Abstract

In a white light cavity (WLC), the group velocity is superluminal over a finite bandwidth. For a WLC-based data buffering system we recently proposed, it is important to visualize the behavior of pulses inside such a cavity. The conventional plane wave transfer functions, valid only over space that is translationally invariant, cannot be used for the space inside WLC or any cavity, which is translationally variant. Here, we develop the plane wave spatio temporal transfer function (PWSTTF) method to solve this problem, and produce visual representations of a Gaussian input pulse incident on a WLC, for all times and positions.

© 2012 OSA

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References

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  1. J. Yu, S. Yuan, J. Y. Gao, and L. Sun, “Optical pulse propagation in a Fabry-Perot etalon: analytical discussion,” J. Opt. Soc. Am. A18(9), 2153–2160 (2001).
    [CrossRef] [PubMed]
  2. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express12(1), 90–103 (2004).
    [CrossRef] [PubMed]
  3. J. E. Heebner, R. W. Boyd, and Q. H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B19(4), 722–731 (2002).
    [CrossRef]
  4. J. E. Heebner and R. W. Boyd, “Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt.49(14−15), 2629–2636 (2002).
    [CrossRef]
  5. J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(3), 036619 (2002).
    [CrossRef] [PubMed]
  6. G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett.99(13), 133601 (2007).
    [CrossRef] [PubMed]
  7. A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müller, and R.-H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun.134(1-6), 431–439 (1997).
    [CrossRef]
  8. R. H. Rinkleff and A. Wicht, “The concept of white light cavities using atomic phase coherence,” Phys. Scr. T118, 85–88 (2005).
    [CrossRef]
  9. R. Fleischhaker and J. Evers, “Four wave mixing enhanced white-light cavity,” Phys. Rev. A78(5), 051802 (2008).
    [CrossRef]
  10. H. N. Yum, M. E. Kim, Y. J. Jang, and M. S. Shahriar, “Distortion free pulse delay system using a pair of tunable white light cavities,” Opt. Express19(7), 6705–6713 (2011).
    [CrossRef] [PubMed]
  11. H. Yum, X. Liu, Y. J. Jang, M. E. Kim, and S. M. Shahriar, “Pulse delay via tunable white light cavities using fiber optic resonators,” J. Lightwave Technol.29(18), 2698–2705 (2011).
    [CrossRef]
  12. L. Levi, “Spatiotemporal transfer function: recent developments,” Appl. Opt.22(24), 4038–4040 (1983).
    [CrossRef] [PubMed]
  13. In modeling a FP, it is customary to consider each mirror to have an anti-reflection coating on one face, and a partially reflecting surface on the other. However, in reality, the partial reflectivity is produced by using layers of dielectric materials. Thus, in order to model properly any potential phase shift, in reflection or transmission at such an interface, it is necessary to take into account the presence of such a layer, which can be considered to be a Bragg grating. When this is done, it is no longer necessary to consider the presence of another surface with an anti-reflection coating.
  14. H. Kogelnik, “Coupled waved theory for thick hologram gratings,” Bell Syst. Tech. J.48(9), 2909–2947 (1969).
  15. A. Yariv and P. Yeh, Optical Waves in crystals (John Wiley & Sons, 1984).
  16. To calculate the phase shift due to the Bragg reflection, we solved the coupled equations presented in Ref. 14 for the case of λ(wavelength) = 1550nm, θ(angle of incidence) = 0, ϕ(slant grating angle) = 0 and σ(conductivity) = 0. The modulated dielectric constant is assumed to be of the form ε=ε0+ε1cos[(2π/Λ)z]where ε0 is the average dielectric permittivity, ε1is the modulation amplitude and Λ is the grating period. Λ is chosen to fulfill the Bragg matched condition. We have found that the phase shift for the reflected beam is a constant of −π/2, independent of the value of ε1, which determines the amplitude of the reflectivity. We also found that there is no phase shift for the transmitted wave. Finally, we assumed that the thickness of each BG is infinitesimally small, so that ΔνBragg≫ΔνFWHMwhere ΔνBraggis a bandwidth of the Bragg reflection and ΔνFWHMis a full width half maximum of the FP cavity. Thus, for a particularε1, the Bragg reflection coefficient can be expressed as exp(−jπ/2)Rwith a constant value of R, within the spectral range ofΔνFWHM.
  17. A. Yariv, Optical Electronics (Oxford University Press, 1990).
  18. L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light Propagation,” Nature406(6793), 277–279 (2000).
    [CrossRef] [PubMed]
  19. R. W. Boyd and D. J. Gauthier, “Slow and fast light,” in Progress in Optics: 43, E. Wolf, ed. (Elsevier, 2002), Chap. 6.
  20. A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett.86(18), 3925–3929 (2001).
    [CrossRef] [PubMed]

2011

2008

R. Fleischhaker and J. Evers, “Four wave mixing enhanced white-light cavity,” Phys. Rev. A78(5), 051802 (2008).
[CrossRef]

2007

G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett.99(13), 133601 (2007).
[CrossRef] [PubMed]

2005

R. H. Rinkleff and A. Wicht, “The concept of white light cavities using atomic phase coherence,” Phys. Scr. T118, 85–88 (2005).
[CrossRef]

2004

2002

J. E. Heebner, R. W. Boyd, and Q. H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B19(4), 722–731 (2002).
[CrossRef]

J. E. Heebner and R. W. Boyd, “Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt.49(14−15), 2629–2636 (2002).
[CrossRef]

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(3), 036619 (2002).
[CrossRef] [PubMed]

2001

J. Yu, S. Yuan, J. Y. Gao, and L. Sun, “Optical pulse propagation in a Fabry-Perot etalon: analytical discussion,” J. Opt. Soc. Am. A18(9), 2153–2160 (2001).
[CrossRef] [PubMed]

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett.86(18), 3925–3929 (2001).
[CrossRef] [PubMed]

2000

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light Propagation,” Nature406(6793), 277–279 (2000).
[CrossRef] [PubMed]

1997

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müller, and R.-H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun.134(1-6), 431–439 (1997).
[CrossRef]

1983

1969

H. Kogelnik, “Coupled waved theory for thick hologram gratings,” Bell Syst. Tech. J.48(9), 2909–2947 (1969).

Boyd, R. W.

J. E. Heebner, R. W. Boyd, and Q. H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B19(4), 722–731 (2002).
[CrossRef]

J. E. Heebner and R. W. Boyd, “Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt.49(14−15), 2629–2636 (2002).
[CrossRef]

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(3), 036619 (2002).
[CrossRef] [PubMed]

Chiao, R. Y.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett.86(18), 3925–3929 (2001).
[CrossRef] [PubMed]

Danzmann, K.

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müller, and R.-H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun.134(1-6), 431–439 (1997).
[CrossRef]

Dogariu, A.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett.86(18), 3925–3929 (2001).
[CrossRef] [PubMed]

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light Propagation,” Nature406(6793), 277–279 (2000).
[CrossRef] [PubMed]

Evers, J.

R. Fleischhaker and J. Evers, “Four wave mixing enhanced white-light cavity,” Phys. Rev. A78(5), 051802 (2008).
[CrossRef]

Fleischhaker, R.

R. Fleischhaker and J. Evers, “Four wave mixing enhanced white-light cavity,” Phys. Rev. A78(5), 051802 (2008).
[CrossRef]

Fleischhauer, M.

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müller, and R.-H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun.134(1-6), 431–439 (1997).
[CrossRef]

Gao, J. Y.

Heebner, J. E.

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(3), 036619 (2002).
[CrossRef] [PubMed]

J. E. Heebner, R. W. Boyd, and Q. H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B19(4), 722–731 (2002).
[CrossRef]

J. E. Heebner and R. W. Boyd, “Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt.49(14−15), 2629–2636 (2002).
[CrossRef]

Huang, Y.

Jang, Y. J.

Kim, M. E.

Kogelnik, H.

H. Kogelnik, “Coupled waved theory for thick hologram gratings,” Bell Syst. Tech. J.48(9), 2909–2947 (1969).

Kuzmich, A.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett.86(18), 3925–3929 (2001).
[CrossRef] [PubMed]

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light Propagation,” Nature406(6793), 277–279 (2000).
[CrossRef] [PubMed]

Levi, L.

Liu, X.

Milonni, P. W.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett.86(18), 3925–3929 (2001).
[CrossRef] [PubMed]

Mookherjea, S.

Müller, G.

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müller, and R.-H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun.134(1-6), 431–439 (1997).
[CrossRef]

Paloczi, G. T.

Park, Q. H.

J. E. Heebner, R. W. Boyd, and Q. H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B19(4), 722–731 (2002).
[CrossRef]

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(3), 036619 (2002).
[CrossRef] [PubMed]

Pati, G. S.

G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett.99(13), 133601 (2007).
[CrossRef] [PubMed]

Poon, J. K. S.

Rinkleff, R. H.

R. H. Rinkleff and A. Wicht, “The concept of white light cavities using atomic phase coherence,” Phys. Scr. T118, 85–88 (2005).
[CrossRef]

Rinkleff, R.-H.

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müller, and R.-H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun.134(1-6), 431–439 (1997).
[CrossRef]

Salit, K.

G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett.99(13), 133601 (2007).
[CrossRef] [PubMed]

Salit, M.

G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett.99(13), 133601 (2007).
[CrossRef] [PubMed]

Scheuer, J.

Scully, M.

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müller, and R.-H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun.134(1-6), 431–439 (1997).
[CrossRef]

Shahriar, M. S.

H. N. Yum, M. E. Kim, Y. J. Jang, and M. S. Shahriar, “Distortion free pulse delay system using a pair of tunable white light cavities,” Opt. Express19(7), 6705–6713 (2011).
[CrossRef] [PubMed]

G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett.99(13), 133601 (2007).
[CrossRef] [PubMed]

Shahriar, S. M.

Sun, L.

Wang, L. J.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett.86(18), 3925–3929 (2001).
[CrossRef] [PubMed]

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light Propagation,” Nature406(6793), 277–279 (2000).
[CrossRef] [PubMed]

Wicht, A.

R. H. Rinkleff and A. Wicht, “The concept of white light cavities using atomic phase coherence,” Phys. Scr. T118, 85–88 (2005).
[CrossRef]

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müller, and R.-H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun.134(1-6), 431–439 (1997).
[CrossRef]

Yariv, A.

Yu, J.

Yuan, S.

Yum, H.

Yum, H. N.

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, “Coupled waved theory for thick hologram gratings,” Bell Syst. Tech. J.48(9), 2909–2947 (1969).

J. Lightwave Technol.

J. Mod. Opt.

J. E. Heebner and R. W. Boyd, “Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt.49(14−15), 2629–2636 (2002).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nature

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light Propagation,” Nature406(6793), 277–279 (2000).
[CrossRef] [PubMed]

Opt. Commun.

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müller, and R.-H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun.134(1-6), 431–439 (1997).
[CrossRef]

Opt. Express

Phys. Rev. A

R. Fleischhaker and J. Evers, “Four wave mixing enhanced white-light cavity,” Phys. Rev. A78(5), 051802 (2008).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(3), 036619 (2002).
[CrossRef] [PubMed]

Phys. Rev. Lett.

G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett.99(13), 133601 (2007).
[CrossRef] [PubMed]

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett.86(18), 3925–3929 (2001).
[CrossRef] [PubMed]

Phys. Scr. T

R. H. Rinkleff and A. Wicht, “The concept of white light cavities using atomic phase coherence,” Phys. Scr. T118, 85–88 (2005).
[CrossRef]

Other

In modeling a FP, it is customary to consider each mirror to have an anti-reflection coating on one face, and a partially reflecting surface on the other. However, in reality, the partial reflectivity is produced by using layers of dielectric materials. Thus, in order to model properly any potential phase shift, in reflection or transmission at such an interface, it is necessary to take into account the presence of such a layer, which can be considered to be a Bragg grating. When this is done, it is no longer necessary to consider the presence of another surface with an anti-reflection coating.

A. Yariv and P. Yeh, Optical Waves in crystals (John Wiley & Sons, 1984).

To calculate the phase shift due to the Bragg reflection, we solved the coupled equations presented in Ref. 14 for the case of λ(wavelength) = 1550nm, θ(angle of incidence) = 0, ϕ(slant grating angle) = 0 and σ(conductivity) = 0. The modulated dielectric constant is assumed to be of the form ε=ε0+ε1cos[(2π/Λ)z]where ε0 is the average dielectric permittivity, ε1is the modulation amplitude and Λ is the grating period. Λ is chosen to fulfill the Bragg matched condition. We have found that the phase shift for the reflected beam is a constant of −π/2, independent of the value of ε1, which determines the amplitude of the reflectivity. We also found that there is no phase shift for the transmitted wave. Finally, we assumed that the thickness of each BG is infinitesimally small, so that ΔνBragg≫ΔνFWHMwhere ΔνBraggis a bandwidth of the Bragg reflection and ΔνFWHMis a full width half maximum of the FP cavity. Thus, for a particularε1, the Bragg reflection coefficient can be expressed as exp(−jπ/2)Rwith a constant value of R, within the spectral range ofΔνFWHM.

A. Yariv, Optical Electronics (Oxford University Press, 1990).

R. W. Boyd and D. J. Gauthier, “Slow and fast light,” in Progress in Optics: 43, E. Wolf, ed. (Elsevier, 2002), Chap. 6.

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Figures (4)

Fig. 1
Fig. 1

Schematic illustration of a typical Fabry-Perot (FP) cavity of length L, containing a dispersive medium. We model the mirrors as Bragg gratings. The medium outside the FP is assumed to be non-dispersive. At t = t1, the peak of a test pulse is located at z = z1, moving in the positive z direction

Fig. 2
Fig. 2

A propagating pulse shown as freeze-frames for (a) t= 250 /c , (b) t= 300 /c , (c) t= 350 /c , (d) t= 390 /c , (e) t= 650 /c . (f) Numerical simulation by the TTF method. The insets present expanded views of the interference patterns. In Fig. 2(e) and 2(f), the reference pulse (blue) propagates together with the first output from the cavity. Note that the left BG is at z=500 z L and the right BG is at z=593 z R . For convenience, we have also defined z2 = P−λ = zL−λ, and z3 = P + L−3 = zR3.

Fig. 3
Fig. 3

Propagation of a pulse shown as freeze-frames, for an intracavity fast-light medium with group index n g =0.2 , for (a) t= 300 /c , (b) t= 350 /c , (c) t= 390 /c , (d) t= 410 /c with a reference pulse(blue). The insets present expanded views over one wavelength. (e) Output pulses and the reference in time domain.

Fig. 4
Fig. 4

Propagation of a pulse shown as freeze-frames, through an intracavity fast-light medium under ideal WLC condition ( n g =0 ) for (a) t= 290 /c , (b) t= 300 /c , (c) t= 310 /c , (d) t= 350 /c , The upper insets illustrate the case of a fast-light medium with n g =0 in free space. The lower insets show the views expanded horizontally close to z4 = 550. (e) Numerical simulations for cavity output (black) and the reference (blue) in time domain, produced via the TTF method.

Equations (15)

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E in ( ω,z,t )=exp( j ϕ in ); ϕ in = k 1 ( z z 1 )ω( t t 1 )
E r1 ( ω,z,t )= R exp( jπ 2 )exp( j ϕ r1 ); ϕ r1 = k 1 (2Pz z 1 )ω(t t 1 )
E f ( ω,z,t )= T exp( j ϕ f ) m=0 [ R m exp( jmπ )exp( 2jm k d L ) ] ; ϕ f = k 1 (P z 1 )+ k d (zP)ω(t t 1 )
E b ( ω,z,t )= T R exp( j ϕ b ) m=0 [ R m exp( jmπ )exp(2jm k d L) ] ; ϕ b = k 1 (P z 1 )+ k d L+ k d (P+Lz)π/2ω(t t 1 )
E r2 ( ω,z,t )=T R exp( j ϕ r2 ) m=0 [ R m exp( jmπ )exp(2jm k d L) ] ; ϕ r2 = k 1 (P z 1 )+2 k d L+ k 1 (Pz)π/2ω(t t 1 )
E out ( ω,z,t )=Texp( j ϕ out ) m=0 [ R m exp( jmπ )exp( 2jm k d L ) ] ; ϕ out = k 1 (P z 1 )+ k d L+ k 1 (zPL)ω(t t 1 )
E f ( ω,z,t )= T 1Rexp( jπ )exp(2j k d L) exp( j ϕ f )
E b ( ω,z,t )= R T 1Rexp( jπ )exp(2j k d L) exp( j ϕ b )
E r2 ( ω,z,t )= R T 1Rexp( jπ )exp(2j k d L) exp( j ϕ r2 )
E r ( ω,z,t )= p=1 2 E rp ( ω,z,t )
E out ( ω,z,t )= T 1Rexp( jπ )exp(2j k d L) exp( j ϕ out )
For 0<z<P, S 1 (z,t)=1/ 2π S ˜ 0 (k)[ i=in,r E i (ω,z,t) ]dk
For P<z<P+L, S 2 (z,t)=1/ 2π S ˜ 0 (k)[ i=f,b E i (ω,z,t) ]dk
For z>P+L, S 3 (z,t)=1/ 2π S ˜ 0 (k) E out (ω,z,t)dk
z L z:E( z )=f( z );z < L z< z R :E( z )=f( z L ); z R z:E( z )=f( z( z R z L ) )

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