Abstract

The small-perturbation method (SPM) for rough surface scattering, originally derived by Rice [Commun. Pure Appl. Math. 4, 361 (1951)], has been applied extensively to problems in optics, remote sensing, and propagation. Typical uses of the theory involve only the first- or second-order scattered fields in surface height, owing to increasing complexity of the SPM equations as order increases. The SPM equations are solved in a systematic manner that permits third order in surface-height terms to be determined apparently for the first time for scattering from a dielectric surface rough in two directions. Sample results for both periodic and nonperiodic surfaces show that third-order field terms can contribute to fourth-order scattered power and also to a third-order specular-reflection coefficient correction for surfaces with nonvanishing bispectra. The latter case is of particular interest in passive remote sensing of the ocean, since these third-order terms contribute to the first prediction of a first azimuthal harmonic of ocean brightness temperatures.

© 1999 Optical Society of America

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References

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  1. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 361–378 (1951).
    [CrossRef]
  2. G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
    [CrossRef]
  3. G. R. Valenzuela, “Scattering of electromagnetic waves from a tilted slightly rough surface,” Radio Sci. 3, 1057–1066 (1968).
  4. D. E. Barrick, “Theory of HF and VHF propagation across the rough sea 1: the effective surface impedance for a slightly rough highly conducting medium at grazing incidence,” Radio Sci. 6, 517–526 (1971).
    [CrossRef]
  5. S. T. Wu, A. K. Fung, “A noncoherent model for microwave emissions and backscattering from the sea surface,” J. Geophys. Res. 77, 5917–5929 (1972).
    [CrossRef]
  6. F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing (Addison Wesley, Reading, Mass., 1982).
  7. M. Nieto-Vesperinas, “Depolarization of electromagnetic waves scattered from slightly rough random surfaces: a study by means of the extinction theorem,” J. Opt. Soc. Am. 72, 539–547 (1982).
    [CrossRef]
  8. J. M. Soto-Crespo, M. Nieto-Vesperinas, A. Friberg, “Scattering from slightly rough random surfaces: a detailed study on the validity of the small perturbation method,” J. Opt. Soc. Am. A 7, 1185–1201 (1990).
    [CrossRef]
  9. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).
  10. E. I. Thorsos, D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86, 261–277 (1989).
    [CrossRef]
  11. A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Boston, Mass., 1994).
  12. A. G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, Berlin, 1994).
  13. S. H. Yueh, R. Kwok, F. K. Li, S. V. Nghiem, W. J. Wilson, “Polarimetric passive remote sensing of ocean wind vectors,” Radio Sci. 29, 799–814 (1994).
    [CrossRef]
  14. R. Deroo, F. T. Ulaby, “Bistatic specular scattering from rough dielectric surfaces,” IEEE Trans. Antennas Propag. 42, 220–231 (1994).
    [CrossRef]
  15. V. G. Irisov, “Small-slope expansion for thermal and reflected radiation from a rough surface,” Waves Random Media 7, 1–10 (1997).
    [CrossRef]
  16. V. G. Irisov, “Microwave radiation from a weakly non-Gaussian surface,” in Proceedings of the 1998 International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 5, pp. 2329–2332.
  17. M. Zhang, J. T. Johnson, “Theoretical studies of ocean polarimetric brightness signatures,” in Proceedings of the 1998 International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 5, pp. 2333–2335.
  18. J. T. Johnson, M. Zhang, “Theoretical study of the small slope approximation for ocean polarimetric thermal emission,” IEEE Trans. Geosci. Remote Sens. (to be published).
  19. K. Pak, L. Tsang, J. T. Johnson, “Numerical simulations and backscattering enhancement of electromagnetic waves from two dimensional dielectric random rough surfaces with sparse matrix canonical grid method,” J. Opt. Soc. Am. A 14, 1515–1529 (1997).
    [CrossRef]
  20. J. T. Johnson, R. T. Shin, J. A. Kong, L. Tsang, K. Pak, “A numerical study of ocean polarimetric thermal emission,” IEEE Trans. Geosci. Remote Sens. 37(1), 8–20 (1999).
    [CrossRef]
  21. J. T. Johnson, R. T. Shin, J. A. Kong, “Scattering and thermal emission from a two dimensional periodic surface,” in Progress in Electromagnetic Research15, J. A. Kong, ed. (EMW, Cambridge, Mass., 1997), Chap. 11.

1999 (1)

J. T. Johnson, R. T. Shin, J. A. Kong, L. Tsang, K. Pak, “A numerical study of ocean polarimetric thermal emission,” IEEE Trans. Geosci. Remote Sens. 37(1), 8–20 (1999).
[CrossRef]

1997 (2)

1994 (2)

S. H. Yueh, R. Kwok, F. K. Li, S. V. Nghiem, W. J. Wilson, “Polarimetric passive remote sensing of ocean wind vectors,” Radio Sci. 29, 799–814 (1994).
[CrossRef]

R. Deroo, F. T. Ulaby, “Bistatic specular scattering from rough dielectric surfaces,” IEEE Trans. Antennas Propag. 42, 220–231 (1994).
[CrossRef]

1990 (1)

1989 (1)

E. I. Thorsos, D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86, 261–277 (1989).
[CrossRef]

1982 (1)

1972 (1)

S. T. Wu, A. K. Fung, “A noncoherent model for microwave emissions and backscattering from the sea surface,” J. Geophys. Res. 77, 5917–5929 (1972).
[CrossRef]

1971 (1)

D. E. Barrick, “Theory of HF and VHF propagation across the rough sea 1: the effective surface impedance for a slightly rough highly conducting medium at grazing incidence,” Radio Sci. 6, 517–526 (1971).
[CrossRef]

1968 (1)

G. R. Valenzuela, “Scattering of electromagnetic waves from a tilted slightly rough surface,” Radio Sci. 3, 1057–1066 (1968).

1967 (1)

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
[CrossRef]

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 361–378 (1951).
[CrossRef]

Barrick, D. E.

D. E. Barrick, “Theory of HF and VHF propagation across the rough sea 1: the effective surface impedance for a slightly rough highly conducting medium at grazing incidence,” Radio Sci. 6, 517–526 (1971).
[CrossRef]

Deroo, R.

R. Deroo, F. T. Ulaby, “Bistatic specular scattering from rough dielectric surfaces,” IEEE Trans. Antennas Propag. 42, 220–231 (1994).
[CrossRef]

Friberg, A.

Fung, A. K.

S. T. Wu, A. K. Fung, “A noncoherent model for microwave emissions and backscattering from the sea surface,” J. Geophys. Res. 77, 5917–5929 (1972).
[CrossRef]

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing (Addison Wesley, Reading, Mass., 1982).

A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Boston, Mass., 1994).

Irisov, V. G.

V. G. Irisov, “Small-slope expansion for thermal and reflected radiation from a rough surface,” Waves Random Media 7, 1–10 (1997).
[CrossRef]

V. G. Irisov, “Microwave radiation from a weakly non-Gaussian surface,” in Proceedings of the 1998 International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 5, pp. 2329–2332.

Jackson, D. R.

E. I. Thorsos, D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86, 261–277 (1989).
[CrossRef]

Johnson, J. T.

J. T. Johnson, R. T. Shin, J. A. Kong, L. Tsang, K. Pak, “A numerical study of ocean polarimetric thermal emission,” IEEE Trans. Geosci. Remote Sens. 37(1), 8–20 (1999).
[CrossRef]

K. Pak, L. Tsang, J. T. Johnson, “Numerical simulations and backscattering enhancement of electromagnetic waves from two dimensional dielectric random rough surfaces with sparse matrix canonical grid method,” J. Opt. Soc. Am. A 14, 1515–1529 (1997).
[CrossRef]

J. T. Johnson, R. T. Shin, J. A. Kong, “Scattering and thermal emission from a two dimensional periodic surface,” in Progress in Electromagnetic Research15, J. A. Kong, ed. (EMW, Cambridge, Mass., 1997), Chap. 11.

M. Zhang, J. T. Johnson, “Theoretical studies of ocean polarimetric brightness signatures,” in Proceedings of the 1998 International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 5, pp. 2333–2335.

J. T. Johnson, M. Zhang, “Theoretical study of the small slope approximation for ocean polarimetric thermal emission,” IEEE Trans. Geosci. Remote Sens. (to be published).

Kong, J. A.

J. T. Johnson, R. T. Shin, J. A. Kong, L. Tsang, K. Pak, “A numerical study of ocean polarimetric thermal emission,” IEEE Trans. Geosci. Remote Sens. 37(1), 8–20 (1999).
[CrossRef]

J. T. Johnson, R. T. Shin, J. A. Kong, “Scattering and thermal emission from a two dimensional periodic surface,” in Progress in Electromagnetic Research15, J. A. Kong, ed. (EMW, Cambridge, Mass., 1997), Chap. 11.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Kwok, R.

S. H. Yueh, R. Kwok, F. K. Li, S. V. Nghiem, W. J. Wilson, “Polarimetric passive remote sensing of ocean wind vectors,” Radio Sci. 29, 799–814 (1994).
[CrossRef]

Li, F. K.

S. H. Yueh, R. Kwok, F. K. Li, S. V. Nghiem, W. J. Wilson, “Polarimetric passive remote sensing of ocean wind vectors,” Radio Sci. 29, 799–814 (1994).
[CrossRef]

Moore, R. K.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing (Addison Wesley, Reading, Mass., 1982).

Nghiem, S. V.

S. H. Yueh, R. Kwok, F. K. Li, S. V. Nghiem, W. J. Wilson, “Polarimetric passive remote sensing of ocean wind vectors,” Radio Sci. 29, 799–814 (1994).
[CrossRef]

Nieto-Vesperinas, M.

Pak, K.

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 361–378 (1951).
[CrossRef]

Shin, R. T.

J. T. Johnson, R. T. Shin, J. A. Kong, L. Tsang, K. Pak, “A numerical study of ocean polarimetric thermal emission,” IEEE Trans. Geosci. Remote Sens. 37(1), 8–20 (1999).
[CrossRef]

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

J. T. Johnson, R. T. Shin, J. A. Kong, “Scattering and thermal emission from a two dimensional periodic surface,” in Progress in Electromagnetic Research15, J. A. Kong, ed. (EMW, Cambridge, Mass., 1997), Chap. 11.

Soto-Crespo, J. M.

Thorsos, E. I.

E. I. Thorsos, D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86, 261–277 (1989).
[CrossRef]

Tsang, L.

J. T. Johnson, R. T. Shin, J. A. Kong, L. Tsang, K. Pak, “A numerical study of ocean polarimetric thermal emission,” IEEE Trans. Geosci. Remote Sens. 37(1), 8–20 (1999).
[CrossRef]

K. Pak, L. Tsang, J. T. Johnson, “Numerical simulations and backscattering enhancement of electromagnetic waves from two dimensional dielectric random rough surfaces with sparse matrix canonical grid method,” J. Opt. Soc. Am. A 14, 1515–1529 (1997).
[CrossRef]

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Ulaby, F. T.

R. Deroo, F. T. Ulaby, “Bistatic specular scattering from rough dielectric surfaces,” IEEE Trans. Antennas Propag. 42, 220–231 (1994).
[CrossRef]

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing (Addison Wesley, Reading, Mass., 1982).

Valenzuela, G. R.

G. R. Valenzuela, “Scattering of electromagnetic waves from a tilted slightly rough surface,” Radio Sci. 3, 1057–1066 (1968).

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
[CrossRef]

Voronovich, A. G.

A. G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, Berlin, 1994).

Wilson, W. J.

S. H. Yueh, R. Kwok, F. K. Li, S. V. Nghiem, W. J. Wilson, “Polarimetric passive remote sensing of ocean wind vectors,” Radio Sci. 29, 799–814 (1994).
[CrossRef]

Wu, S. T.

S. T. Wu, A. K. Fung, “A noncoherent model for microwave emissions and backscattering from the sea surface,” J. Geophys. Res. 77, 5917–5929 (1972).
[CrossRef]

Yueh, S. H.

S. H. Yueh, R. Kwok, F. K. Li, S. V. Nghiem, W. J. Wilson, “Polarimetric passive remote sensing of ocean wind vectors,” Radio Sci. 29, 799–814 (1994).
[CrossRef]

Zhang, M.

J. T. Johnson, M. Zhang, “Theoretical study of the small slope approximation for ocean polarimetric thermal emission,” IEEE Trans. Geosci. Remote Sens. (to be published).

M. Zhang, J. T. Johnson, “Theoretical studies of ocean polarimetric brightness signatures,” in Proceedings of the 1998 International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 5, pp. 2333–2335.

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 361–378 (1951).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
[CrossRef]

R. Deroo, F. T. Ulaby, “Bistatic specular scattering from rough dielectric surfaces,” IEEE Trans. Antennas Propag. 42, 220–231 (1994).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

J. T. Johnson, R. T. Shin, J. A. Kong, L. Tsang, K. Pak, “A numerical study of ocean polarimetric thermal emission,” IEEE Trans. Geosci. Remote Sens. 37(1), 8–20 (1999).
[CrossRef]

J. Acoust. Soc. Am. (1)

E. I. Thorsos, D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86, 261–277 (1989).
[CrossRef]

J. Geophys. Res. (1)

S. T. Wu, A. K. Fung, “A noncoherent model for microwave emissions and backscattering from the sea surface,” J. Geophys. Res. 77, 5917–5929 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Radio Sci. (3)

S. H. Yueh, R. Kwok, F. K. Li, S. V. Nghiem, W. J. Wilson, “Polarimetric passive remote sensing of ocean wind vectors,” Radio Sci. 29, 799–814 (1994).
[CrossRef]

G. R. Valenzuela, “Scattering of electromagnetic waves from a tilted slightly rough surface,” Radio Sci. 3, 1057–1066 (1968).

D. E. Barrick, “Theory of HF and VHF propagation across the rough sea 1: the effective surface impedance for a slightly rough highly conducting medium at grazing incidence,” Radio Sci. 6, 517–526 (1971).
[CrossRef]

Waves Random Media (1)

V. G. Irisov, “Small-slope expansion for thermal and reflected radiation from a rough surface,” Waves Random Media 7, 1–10 (1997).
[CrossRef]

Other (8)

V. G. Irisov, “Microwave radiation from a weakly non-Gaussian surface,” in Proceedings of the 1998 International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 5, pp. 2329–2332.

M. Zhang, J. T. Johnson, “Theoretical studies of ocean polarimetric brightness signatures,” in Proceedings of the 1998 International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 5, pp. 2333–2335.

J. T. Johnson, M. Zhang, “Theoretical study of the small slope approximation for ocean polarimetric thermal emission,” IEEE Trans. Geosci. Remote Sens. (to be published).

A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Boston, Mass., 1994).

A. G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, Berlin, 1994).

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing (Addison Wesley, Reading, Mass., 1982).

J. T. Johnson, R. T. Shin, J. A. Kong, “Scattering and thermal emission from a two dimensional periodic surface,” in Progress in Electromagnetic Research15, J. A. Kong, ed. (EMW, Cambridge, Mass., 1997), Chap. 11.

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Figures (3)

Fig. 1
Fig. 1

In plane, HH bistatic scattering cross sections per unit area for a Gaussian correlation function surface with h=0.06λ and l=λ, θi=10°, and =3.

Fig. 2
Fig. 2

Comparison of SPM, SSA, and MOM bistatic scattering cross sections per unit area for a Gaussian correlation function surface with h=0.06λ and l=λ, θi=10°, and =3. (a) HH, (b) VV.

Fig. 3
Fig. 3

Asymmetric pyramidal surface.

Tables (2)

Tables Icon

Table 1 HH Percent Reflectivities in Near-Specular Modes for an Asymmetric Pyramidal Grating

Tables Icon

Table 2 VH Percent Reflectivities in Near-Specular Modes for an Asymmetric Pyramidal Grating

Equations (159)

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

f(x, y)=n=-m=- expi 2πnxPxexpi 2πmyPyhn,m,
hn,m=1Px 1Py 0Pxdx0Pydy exp-i 2πnxPx×exp-i 2πmyPyf(x, y).
fx=nm expi 2πnxPxexpi 2πmyPyi 2πnPx hn,m,
fy=nm expi 2πnxPxexpi 2πmyPyi 2πmPy hn,m.
F{f(x, y)}(n, m)=1Px 1Py 0Pxdx0Pydy exp-i 2πnxPx×exp-i 2πmyPyf(x, y),
F{f(x, y)N}(n, m)
=n1m1n2m2()nN-1mN-1hn1,m1hn2,m2hnN-1,mN-1hn-n1-n2--nN-1,m-m1-m2--mN-1
Ffx f(x, y)N-1(n, m)
=n1m1n2m2()nN-1mN-1hn1,m1hn2,m2hnN-1,mN-1hn-n1-n2--nN-1,m-m1-m2--mN-1
×i 2πPx (n-n1-n2--nN-1)
Ffy f(x, y)N-1(n, m)
=n1m1n2m2()nN-1mN-1hn1,m1hn2,m2hnN-1,mN-1hn-n1-n2--nN-1,m-m1-m2-mN-1×i 2πPy (m-m1-m2--mN-1).
Ei=e^i exp(ikir),
Hi=k^i×e^iη0 exp(ikir),
ki=k0k^i=xˆkxi+yˆkyi-zˆkzi
r=xˆx+yˆy+zˆz
Es=mn[h^sn,mαn,m+v^sn,mβn,m]exp(iksn,mr),
Hs=1η0 mn[-v^sn,mαn,m+h^sn,mβn,m]exp(iksn,mr),
Et=mn[h^sn,mγn,m+v^tn,mδn,m]exp(iktn,mr),
Ht=1η1 mn[-v^tn,mγn,m+h^sn,mδn,m]exp(iktn,mr),
ksn,m=xˆkxn+yˆkym+zˆkznm,
ktn,m=xˆkxn+yˆkym-zˆkz1nm,
kxn=kxi+2πnPx,
kym=kyi+2πmPy,
kρnm=kxn2+kym2,
kznm=k02-kρnm2,
kz1nm=k02-kρnm2.
h^i=xˆ kyikρi-yˆ kxikρi,
h^sn,m=xˆ kymkρnm-yˆ kxnkρnm,
h^tn,m=h^sn,m,
v^i=xˆ kxikzik0kρi+yˆ kyikzik0kρi+zˆ kρik0,
v^sn,m=-xˆ kxnkznmk0kρnm-yˆ kymkznmk0kρnm+zˆ kρnmk0,
v^tn,m=xˆ kxnkz1nmk1kρnm+yˆ kymkz1nmk1kρnm+zˆ kρnmk1,
(zˆ-f)×(Ei+Es)=(zˆ-f)×Et,
(zˆ-f)×(Hi+Hs)=(zˆ-f)×Ht,
zˆ×Es-zˆ×Et
=-zˆ×Ei+f×Ei+f×Es-f×Et,
zˆ×Hs-zˆ×Ht
=-zˆ×Hi+f×Hi+f×Hs-f×Ht.
mn expi 2πnxPxexpi 2πmyPy(zˆ×h^sn,m)[αn,m exp(ikznmz)-γn,m exp(-ikz1nmz)]
+(zˆ×v^sn,m)βn,m exp(ikznmz)+k0kz1nmk1kznm δn,m exp(-ikz1nmz)
=-(zˆ×e^i)exp(-ikziz)+(f×e^i)exp(-ikziz)+mn expi 2πnxPxexpi 2πmyPy
×(f×v^sn,m)βn,m exp(ikznmz)-k0k1 δn,m exp(-ikz1nmz),
mn expi 2πnxPxexpi 2πmyPy(-zˆ×v^sn,m)αn,m exp(ikznmz)+kz1nmkznm γn,m exp(-ikz1nmz)
+(zˆ×h^sn,m)βn,m exp(ikznmz)-k1k0 δn,m exp(-ikz1nmz)
=-(zˆ×k^i×e^i)exp(-ikziz)+(f×k^i×e^i)exp(-ikziz)+mn expi 2πnxPxexpi 2πmyPy
×{(-f×v^sn,m)[αn,m exp(ikznmz)-γn,m exp(-ikz1nmz)]}.
exp(±ikzz)=q=0 (±ikzz)qq!
αn,m=αn,m(0)+αn,m(1)+=l=0αn,m(l)
mn expi 2πnxPxexpi 2πmyPy(zˆ×h^sn,m)(αn,m(N)-γn,m(N))+(zˆ×v^sn,m)βn,m(N)+k0kz1nmk1kznm δn,m(N)
=-(zˆ×e^i) (-ikziz)NN!+(f×e^i) (-ikziz)N-1(N-1)!-l=0N-1 (iz)N-l(N-l)! mn expi 2πnxPxexpi 2πmyPy×(zˆ×h^sn,m)[αn,m(l)(kznm)N-l-γn,m(l)(-kz1nm)N-l]+(zˆ×v^sn,m)βn,m(l)(kznm)N-l+k0kz1nmk1kznm δn,m(l)(-kz1nm)N-l+l=0N-1 (iz)N-l-1(N-l-1)! mn expi 2πnxPxexpi 2πmyPy(f×v^sn,m)βn,m(l)(kznm)N-l-1-k0k1 δn,m(l)(-kz1nm)N-l-1,
mn expi 2πnxPxexpi 2πmyPy(-zˆ×v^sn,m)αn,m(N)+kz1nmkznm γn,m(N)+(zˆ×h^sn,m)βn,m(N)-k1k0 δn,m(N)
=-(zˆ×k^i×e^i) (-ikziz)NN!+(f×k^i×e^i) (-ikziz)N-1(N-1)!-l=0N-1 (iz)N-l(N-l)! mn expi 2πnxPxexpi 2πmyPy×-(zˆ×v^sn,m)αn,m(l)(kznm)N-l+kz1nmkznm γn,m(l)(-kz1nm)N-l+(zˆ×h^sn,m)βn,m(l)(kznm)N-l-k1k0 δn,m(l)(-kz1nm)N-l+l=0N-l (iz)N-l-1(N-l-1)! mn expi 2πnxPxexpi 2πmyPy{(-f×v^sn,m)[αn,m(l)(kznm)N-l-1-γn,m(l)(-kz1nm)N-l-1]}.
mn expi 2πnxPxexpi 2πmyPy
×(zˆ×h^sn,m)(αn,m(N)-γn,m(N))+(zˆ×v^sn,m)
×βn,m(N)+k0kz1nmk1kznm δn,m(N)=SE(N)(x, y),
mn expi 2πnxPxexpi 2πmyPy
×(-zˆ×v^sn,m)αn,m(N)+kz1nmkznm γn,m(N)+(zˆ×h^sn,m)
×βn,m(N)-k1k0 δn,m(N)=SH(N)(x, y).
(zˆ×h^sn,m)(αn,m(N)-γn,m(N))+(zˆ×v^sn,m)
×βn,m(N)+k0kz1nmk1kznm δn,m(N)
=F{SE(N)(x, y)}(n, m),
(-zˆ×v^sn,m)αn,m(N)+kz1nmkznm γn,m(N)+(zˆ×h^sn,m)
×βn,m(N)-k1k0 δn,m(N)
=F{SH(N)(x, y)}(n, m),
αn,m(N)=-k0kznm+kz1nmv^sn,mF{SE(N)(x, y)}×kz1nmkznm+h^sn,mF{SH(N)(x, y)},
βn,m(N)=k0kznm+kz1nmh^sn,mF{SE(N)(x, y)}-v^sn,mF{SH(N)(x, y)} kz1nmkznm,
γn,m(N)=k0kznm+kz1nm(v^sn,mF{SE(N)(x, y)}-h^sn,mF{SH(N)(x, y)}),
δn,m(N)=k1kznm+kz1nm(h^sn,mF{SE(N)(x, y)}+v^sn,mF{SH(N)(x, y)}),
SE(0)=-zˆ×e^i,
SH(0)=-zˆ×k^i×e^i.
α0,0(0)=kzi-kz1ikzi+kz1i=ΓH,
β0,0(0)=0,
γ0,0(0)=1+ΓH,
δ0,0(0)=0,
α0,0(0)=0,
β0,0(0)=kzi-kz1ikzi+kz1i=ΓV,
γ0,0(0)=0,
δ0,0(0)=k0k1 (1+ΓV),
αn,m(N, 0)=iNN! -k0kznm+kz1nm-F{zN}ci,n-kz1nmk0 R1h-kzik0 R2h+kymkρnm Ffy zN-1+kxnkρnm Ffx zN-1(-iN)R3h,
βn,m(N, 0)=iNN! k0kznm+kz1nm-F{zN}si,nR1h+kzikz1nmk02 R2h-kz1nmk0 kymkρnm Ffx zN-1-kxnkρnm Ffy zN-1(-iN)R3h,
γn,m(N, 0)=iNN! k0kznm+kz1nm-F{zN}ci,n-kznmk0 R1h+kzik0 R2h-kymkρnm Ffy zN-1+kxnkρnm Ffx zN-1(-iN)R3h,
δn,m(N, 0)=iNN! k1kznm+kz1nm-F{zN}si,nR1h-kzikznmk02 R2h+kznmk0 kymkρnm Ffx zN-1-kxnkρnm Ffy zN-1(-iN)R3h,
R1h=ΓH(kzi)N+(-kzi)N-(1+ΓH)(-kz1i)N,
R2h=ΓH(kzi)N-(-kzi)N+kz1ikzi (1+ΓH)(-kz1i)N,
R3h=-kρik0 ((-kzi)N-1+ΓH(kzi)N-1-(1+ΓH)(-kz1i)N-1),
ci,n=kxikxn+kyikymkρikρnm,
si,n=kxikym-kyikxnkρikρnm.
αn,m(N, 0)=iNN! -k0kznm+kz1nm-F{zN}si,nkzikz1nmk02 R1v+R2v+kz1nmk0 kymkρnm Ffx zN-1-kxnkρnm Ffy zN-1(-iN)R3v,
βn,m(N, 0)=iNN! k0kznm+kz1nm-F{zN}ci,nkzik0 R1v+kz1nmk0 R2v+kymkρnm Ffy zN-1+kxnkρnm Ffx zN-1(-iN)R3v,
γn,m(N, 0)=iNN! k0kznm+kz1nm-F{zN}si,nkzikznmk02 R1v-R2v+kznmk0 kymkρnm Ffx zN-1-kxnkρnm Ffy zN-1(-iN)R3v,
δn,m(N, 0)=iNN! k1kznm+kz1nm-F{zN}ci,nkzik0 R1v-kznmk0 R2v+kymkρnm Ffy zN-1+kxnkρnm Ffx zN-1(-iN)R3v,
R1v=ΓV(kzi)N-(-kzi)N+kz1ikzi (1+ΓV)(-kz1i)N,
R2v=ΓV(kzi)N+(-kzi)N-(1+ΓV)(-kz1i)N,
R3v=kρik0 (-kzi)N-1+ΓV(kzi)N-1-1+ΓV (-kz1i)N-1.
ζn,m(1)=hn,mgζ(1)(kxn, kym),
gα(1)=-2ikzi(k02-k12)(kznm+kz1nm)(kzi+kz1i) ci,n,
gβ(1)=-2ikzi(k02-k12)(kznm+kz1nm)(kzi+kz1i) kz1nmk0si,n,
gγ(1)=gα(1),
gδ(1)=-gβ(1)kznmk1kz1nmk0,
gα(1)=-2ikzi(k02-k12)(kznm+kz1nm)(kzi+kz1i) kz1ik0si,n,
gβ(1)=-2ikzi(k02-k12)(kznm+kz1nm)(kzi+kz1i)×kρikρnmk02-kz1ikz1nmk02 ci,n,
gγ(1)=gα(1),
gδ(1)=-2ikzi(k02-k12)(kznm+kz1nm)(kzi+kz1i)×kρikρnmk02+kz1ikznmk02 ci,n,
αn,m(N, r)=-k0kznm+kz1nm -l=1N-1 iN-l(N-l)! mnF{zN-l}(n-n, m-m)×cn,n-αn,m(l)(kznm)N-lkz1nm+kznmk0+γn,m(l)(-kz1nm)N-lkz1nm-kz1nmk0+sn,nβn,m(l)(kznm)N-lk02+kz1nmkznmk02+δn,m(l)(-kz1nm)N-l-k12+kz1nmkz1nmk1k0+l=1N-1 iN-l-1(N-l-1)! mn kρnmk0kymkρnm Ffx zN-l-1(n-n, m-m)-kxnkρnm Ffy zN-l-1(n-n, m-m)kz1nmk0βn,m(l)(kznm)N-l-1-k0k1 δn,m(l)(-kz1nm)N-l-1-kymkρnm Ffy zN-l-1(n-n, m-m)+kxnkρnm Ffx zN-l-1(n-n, m-m)×[αn,m(l)(kznm)N-l-1-γn,m(l)(-kz1nm)N-l-1],
βn,m(N, r)=k0kznm+kz1nm -l=1N-1 iN-1(N-l)! mnF{zN-1}(n-n, m-m)×sn,nαn,m(l)(kznm)N-lk12+kz1nmkznmk02+γn,m(l)(-kz1nm)N-l-k12+kz1nmkz1nmk02+cn,nβn,m(l)(kznm)N-lkznm+kz1nmk0+δn,m(l)(-kz1nm)N-l kz1nm-kz1nmk1+l=1N-1 iN-l-1(N-l-1)! mn kρnmk0kymkρnm Ffx zN-l-1(n-n, m-m)-kxnkρnm Ffy zN-l-1(n-n, m-m) kz1nmk0 [αn,m(l)(kznm)N-l-1-γn,m(l)(-kz1nm)N-l-1]+kymkρnm F fy zN-l-1(n-n,m-m)+kxnkρnm Ffx zN-l-1(n-n, m-m)×βn,m(l)(kznm)N-l-1-k0k1 δn,m(l)(-kz1nm)N-l-1,
γn,m(N, r)=k0kznm+kz1nm(-l=1N-1 iN-l(N-l)! mnF{zN-l}(n-n,m-m)×cn,nαn,m(l)(kznm)N-lkznm-kznmk0+γn,m(l)(-kz1nm)N-lkznm+kz1nmk0+sn,nβn,m(l)(kznm)N-l-k02+kznmkznmk02+δn,m(l)(-kz1nm)N-lk12+kznmkz1nmk1k0+l=1N-1 iN-l-1(N-l-1)! mn kρnmk0kymkρnm Ffx zN-l-1(n-n, m-m)-kxnkρnm Ffy zN-l-1(n-n, m-m)kznmk0βn,m(l)(kznm)N-l-1-k0k1 δn,m(l)(-kz1nm)N-l-1
+kymkρnm Ffy zN-l-1(n-n, m-m)+kxnkρnm Ffx zN-l-1(n-n, m-m)
×[αn,m(l)(kznm)N-l-1-γn,m(l)(-kz1nm)N-l-1]),
δn,m(N, r)=k1kznm+kz1nm -l=1N-1 iN-l(N-l)! mnF{zN-l}(n-n, m-m)×sn,nαn,m(l)(kznm)N-lk02-kznmkznmk02-γn,m(l)(-kz1nm)N-lk02+kznmkz1nmk02+cn,nβn,m(l)(kznm)N-lkznm-kznmk0+δn,m(l)(-kz1nm)N-lkz1nm+kznmk1+l=1N-1 iN-l-1(N-l-1)! mn kρnmk0 kymkρnm Ffx zN-l-1(n-n, m-m)-kxnkρnm Ffy zN-l-1(n-n, m-m)-kznmk0[αn,m(l)(kznm)N-l-1-γn,m(l)(-kz1nm)N-l-1]+kymkρnm Ffy zN-l-1(n-n, m-m)+kxnkρnm Ffx zN-l-1(n-n, m-m)×βn,m(l)(kznm)N-l-1-k0k1 δn,m(l)(-kz1nm)N-l-1,
ζn,m(2)=mnhn-n,m-mhn,mgζ(2)(kxn, kym, kxn, kym)
gα(2)=-2kzi(k02-k12)(kznm+kz1nm)(kzi+kz1i) cn,ncn,i(κ1nm+kz1nm)+sn,nsn,i(kz1nm+κ2nm)+cn,i2 (kz1i-kz1nm),
gβ(2)=-2kzi(k02-k12)(kznm+kz1nm)(kzi+kz1i) sn,ncn,ik0+kz1nmk0 κ1nm-cn,nsn,ik0+kz1nmk0 κ2nm+sn,i kρnmk0 κ3nm+sn,i2 k0-kz1nmk0 kz1i,
gγ(2)=-2kzi(k02-k12)(kznm+kz1nm)(kzi+kz1i) cn,ncn,i(κ1nm-kznm)+sn,nsn,i(-kznm+κ2nm)+cn,i2 (kz1i+kznm),
gδ(2)=-2kzi(k02-k12)(kznm+kz1nm)(kzi+kz1i) sn,ncn,ik0-kznmk0 κ1nm-cn,nsn,ik0-kznmk0 κ2nm+sn,i kρnmk0 κ3nm+sn,i2 k0+kznmk0 kz1i,
κ1nm=kznm-kz1nm,
κ2nm=kznmkz1nm(k02-k12)k02(kznm+kz1nm),
κ3nm=kρnmk02kρnm2+kznmkz1nm.
gα(2)=-2kzi(k02-k12)(kznm+kz1nm)(kzi+kz1i) -cn,nsn,ikz1ik0(κ1nm+kz1nm)+sn,ncn,ikz1ik0(kz1nm+κ2nm)-sn,nkρik0κ3nm-sn,i2 k0-kz1nmk0 kz1i,
gβ(2)=-2kzi(k02-k12)(kznm+kz1nm)(kzi+kz1i) -sn,nsn,ikz1ik0k0+kz1nmk0 κ1nm-cn,ncn,ikz1ik0k0+kz1nmk0 κ2nm+cn,nkρikz1nmk02κ3nm+kρnmκ3nmk02kz1icn,i+kρnmkρik02 κ1nm+cn,i2 (kz1i-kz1nm),
gγ(2)=-2kzi(k02-k12)(kznm+kz1nm)(kzi+kz1i) -cn,nsn,ikz1ik0(κ1nm-kznm)+sn,ncn,ikz1ik0(-kznm+κ2nm)-sn,nkρik0κ3nm-sn,i2 k0+kznmk0 kz1i,
gδ(2)=-2kzi(k02-k12)(kznm+kz1nm)(kzi+kz1i) -sn,nsn,ikz1ik0k0-kznmk0 κ1nm-cn,ncn,ikz1ik0k0-kznmk0 κ2nm-cn,nkρikznmk02κ3nm+kρnmκ3nmk02kz1icn,i+kρnmkρik02 κ1nm+cn,i2 (kz1i+kznm).
ζn,m(3)=mnm1n1hn,mhn1,m1hn-n-n1,m-m-m1gζ(3)(kxn, kym, kxn, kym, kxn1, kym1),
gα(3)=ikzmn+kz1nmkzi(k02-k12)kzi+kz1i-cn,ncn,i(kz1nmκ1nm+k0κ4nm)-sn,nsn,iκ2nm(κ5nm+kz1nm)+(2cn,ikρnm-kρi)(kρnm-kρnmcn,n)-cn,i3 (kz1nmkz1i-kzi2-kz1i2)+kρnmkρnm(gα(2)-gγ(2))+cn,n(k0G1h(2)-kz1nmG2h(2))-sn,n(k0G3h(2)+kz1nmG4h(2)),
gβ(3)=-ikzmn+kz1nmkzi(k02-k12)kzi+kz1isn,ncn,i(k0κ1nm+kz1nmκ4nm)-cn,nsn,iκ2nmk0+kz1nmk0 κ5nm-sn,n(2cn,ikρnm-kρi)kρnmkz1nmk0-sn,i3 k12kz1i-kz1nm(kzi2+kz1i2)k0-kρnmkρnm(gβ(2)-gδ(2)/)+sn,n(kz1nmG1h(2)-k0G2h(2))+cn,n(kz1nmG3h(2)+k0G4h(2)),
gγ(3)=-ikzmn+kz1nmkzi(k02-k12)kzi+kz1i-cn,ncn,i(kznmκ1nm-k0κ4nm)-sn,nsn,iκ2nm(-κ5nm+kznm)-(2cn,ikρnm-kρi)(kρnm-kρnmcn,n)-cn,i3 (kznmkz1i+kzi2+kz1i2)-kρnmkρnm(gα(2)-gγ(2))+cn,n(-k0G1h(2)-kznmG2h(2))-sn,n(-k0G3h(2)+kznmG4h(2)),
gδ(3)=-ikzmn+kz1nmkzi(k02-k12)kzi+kz1isn,ncn,i(k0κ1nm-kznmκ4nm)-cn,nsn,iκ2nmk0-kznmk0 κ5nm+sn,n(2cn,ikρnm-kρi)kρnmkznmk0-sn,i3 k02kz1i+kznm(kzi2+kz1i2)k0-kρnmkρnm(gβ(2)-gδ(2)/)+sn,n(-kznmG1h(2)-k0G2h(2))+cn,n(-kznmG3h(2)+k0G4h(2)),
κ4nm=kznm2-kznmkz1nm+kz1nm2k0,
κ5nm=kznm+kz1nm1-,
G1h(2)=k0(gγ(2)-gα(2)),
G2h(2)=kznmgα(2)+kz1nmgγ(2),
G3h(2)=kznmgβ(2)+kz1nmgδ(2),
G4h(2)=k0(gβ(2)-gδ(2)),
gζ(2)(kxn+kxn1-kxi, kym+kym1-kyi, kxn1, kym1)
gα(3)=ikzmn+kz1nmkzi(k02-k12)kzi+kz1icn,nsn,ikz1ik0(kz1nmκ1nm+k0κ4nm)-sn,ncn,iκ2nmkz1ik0(κ5nm+kz1nm)+sn,nkρikρnmk0k12(k02-k12)+kz1nm(kznm3+kz1nm3)k02(kznm+kz1nm)-2sn,ikz1ikρnmk0(kρnm-kρnmcn,n)-sn,nkρnmkz1nmkρikρnm2k03+sn,ik03(k02-k12) kz1nm(kzi4-kz1i4)k02+kz1i(kz1i2-kzi2)+kρnmkρnm(ga(2)-gγ(2))+cn,n(k0G1v(2)-kz1nmG2v(2))-sn,n(k0G3v(2)+kz1nmG4v(2)),
gβ(3)=-ikzmn+kz1nmkzi(k02-k12)kzi+kz1i-sn,nsn,ikz1ik0(k0κ1nm+kz1nmκ4nm)-cn,ncn,iκ2nmkz1ik0k0+kz1nmk0 κ5nm+cn,nkρikρnmk02kznm3+kz1nm3+kz1nm(k02-k12)kznm+kz1nm
+2sn,isn,nkρnmkρnmkz1nmkz1ik02-kρnm2kρik02(kρnm-cn,nkρnm)
-cn,i3(k02-k12) ((kzi4-kz1i4)+kz1nmkz1i(kz1i2-kzi2))-kρnmkρnm(gβ(2)-gδ(2)/)
+sn,n(kz1nmG1v(2)-k0G2v(2))+cn,n(kz1nmG3v(2)+k0G4v(2)),
gγ(3)=-ikzmn+kz1nmkzi(k02-k12)kzi+kz1icn,nsn,ikz1ik0(kznmκ1nm-k0κ4nm)-sn,ncn,iκ2nmkz1ik0(-κ5nm+kznm)+sn,nkρikρnmk0k12(k12-k02)+kznm(kznm3+kz1nm3)k02(kznm+kz1nm)+2sn,ikz1ikρnmk0(kρnm-kρnmcn,n)-sn,nkρnmkznmkρikρnm2k03+sn,ik03(k02-k12) kznm(kzi4-kz1i4)k02-kz1i(kz1i2-kzi2)-kρnmkρnm(gα(2)-gγ(2))+cn,n(-k0G1v(2)-kznmG2v(2))-sn,n(-k0G3v(2)+kznmG4v(2)),
gδ(3)=-ikzmn+kz1nmkzi(k02-k12)kzi+kz1i-sn,nsn,ikz1ik0(k0κ1nm-kznmκ4nm)-cn,ncn,iκ2nmkz1ik0k0-kznmk0 κ5nm+cn,nkρikρnmk02kznm3+kz1nm3-kznm(k02-k12)kznm+kz1nm-2sn,isn,nkρnmkρnmkznmkz1ik02-kρnm2kρik02(kρnm-cn,nkρnm)-cn,i3(k02-k12) (kzi4-kz1i4-kznmkz1i(kz1i2-kzi2))-kρnmkρnm(gβ(2)-gδ(2)/)-sn,n(kznmG1v(2)+k0G2v(2))+cn,n(-kznmG3v(2)+k0G4v(2)),
gζ(2)(kxn+kxn1-kxi, kym+kym1-kyi, kxn1, kym1)
|ζn,m|2=|ζn,m(0)+ζn,m(1)+ζn,m(2)+ζn,m(3)+|2,
|ζn,m|2=(|ζn,m(0)|2)+(2 Re{ζn,m(0)*ζn,m(1)})+(|ζn,m(1)|2+2 Re{ζn,m(0)*ζn,m(2)})+(2 Re{ζn,m(1)*ζn,m(2)}+2 Re{ζn,m(0)*ζn,m(3)})+(|ζn,m(2)|2+2 Re{ζn,m(1)*ζn,m(3)}+2 Re{ζn,m(0)*ζn,m(4)})+
Rekznmkzi|ζ|2,
Rekz1nmkzi|ζ|2,
|ζ|2=|Γζeff|2
Γζeff=Γζ+mn|hn,m|2gζ(2)(kxi, kyi, kxn, kym)+mnm1n1hn,mhn1,m1h-n-n1,-m-m1×gζ(3)(kxi, kyi, kxn, kym, kxn1, kym1)
|ζn,m-ζn,m|2
=|ζn,m(1)|2+2 Re{ζn,m(1)*ζn,m(2)}+|ζn,m(2)-ζn,m(2)|2+2 Re{ζn,m(1)*ζn,m(3)}
|ζn,m-ζn,m|2=|gζ(1)(kxn, kym)|2|hn,m|2+2 Remnhn,m*hn,mhn-n,m-mgζ(1)*(kxn, kym)gζ(2)(kxn, kym, kxn, kym)+mnm1n1[hn-n,m-mhn,mhn-n1,m-m1*hn1,m1*-hn-n,m-mhn,m×hn-n1,m-m1*hn1,m1*]gζ(2)(kxn, kym, kxn, kym)gζ(2)*(kxn, kym, kxn1, kym1)+2 Remnm1n1hn,mhn1,m1hn-n-n1,m-m-m1h-n,-m×gζ(3)(kxn, kym, kxn, kym, kxn1, kym1)gζ(1)*(kxn, kym),
σζ=4πk02 cos2 θs |ζ|2δkxδky,
|hn,m|2δkxδky
=W(kxn-kxi, kym-kyi),
hn,mhn,mh-n-n,-m-m(δkx)2(δky)2
=B(kxn-kxi, kym-kyi, kxn-kxi, kym-kyi),
hn,mhn,mhn1,m1h-n-n-n1,-m-m-m1(δkx)3(δky)3
=T(kxn-kxi, kym-kyi, kxn-kxi,kym-kyi, kxn1-kxi, kym1-kyi),
σζ(kxn, kym)=4πk02 cos2 θs|gζ(1)(kxn, kym)|2W(kxn-kxi, kym-kyi)+-dkxn-dkym{W(kxn-kxi, kym-kyi)W(kxn-kxn, kym-kym)[|gζ(2)(kxn, kym, kxn, kym)|2+gζ(2)(kxn, kym, kxn, kym)gζ(2)*(kxn, kym, kxn-kxn+kxi, kym-kym+kyi)]}+2 ReW(kxn-kxi, kym-kyi)gζ(1)*(kxn, kym)-dkxn-dkymW(kxn-kxi, kym-kyi)×[gζ(3)(kxn, kym, kxn, kym, kxn, kym)+gζ(3)(kxn, kym, kxn, kym, kxn, kym)×gζ(3)(kxn, kym,kxn, kym, 2kxi-kxn, 2kyi-kym)].

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