Abstract

In previous investigations of optical data storage in persistent spectral-hole-burning materials, either the spectral or the temporal positions of the data were used as memory addresses. We suggest having the address represented by the spectral shape of the writing field and retrieving information with the aid of pattern recognition of this spectral shape. This procedure is expected to accelerate access to the stored data. To evaluate the validity of the method, we consider various physical effects that may give rise to cross talk among the different addresses. A numerical simulation under realistic conditions is presented.

© 1995 Optical Society of America

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    [CrossRef]
  36. S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, “Holography in frequency selective media. II: Controlling the diffraction efficiency,” J. Lumin. 53, 215–218 (1992).
    [CrossRef]
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    [CrossRef]

1994 (1)

1993 (2)

N. Nakatsuka, A. Wakamiya, K. M. Abedin, and T. Hattori, “Accumulated photon echoes by using a nonlaser light source,” Opt. Lett. 18, 832–834 (1993).
[CrossRef] [PubMed]

X. A. Shen and R. Kachru, “Time domain optical memory for image storage and high speed image processing,” Appl. Opt. 32, 5811–5815 (1993).
[CrossRef]

1992 (6)

1991 (7)

1990 (1)

1989 (3)

A. J. Meixner, A. Renn, and U. P. Wild, “Spectral hole-burning and holography. I. Transmission and holographic detection of spectral holes,” J. Chem. Phys. 91, 6728–6736 (1989).
[CrossRef]

A. Débarre, J.-C. Keller, J.-L. Le Gouët, A. Richard, and P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
[CrossRef]

A. Rebane, J. Aaviksoo, and J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
[CrossRef]

1988 (1)

W. R. Babbitt and T. W. Mossberg, “Time-domain frequency-selective optical data storage in a solid-state material,” Opt. Commun. 65, 185–188 (1988).
[CrossRef]

1986 (1)

1985 (1)

1984 (1)

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2536 (1984).
[CrossRef]

1983 (1)

A. Rebane, R. Kaarli, P. Saari, A. Anijalg, and K. Timpmann, “Photochemical time-domain holography of weak picosecond pulses,” Opt. Commun. 47, 173–176 (1983).
[CrossRef]

1982 (1)

1979 (2)

T. W. Mossberg, A. Flusberg, R. Kachru, and S. R. Hartmann, “Total scattering cross-section for Na on He measured by stimulated photon echo,” Phys. Rev. Lett. 42, 1665–1669 (1979).
[CrossRef]

W. H. Hesselink and D. A. Wiersma, “Picosecond photon echoes stimulated from an accumulated grating,” Phys. Rev. Lett. 43, 1991–1994 (1979).
[CrossRef]

Aaviksoo, J.

A. Rebane, J. Aaviksoo, and J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
[CrossRef]

Abedin, K. M.

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Anijalg, A.

A. Rebane, R. Kaarli, P. Saari, A. Anijalg, and K. Timpmann, “Photochemical time-domain holography of weak picosecond pulses,” Opt. Commun. 47, 173–176 (1983).
[CrossRef]

Babbitt, W. R.

W. R. Babbitt and T. W. Mossberg, “Time-domain frequency-selective optical data storage in a solid-state material,” Opt. Commun. 65, 185–188 (1988).
[CrossRef]

Bai, Y. S.

Bernet, S.

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, “Holography in frequency selective media. II: Controlling the diffraction efficiency,” J. Lumin. 53, 215–218 (1992).
[CrossRef]

B. Kohler, S. Bernet, A. Renn, and U. P. Wild, “Holographic optical data storage of 2000 images for photochemical hole burning,” in Persistent Spectral Hole Burning: Science and Application, Vol. 16 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 46–49.

Castro, G.

G. Castro, D. Haarer, R. Morton, R. M. McFarlane, and H. P. Trommsdorff, “Frequency selective optical data storage system,” U.S. patent,101,976 (July18, 1978).

Chen, C.

Chen, H.

Chen, Y.

Débarre, A.

Denz, C.

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Potentialities and limitations of hologram multiplexing by using the phase-encoding technique,” Appl. Opt. 31, 5700–5705 (1992).
[CrossRef] [PubMed]

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Volume hologram multiplexing using a deterministic phase encoding method,” Opt. Commun. 85, 171–176 (1991).
[CrossRef]

Dilworth, D.

Eberly, J. H.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Fainman, Y.

Flusberg, A.

T. W. Mossberg, A. Flusberg, R. Kachru, and S. R. Hartmann, “Total scattering cross-section for Na on He measured by stimulated photon echo,” Phys. Rev. Lett. 42, 1665–1669 (1979).
[CrossRef]

Ford, J. E.

Galaup, J.-P.

Gauthier, D. J.

Haarer, D.

G. Castro, D. Haarer, R. Morton, R. M. McFarlane, and H. P. Trommsdorff, “Frequency selective optical data storage system,” U.S. patent,101,976 (July18, 1978).

Hartmann, S. R.

T. W. Mossberg, A. Flusberg, R. Kachru, and S. R. Hartmann, “Total scattering cross-section for Na on He measured by stimulated photon echo,” Phys. Rev. Lett. 42, 1665–1669 (1979).
[CrossRef]

Hattori, T.

Heritage, J. P.

Hesselink, W. H.

W. H. Hesselink and D. A. Wiersma, “Picosecond photon echoes stimulated from an accumulated grating,” Phys. Rev. Lett. 43, 1991–1994 (1979).
[CrossRef]

Hjelmstad, J. F.

R. Skaug and J. F. Hjelmstad, Spread Spectrum in Communication (Peregrinus, London, 1985).
[CrossRef]

Huang, J.

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N. J., 1989).

Kaarli, R.

P. Saari, R. Kaarli, and A. Rebane, “Picosecond time- and space-domain holography by photochemical hole burning,” J. Opt. Soc. Am. B 3, 527–534 (1986).
[CrossRef]

A. Rebane, R. Kaarli, P. Saari, A. Anijalg, and K. Timpmann, “Photochemical time-domain holography of weak picosecond pulses,” Opt. Commun. 47, 173–176 (1983).
[CrossRef]

Kachru, R.

X. A. Shen and R. Kachru, “Time domain optical memory for image storage and high speed image processing,” Appl. Opt. 32, 5811–5815 (1993).
[CrossRef]

A. Shen, Y. S. Bai, and R. Kachru, “Reprogrammable optical matched filter for biphase-coded pulse compression,” Opt. Lett. 17, 1079–1081 (1992).
[CrossRef] [PubMed]

T. W. Mossberg, A. Flusberg, R. Kachru, and S. R. Hartmann, “Total scattering cross-section for Na on He measured by stimulated photon echo,” Phys. Rev. Lett. 42, 1665–1669 (1979).
[CrossRef]

Keller, J.-C.

Kohler, B.

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, “Holography in frequency selective media. II: Controlling the diffraction efficiency,” J. Lumin. 53, 215–218 (1992).
[CrossRef]

B. Kohler, S. Bernet, A. Renn, and U. P. Wild, “Holographic optical data storage of 2000 images for photochemical hole burning,” in Persistent Spectral Hole Burning: Science and Application, Vol. 16 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 46–49.

Kuhl, J.

A. Rebane, J. Aaviksoo, and J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
[CrossRef]

Le Gouët, J.-L.

Lee, S. H.

Leiard, D. E.

A. M. Weiner, D. E. Leiard, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

A. M. Weiner and D. E. Leiard, “Generation of terahertzrate trains of femtosecond pulses by phase-only filtering,” Opt. Lett. 15, 51–53 (1990).
[CrossRef] [PubMed]

Leith, E.

Lopez, J.

Ma, J.

Masri, R.

McFarlane, R. M.

G. Castro, D. Haarer, R. Morton, R. M. McFarlane, and H. P. Trommsdorff, “Frequency selective optical data storage system,” U.S. patent,101,976 (July18, 1978).

Meixner, A. J.

A. J. Meixner, A. Renn, and U. P. Wild, “Spectral hole-burning and holography. I. Transmission and holographic detection of spectral holes,” J. Chem. Phys. 91, 6728–6736 (1989).
[CrossRef]

Mitsunaga, M.

Morita, N.

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2536 (1984).
[CrossRef]

Morton, R.

G. Castro, D. Haarer, R. Morton, R. M. McFarlane, and H. P. Trommsdorff, “Frequency selective optical data storage system,” U.S. patent,101,976 (July18, 1978).

Mossberg, T. W.

J. M. Zhang, D. J. Gauthier, J. Huang, and T. W. Mossberg, “Use of phase-noisy laser fields in the storage of optical pulse shape in inhomogeneously broadened absorbers,” Opt. Lett. 16, 103–105 (1991).
[CrossRef] [PubMed]

W. R. Babbitt and T. W. Mossberg, “Time-domain frequency-selective optical data storage in a solid-state material,” Opt. Commun. 65, 185–188 (1988).
[CrossRef]

T. W. Mossberg, “Time-domain frequency-selective optical data storage,” Opt. Lett. 7, 77–79 (1982).
[CrossRef] [PubMed]

T. W. Mossberg, A. Flusberg, R. Kachru, and S. R. Hartmann, “Total scattering cross-section for Na on He measured by stimulated photon echo,” Phys. Rev. Lett. 42, 1665–1669 (1979).
[CrossRef]

Nakatsuka, N.

Paek, E. G.

A. M. Weiner, D. E. Leiard, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

Pauliat, G.

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Potentialities and limitations of hologram multiplexing by using the phase-encoding technique,” Appl. Opt. 31, 5700–5705 (1992).
[CrossRef] [PubMed]

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Volume hologram multiplexing using a deterministic phase encoding method,” Opt. Commun. 85, 171–176 (1991).
[CrossRef]

Rebane, A.

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, “Holography in frequency selective media. II: Controlling the diffraction efficiency,” J. Lumin. 53, 215–218 (1992).
[CrossRef]

A. Rebane, J. Aaviksoo, and J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
[CrossRef]

P. Saari, R. Kaarli, and A. Rebane, “Picosecond time- and space-domain holography by photochemical hole burning,” J. Opt. Soc. Am. B 3, 527–534 (1986).
[CrossRef]

A. Rebane, R. Kaarli, P. Saari, A. Anijalg, and K. Timpmann, “Photochemical time-domain holography of weak picosecond pulses,” Opt. Commun. 47, 173–176 (1983).
[CrossRef]

Reitze, D. H.

A. M. Weiner, D. E. Leiard, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

Renn, A.

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, “Holography in frequency selective media. II: Controlling the diffraction efficiency,” J. Lumin. 53, 215–218 (1992).
[CrossRef]

A. J. Meixner, A. Renn, and U. P. Wild, “Spectral hole-burning and holography. I. Transmission and holographic detection of spectral holes,” J. Chem. Phys. 91, 6728–6736 (1989).
[CrossRef]

B. Kohler, S. Bernet, A. Renn, and U. P. Wild, “Holographic optical data storage of 2000 images for photochemical hole burning,” in Persistent Spectral Hole Burning: Science and Application, Vol. 16 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 46–49.

Richard, A.

A. Débarre, J.-C. Keller, J.-L. Le Gouët, A. Richard, and P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
[CrossRef]

Roosen, G.

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Potentialities and limitations of hologram multiplexing by using the phase-encoding technique,” Appl. Opt. 31, 5700–5705 (1992).
[CrossRef] [PubMed]

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Volume hologram multiplexing using a deterministic phase encoding method,” Opt. Commun. 85, 171–176 (1991).
[CrossRef]

Rudd, J.

Saari, P.

Sarapuu, R.

R. Sarapuu, “Theoretical problems of space-and-time domain holography in photochromic media with spectral and polarization selectivity,” Ph.D. dissertation (Institute of Physics, Tartu, Estonia, 1989).

Sasaki, H.

Shen, A.

Shen, X. A.

X. A. Shen and R. Kachru, “Time domain optical memory for image storage and high speed image processing,” Appl. Opt. 32, 5811–5815 (1993).
[CrossRef]

Skaug, R.

R. Skaug and J. F. Hjelmstad, Spread Spectrum in Communication (Peregrinus, London, 1985).
[CrossRef]

Sõnajalg, H.

Sun, P.-C.

Szabo, A.

A. Szabo, “Frequency selective optical memory,” U.S. patent3,896,420 (July22, 1975).

Taketomi, Y.

Tchénio, P.

Thurston, R. N.

Timpmann, K.

A. Rebane, R. Kaarli, P. Saari, A. Anijalg, and K. Timpmann, “Photochemical time-domain holography of weak picosecond pulses,” Opt. Commun. 47, 173–176 (1983).
[CrossRef]

Trommsdorff, H. P.

G. Castro, D. Haarer, R. Morton, R. M. McFarlane, and H. P. Trommsdorff, “Frequency selective optical data storage system,” U.S. patent,101,976 (July18, 1978).

Tschudi, T.

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Potentialities and limitations of hologram multiplexing by using the phase-encoding technique,” Appl. Opt. 31, 5700–5705 (1992).
[CrossRef] [PubMed]

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Volume hologram multiplexing using a deterministic phase encoding method,” Opt. Commun. 85, 171–176 (1991).
[CrossRef]

Uesugi, N.

Valdmanis, J.

Vossler, G.

Wakamiya, A.

Weiner, A. M.

Wiersma, D. A.

W. H. Hesselink and D. A. Wiersma, “Picosecond photon echoes stimulated from an accumulated grating,” Phys. Rev. Lett. 43, 1991–1994 (1979).
[CrossRef]

Wild, U. P.

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, “Holography in frequency selective media. II: Controlling the diffraction efficiency,” J. Lumin. 53, 215–218 (1992).
[CrossRef]

A. J. Meixner, A. Renn, and U. P. Wild, “Spectral hole-burning and holography. I. Transmission and holographic detection of spectral holes,” J. Chem. Phys. 91, 6728–6736 (1989).
[CrossRef]

B. Kohler, S. Bernet, A. Renn, and U. P. Wild, “Holographic optical data storage of 2000 images for photochemical hole burning,” in Persistent Spectral Hole Burning: Science and Application, Vol. 16 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 46–49.

Yajima, T.

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2536 (1984).
[CrossRef]

Yano, R.

Zhang, J. M.

Appl. Opt. (4)

Appl. Phys. Lett. (1)

A. Rebane, J. Aaviksoo, and J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. M. Weiner, D. E. Leiard, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

J. Chem. Phys. (1)

A. J. Meixner, A. Renn, and U. P. Wild, “Spectral hole-burning and holography. I. Transmission and holographic detection of spectral holes,” J. Chem. Phys. 91, 6728–6736 (1989).
[CrossRef]

J. Lumin. (1)

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, “Holography in frequency selective media. II: Controlling the diffraction efficiency,” J. Lumin. 53, 215–218 (1992).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (4)

Opt. Commun. (4)

W. R. Babbitt and T. W. Mossberg, “Time-domain frequency-selective optical data storage in a solid-state material,” Opt. Commun. 65, 185–188 (1988).
[CrossRef]

A. Rebane, R. Kaarli, P. Saari, A. Anijalg, and K. Timpmann, “Photochemical time-domain holography of weak picosecond pulses,” Opt. Commun. 47, 173–176 (1983).
[CrossRef]

A. Débarre, J.-C. Keller, J.-L. Le Gouët, A. Richard, and P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
[CrossRef]

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Volume hologram multiplexing using a deterministic phase encoding method,” Opt. Commun. 85, 171–176 (1991).
[CrossRef]

Opt. Lett. (8)

Phys. Rev. A (1)

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2536 (1984).
[CrossRef]

Phys. Rev. Lett. (2)

T. W. Mossberg, A. Flusberg, R. Kachru, and S. R. Hartmann, “Total scattering cross-section for Na on He measured by stimulated photon echo,” Phys. Rev. Lett. 42, 1665–1669 (1979).
[CrossRef]

W. H. Hesselink and D. A. Wiersma, “Picosecond photon echoes stimulated from an accumulated grating,” Phys. Rev. Lett. 43, 1991–1994 (1979).
[CrossRef]

Other (9)

O. Sild and K. Haller, eds., Zero-Phonon Lines and Spectral Hole Burning in Spectroscopy and Photochemistry (Springer-Verlag, Berlin, 1988).
[CrossRef]

W. E. Moerner, ed., Persistent Spectral Hole-Burning: Science and Applications (Springer-Verlag, Berlin, 1988).
[CrossRef]

B. Kohler, S. Bernet, A. Renn, and U. P. Wild, “Holographic optical data storage of 2000 images for photochemical hole burning,” in Persistent Spectral Hole Burning: Science and Application, Vol. 16 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 46–49.

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A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N. J., 1989).

R. Skaug and J. F. Hjelmstad, Spread Spectrum in Communication (Peregrinus, London, 1985).
[CrossRef]

R. Sarapuu, “Theoretical problems of space-and-time domain holography in photochromic media with spectral and polarization selectivity,” Ph.D. dissertation (Institute of Physics, Tartu, Estonia, 1989).

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Figures (5)

Fig. 1
Fig. 1

Schematic of the experimental configuration.

Fig. 2
Fig. 2

Nonrefractive component of the cross talk as a function of Δinhol, where the nonuniform distribution of oscillators is represented by a Gaussian function. The laser spectral density is assumed to be uniform over the Δhol-wide storage domain. The initial optical density at the center of the absorption band is (a) 0.05 and (b) 0.5. The exposure Nf0 is set equal to 0.5.

Fig. 3
Fig. 3

Refractive component of the cross talk as a function of the exposure Nf0. The ratio Δinhol is set equal to 0.5. The laser spectrum is flat over the storage interval. The initial optical density is D0 = 0.5.

Fig. 4
Fig. 4

Influence of the refractive-index grating: functions (a) |1 − C0|, (b) |C1|, and (c) |C2| [see Eq. (4.14)]; Δ is the width of the spectral element and τ is the time separation of the recording pulses.

Fig. 5
Fig. 5

Histograms of the restored data. The computation is effected for 400 different recordings of 32 data each. The 32 addresses of the memory are read out for each recording. The exposure, the inhomogeneous width, and the laser width, respectively, are given by Nf0 = 1.0, Δinhol = 2.27, and Δlasin = 1.09. The small density limit (D0 = 0.05) is considered in (a). The density is set equal to 0.5 for the computation of curve (b).

Tables (3)

Tables Icon

Table 1 Cross Talk That Is Due to the Finite Delay between the Pulses

Tables Icon

Table 2 Cross Talk in a Nearly Transparent Sample (D0 = 0.05)

Tables Icon

Table 3 Cross Talk in an Absorbing Sample (D0 = 0.5)

Equations (50)

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r i = [ r i ( 1 - N / 2 ) , r i ( 2 - N / 2 ) , r i ( 3 - N / 2 ) , , r i ( N / 2 ) ] , [ r i ( p ) = + 1 or - 1 ] ,
p = - ( N / 2 - 1 ) N / 2 r k ( p ) × r 1 ( p ) = N δ k 1 .
R i ( ω ) = p = - ( N / 2 - 1 ) N / 2 r i ( p ) rect [ ω - ( ω 0 + p Δ - Δ 2 ) Δ ] .
E 1 ( j , ω ) α exp [ i ω ( τ + τ read ) ] R j ( ω ) i = 1 N R i * ( ω ) S i .
E ˜ 1 ( j , t ) = E 1 ( j , ω ) exp ( - i ω t ) d ω .
E ˜ 1 ( j , t ) sinc [ Δ ( τ + τ read - t ) / 2 ] × i = 1 N S i p = - ( N / 2 - 1 ) N / 2 r i ( p ) × r j ( p ) × exp [ i ( ω o + p Δ - Δ 2 ) ( τ + τ read - t ) ] .
E ˜ 1 ( j , τ + τ read ) { N if S j = 1 0 if S j = 0 .
g ( T ) = d t E ˜ G ( t - T ) E ˜ S * ( t ) / d t E ˜ G ( t 2 = d ω E G ( ω ) E S * ( ω ) exp ( i ω T ) / d ω E G ( ω ) 2 .
g ( j , T ) = d ω exp [ - i ω ( τ + τ read - T ) ] × R j * ( ω ) i = 1 N R i ( ω ) S i * / ( N Δ ) .
n ( r , ω , ν ) = n ( ω ) exp [ - cos 2 ν F ( r , ω ) k ( ω ) ] ,
F i ( r , ω ) = f 0 R i ( ω ) + S i exp ( i K x + i ω τ ) 2 = f 0 [ R i ( ω ) 2 + S i 2 × R i * ( ω ) S i exp ( i K x + i ω τ ) + c . c . ] ,
E 1 ( j , ω ) = f 0 E 0 ( ω 0 ) b 2 ( ω ) R j ( ω ) [ W ( ω ) ] 1 / 2 × ( 1 - i H ^ ) { W ( ω ) D ( ω ) i = 1 N S i [ R i * ( ω ) exp ( i ω τ ) ] k ( ω ) k ( ω ) } exp { - ( 1 - i H ^ ) [ b 1 ( ω ) D ( ω ) ] } × exp ( i ω τ read ) .
H ^ [ i = 1 N R i * ( ω ) S i exp ( i ω τ ) ] = i i = 1 N R i * ( ω ) S i exp ( i ω τ ) .
E 1 ( j , ω ) = 2 f 0 E ( ω 0 ) D 0 b 2 R j ( ω ) i = 1 N R i * ( ω ) S i × exp ( - b 1 D 0 ) exp [ i ω ( τ + τ read ) ] .
E ˜ 1 ( j , τ + τ read ) / E ˜ ( τ read ) = 2 F D 0 b 2 exp ( - b 1 D 0 ) / [ N ( 1 + α ) ] ,
b n = ( 3 / 2 ) ln ( 10 ) cos 2 n ν exp ( - F cos 2 ν ) .
X = g i = j / g i j .
E 1 ( i , ω ) = 2 f 0 E 0 ( ω 0 ) b 2 ( ω ) R i ( ω ) [ W ( ω ) ] 3 / 2 × D ( ω ) k j N R k * exp { - ( 1 - i H ^ ) [ b 1 ( ω ) D ( ω ) ] } × exp [ i ω ( τ + τ read ) ] .
g i = 1 N p = 1 - N / 2 p = N / 2 W 2 ( ω p ) b 2 ( ω p ) D ( ω p ) r i ( p ) k j r k r ( p ) × exp { - ( 1 - i H ^ ) [ b 1 ( ω p ) D ( ω p ) ] } ,
k k = δ k k
k = 1 N r k ( p ) r k ( p ) = N δ p p ,
ϕ ( ω ) = H ^ [ b 1 ( ω ) D ( ω ) ] E 1 ( ω ) = rect ( ω - ω 0 N Δ ) f 0 b 2 ( ω ) W 2 ( ω ) D ( ω ) × exp [ - b 1 ( ω ) D ( ω ) ] .
X 2 = N Δ σ 2 σ 1 2 - 1 ,
σ 1 = | d ω E 1 ( ω ) exp [ i ϕ ( ω ) ] | ,             σ 2 = d ω E 1 ( ω ) 2 .
X 2 nr = N Δ σ 2 [ d ω E 1 ( ω ) ] 2 - 1 ,
X 2 r = 2 d ω E 1 ( ω ) ( 1 - cos { H ^ [ b 1 ( ω ) D ( ω ) ] } ) d ω E 1 ( ω ) .
X 2 = 2 N Δ d ω rect ( ω - ω 0 N Δ ) ( 1 - cos { D 0 H ^ [ b 1 ( ω ) ] } ) ,
E 1 ( i , ω ) = f 0 E 0 ( ω 0 ) b 2 D 0 exp ( - b 1 D 0 ) × R i ( ω ) ( 1 - i H ^ ) [ k j N R k * ( ω ) exp ( i ω τ ) ] × exp ( i ω τ read ) .
X = | p = 1 - N / 2 N / 2 p = 1 - N / 2 N / 2 r j ( p ) k j k r k ( p ) C p - p / [ N ( 1 - C 0 ) ] | ,
C n = i d ω Δ rect ( ω Δ ) H ^ [ rect ( ω Δ + n ) exp ( i ω τ ) ] × exp ( - i ω τ ) .
X 2 τ = [ 2 C 1 2 ( 1 - S 1 2 ) + ( C 1 2 + C 1 * 2 ) ( S 2 - S 1 2 ) ] / 1 - C 0 2 ,
S n = p = - N / 2 + n N / 2 r j ( p ) r j ( p ) / N .
X 2 τ = [ S 1 ( 1 - S 1 ) ( C 1 + C 1 * ) 2 - ( 1 - S 1 ) ( C 1 - C 1 * ) 2 ] / 1 - C 0 2 .
E 1 ( j , ω ) = 2 f 0 E 0 ( ω 0 ) b 2 ( ω ) R j ( ω ) [ W ( ω ) ] 3 / 2 × D ( ω ) i = 1 N R i * ( ω ) S i exp ( i ω τ ) k ( ω ) k ( ω ) × exp { - ( 1 - i H ^ ) [ b 1 ( ω ) D ( ω ) ] } exp ( i ω τ read ) .
E 1 ( j , ω ) = 2 f 0 E 0 ( ω 0 ) b 2 R j ( ω ) D ( ω ) i = 1 N R i * ( ω ) S i × exp [ i ω ( τ + τ read ) ] exp { - ( 1 - i H ^ ) × [ b 1 ( ω ) D ( ω ) ] } exp ( - 2 Γ τ ) .
k ( ω ) = α DW k ( ω ) + ( 1 - α DW ) k p ( ω ) ,
k p ( ω ) = [ ω Γ p 2 exp ( - ω Γ p ) ω 0 0 ω > 0 ,
Δ E ( r , ω ) + ω 2 c 2 ( r , ω ) E ( r , ω ) = 0.
( ω ) = 0 [ 1 + α 0 π k 0 n ( ω ) d ω ω - ω - i Γ ] ,
( ω ) = 0 { 1 + i α 0 k 0 ( 1 - i H ^ ) [ n ( ω ) k ( ω ) ] } .
( r , ω ) = 0 { 1 + i 3 α 0 k 0 ( 1 - i H ^ ) [ n ( r , ω , ν ) cos 2 ν k ( ω ) ] } .
n ( r , ω , ν ) = n ( ω ) [ 1 - G ( z , ω , ν ) exp ( i K x ) - c . c . ] × exp [ - B ( z , ω , ν ) ] ,
G ( z , ω , ν ) = f 0 cos 2 ( ν ) i = 1 N [ R i * ( z , ω ) S i ( z , ω ) exp ( i ω τ ) ] k ( ω ) , B ( z , ω , ν ) = f 0 cos 2 ( ν ) i = 1 N [ R i ( z , ω ) 2 + S i ( z , ω ) 2 ] k ( ω ) .
2 z E 1 ( z , ω ) + 3 α 0 ( 1 - i H ^ ) [ n 0 ( z , ω , ν ) cos 2 ν k ( ω ) ] × E 1 ( z , ω ) = - 3 α 0 ( 1 - i H ^ ) [ n 1 ( z , ω , ν ) cos 2 ν k ( ω ) ] E 0 ( z , ω ) , 2 z E 0 ( z , ω ) + 3 α 0 ( 1 - i H ^ ) [ n 0 ( z , ω , ν ) cos 2 ν k ( ω ) ] E 0 ( z , ω ) = 0 ,
E ( r , ω ) = exp ( i k 0 z ) p E p ( z , ω ) exp ( i p K x ) ,
E p ( 0 , ω ) = δ 0 p E ( 0 , ω )
n 0 ( z , ω , ν ) = n ( ω ) exp [ - B ( z , ω , ν ) ] , n 1 ( z , ω , ν ) = - n 0 ( z , ω , ν ) G ( z , ω , ν ) .
E 1 ( j , ω ) = ( 3 / 2 ) α 0 R j ( ω ) E 0 ( ω ) ( 1 - i H ^ ) × [ 0 L d z G ( z , ω , ν ) n 0 ( z , ω , ν ) cos 2 ν k ( ω ) ] × exp { - ( 3 / 2 α 0 ( - i H ^ ) ) × [ 0 L d z n 0 ( z , ω , ν ) cos 2 ν ] } exp ( i ω τ read ) ,
B 0 ( ω , ν ) = rect ( ω - ω 0 N Δ ) cos 2 ν N f 0 W ( ω ) ( 1 + α ) ,
E 1 ( j , ω ) = f 0 E 0 ( ω 0 ) b 2 ( ω ) R j [ W ( ω ) ] 1 / 2 ( 1 - i H ^ ) × { W ( ω ) D ( ω ) i = 1 N S i [ R i * ( ω ) exp ( i ω τ ) ] k ( ω ) k ( ω ) } exp { - ( 1 - i H ^ ) [ b 1 ( ω ) D ( ω ) ] } × exp ( i ω τ read ) ,

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