Abstract

Periodic thin-film structures are widely used as absorptive structures for electromagnetic radiation. We show that the absorption behavior for partially coherent illumination can be fully characterized by a set of characteristic functions in wavenumber space. We discuss the prediction of these functions using electromagnetic solvers based on periodic boundary conditions, and their measurement experimentally using Energy Absorption Interferometry (EAI). The theory is developed here for the case of 2D absorbers with TE illumination and arbitrary material properties in the plane of the problem, except for the resistivity, which is assumed isotropic. Numerical examples are given for the case of absorbing strips printed on a semi-infinite substrate. We derive rules for the convergence of the representation as a function of the number of characteristic functions used, as well as conditions for sampling in EAI experiments.

© 2014 Optical Society of America

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  1. T. Perera, T. Downes, S. Meyer, T. Crawford, E. Cheng, T. Chen, D. Cottingham, E. Sharp, R. Silverberg, F. Finkbeiner, D. Fixsen, D. Logan, and G. Wilson, “Optical performance of frequency-selective bolometers,” Appl. Opt. 45, 7643–7651 (2006).
    [CrossRef]
  2. P. Mauskopf, J. Bock, H. Del Castillo, W. Holzapfel, and A. Lange, “Composite infrared bolometers with Si3N4 micromesh absorbers,” Appl. Opt. 36, 765–771 (1997).
    [CrossRef]
  3. J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. P. Drupp, and L. Li, “The design and fabrication of planar multiband metallodielectric frequency selective surfaces for infrared applications,” IEEE Trans. Antennas Propag. 54, 1265–1276 (2006).
    [CrossRef]
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  5. G. Saklatvala, S. Withington, and M. P. Hobson, “Coupled-mode theory for infrared and submillimeter wave detectors,” J. Opt. Soc. Am. A 24, 764–775 (2007).
    [CrossRef]
  6. S. Withington and G. Saklatvala, “Characterizing the behavior of partially coherent detectors through spatio-temporal modes,” J. Opt. A 9, 626–633 (2007).
    [CrossRef]
  7. C. N. Thomas and S. Withington, “Electromagnetic simulations of the partially coherent optical behavior of resistive film TES detectors,” in Proceedings of the 21st International Symposium on Space Terahertz Technology, Oxford, UK, March23–25, 2010.
  8. C. N. Thomas and S. Withington, “Experimental demonstration of an interferometric technique for characterizing the full optical behavior of multi-mode power detectors,” IEEE Trans. Terahertz Sci. Technol. 2, 50–60 (2012).
    [CrossRef]
  9. S. Withington, C. N. Thomas, and C. Craeye, “Determining the natural absorption and radiation modes of lossy periodic structures using energy absorption interferometry,” in Proceedings of the International Conference on Electromagnetics in Advanced Applications (IEEE, 2011), pp. 155–158.
  10. B. Munk and G. Burrell, “Plane-wave expansion for arrays of arbitrarily oriented piecewise linear elements and its application in determining the impedance of a single linear antenna in a lossy half-space,” IEEE Trans. Antennas Propag. 27, 331–343 (1979).
    [CrossRef]
  11. E. Wolf, “Coherent-mode propagation in spatially band-limited wave fields,” J. Opt. Soc. Am. A 3, 1920–1924 (1986).
    [CrossRef]
  12. S. Withington and C. N. Thomas, “Probing the dynamical behavior of surface dipoles through energy-absorption interferometry,” Phys. Rev. A 86, 043835 (2012).
    [CrossRef]
  13. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).
  14. C. Craeye and D. González-Ovejero, “A review on array mutual coupling analysis,” Radio Sci. 46, RS2012 (2011).
    [CrossRef]
  15. F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
    [CrossRef]
  16. R. F. Harrington and J. L. Harrington, Field Computation by Moment Methods (Oxford University, 1996).
  17. C. Craeye, X. Radu, F. Capolino, and A. Schuchinsky, “Fundamentals of method of moments for metamaterials,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (Taylor & Francis, 2009), pp. 5–1–5–31.
  18. C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).

2012 (2)

C. N. Thomas and S. Withington, “Experimental demonstration of an interferometric technique for characterizing the full optical behavior of multi-mode power detectors,” IEEE Trans. Terahertz Sci. Technol. 2, 50–60 (2012).
[CrossRef]

S. Withington and C. N. Thomas, “Probing the dynamical behavior of surface dipoles through energy-absorption interferometry,” Phys. Rev. A 86, 043835 (2012).
[CrossRef]

2011 (1)

C. Craeye and D. González-Ovejero, “A review on array mutual coupling analysis,” Radio Sci. 46, RS2012 (2011).
[CrossRef]

2007 (3)

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[CrossRef]

S. Withington and G. Saklatvala, “Characterizing the behavior of partially coherent detectors through spatio-temporal modes,” J. Opt. A 9, 626–633 (2007).
[CrossRef]

G. Saklatvala, S. Withington, and M. P. Hobson, “Coupled-mode theory for infrared and submillimeter wave detectors,” J. Opt. Soc. Am. A 24, 764–775 (2007).
[CrossRef]

2006 (2)

T. Perera, T. Downes, S. Meyer, T. Crawford, E. Cheng, T. Chen, D. Cottingham, E. Sharp, R. Silverberg, F. Finkbeiner, D. Fixsen, D. Logan, and G. Wilson, “Optical performance of frequency-selective bolometers,” Appl. Opt. 45, 7643–7651 (2006).
[CrossRef]

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. P. Drupp, and L. Li, “The design and fabrication of planar multiband metallodielectric frequency selective surfaces for infrared applications,” IEEE Trans. Antennas Propag. 54, 1265–1276 (2006).
[CrossRef]

1997 (1)

1986 (1)

1979 (1)

B. Munk and G. Burrell, “Plane-wave expansion for arrays of arbitrarily oriented piecewise linear elements and its application in determining the impedance of a single linear antenna in a lossy half-space,” IEEE Trans. Antennas Propag. 27, 331–343 (1979).
[CrossRef]

Bock, J.

Bossard, J. A.

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. P. Drupp, and L. Li, “The design and fabrication of planar multiband metallodielectric frequency selective surfaces for infrared applications,” IEEE Trans. Antennas Propag. 54, 1265–1276 (2006).
[CrossRef]

Burrell, G.

B. Munk and G. Burrell, “Plane-wave expansion for arrays of arbitrarily oriented piecewise linear elements and its application in determining the impedance of a single linear antenna in a lossy half-space,” IEEE Trans. Antennas Propag. 27, 331–343 (1979).
[CrossRef]

Capolino, F.

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[CrossRef]

C. Craeye, X. Radu, F. Capolino, and A. Schuchinsky, “Fundamentals of method of moments for metamaterials,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (Taylor & Francis, 2009), pp. 5–1–5–31.

Chen, T.

Cheng, E.

Cottingham, D.

Craeye, C.

C. Craeye and D. González-Ovejero, “A review on array mutual coupling analysis,” Radio Sci. 46, RS2012 (2011).
[CrossRef]

C. Craeye, X. Radu, F. Capolino, and A. Schuchinsky, “Fundamentals of method of moments for metamaterials,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (Taylor & Francis, 2009), pp. 5–1–5–31.

S. Withington, C. N. Thomas, and C. Craeye, “Determining the natural absorption and radiation modes of lossy periodic structures using energy absorption interferometry,” in Proceedings of the International Conference on Electromagnetics in Advanced Applications (IEEE, 2011), pp. 155–158.

Crawford, T.

Del Castillo, H.

Downes, T.

Drupp, R. P.

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. P. Drupp, and L. Li, “The design and fabrication of planar multiband metallodielectric frequency selective surfaces for infrared applications,” IEEE Trans. Antennas Propag. 54, 1265–1276 (2006).
[CrossRef]

Felsen, L. B.

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Finkbeiner, F.

Fixsen, D.

González-Ovejero, D.

C. Craeye and D. González-Ovejero, “A review on array mutual coupling analysis,” Radio Sci. 46, RS2012 (2011).
[CrossRef]

Harrington, J. L.

R. F. Harrington and J. L. Harrington, Field Computation by Moment Methods (Oxford University, 1996).

Harrington, R. F.

R. F. Harrington and J. L. Harrington, Field Computation by Moment Methods (Oxford University, 1996).

Hobson, M. P.

Holzapfel, W.

Jackson, D. R.

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[CrossRef]

Lange, A.

Li, L.

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. P. Drupp, and L. Li, “The design and fabrication of planar multiband metallodielectric frequency selective surfaces for infrared applications,” IEEE Trans. Antennas Propag. 54, 1265–1276 (2006).
[CrossRef]

Logan, D.

Mandel, L.

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

Mauskopf, P.

Mayer, T. S.

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. P. Drupp, and L. Li, “The design and fabrication of planar multiband metallodielectric frequency selective surfaces for infrared applications,” IEEE Trans. Antennas Propag. 54, 1265–1276 (2006).
[CrossRef]

Meyer, S.

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Munk, B.

B. Munk and G. Burrell, “Plane-wave expansion for arrays of arbitrarily oriented piecewise linear elements and its application in determining the impedance of a single linear antenna in a lossy half-space,” IEEE Trans. Antennas Propag. 27, 331–343 (1979).
[CrossRef]

Perera, T.

Radu, X.

C. Craeye, X. Radu, F. Capolino, and A. Schuchinsky, “Fundamentals of method of moments for metamaterials,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (Taylor & Francis, 2009), pp. 5–1–5–31.

Saklatvala, G.

S. Withington and G. Saklatvala, “Characterizing the behavior of partially coherent detectors through spatio-temporal modes,” J. Opt. A 9, 626–633 (2007).
[CrossRef]

G. Saklatvala, S. Withington, and M. P. Hobson, “Coupled-mode theory for infrared and submillimeter wave detectors,” J. Opt. Soc. Am. A 24, 764–775 (2007).
[CrossRef]

Schuchinsky, A.

C. Craeye, X. Radu, F. Capolino, and A. Schuchinsky, “Fundamentals of method of moments for metamaterials,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (Taylor & Francis, 2009), pp. 5–1–5–31.

Sharp, E.

Silverberg, R.

Smith, J. A.

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. P. Drupp, and L. Li, “The design and fabrication of planar multiband metallodielectric frequency selective surfaces for infrared applications,” IEEE Trans. Antennas Propag. 54, 1265–1276 (2006).
[CrossRef]

Tai, C.-T.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).

Tang, Y. U.

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. P. Drupp, and L. Li, “The design and fabrication of planar multiband metallodielectric frequency selective surfaces for infrared applications,” IEEE Trans. Antennas Propag. 54, 1265–1276 (2006).
[CrossRef]

Thomas, C. N.

S. Withington and C. N. Thomas, “Probing the dynamical behavior of surface dipoles through energy-absorption interferometry,” Phys. Rev. A 86, 043835 (2012).
[CrossRef]

C. N. Thomas and S. Withington, “Experimental demonstration of an interferometric technique for characterizing the full optical behavior of multi-mode power detectors,” IEEE Trans. Terahertz Sci. Technol. 2, 50–60 (2012).
[CrossRef]

C. N. Thomas and S. Withington, “Electromagnetic simulations of the partially coherent optical behavior of resistive film TES detectors,” in Proceedings of the 21st International Symposium on Space Terahertz Technology, Oxford, UK, March23–25, 2010.

S. Withington, C. N. Thomas, and C. Craeye, “Determining the natural absorption and radiation modes of lossy periodic structures using energy absorption interferometry,” in Proceedings of the International Conference on Electromagnetics in Advanced Applications (IEEE, 2011), pp. 155–158.

Werner, D. H.

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. P. Drupp, and L. Li, “The design and fabrication of planar multiband metallodielectric frequency selective surfaces for infrared applications,” IEEE Trans. Antennas Propag. 54, 1265–1276 (2006).
[CrossRef]

Wilson, G.

Wilton, D. R.

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[CrossRef]

Withington, S.

S. Withington and C. N. Thomas, “Probing the dynamical behavior of surface dipoles through energy-absorption interferometry,” Phys. Rev. A 86, 043835 (2012).
[CrossRef]

C. N. Thomas and S. Withington, “Experimental demonstration of an interferometric technique for characterizing the full optical behavior of multi-mode power detectors,” IEEE Trans. Terahertz Sci. Technol. 2, 50–60 (2012).
[CrossRef]

S. Withington and G. Saklatvala, “Characterizing the behavior of partially coherent detectors through spatio-temporal modes,” J. Opt. A 9, 626–633 (2007).
[CrossRef]

G. Saklatvala, S. Withington, and M. P. Hobson, “Coupled-mode theory for infrared and submillimeter wave detectors,” J. Opt. Soc. Am. A 24, 764–775 (2007).
[CrossRef]

S. Withington, C. N. Thomas, and C. Craeye, “Determining the natural absorption and radiation modes of lossy periodic structures using energy absorption interferometry,” in Proceedings of the International Conference on Electromagnetics in Advanced Applications (IEEE, 2011), pp. 155–158.

C. N. Thomas and S. Withington, “Electromagnetic simulations of the partially coherent optical behavior of resistive film TES detectors,” in Proceedings of the 21st International Symposium on Space Terahertz Technology, Oxford, UK, March23–25, 2010.

Wolf, E.

E. Wolf, “Coherent-mode propagation in spatially band-limited wave fields,” J. Opt. Soc. Am. A 3, 1920–1924 (1986).
[CrossRef]

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

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (3)

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[CrossRef]

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. P. Drupp, and L. Li, “The design and fabrication of planar multiband metallodielectric frequency selective surfaces for infrared applications,” IEEE Trans. Antennas Propag. 54, 1265–1276 (2006).
[CrossRef]

B. Munk and G. Burrell, “Plane-wave expansion for arrays of arbitrarily oriented piecewise linear elements and its application in determining the impedance of a single linear antenna in a lossy half-space,” IEEE Trans. Antennas Propag. 27, 331–343 (1979).
[CrossRef]

IEEE Trans. Terahertz Sci. Technol. (1)

C. N. Thomas and S. Withington, “Experimental demonstration of an interferometric technique for characterizing the full optical behavior of multi-mode power detectors,” IEEE Trans. Terahertz Sci. Technol. 2, 50–60 (2012).
[CrossRef]

J. Opt. A (1)

S. Withington and G. Saklatvala, “Characterizing the behavior of partially coherent detectors through spatio-temporal modes,” J. Opt. A 9, 626–633 (2007).
[CrossRef]

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

Phys. Rev. A (1)

S. Withington and C. N. Thomas, “Probing the dynamical behavior of surface dipoles through energy-absorption interferometry,” Phys. Rev. A 86, 043835 (2012).
[CrossRef]

Radio Sci. (1)

C. Craeye and D. González-Ovejero, “A review on array mutual coupling analysis,” Radio Sci. 46, RS2012 (2011).
[CrossRef]

Other (7)

R. F. Harrington and J. L. Harrington, Field Computation by Moment Methods (Oxford University, 1996).

C. Craeye, X. Radu, F. Capolino, and A. Schuchinsky, “Fundamentals of method of moments for metamaterials,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (Taylor & Francis, 2009), pp. 5–1–5–31.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

S. Withington, C. N. Thomas, and C. Craeye, “Determining the natural absorption and radiation modes of lossy periodic structures using energy absorption interferometry,” in Proceedings of the International Conference on Electromagnetics in Advanced Applications (IEEE, 2011), pp. 155–158.

C. N. Thomas and S. Withington, “Electromagnetic simulations of the partially coherent optical behavior of resistive film TES detectors,” in Proceedings of the 21st International Symposium on Space Terahertz Technology, Oxford, UK, March23–25, 2010.

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

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Figures (11)

Fig. 1.
Fig. 1.

Partially coherent field incident on arbitrary semi-infinite medium, leading to induced current density K(x,y).

Fig. 2.
Fig. 2.

Partially coherent field incident on semi-infinite medium with period b along y coordinate.

Fig. 3.
Fig. 3.

Structure of cross-spectral power density in sum-and-difference spectral coordinates: discrete spectrum characterized by Hv(k+) functions.

Fig. 4.
Fig. 4.

Basic operating principle of energy absorption interferometry.

Fig. 5.
Fig. 5.

EAI above a periodic surface.

Fig. 6.
Fig. 6.

General geometry of simulated structure.

Fig. 7.
Fig. 7.

Characteristic functions Hv for 2v2, for a height of h=0.25λ from the surface.

Fig. 8.
Fig. 8.

H˜v functions for a height of h=0.25λ from the surface.

Fig. 9.
Fig. 9.

Hv functions for a height of h=0.25λ from the surface.

Fig. 10.
Fig. 10.

Hv functions for a height of 0.5λ from the surface.

Fig. 11.
Fig. 11.

Decay of Hv(0) according to simulation (bullets) and according to model (solid), for h=λ/2.

Equations (53)

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

Ei(y)=12πE˜i(ky)exp(jkyy)dky.
K(y,z)=12πE˜i(ky)K(y,z|ky)dky,
P=120ρ(y,z)|K(y,z)|2dydz,
P=121(2π)2ky1ky2E˜i(ky1,ky2)P(ky1,ky2)dky1dky2,
E˜i(ky1,ky2)=E˜i(ky1)E˜i,(ky2)
P(ky1,ky2)=SρK(y,z|ky1)K,(y,z|ky2)dS,
E˜i(ky1,ky2)=nγnϕ˜n(ky1)ϕ˜n(ky2),
P=18π2nγnky1ky2ϕ˜n(ky1)P(ky1,ky2)ϕ˜n(ky2)dky1dky2,
ρ(y,z)K(y,z|ky1)K,(y,z|ky2)=12πej(ky1ky2)yv=Qv(z|ky1,ky2)ejvy2π/b
Qv(z|ky1,ky2)=2πb0bρ(y,z)K(y,z|ky1)K,(y,z|ky2)ej(ky1ky2)yejvy2π/bdy.
P(ky1,ky2)=v=Hv(k+)δ(kv2πb)
k+=(ky1+ky2)/2andk=ky1ky2
Hv(k+)=2πbSρK(y,z|k++vπb)K,(y,z|k+vπb)dS,
P=18π2nγnv=k+ϕ˜n(k++vπb)ϕ˜n(k+vπb)Hv(k+)dk+.
Pl=12ρ|Kl|2dS,
Pl(y1,y2)=12(α11+α22+2α12cos(ϕβ12))
Cij(y1,y2)=αijejβij,
=0ρKil(y,z)Kjl,(y,z)dydz,
=ρKilKjl,dS,
Ei(y)=Ikη2πejky(yys)ejkzh2kzdky
ky2+kz2=k2=ω2/c2,
Kil(y,z)=Ikη2πK(y,z|ky)ejkyys,iejkzh2kzdky.
C12(y1,y2)=k2η2|I|24(2π)2ky1ky2P(ky1,ky2)ej(ky1y1ky2y2)ej(kz1±kz2)hkz1kz2dky1dky2,
r=(y1+y2)/2,s=y1y2.
C12(y1,y2)=k2η2|I|24(2π)2k+kP(ky1,ky2)ej(k+s+kr)ej(kz1±kz2)hkz1kz2dk+dk,
P(ky1,ky2)=4kz1kz2k2η2|I|2ej(kz1±kz2)hrsC12(r,s)ej(k+s+kr)drds,
C12(y1,y2)=C12(r,s)=12πv=ej2πbvrHv(s).
Hv(s)=2πb0bC12(r,s)ej2πbvrdr.
rsC12(r,s)ej(k+s+kr)drds=12πvsHv(s)ejk+srejv2πbrejkrdrds,
P(ky1,ky2)=4kz1kz2k2η2|I|2ej(kz1±kz2)hvsHv(s)ejk+sdsδ(kv2πb).
P=12π21k2η2|I|2nγnvk+kz1kz2ej(kz1±kz2)hϕ˜n(ky1)ϕ˜n(ky2)sHv(s)ejk+sdsdk+,
ky1=k++vπ/bandky2=k+vπ/b.
P=M0nγnvkz1kz2ej(kz1±kz2)hϕ˜n(ky1)ϕ˜n(ky2)H˜v(k+)dk+,
Hv(k+)=ej(kz1±kz2)hH˜v(k+)4kz1kz2k2η2|I|2.
D=|ej(kz1±kz2)(hsh)|,
De|v|2πb(hhs),
DdB20πln10hshb|v|.
Kl(y+nb,z)=12π02πK(y,z|ψ)|ysejnψdψ,
P=12n=00bρ|Kl(y+nb,z)|2dydz,
=12(2π)2n=Sρ|02πK(y,z|ψ)ejnψdψ|2dS,
=14π02πSρ|K(y,z|ψ)|2dSdψ,
P=n=12Sρ|K1l(y+nb,z)+K2l(y+nb,z)ejϕ|2dS,
=14π02πSρ|K1(y+nb,z|ψ)+K2(y+nb,z|ψ)ejϕ|2dSdψ,
=12(α11+α22+2α12cos(ϕβ12))
Cij=αijejβij=12π02πSρKi(y,z|ψ)Kj,(y,z|ψ)dSdψ.
P(ky1,ky2)=H0(k+)δ(ky1ky2)
H0(k+)=2π0ρ|K(z|k+)|2dz.
P=18π2nγn|ϕ˜n(k+)|2H0(k+)dk+.
P(ky1,ky2)=4kz1kz2k2η2|I|2ej(kz1±kz2)hsC12(s)ejk+sdsδ(ky1ky2)
C12(s)=H0(s)/(2π).
H0(k+)=8πkz1kz2k2η2|I|2ej(kz1±kz2)hC˜12(k+)
C˜12(k+)=C12(s)ejk+sds.
n=|02πK(ψ)ejnψdψ|2=02π02πK(ψ1)K,(ψ2)(n=ejn(ψ1ψ2))dψ1dψ2,

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