Abstract

We derive a method for predicting the efficiency of stimulated photon echoes in optically thick media. Analytic solutions to the Maxwell–Bloch equations are derived for multiple pulse sequences in which the pulses are either temporally brief or have a low pulse area. This method retains the temporal and spatial information of the recalled waveform, and it provides a nonintensive computational method for determining the recall efficiency of stimulated photon echoes. The case considered is when the medium is optically thick and the recall efficiency is high, but with minimal distortion of the data. The results are in good agreement with a full space, time, and frequency numerical integration of the Maxwell–Bloch equations.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Sonajalg and P. Saari, “Diffraction efficiency in space-and-time-domain holography,” J. Opt. Soc. Am. B 11, 372-379 (1994).
    [CrossRef]
  2. M. Tian, J. Zhang, I. Lorgere, J.-P. Galaup, and J.-L. Le Gouet, “Photoautomerization and broadband spectral holography,” J. Opt. Soc. Am. B 15, 2216-2225 (1998).
    [CrossRef]
  3. T. Wang, C. Greiner, J. R. Bochinski, and T. W. Mossberg, “Experimental study of photon-echo size in optically thick media,” Phys. Rev. A 60, R757-R760 (1999).
    [CrossRef]
  4. O. M. Fedotava, T. V. Smirnova, and O. K. Khasanov, “Non-collinear excitation of photon echo in optically dense media,” J. Luminescence 87-89, 880-882 (2000).
    [CrossRef]
  5. T. W. Mossberg, “Time-domain frequency-selective optical data storage,” Opt. Lett. 7, 77-79 (1982).
    [CrossRef] [PubMed]
  6. W. R. Babbitt and J. A. Bell, “Coherent transient continuous optical processor,” Appl. Opt. 33, 1538-1548 (1994).
    [CrossRef] [PubMed]
  7. W. R. Babbitt and T. W. Mossberg, “Spatial routing of optical beams through time-domain spatial-spectral filtering,” Opt. Lett. 20, 910-912 (1995).
    [CrossRef] [PubMed]
  8. K. D. Merkel and W. R. Babbitt, “Optical coherent-transient true-time-delay regenerator,” Opt. Lett. 21, 1102-1104 (1996).
    [CrossRef] [PubMed]
  9. L. Menager, I. Lorgere, J.-L. Le Gouet, D. Dolfi, and J.-P. Huignard, “Demonstration of a radio-frequency spectrum analyzer based on spectral hole burning,” Opt. Lett. 26, 1245-1247 (2001).
    [CrossRef]
  10. L. Menager, J.-L. Le Gouet, and I. Lorgere, “Time-to-frequency Fourier transformation with photon echoes,” Opt. Lett. 26, 1397-1399 (2001).
    [CrossRef]
  11. M. Azadeh, C. Sjaarda Cornish, W. R. Babbitt, and L. Tsang, “Efficient photon echoes in optically thick media,” Phys. Rev. A 57, 4662-4668 (1998).
    [CrossRef]
  12. C. Sjaarda Cornish, M. Azadeh, W. R. Babbitt, and L. Tsang, “Efficient waveform recall in absorbing media,” Proc. SPIE 3468, 174-181 (1998).
    [CrossRef]
  13. C. Sjaarda Cornish, W. R. Babbitt, and L. Tsang, “Demonstration of highly efficient photon echoes in an absorbing medium,” Opt. Lett. 25, 1276-1278 (2000).
    [CrossRef]
  14. D. Grischkowsky and J. A. Armstrong, “Self-defocusing of light by adiabatic following in rubidium vapor,” Phys. Rev. A 6, 1566-1570 (1972).
    [CrossRef]
  15. M. D. Crisp, “Adiabatic-following approximation,” Phys. Rev. A 8, 2128-2135 (1973).
    [CrossRef]
  16. L. Mandel and E. Wolf, Optical Coherences and Quantum Optics (Cambridge University, Cambridge, UK, 1995).
  17. S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457-485 (1969).
    [CrossRef]
  18. C. Sjaarda Cornish, “High efficient photon echo generation,” Ph.D. dissertation (Department of Electrical Engineering, University of Washington, Seattle, Washington, 2000).
  19. O. Kh. Khasanov, T. V. Smirnova, and O. M. Fedotova, “Accumulated photon echo properties in extended LaF3:Pr3+ crystals,” J. Luminescence 72–74, 851–853 (1997).
    [CrossRef]

2001

2000

C. Sjaarda Cornish, W. R. Babbitt, and L. Tsang, “Demonstration of highly efficient photon echoes in an absorbing medium,” Opt. Lett. 25, 1276-1278 (2000).
[CrossRef]

O. M. Fedotava, T. V. Smirnova, and O. K. Khasanov, “Non-collinear excitation of photon echo in optically dense media,” J. Luminescence 87-89, 880-882 (2000).
[CrossRef]

1999

T. Wang, C. Greiner, J. R. Bochinski, and T. W. Mossberg, “Experimental study of photon-echo size in optically thick media,” Phys. Rev. A 60, R757-R760 (1999).
[CrossRef]

1998

M. Tian, J. Zhang, I. Lorgere, J.-P. Galaup, and J.-L. Le Gouet, “Photoautomerization and broadband spectral holography,” J. Opt. Soc. Am. B 15, 2216-2225 (1998).
[CrossRef]

M. Azadeh, C. Sjaarda Cornish, W. R. Babbitt, and L. Tsang, “Efficient photon echoes in optically thick media,” Phys. Rev. A 57, 4662-4668 (1998).
[CrossRef]

C. Sjaarda Cornish, M. Azadeh, W. R. Babbitt, and L. Tsang, “Efficient waveform recall in absorbing media,” Proc. SPIE 3468, 174-181 (1998).
[CrossRef]

1997

O. Kh. Khasanov, T. V. Smirnova, and O. M. Fedotova, “Accumulated photon echo properties in extended LaF3:Pr3+ crystals,” J. Luminescence 72–74, 851–853 (1997).
[CrossRef]

1996

1995

1994

1982

1973

M. D. Crisp, “Adiabatic-following approximation,” Phys. Rev. A 8, 2128-2135 (1973).
[CrossRef]

1972

D. Grischkowsky and J. A. Armstrong, “Self-defocusing of light by adiabatic following in rubidium vapor,” Phys. Rev. A 6, 1566-1570 (1972).
[CrossRef]

1969

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457-485 (1969).
[CrossRef]

Armstrong, J. A.

D. Grischkowsky and J. A. Armstrong, “Self-defocusing of light by adiabatic following in rubidium vapor,” Phys. Rev. A 6, 1566-1570 (1972).
[CrossRef]

Azadeh, M.

C. Sjaarda Cornish, M. Azadeh, W. R. Babbitt, and L. Tsang, “Efficient waveform recall in absorbing media,” Proc. SPIE 3468, 174-181 (1998).
[CrossRef]

M. Azadeh, C. Sjaarda Cornish, W. R. Babbitt, and L. Tsang, “Efficient photon echoes in optically thick media,” Phys. Rev. A 57, 4662-4668 (1998).
[CrossRef]

Babbitt, W. R.

Bell, J. A.

Bochinski, J. R.

T. Wang, C. Greiner, J. R. Bochinski, and T. W. Mossberg, “Experimental study of photon-echo size in optically thick media,” Phys. Rev. A 60, R757-R760 (1999).
[CrossRef]

Cornish, C. Sjaarda

C. Sjaarda Cornish, W. R. Babbitt, and L. Tsang, “Demonstration of highly efficient photon echoes in an absorbing medium,” Opt. Lett. 25, 1276-1278 (2000).
[CrossRef]

M. Azadeh, C. Sjaarda Cornish, W. R. Babbitt, and L. Tsang, “Efficient photon echoes in optically thick media,” Phys. Rev. A 57, 4662-4668 (1998).
[CrossRef]

C. Sjaarda Cornish, M. Azadeh, W. R. Babbitt, and L. Tsang, “Efficient waveform recall in absorbing media,” Proc. SPIE 3468, 174-181 (1998).
[CrossRef]

Crisp, M. D.

M. D. Crisp, “Adiabatic-following approximation,” Phys. Rev. A 8, 2128-2135 (1973).
[CrossRef]

Dolfi, D.

Fedotava, O. M.

O. M. Fedotava, T. V. Smirnova, and O. K. Khasanov, “Non-collinear excitation of photon echo in optically dense media,” J. Luminescence 87-89, 880-882 (2000).
[CrossRef]

Fedotova, O. M.

O. Kh. Khasanov, T. V. Smirnova, and O. M. Fedotova, “Accumulated photon echo properties in extended LaF3:Pr3+ crystals,” J. Luminescence 72–74, 851–853 (1997).
[CrossRef]

Galaup, J.-P.

Greiner, C.

T. Wang, C. Greiner, J. R. Bochinski, and T. W. Mossberg, “Experimental study of photon-echo size in optically thick media,” Phys. Rev. A 60, R757-R760 (1999).
[CrossRef]

Grischkowsky, D.

D. Grischkowsky and J. A. Armstrong, “Self-defocusing of light by adiabatic following in rubidium vapor,” Phys. Rev. A 6, 1566-1570 (1972).
[CrossRef]

Hahn, E. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457-485 (1969).
[CrossRef]

Huignard, J.-P.

Khasanov, O. K.

O. M. Fedotava, T. V. Smirnova, and O. K. Khasanov, “Non-collinear excitation of photon echo in optically dense media,” J. Luminescence 87-89, 880-882 (2000).
[CrossRef]

Khasanov, O. Kh.

O. Kh. Khasanov, T. V. Smirnova, and O. M. Fedotova, “Accumulated photon echo properties in extended LaF3:Pr3+ crystals,” J. Luminescence 72–74, 851–853 (1997).
[CrossRef]

Le Gouet, J.-L.

Lorgere, I.

McCall, S. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457-485 (1969).
[CrossRef]

Menager, L.

Merkel, K. D.

Mossberg, T. W.

Saari, P.

Smirnova, T. V.

O. M. Fedotava, T. V. Smirnova, and O. K. Khasanov, “Non-collinear excitation of photon echo in optically dense media,” J. Luminescence 87-89, 880-882 (2000).
[CrossRef]

O. Kh. Khasanov, T. V. Smirnova, and O. M. Fedotova, “Accumulated photon echo properties in extended LaF3:Pr3+ crystals,” J. Luminescence 72–74, 851–853 (1997).
[CrossRef]

Sonajalg, H.

Tian, M.

Tsang, L.

C. Sjaarda Cornish, W. R. Babbitt, and L. Tsang, “Demonstration of highly efficient photon echoes in an absorbing medium,” Opt. Lett. 25, 1276-1278 (2000).
[CrossRef]

M. Azadeh, C. Sjaarda Cornish, W. R. Babbitt, and L. Tsang, “Efficient photon echoes in optically thick media,” Phys. Rev. A 57, 4662-4668 (1998).
[CrossRef]

C. Sjaarda Cornish, M. Azadeh, W. R. Babbitt, and L. Tsang, “Efficient waveform recall in absorbing media,” Proc. SPIE 3468, 174-181 (1998).
[CrossRef]

Wang, T.

T. Wang, C. Greiner, J. R. Bochinski, and T. W. Mossberg, “Experimental study of photon-echo size in optically thick media,” Phys. Rev. A 60, R757-R760 (1999).
[CrossRef]

Zhang, J.

Appl. Opt.

J. Luminescence

O. M. Fedotava, T. V. Smirnova, and O. K. Khasanov, “Non-collinear excitation of photon echo in optically dense media,” J. Luminescence 87-89, 880-882 (2000).
[CrossRef]

O. Kh. Khasanov, T. V. Smirnova, and O. M. Fedotova, “Accumulated photon echo properties in extended LaF3:Pr3+ crystals,” J. Luminescence 72–74, 851–853 (1997).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Rev.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457-485 (1969).
[CrossRef]

Phys. Rev. A

T. Wang, C. Greiner, J. R. Bochinski, and T. W. Mossberg, “Experimental study of photon-echo size in optically thick media,” Phys. Rev. A 60, R757-R760 (1999).
[CrossRef]

M. Azadeh, C. Sjaarda Cornish, W. R. Babbitt, and L. Tsang, “Efficient photon echoes in optically thick media,” Phys. Rev. A 57, 4662-4668 (1998).
[CrossRef]

D. Grischkowsky and J. A. Armstrong, “Self-defocusing of light by adiabatic following in rubidium vapor,” Phys. Rev. A 6, 1566-1570 (1972).
[CrossRef]

M. D. Crisp, “Adiabatic-following approximation,” Phys. Rev. A 8, 2128-2135 (1973).
[CrossRef]

Proc. SPIE

C. Sjaarda Cornish, M. Azadeh, W. R. Babbitt, and L. Tsang, “Efficient waveform recall in absorbing media,” Proc. SPIE 3468, 174-181 (1998).
[CrossRef]

Other

L. Mandel and E. Wolf, Optical Coherences and Quantum Optics (Cambridge University, Cambridge, UK, 1995).

C. Sjaarda Cornish, “High efficient photon echo generation,” Ph.D. dissertation (Department of Electrical Engineering, University of Washington, Seattle, Washington, 2000).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Pulse train for a stimulated photon echo. The first three pulses are input pulses. The fourth pulse is the stimulated echo emitted by the medium. Σi is the pulse envelope for the jth pulse, centered at η0j.

Fig. 2
Fig. 2

Arbitrary data pulse with known time content.

Fig. 3
Fig. 3

Brief pulse. Pulse duration is one-third the duration of the shortest data subpulse. The brief pulse has no time content for information storage.

Fig. 4
Fig. 4

Pulse train for a two-pulse photon echo. The time window for each pulse is noted.

Fig. 5
Fig. 5

Analytic solutions derived in this paper are compared with the numerical simulations obtained from a full space, time, and frequency integration of the Maxwell–Bloch equations. The echo power efficiency (log scale) is plotted versus absorption length. The analytic solutions (dotted curves) match the Maxwell–Bloch simulations (solid curves) to within a few percent for brief-pulse areas up to 0.6π and to within 15% for brief-pulse areas up to 0.7π.

Equations (86)

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

ρab(Δ, z, t)t=-iΔρab(Δ, z, t)+i2 Ω(z, t)[1-2ρaa(Δ, z, t)],
ρaa(Δ, z, t)t=-i2 Ω(z, t)[ρab(Δ, z, t)-ρba(Δ, z, t)],
E(z, t)=hμ Ω(z, t)cos(ω0t-kz),
r1=2 Re ρab(Δ, z, t),
r2=2 Im ρab(Δ, z, t),
r3=1-2ρaa(Δ, z, t),
r1(Δ, z, t)t=Δr2(Δ, z, t),
r2(Δ, z, t)t=-Δr1(Δ, z, t)+Ω(z, t)r3(Δ, z, t),
r3(Δ, z, t)t=-Ω(z, t)r2(Δ, z, t).
Ω(z, t)z=απ-g(Δ)Im ρab(Δ, z, t)dΔ,
ρab(Δ, z, t)=ρab(Δ, z, ti)exp[iΔ(ti-t)]+titdti(t-η)2exp[iΔ(t-t)]×[1-2ρaa(Δ, z, t)],
ρaa(Δ, z, t)=ρaa(Δ, z, ti)-titdti(t-η)2 [ρab(Δ, z, t)-ρba(Δ, z, t)].
ρab(Δ, z, t)=ρab(Δ, z, ti)exp(iΔti-iΔt)+[1-2ρaa(Δ, z, ti)]×titi2(z, t-η)exp[iΔ(t-t)]dt.
ρaa(Δ, z, t)=ρaa(Δ, z, ti)-Reiρab(Δ, z, ti)exp(iΔti)×tit(z, t-η)exp(-iΔt)dt.
ρab(Δ, z, tf)=cos2A(z)2ρab(Δ, z, ti)exp[iΔ(ti-tf)]+sin2A(z)2ρba(Δ, z, ti)exp(-iΔti+2iΔη)exp(-iΔtf)+i2 [1-2ρaa(Δ, z, ti)]×exp(iΔη-iΔtf)sin[A(z)],
ρaa(Δ, z, tf)=sin2A(z)2+ρaa(Δ, z, ti)cos[A(z)]+12sin[A(z)][iρba(Δ, z, ti)×exp(-iΔti+iΔη)-iρab(Δ, z, ti)×exp(iΔti-iΔη),
A(z)=titf(z, t-η)dt.
A(z)z=απ-dΔtitfdt 1Δt [Re ρab(Δ, z, t)],
A(z)z=απ-dΔ 1Δ [Re ρab(Δ, z, tf)-Re ρab(Δ, z, ti)].
A(z)z=απ-dΔ 1ΔRecos2A(z)2ρab(Δ, z, ti)×exp[iΔ(ti-tf)]+sin2A(z)2ρba(Δ, z, ti)×exp(-iΔti+2iΔη-iΔtf)+i2 [1-2ρaa(Δ, z, ti)]exp(iΔη-iΔtf)sin[A(z)]-Re ρab(Δ, z, ti).
ρaa(Δ, z, t)=ρaa(Δ, z, t1i)=1,
ρab(Δ, z, t)=-i2t1it1(t-η1)×exp(iΔt)exp(-iΔt)dt.
Ω1(z, t)z=-απIm-t1iti21(t-η1)exp(iΔt)×exp(-iΔt)dtdΔ.
Ω1(z, t)z=-α Imt1iti1(t-η1)δ(t-t)dt.
Ω1(z, t)z=-α21(t-η1),
Ω1(z, t)=01(t-η1)exp-αz2.
ρaa(Δ, z, t2i)=ρaa(Δ, z, t1f)=1.
ρab(Δ, z, t1f)=-i2exp(-iΔt1f)t1it1f1(t-η1)×exp(iΔt)dt.
˜1(Δ)=-1(t)exp(iΔt),
ρab(Δ, z, t2i)=ρab(Δ, z, t1f)=-i2exp[-iΔ(t1f-η1)]˜1(z, Δ).
ρab(Δ, z, t2f)=cos2A2(z)2ρab(Δ, z, t2i)×exp[iΔ(t2i-t2f)]+sin2A2(z)2ρba(Δ, z, t2i)×exp[-iΔ(t2i+t2f-η2)]+i2 [1-2ρaa(Δ, z, t2i)]×exp[iΔ(η2-t2f)]sin[A2(z)],
ρab(Δ, z, t2f)=exp(-iΔt2f)cos2A2(z)2×-i2exp(iΔη1)˜1(z, Δ)+sin2A2(z)2i2exp[-iΔ(η1-2η2+t2f)]˜1*(z, Δ)-i2exp[iΔ(η2-t2f)]sin[A2(z)],
ρaa(z, t2f)=cos2A2(z)2-12sin[A2(z)]×Re[exp[iΔ(η1-η2)]˜1(z, Δ)].
A2(z)z=απ-1ΔRe-i2cos2A2(z)2exp[-iΔ(t2f-η1)]˜1(z, Δ)+i2sin2A2(z)2×exp[-iΔ(t2f+η1-2η2)]˜1*(z, Δ)-i2sin[A2(z)]exp[iΔ(η2-t2f)]-Re-i2exp[-iΔ(t2i-η1)]˜1(z, Δ)dΔ.
A2(z)z=α2πsin[A2(z)]-1Δsin[(η2-t2f)Δ]dΔ.
A2(z)z=-α2sin[A2(z)],
ρab(Δ, z, t)=ρab(Δ, z, t3i)exp[iΔ(t3i-t)]+[1-2ρaa(z, t3i)]×t3iti23(t-η3)exp[iΔ(t-t)]dt,
ρab(Δ, z, t)=-i2exp[-iΔ(t-η1)]cos2A2(z)2˜1(Δ)+i2exp[-iΔ(t+η1-2η2)]×sin2A2(z)2˜1*(z, Δ)-i2exp[-iΔ(t-η2)]sin[A2(z)]-i2cos[A2(z)]t3it3(t-η3)×exp[iΔ(t-t)]dt.
Ω3(z, t)z=απIm--i2exp[-iΔ(t-η1)]×cos2A2(z)2˜1(z, Δ)+i2exp[-iΔ(t-2η2+η1)]×sin2A2(z)2˜1*(z, Δ)-i2exp[-iΔ(t-η2)]sin[A2(z)]-i2cos[A2(z)]t3it3(t-η3)×exp[iΔ(t-t)]dtdΔ.
Ω3(z, t)z=-α2cos[A2(z)]Ω3(z, t)+α sin2A2(z)21(2η2-η1-t).
ρab(Δ, z, t1f)=-i2exp[-iΔ(t1f-η1)]sin[A1(z)].
ρaa(Δ, z, t1f)=sin2A1(z)2+cos[A1(z)]=cos2A1(z)2.
A1(z)z=απsin[A1(z)]×Re--i2Δexp[iΔ(η1-t1f)]dΔ,
dA1(z)dz=-α2sin[A1(z)].
ρab(Δ, z, t)=-i2sin[A1(z)]exp[-iΔ(t-η1)]
-cos[A1(z)]exp(-iΔt)×t2iti22(t-η2)exp(iΔt)dt.
Ω2(z, t)z=απIm--i2sin[A1(z)]exp[-iΔ(t-η1)]-cos[A1(z)]exp(-iΔt)×t2itdti22(t-η2)exp(iΔt)dΔ.
Ω2(z, t)z=απIm{-iπ sin[A1(z)]δ(t-η1)}-απcos[A1(z)]×Imt2itiπ2(t-η2)δ(t-t)dt.
Ω2(z, t)z=-α2cos[A1(z)]Ω2(z, t).
Ω2(z, t)=02(t-η2)exp-αz2cos[A1(z)].
ρaa(Δ, z, t2f)=cos2A1(z)2-Resin[A1(z)]2exp[-iΔ(η2-η1)]˜2*(z, Δ),
˜2(z, Δ)=02(z=0, Δ)exp-αz2cos[A1(z)].
ρab(Δ, z, t3i)=0,
ρaa(Δ, z, t3i)=ρaa(Δ, z, t2f)=1-ρbb(Δ, z, t3i).
ρab(Δ, z, t3f)=i2-cos[A1(z)]+Resin[A1(z)]×exp[-iΔ(η2-η1)]˜2*(z, Δ)×sin[A3(z)]exp[iΔ(η3-t3f)].
ρaa(Δ, z, t3f)=sin2A3(z)2+cos[A3(z)]×cos2A1(z)2-Resin[A1(z)]2×exp[-iΔ(η2-η1)]¯2*(z, Δ).
A3(z)z=απ-1ΔRei2 [1-2ρaa(Δ, z, t3i)]exp[iΔ(η3-t3f)]sin[A3(z)]dΔ.
A3(z)z=απ-1ΔRei2exp(iΔη3-iΔt3f)sin[A3(z)]×1-2 cos2A1(z)2+Resin[A1(z)]exp[-iΔ(η2-η1)]˜2*(z, Δ)dΔ.
A3(z)z=-α2cos[A1(z)]sin[A3(z)].
ρab(Δ, z, t)=exp(-iΔt) i2sin[A3(z)]exp(iΔη3)×-cos[A1(z)]Re{sin[A1(z)]×exp[-iΔ(η2-η1)]˜2*(z, Δ)}+1-2 sin2A1(z)2-2 cos[A3(z)]×cos2A1(z)2-Resin[A1(z)]2exp[-iΔ(η2-η1)]˜2*(z, Δ)t4iti24(t-η4)exp(-iΔt+iΔt)dt.
ρab(Δ, z, t)=i2exp[-Δ(t-η3)]sin[A3(z)]×-cos[A1(z)]+sin[A1(z)]2exp[-iΔ(η2-η1)]ט2*(z, Δ)+sin[A1(z)]2×exp[iΔ(η2-η1)]˜2(z, Δ)-cos[A1(z)]cos[A3(z)]×t4iti24(t-η4)×exp[-iΔ(t-t)]dt.
Ω4(z, t)z=α Im(iδ(t-η3)sin[A3(z)]{-cos[A1(z)]})+α2sin[A1(z)]sin[A3(z)]×Im[i2(-t+η3-η2+η1)]+α2sin[A1(z)]sin[A3(z)]×Im[i2(t-η3-η2+η1)]
-α cos[A1(z)]cos[A3(z)]Imit4it4(t-η4)
×δ(t-t)dt.
Ω4(z, t)z=α2sin[A1(z)]sin[A3(z)]×2(t-η3-η2+η1)-α2cos[A1(z)]cos[A3(z)]4(t-η4).
Ω4(z, t)z=α2sin[A1(z)]sin[A3(z)]×20(t-η3-η2+η1)×exp-αz2cos[A1(z)]-α2×cos[A1(z)]cos[A3(z)]Ω4(z, t),
ρab(Δ, z, t)=ρab(Δ, z, t)exp(iΔt).
ρab(Δ, z, t)t=i2(t)exp(iΔη)×exp(iΔt)[1-2ρaa(t)],
ρaa(Δ, z, t)t=-i2(t)×[ρab(Δ, z, t)exp(-iΔt)exp(-iΔη)-ρba(Δ, z, t)exp(iΔt)exp(iΔη)].
ρab(Δ, z, t)t=i2(t)exp(iΔη)[1-2ρaa(Δ, z, t)],
ρaa(Δ, z, t)t=-i2(t)[ρab(Δ, z, t)exp(-iΔη)-ρba(Δ, z, t)exp(iΔη)].
w(Δ, z, t)=1-2ρaa(Δ, z, t).
θ(z, t)=ti-ηt-ηdt(z, t)
ρab(Δ, z, t)θ=i2exp(iΔη)w,
-w(Δ, z, t)θ=-i[ρab(Δ, z, t)exp(-iΔη)-ρba(Δ, z, t)exp(iΔη)].
θ {2 Re[ρab(Δ, z, t)exp(-iΔη)]}=0,
θ {2 Im[ρab(Δ, z, t)exp(-iΔη)]}=w,
-w(Δ, z, t)θ=2 Im[ρab(Δ, z, t)exp(-iΔη)].
w(Δ, z, t)=[1-2ρaa(Δ, z, ti)]cos(θ)-2 Im[ρab(Δ, z, ti)exp(-iΔη)]sin(θ).
ρaa(Δ, z, t)=ρbb(Δ, z, ti)sin2θ2+ρaa(ti)cos2θ2+12sin(θ)[iρba(Δ, z, ti)exp(iΔη)-iρab(Δ, z, ti)exp(-iΔη)].
2 Im[ρab(Δ, z, t)exp(-iΔη)]
=[1-2ρaa(Δ, z, ti)]sin(θ)
+2 Im[ρab(Δ, z, ti)exp(-iΔη)]cos(θ).
θIm2ρab(Δ, z, t)iexp(-iΔη)=0,
ρab(Δ, z, t)exp(-iΔη)i+ρba(Δ, z, t)iexp(iΔη)=ρab(Δ, z, ti)exp(-iΔη)i+ρba(Δ, z, ti)iexp(iΔη).
ρab(Δ, z, t)=cos2θ2ρab(Δ, z, ti)+sin2θ2ρba(Δ, z, ti)exp(2iΔη)+i2 [1-2ρaa(Δ, z, ti)]exp(iΔη)sin(θ).

Metrics