Abstract

Under the oblique incidence condition, the multiple reflection of wave packets in a layered film structure will have a lateral shift increasing with the film thickness. In the analysis of the spatial interference with consideration of the lateral shift effect, a set of new analytic formulae to normalize the intensity of the s-and p-polarized wave packet was obtained to reduce the error of the ellipsometry parameters significantly as the optical path difference δ is close to mπ. The principle and method developed in this work also can be applied to other film structures in more general applications.

©2010 Optical Society of America

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References

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2010 (1)

2007 (2)

2006 (1)

2005 (3)

D. Guo, R. Lin, and W. Wang, “Gaussian-optics-based optical modeling and characterization of a Fabry–Perot microcavity for sensing applications,” J. Opt. Soc. Am. A 22(8), 1577–1589 (2005).
[Crossref]

L.-G. Wang and S.-Y. Zhu, “Large negative lateral shifts from the Kretschmann–Raether configuration with left-handed materials,” Appl. Phys. Lett. 87(22), 221102 (2005).
[Crossref]

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of superresolution in reconstruction of optical objects,” Phys. Rev. A 71(4), 043802 (2005).
[Crossref]

2004 (1)

S. Feng and O. Pfister, “Quantum interference of ultrastable twin optical beams,” Phys. Rev. Lett. 92(20), 203601 (2004).
[Crossref] [PubMed]

2003 (1)

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
[Crossref] [PubMed]

2000 (2)

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85(18), 3789–3792 (2000).
[Crossref] [PubMed]

1999 (3)

L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. 71(2), S274–S282 (1999).
[Crossref]

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71(5), 1539–1589 (1999).
[Crossref]

Y. A. Vlasov, S. Petit, G. Klein, B. Hönerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 1030–1035 (1999).
[Crossref]

1997 (1)

1996 (1)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

1995 (2)

1994 (1)

1992 (1)

1991 (1)

G. E. Jellison, “Examination of thin SiO2 films on Si using spectroscopic polarization modulation ellipsometry,” J. Appl. Phys. 69(11), 7627–7634 (1991).
[Crossref]

1990 (1)

1986 (1)

1971 (1)

1948 (1)

K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
[Crossref]

1947 (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
[Crossref]

1871 (1)

W. Sellmeier, “Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Annalen der Physik und Chemie 219(6), 272–282 (1871).
[Crossref]

Artmann, K.

K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
[Crossref]

Aydin, K.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
[Crossref] [PubMed]

Babnik, A.

Bertoni, H. L.

Beskrovnyy, V. N.

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of superresolution in reconstruction of optical objects,” Phys. Rev. A 71(4), 043802 (2005).
[Crossref]

Blanco, A.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Bretenaker, F.

Burger, S.

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Chen, L. Y.

Chen, Y. R.

Chomski, E.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Cotteverte, J. C.

Cotteverte, J.-C.

Cubukcu, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
[Crossref] [PubMed]

Dolling, G.

Fabre, C.

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85(18), 3789–3792 (2000).
[Crossref] [PubMed]

Falco, F.

Fan, S.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Feng, S.

S. Feng and O. Pfister, “Quantum interference of ultrastable twin optical beams,” Phys. Rev. Lett. 92(20), 203601 (2004).
[Crossref] [PubMed]

Feng, S. Z.

Feng, X. W.

Floch, A. L.

Foteinopolou, S.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
[Crossref] [PubMed]

Foteinopoulou, S.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
[Crossref] [PubMed]

Gonzalez, F.

Goos, F.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
[Crossref]

Grabtchak, S.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Gregorcic, P.

Guo, D.

Han, Y.

Hänchen, H.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
[Crossref]

Hirlimann, C.

Y. A. Vlasov, S. Petit, G. Klein, B. Hönerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 1030–1035 (1999).
[Crossref]

Hönerlage, B.

Y. A. Vlasov, S. Petit, G. Klein, B. Hönerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 1030–1035 (1999).
[Crossref]

Ibisate, M.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Jellison, G. E.

G. E. Jellison, “Examination of thin SiO2 films on Si using spectroscopic polarization modulation ellipsometry,” J. Appl. Phys. 69(11), 7627–7634 (1991).
[Crossref]

Joannopoulos, J. D.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

John, S.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Kettner, B.

Klein, G.

Y. A. Vlasov, S. Petit, G. Klein, B. Hönerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 1030–1035 (1999).
[Crossref]

Klein, M. W.

Kolobov, M. I.

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of superresolution in reconstruction of optical objects,” Phys. Rev. A 71(4), 043802 (2005).
[Crossref]

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85(18), 3789–3792 (2000).
[Crossref] [PubMed]

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71(5), 1539–1589 (1999).
[Crossref]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Lanternier, T.

Le Floch, A.

Lee, B. J.

Leonard, S. W.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Lin, R.

Linden, S.

Lopez, C.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Ma, H. Z.

Mandel, L.

L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. 71(2), S274–S282 (1999).
[Crossref]

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Meseguer, F.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Miguez, H.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Mondia, J. P.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Moreno, F.

Možina, J.

Nichelatti, E.

Ozbay, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
[Crossref] [PubMed]

Ozin, G. A.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Petit, S.

Y. A. Vlasov, S. Petit, G. Klein, B. Hönerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 1030–1035 (1999).
[Crossref]

Pfister, O.

S. Feng and O. Pfister, “Quantum interference of ultrastable twin optical beams,” Phys. Rev. Lett. 92(20), 203601 (2004).
[Crossref] [PubMed]

Poirson, J.

Qian, Y. H.

Salvetti, G.

Schädle, A.

Sellmeier, W.

W. Sellmeier, “Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Annalen der Physik und Chemie 219(6), 272–282 (1871).
[Crossref]

Sheng, M. Y.

Soukoulis, C. M.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
[Crossref] [PubMed]

Su, Y.

Tamir, T.

Toader, O.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

van Driel, H. M

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Villeneuve, P. R.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Vlasov, Y. A.

Y. A. Vlasov, S. Petit, G. Klein, B. Hönerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 1030–1035 (1999).
[Crossref]

Wang, L.-G.

L.-G. Wang and S.-Y. Zhu, “Large negative lateral shifts from the Kretschmann–Raether configuration with left-handed materials,” Appl. Phys. Lett. 87(22), 221102 (2005).
[Crossref]

Wang, W.

Wegener, M.

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Figures (4)

Fig. 1
Fig. 1 The Schematic diagram of multiple-reflected wave packets propagating in the film structure. The initial light I0 is incident at the angle θ0 onto the air/SiO2/Si film structure with the refraction angle θ1 at the film side, where d is the thickness of the film; n0, n1 and ñ2 are the refractive index of the medium of air, SiO2 film and Si substrate, respectively.
Fig. 2
Fig. 2 The incidence angle θmax, at which the maximum lateral shift occurs, decreases with refractive index increasing for the single-layered film with a given thickness. The inset shows the normalized lateral shift Δx/w changing with the incident angle under the condition in which the thickness of the film d = 400nm and the refractive index of the SiO2 film n1 = 1.478, the incident wavelength λ = 640nm with the assumption that the width of wave packet w = λ.
Fig. 3
Fig. 3 Normalized reflected intensity changing with optical path difference for s- and p-polarized wave packets with and without consideration of the lateral shift effect. In calculation, assuming that the incident angle θ0 = 70°, the refractive index of the SiO2 film n = 1.478 at the wavelength λ = 640nm and wave packets with the width w = λ and w = 3λ, respectively.
Fig. 4
Fig. 4 Comparisons of experiment (blue square dot) and simulated data by considering the interference with (red dash line) and without (green line) lateral shift effect. The optical path difference in the 0.95-1.1π (rad.) range is corresponding to the incidence angle changing from 50° to 75° at the incident wavelength λ = 640nm for sample A, in the 1.9-2.2π (rad.) and 2.8-3.2π (rad.) ranges is corresponding to the incident angle changing from 50° to 75° at the wavelengths λ = 440nm and 310nm for sample B, in the 3.8-4.4π (rad.) range is corresponding to the incident angle changing from 50° to 75° at the wavelength λ = 430nm for sample C, and in the 4-12π (rad.) range is corresponding to the incidence angle 50° (sample D) and 65° (sample E) and in the 275.5-413.3nm wavelength range. The width of the wave packet is kept constant with the best fitting value of w = 5λ for all samples in the δ range of 0.9-12π (rad).

Equations (51)

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E(x,y,z,t)=E0ex2y2w2ei(kzz2πνt)
Δx=dsin(2θ0)n12sin2θ0.
Es,p(x,y)=E0r01s,pex2y2w2+E0[1(r01s,p)2]r12s,peiδn=0[r01s,pr12s,peiδ]ne[x(n+1)Δx]2y2w2
Is,p=++Es,pEs,pdxdy++EEdxdy.
Is,p=(R1s,p)2+(R2s,p)2k=0(R3s,p)2k+2R1s,pR2s,pn=0(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)e(n+1)2Δx22w2+2(R2s,p)2k=0q=1k(R3s,p)2kcos(2qϕR3s,p)e(2q)2Δx22w2+2(R2s,p)2k=0q=0k(R3s,p)2k+1cos[(2q+1)ϕR3s,p]e(2q+1)2Δx22w2
R1s,peiϕ1s,p=r01s,p,R2s,peiϕ2s,p=[1(r01s,p)2]r12s,peiδ,R3s,peiϕ3s,p=r01s,pr12s,peiδ
Is,p=(R1s,p)2+(R2s,p)2+2R1s,pR2s,pcos(ϕR1s,pϕR2s,p)2R1s,pR2s,pR3s,pcos(ϕR1s,pϕR2s,p+ϕR3s,p)12R3s,pcos(ϕR3s,p)+(R3s,p)2
Is,p=(R1s,p)2+(R2s,p)21(R3s,p)2.
rs,p=r01s,p+r12s,peiδ1+r01s,pr12s,peiδ,rprs=tanΨeiΔ.
tanΨshift=IpIs,cosΔshift=IpsIsIp.
Ips=++Re[Es(Ep)]dxdy++EEdxdy.
M=(ΨexpΨcal)2+(ΔexpΔcal)2.
Ips=R1sR1pcos(ϕR1sϕR1p)+R1sR2pn=0(R3p)ncos(ϕR1sϕR2pnϕR3p)e(n+1)2Δx22w2+R1pR2sn=0(R3s)ncos(ϕR1pϕR2snϕR3s)e(n+1)2Δx22w2+R2sR2pk=0(R3sR3p)kcos[ϕR2sϕR2p+k(ϕR3sϕR3p)]+R2sR2pk=0q=1k(R3s)kq(R3p)k+qcos[(ϕR2sϕR2p)+(kq)ϕR3s(k+q)ϕR3p]e(2q)2Δx22w2+R2sR2pk=0q=1k(R3s)k+q(R3p)kqcos[(ϕR2pϕR2s)+(kq)ϕR3p(k+q)ϕR3s]e(2q)2Δx22w2+R2sR2pk=0q=0k(R3s)k+q+1(R3p)kqcos[(ϕR2pϕR2s)(k+q+1)ϕR3s+(kq)ϕR3p]e(2q+1)2Δx22w2+R2sR2pk=0q=0k(R3s)kq(R3p)k+q+1cos[(ϕR2sϕR2p)(k+q+1)ϕR3p+(kq)ϕR3s]e(2q+1)2Δx22w2
E(x,y,z,t)=E0ex2+y2w2ei(kzz2πνt)
Iin=++EEdxdy=I0w2π/2
I0=++E0E0dxdy
Es,p(x,y)=E0r01s,pex2y2w2+E0n=0[1(r01s,p)2]r12s,peiδ[r01s,pr12s,peiδ]ne[x(n+1)Δx]2y2w2
Is,p=1Iin++Es,p(Es,p)dxdy,Ips=1Iin++Re[Es(Ep)]dxdy
E1s,p=E0r01s,pex2y2w2E2s,p=E0n=0[1(r01s,p)2]r12s,peiδ[r01s,pr12s,peiδ]ne[x(n+1)Δx]2y2w2
Es,p=E1s,p+E2s,p
Es,pEs,p*=|E1s,p|2+E1s,p(E2s,p)+(E1s,p)*E2s,p+|E2s,p|2
Re[Es(Ep)]=12[Es(Ep)+(Es)Ep]
Re[Es(Ep)]=12[E1s(E1p)+(E1s)E1p+E2s(E1p)+(E2s)E1p)]+12[E1s(E2p)+(E1s)E2p+E2s(E2p)+(E2s)E2p)]
R1s,peiϕ1s,p=r01s,p,R2s,peiϕ2s,p=[1(r01s,p)2]r12s,peiδ,R3s,peiϕ3s,p=r01s,pr12s,peiδ
|E1s,p|2=(R1s,p)2e2(x2+y2)w2
E1s,pE2s,p*+E1s,p*E2s,p=ex2+y2w2n=02R1s,pR2s,p(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)e[x(n+1)Δx]2y2w2
NL=R2s,pexp(iϕR2s,p)[R3s,pexp(iϕR3s,p)]Le(x(L+1)Δx)2y2w2
|E2s,p|2=L=0NL×q=0Nq
L=0NL×q=0Nq=k=0NkNk+k=0q=1k(NkqNk+q+Nk+qNkq)+k=0q=0k(Nk+1+qNkq+NkqNk+q+1)
|E2s,p|2=k=0(R2s,p)2(R3s,p)2ke2[x(k+1)Δx]2+2y2w2+2(R2s,p)2k=0q=1k(R3s,p)2kcos(2qϕR3s,p)e[x(kq+1)Δx]2+y2w2e[x(k+q+1)Δx]2+y2w2+2(R2s,p)2k=0p=0k(R3s,p)2k+1cos[(2q+1)ϕR3s,p]e[x(k+q+2)Δx]2+y2w2e[x(kq+1)Δx]2+y2w2]
E1sE1p+E1sE1p=2R1sR1pe2(x2+y2)w2cos(ϕR1sϕR1p)
E2sE1p+E2sE1p=2R1pR2sex2+y2w2n=0(R3s)ncos[ϕR1p+ϕR2s+nϕR3s]e[x(n+1)Δx]2+y2w2
E1sE2p+E1sE2p=2R1sR2pex2+y2w2n=0(R3p)ncos(ϕR1sϕR2pnϕR3p)e[x(n+1)Δx]2+y2w2
ML=R2sexp(iϕR2s)[R3sexp(iϕR3s)]Le[x(L+1)Δx]2+y2w2Nq=R2pexp(iϕR2p)[R3pexp(iϕR3p)]qe[x(q+1)Δx]2+y2w2
E2sE2p+E2sE2p=L=0ML×q=0Nq*+L=0ML×q=0Nq
L=0ML×q=0Nq*=k=0[MkNk*]+k=0q=1k(MkqNk+q+Mk+qNkq)+k=0q=0k(Mk+q+1Nkq+MkqNk+q+1)L=0ML×q=0Nq=k=0[NkMk*]+k=0q=0k(NkqMk+q+Nk+qMkq)+k=0q=0k(Nk+1+qMkq+NkqMk+q+1)
E2sE2p+E2sE2p=k=02R2sR2p(R3sR3p)kcos[ϕR2sϕR2p+k(ϕR3sϕR3p)]e2[x(k+1)Δx]2+2y2w2+2R2sR2pk=0q=1k{(R3s)kq(R3p)k+qcos[(ϕR2sϕR2p)+(kq)ϕR3s(k+q)ϕR3p]×e[x(kq+1)Δx]2+y2w2e[x(k+q+1)Δx]2+y2w2}+2R2sR2pk=0q=1k{(R3s)k+q(R3p)kqcos[(ϕR2pϕR2s)+(kq)ϕR3p(k+q)ϕR3s]×e[x(kq+1)Δx]2+y2w2e[x(k+q+1)Δx]2+y2w2+2R2sR2pk=0q=0k(R3s)k+q+1(R3p)kqcos[(ϕR2pϕR2s)(k+q+1)ϕR3s+(kq)ϕR3p]×e(x(kq+1)Δx)2+y2w2e(x(k+q+2)Δx)2+y2w2+2R2sR2pk=0q=0k(R3p)k+p+1(R3s)kpcos[(ϕR2sϕR2p)(k+q+1)ϕR3p+(kq)ϕR3s]×e(x(kq+1)Δx)2+y2w2e(x(k+q+2)Δx)2+y2w2
Is,p=1Iin[++|E1s,p|2dxdy+++(E1s,pE2s,p+E1s,pE2s,p)dxdy+++|E2s,p|2dxdy]Ireps=1Iin[++(E1sE1p+E1sE1p)dxdy+++(E2sE1p+E2sE1p)dxdy++++(E1sE2p+E1sE2p)dxdy+++(E2sE2p+E2sE2p)dxdy]
exp(2x2w2)dx=wπ2exp(ax2)cos(2bx)dx=πaexp(b2a)exp(ax2)sin(2bx)dx=0exp(2x2+y2w2)dxdy=w2π2
Is,p=(R1s,p)2+(R2s,p)2k=0(R3s,p)2k+2R1s,pR2s,pn=0(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)e(n+1)2Δx22w2+2(R2s,p)2k=0q=1k(R3s,p)2kcos(2qϕR3s,p)e(2q)2Δx22w2+2(R2s,p)2k=0q=0k(R3s,p)2k+1cos[(2q+1)ϕR3s,p]e(2q+1)2Δx22w2
Ips=R1sR1pcos(ϕR1sϕR1p)+R1sR2pn=0(R3p)ncos(ϕR1sϕR2pnϕR3p)e(n+1)2Δx22w2+R1pR2sn=0(R3s)ncos(ϕR1pϕR2snϕR3s)e(n+1)2Δx22w2+R2sR2pk=0(R3sR3p)kcos[ϕR2sϕR2p+k(ϕR3sϕR3p)]+R2sR2pk=0q=1k(R3s)kq(R3p)k+qcos[(ϕR2sϕR2p)+(kq)ϕR3s(k+q)ϕR3p]e(2q)2Δx22w2+R2sR2pk=0q=1k(R3s)k+q(R3p)kqcos[(ϕR2pϕR2s)+(kq)ϕR3p(k+q)ϕR3s]e(2q)2Δx22w2+R2sR2pk=0q=1k(R3s)k+q+1(R3p)kqcos[(ϕR2pϕR2s)(k+q+1)ϕR3s+(kq)ϕR3p]e(2q+1)2Δx22w2+R2sR2pk=0q=1k(R3s)kq(R3p)k+q+1cos[(ϕR2sϕR2p)(k+q+1)ϕR3p+(kq)ϕR3s]e(2q+1)2Δx22w2
Is,p=(R1s,p)2+(R2s,p)2k=0(R3s,p)2k.
k=0(R3s,p)2k=11(R3s,p)2,
Is,p=(R1s,p)2+(R2s,p)21(R3s,p)2.
Is,p=(R1s,p)2+2R1s,pR2s,pn=0(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)+(R2s,p)2k=0[(R3s,p)2k+2p=1k(R3s,p)2kcos(2pϕR3s,p)+2p=0k(R3s,p)2k+1cos[(2p+1)ϕR3s,p]
n=0(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)=12n=0(R3s,p)n[exp[i(ϕR1s,pϕR2s,pnϕR3s,p)]+12n=0(R3s,p)n[exp[i(ϕR1s,pϕR2s,pnϕR3s,p)]
n=0(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)]=cos(ϕR1s,pϕR2s,p)R3s,pcos(ϕR1s,pϕR2s,p+ϕR3s,p)12R3s,pcos(ϕR3s,p)+(R3s,p)2
n=0[R3s,pexp(iϕR3s,p)]n×m=0[R3s,pexp(iϕR3s,p)]m=k=0[(R3s,p)2k+2p=1k(R3s,p)2kcos(2pϕR3s,p)]+k=0[2p=0k(R3s,p)2k+1cos[(2p+1)ϕR3s,p]
k=0[(R3s,p)2k+2p=1k(R3s,p)2kcos(2pϕR3s,p)+2p=0k(R3s,p)2k+1cos[(2p+1)ϕR3s,p]=112R3s,pcos(ϕR3s,p)+(R3s,p)2
Is,p=(R1s,p)2+(R2s,p)2+2R1s,pR2s,pcos(ϕR1s,pϕR2s,p)2R1s,pR2s,pR3s,pcos(ϕR1s,pϕR2s,p+ϕR3s,p)12R3s,pcos(ϕR3s,p)+(R3s,p)2
rs,p=R1s,pexp(iϕR1s,p)R1s,pR3s,pexp[i(ϕR1s,p+ϕR3s,p)]+R2s,pexp(iϕR2s,p)1R3s,pexp(iϕR3s,p).

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