Abstract

A general theory of two-wave mixing (TWM) in nonlinear media is presented. Assuming a gain (or absorption) grating and a refractive index grating are generated because of the nonlinear process in a nonlinear medium, the coupled-wave equations of TWM are derived based on Maxwell’s wave equation. The coupled-wave equations can be decoupled as coupled equations for the intensity and coupled equations for the phase of both beams, and these two sets of coupled equations can be solved analytically by using average total intensity in the medium instead of using the total intensity. Compared with the previous theory of TWM, the theory presented here is more general, and the applications of the theory to photorefractive materials, Kerr media, and semiconductor broad-area amplifiers are described.

© 2009 Optical Society of America

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  1. A. Marrakchi, J.-P. Huignard, and P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131-138 (1981).
    [CrossRef]
  2. J.-P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals,” Opt. Commun. 38, 249-254 (1981).
    [CrossRef]
  3. P. Yeh, “Contra-directional two-wave mixing in photorefractive media,” Opt. Commun. 45, 323-326 (1983).
    [CrossRef]
  4. P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484-519 (1989).
    [CrossRef]
  5. Y. Silberberg and I. Bar-Joseph, “Instabilities, self-oscillation, and chaos in a simple nonlinear optical interaction,” Phys. Rev. Lett. 48, 1541-1543 (1982).
    [CrossRef]
  6. Y. Silberberg and I. Bar-Joseph, “Optical instabilities in a nonlinear Kerr medium,” J. Opt. Soc. Am. B 1, 662-670 (1984).
    [CrossRef]
  7. D. Grandclément, G. Grynberg, and M. Pinard, “Observation of continuous-wave self-oscillation due to pressure-induced two-wave mixing in sodium,” Phys. Rev. Lett. 59, 40-43 (1987).
    [CrossRef] [PubMed]
  8. R. McGraw, “Light-scattering-noise limits to two-wave-mixing gain in Kerr media,” J. Opt. Soc. Am. B 9, 98-103 (1992).
    [CrossRef]
  9. M. Chi, S. B. Jensen, J.-P. Huignard, and P. M. Petersen, “Two-wave mixing in a broad-area semiconductor amplifier,” Opt. Express 14, 12373-12379 (2006).
    [CrossRef] [PubMed]
  10. M. Chi, J.-P. Huignard, and P. M. Petersen, “Nonlinear gain amplification due to two-wave mixing in a broad-area semiconductor amplifier with moving gratings,” Opt. Express 16, 5565-5571 (2008).
    [CrossRef] [PubMed]
  11. P.Günter and J.-P.Huignard, eds., Photorefractive Materials and Their Applications I (Springer-Verlag, 1988).
  12. P.Günter and J.-P.Huignard, eds., Photorefractive Materials and Their Applications II (Springer-Verlag, 1989).
  13. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in eletrooptic crystal. I. steady state,” Ferroelectrics 22, 949-960 (1979).
    [CrossRef]
  14. P. Tayebati and D. Mahgerefteh, “Theory of the photorefractive effect for Bi12SiO20 and BaTiO3 with shallow traps,” J. Opt. Soc. Am. B 8, 1053-1064 (1991).
    [CrossRef]
  15. M. H. Garrett, J. Y. Chang, H. P. Jenssen, and C. Warde, “High beam-coupling gain and deep- and shallow-trap effects in cobalt-doped barium titanate, BaTiO3:Co,” J. Opt. Soc. Am. B 9, 1407-1415 (1992).
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  16. K. Buse, S. Loheide, D. Sabbert, and E. Krätzig, “Photorefractive properties of tetragonal KTa0.52Nb0.48O3:Fe crystals and explanation by the three-valence charge-transport model,” J. Opt. Soc. Am. B 13, 2644-2651 (1996).
    [CrossRef]
  17. H. Nakajima and R. Frey, “Collinear nearly degenerate four-wave mixing in intracavity amplifying media,” IEEE J. Quantum Electron. 22, 1349-1354 (1986).
    [CrossRef]
  18. A. Brignon and J.-P. Huignard, “Two-wave mixing in Nd:YAG by gain saturation,” Opt. Lett. 18, 1639-1641 (1993).
    [CrossRef] [PubMed]
  19. S. Sternklar, Y. Glick, and S. Jackel, “Noise limitations of Brillouin two-beam coupling: theory and experiment,” J. Opt. Soc. Am. B 9, 391-397 (1992).
    [CrossRef]

2008

2006

1996

1993

1992

1991

1989

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484-519 (1989).
[CrossRef]

1987

D. Grandclément, G. Grynberg, and M. Pinard, “Observation of continuous-wave self-oscillation due to pressure-induced two-wave mixing in sodium,” Phys. Rev. Lett. 59, 40-43 (1987).
[CrossRef] [PubMed]

1986

H. Nakajima and R. Frey, “Collinear nearly degenerate four-wave mixing in intracavity amplifying media,” IEEE J. Quantum Electron. 22, 1349-1354 (1986).
[CrossRef]

1984

1983

P. Yeh, “Contra-directional two-wave mixing in photorefractive media,” Opt. Commun. 45, 323-326 (1983).
[CrossRef]

1982

Y. Silberberg and I. Bar-Joseph, “Instabilities, self-oscillation, and chaos in a simple nonlinear optical interaction,” Phys. Rev. Lett. 48, 1541-1543 (1982).
[CrossRef]

1981

A. Marrakchi, J.-P. Huignard, and P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131-138 (1981).
[CrossRef]

J.-P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals,” Opt. Commun. 38, 249-254 (1981).
[CrossRef]

1979

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in eletrooptic crystal. I. steady state,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

Bar-Joseph, I.

Y. Silberberg and I. Bar-Joseph, “Optical instabilities in a nonlinear Kerr medium,” J. Opt. Soc. Am. B 1, 662-670 (1984).
[CrossRef]

Y. Silberberg and I. Bar-Joseph, “Instabilities, self-oscillation, and chaos in a simple nonlinear optical interaction,” Phys. Rev. Lett. 48, 1541-1543 (1982).
[CrossRef]

Brignon, A.

Buse, K.

Chang, J. Y.

Chi, M.

Frey, R.

H. Nakajima and R. Frey, “Collinear nearly degenerate four-wave mixing in intracavity amplifying media,” IEEE J. Quantum Electron. 22, 1349-1354 (1986).
[CrossRef]

Garrett, M. H.

Glick, Y.

Grandclément, D.

D. Grandclément, G. Grynberg, and M. Pinard, “Observation of continuous-wave self-oscillation due to pressure-induced two-wave mixing in sodium,” Phys. Rev. Lett. 59, 40-43 (1987).
[CrossRef] [PubMed]

Grynberg, G.

D. Grandclément, G. Grynberg, and M. Pinard, “Observation of continuous-wave self-oscillation due to pressure-induced two-wave mixing in sodium,” Phys. Rev. Lett. 59, 40-43 (1987).
[CrossRef] [PubMed]

Günter, P.

A. Marrakchi, J.-P. Huignard, and P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131-138 (1981).
[CrossRef]

Huignard, J.-P.

Jackel, S.

Jensen, S. B.

Jenssen, H. P.

Krätzig, E.

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in eletrooptic crystal. I. steady state,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

Loheide, S.

Mahgerefteh, D.

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in eletrooptic crystal. I. steady state,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

Marrakchi, A.

J.-P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals,” Opt. Commun. 38, 249-254 (1981).
[CrossRef]

A. Marrakchi, J.-P. Huignard, and P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131-138 (1981).
[CrossRef]

McGraw, R.

Nakajima, H.

H. Nakajima and R. Frey, “Collinear nearly degenerate four-wave mixing in intracavity amplifying media,” IEEE J. Quantum Electron. 22, 1349-1354 (1986).
[CrossRef]

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in eletrooptic crystal. I. steady state,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

Petersen, P. M.

Pinard, M.

D. Grandclément, G. Grynberg, and M. Pinard, “Observation of continuous-wave self-oscillation due to pressure-induced two-wave mixing in sodium,” Phys. Rev. Lett. 59, 40-43 (1987).
[CrossRef] [PubMed]

Sabbert, D.

Silberberg, Y.

Y. Silberberg and I. Bar-Joseph, “Optical instabilities in a nonlinear Kerr medium,” J. Opt. Soc. Am. B 1, 662-670 (1984).
[CrossRef]

Y. Silberberg and I. Bar-Joseph, “Instabilities, self-oscillation, and chaos in a simple nonlinear optical interaction,” Phys. Rev. Lett. 48, 1541-1543 (1982).
[CrossRef]

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in eletrooptic crystal. I. steady state,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

Sternklar, S.

Tayebati, P.

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in eletrooptic crystal. I. steady state,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

Warde, C.

Yeh, P.

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484-519 (1989).
[CrossRef]

P. Yeh, “Contra-directional two-wave mixing in photorefractive media,” Opt. Commun. 45, 323-326 (1983).
[CrossRef]

Appl. Phys.

A. Marrakchi, J.-P. Huignard, and P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131-138 (1981).
[CrossRef]

Ferroelectrics

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in eletrooptic crystal. I. steady state,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

IEEE J. Quantum Electron.

H. Nakajima and R. Frey, “Collinear nearly degenerate four-wave mixing in intracavity amplifying media,” IEEE J. Quantum Electron. 22, 1349-1354 (1986).
[CrossRef]

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484-519 (1989).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

J.-P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals,” Opt. Commun. 38, 249-254 (1981).
[CrossRef]

P. Yeh, “Contra-directional two-wave mixing in photorefractive media,” Opt. Commun. 45, 323-326 (1983).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

Y. Silberberg and I. Bar-Joseph, “Instabilities, self-oscillation, and chaos in a simple nonlinear optical interaction,” Phys. Rev. Lett. 48, 1541-1543 (1982).
[CrossRef]

D. Grandclément, G. Grynberg, and M. Pinard, “Observation of continuous-wave self-oscillation due to pressure-induced two-wave mixing in sodium,” Phys. Rev. Lett. 59, 40-43 (1987).
[CrossRef] [PubMed]

Other

P.Günter and J.-P.Huignard, eds., Photorefractive Materials and Their Applications I (Springer-Verlag, 1988).

P.Günter and J.-P.Huignard, eds., Photorefractive Materials and Their Applications II (Springer-Verlag, 1989).

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Figures (5)

Fig. 1
Fig. 1

Configuration of the codirectional TWM in a nonlinear medium. K shows the direction of the grating vector.

Fig. 2
Fig. 2

Relative position of the interference pattern of the two coherent beams, the phase grating, and the gain (or absorption) grating induced in the nonlinear medium. θ is the phase difference between the phase grating and the interference pattern, and φ is the phase difference between the gain (or absorption) grating and the interference pattern.

Fig. 3
Fig. 3

TWM gain g TWM versus the parameter δ τ pr in photorefractive materials when m 1 .

Fig. 4
Fig. 4

TWM gain g TWM versus the parameter δ τ k in Kerr media when m 1 .

Fig. 5
Fig. 5

TWM gain g TWM versus the parameter δ τ s in semiconductor broad-area amplifiers when m 1 . D τ s = 4 μ m 2 , K = 0.5 μ m 1 , I 0 ¯ P s = 0.2 and β = 5 . Assuming the amplifier is operated with the current above the transparent current.

Tables (2)

Tables Icon

Table 1 The Contribution of Phase Grating to the Intensity and Phase Coupling with Different Phase Difference θ

Tables Icon

Table 2 The Contribution of Gain Grating to the Intensity and Phase Coupling with Different Phase Difference φ

Equations (54)

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n 2 n 0 2 + 2 n 0 Δ n i 2 n 0 ( α 0 + Δ α ) k 0 .
E = A 1 exp [ i ( K 1 r ω 1 t ) ] + A 2 exp [ i ( K 2 r ω 2 t ) ] ,
Δ n = n 1 + { Δ n 1 A 1 A 2 * exp [ i ( K x + δ t + θ ) ] + c.c. } ,
Δ α = α 1 + { Δ α 1 A 1 A 2 * exp [ i ( K x + δ t + φ ) ] + c.c. } ,
2 E 1 c 2 2 ( n 2 E ) t 2 = 0 .
cos θ 1 A 1 z ( α 0 + α 1 + i k 0 n 1 ) A 1 ( Δ α 1 e i φ + i k 0 Δ n 1 e i θ ) A 2 2 A 1 = 0 ,
cos θ 2 A 2 z ( α 0 + α 1 + i k 0 n 1 ) A 2 ( Δ α 1 e i φ + i k 0 Δ n 1 e i θ ) A 1 2 A 2 = 0 .
A 1 = I 1 exp ( i ψ 1 ) ,
A 2 = I 2 exp ( i ψ 2 ) ,
cos θ 1 I 1 z 2 ( α 0 + α 1 ) I 1 2 ( Δ α 1 cos φ k 0 Δ n 1 sin θ ) I 1 I 2 = 0 ,
cos θ 2 I 2 z 2 ( α 0 + α 1 ) I 2 2 ( Δ α 1 cos φ + k 0 Δ n 1 sin θ ) I 1 I 2 = 0 ,
cos θ 1 ψ 1 z k 0 n 1 ( k 0 Δ n 1 cos θ + Δ α 1 sin φ ) I 2 = 0 ,
cos θ 2 ψ 2 z k 0 n 1 ( k 0 Δ n 1 cos θ Δ α 1 sin φ ) I 1 = 0 .
I 1 ( z ) = ( m b a ) I 1 ( 0 ) exp ( α 2 z ) m b a exp { c [ exp ( α 2 z ) 1 ] α 2 } ,
I 2 ( z ) = ( m b a ) exp { c [ exp ( α 2 z ) 1 ] α 2 } I 2 ( 0 ) exp ( α 2 z ) m b a exp { c [ exp ( α 2 z ) 1 ] α 2 } ,
m = I 1 ( 0 ) I 2 ( 0 ) is the input intensity ratio ,
α 2 = 2 ( α 0 + α 1 ¯ ) cos θ 1 ,
a = 2 ( Δ α 1 ¯ cos φ ¯ k 0 Δ n 1 ¯ sin θ ¯ ) cos θ 1 ,
b = 2 ( Δ α 1 ¯ cos φ ¯ + k 0 Δ n 1 ¯ sin θ ¯ ) cos θ 2 ,
c = b I 1 ( 0 ) a I 2 ( 0 ) .
ψ 1 ( z ) = ψ 1 ( 0 ) + β z + d a ln m b a m b a exp { c [ exp ( α 2 z ) 1 ] α 2 } ,
ψ 2 ( z ) = ψ 2 ( 0 ) + β z + e b ln ( m b a ) exp { c [ exp ( α 2 z ) 1 ] α 2 } m b a exp { c [ exp ( α 2 z ) 1 ] α 2 } ,
β = k 0 n 1 ¯ cos θ 1 ,
d = ( k 0 Δ n 1 ¯ cos θ ¯ + Δ α 1 ¯ sin φ ¯ ) cos θ 1 ,
e = ( k 0 Δ n 1 ¯ cos θ ¯ Δ α 1 ¯ sin φ ¯ ) cos θ 2 .
g TWM = ln ( I 2 ( z 0 ) coherent pump I 2 ( z 0 ) noncoherent pump ) = ln ( m b a ) exp { c [ exp ( α 2 z ) 1 ] α 2 } m b a exp { c [ exp ( α 2 z ) 1 ] α 2 } b [ I 1 ( z 0 ) I 1 ( 0 ) ] α 2 b I 1 ¯ z 0 , ( m 1 ) ,
Δ n = i γ eo cos θ 1 2 k 0 ( 1 + i δ τ pr ) I 0 A 1 A 2 * exp [ i ( K x + δ t ) ] + c.c. ,
Δ α = α light induced + { γ abs 2 ( 1 + i δ τ pr ) I 0 A 1 A 2 * exp [ i ( K x + δ t ) ] + c.c. } ,
I 1 z = 2 ( α 0 + α 1 ) I 1 cos θ 1 + γ abs cos θ 1 γ eo [ 1 + ( δ τ pr ) 2 ] I 0 I 1 I 2 ,
I 2 z = 2 ( α 0 + α 1 ) I 2 cos θ 1 + γ abs cos θ 1 + γ eo [ 1 + ( δ τ pr ) 2 ] I 0 I 1 I 2 .
g TWM b I 1 ¯ z 0 = γ abs ¯ cos θ 1 + γ eo ¯ [ 1 + ( δ τ pr ) 2 ] I 0 ¯ I 1 ¯ z 0 γ abs ¯ cos θ 1 + γ eo ¯ [ 1 + ( δ τ pr ) 2 ] z 0 .
Δ n = n 2 ( I 0 + { A 1 A 2 * 1 + i δ τ k exp [ i ( K x + δ t ) ] + c.c. } ) ,
I 1 z = 2 α 0 cos θ 1 I 1 + 2 k 0 n 2 δ τ k [ 1 + ( δ τ k ) 2 ] cos θ 1 I 1 I 2 ,
I 2 z = 2 α 0 cos θ 1 I 2 2 k 0 n 2 δ τ k [ 1 + ( δ τ k ) 2 ] cos θ 1 I 1 I 2 .
I 1 ( z ) = ( 1 + m 1 ) I 1 ( 0 ) exp ( α 2 z ) 1 + m 1 exp { c [ exp ( α 2 z ) 1 ] α 2 } ,
I 2 ( z ) = ( 1 + m ) I 2 ( 0 ) exp ( α 2 z ) 1 + m exp { c [ exp ( α 2 z ) 1 ] α 2 } .
g TWM b I 1 ¯ z 0 = 2 k 0 n 2 δ τ k [ 1 + ( δ τ k ) 2 ] cos θ 1 I 1 ¯ z 0 .
ψ 1 z = k 0 n 1 cos θ 1 + k 0 n 2 [ 1 + ( δ τ k ) 2 ] cos θ 1 I 2 ,
ψ 2 z = k 0 n 1 cos θ 1 + k 0 n 2 [ 1 + ( δ τ k ) 2 ] cos θ 1 I 1 .
ψ 1 ( z ) = ψ 1 ( 0 ) + β z + 1 2 δ τ k ln 1 + m 1 1 + m 1 exp { c [ exp ( α 2 z ) 1 ] α 2 } ,
ψ 2 ( z ) = ψ 2 ( 0 ) + β z 1 2 δ τ k ln 1 + m 1 + m exp { c [ exp ( α 2 z ) 1 ] α 2 } .
Δ α = α s 1 + I 0 P s ( 1 + { A 1 A 2 * P s ( 1 + D τ s K 2 + I 0 P s ) 2 + ( δ τ s ) 2 exp [ i ( K x + δ t + π θ s ) ] + c.c. } ) ,
α 1 = α s ( 1 + I 0 P s ) , Δ α 1 = α s { ( I 0 + P s ) [ ( 1 + D τ s K 2 + I 0 P s ) 2 + ( δ τ s ) 2 ] 1 2 } .
Δ n = β s α s k 0 ( 1 + I 0 P s ) ( 1 + { A 1 A 2 * P s ( 1 + D τ s K 2 + I 0 P s ) 2 + ( δ τ s ) 2 exp [ i ( K x + δ t θ s ) ] + c.c. } ) ,
n 1 = β s α s [ k 0 ( 1 + I 0 P s ) ] , Δ n 1 = β s α s { k 0 ( I 0 + P s ) [ ( 1 + D τ s K 2 + I 0 P s ) 2 + ( δ τ s ) 2 ] 1 2 } ,
cos θ 1 A 1 z i [ α s ( β s + i ) 1 + I 0 P s ] ( 1 A 2 2 P s 1 + D τ s K 2 + I 0 P s + i δ τ s ) A 1 = 0 ,
cos θ 2 A 2 z i [ α s ( β s + i ) 1 + I 0 P s ] ( 1 A 1 2 P s 1 + D τ s K 2 + I 0 P s i δ τ s ) A 2 = 0 .
α 2 = 2 α s ( 1 + I 0 ¯ P s ) cos θ 1 ,
a = 2 α s ( P s + I 0 ¯ ) cos θ 1 1 + D τ s K 2 + I 0 ¯ P s β s δ τ s ( 1 + D τ s K 2 + I 0 ¯ P s ) 2 + ( δ τ s ) 2 ,
b = 2 α s ( P s + I 0 ¯ ) cos θ 2 1 + D τ s K 2 + I 0 ¯ P s + β s δ τ s ( 1 + D τ s K 2 + I 0 ¯ P s ) 2 + ( δ τ s ) 2 ,
β = β s α s ( 1 + I 0 ¯ P s ) cos θ 1 ,
d = α s ( P s + I 0 ¯ ) cos θ 1 ( 1 + D τ s K 2 + I 0 ¯ P s ) β s δ τ s ( 1 + D τ s K 2 + I 0 ¯ P s ) 2 + ( δ τ s ) 2 ,
e = α s ( P s + I 0 ¯ ) cos θ 2 ( 1 + D τ s K 2 + I 0 ¯ P s ) β s + δ τ s ( 1 + D τ s K 2 + I 0 ¯ P s ) 2 + ( δ τ s ) 2 .
g TWM b ( I 1 ( z 0 ) I 1 ( 0 ) ) α 2 = ( I 1 ( z 0 ) I 1 ( 0 ) ) P s 1 + D τ s K 2 + I 0 ¯ P s + δ τ s β s ( 1 + D τ s K 2 + I 0 ¯ P s ) 2 + ( δ τ s ) 2 .

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