Abstract

A fully vectorial treatment of the problem of scattering by singly periodic interfaces between chiral and achiral media has been presented. By means of an improved T-matrix method of solution, this treatment is able to bypass the restrictions imposed by the Rayleigh hypothesis. In addition, solutions to several other related problems can be obtained by the method presented here. The specific features of the solution procedure allow it to be converted easily for doubly periodic interfaces. The analysis may be useful for developing new surface-relief gratings made using chiral materials.

© 1989 Optical Society of America

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References

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  1. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
    [CrossRef]
  2. B. A. Lippmann, “Note on the theory of gratings,”J. Opt. Soc. Am. 43, 408 (1953).
    [CrossRef]
  3. L. Fortuin, “Survey of literature on reflection and scattering of sound waves at the sea surface,”J. Acoust. Soc. Am. 47, 1209–1228 (1970).
    [CrossRef]
  4. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (Elsevier, Amsterdam, 1984).
    [CrossRef]
  5. R. Petit, ed., Electromagnetic Theory Of Gratings (Springer-Verlag, Heidelberg, 1980).
    [CrossRef]
  6. R. A. Depine, J. M. Simon, “Comparison between the differential and integral methods used to solve the grating problem in the H||case,” J. Opt. Soc. Am. A 4, 834–838 (1987).
    [CrossRef]
  7. P. C. Waterman, “Scattering by periodic surfaces,”J. Acoust. Soc. Am. 57, 791–802 (1975).
    [CrossRef]
  8. W. N. Cain, V. K. Varadan, V. V. Varadan, A. Lakhtakia, “Reflection and transmission characteristics of a slab with periodically varying surfaces,”IEEE Trans. Antennas Propag. AP-34, 1159–1163 (1986).
    [CrossRef]
  9. S.-L. Chuang, J. A. Kong, “Scattering from periodic surfaces,” Proc. IEEE 69, 1132–1144 (1981).
    [CrossRef]
  10. P. M. van den Berg, “Rigorous diffraction theory of optical reflection and transmission gratings,” thesis report 1971-16 (Delft University of Technology, Delft, The Netherlands, 1971).
  11. K. C. Chang, V. Shah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,”J. Opt. Soc. Am. 70, 804–813 (1980).
    [CrossRef]
  12. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,”J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  13. A. Lakhtakia, V. V. Varadan, V. K. Varadan, “Field equations, Huygens’s principle, integral equations, and theorems for radiation and scattering of electromagnetic waves in isotropic chiral media,” J. Opt. Soc. Am. A 5, 175–184 (1988).
    [CrossRef]
  14. S. Bassiri, N. Engheta, C. H. Papas, “Dyadic Green’s function and dipole radiation in chiral media,” Alta Freq. 55, 83–88 (1986).
  15. W. Weiglhofer, “Isotropic chiral media and scalar Hertz potentials,”J. Phys. A 21, 2249–2251 (1988).
    [CrossRef]
  16. R. I. Masel, R. P. Merrill, W. H. Miller, “Quantum scattering from a sinusoidal hard wall: atomic diffraction from solid surfaces,” Phys. Rev. B 12, 5545–5551 (1975).
    [CrossRef]
  17. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  18. A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Regarding the sources of radiation fields in an isotropic chiral medium,”J. Wave Mater. Interact. 2, 183–189 (1987).
  19. A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Excitation of layered media having rough interfaces by line sources,”IEEE Trans. Antennas Propag. AP-35, 462–466 (1987).
    [CrossRef]
  20. A. Lakhtakia, V. K. Varadan, V. V. Varadan, “On the acoustic response of a deeply corrugated periodic surface—a hybrid T-matrix approach,”J. Acoust. Soc. Am. 78, 2100–2104 (1985).
    [CrossRef]
  21. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1956).
  22. S. Bassiri, C. H. Papas, N. Engheta, “Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450–1459 (1988).
    [CrossRef]
  23. A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Elastodynamic radiation from line sources buried in layered media having periodically varying interfaces,” ASME J. Vib. Acoust. Stress Reliab. Design 109, 69–74 (1987).
    [CrossRef]
  24. A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Scattering by a partially illuminated, doubly periodic, doubly infinite surface,”J. Acoust. Soc. Am. 77, 1999–2004 (1985).
    [CrossRef]

1988 (3)

1987 (4)

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Elastodynamic radiation from line sources buried in layered media having periodically varying interfaces,” ASME J. Vib. Acoust. Stress Reliab. Design 109, 69–74 (1987).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Regarding the sources of radiation fields in an isotropic chiral medium,”J. Wave Mater. Interact. 2, 183–189 (1987).

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Excitation of layered media having rough interfaces by line sources,”IEEE Trans. Antennas Propag. AP-35, 462–466 (1987).
[CrossRef]

R. A. Depine, J. M. Simon, “Comparison between the differential and integral methods used to solve the grating problem in the H||case,” J. Opt. Soc. Am. A 4, 834–838 (1987).
[CrossRef]

1986 (2)

W. N. Cain, V. K. Varadan, V. V. Varadan, A. Lakhtakia, “Reflection and transmission characteristics of a slab with periodically varying surfaces,”IEEE Trans. Antennas Propag. AP-34, 1159–1163 (1986).
[CrossRef]

S. Bassiri, N. Engheta, C. H. Papas, “Dyadic Green’s function and dipole radiation in chiral media,” Alta Freq. 55, 83–88 (1986).

1985 (3)

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “On the acoustic response of a deeply corrugated periodic surface—a hybrid T-matrix approach,”J. Acoust. Soc. Am. 78, 2100–2104 (1985).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Scattering by a partially illuminated, doubly periodic, doubly infinite surface,”J. Acoust. Soc. Am. 77, 1999–2004 (1985).
[CrossRef]

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

1982 (1)

1981 (1)

S.-L. Chuang, J. A. Kong, “Scattering from periodic surfaces,” Proc. IEEE 69, 1132–1144 (1981).
[CrossRef]

1980 (1)

1975 (2)

P. C. Waterman, “Scattering by periodic surfaces,”J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

R. I. Masel, R. P. Merrill, W. H. Miller, “Quantum scattering from a sinusoidal hard wall: atomic diffraction from solid surfaces,” Phys. Rev. B 12, 5545–5551 (1975).
[CrossRef]

1970 (1)

L. Fortuin, “Survey of literature on reflection and scattering of sound waves at the sea surface,”J. Acoust. Soc. Am. 47, 1209–1228 (1970).
[CrossRef]

1953 (1)

1907 (1)

Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Bassiri, S.

S. Bassiri, C. H. Papas, N. Engheta, “Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450–1459 (1988).
[CrossRef]

S. Bassiri, N. Engheta, C. H. Papas, “Dyadic Green’s function and dipole radiation in chiral media,” Alta Freq. 55, 83–88 (1986).

Cain, W. N.

W. N. Cain, V. K. Varadan, V. V. Varadan, A. Lakhtakia, “Reflection and transmission characteristics of a slab with periodically varying surfaces,”IEEE Trans. Antennas Propag. AP-34, 1159–1163 (1986).
[CrossRef]

Chang, K. C.

Chuang, S.-L.

S.-L. Chuang, J. A. Kong, “Scattering from periodic surfaces,” Proc. IEEE 69, 1132–1144 (1981).
[CrossRef]

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1956).

Depine, R. A.

Engheta, N.

S. Bassiri, C. H. Papas, N. Engheta, “Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450–1459 (1988).
[CrossRef]

S. Bassiri, N. Engheta, C. H. Papas, “Dyadic Green’s function and dipole radiation in chiral media,” Alta Freq. 55, 83–88 (1986).

Fortuin, L.

L. Fortuin, “Survey of literature on reflection and scattering of sound waves at the sea surface,”J. Acoust. Soc. Am. 47, 1209–1228 (1970).
[CrossRef]

Gaylord, T. K.

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

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,”J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

Kong, J. A.

S.-L. Chuang, J. A. Kong, “Scattering from periodic surfaces,” Proc. IEEE 69, 1132–1144 (1981).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, V. V. Varadan, V. K. Varadan, “Field equations, Huygens’s principle, integral equations, and theorems for radiation and scattering of electromagnetic waves in isotropic chiral media,” J. Opt. Soc. Am. A 5, 175–184 (1988).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Elastodynamic radiation from line sources buried in layered media having periodically varying interfaces,” ASME J. Vib. Acoust. Stress Reliab. Design 109, 69–74 (1987).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Regarding the sources of radiation fields in an isotropic chiral medium,”J. Wave Mater. Interact. 2, 183–189 (1987).

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Excitation of layered media having rough interfaces by line sources,”IEEE Trans. Antennas Propag. AP-35, 462–466 (1987).
[CrossRef]

W. N. Cain, V. K. Varadan, V. V. Varadan, A. Lakhtakia, “Reflection and transmission characteristics of a slab with periodically varying surfaces,”IEEE Trans. Antennas Propag. AP-34, 1159–1163 (1986).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “On the acoustic response of a deeply corrugated periodic surface—a hybrid T-matrix approach,”J. Acoust. Soc. Am. 78, 2100–2104 (1985).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Scattering by a partially illuminated, doubly periodic, doubly infinite surface,”J. Acoust. Soc. Am. 77, 1999–2004 (1985).
[CrossRef]

Lippmann, B. A.

Masel, R. I.

R. I. Masel, R. P. Merrill, W. H. Miller, “Quantum scattering from a sinusoidal hard wall: atomic diffraction from solid surfaces,” Phys. Rev. B 12, 5545–5551 (1975).
[CrossRef]

Maystre, D.

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (Elsevier, Amsterdam, 1984).
[CrossRef]

Merrill, R. P.

R. I. Masel, R. P. Merrill, W. H. Miller, “Quantum scattering from a sinusoidal hard wall: atomic diffraction from solid surfaces,” Phys. Rev. B 12, 5545–5551 (1975).
[CrossRef]

Miller, W. H.

R. I. Masel, R. P. Merrill, W. H. Miller, “Quantum scattering from a sinusoidal hard wall: atomic diffraction from solid surfaces,” Phys. Rev. B 12, 5545–5551 (1975).
[CrossRef]

Moharam, M. G.

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

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,”J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

Papas, C. H.

S. Bassiri, C. H. Papas, N. Engheta, “Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450–1459 (1988).
[CrossRef]

S. Bassiri, N. Engheta, C. H. Papas, “Dyadic Green’s function and dipole radiation in chiral media,” Alta Freq. 55, 83–88 (1986).

Rayleigh,

Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Shah, V.

Simon, J. M.

Tamir, T.

van den Berg, P. M.

P. M. van den Berg, “Rigorous diffraction theory of optical reflection and transmission gratings,” thesis report 1971-16 (Delft University of Technology, Delft, The Netherlands, 1971).

Varadan, V. K.

A. Lakhtakia, V. V. Varadan, V. K. Varadan, “Field equations, Huygens’s principle, integral equations, and theorems for radiation and scattering of electromagnetic waves in isotropic chiral media,” J. Opt. Soc. Am. A 5, 175–184 (1988).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Excitation of layered media having rough interfaces by line sources,”IEEE Trans. Antennas Propag. AP-35, 462–466 (1987).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Regarding the sources of radiation fields in an isotropic chiral medium,”J. Wave Mater. Interact. 2, 183–189 (1987).

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Elastodynamic radiation from line sources buried in layered media having periodically varying interfaces,” ASME J. Vib. Acoust. Stress Reliab. Design 109, 69–74 (1987).
[CrossRef]

W. N. Cain, V. K. Varadan, V. V. Varadan, A. Lakhtakia, “Reflection and transmission characteristics of a slab with periodically varying surfaces,”IEEE Trans. Antennas Propag. AP-34, 1159–1163 (1986).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Scattering by a partially illuminated, doubly periodic, doubly infinite surface,”J. Acoust. Soc. Am. 77, 1999–2004 (1985).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “On the acoustic response of a deeply corrugated periodic surface—a hybrid T-matrix approach,”J. Acoust. Soc. Am. 78, 2100–2104 (1985).
[CrossRef]

Varadan, V. V.

A. Lakhtakia, V. V. Varadan, V. K. Varadan, “Field equations, Huygens’s principle, integral equations, and theorems for radiation and scattering of electromagnetic waves in isotropic chiral media,” J. Opt. Soc. Am. A 5, 175–184 (1988).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Elastodynamic radiation from line sources buried in layered media having periodically varying interfaces,” ASME J. Vib. Acoust. Stress Reliab. Design 109, 69–74 (1987).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Regarding the sources of radiation fields in an isotropic chiral medium,”J. Wave Mater. Interact. 2, 183–189 (1987).

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Excitation of layered media having rough interfaces by line sources,”IEEE Trans. Antennas Propag. AP-35, 462–466 (1987).
[CrossRef]

W. N. Cain, V. K. Varadan, V. V. Varadan, A. Lakhtakia, “Reflection and transmission characteristics of a slab with periodically varying surfaces,”IEEE Trans. Antennas Propag. AP-34, 1159–1163 (1986).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “On the acoustic response of a deeply corrugated periodic surface—a hybrid T-matrix approach,”J. Acoust. Soc. Am. 78, 2100–2104 (1985).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Scattering by a partially illuminated, doubly periodic, doubly infinite surface,”J. Acoust. Soc. Am. 77, 1999–2004 (1985).
[CrossRef]

Waterman, P. C.

P. C. Waterman, “Scattering by periodic surfaces,”J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

Weiglhofer, W.

W. Weiglhofer, “Isotropic chiral media and scalar Hertz potentials,”J. Phys. A 21, 2249–2251 (1988).
[CrossRef]

Alta Freq. (1)

S. Bassiri, N. Engheta, C. H. Papas, “Dyadic Green’s function and dipole radiation in chiral media,” Alta Freq. 55, 83–88 (1986).

ASME J. Vib. Acoust. Stress Reliab. Design (1)

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Elastodynamic radiation from line sources buried in layered media having periodically varying interfaces,” ASME J. Vib. Acoust. Stress Reliab. Design 109, 69–74 (1987).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Excitation of layered media having rough interfaces by line sources,”IEEE Trans. Antennas Propag. AP-35, 462–466 (1987).
[CrossRef]

W. N. Cain, V. K. Varadan, V. V. Varadan, A. Lakhtakia, “Reflection and transmission characteristics of a slab with periodically varying surfaces,”IEEE Trans. Antennas Propag. AP-34, 1159–1163 (1986).
[CrossRef]

J. Acoust. Soc. Am. (4)

P. C. Waterman, “Scattering by periodic surfaces,”J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

L. Fortuin, “Survey of literature on reflection and scattering of sound waves at the sea surface,”J. Acoust. Soc. Am. 47, 1209–1228 (1970).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “On the acoustic response of a deeply corrugated periodic surface—a hybrid T-matrix approach,”J. Acoust. Soc. Am. 78, 2100–2104 (1985).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Scattering by a partially illuminated, doubly periodic, doubly infinite surface,”J. Acoust. Soc. Am. 77, 1999–2004 (1985).
[CrossRef]

J. Opt. Soc. Am. (3)

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

J. Phys. A (1)

W. Weiglhofer, “Isotropic chiral media and scalar Hertz potentials,”J. Phys. A 21, 2249–2251 (1988).
[CrossRef]

J. Wave Mater. Interact. (1)

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Regarding the sources of radiation fields in an isotropic chiral medium,”J. Wave Mater. Interact. 2, 183–189 (1987).

Phys. Rev. B (1)

R. I. Masel, R. P. Merrill, W. H. Miller, “Quantum scattering from a sinusoidal hard wall: atomic diffraction from solid surfaces,” Phys. Rev. B 12, 5545–5551 (1975).
[CrossRef]

Proc. IEEE (2)

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

S.-L. Chuang, J. A. Kong, “Scattering from periodic surfaces,” Proc. IEEE 69, 1132–1144 (1981).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Other (4)

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (Elsevier, Amsterdam, 1984).
[CrossRef]

R. Petit, ed., Electromagnetic Theory Of Gratings (Springer-Verlag, Heidelberg, 1980).
[CrossRef]

P. M. van den Berg, “Rigorous diffraction theory of optical reflection and transmission gratings,” thesis report 1971-16 (Delft University of Technology, Delft, The Netherlands, 1971).

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1956).

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Figures (5)

Fig. 1
Fig. 1

Specular reflection and transmission coefficients computed for the sinusoidal interface z = h cos(2πx/L) as functions of the normalized frequency k0L, for h/L = 0.1, θinc = 30°, /0 = 5, and β/L = 0: (—) R11,0, (- - -) R22,0, (· · · ·) TL1,0 = TR1,0, and (·-·-·-·) TL2,0 = TR2,0. For all n, R12,n = R21,n ≡ 0.

Fig. 2
Fig. 2

Specular reflection and transmission coefficients computed for the sinusoidal interface z = h cos(2πx/L) as functions of the normalized frequency k0L, for h/L = 0.1, θinc = 30°, /0 = 5, and β/L = 10−3: (—) R11,0, (- - -) TR1,0, and (·-·-·-·) TL1,0. The depolarized component R21,0 is of the order of 10−6–10−7.

Fig. 3
Fig. 3

Specular reflection and transmission coefficients computed for the sinusoidal interface z = h cos(2πx/L) as functions of the normalized frequency k0L, for h/L = 0.1, θinc = 30°, /0 = 5, and β/L = 10−3: (—) R22,0, (- - -) TR2,0, and (·-·-·-·) TL2,0. The depolarized component R12,0 is of the order of 10−6–10−7.

Fig. 4
Fig. 4

Specular reflection and transmission coefficients computed for the sinusoidal interface z = h cos(2πx/L) as functions of the angle of incidence θinc, for h/L = 0.1, k0L = 10, /0 = 5, and β/L = 10−3: (—) R11,0, (- - -) TR1,0, and (·-·-·-·) TL1,0.

Fig. 5
Fig. 5

Specular reflection and transmission coefficients computed for the sinusoidal interface z = h cos(2πx/L) as functions of the angle of incidence θinc for h/L = 0.1, k0L = 10, /0 = 5, and β/L = 10−3: (—) R22,0, (- - -) TR2,0, and (·-·-·-·) TL2,0.

Tables (2)

Tables Icon

Table 1 Comparison of Results of the Method Presented with Those of Integral Equation Methods for the Achiral–Achiral Sinusoidal Interface z = h cos(2πx/L) and a TE-Polarized Incident Plane Wavea

Tables Icon

Table 2 Comparison of Results of the Method Presented with Those of Integral Equation Methods for the Achiral–Achiral Sinusoidal Interface z = h cos(2πx/L) and a TM-Polarized Incident Plane Wavea

Equations (52)

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D = ( E + β × E ) ,
B = μ ( H × β × H ) ,
± E f ± ( r ) = S d s { i ω μ 0 G 0 ( R ) · [ n ^ ( r ) × H + ( r ) ] + [ × G 0 ( R ) ] · [ n ^ ( r ) × E + ( r ) ] } ,
G 0 ( R ) = ( i / 4 ) ( J + / k 0 2 ) H 0 ( k 0 R ) .
H 0 ( k 0 R ) = ( 1 / π ) - d κ ( k 0 2 - κ 2 ) - 1 / 2 × exp { i [ κ e ^ x ± ( k 0 2 - κ 2 ) 1 / 2 e ^ z ] · R } ,
H + ( r + m L e ^ x ) = H + ( r ) exp ( i m κ 0 L ) , m = 0 , ± 1 , ± 2 , , r S , r + m L e ^ x S ,    
E + ( r + m L e ^ x ) = E + ( r ) exp ( i m κ 0 L ) , m = 0 , ± 1 , ± 2 , , r S , r + m L e ^ x S ,    
± 2 L E f ± ( r ) = - ω μ 0 n α n - 1 S d s exp [ i ( κ n e x ± α n e ^ z ) · R ] × [ n ^ ( r ) × H + ( r ) ] - n α n - 1 S d s exp [ i ( κ n e x ± α n e ^ z ) · R ] × [ J × ( κ n e ^ x ± α n e ^ z ) ] · [ n ^ ( r ) × E + ( r ) ] .
κ n = κ 0 + 2 n π / L ,             n = 0 , ± 1 , ± 2 , ,
α n = + ( k 0 2 - κ n 2 ) 1 / 2 .
E f ± ( r ) = n [ A n ± e ^ y - B n ± ( ± α n e ^ x - κ n e ^ z ) / k 0 ] ( k 0 / α n ) × exp [ i ( κ n x ± α n z ) ] ,
± A n ± = - ( ω μ 0 / 2 k 0 L ) S d s exp [ - i ( κ n e ^ x ± α n e ^ z ) · r ] × e ^ y · [ n ^ ( r ) × H + ( r ) ] - ( 1 / 2 k 0 L ) S d s exp [ - i ( κ n e ^ x ± α n e ^ z ) · r ] × ( ± α n e ^ x - κ n e ^ z ) · [ n ^ ( r ) × E + ( r ) ] ,
± B n ± = ( 1 / 2 ω 0 L ) S d s exp [ - i ( κ n e ^ x ± α n e ^ z ) · r ] × ( ± α n e ^ x - κ n e ^ z ) · [ n ^ ( r ) × H + ( r ) ] - ( 1 / 2 L ) S d s exp [ - i ( κ n e ^ x ± α n e z ) · r ] × e ^ y · [ n ^ ( r ) × E + ( r ) ] .
± E c ± ( r ) = ( γ 1 γ 2 / k 2 ) S d s { G 1 ( R ) · n ^ ( r ) × [ i ω μ H - ( r ) + k E - ( r ) } + ( γ 1 γ 2 / k 2 ) × S d s { G 2 ( R ) · n ^ ( r ) × [ i ω μ H - ( r ) - k E - ( r ) ] } ,
γ 1 = k / ( 1 - k β ) ,
γ 2 = k / ( 1 - k β ) ,
G 1 ( R ) = ( i k / 8 γ 1 γ 2 ) ( γ 1 J + / γ 1 + × J ) H 0 ( γ 1 R ) ,
G 2 ( R ) = ( i k / 8 γ 1 γ 2 ) ( γ 2 J + / γ 2 - × J ) H 0 ( γ 2 R ) ,
± E c ± ( r ) = ( i γ 1 / 4 k L ) n ξ 1 n - 1 [ J + i J × ( κ n e ^ x ± ξ 1 n e ^ z ) / γ 1 ] · S d s exp [ i ( κ n e ^ x ± ξ 1 n e ^ z ) · R ] n ^ ( r ) × [ i ω μ H - ( r ) + k E - ( r ) ] + ( i γ 2 / 4 k L ) n ξ 2 n - 1 × [ I - i I × ( κ n e ^ x ± ξ 2 n e ^ z ) / γ 2 ] · S d s exp [ i ( κ n e ^ x ± ξ 2 n e ^ z ) · R ] × n ^ ( r ) × [ i ω μ H - ( r ) - k E - ( r ) ] ,
ξ 1 n = + ( γ 1 2 - κ n 2 ) 1 / 2 ,
ξ 2 n = + ( γ 2 2 - κ n 2 ) 1 / 2 .
E c ± = n C n ± e 1 n ± exp [ i ( κ n x ± ξ 1 n z ) ] ( γ 1 / ξ 1 n ) + n D n ± a R e 2 n ± exp [ i ( κ n x ± ξ 2 n z ) ] ( γ 2 / ξ 2 n ) ,
e 1 n ± = ( 1 / 2 ) [ e ^ y - i ( ± ξ 1 n e ^ x - κ n e ^ z ) / γ 1 ] ,
e 2 n ± = ( 1 / 2 ) [ e ^ y + i ( ± ξ 2 n e ^ x - κ n e ^ z ) / γ 2 ] .
± C n ± e 1 n ± = ( i / 4 k L ) [ I + i I × ( κ n e ^ x ± ξ 1 n e ^ z ) / γ 1 ] · S d s exp [ - i ( κ n e ^ x ± ξ 1 n e ^ z ) · r ] n ^ ( r ) × [ i ω μ H - ( r ) + k E - ( r ) ]
± a R D n ± e 2 n ± = ( i / 4 k L ) [ J - i J × ( κ n e ^ x ± ξ 2 n e ^ z ) / γ 2 ] · S d s exp [ - i ( κ n e ^ x ± ξ 2 n e ^ z ) · r ] n ^ ( r ) × [ i ω μ H - ( r ) - k E - ( r ) ] .
± C n ± = ( i / k L 8 ) S d s exp [ - i ( κ n e ^ x ± ξ 1 n e ^ z ) · r ] × [ e ^ y + i ( ± ξ 1 n e ^ x - κ n e ^ z ) / γ 1 ] · n ^ ( r ) × [ i ω μ H - ( r ) + k E - ( r ) ] ,
± a R D n ± = ( i / k L 8 ) S d s exp [ - i ( κ n e x ± ξ 2 n e z ) · r ] × [ e ^ y - i ( ± ξ 2 n e x - κ n e z ) / γ 2 ] · n ^ ( r ) × [ i ω μ H - ( r ) - k E - ( r ) ] .
n ^ d s × H + = n ^ d s × H - = d x m [ P 1 m ( e ^ x + F e ^ z ) + P 2 m e ^ y ] exp ( i κ m x ) ,
n ^ d s × E + = n ^ d s × E - = d x m [ Q 1 m ( e ^ x + F e ^ z ) + Q 2 m e ^ y ] exp ( i κ m x ) ,
[ A n - B n - C n + a R D n + ] = [ 0 ( ω μ 0 / 2 k 0 ) · I n m ( α n ) ( 1 / 2 k 0 ) · K n m ( α n ) 0 ( - 1 / 2 ω 0 ) · K n m ( α n ) 0 0 ( 1 / 2 ) I n m ( α n ) ( - i ω μ / k γ 1 ) · K n m ( - ξ 1 n ) / 8 ( - ω μ / k ) · I n m ( - ξ 1 n ) / 8 ( - 1 / γ 1 ) · K n m ( - ξ 1 n ) / 8 ( i / 8 ) I n m ( - ξ 1 n ) ( i ω μ / k γ 2 ) · K n m ( - ξ 2 n ) / 8 ( - ω μ / k ) · I n m ( - ξ 2 n ) / 8 ( - 1 / γ 2 ) · K n m ( - ξ 2 n ) / 8 ( - i / 8 ) I n m ( - ξ 2 n ) ] [ P 1 m P 2 m Q 1 m Q 2 m ]
[ A n + B n + C n - a R D n - ] = [ 0 ( - ω μ 0 / 2 k 0 ) · I n m ( - α n ) ( - 1 / 2 k 0 ) · K n m ( - α n ) 0 ( - 1 / 2 ω 0 ) · K n m ( - α n ) 0 0 ( - 1 / 2 ) I n m ( - α n ) ( i ω μ / k γ 1 ) · K n m ( ξ 1 n ) / 8 ( ω μ / k ) · I n m ( ξ 1 n ) / 8 ( 1 / γ 1 ) · K n m ( ξ 1 n ) / 8 ( - i / 8 ) I n m ( ξ 1 n ) ( - i ω μ / k γ 2 ) · K n m ( ξ 2 n ) / 8 ( ω μ / k ) · I n m ( ξ 2 n ) / 8 ( 1 / γ 2 ) · K n m ( ξ 2 n ) / 8 ( i / 8 ) I n m ( ξ 2 n ) ] [ P 1 m P 2 m Q 1 m Q 2 m ] .
I n m ( Ω ) = - L / 2 L / 2 ( d x / L ) exp [ - i ( κ n - κ m ) x - i Ω F ( x ) ]
K n m ( Ω ) = - L / 2 L / 2 ( d x / L ) ( Ω - κ n F ) exp [ - i ( κ n - κ m ) x - i Ω F ( x ) ] .
I n m ( Ω ) = ( - i ) n - m J n - m ( Ω h ) ,
K n m ( Ω 0 ) = [ Ω + κ n ( κ n - κ m ) / Ω ] ( - i ) n - m J n - m ( Ω h ) ,
K n m ( Ω = 0 ) = - i π κ n ( h / L ) δ n - m , 1 ,
I n m ( Ω ) = exp ( - i Ω F 0 ) δ n , m ,
K n m ( Ω ) = Ω exp ( - i Ω F 0 ) δ n , m ,
κ 0 = k 0 sin θ inc ,
A n - = B n - 0 n 0 ,
C n + = D n + 0 n .
R 11 , n = ( α 0 / α n ) A n + 2 A 0 - - 2 ,
R 21 , n = ( α 0 / α n ) B n + 2 A 0 - - 2 ,
R 12 , n = ( α 0 / α n ) A n + 2 B 0 - - 2 ,
R 22 , n = ( α 0 / α n ) B n + 2 B 0 - - 2 ;
T L 1 , n = ( / μ ) 1 / 2 ( α 0 / ω 0 ) ( γ 1 / ξ 1 n ) C n - 2 A 0 - - 2 ,
T L 2 , n = ( / μ ) 1 / 2 ( α 0 / ω 0 ) ( γ 1 / ξ 1 n ) C n - 2 B 0 - - 2 .
T R 1 , n = ( μ / ) 1 / 2 ( α 0 / ω 0 ) ( γ 2 / ξ 2 n ) D n - 2 A 0 - - 2 ,
T R 2 , n = ( μ / ) 1 / 2 ( α 0 / ω 0 ) ( γ 2 / ξ 2 n ) D n - 2 B 0 - - 2
n ( R 11 , n + R 21 , n + T L 1 , n + T R 1 , n ) = 1.0 ,
n ( R 12 , n + R 22 , n + T L 2 , n + T R 2 , n ) = 1.0.

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