Abstract

We study the transient pulse propagation in one-dimensional photonic crystal. Two new effects are identified: double reflection and slow-light ringing of transmitted and reflected pulses. We analyze these effects in superconductor-dielectric photonic crystal around the polariton resonance region of the dielectric material. Distinct double-polariton dispersions and narrow peaks in the transmission and reflection spectra are found when the plasma gap and the polariton gap of the dielectric overlap. Potential applications of these effect are discussed.

© 2010 Optical Society of America

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References

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  1. C. M. Soukoulis, Photonic Band Gap Materials (Kluwer, 1996).
  2. C. Kittel, Introduction to Solid State Physics, 6th ed. (Wiley, 1986).
  3. H. Takeda and K. Yoshino, “Properties of Abrikosov lattices as photonic crystals,” Phys. Rev. B 70, 085109 (2004).
    [CrossRef]
  4. C. H. Raymond Ooi and T. C. Au Yeung “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Lett. A 259, 413-419 (1999).
    [CrossRef]
  5. C. H. Raymond Ooi, T. C. Au Yeung, C. H. Kam, and T. K. Lim, “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Rev. B 61, 5920-5923 (2000).
    [CrossRef]
  6. L. Feng, X. P. Liu, Y. F. Tang, Y. F. Chen, J. Zi, S. N. Zhu, and Y. Y. Zhu, “Tunable negative refractions in two-dimensional photonic crystals with superconductor constituents,” J. Appl. Phys. 97, 073104 (2005).
    [CrossRef]
  7. A. Pimenov, A. Loidl, P. Przyslupski, and B. Dabrowski “Negative refraction in ferromagnet-superconductor superlattices,” Phys. Rev. Lett. 95, 247009 (2005).
    [CrossRef] [PubMed]
  8. C. H. Raymond Ooi, T. C. Au Yeung, T. K. Lim, and C. H. Kam “Two-dimensional superconductor-dielectric photonic crystal,” in Photonics Technology into the 21st Century: Semiconductors, Microstructures, and Nanostructures, Proc. SPIE 3899, 278-287 (1999).
  9. O. L. Berman, Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, “Superconducting photonic crystals: Numerical calculations of the band structure,” Phys. Rev. B 74, 092505 (2006).
    [CrossRef]
  10. M. G. Cottam and D. R. Tilley, Introduction to Surface and Superlattice Excitations (University Press, Cambridge, 1989).
    [CrossRef]
  11. T. van Duzer and C. W. Turner, Principles of Superconductive Devices and Circuits (London, 1981).
  12. The problem of negative ns/n0=1−(T/Tc)ζ, at T>Tc, can be overcome by using expressions obtained from self-consistent calculations, as was done in A. Jordan, C. H. Raymond Ooi, and A. Svidzinsky “Fluctuation statistics of mesoscopic Bose-Einstein condensates: Reconciling the master equation with the partition function to reexamine the Uhlenbeck-Einstein dilemma,” Phys. Rev. A 74, 032506 (2006).
    [CrossRef]
  13. H. E. Portceanu, K. Karray, R. Seifert, F. Koch, P. Berberich, and H. Kinder, “Microwave Fabry-Pérot transmission through YBaCuO superconducting thin films,” Phys. Rev. Lett. 75, 3934-3937 (1995).
    [CrossRef]
  14. C. G. Ribbing, H. Högström, and A. Rung “Studies of polaritonic gaps in photonic crystals,” Appl. Opt. 45, 1575-1582 (2006).
    [CrossRef] [PubMed]
  15. M. Born and E. Wolf, Principles of Optics7th ed. (Pergamon Press, 1999).
  16. H. Takeda and K. Yoshino, “Tunable photonic band schemes in two-dimensional photonic crystals composed of copper oxide high-temperature superconductors,” Phys. Rev. B 67, 245109 (2003).
    [CrossRef]

2006

O. L. Berman, Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, “Superconducting photonic crystals: Numerical calculations of the band structure,” Phys. Rev. B 74, 092505 (2006).
[CrossRef]

The problem of negative ns/n0=1−(T/Tc)ζ, at T>Tc, can be overcome by using expressions obtained from self-consistent calculations, as was done in A. Jordan, C. H. Raymond Ooi, and A. Svidzinsky “Fluctuation statistics of mesoscopic Bose-Einstein condensates: Reconciling the master equation with the partition function to reexamine the Uhlenbeck-Einstein dilemma,” Phys. Rev. A 74, 032506 (2006).
[CrossRef]

C. G. Ribbing, H. Högström, and A. Rung “Studies of polaritonic gaps in photonic crystals,” Appl. Opt. 45, 1575-1582 (2006).
[CrossRef] [PubMed]

2005

L. Feng, X. P. Liu, Y. F. Tang, Y. F. Chen, J. Zi, S. N. Zhu, and Y. Y. Zhu, “Tunable negative refractions in two-dimensional photonic crystals with superconductor constituents,” J. Appl. Phys. 97, 073104 (2005).
[CrossRef]

A. Pimenov, A. Loidl, P. Przyslupski, and B. Dabrowski “Negative refraction in ferromagnet-superconductor superlattices,” Phys. Rev. Lett. 95, 247009 (2005).
[CrossRef] [PubMed]

2004

H. Takeda and K. Yoshino, “Properties of Abrikosov lattices as photonic crystals,” Phys. Rev. B 70, 085109 (2004).
[CrossRef]

2003

H. Takeda and K. Yoshino, “Tunable photonic band schemes in two-dimensional photonic crystals composed of copper oxide high-temperature superconductors,” Phys. Rev. B 67, 245109 (2003).
[CrossRef]

2000

C. H. Raymond Ooi, T. C. Au Yeung, C. H. Kam, and T. K. Lim, “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Rev. B 61, 5920-5923 (2000).
[CrossRef]

1999

C. H. Raymond Ooi and T. C. Au Yeung “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Lett. A 259, 413-419 (1999).
[CrossRef]

1995

H. E. Portceanu, K. Karray, R. Seifert, F. Koch, P. Berberich, and H. Kinder, “Microwave Fabry-Pérot transmission through YBaCuO superconducting thin films,” Phys. Rev. Lett. 75, 3934-3937 (1995).
[CrossRef]

Au Yeung, T. C.

C. H. Raymond Ooi, T. C. Au Yeung, C. H. Kam, and T. K. Lim, “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Rev. B 61, 5920-5923 (2000).
[CrossRef]

C. H. Raymond Ooi and T. C. Au Yeung “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Lett. A 259, 413-419 (1999).
[CrossRef]

C. H. Raymond Ooi, T. C. Au Yeung, T. K. Lim, and C. H. Kam “Two-dimensional superconductor-dielectric photonic crystal,” in Photonics Technology into the 21st Century: Semiconductors, Microstructures, and Nanostructures, Proc. SPIE 3899, 278-287 (1999).

Berberich, P.

H. E. Portceanu, K. Karray, R. Seifert, F. Koch, P. Berberich, and H. Kinder, “Microwave Fabry-Pérot transmission through YBaCuO superconducting thin films,” Phys. Rev. Lett. 75, 3934-3937 (1995).
[CrossRef]

Berman, O. L.

O. L. Berman, Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, “Superconducting photonic crystals: Numerical calculations of the band structure,” Phys. Rev. B 74, 092505 (2006).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics7th ed. (Pergamon Press, 1999).

Chen, Y. F.

L. Feng, X. P. Liu, Y. F. Tang, Y. F. Chen, J. Zi, S. N. Zhu, and Y. Y. Zhu, “Tunable negative refractions in two-dimensional photonic crystals with superconductor constituents,” J. Appl. Phys. 97, 073104 (2005).
[CrossRef]

Coalson, R. D.

O. L. Berman, Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, “Superconducting photonic crystals: Numerical calculations of the band structure,” Phys. Rev. B 74, 092505 (2006).
[CrossRef]

Cottam, M. G.

M. G. Cottam and D. R. Tilley, Introduction to Surface and Superlattice Excitations (University Press, Cambridge, 1989).
[CrossRef]

Dabrowski, B.

A. Pimenov, A. Loidl, P. Przyslupski, and B. Dabrowski “Negative refraction in ferromagnet-superconductor superlattices,” Phys. Rev. Lett. 95, 247009 (2005).
[CrossRef] [PubMed]

Eiderman, S. L.

O. L. Berman, Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, “Superconducting photonic crystals: Numerical calculations of the band structure,” Phys. Rev. B 74, 092505 (2006).
[CrossRef]

Feng, L.

L. Feng, X. P. Liu, Y. F. Tang, Y. F. Chen, J. Zi, S. N. Zhu, and Y. Y. Zhu, “Tunable negative refractions in two-dimensional photonic crystals with superconductor constituents,” J. Appl. Phys. 97, 073104 (2005).
[CrossRef]

Högström, H.

Jordan, A.

The problem of negative ns/n0=1−(T/Tc)ζ, at T>Tc, can be overcome by using expressions obtained from self-consistent calculations, as was done in A. Jordan, C. H. Raymond Ooi, and A. Svidzinsky “Fluctuation statistics of mesoscopic Bose-Einstein condensates: Reconciling the master equation with the partition function to reexamine the Uhlenbeck-Einstein dilemma,” Phys. Rev. A 74, 032506 (2006).
[CrossRef]

Kam, C. H.

C. H. Raymond Ooi, T. C. Au Yeung, C. H. Kam, and T. K. Lim, “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Rev. B 61, 5920-5923 (2000).
[CrossRef]

C. H. Raymond Ooi, T. C. Au Yeung, T. K. Lim, and C. H. Kam “Two-dimensional superconductor-dielectric photonic crystal,” in Photonics Technology into the 21st Century: Semiconductors, Microstructures, and Nanostructures, Proc. SPIE 3899, 278-287 (1999).

Karray, K.

H. E. Portceanu, K. Karray, R. Seifert, F. Koch, P. Berberich, and H. Kinder, “Microwave Fabry-Pérot transmission through YBaCuO superconducting thin films,” Phys. Rev. Lett. 75, 3934-3937 (1995).
[CrossRef]

Kinder, H.

H. E. Portceanu, K. Karray, R. Seifert, F. Koch, P. Berberich, and H. Kinder, “Microwave Fabry-Pérot transmission through YBaCuO superconducting thin films,” Phys. Rev. Lett. 75, 3934-3937 (1995).
[CrossRef]

Kittel, C.

C. Kittel, Introduction to Solid State Physics, 6th ed. (Wiley, 1986).

Koch, F.

H. E. Portceanu, K. Karray, R. Seifert, F. Koch, P. Berberich, and H. Kinder, “Microwave Fabry-Pérot transmission through YBaCuO superconducting thin films,” Phys. Rev. Lett. 75, 3934-3937 (1995).
[CrossRef]

Lim, T. K.

C. H. Raymond Ooi, T. C. Au Yeung, C. H. Kam, and T. K. Lim, “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Rev. B 61, 5920-5923 (2000).
[CrossRef]

C. H. Raymond Ooi, T. C. Au Yeung, T. K. Lim, and C. H. Kam “Two-dimensional superconductor-dielectric photonic crystal,” in Photonics Technology into the 21st Century: Semiconductors, Microstructures, and Nanostructures, Proc. SPIE 3899, 278-287 (1999).

Liu, X. P.

L. Feng, X. P. Liu, Y. F. Tang, Y. F. Chen, J. Zi, S. N. Zhu, and Y. Y. Zhu, “Tunable negative refractions in two-dimensional photonic crystals with superconductor constituents,” J. Appl. Phys. 97, 073104 (2005).
[CrossRef]

Loidl, A.

A. Pimenov, A. Loidl, P. Przyslupski, and B. Dabrowski “Negative refraction in ferromagnet-superconductor superlattices,” Phys. Rev. Lett. 95, 247009 (2005).
[CrossRef] [PubMed]

Lozovik, Y. E.

O. L. Berman, Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, “Superconducting photonic crystals: Numerical calculations of the band structure,” Phys. Rev. B 74, 092505 (2006).
[CrossRef]

Pimenov, A.

A. Pimenov, A. Loidl, P. Przyslupski, and B. Dabrowski “Negative refraction in ferromagnet-superconductor superlattices,” Phys. Rev. Lett. 95, 247009 (2005).
[CrossRef] [PubMed]

Portceanu, H. E.

H. E. Portceanu, K. Karray, R. Seifert, F. Koch, P. Berberich, and H. Kinder, “Microwave Fabry-Pérot transmission through YBaCuO superconducting thin films,” Phys. Rev. Lett. 75, 3934-3937 (1995).
[CrossRef]

Przyslupski, P.

A. Pimenov, A. Loidl, P. Przyslupski, and B. Dabrowski “Negative refraction in ferromagnet-superconductor superlattices,” Phys. Rev. Lett. 95, 247009 (2005).
[CrossRef] [PubMed]

Raymond Ooi, C. H.

The problem of negative ns/n0=1−(T/Tc)ζ, at T>Tc, can be overcome by using expressions obtained from self-consistent calculations, as was done in A. Jordan, C. H. Raymond Ooi, and A. Svidzinsky “Fluctuation statistics of mesoscopic Bose-Einstein condensates: Reconciling the master equation with the partition function to reexamine the Uhlenbeck-Einstein dilemma,” Phys. Rev. A 74, 032506 (2006).
[CrossRef]

C. H. Raymond Ooi, T. C. Au Yeung, C. H. Kam, and T. K. Lim, “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Rev. B 61, 5920-5923 (2000).
[CrossRef]

C. H. Raymond Ooi and T. C. Au Yeung “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Lett. A 259, 413-419 (1999).
[CrossRef]

C. H. Raymond Ooi, T. C. Au Yeung, T. K. Lim, and C. H. Kam “Two-dimensional superconductor-dielectric photonic crystal,” in Photonics Technology into the 21st Century: Semiconductors, Microstructures, and Nanostructures, Proc. SPIE 3899, 278-287 (1999).

Ribbing, C. G.

Rung, A.

Seifert, R.

H. E. Portceanu, K. Karray, R. Seifert, F. Koch, P. Berberich, and H. Kinder, “Microwave Fabry-Pérot transmission through YBaCuO superconducting thin films,” Phys. Rev. Lett. 75, 3934-3937 (1995).
[CrossRef]

Soukoulis, C. M.

C. M. Soukoulis, Photonic Band Gap Materials (Kluwer, 1996).

Svidzinsky, A.

The problem of negative ns/n0=1−(T/Tc)ζ, at T>Tc, can be overcome by using expressions obtained from self-consistent calculations, as was done in A. Jordan, C. H. Raymond Ooi, and A. Svidzinsky “Fluctuation statistics of mesoscopic Bose-Einstein condensates: Reconciling the master equation with the partition function to reexamine the Uhlenbeck-Einstein dilemma,” Phys. Rev. A 74, 032506 (2006).
[CrossRef]

Takeda, H.

H. Takeda and K. Yoshino, “Properties of Abrikosov lattices as photonic crystals,” Phys. Rev. B 70, 085109 (2004).
[CrossRef]

H. Takeda and K. Yoshino, “Tunable photonic band schemes in two-dimensional photonic crystals composed of copper oxide high-temperature superconductors,” Phys. Rev. B 67, 245109 (2003).
[CrossRef]

Tang, Y. F.

L. Feng, X. P. Liu, Y. F. Tang, Y. F. Chen, J. Zi, S. N. Zhu, and Y. Y. Zhu, “Tunable negative refractions in two-dimensional photonic crystals with superconductor constituents,” J. Appl. Phys. 97, 073104 (2005).
[CrossRef]

Tilley, D. R.

M. G. Cottam and D. R. Tilley, Introduction to Surface and Superlattice Excitations (University Press, Cambridge, 1989).
[CrossRef]

Turner, C. W.

T. van Duzer and C. W. Turner, Principles of Superconductive Devices and Circuits (London, 1981).

van Duzer, T.

T. van Duzer and C. W. Turner, Principles of Superconductive Devices and Circuits (London, 1981).

Wolf, E.

M. Born and E. Wolf, Principles of Optics7th ed. (Pergamon Press, 1999).

Yoshino, K.

H. Takeda and K. Yoshino, “Properties of Abrikosov lattices as photonic crystals,” Phys. Rev. B 70, 085109 (2004).
[CrossRef]

H. Takeda and K. Yoshino, “Tunable photonic band schemes in two-dimensional photonic crystals composed of copper oxide high-temperature superconductors,” Phys. Rev. B 67, 245109 (2003).
[CrossRef]

Zhu, S. N.

L. Feng, X. P. Liu, Y. F. Tang, Y. F. Chen, J. Zi, S. N. Zhu, and Y. Y. Zhu, “Tunable negative refractions in two-dimensional photonic crystals with superconductor constituents,” J. Appl. Phys. 97, 073104 (2005).
[CrossRef]

Zhu, Y. Y.

L. Feng, X. P. Liu, Y. F. Tang, Y. F. Chen, J. Zi, S. N. Zhu, and Y. Y. Zhu, “Tunable negative refractions in two-dimensional photonic crystals with superconductor constituents,” J. Appl. Phys. 97, 073104 (2005).
[CrossRef]

Zi, J.

L. Feng, X. P. Liu, Y. F. Tang, Y. F. Chen, J. Zi, S. N. Zhu, and Y. Y. Zhu, “Tunable negative refractions in two-dimensional photonic crystals with superconductor constituents,” J. Appl. Phys. 97, 073104 (2005).
[CrossRef]

Appl. Opt.

J. Appl. Phys.

L. Feng, X. P. Liu, Y. F. Tang, Y. F. Chen, J. Zi, S. N. Zhu, and Y. Y. Zhu, “Tunable negative refractions in two-dimensional photonic crystals with superconductor constituents,” J. Appl. Phys. 97, 073104 (2005).
[CrossRef]

Phys. Lett. A

C. H. Raymond Ooi and T. C. Au Yeung “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Lett. A 259, 413-419 (1999).
[CrossRef]

Phys. Rev. A

The problem of negative ns/n0=1−(T/Tc)ζ, at T>Tc, can be overcome by using expressions obtained from self-consistent calculations, as was done in A. Jordan, C. H. Raymond Ooi, and A. Svidzinsky “Fluctuation statistics of mesoscopic Bose-Einstein condensates: Reconciling the master equation with the partition function to reexamine the Uhlenbeck-Einstein dilemma,” Phys. Rev. A 74, 032506 (2006).
[CrossRef]

Phys. Rev. B

H. Takeda and K. Yoshino, “Properties of Abrikosov lattices as photonic crystals,” Phys. Rev. B 70, 085109 (2004).
[CrossRef]

H. Takeda and K. Yoshino, “Tunable photonic band schemes in two-dimensional photonic crystals composed of copper oxide high-temperature superconductors,” Phys. Rev. B 67, 245109 (2003).
[CrossRef]

C. H. Raymond Ooi, T. C. Au Yeung, C. H. Kam, and T. K. Lim, “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Rev. B 61, 5920-5923 (2000).
[CrossRef]

O. L. Berman, Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, “Superconducting photonic crystals: Numerical calculations of the band structure,” Phys. Rev. B 74, 092505 (2006).
[CrossRef]

Phys. Rev. Lett.

A. Pimenov, A. Loidl, P. Przyslupski, and B. Dabrowski “Negative refraction in ferromagnet-superconductor superlattices,” Phys. Rev. Lett. 95, 247009 (2005).
[CrossRef] [PubMed]

H. E. Portceanu, K. Karray, R. Seifert, F. Koch, P. Berberich, and H. Kinder, “Microwave Fabry-Pérot transmission through YBaCuO superconducting thin films,” Phys. Rev. Lett. 75, 3934-3937 (1995).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics7th ed. (Pergamon Press, 1999).

C. H. Raymond Ooi, T. C. Au Yeung, T. K. Lim, and C. H. Kam “Two-dimensional superconductor-dielectric photonic crystal,” in Photonics Technology into the 21st Century: Semiconductors, Microstructures, and Nanostructures, Proc. SPIE 3899, 278-287 (1999).

M. G. Cottam and D. R. Tilley, Introduction to Surface and Superlattice Excitations (University Press, Cambridge, 1989).
[CrossRef]

T. van Duzer and C. W. Turner, Principles of Superconductive Devices and Circuits (London, 1981).

C. M. Soukoulis, Photonic Band Gap Materials (Kluwer, 1996).

C. Kittel, Introduction to Solid State Physics, 6th ed. (Wiley, 1986).

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Figures (4)

Fig. 1
Fig. 1

All dielectric superlattice, ε d = 1 , ε s = 5 and N = 10 . Time evolution of the reflected and transmitted pulse versus central frequency ω o of input Gaussian pulse, showing double reflections with temporal separation that increases with the number of layers N. For larger bandwidth Δ ω , transient ringing is found for ω o within the bandgap. This is due to multimode interference of slow light modes.

Fig. 2
Fig. 2

(Top) Band structure ( ω vs. K ) , reflection R ( ω ) , and transmission T ( ω ) spectra for superlattice with constant dielectric ε d = 1 , λ L = 1.2 μ m , a = 0.5 λ L , d = 0.82 , and n = 30 . The low-frequency gap Δ l f and superpolariton gap Δ sup are due to the combined effects of periodicity and the plasma gap of superconductor. (Bottom) 3D plots show the time evolution of the reflected and transmitted pulse versus central frequency ω o of input Gaussian pulse showing (i) double reflection (square box) (ii) slow light at the edges of superpolariton gap (ellipse), (iii) ringing due to multimode interference of slow-light modes in input pulse with larger bandwidth Δ ω .

Fig. 3
Fig. 3

Dispersions and absorption of the superconductor (thick curves) and dielectric (thin curves), band structures, reflection, and transmission spectra for superlattice with dispersive dielectric, such that polariton gap Δ p o is within plasma gap Δ p l . The structural and superconductor parameters are the same as in Fig. 2: λ L = 1.2 μ m , a = 0.5 λ L , d = 0.8 a for n = 30 . Normal fluid proportion is zero. Parameters for dispersive dielectric are ω T = 87 meV or 2 π 2.103 6 × 10 13 s 1 with ϵ 0 = 7.65 , ϵ = 2.99 and γ = 2 π 3.651 2 × 10 11 s 1 . Note the presence of an additional superpolariton gap, giving double superpolariton gaps, Δ sup and Δ sup . The 3D plots show the temporal evolution of the reflected and transmitted pulse versus central frequency of the pulse. Note the arc feature highlighted by a circle, the transient beating due to interference and group delay effects. The bandwidth of the input pulse is Δ ω .

Fig. 4
Fig. 4

Same as Fig. 3 except that λ L = 1.2 μ m , such that the polariton gap and plasma gap overlap. Note the presence of a narrow and high transmission peak around the superconductor frequency, ω a 2 π c = 0.08 .

Equations (7)

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

σ ( ω ) = σ [ f n 1 + ( ω τ ) 2 + i ( f n ω τ 1 + ( ω τ ) 2 + f s ω τ ) ] ,
cos ( K a ) = cos ( k s z s ) cos ( k d z d ) 1 2 ( r p + 1 r p ) sin ( k s z s ) sin ( k d z d ) ,
r = ( M 11 + M 12 p f ) p i ( M 21 + M 22 p f ) ( M 11 + M 12 p f ) p i + ( M 21 + M 22 p f ) ,
t = s i f 2 p i ( M 11 + M 12 p f ) p i + ( M 21 + M 22 p f ) ,
M = ( m 11 U N 1 U N 2 m 12 U N 1 m 21 U N 1 m 22 U N 1 U N 2 ) ,
E r ( t ) = r ( ω ) E in ( ω ) e i ω t d ω ,
E t ( t ) = t ( ω ) E in ( ω ) e i ω t d ω .

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