Abstract

The interference of light has been analyzed for a film structure by considering that a spatial separation exists for the two neighboring light beams to be interfered in the space. There is a significant difference between the situations of the interference with or without consideration of the spatial effect, especially around the region where the phase delay δ=π and 2π by taking the example of the one-layered SiO2/Si structure. It is reasonable to extract the optical parameters by neglecting the spatial effect only for the thinner film with a thickness much smaller than a wavelength, which satisfies the condition that δ<π; otherwise, the film equation used for the periodic or nonperiodic structures will be modified by including the spatial effect in the data analysis and applications.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2005

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

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

2004

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

2003

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

2000

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

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

1999

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

Y. A. Vlasov, S. Petit, G. Klein, B. Honerlage, C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E 60, 1030–1035 (1999).
[CrossRef]

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

1996

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

1995

1994

1992

1991

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

Aydin, K.

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

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1985).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1985).

Beskrovnyy, V. N.

V. N. Beskrovnyy, M. I. Kolobov, “Quantum limits of superresolution in reconstruction of optical objects,” Phys. Rev. A 71, 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. Mondia, G. A. Ozin, O. Toader, H. M. Van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometers,” Nature 405, 437–440 (2000).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chaps. 1 and 7.

Chen, J. C.

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

Chen, L. Y.

Chomski, E.

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

Cotteverte, J. C.

Cubukcu, E.

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

Fabre, C.

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

Fan, S.

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

Feng, S.

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

Feng, X. W.

Foteinopoulou, S.

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

Gonzalez, F.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005), Chap. 3.

Grabtchak, S.

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

Guo, D.

Han, Y.

Hirlimann, C.

Y. A. Vlasov, S. Petit, G. Klein, B. Honerlage, C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E 60, 1030–1035 (1999).
[CrossRef]

Honerlage, B.

Y. A. Vlasov, S. Petit, G. Klein, B. Honerlage, C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E 60, 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. Mondia, G. A. Ozin, O. Toader, H. M. Van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometers,” Nature 405, 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, 7627–7634 (1991).
[CrossRef]

Joannopoulos, J. D.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 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. Mondia, G. A. Ozin, O. Toader, H. M. Van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometers,” Nature 405, 437–440 (2000).
[CrossRef] [PubMed]

Klein, G.

Y. A. Vlasov, S. Petit, G. Klein, B. Honerlage, C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E 60, 1030–1035 (1999).
[CrossRef]

Klein, M. V.

M. V. Klein, Optics (Wiley, 1970), Chap. 5.

Kolobov, M. I.

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

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

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

Kurland, I.

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

Lanternier, T.

Leonard, S. W.

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

Levin, F. S.

F. S. Levin, An Introduction to Quantum Theory (Cambridge U. Press, 2002), Chap. 7.

Lin, R.

Lopez, C.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, H. M. Van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometers,” Nature 405, 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, S274–S282 (1999).
[CrossRef]

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 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. Mondia, G. A. Ozin, O. Toader, H. M. Van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometers,” Nature 405, 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. Mondia, G. A. Ozin, O. Toader, H. M. Van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometers,” Nature 405, 437–440 (2000).
[CrossRef] [PubMed]

Mondia, J.

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

Moreno, F.

Nichelatti, E.

Ozbay, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91, 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. Mondia, G. A. Ozin, O. Toader, H. M. Van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometers,” Nature 405, 437–440 (2000).
[CrossRef] [PubMed]

Petit, S.

Y. A. Vlasov, S. Petit, G. Klein, B. Honerlage, C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E 60, 1030–1035 (1999).
[CrossRef]

Pfister, O.

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

Poirson, J.

Qian, Y. H.

Salvetti, G.

Soukoulis, C. M.

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

Su, Y.

Toader, O.

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

Villeneuve, P. R.

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

Vlasov, Y. A.

Y. A. Vlasov, S. Petit, G. Klein, B. Honerlage, C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E 60, 1030–1035 (1999).
[CrossRef]

Wang, W.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chaps. 1 and 7.

Appl. Opt.

J. Appl. Phys.

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

J. Opt. Soc. Am. A

Nature

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

Phys. Rev. A

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

Phys. Rev. E

Y. A. Vlasov, S. Petit, G. Klein, B. Honerlage, C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E 60, 1030–1035 (1999).
[CrossRef]

Phys. Rev. Lett.

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

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

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

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

Rev. Mod. Phys.

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

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

Other

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chaps. 1 and 7.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005), Chap. 3.

F. S. Levin, An Introduction to Quantum Theory (Cambridge U. Press, 2002), Chap. 7.

M. V. Klein, Optics (Wiley, 1970), Chap. 5.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1985).

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Figures (4)

Fig. 1
Fig. 1

(Color online) Two photons propagate along the z axis and have the same amplitude of the electrical field ( | E 1 | = | E 2 | ) with the same phase (1A) or a phase difference of 180° (1B) in assumption. They arrive at the position where y 1 = y 2 , z 1 = z 2 , but x 1 x 2 . With respect to the interference effect, E = E 1 + E 2 , the two photons are: (a) fully overlapped in the space to have a full interference; (b) partly overlapped; (c) totally separated and fully deinterfered in the space.

Fig. 2
Fig. 2

(Color online) Schematic showing multiple reflections of the light propagating in the film structure. The initial light I 0 is incident at the angle θ 0 onto the air∕ S i O 2 / S i film structure with the refraction angle θ 1 at the film side, where n 0 , n 1 , and n ˜ 2 are the refractive index of the medium of air, SiO 2 film, and Si substrate, respectively. The multiple reflections of the light I 1 , I 2 , I 3 , , I N with different phases and amplitudes will result as the light travels through the SiO 2 film with the thickness d.

Fig. 3
Fig. 3

(Color online) Normalized spatial separation ( a / d ) varies with the refractive index of the film and the incidence angle. The separation maxima will occur at the condition of cos 2 θ 0 = 2 n 1 [ ( n 1 2 1 ) 1 / 2 n 1 ] + 1 . The inset shows the interference factor of ξ as the function of aw.

Fig. 4
Fig. 4

(Color online) Standard deviation M analyzed with (solid curve) or without (square-dotted curve) consideration of the spatial effect. The phase delay is in the 0.75 2.41 π (rad) range, which corresponds to the wavelength λ = 800 , 720, 640, 570, 510, and 460   nm for sample A in the range of 1–6 and corresponds to the wavelength λ = 490 , 440, 390, and 350   nm for sample B in the range of 7–10.

Equations (14)

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

E ( x , y , z ; t ) = E 0 exp [ i ( k r 2 π ν t ) ] .
E ( x , y , z ; t ) = E 0 exp [ ( x 2 + y 2 ) / ( 2 w 2 ) ] × exp [ i ( k z z 2 π ν t ) ] .
E r = m = 1 N E m .
E r = E r 1 + E r 2 = ( r 1 + r 2 e i δ ) E 0 ,
R = E r 2 / E 0 2 = r 1 2 + r 2 2 + 2 r 1 r 2 cos δ .
E r ( x ) = E r 1 ( x ) + E r 2 ( x ) = ( r 1 e x 2 + r 2 e ( x a ) 2 e i δ ) E 0 ,
R = E r ( x ) 2 d x / E 0 ( x ) 2 d x = r 1 2 + r 2 2 + 2 r 1 r 2 ξ cos δ ,
r p , s = r 01 p , s + [ 1 ( r 01 p , s ) 2 ] r 12 p , s e i δ n = 2 ( r 01 p , s r 12 p , s e i δ ) n 2 ,
r s p , s = r 01 p , s g 1 + [ 1 ( r 01 p , s ) 2 ] r 12 p , s e i δ n = 2 ( r 01 p , s r 12 p , s e i δ ) n 2 g n ,
g n ( x ) = exp { [ x ( n 1 ) a ] 2 2 w 2 } .
I s p , s = I 0 a [ r s p , s ( x ) ] [ r s p , s ( x ) ] * d x ,
I s p s = I 0 a Re { [ r s p ( x ) ] [ r s s ( x ) ] * } d x ,
tan Ψ s = ( I s p I s s ) 1 / 2 , cos Δ s = I s p s ( I s s I s p ) 1 / 2 .
M = ( Ψ exp Ψ cal ) 2 + ( Δ exp Δ cal ) 2 ,

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