Abstract

We present a plane-wave analysis of a recently demonstrated self-pumped phase conjugator. This device uses four-wave mixing to produce the phase-conjugate replica of an incident optical wave. All the waves are derived from the single incident wave: there are no externally supplied pumping beams. We consider the case of four-wave mixing in two interaction regions coupled by simple reflection. We calculate the phase-conjugate reflectivity as a function of coupling strength, taking into account imperfect coupling between the two interaction regions, and show that there is a threshold coupling strength below which the reflectivity is zero and above which the reflectivity is multiple valued. We also compute the coupling strength per unit length for a photorefractive crystal of barium titanate.

© 1983 Optical Society of America

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  1. A recent review of phase conjugation is featured in Opt. Electron. 21, 155–283 (1982).
  2. A. Yariv, “Four-wave nonlinear optical mixing as real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
    [Crossref]
  3. J. P. Huignard, J. P. Herriau, P. Aubourg, and E. Spitz, “Phase-conjugate wave-front generation via real-time holography in Bi12SiO20crystals,” Opt. Lett. 4, 21–23 (1979); J. Feinberg and R. W. Hellwarth, “Phase-conjugate mirror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980); erratum 6, 257 (1981).
    [Crossref]
  4. S. M. Jensen and R. W. Hellwarth, “Generation of time-reversed waves by nonlinear refraction in a waveguide,” Appl. Phys. Lett. 33, 404–405 (1978).
    [Crossref]
  5. J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
    [Crossref]
  6. M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “A passive (self-pumped) phase-conjugate mirror: a theoretical and experimental investigation,” Appl. Phys. Lett. 41, 689–691 (1982).
    [Crossref]
  7. J. Feinberg, “Self-pumped, continuous-wave phase-conjugator using internal reflection,” Opt. Lett. 7, 486–488 (1982).
    [Crossref] [PubMed]
  8. R. W. Hellwarth, “Theory of phase-conjugation by four-wave mixing in a waveguide,” IEEE J. Quantum Electron. QE-15, 101–109 (1979).
    [Crossref]
  9. M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, “Exact solution of a nonlinear model of four-wave mixing and phase conjugation,” Opt. Lett. 7, 313–315 (1982).
    [Crossref] [PubMed]
  10. B. Fischer, M. Cronin-Golomb, J. O. White, and A. Yariv, “Amplified reflection, transmission, and self-oscillation in real-time holography,” Opt. Lett. 6, 519–521 (1981).
    [Crossref] [PubMed]
  11. In Ref. 10 the coupled wave equations are derived with the assumption that the charges involved in the photorefractive effect (and consequently the sign of the coupling parameter γ) are negative. Since these charges are positive in barium titanate,13 we consistently take γ to have the opposite sign.
  12. The coupling strength γ is generally complex; in barium titanate γ is real in the absence of any uniform dc electric field (either applied or intrinsic).13 We take γ to be real here and will consider the more-general case in a future publication.
  13. J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); erratum 52, 537 (1981).
    [Crossref]
  14. J. Feinberg, “Asymmetric self-defocusing of an optical beam from the photorefractive effect,” J. Opt. Soc. Am. 72, 46–51 (1982).
    [Crossref]
  15. S. H. Wemple, D. Didomenico, and I. Camlibel, “Dielectric optical properties of melt-grown BaTiO3,” J. Phys. Chem. Solids 29, 1797–1806 (1968).
    [Crossref]
  16. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).
  17. In addition to the threshold in the coupling strength, there is a small intensity threshold (of the order of 10−2W/cm2). This is because the input wave intensity must be high enough that the grating-formation rate (which is proportional to the intensity) exceeds the dark leakage rate.

1982 (6)

A recent review of phase conjugation is featured in Opt. Electron. 21, 155–283 (1982).

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[Crossref]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “A passive (self-pumped) phase-conjugate mirror: a theoretical and experimental investigation,” Appl. Phys. Lett. 41, 689–691 (1982).
[Crossref]

J. Feinberg, “Self-pumped, continuous-wave phase-conjugator using internal reflection,” Opt. Lett. 7, 486–488 (1982).
[Crossref] [PubMed]

M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, “Exact solution of a nonlinear model of four-wave mixing and phase conjugation,” Opt. Lett. 7, 313–315 (1982).
[Crossref] [PubMed]

J. Feinberg, “Asymmetric self-defocusing of an optical beam from the photorefractive effect,” J. Opt. Soc. Am. 72, 46–51 (1982).
[Crossref]

1981 (1)

1980 (1)

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); erratum 52, 537 (1981).
[Crossref]

1979 (2)

1978 (2)

S. M. Jensen and R. W. Hellwarth, “Generation of time-reversed waves by nonlinear refraction in a waveguide,” Appl. Phys. Lett. 33, 404–405 (1978).
[Crossref]

A. Yariv, “Four-wave nonlinear optical mixing as real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[Crossref]

1968 (1)

S. H. Wemple, D. Didomenico, and I. Camlibel, “Dielectric optical properties of melt-grown BaTiO3,” J. Phys. Chem. Solids 29, 1797–1806 (1968).
[Crossref]

Aubourg, P.

Camlibel, I.

S. H. Wemple, D. Didomenico, and I. Camlibel, “Dielectric optical properties of melt-grown BaTiO3,” J. Phys. Chem. Solids 29, 1797–1806 (1968).
[Crossref]

Cronin-Golomb, M.

M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, “Exact solution of a nonlinear model of four-wave mixing and phase conjugation,” Opt. Lett. 7, 313–315 (1982).
[Crossref] [PubMed]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “A passive (self-pumped) phase-conjugate mirror: a theoretical and experimental investigation,” Appl. Phys. Lett. 41, 689–691 (1982).
[Crossref]

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[Crossref]

B. Fischer, M. Cronin-Golomb, J. O. White, and A. Yariv, “Amplified reflection, transmission, and self-oscillation in real-time holography,” Opt. Lett. 6, 519–521 (1981).
[Crossref] [PubMed]

Didomenico, D.

S. H. Wemple, D. Didomenico, and I. Camlibel, “Dielectric optical properties of melt-grown BaTiO3,” J. Phys. Chem. Solids 29, 1797–1806 (1968).
[Crossref]

Feinberg, J.

J. Feinberg, “Asymmetric self-defocusing of an optical beam from the photorefractive effect,” J. Opt. Soc. Am. 72, 46–51 (1982).
[Crossref]

J. Feinberg, “Self-pumped, continuous-wave phase-conjugator using internal reflection,” Opt. Lett. 7, 486–488 (1982).
[Crossref] [PubMed]

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); erratum 52, 537 (1981).
[Crossref]

Fischer, B.

M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, “Exact solution of a nonlinear model of four-wave mixing and phase conjugation,” Opt. Lett. 7, 313–315 (1982).
[Crossref] [PubMed]

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[Crossref]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “A passive (self-pumped) phase-conjugate mirror: a theoretical and experimental investigation,” Appl. Phys. Lett. 41, 689–691 (1982).
[Crossref]

B. Fischer, M. Cronin-Golomb, J. O. White, and A. Yariv, “Amplified reflection, transmission, and self-oscillation in real-time holography,” Opt. Lett. 6, 519–521 (1981).
[Crossref] [PubMed]

Heiman, D.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); erratum 52, 537 (1981).
[Crossref]

Hellwarth, R. W.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); erratum 52, 537 (1981).
[Crossref]

R. W. Hellwarth, “Theory of phase-conjugation by four-wave mixing in a waveguide,” IEEE J. Quantum Electron. QE-15, 101–109 (1979).
[Crossref]

S. M. Jensen and R. W. Hellwarth, “Generation of time-reversed waves by nonlinear refraction in a waveguide,” Appl. Phys. Lett. 33, 404–405 (1978).
[Crossref]

Herriau, J. P.

Huignard, J. P.

Jensen, S. M.

S. M. Jensen and R. W. Hellwarth, “Generation of time-reversed waves by nonlinear refraction in a waveguide,” Appl. Phys. Lett. 33, 404–405 (1978).
[Crossref]

Spitz, E.

Tanguay, A. R.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); erratum 52, 537 (1981).
[Crossref]

Wemple, S. H.

S. H. Wemple, D. Didomenico, and I. Camlibel, “Dielectric optical properties of melt-grown BaTiO3,” J. Phys. Chem. Solids 29, 1797–1806 (1968).
[Crossref]

White, J. O.

M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, “Exact solution of a nonlinear model of four-wave mixing and phase conjugation,” Opt. Lett. 7, 313–315 (1982).
[Crossref] [PubMed]

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[Crossref]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “A passive (self-pumped) phase-conjugate mirror: a theoretical and experimental investigation,” Appl. Phys. Lett. 41, 689–691 (1982).
[Crossref]

B. Fischer, M. Cronin-Golomb, J. O. White, and A. Yariv, “Amplified reflection, transmission, and self-oscillation in real-time holography,” Opt. Lett. 6, 519–521 (1981).
[Crossref] [PubMed]

Yariv, A.

M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, “Exact solution of a nonlinear model of four-wave mixing and phase conjugation,” Opt. Lett. 7, 313–315 (1982).
[Crossref] [PubMed]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “A passive (self-pumped) phase-conjugate mirror: a theoretical and experimental investigation,” Appl. Phys. Lett. 41, 689–691 (1982).
[Crossref]

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[Crossref]

B. Fischer, M. Cronin-Golomb, J. O. White, and A. Yariv, “Amplified reflection, transmission, and self-oscillation in real-time holography,” Opt. Lett. 6, 519–521 (1981).
[Crossref] [PubMed]

A. Yariv, “Four-wave nonlinear optical mixing as real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[Crossref]

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).

Appl. Phys. Lett. (3)

S. M. Jensen and R. W. Hellwarth, “Generation of time-reversed waves by nonlinear refraction in a waveguide,” Appl. Phys. Lett. 33, 404–405 (1978).
[Crossref]

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[Crossref]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “A passive (self-pumped) phase-conjugate mirror: a theoretical and experimental investigation,” Appl. Phys. Lett. 41, 689–691 (1982).
[Crossref]

IEEE J. Quantum Electron. (2)

R. W. Hellwarth, “Theory of phase-conjugation by four-wave mixing in a waveguide,” IEEE J. Quantum Electron. QE-15, 101–109 (1979).
[Crossref]

A. Yariv, “Four-wave nonlinear optical mixing as real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[Crossref]

J. Appl. Phys. (1)

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); erratum 52, 537 (1981).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. Chem. Solids (1)

S. H. Wemple, D. Didomenico, and I. Camlibel, “Dielectric optical properties of melt-grown BaTiO3,” J. Phys. Chem. Solids 29, 1797–1806 (1968).
[Crossref]

Opt. Electron. (1)

A recent review of phase conjugation is featured in Opt. Electron. 21, 155–283 (1982).

Opt. Lett. (4)

Other (4)

In Ref. 10 the coupled wave equations are derived with the assumption that the charges involved in the photorefractive effect (and consequently the sign of the coupling parameter γ) are negative. Since these charges are positive in barium titanate,13 we consistently take γ to have the opposite sign.

The coupling strength γ is generally complex; in barium titanate γ is real in the absence of any uniform dc electric field (either applied or intrinsic).13 We take γ to be real here and will consider the more-general case in a future publication.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).

In addition to the threshold in the coupling strength, there is a small intensity threshold (of the order of 10−2W/cm2). This is because the input wave intensity must be high enough that the grating-formation rate (which is proportional to the intensity) exceeds the dark leakage rate.

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Figures (8)

Fig. 1
Fig. 1

Pictorial summary of optical phase-conjugation techniques. The phase-conjugate output is represented by a wavy arrow. From top to bottom: In stimulated scattering, a strong monochromatic wave incident upon a suitable medium provides exponential gain for light that is shifted by a transition energy. This scattered light forms the phase conjugate of the incident wave. Another method of producing a phase-conjugate wave is by four-wave mixing in a nonlinear medium, traditionally by providing two counterpropagating pumping beams in addition to the input wave. If the nonlinear medium is enclosed in an optical waveguide, only one pumping beam is necessary since the input wave can also act as the second pumping beam. Two devices in which the pumping beams are internally generated have been demonstrated. In one, the pumping beams are supported in a resonant cavity consisting of external mirrors; in another, the pumping beams are fed back by internal reflection. The last phase conjugator is described in more detail in the text.

Fig. 2
Fig. 2

Two self-pumped phase conjugators having (a) one interaction region or (b) two interaction regions. The image-bearing beam is shown entering from the top, and the direction of its phase-conjugate wave is shown by the dashed arrow. The self-generated pumping beams are reflected by an external mirror M. For the case of one interaction region, the counterpropagating beams must match the radius of curvature of the mirror. In the case of two interaction regions, the counterpropagating pumping beams need not match the curvature of the mirror.

Fig. 3
Fig. 3

Four-wave mixing geometry. The coordinates defining the limits of an interaction region are indicated schematically. Beam 4 is the incident, image-bearing beam, and beam 3 is the phase-conjugate replica. Beams 1 and 2 are the pumping beams.

Fig. 4
Fig. 4

Phase conjugator with two interaction regions coupled by internal reflection. The single crystal of barium titanate and the direction of its c axis are shown. The pumping beams 1 and 1′ generated from the input wave 4 and beam 4′ are internally reflected from adjacent crystal faces near an edge to form the incoming pumping beams 2′ and 2, linking the two interaction regions. Each interaction region contributes to the phase-conjugate output wave 3. If α2 = α2′, the pumping beams align themselves at the angle α2 such that maximum beam-coupling strength for the input angle α1 is achieved in each region.

Fig. 5
Fig. 5

Maximum available coupling strength per millimeter versus input beam angle α1. The curve shown is for a crystal of barium titanate with a density of mobile charges N = 2 × 1016/cm3. The input beam incident at an angle α1 to the crystal c axis has extraordinary polarization. For a given input angle α1, the angle α2 of the pumping beams with the c axis determines the coupling per unit length.

Fig. 6
Fig. 6

Phase-conjugate reflectivity R versus coupling strength γl. In this figure, we have assumed that the coupling strength is the same in the two interaction regions (γl = γl′). L is the intensity lost by a pumping beam as it reflects from one interaction region into the other. Note the threshold in the coupling strength and the multiple-valuedness of the reflectivity.

Fig. 7
Fig. 7

Threshold coupling strength versus coupling loss L. For a coupling strength smaller than the threshold coupling strength, the phase-conjugate reflectivity is zero. The threshold coupling strength is seen to increase as the loss increases.

Fig. 8
Fig. 8

Phase-conjugate reflectivity at threshold versus reflectivity loss L.

Equations (35)

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d A 1 d z = γ I 0 ( A 1 A 4 * + A 2 * A 3 ) A 4 , d A 2 * d z = γ I 0 ( A 1 A 4 * + A 2 * A 3 ) A 3 * , d A 3 d z = - γ I 0 ( A 1 A 4 * + A 2 * A 3 ) A 2 , d A 4 * d z = - γ I 0 ( A 1 A 4 * + A 2 * A 3 ) A 1 * ,
d 1 = I 1 + I 4 d 2 = I 2 + I 3 , c = A 1 A 2 + A 3 A 4 .
A 1 A 2 * ( z ) = - [ ( Δ - r ) D e μ z - ( Δ + r ) e - μ z 2 c * ( D e μ z - e - μ z ) ] , A 3 A 4 * ( z ) = [ ( Δ - r ) E e μ z - ( Δ + r ) e - μ z 2 c * ( E e μ z - e - μ z ) ] ,
A 4 ( l 2 ) = 0             ( no input phase - conjugate wave ) ,
A 1 ( l 1 ) = 0 A 1 ( l 1 ) = 0 }             ( no intut pumping beams ) ,
A 2 ( l 2 ) = ( 1 - L ) 1 / 2 e i θ A 1 ( l 2 ) A 2 ( l 2 ) = ( 1 - L ) 1 / 2 e i θ A 1 ( l 2 ) }             ( reflection connecting the two interaction regions ) ,
A 3 ( l 2 ) = e i ϕ A 3 ( l 1 ) A 4 ( l 1 ) = e i ϕ A 4 ( l 2 ) }             ( continuity ) ,
c = 2 c ,
d 1 = d 1 + d 2 / ( 1 - L ) .
tanh ( γ l 2 ξ ) = ξ
[ ( 1 + ξ ) a ( r ) b ( r ) + ( 1 - ξ ) ( 1 - L ) ] [ ( 1 + ξ ) ( 1 - L ) b ( r ) a ( r ) + ( 1 - ξ ) ] = 4 [ r - tan ( γ l 2 r ) ] 2 ( 1 - r 2 ) tanh 2 ( γ l 2 r ) .
R | A 3 ( l 1 ) A 4 * ( l 1 ) | 2 = - a ( r ) b ( r ) 4 ( 1 - r 2 ) tanh 4 ( γ l 2 r ) ,
a ( r ) = r 2 [ 1 - tanh 2 ( γ l 2 r ) ] , b ( r ) = [ r - 2 tanh ( γ l 2 r ) ] 2 - r 2 tanh 2 ( γ l 2 r ) ,
γ = ω 2 n c r eff E cos ( α 1 - α 2 2 ) .
E = k B T q k g 1 + ( k g / k 0 ) 2 ê 1 · ê 2 * .
k 0 = ( N q 2 0 k B T ) 1 / 2 .
r eff = n o 4 r 13 sin ( α 1 + α 2 2 )
r eff = [ n o 4 r 13 cos α 1 cos α 2 + 2 n e 2 n o 2 r 42 cos 2 ( α 1 + α 2 2 ) + n e 4 r 33 sin α 1 sin α 2 ] sin ( α 1 + α 2 2 ) ,
A 1 ( z ) A 2 * ( z ) = - [ ( Δ - r ) D e μ z - ( Δ + r ) e - μ z 2 c * ( D e μ z - e - μ z ) ] , A 3 ( z ) A 4 * ( z ) = [ ( Δ - r ) E e μ z - ( Δ + r ) e - μ z 2 c * ( E e μ z - e - μ z ) ] , A 1 ( z ) A 2 * ( z ) = - [ ( Δ - r ) D e μ z - ( Δ + r ) e - μ z 2 c * ( D e μ z - e - μ z ) ] , A 3 ( z ) A 4 * ( z ) = [ ( Δ - r ) E e μ z - ( Δ + r ) e - μ z 2 c * ( E e μ z - e - μ z ) ] ,
A 1 ( z ) A 2 * ( z ) = 2 c tanh [ μ ( z - l 1 ) ] Δ tanh [ μ ( z - l 1 ) ] + r ,
A 1 ( z ) A 2 * ( z ) = 2 c tanh [ μ ( z - l 1 ) ] Δ tanh [ μ ( z - l 1 ) ] + r ,
A 3 ( z ) A 4 * ( z ) ( z ) = - 2 c tanh [ μ ( z - l 2 ) ] Δ tanh [ μ ( z - l 2 ) ] + r .
A 3 ( l 1 ) A 4 * ( l 1 ) = c d 1 = 2 c 1 - Δ ,
A 3 A 4 * ( z ) = 2 c { tanh [ μ ( z - l 1 ) ] - r } ( r 2 - Δ ) tanh [ μ ( z - l 1 ) ] - r ( 1 - Δ ) .
A 3 A 4 * ( l 1 ) = c d 1 = ( 2 - L ) c ( 1 - L ) ( 1 - Δ ) - 2 Δ ,
2 - L ( 1 - L ) ( 1 - Δ ) - 2 Δ = tanh ( μ l ) - Δ tanh ( μ l ) + r .
= 2 [ tanh ( μ l ) - r ] ( r 2 - Δ ) tanh ( μ l ) - r ( 1 - Δ ) ,
2 c 2 ( 1 - L ) tanh ( μ l ) Δ tanh ( μ l ) + r = Δ tanh ( μ l ) + r tanh ( μ l ) ,
μ = γ r 2 I 0
tanh ( γ l 2 ξ ) = ξ ,
ξ r I 0 = ( 2 - L ) ( Δ 2 + c 2 ) 1 / 2 ( 1 - L ) ( 1 - Δ ) - L Δ .
Δ = - L ( 1 - Δ ) 4 ± ( 2 - L ) 4 [ ( 1 - r 2 ) ( 1 - Δ ) 1 + Δ ] 1 / 2 .
{ ( 1 + ξ ) ( 1 - L ) [ 1 - ( 1 - r 2 1 - Δ 2 ) 1 / 2 ] - ( 1 - ξ ) [ 1 + ( 1 - r 2 1 - Δ 2 ) 1 / 2 ] } × { ( 1 + ξ ) [ 1 + ( 1 - r 2 1 - Δ 2 ) 1 / 2 ] - ( 1 - ξ ) ( 1 - L ) × [ 1 - ( 1 - r 2 1 - Δ 2 ) ] 1 / 2 = 4 r 2 - Δ 2 ( 1 - Δ ) 2
Δ = [ r - tanh ( γ l 2 r ) ] 2 - ( 1 - r 2 ) tanh 2 ( γ l 2 r ) [ r - tanh ( γ l 2 r ) ] 2 + ( 1 - r 2 ) tanh 2 ( γ l 2 r ) .
R | A 3 A 4 * ( l 1 ) | 2 = | c d 1 | 2 = r 2 - Δ 2 ( 1 - Δ ) 2 .