter—factors affecting contrast ratio,” IEEE J. Quantum Electron. 28, 501–506 (1992).
    [CrossRef]
  10. D. W. Berreman, “Optics in smoothly varying anisotropic planar structures: application to liquid-crystal twist cells,”J. Opt. Soc. Am. 63, 1374–1379 (1973).
    [CrossRef]
  11. C. H. Kwak, J. T. Kim, S. S. Lee, “Nonlinear optical image processing in photoanisotropic amorphous As2S3thin film,” Appl. Opt. 28, 737–739 (1989).
    [CrossRef] [PubMed]
  12. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 6, pp. 205–208.
  13. T. A. Maldonado, T. K. Gaylord, “Electro-optic effect calculations: a simplified procedure for arbitrary cases,” Appl. Opt. 27, 5051–5066 (1988).
    [CrossRef] [PubMed]
  14. See, for example, Organic Thin Films for Photonic Applications, 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995).
  15. D. W. Berreman, “Optics in stratified and anisotropic media,”J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  16. P. Yeh, “Electromagnetic propagation in birefringent layered media,”J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  17. K. Hano, “Zigzag ray model of hybrid modes in thin-film optical waveguides with uniaxial anisotropic substrates,” J. Opt. Soc. Am. A 4, 1887–1894 (1987).
    [CrossRef]
  18. A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals: surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
    [CrossRef]
  19. T. A. Maldonado, T. K. Gaylord, “Hybrid guided modes in biaxial slab waveguides,” IEEE J. Lightwave Technol. 14, 486–499 (1996).
    [CrossRef]
  20. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 4, pp. 86–97.
  21. R. M. A. Azzam, N. H. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  22. G. D. Landry, T. A. Maldonado, “Complete method to determine transmission and reflection characteristics at a planar interface between arbitrarily oriented biaxial media,” J. Opt. Soc. Am. A 12, 2048–2063 (1995).
    [CrossRef]
  23. J. H. Jellet, S. Haughton, eds., The Collected Works of James MacCullagh (Hodges, Figgis, Dublin, 1880).

1996 (1)

T. A. Maldonado, T. K. Gaylord, “Hybrid guided modes in biaxial slab waveguides,” IEEE J. Lightwave Technol. 14, 486–499 (1996).
[CrossRef]

1995 (1)

1993 (1)

1992 (2)

X. Wang, J. Yao, “Transmitted and tuning characteristics of birefringent filters,” Appl. Opt. 31, 4505–4508 (1992).
[CrossRef] [PubMed]

J. Staromlynska, “A double-element broad-band liquid crystal tunable filter—factors affecting contrast ratio,” IEEE J. Quantum Electron. 28, 501–506 (1992).
[CrossRef]

1989 (2)

1988 (4)

1987 (2)

K. Hano, “Zigzag ray model of hybrid modes in thin-film optical waveguides with uniaxial anisotropic substrates,” J. Opt. Soc. Am. A 4, 1887–1894 (1987).
[CrossRef]

R. S. Weis, T. K. Gaylord, “Electromagnetic transmission and reflection characteristics of anisotropic multilayered structures,” J. Opt. Soc. Am. A. 4, 1720–1739 (1987).
[CrossRef]

1985 (1)

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals: surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

1979 (1)

1973 (1)

1972 (1)

Azzam, R. M. A.

R. M. A. Azzam, N. H. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. H.

R. M. A. Azzam, N. H. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Berreman, D. W.

Endelema, D.

Flory, F.

Gaylord, T. K.

T. A. Maldonado, T. K. Gaylord, “Hybrid guided modes in biaxial slab waveguides,” IEEE J. Lightwave Technol. 14, 486–499 (1996).
[CrossRef]

R. S. Weis, T. K. Gaylord, “Magnetooptic multilayered memory structure with a birefringent superstate: a rigorous analysis,” Appl. Opt. 28, 1926–1930 (1989).
[CrossRef] [PubMed]

R. S. Weis, T. K. Gaylord, “Fabry–Perot/Solč filter with distributed Bragg reflectors: a narrow-band electro-optically tunable spectral filter,” J. Opt. Soc. Am. A 5, 1565–1570 (1988).
[CrossRef]

T. A. Maldonado, T. K. Gaylord, “Electro-optic effect calculations: a simplified procedure for arbitrary cases,” Appl. Opt. 27, 5051–5066 (1988).
[CrossRef] [PubMed]

R. S. Weis, T. K. Gaylord, “Electromagnetic transmission and reflection characteristics of anisotropic multilayered structures,” J. Opt. Soc. Am. A. 4, 1720–1739 (1987).
[CrossRef]

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals: surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Hano, K.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991). See, for example, P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Hodgkinson, I.

Kim, J. T.

Knoesen, A.

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals: surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Kwak, C. H.

Landry, G. D.

Lee, S. S.

Li, Z. M.

Maldonado, T. A.

Moharam, M. G.

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals: surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Parsons, R. R.

Pelletier, E.

Staromlynska, J.

J. Staromlynska, “A double-element broad-band liquid crystal tunable filter—factors affecting contrast ratio,” IEEE J. Quantum Electron. 28, 501–506 (1992).
[CrossRef]

Sullivan, B. T.

Thylén, L.

See, for example, L. Thylén, “Integrated optics in LiNbO3: recent developments in devices in telecommunications,” J. Lightwave Technol. 6, 847–861 (1988).
[CrossRef]

Wang, X.

Weis, R. S.

Yao, J.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 6, pp. 205–208.

Yeh, P.

P. Yeh, “Electromagnetic propagation in birefringent layered media,”J. Opt. Soc. Am. 69, 742–756 (1979).
[CrossRef]

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 6, pp. 205–208.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 4, pp. 86–97.

Appl. Opt. (6)

Appl. Phys. B (1)

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals: surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

IEEE J. Lightwave Technol. (1)

T. A. Maldonado, T. K. Gaylord, “Hybrid guided modes in biaxial slab waveguides,” IEEE J. Lightwave Technol. 14, 486–499 (1996).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Staromlynska, “A double-element broad-band liquid crystal tunable filter—factors affecting contrast ratio,” IEEE J. Quantum Electron. 28, 501–506 (1992).
[CrossRef]

J. Lightwave Technol. (1)

See, for example, L. Thylén, “Integrated optics in LiNbO3: recent developments in devices in telecommunications,” J. Lightwave Technol. 6, 847–861 (1988).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. A. (1)

R. S. Weis, T. K. Gaylord, “Electromagnetic transmission and reflection characteristics of anisotropic multilayered structures,” J. Opt. Soc. Am. A. 4, 1720–1739 (1987).
[CrossRef]

Other (6)

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991). See, for example, P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 6, pp. 205–208.

See, for example, Organic Thin Films for Photonic Applications, 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 4, pp. 86–97.

R. M. A. Azzam, N. H. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

J. H. Jellet, S. Haughton, eds., The Collected Works of James MacCullagh (Hodges, Figgis, Dublin, 1880).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Coordinate system definition for a single biaxial layer between two isotropic regions. The two interfaces between the three regions lie in the (x, y) plane, and the plane of incidence is the (x, z) plane. The incident, the reflected, and the transmitted wave vectors are denoted by ki, kr, and kt, respectively. The vectors normal to the wave vectors are the electric displacement polarization vectors, D, which are parallel to the electric-field vectors, E, in the isotropic regions. The indices of refraction for the input and the output regions are ni and nt, respectively. The index of refraction for the slab is a tensor.

Fig. 2
Fig. 2

Isotropic layer wave vector geometry. All the angles of reflection inside the layer are equal to the incident refracted angle. All the transmitted waves propagate at the same angle, νt. All the reflected waves transmit at the same angle, νr.

Fig. 3
Fig. 3

Ellipsometric parameters for an isotropic slab (n2 = 1.7) in air versus the incidence angle νi, with λ0 = 1.064 μm, d = 100 μm, and θi = 75°. (a) Transmitted ellipticity; (b) transmitted rotated major axis, Δθ(deg); (c) transmitted intensity (normalized to the input intensity); (d) reflected ellipticity; (e) reflected rotated major axis, Δθ(deg); (f) reflected intensity (normalized to the input intensity).

Fig. 4
Fig. 4

(a) Reflected and (b) transmitted intensity for an isotropic slab (n2 = 1.7) in air versus νi, with λ0 = 1.064, d = 100 μm, θi = 75°. The results presented consider the following contributions: only zigzag 0, through zigzag 1, and all the zigzags.

Fig. 5
Fig. 5

Zigzag behavior of the wave vectors for four e waves in a biaxial slab. Within the slab, the two e waves A and B interfere with the two e waves C and D. One finds these four wave vectors by solving the Booker quartic as discussed in Ref. 22.

Fig. 6
Fig. 6

Polarization angle relationships of (a) the transmitted and (b) the reflected waves. The angles θA and θB for the transmitted A and B waves are not equal in general. The projections of these fields onto TE and TM directions can be used to find ellipsometric parameters for (a) transmitted and (b) reflected waves.

Fig. 7
Fig. 7

Amplitudes of the transmission coefficients tA and tB for the A and the B waves and reflection coefficients rC and rD for the C and the D waves, shown as a function of zigzag number for the example presented in Section 6. This example uses a 1-mm-thick slab of KTP oriented at the x-convention Euler angles of ϕ = 40°, θ = 80°, and ψ = 10°, with input beam parameters of λ0 = 1.064 μm, νi = 75°, and θi = 45°. The maximum magnitude for a given wave type can occur at any zigzag number, as is shown for tA and rD in this example.

Fig. 8
Fig. 8

Ellipsometric parameters versus zigzag number for the example presented in Section 6. All the plots are cumulative with respect to zigzag number. The dashed lines represent the complete solutions found by the 4 × 4 matrix method given by Berreman.15 This example uses a 1-mm-thick slab of KTP oriented at the x-convention Euler angles of ϕ = 40°, θ = 80°, and ψ = 10°, with input beam parameters of λ0 = 1.064 μm, νi = 75°, and θi = 45°. (a) Transmitted ellipticity; (b) transmitted rotated major axis, Δθ(deg); (c) transmitted intensity (normalized to the input intensity); (d) reflected ellipticity; (e) reflected rotated major axis, Δθ(deg); (f) reflected intensity (normalized to the input intensity).

Tables (1)

Tables Icon

Table 1 Calculated Wave Characteristics for the Example Presented in Section 6

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

t 0 α = t 12 α t 23 α exp ( j ϕ ) ,
ϕ = k 0 d n 2 cos ( υ 2 ) = k 0 d [ n 2 2 n i 2 sin 2 ( υ i ) ] 1 / 2 .
t TOT α = t 12 α t 23 α exp ( j ϕ ) + t 12 α ( r 23 α r 21 α ) t 23 α exp ( j 3 ϕ ) + t 12 α ( r 23 α r 21 α ) 2 t 23 α exp ( j 5 ϕ ) + = t 12 α t 23 α exp ( j ϕ ) 1 r 23 α r 21 α exp ( j 2 ϕ ) = t 12 α t 23 α exp ( j ϕ ) 1 + r 12 α r 23 α exp ( j 2 ϕ ) ,
r TOT α = r 12 α + t 12 α r 23 α t 21 α exp ( j 2 ϕ ) + t 12 α ( r 23 α r 21 α ) r 23 α t 21 α exp ( j 4 ϕ ) + t 12 α ( r 23 α r 21 α ) 2 r 23 α t 21 α exp ( j 6 ϕ ) + = r 12 α + t 12 α r 23 α t 21 α exp ( j 2 ϕ ) 1 r 23 α r 21 α exp ( j 2 ϕ ) = r 12 α + r 23 α exp ( j 2 ϕ ) 1 + r 12 α r 23 α exp ( j 2 ϕ ) .
t TOT = t TOT TE cos ( θ i ) + t TOT TM sin ( θ i ) , r TOT = r TOT TE cos ( θ i ) + r TOT TM sin ( θ i ) .
N A waves + N B waves = 2 2 z + 2 2 z = 2 2 z + 1 ,
ν t = sin 1 [ n i n t sin ( υ i ) ] ,
N C waves + N D waves = 2 2 z + 1 + 2 2 z + 1 = 2 2 z + 1 ,
θ new = π θ old .
t 0 A = t 12 A t 23 A exp ( j ϕ A ) ,
ϕ α = k 0 d n α cos ( ν α ) = k 0 d [ n α 2 n i 2 sin 2 ( ν i ) ] 1 / 2 ,
t 1 A = t 12 A exp ( j ϕ A ) [ r 23 A C exp ( j ϕ C ) r 21 C A exp ( j ϕ A ) ] + r 23 A D exp ( j ϕ D ) r 21 D A exp ( j ϕ A ) ] t 23 A + t 12 B exp ( j ϕ B ) [ r 23 B C exp ( j ϕ C ) r 21 C A exp ( j ϕ A ) ] + r 23 B D exp ( j ϕ D ) r 21 D A exp ( j ϕ A ) ] t 23 A = t 21 A exp ( j ϕ A ) κ A A t 23 A + t 12 B exp ( j ϕ B ) κ B A t 23 A ,
κ α β = exp ( j ϕ β ) [ r 23 α C r 21 C β exp ( j ϕ C ) + r 23 α D r 21 D β exp ( j ϕ D ) ] .
t 2 A = t 12 A exp ( j ϕ A ) κ A A κ A A t 23 A + t 12 A exp ( j ϕ A ) κ A B κ B A t 23 A + t 12 B exp ( j ϕ B ) κ B A κ A A t 23 A + t 12 B exp ( j ϕ B ) κ B B κ B A t 23 A ,
t 0 A + t 1 A + t 2 A = t 12 A t 23 A exp ( j ϕ A ) + t 12 A exp ( j ϕ A ) κ A A t 23 A + t 12 B exp ( j ϕ B ) κ B A t 23 A + t 12 A exp ( j ϕ A ) κ A A κ A A t 23 A + t 12 B exp ( j ϕ B ) κ B A κ A A t 23 A + t 12 A exp ( j ϕ A ) κ A B κ B A t 23 A + t 12 B exp ( j ϕ B ) κ B B κ B A t 23 A .
t TOT A = X A A K A A + X B A K B A ,
X A A = t 12 A exp ( j ϕ A ) t 23 A , X B A = t 12 B exp ( j ϕ B ) t 23 A κ B A , K A A = p = 0 κ A A p + m = 1 n = 0 q = 0 a m n q ( κ A B κ B A ) m κ A A n κ B B q , K B A = p = 0 κ A A p + m = 1 n = 0 q = .0 b m n q ( κ A B κ B A ) m κ A A n κ B B q ,
C m + n , m = ( m + n m ) = ( m + n ) ! m ! n ! .
C m + q 1 , q = ( m + q 1 q ) = ( m + q 1 ) ! q ! ( m 1 ) ! .
a m n q = C m + n , m C m + q 1 , q = ( m + n ) ! ( m + q 1 ) ! m ! n ! q ! ( m 1 ) ! .
b m n q = C m + n , m C m + q , q = ( m + n ) ! ( m + q ) ! m ! n ! q ! m ! .
t TOT A = X A A κ A A z + u ( z 1 ) X B A n = 0 z 1 κ A A n κ B B z n 1 + u ( z 2 ) X A A α = 1 z / 2 β = 0 z 2 α a α , β , z 2 α β × ( κ A B κ B A ) α κ A A β κ B B z 2 α β + u ( z 3 ) X B A α = 1 ( z 1 ) / 2 β = 0 z 1 2 α b α , β , z 2 α β 1 × ( κ A B κ B A ) α κ A A β κ B B z 2 α β 1 ,
t TOT B = X B B K B B + X A B K A B ,
X B B = t 12 B exp ( j ϕ B ) t 23 B , X A B = t 12 A exp ( j ϕ A ) t 23 B κ A B , K B B = p = 0 κ B B p + m = 1 n = 0 q = 0 a m n q ( κ B A κ A B ) m κ B B n κ A A q , K A B = p = 0 κ B B p + m = 1 n = 0 q = 0 b m n q ( κ B A κ A B ) m κ B B n κ A A q .
t TOT B = X B B κ B B z + u ( z 1 ) X A B n = 0 z 1 κ B B n κ A A z n 1 + u ( z 2 ) X B B α = 1 z / 2 β = 0 z 2 α a α , β , z 2 α β × ( κ B A κ A B ) α κ B B β κ A A z 2 α β + u ( z 3 ) X A B × α = 1 ( z 1 ) / 2 β = 0 z 1 2 α b α , β , z 2 α β 1 × ( κ B A κ A B ) α κ B B β κ A A z 2 α β 1 .
t TE = t TOT A cos ( θ A ) + t TOT B cos ( θ B ) , t TM = t TOT A sin ( θ A ) + t TOT B sin ( θ B ) .
r 0 C = t 12 A exp ( j ϕ A ) r 23 A C exp ( j ϕ C ) t 21 C + t 12 B exp ( j ϕ B ) r 23 B C exp ( j ϕ C ) t 21 C .
r 1 C = t 12 A exp ( j ϕ A ) exp ( j ϕ C ) t 21 C × { r 23 A C exp ( j ϕ C ) [ r 21 C A exp ( j ϕ A ) r 23 A C + r 21 C B exp ( j ϕ B ) r 23 B C ] + r 23 A D exp ( j ϕ D ) [ r 21 D A exp ( j ϕ A ) r 23 A C + r 21 D B exp ( j ϕ B ) r 23 B C ] } + t 12 B exp ( j ϕ B ) exp ( j ϕ C ) t 21 C × { r 23 B C exp ( j ϕ C ) [ r 21 C A exp ( j ϕ A ) r 23 A C + r 21 C B exp ( j ϕ B ) r 23 B C ] + r 23 B D exp ( j θ D ) [ r 21 D A exp ( j ϕ A ) r 23 A C + r 21 D B exp ( j ϕ B ) r 23 B C ] } = t 12 A exp ( j ϕ A ) t 21 C [ r 23 A C exp ( j ϕ C ) κ C C + r 23 A D exp ( j ϕ D ) κ D C ] + t 12 B exp ( j ϕ B ) t 21 C [ r 23 B C exp ( j ϕ C ) κ C C + r 23 A D exp ( j ϕ D ) κ D C ] ,
κ α β = exp ( j ϕ β ) [ r 21 σ A r 23 A β exp ( j ϕ A ) + r 21 α B r 23 B β exp ( j ϕ β ) ] .
r TOT C = X C C K C C + X D C K D C ,
X C C = [ t 12 A r 23 A C exp ( j ϕ A ) + t 12 B r 23 B C exp ( j ϕ B ) ] exp ( j ϕ C ) t 21 C , X D C = [ t 12 A r 23 A D exp ( j ϕ A ) + t 12 B r 23 B D exp ( j ϕ B ) ] exp ( j ϕ D ) t 21 C κ D C , K C C = p = 0 κ C C p + m = 1 n = 0 q = 0 a m n q ( κ C D κ D C ) m κ C C n κ D D q , K D C = p = 0 κ C C p + m = 1 n = 0 q = 0 b m n q ( κ C D κ D C ) m κ C C n κ D D q .
r TOT C = X C C κ C C z + u ( z 1 ) X D C n = 0 z 1 κ C C n κ D D z n 1 + u ( z 2 ) X C C α = 1 z / 2 β = 0 z 2 α α α , β , z 2 α β ( κ C D κ D C ) α × κ C C β κ D D z 2 α β + u ( z 3 ) X D C α = 1 ( z 1 ) / 2 β = 0 z 1 2 α b α , β , z 2 α β 1 × ( κ C D κ D c ) α κ C C β κ D D z 2 α β 1 .
r TOT D = X D D K D D + X C D K C D ,
X D D = [ t 12 A r 23 A D exp ( j ϕ A ) + t 12 B r 23 B D exp ( j ϕ B ) ] exp ( j ϕ D ) t 21 D , X C D = [ t 12 A r 23 A C exp ( j ϕ A ) + t 12 B r 23 B C exp ( j ϕ B ) ] exp ( j ϕ C ) t 21 D κ C D , K D D = p = 0 κ D D p + m = 1 n = 0 q = 0 a m n q ( κ D C κ C D ) m κ D D n κ C C q , K C D = p = 0 κ D D p + m = 1 n = 0 q = 0 b m n q ( κ D C κ C D ) m κ D D n κ C C q ,
r TOT D = X D D κ D D z + u ( z 1 ) X C D n = 0 z 1 κ D D n κ C C z n 1 + u ( z 2 ) X D D α = 1 z / 2 β = 0 z 2 α a α , β , z 2 α β × ( κ D C κ C D ) α κ D D β κ C C z 2 α β + u ( z 3 ) X C D α = 1 ( z 1 ) / 2 β = 0 z 1 2 α b α , β , z 2 α β 1 × ( κ D C κ C D ) α κ D D β κ C C z 2 α β 1 .
r TE = r 12 cos ( θ r ) + r TOT C cos ( θ C ) + r TOT D cos ( θ D ) , r TM = r 12 sin ( θ r ) + r TOT C sin ( θ C ) + r TOT D sin ( θ D ) .
κ A A = 0.36798 j 0.043638 , κ A B = 0.18120 + j 0.0078612 , κ B A = 0.16970 j 0.046286 , κ B B = 0.19111 + j 0.00083247 , κ C C = 0.33120 + j 0.047433 , κ C D = 0.19230 + j 0.080115 , κ D C = 0.18741 j 0.057515 , κ D D = 0.22789 j 0.090238.
X A A = 0.019584 j 0.061206 , X B A = 0.38616 + j 0.54192 , X A B = 0.43777 + j 0.61435 , X B B = 0.022202 j 0.069386 , X C C = 0.32227 j 0.010134 , X D C = 0.047283 j 0.025225 , X C D = 0.049000 j 0.026141 , X D D = 0.33398 j 0.010502.

Metrics