Abstract

We study the diffraction produced by a slab of purely reflective PT-symmetric volume Bragg grating that combines modulations of refractive index and gain/loss of the same periodicity with a quarter-period shift between them. Such a complex grating has a directional coupling between the different diffraction orders, which allows us to find an analytic solution for the first three orders of the full Maxwell equations without resorting to the paraxial approximation. This is important, because only with the full equations can the boundary conditions, allowing for the reflections, be properly implemented. Using our solution we analyze unidirectional invisibility of such a grating in a wide variety of configurations.

© 2015 Optical Society of America

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References

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  1. S. Bernet, S. B. Altner, F. R. Graf, E. S. Maniloff, A. Renn, and U. P. Wild, “Frequency and phase swept holograms in spectral hole-burning materials,” Appl. Opt. 34(22), 4674–4684 (1995).
    [Crossref] [PubMed]
  2. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
    [Crossref] [PubMed]
  3. M. Kulishov, J. Laniel, N. Bélanger, J. Azaña, and D. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005).
    [Crossref] [PubMed]
  4. C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored complex potentials and Friedel’s law in atom optics,” Phys. Rev. Lett. 79(18), 3327–3330 (1997).
    [Crossref]
  5. M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31(15), 3493–3502 (1998).
    [Crossref]
  6. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
    [Crossref]
  7. C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
    [Crossref] [PubMed]
  8. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
    [Crossref] [PubMed]
  9. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
    [Crossref] [PubMed]
  10. Y. Yan and N. C. Giebink, “Passive PT symmetry in organic composite films via complex refractive index modulation,” Adv. Opt. Mater. 2(5), 423–427 (2014).
    [Crossref]
  11. L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22(2), 1760–1767 (2014).
    [Crossref] [PubMed]
  12. S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A 44(48), 485302 (2011).
    [Crossref]
  13. X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, “One-way invisible cloak using parity-time symmetric transformation optics,” Opt. Lett. 38(15), 2821–2824 (2013).
    [Crossref] [PubMed]
  14. C. Huang, F. Ye, Y. V. Kartashov, D. A. Malomed, and X. Chen, “PT-symmetry in optics beyond the paraxial approximation,” Opt. Lett. 39(18), 5443–5446 (2014).
    [Crossref]
  15. T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28(1), 1–14 (1982).
    [Crossref]
  16. T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
    [Crossref]
  17. M. Kulishov, H. F. Jones, and B. Kress, “Analysis of PT -symmetric volume gratings beyond the paraxial approximation,” Opt. Express 23(7), 9347–9362 (2015).
    [Crossref] [PubMed]

2015 (1)

2014 (3)

2013 (1)

2012 (1)

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

2011 (2)

S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A 44(48), 485302 (2011).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

2005 (1)

2002 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref] [PubMed]

1998 (2)

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31(15), 3493–3502 (1998).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

1997 (1)

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored complex potentials and Friedel’s law in atom optics,” Phys. Rev. Lett. 79(18), 3327–3330 (1997).
[Crossref]

1996 (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[Crossref] [PubMed]

1995 (1)

1985 (1)

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

1982 (1)

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28(1), 1–14 (1982).
[Crossref]

Abfalterer, R.

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored complex potentials and Friedel’s law in atom optics,” Phys. Rev. Lett. 79(18), 3327–3330 (1997).
[Crossref]

Almeida, V. R.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Altner, S. B.

Azaña, J.

Bélanger, N.

Bender, C. M.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref] [PubMed]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Bernet, S.

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored complex potentials and Friedel’s law in atom optics,” Phys. Rev. Lett. 79(18), 3327–3330 (1997).
[Crossref]

S. Bernet, S. B. Altner, F. R. Graf, E. S. Maniloff, A. Renn, and U. P. Wild, “Frequency and phase swept holograms in spectral hole-burning materials,” Appl. Opt. 34(22), 4674–4684 (1995).
[Crossref] [PubMed]

Berry, M. V.

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31(15), 3493–3502 (1998).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref] [PubMed]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Chen, X.

Chen, Y.-F.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Christodoulides, D. N.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Fegadolli, W. S.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Feng, L.

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28(1), 1–14 (1982).
[Crossref]

Giebink, N. C.

Y. Yan and N. C. Giebink, “Passive PT symmetry in organic composite films via complex refractive index modulation,” Adv. Opt. Mater. 2(5), 423–427 (2014).
[Crossref]

Graf, F. R.

Huang, C.

Jones, H. F.

Kartashov, Y. V.

Keller, C.

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored complex potentials and Friedel’s law in atom optics,” Phys. Rev. Lett. 79(18), 3327–3330 (1997).
[Crossref]

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Kress, B.

Kulishov, M.

Laniel, J.

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Longhi, S.

S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A 44(48), 485302 (2011).
[Crossref]

Lu, M.-H.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Malomed, D. A.

Maniloff, E. S.

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28(1), 1–14 (1982).
[Crossref]

Oberthaler, M. K.

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored complex potentials and Friedel’s law in atom optics,” Phys. Rev. Lett. 79(18), 3327–3330 (1997).
[Crossref]

Oliveira, J. E.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Plant, D.

Poladian, L.

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[Crossref] [PubMed]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Renn, A.

Scherer, A.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Schmiedmayer, J.

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored complex potentials and Friedel’s law in atom optics,” Phys. Rev. Lett. 79(18), 3327–3330 (1997).
[Crossref]

Wang, Y.

Wild, U. P.

Xu, Y.-L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Yan, Y.

Y. Yan and N. C. Giebink, “Passive PT symmetry in organic composite films via complex refractive index modulation,” Adv. Opt. Mater. 2(5), 423–427 (2014).
[Crossref]

Yang, S.

Ye, F.

Yin, X.

Zeilinger, A.

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored complex potentials and Friedel’s law in atom optics,” Phys. Rev. Lett. 79(18), 3327–3330 (1997).
[Crossref]

Zhang, P.

Zhang, X.

Zhu, H.

Zhu, X.

Adv. Opt. Mater. (1)

Y. Yan and N. C. Giebink, “Passive PT symmetry in organic composite films via complex refractive index modulation,” Adv. Opt. Mater. 2(5), 423–427 (2014).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28(1), 1–14 (1982).
[Crossref]

J. Phys. A (1)

S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A 44(48), 485302 (2011).
[Crossref]

J. Phys. Math. Gen. (1)

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31(15), 3493–3502 (1998).
[Crossref]

Nat. Mater. (1)

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[Crossref] [PubMed]

Phys. Rev. Lett. (4)

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored complex potentials and Friedel’s law in atom optics,” Phys. Rev. Lett. 79(18), 3327–3330 (1997).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Proc. IEEE (1)

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1 Planar purely reflective grating of the index (black color fringes) and gain/loss (red color fringes) modulation.
Fig. 2
Fig. 2 Filled-space configuration ( ε 1 = ε 2 = ε 2 =2.4 ): (left side) zeroth (blue, dashed), first order (red, solid) (a) and second (d) order reflections, as well as the combined coherent reflection (g) and zeroth, first (b) and second (e) orders and the combined coherent transmission (h) along with (right side) zeroth, first order (c), second order (f) and the combined coherent reflection (i) as a function of the internal angle of incidence, θ, for Λ = 0.42 µm (red, solid). The other parameters are d = 8.4 µm, λ0 = 0.633 µm, ξ = 0.02.
Fig. 3
Fig. 3 Prominent modes of the PT-symmetric grating for incidence at different angles and from different sides: (a) from the left near the first Bragg angle θB; (b) from the left near -θB; (c) from the right near θB; (d) from the right near -θB.
Fig. 4
Fig. 4 Filled-space configuration ( ε 1 = ε 2 = ε 2 =2.4 ): the combined coherent zeroth, first and second orders in reflection (red solid curve) for left-side incidence and in transmission (blue, dashed curve) as functions of the internal angle of incidence θ for Λ = 0.25 µm (a) Λ = 0.205 (b). The other parameters are d = 8 µm, λ0 = 0.633 µm, ξ = 0.02.
Fig. 5
Fig. 5 The slab in air ( ε 1 = ε 3 =1 ε 2 =2.4 ). Left-side incidence: zeroth (blue), first order (red) (a), second (d) order reflection, as well as the combined coherent reflection (g) and zeroth (blue), first (red) (b) and second (d) order transmission, along with combined coherent transmission (h). Right side incidence: zeroth (blue), first (red) (c), second (f) order reflection and combined coherent reflection (i) as functions of the internal angle of incidence for Λ = 0.23 µm. The other parameters are d = 36Λ = 8.28 µm, λ0 = 0.633 µm, ξ = 0.02; θB = 27.35°.
Fig. 6
Fig. 6 The slab in air ( ε 1 = ε 3 =1 ε 2 =2.4 ): the grating visibility factor in transmission (a) and in reflection (b) for right-side incidence (from non-reflective side) as functions of the internal angle of incidence for Λ = 0.23 µm (red, solid). The other parameters are d = 36Λ = 8.28 µm, λ0 = 0.633 µm, ξ = 0.02; θB = 27.35°.
Fig. 7
Fig. 7 Non-symmetrical configurations of the PT-symmetric grating on a substrate.
Fig. 8
Fig. 8 The slab attached to the left of the substrate:( ε 1 =1 ε 2 =2.4 ε 3 =2.0 ): Light incident from air side: zeroth (blue), first (red) (a) and second (d) order reflection, as well as the combined coherent reflection (g), and zeroth (blue), first (red) (b) and second (e) order transmission, along with the combined coherent transmission (h). Light incident from the substrate side: zeroth (blue), first (red) (c) and second (f) order reflection, along with the combined coherent reflection (i) as functions of the internal angle of incidence for Λ = 0.23 µm. The other parameters are d = 8.28 µm, λ0 = 0.633 µm, ξ = 0.02.
Fig. 9
Fig. 9 The grating visibility when it is attached to the left of the substrate ( ε 1 =1; ε 3 =2 ε 2 =2.4 ) in transmission (a) and reflection (b) for right-side incidence (from non-reflective side). The other parameters are d = 36Λ = 8.28 µm, λ0 = 0.633 µm, ξ = 0.02; θB = 27.35°.
Fig. 10
Fig. 10 The slab attached to the right of the substrate ( ε 1 =2.0 ε 2 =2.4 ε 3 =1.0 ). Light incident from substrate side: zeroth (blue), first (red) (a) and second (d) order reflection, as well as the combined coherent reflection (g), and zeroth (blue) and first (red) (b) and second (e) order along with the combined coherent transmission (h). Light incident from the air side: zeroth (blue), first (red) (c) and second (f) order reflection, along with the combined coherent reflection (i) as functions of the internal angle of incidence for Λ = 0.23 µm . The other parameters are d = 8.28 µm, λ0 = 0.633 µm, ξ = 0.02.
Fig. 11
Fig. 11 The grating visibility when it is attached to the right of the substrate ( ε 1 =2; ε 3 =1 ε 2 =2.4 ) in transmission (a) and reflection (b) for right-side incidence (from non-reflective side). The other parameters are d = 36Λ = 8.28 µm, λ0 = 0.633 µm, ξ = 0.02; θB = 27.35°.

Equations (54)

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

ε(x,z)= ε 2 +Δεcos( Kz )
σ(x,z)=Δσsin( Kz )
k 2 (x,z)= k 0 2 ε(x,z)jωμσ(x,z)
k 2 (x,z)= k 2 2 +2 k 2 κ exp(j K r )+2 k 2 κ + exp(j K r )
κ ± = 1 4 ( ε 2 ) 1/2 ( k 0 Δε±cμΔσ )
E 1 (x,z)=exp(jk (хsin( θ )+zcos( θ )) 1 + m= m=+ R m exp[j{ k 2 xsinθ [ k 1 2 k 2 2 sin 2 θ] 1/2 z}]
E 3 (x,z)= m= m=+ Т m exp[j{ k 2 xsinθ+ [ k 3 2 k 2 2 sin 2 θ] 1/2 (zd)}]
E 2 (x,z)= m= m=+ S m (z)exp(j k 2 xsinθ)
2 E 2 (x,z)+ k 0 2 ε(x,z) E 2 (x,z)=0
d 2 S m (z) d z 2 k 2 2 cos 2 θ S m (z)+2 k 2 [ κ exp( jKz ) S m+1 (z)+ κ + exp( jKz ) S m1 (z) ]=0
d 2 S m (z) d u 2 + cos 2 θ S m (z)+ξexp(jγu) S m1 (z)=0
d 2 S m (u) d u 2 + cos 2 θ S m (u)+ξexp( jγu ) S m+1 (u)=0
δ 0m + R m (0)= S m (0);
d S m (0) du du dz =j ( k 1 2 k 2 2 sin 2 θ) 1/2 ( R m δ 0m );
Т m (d)= S m (d)
d S m (d) du du dz =j ( k 3 2 k 2 2 sin 2 θ) 1/2 T m .
d 2 S 0 (u) d u 2 + η 0 2 S 0 (u)=0
S 0 (u)= A 0 exp(ju η 0 )+ B 0 exp(ju η 0 )
Т 0 = 4 α 0 η 0 ( α 0 + η 0 )( β 0 + η 0 )exp(j u d η 0 )( α 0 η 0 )( β 0 η 0 )exp(j u d η 0 )
R 0 = ( η 0 + α 0 )( η 0 β 0 )exp(j u d η 0 )( η 0 α 0 )( η 0 + β 0 )exp(j u d η 0 ) ( η 0 + α 0 )( η 0 + β 0 )exp(j u d η 0 )( η 0 α 0 )( η 0 β 0 )exp(j u d η 0 )
А 0 = Т 0 2 ( η 0 β 0 β 0 )exp(j u d η 0 ) B 0 = Т 0 2 ( η 0 + β 0 β 0 )exp(j u d η 0 )
d 2 S 1 (u) d u 2 + η 0 2 S 1 (u)+ξexp( jγu ) S 0 (u)=0
d 2 S 1 (u) d u 2 + η 0 2 S 1 (u)+ξexp( jγu ) S 0 (u)=0
( S ±1 (u)) H = А ±1 exp( j η 0 u )+ B ±1 exp( j η 0 u )
( S ±1 (u)) P =ξ A 0 X 0 (±) exp(j η 0 u)+ξ B 0 Y 0 (±) exp(j η 0 u)
Х 0 (±) = 1 γ(2 η 0 ±γ) Y 0 (±) = 1 γ(γ2 η 0 )
R ±1 = A 0 ξ[ η 0 + β 0 ±γ ( е j η 0 u d е j( η 0 ±γ) u d ) η 0 β 0 2 η 0 ±γ ( е j( η 0 ±γ) u d е j η 0 u d ) ] ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d ) + + B 0 ξ[ η 0 + β 0 ±γ2 η 0 ( е j η 0 u d е j(±γ η 0 ) u d ) η 0 β 0 ±γ ( е j(±γ η 0 ) u d е j η 0 u d ) ] ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d )
Т ±1 = A 0 ξ[ η 0 α 0 ±γ (1 е ±jγ u d )+ η 0 + α 0 2 η 0 ±γ (1 е j(2 η 0 ±γ) u d ) ] ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d ) + + B 0 ξ[ η 0 α 0 ±γ2 η 0 (1 е j(2 η 0 γ) u d )+ η 0 + α 0 ±γ (1 е ±jγ u d ) ] ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d )
A ±1 = Т ±1 2 ( η 0 β 0 η 0 ) e j η 0 u d ± ξ А 0 2γ η 0 e ±jγ u d ξ B 0 2 η 0 (±γ2 η 0 ) e j(±γ η 0 ) u d
B ±1 = Т ±1 2 ( η 0 + β 0 η 0 ) e +j η 0 u d + ξ A 0 2 η 0 (±γ+2 η 0 ) e j(±γ+ η 0 ) u d ± ξ B 0 2γ η 0 e ±jγ u d
d 2 S ±2 (u) d u 2 + η 0 2 S ±2 (u)+ξexp( ±jγu ) S ±1 (u)=0
( S ±2 (u)) H = А ±2 exp( j η 0 u )+ B ±2 exp( j η 0 u )
( S ±2 (u)) P = A ±2 exp(j( η 0 ±γ)u)+ B ±2 exp(j( η 0 ±γ)u)+ + X 1 (±) exp(j( η 0 ±2γ)u)+ Y 1 (±) exp(j( η 0 ±2γ)u)
A ±2 = ξ A ±1 ±γ(±γ+ η 0 ) B ±2 = ξ B ±1 ±γ(±γ η 0 )
X 1 (±) = ξ 2 А 0 4 γ 2 (±γ+2 η 0 )(±γ+ η 0 ) ; Y 1 (±) = ξ 2 B 0 4 γ 2 (±γ2 η 0 )(±γ η 0 ) ;
R ±2 = A ±2 [ (±γ+2 η 0 )( η 0 + β 0 )( е j η 0 u d е j(±γ+ η 0 ) u d )±γ( η 0 β 0 )( е j η 0 u d е j(±γ+ η 0 ) u d ) ] ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d ) + + B ±2 [ ±γ( η 0 + β 0 )( е j η 0 u d е j(±γ η 0 ) u d )+(±γ2 η 0 )( η 0 β 0 )( е j η 0 u d е j(±γ η 0 ) u d ) ] ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d ) + + 2 X 1 (±) [ (±γ+ η 0 )( η 0 + β 0 )( е j η 0 u d е j(±2γ+ η 0 ) u d )±γ( η 0 β 0 )( е j η 0 u d е j(±2γ+ η 0 ) u d ) ] ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d ) + + 2 Y 1 (±) [ ±γ( η 0 + β 0 )( е j η 0 u d е j(±2γ η 0 ) u d )+(±γ η 0 )( η 0 β 0 )( е j η 0 u d е j(±γ η 0 ) u d ) ] ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d )
Т ±2 = A ±2 [ (±γ+2 η 0 )( η 0 α 0 )(1 е ±jγ u d )±γ( η 0 + α 0 )(1 е j(±γ+2 η 0 ) u d ) ] ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d ) + + B ±2 [ ±γ( η 0 α 0 )(1 е j(±γ2 η 0 ) u d )+(±γ2 η 0 )( η 0 + α 0 )(1 е ±jγ u d ) ] ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d ) + + 2 X 1 (±) [ (±γ+ η 0 )( η 0 α 0 )(1 е ±j2γ u d )±γ( η 0 + α 0 )(1 е j2(±γ+ η 0 ) u d ) ] ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d ) + + 2 Y 1 (±) [ ±γ( η 0 α 0 )(1 е j2(±γ η 0 ) u d )+(±γ η 0 )( η 0 + α 0 )(1 е ±j2γ u d ) ] ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d )
R 1 =j ξ 2 sin[ (cos θ B cosθ) u d ] cosθ(cos θ B cosθ) exp(j(cos θ B cosθ) u d )
R 1 =j ξ 2 sin[ (cos θ B +cosθ) u d ] cosθ(cos θ B +cosθ) exp(j(cos θ B +cosθ) u d )
Т 1 = ξ 4 [ exp(2jcos θ B u d )1 cosθcos θ B ]exp(j u d cosθ)
Т 1 = ξ 4 [ exp(2jcos θ B u d )1 cosθcos θ B ]exp(j u d cosθ)
R 2 =j ξ 2 exp(j(cos θ B cosθ) u d ) 4 cos 2 θcos θ B [ sin[ (2cos θ B cosθ) u d ] 2cos θ B cosθ exp(jcos θ B u d ) sin[ (cos θ B cosθ) u d ] cos θ B cosθ ]
R 2 =j ξ 2 exp(j(cos θ B +cosθ) u d ) 4 cos 2 θcos θ B [ sin[ (2cos θ B +cosθ) u d ] 2cos θ B +cosθ exp(jcos θ B u d ) sin[ (cos θ B +cosθ) u d ] cos θ B +cosθ ]
T 2 =j ξ 2 exp(j(cos θ B cosθ) u d ) 8 cos 2 θ [ sin[ (cos θ B cosθ) u d ] cos 2 θ B cos 2 θ exp(jcosθ u d )+ + sin(cos θ B u d ) cos 2 θ B sin(2cos θ B u d ) 2 cos 2 θ B (2cos θ B +cosθ) (cos θ B +cosθ) exp(jcos θ B u d ) ]
T 2 =j ξ 2 exp(j(cos θ B +cosθ) u d ) 8 cos 2 θ [ sin[ (cos θ B +cosθ) u d ] cos 2 θ B cos 2 θ exp(cosθ u d )+ + sin(cos θ B u d ) cos 2 θ B sin(2cos θ B u d ) 2 cos 2 θ B (2cos θ B cosθ) (cos θ B cosθ) exp(jcos θ B u d ) ]
R ±1 = jξsin( η 0 u d ) ( η 0 ±γ/2) [ A 0 ( η 0 β 0 ) B 0 ( η 0 + β 0 ) ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d ) ]
Т ±1 =ξ (1 е j2 η 0 u d ) 2 η 0 ±γ [ A 0 ( η 0 + α 0 ) B 0 ( η 0 α 0 ) ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d ) ]
R ±2 =±γ [ ( B ±2 +2 Y 1 (±) )( η 0 + β 0 )( A ±2 +2 X 1 (±) )( η 0 β 0 ) ]( е j η 0 u d е j η 0 u d ) ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d )
Т ±2 =±γ ( A ±2 +2 X 1 (±) )( η 0 + α 0 )(1 е j2 η 0 u d )+( B ±2 +2 Y 1 (±) )( η 0 α 0 )(1 е j2 η 0 u d ) ( η 0 + α 0 )( η 0 + β 0 )exp(j η 0 u d )( η 0 α 0 )( η 0 β 0 )exp(j η 0 u d )
R 1 =j ξ u d 2cos θ B ; T 1 =j ξsin( u d cos θ B ) 2 cos 2 θ B
R 1 =j ξ 4 sin(2cos θ B u d ) cos 2 θ B exp(2jcos θ B u d ); Т 1 =j ξ 2 sin(cos θ B u d ) cos 2 θ B exp(2jcos θ B u d )
R 2 =j ξ 2 4 cos 4 θ B [ sin(cos θ B u d )exp(jcos θ B u d ) u d ]
R 2 =j ξ 2 exp(2jcos θ B u d ) 4 cos 4 θ B [ 1 3 sin(3cos θ B u d )exp(jcos θ B u d ) 1 2 sin(3cos θ B u d ) ]
T 2 =j ξ 2 8 cos 3 θ B [ u d 2 exp(jcosθ u d )+ sin(cos θ B u d ) cos θ B 3 4 sin(2cos θ B u d ) cos θ B exp(jcos θ B u d ) ]

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