Abstract

Birefringence effects can have a significant influence on the polarization state as well as on the transversal mode structure of laser resonators. This work introduces a flexible, fast and fully vectorial algorithm for the analysis of resonators containing homogeneous, anisotropic optical components. It is based on a generalization of the Fox and Li algorithm by field tracing, enabling the calculation of the dominant transversal resonator eigenmode. For the simulation of light propagation through the anisotropic media, a fast Fourier Transformation (FFT) based angular spectrum of plane waves approach is introduced. Besides birefringence effects, this technique includes intra-crystal diffraction and interface refraction at crystal surfaces. The combination with numerically efficient eigenvalue solvers, namely vector extrapolation methods, ensures the fast convergence of the method. Furthermore a numerical example is presented which is in good agreement to experimental measurements.

© 2015 Optical Society of America

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References

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2015 (1)

D. Asoubar, M. Kuhn, and F. Wyrowski, “Fully vectorial laser resonator modeling by vector extrapolation methods,” Proc. SPIE 9342, 934214(2015).

2014 (2)

D. Asoubar, S. Zhang, F. Wyrowski, and M. Kuhn, “Laser resonator modeling by field tracing: a flexible approach for fully vectorial transversal eigenmode calculation,” J. Opt. Soc. Am. B 31(11), 2565–2573 (2014).
[Crossref]

D. Asoubar, F. Wyrowski, H. Schweitzer, C. Hellmann, and M. Kuhn, “Resonator modeling by field tracing: a flexible approach for fully vectorial laser resonator modeling,” Proc. SPIE 9135, 91350B (2014).
[Crossref]

2011 (1)

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5–6), 449–466 (2011).
[Crossref]

2008 (1)

A. Sidi, “Vector extrapolation methods with applications to solution of large systems of equations and to Page-Rank computations,” Comput. Math. Appl. 56, 1–24 (2008).
[Crossref]

2007 (2)

2006 (1)

2003 (2)

2001 (1)

G. P. Bava, P. Debernardi, and L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63(2), 023816 (2001).
[Crossref]

2000 (1)

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[Crossref]

1999 (2)

B. Demeulenaere, P. Bienstman, B. Dhoedt, and R. G. Baets, “Detailed study of AlAs-oxidized apertures in VCSEL cavities for optimized modal performance,” IEEE J. Quantum Electron. 35(3), 358–367 (1999).
[Crossref]

P. Bienstman, B. Demeulenaere, B. Dhoedt, and R. Baets, “Simulation results of transverse-optical confinement in airpost, regrown, and oxidized vertical-cavity surface-emitting laser structures,” J. Opt. Soc. Am. B 16(11), 2055–2059 (1999).
[Crossref]

1997 (3)

D. Burak and R. Binder, “Cold-cavity vectorial eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. 33(7), 1205–1215 (1997).
[Crossref]

Q. Deng and D. G. Deppe, “Self-consistent calculation of the lasing eigenmode of the dielectrically apertured Fabry-Perot microcavity with idealized or distributed Bragg reflectors,” IEEE J. Quantum Electron. 33(12), 2319–2326 (1997).
[Crossref]

N. V. Kuleshov, A. A. Lagatsky, V. G. Shcherbitsky, V. P. Mikhailov, E. Heumann, T. Jensen, A. Diening, and G. Huber, “CW laser performance of Yb and Er, Yb doped tungstates,” Appl. Phys. B 64(4), 409–413 (1997).
[Crossref]

1995 (1)

1994 (1)

L. Xu, P. Huang, J. Chrostowski, and S. K. Chaudhuri, “Full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12(11), 1926–1931 (1994).
[Crossref]

1991 (2)

T. Taira, A. Mukai, Y. Nozawa, and T. Kobayashi, “Single-mode oscillation of laser-diode-pumped Nd:YVO4 microchip lasers,” Opt. Lett. 16(24), 1955–1957 (1991).
[Crossref] [PubMed]

A. Sidi, “Efficient implementation of minimal polynomial and reduced rank extrapolation methods,” J. Comput. Appl. Math. 36(3), 305–337 (1991).
[Crossref]

1986 (1)

1983 (1)

1982 (2)

L. Thylen and D. Yevick, “Beam propagation method in anisotropic media,” Appl. Opt. 21(15), 2751–2754 (1982).
[Crossref] [PubMed]

H. C. Chen, “A coordinate-free approach to wave propagation in anisotropic media,” J. Appl. Phys. 53(7), 4606–4609 (1982).
[Crossref]

1980 (1)

1979 (1)

1975 (1)

1974 (2)

1965 (1)

1964 (1)

W. M. Doyle and M. B. White, “Frequency splitting and mode competition in a dual-polarization He-Ne gas laser,” Appl. Phys. Lett. 5(10), 193–195 (1964).
[Crossref]

1942 (1)

Asoubar, D.

D. Asoubar, M. Kuhn, and F. Wyrowski, “Fully vectorial laser resonator modeling by vector extrapolation methods,” Proc. SPIE 9342, 934214(2015).

D. Asoubar, F. Wyrowski, H. Schweitzer, C. Hellmann, and M. Kuhn, “Resonator modeling by field tracing: a flexible approach for fully vectorial laser resonator modeling,” Proc. SPIE 9135, 91350B (2014).
[Crossref]

D. Asoubar, S. Zhang, F. Wyrowski, and M. Kuhn, “Laser resonator modeling by field tracing: a flexible approach for fully vectorial transversal eigenmode calculation,” J. Opt. Soc. Am. B 31(11), 2565–2573 (2014).
[Crossref]

Baets, R.

Baets, R. G.

B. Demeulenaere, P. Bienstman, B. Dhoedt, and R. G. Baets, “Detailed study of AlAs-oxidized apertures in VCSEL cavities for optimized modal performance,” IEEE J. Quantum Electron. 35(3), 358–367 (1999).
[Crossref]

Bass, M.

Bava, G. P.

G. P. Bava, P. Debernardi, and L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63(2), 023816 (2001).
[Crossref]

Bienstman, P.

B. Demeulenaere, P. Bienstman, B. Dhoedt, and R. G. Baets, “Detailed study of AlAs-oxidized apertures in VCSEL cavities for optimized modal performance,” IEEE J. Quantum Electron. 35(3), 358–367 (1999).
[Crossref]

P. Bienstman, B. Demeulenaere, B. Dhoedt, and R. Baets, “Simulation results of transverse-optical confinement in airpost, regrown, and oxidized vertical-cavity surface-emitting laser structures,” J. Opt. Soc. Am. B 16(11), 2055–2059 (1999).
[Crossref]

Binder, R.

D. Burak and R. Binder, “Cold-cavity vectorial eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. 33(7), 1205–1215 (1997).
[Crossref]

Blit, S.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[Crossref]

Bomzon, Z.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[Crossref]

Burak, D.

D. Burak and R. Binder, “Cold-cavity vectorial eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. 33(7), 1205–1215 (1997).
[Crossref]

Byer, R. L.

Chaudhuri, S. K.

L. Xu, P. Huang, J. Chrostowski, and S. K. Chaudhuri, “Full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12(11), 1926–1931 (1994).
[Crossref]

Chen, H. C.

H. C. Chen, “A coordinate-free approach to wave propagation in anisotropic media,” J. Appl. Phys. 53(7), 4606–4609 (1982).
[Crossref]

Christ, A.

Chrostowski, J.

L. Xu, P. Huang, J. Chrostowski, and S. K. Chaudhuri, “Full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12(11), 1926–1931 (1994).
[Crossref]

Davidson, N.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[Crossref]

Debernardi, P.

G. P. Bava, P. Debernardi, and L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63(2), 023816 (2001).
[Crossref]

Demeulenaere, B.

B. Demeulenaere, P. Bienstman, B. Dhoedt, and R. G. Baets, “Detailed study of AlAs-oxidized apertures in VCSEL cavities for optimized modal performance,” IEEE J. Quantum Electron. 35(3), 358–367 (1999).
[Crossref]

P. Bienstman, B. Demeulenaere, B. Dhoedt, and R. Baets, “Simulation results of transverse-optical confinement in airpost, regrown, and oxidized vertical-cavity surface-emitting laser structures,” J. Opt. Soc. Am. B 16(11), 2055–2059 (1999).
[Crossref]

Deng, Q.

Q. Deng and D. G. Deppe, “Self-consistent calculation of the lasing eigenmode of the dielectrically apertured Fabry-Perot microcavity with idealized or distributed Bragg reflectors,” IEEE J. Quantum Electron. 33(12), 2319–2326 (1997).
[Crossref]

Deppe, D. G.

Q. Deng and D. G. Deppe, “Self-consistent calculation of the lasing eigenmode of the dielectrically apertured Fabry-Perot microcavity with idealized or distributed Bragg reflectors,” IEEE J. Quantum Electron. 33(12), 2319–2326 (1997).
[Crossref]

Dhoedt, B.

B. Demeulenaere, P. Bienstman, B. Dhoedt, and R. G. Baets, “Detailed study of AlAs-oxidized apertures in VCSEL cavities for optimized modal performance,” IEEE J. Quantum Electron. 35(3), 358–367 (1999).
[Crossref]

P. Bienstman, B. Demeulenaere, B. Dhoedt, and R. Baets, “Simulation results of transverse-optical confinement in airpost, regrown, and oxidized vertical-cavity surface-emitting laser structures,” J. Opt. Soc. Am. B 16(11), 2055–2059 (1999).
[Crossref]

Diening, A.

N. V. Kuleshov, A. A. Lagatsky, V. G. Shcherbitsky, V. P. Mikhailov, E. Heumann, T. Jensen, A. Diening, and G. Huber, “CW laser performance of Yb and Er, Yb doped tungstates,” Appl. Phys. B 64(4), 409–413 (1997).
[Crossref]

Doyle, W. M.

W. M. Doyle and M. B. White, “Properties of an anisotropic Fabry-Perot resonator,” J. Opt. Soc. Am. 55(10), 1221–1225 (1965).
[Crossref]

W. M. Doyle and M. B. White, “Frequency splitting and mode competition in a dual-polarization He-Ne gas laser,” Appl. Phys. Lett. 5(10), 193–195 (1964).
[Crossref]

Feit, M. D.

Fichtner, W.

Fleck, J. A.

Fratta, L.

G. P. Bava, P. Debernardi, and L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63(2), 023816 (2001).
[Crossref]

Friesem, A. A.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[Crossref]

Giuliani, G.

Harter, D. J.

Hasman, E.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[Crossref]

Hellmann, C.

D. Asoubar, F. Wyrowski, H. Schweitzer, C. Hellmann, and M. Kuhn, “Resonator modeling by field tracing: a flexible approach for fully vectorial laser resonator modeling,” Proc. SPIE 9135, 91350B (2014).
[Crossref]

Heumann, E.

N. V. Kuleshov, A. A. Lagatsky, V. G. Shcherbitsky, V. P. Mikhailov, E. Heumann, T. Jensen, A. Diening, and G. Huber, “CW laser performance of Yb and Er, Yb doped tungstates,” Appl. Phys. B 64(4), 409–413 (1997).
[Crossref]

Hodgson, N.

N. Hodgson and H. Weber, Optical Resonators: Fundamentals, Advanced Concepts, Applications (Springer Science, 2005).

Huang, P.

L. Xu, P. Huang, J. Chrostowski, and S. K. Chaudhuri, “Full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12(11), 1926–1931 (1994).
[Crossref]

Huber, G.

N. V. Kuleshov, A. A. Lagatsky, V. G. Shcherbitsky, V. P. Mikhailov, E. Heumann, T. Jensen, A. Diening, and G. Huber, “CW laser performance of Yb and Er, Yb doped tungstates,” Appl. Phys. B 64(4), 409–413 (1997).
[Crossref]

Jensen, T.

N. V. Kuleshov, A. A. Lagatsky, V. G. Shcherbitsky, V. P. Mikhailov, E. Heumann, T. Jensen, A. Diening, and G. Huber, “CW laser performance of Yb and Er, Yb doped tungstates,” Appl. Phys. B 64(4), 409–413 (1997).
[Crossref]

Jones, R. C.

Junghans, J.

Keller, M.

Kliger, D. S.

D. S. Kliger and J. W. Lewis, Polarized Light in Optics and Spectroscopy (Elsevier, 1990).

Kobayashi, T.

Kozawa, Y.

Kuhn, M.

D. Asoubar, M. Kuhn, and F. Wyrowski, “Fully vectorial laser resonator modeling by vector extrapolation methods,” Proc. SPIE 9342, 934214(2015).

D. Asoubar, F. Wyrowski, H. Schweitzer, C. Hellmann, and M. Kuhn, “Resonator modeling by field tracing: a flexible approach for fully vectorial laser resonator modeling,” Proc. SPIE 9135, 91350B (2014).
[Crossref]

D. Asoubar, S. Zhang, F. Wyrowski, and M. Kuhn, “Laser resonator modeling by field tracing: a flexible approach for fully vectorial transversal eigenmode calculation,” J. Opt. Soc. Am. B 31(11), 2565–2573 (2014).
[Crossref]

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5–6), 449–466 (2011).
[Crossref]

Kuleshov, N. V.

N. V. Kuleshov, A. A. Lagatsky, V. G. Shcherbitsky, V. P. Mikhailov, E. Heumann, T. Jensen, A. Diening, and G. Huber, “CW laser performance of Yb and Er, Yb doped tungstates,” Appl. Phys. B 64(4), 409–413 (1997).
[Crossref]

Kuster, N.

Lagatsky, A. A.

N. V. Kuleshov, A. A. Lagatsky, V. G. Shcherbitsky, V. P. Mikhailov, E. Heumann, T. Jensen, A. Diening, and G. Huber, “CW laser performance of Yb and Er, Yb doped tungstates,” Appl. Phys. B 64(4), 409–413 (1997).
[Crossref]

Landry, G. D.

Lewis, J. W.

D. S. Kliger and J. W. Lewis, Polarized Light in Optics and Spectroscopy (Elsevier, 1990).

Maldonado, T. A.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Mikhailov, V. P.

N. V. Kuleshov, A. A. Lagatsky, V. G. Shcherbitsky, V. P. Mikhailov, E. Heumann, T. Jensen, A. Diening, and G. Huber, “CW laser performance of Yb and Er, Yb doped tungstates,” Appl. Phys. B 64(4), 409–413 (1997).
[Crossref]

Mukai, A.

Nozawa, Y.

Nyakas, P.

Oron, R.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[Crossref]

Parkt, Y. K.

Pfeiffer, M.

M. Streiff, A. Witzig, M. Pfeiffer, P. Royo, and W. Fichtner, “A comprehensive VCSEL device simulator,” IEEE J. Sel. Topics Quantum Electron. 9(3), 879–891 (2003).
[Crossref]

Royo, P.

M. Streiff, A. Witzig, M. Pfeiffer, P. Royo, and W. Fichtner, “A comprehensive VCSEL device simulator,” IEEE J. Sel. Topics Quantum Electron. 9(3), 879–891 (2003).
[Crossref]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley Interscience, 2007).

Sato, S.

Schweitzer, H.

D. Asoubar, F. Wyrowski, H. Schweitzer, C. Hellmann, and M. Kuhn, “Resonator modeling by field tracing: a flexible approach for fully vectorial laser resonator modeling,” Proc. SPIE 9135, 91350B (2014).
[Crossref]

Shcherbitsky, V. G.

N. V. Kuleshov, A. A. Lagatsky, V. G. Shcherbitsky, V. P. Mikhailov, E. Heumann, T. Jensen, A. Diening, and G. Huber, “CW laser performance of Yb and Er, Yb doped tungstates,” Appl. Phys. B 64(4), 409–413 (1997).
[Crossref]

Shu, H.

Sidi, A.

A. Sidi, “Vector extrapolation methods with applications to solution of large systems of equations and to Page-Rank computations,” Comput. Math. Appl. 56, 1–24 (2008).
[Crossref]

A. Sidi, “Efficient implementation of minimal polynomial and reduced rank extrapolation methods,” J. Comput. Appl. Math. 36(3), 305–337 (1991).
[Crossref]

Siegman, A. E.

Streiff, M.

Sziklas, E. A.

Taira, T.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley Interscience, 2007).

Thylen, L.

Walling, J. C.

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Figures (8)

Fig. 1
Fig. 1 Example for single round trip of a resonator including several isotropic and anisotropic optical elements: round trip operator consists of a micro structure component operator in forward ( C 1) and backward ( C 7) direction, an operator for a lens component ( C 2 and C 6), an anisotropic crystal operator in forward ( C 3) and backward direction ( C 5), component operators due to light reflection at the cavity mirrors ( C 4 and C 10), intracavity aperture operators ( C 8 and C 12), anisotropic operators for a Brewster window ( C 9 and C 11) as well as free-space propagation operators between the optical components ( P 1 , 0 to P 12 , 11). In this example the dominant transversal resonator mode V(x, y, z0) is calculated in the aperture plane.
Fig. 2
Fig. 2 Illustration of the simulation task for propagating light through an arbitrarily oriented anisotropic medium with dielectric tensor ε ¯. The anisotropic medium is embedded into two isotropic media with dielectric constants εi and εt respectively. Therefore for the unique representation of the incident field V i ( ρ , z in ) and the transmitted field V t ( ρ , z out ), only two field components = 1, 2 are necessary.
Fig. 3
Fig. 3 Example workflow on solving the refraction problem at a plane interface from εi into ε ¯. The explicit process of Step II is shown. For example, with root k Z , 1 t from the quartic equation, a wavevector k 1 t and a refractive index n 1 t is determined. Using k 1 t and ε ¯, a 3 × 3 matrix Q ¯ 1 is built up and three eigenvalues Λ 1, Λ 1 and Λ 1 are found. Amongst them only Λ 1 = 1 / ( n 1 t ) 2 while the other two cases are terminated. The termination is denoted by the × symbol. Using the eigenvector ε ^ 1 t for Λ 1, the time-averaged Poynting vector S 1 t is calculated. Because of S 1 Z t > 0 we define k α t : = k 1 t and ε ^ α t : = ε ^ 1 t, and return them. Otherwise this process is terminated, as shown for S 2 t and S 4 t.
Fig. 4
Fig. 4 Refraction at a plane interface between an isotropic medium (left) and an arbitrarily oriented uniaxial crystal (right). The dispersion relation of the isotropic medium appears as a semi-sphere on the left side; for the uniaxial crystal on the right side, its dispersion relations are presented as two surfaces, a partial ellipsoid for the extraordinary wave and a semi-sphere for the ordinary waves. Due to the phase matching condition at the interface, the transverse components κ of the wavevector k must be equal for the incident and the three resulting plane waves.
Fig. 5
Fig. 5 Schematic of a laser resonator for generation of a radially polarized beam. It is shown that the ordinary and extraordinary beams take different optical paths in the Nd:YVO4 rod because of the different refractive indices [1]. Other used parameters can be found in Table 1.
Fig. 6
Fig. 6 Convergence velocity of different eigenvalue solvers for the nonlinear eigenvalue problem given by the example laser resonator. The evolution of the deviation σ ( j ) between adjacent iteration results for the coupled Ex (a) and Ey (b) field components is shown. Please note that, for better illustration, the vertical axes were scaled logarithmically. This example clearly shows that the MPE algorithm has a much faster convergence velocity than the iterative power method.
Fig. 7
Fig. 7 Intensity distributions of the dominant transversal resonator mode in the plane of the outcoupling mirror M2. (a) Overall intensity distribution. (b)–(e) Intensity distributions after the mode passes through a linear polarizer with different directions. The arrows indicate the directions of the polarizer. (f) Intensity profile along the vertical line intersecting the center of (a). All of the simulation results are in good agreement to measured results given by Yonezawa et al. [1].
Fig. 8
Fig. 8 Intensity distribution of the V1 (a) and V2 (b) components of the resonator setup with reduced effective cavity length obtained by the MPE algorithm after 120 round trip iterations. The different transversal shapes are caused by nonlinear mode competition inside the active medium. Clearly the TEM00-like mode in (a) is stronger than the higher order mode in (b).

Tables (2)

Tables Icon

Table 1 Other parameters used for the resonator setup are given in Fig. 5. Most of the parameters are based on the description of the experiment performed by Yonezawa et al. [1]. For parameters which where not given explicitly in [1], useful suggestions were made.

Tables Icon

Table 2 Component operators used in different domains of the resonator. The corresponding equations or references for the component operators are given in brackets.

Equations (38)

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ε ¯ = ( ε X X ε X Y ε X Z ε Y X ε Y Y ε Y Z ε Z X ε Z Y ε Z Z ) .
2 i k 0 n 0 V ˜ 1 z = ( 2 + k 0 2 ε X X ) V ˜ 1 k 0 2 ε X Y V ˜ 2
2 i k 0 n 0 V ˜ 2 z = ( 2 + k 0 2 ε Y Y ) V ˜ 2 k 0 2 ε Y X V ˜ 1
V = V ˜ exp ( i k 0 n 0 z ) for = 1 , 2.
( γ 1 0 0 γ 2 ) ( V 1 V 2 ) = R ( V 1 V 2 ) = ( R 11 R 12 R 21 R 22 ) ( V 1 V 2 ) .
R = m = 1 n ( C m P m , m 1 ) .
C = ( t 11 exp ( i Φ 11 ) p 12 p 21 t 22 exp ( i Φ 22 ) ) .
( V 1 t ( ρ , z out ) V 2 t ( ρ , z out ) ) = C aniso ( V 1 i ( ρ , z in ) V 2 i ( ρ , z in ) ) ,
A i ( κ ) = FT { V i ( ρ , z in ) } = 1 2 π + V i ( ρ , z in ) exp ( i κ ρ ) d X d Y ,
V i ( ρ , z in ) = FT 1 { A i ( κ ) } = 1 2 π + A i ( κ ) exp ( i κ ρ ) d k X d k Y
V t ( ρ , z out ) = FT 1 { A t ( κ ) } = 1 2 π + A t ( κ ) exp ( i κ ρ ) d k X d k Y .
E i ( r ) = ε i exp ( i k i r ) ,
k Z i = [ ε i k 0 2 k X i 2 k Y i 2 ] 1 / 2 ,
ε Z i = k X i ε X i + k Y i ε Y i k Z i ,
E r ( r ) = ε r exp ( i k r r )
E t ( r ) = α t ε ^ α t exp ( i k α t r ) + β t ε ^ β t exp ( i k β t r ) .
κ i = κ r = κ t = ( k X , k Y ) T
a k Z t 4 + b k Z t 3 + c k Z t 2 + d k Z t + e = 0 ,
a = ε Z Z , b = 2 ( ε X Z k X + ε Y Z k Y ) , c = ( ε X X + ε Z Z ) k X 2 + ( ε Y Y + ε Z Z ) k Y 2 + k 0 2 [ ε X Z 2 + ε Y Z 2 ε Z Z ( ε X X + ε Y Y ) ] , d = 2 [ ε X Z k X 3 + ε Y Z k Y 3 + ε X Z k X k Y 2 + ε Y Z k X 2 k Y + k 0 2 k X ( ε X Y ε Y Z ε X Z ε Y Y ) + k 0 2 k Y ( ε X Y ε X Z ε X X ε Y Z ) ] , e = k 0 4 ( ε X X ε Y Y ε Z Z + 2 ε X Z ε X Y ε Y Z ε X X ε Y Z 2 ε Y Y ε X Z 2 ε Z Z ε X Y 2 ) + 2 k 0 2 k X k Y ( ε X Z ε Y Z ε X Y ε Z Z ) + k 0 2 k X 2 ( ε X Z 2 ε X X ε Z Z ε X X ε Y Y + ε X Y 2 ) + k 0 2 k Y 2 ( ε Y Z 2 ε Y Y ε Z Z ε X X ε Y Y + ε X Y 2 ) + k X 2 k Y 2 ( ε X X + ε Y Y ) + 2 k X k Y ( k X 2 + k Y 2 ) ε X Y + k X 4 ε X X + k Y 4 ε Y Y .
Q ¯ j ε j t = Λ j ε j t and Λ j = ! 1 ( n j t ) 2 ,
Q ¯ j = ε ¯ 1 ( k ^ Y 2 ( k ^ Z , j t ) 2 k ^ X k ^ Y k ^ X k ^ Z , j t k ^ X k ^ Y k ^ X 2 ( k ^ Z , j t ) 2 k ^ Y k ^ Z , j t k ^ X k ^ Z , j t k ^ Y k ^ Z , j t k ^ X 2 k ^ Y 2 )
S j t ε ^ j t × ( k j t × ε ^ j t )
E X i ( X , Y , 0 ) + E X r ( X , Y , 0 ) = E α X t ( X , Y , 0 ) + E β X t ( X , Y , 0 ) , E Y i ( X , Y , 0 ) + E Y r ( X , Y , 0 ) = E α Y t ( X , Y , 0 ) + E β Y t ( X , Y , 0 ) , H X i ( X , Y , 0 ) + H X r ( X , Y , 0 ) = H α X t ( X , Y , 0 ) + H β X t ( X , Y , 0 ) , H Y i ( X , Y , 0 ) + H Y r ( X , Y , 0 ) = H α Y t ( X , Y , 0 ) + H β Y t ( X , Y , 0 ) ,
H ( r ) = 1 ω μ 0 k × E ( r ) ,
M ¯ ( ε X r ε Y r α t β t ) = ( ε Y i ε X i ( k Z i + k X 2 k Z i ) ε X i + k X k Y k Z i ε Y i k X k Y k Z i ε X i + ( k Z i + k Y 2 k Z i ) ε Y i ) ,
M ¯ = ( 0 1 ε ^ α Y t ε ^ β Y t 1 0 ε ^ β X t ε ^ β X t k Z i + k X 2 k Z i k X k Y k Z i k α Z t ε ^ α X t k X ε ^ α Z t k β Z t ε ^ β X t k X ε ^ β Z t k X k Y k Z i k Z i + k Y 2 k Z i k α Z t ε ^ α Y t k Y ε ^ α Z t k β Z t ε ^ β Y t k Y ε ^ β Z t ) .
( ε X r ε Y r α t β t ) = M ¯ 1 ( ε Y i ε X i ( k Z i + k X 2 k Z i ε X i ) + k X k Y k Z i ε Y i k X k Y k Z i ε X i + ( k Z i + k Y 2 k Z i ) ε Y i ) ,
E t ( ρ , d ) = FT 1 { ε α t ( κ , d ) + ε β t ( κ , d ) } = 1 2 π + [ α t ( κ ) ε ^ α t ( κ ) exp ( i k α Z t d ) + β t ( κ ) ε ^ β t ( κ ) exp ( i k β Z t d ) ] exp ( i κ ρ ) d k X d k Y
E i ( r ) = α i ε ^ α i exp ( i k α i r ) + β i ε ^ β i exp ( i k β i r ) ,
E r ( r ) = α r ε ^ α r exp ( i k α r r ) + β r ε ^ β r exp ( i k β r r ) ,
E t ( r ) = ε t exp ( i k t r ) .
N ¯ ( α r β r ε X t ε Y t ) = ( α i ε ^ α Y i + β i ε ^ β Y i α i ε ^ α X i + β i ε ^ β X i α i ( k α Z i ε ^ α X i k X ε ^ α Z i ) + β i ( k β Z i ε ^ β X i k X ε ^ β Z i ) α i ( k α Z i ε ^ α Y i k Y ε ^ α Z i ) + β i ( k β Z i ε ^ β Y i k Y ε ^ β Z i ) ) ,
N ¯ = ( ε ^ α Y r ε ^ β Y r 0 1 ε ^ α X r ε ^ β X r 1 0 k X ε ^ α Z r k α Z r ε ^ α X r k X ε ^ β Z r k β Z r ε ^ β X r k Z t + k X 2 k Z t k X k Y k Z t k Y ε ^ α Z r k α Z r ε ^ α Y r k Y ε ^ β Z r k β Z r ε ^ β Y r k X k Y k Z t k Z t + k Y 2 k Z t ) .
( α r β r ε X t ε Y t ) = N ¯ 1 ( α i ε ^ α Y i + β i ε ^ β Y i α i ε ^ α X i + β i ε ^ β X i α i ( k α Z i ε ^ α X i k X ε ^ α Z i ) + β i ( k β Z i ε ^ β X i k X ε ^ β Z i ) α i ( k α Z i ε ^ α Y i k Y ε ^ α Z i ) + β i ( k β Z i ε ^ β Y i k Y ε ^ β Z i ) ) .
C AM = ( C AM 0 0 C AM )
C AM V = exp ( g 0 1 + g 1 1 ( = 1 , 2 | V | 2 ) ) V
Γ ( j ) = arg [ ( V ( j ) , V ( j 1 ) ) ( V ( j 1 ) , V ( j 1 ) ) ]
σ ( j ) : = | V ( j 1 ) V ( j ) exp [ i Γ ( j ) ] | 2 d x d y | V ( j 1 ) | 2 d x d y

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