Abstract

A theoretical investigation is made of third-order nonlinear processes, including Raman scattering and the parametric process in optical fibers. The analytical expressions are obtained for the gain coefficient and the required index mismatch for maximum gain, with the contributions from both the real and the imaginary parts of the third-order nonlinear susceptibility taken into account. This study shows that, as a result of the simultaneous occurrence of these two processes, the Raman–Stokes wave, the signal, and the idler (resulting from the parametric process) are allocated specific amounts of power at the end of a long fiber and that the amount of power in each wave depends on the phase-matching state of the parametric process. Our results also show that the threshold for a Raman–Stokes wave is greatly enhanced when the index mismatch among the waves for the parametric process reaches its optimum value.

© 1990 Optical Society of America

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References

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  1. R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
    [CrossRef]
  2. J. M. Dziedzic, R. H. Stolen, and A. Ashkin, “Optical Kerr effect in long fibers,” Appl. Opt. 20, 1403–1406 (1981).
    [CrossRef] [PubMed]
  3. R. H. Stolen and W. N. Leibolt, “Optical fiber modes using stimulated four photon mixing,” Appl. Opt. 15, 239–243 (1976).
    [CrossRef] [PubMed]
  4. R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. QE-11, 100–103 (1975).
    [CrossRef]
  5. C. Lin, W. A. Reed, A. D. Pearson, and H. T. Shang, “Phase matching in the minimum-chromatic-dispersion region of single-mode fibers for stimulated four-photon mixing,” Opt. Lett. 6, 493–495 (1981).
    [CrossRef] [PubMed]
  6. R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
    [CrossRef]
  7. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
    [CrossRef]
  8. Y. Chen and A. W. Snyder, “Four photon mixing: effect of pump depletion,” Opt. Lett. 14, 87–89 (1989).
    [CrossRef] [PubMed]
  9. R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2489–2494 (1972).
    [CrossRef] [PubMed]
  10. J. Anyeung and A. Yariv, “Spontaneous and stimulated Raman scattering in long low loss fibers,” IEEE J. Quantum Electron. QE-14, 347–352 (1978).
    [CrossRef]
  11. R. H. Stolen, C. Lee, and R. K. Jain, “Development of the stimulated Raman spectrum in single-mode silica fibers,” J. Opt. Soc. Am. B 1, 652–657 (1984).
    [CrossRef]
  12. J. T. Manassam, “Induced phase modulation of the stimulated Raman pulse in optical fibers,” Appl. Opt. 26, 3747–3752 (1987).
    [CrossRef]
  13. S. J. Garth and C. Pask, “Four-photon mixing and dispersion in single-mode fibers,” Opt. Lett. 11, 380–380 (1986).
    [CrossRef] [PubMed]
  14. A. Vatarescu, “Light conversion in nonlinear monomode optical fiber,” IEEE J. Lightwave Technol. LT-5, 1652–1659 (1987).
    [CrossRef]
  15. K. Kitayama, M. Ohashi, and T. K. Gustafson, “Sequence-frequency three-wave mixing emissions in a birefringent optical fiber,” IEEE J. Quantum Electron. QE-21, 1689–1700 (1985).
    [CrossRef]
  16. A. R. Chraplyvy, D. Marcuse, and P. S. Henry, “Carrier-induced phase noise in angle-modulated optical fiber systems,” IEEE J. Lightwave Technol. LT-2, 6–10 (1984).
    [CrossRef]
  17. T. Nakashima, S. Seikai, and M. Nakazawa, “Dependence of Raman gain on relative index difference for GeO2-doped single-mode fibers,” Opt. Lett. 10, 420–422 (1985).
    [CrossRef] [PubMed]
  18. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).
  19. R. H. Stolen, J. Botineau, and A. Ashkin, “Intensity discrimination of an optical pulse with birefringent fibers,” Opt. Lett. 7, 512–514 (1982).
    [CrossRef] [PubMed]

1989 (1)

1987 (2)

J. T. Manassam, “Induced phase modulation of the stimulated Raman pulse in optical fibers,” Appl. Opt. 26, 3747–3752 (1987).
[CrossRef]

A. Vatarescu, “Light conversion in nonlinear monomode optical fiber,” IEEE J. Lightwave Technol. LT-5, 1652–1659 (1987).
[CrossRef]

1986 (1)

1985 (2)

K. Kitayama, M. Ohashi, and T. K. Gustafson, “Sequence-frequency three-wave mixing emissions in a birefringent optical fiber,” IEEE J. Quantum Electron. QE-21, 1689–1700 (1985).
[CrossRef]

T. Nakashima, S. Seikai, and M. Nakazawa, “Dependence of Raman gain on relative index difference for GeO2-doped single-mode fibers,” Opt. Lett. 10, 420–422 (1985).
[CrossRef] [PubMed]

1984 (2)

A. R. Chraplyvy, D. Marcuse, and P. S. Henry, “Carrier-induced phase noise in angle-modulated optical fiber systems,” IEEE J. Lightwave Technol. LT-2, 6–10 (1984).
[CrossRef]

R. H. Stolen, C. Lee, and R. K. Jain, “Development of the stimulated Raman spectrum in single-mode silica fibers,” J. Opt. Soc. Am. B 1, 652–657 (1984).
[CrossRef]

1982 (2)

R. H. Stolen, J. Botineau, and A. Ashkin, “Intensity discrimination of an optical pulse with birefringent fibers,” Opt. Lett. 7, 512–514 (1982).
[CrossRef] [PubMed]

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

1981 (2)

1978 (2)

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
[CrossRef]

J. Anyeung and A. Yariv, “Spontaneous and stimulated Raman scattering in long low loss fibers,” IEEE J. Quantum Electron. QE-14, 347–352 (1978).
[CrossRef]

1976 (1)

1975 (1)

R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. QE-11, 100–103 (1975).
[CrossRef]

1973 (1)

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

1972 (1)

Anyeung, J.

J. Anyeung and A. Yariv, “Spontaneous and stimulated Raman scattering in long low loss fibers,” IEEE J. Quantum Electron. QE-14, 347–352 (1978).
[CrossRef]

Ashkin, A.

Bjorkholm, J. E.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

Bloembergen, N.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Botineau, J.

Chen, Y.

Chraplyvy, A. R.

A. R. Chraplyvy, D. Marcuse, and P. S. Henry, “Carrier-induced phase noise in angle-modulated optical fiber systems,” IEEE J. Lightwave Technol. LT-2, 6–10 (1984).
[CrossRef]

Dziedzic, J. M.

Garth, S. J.

Gustafson, T. K.

K. Kitayama, M. Ohashi, and T. K. Gustafson, “Sequence-frequency three-wave mixing emissions in a birefringent optical fiber,” IEEE J. Quantum Electron. QE-21, 1689–1700 (1985).
[CrossRef]

Henry, P. S.

A. R. Chraplyvy, D. Marcuse, and P. S. Henry, “Carrier-induced phase noise in angle-modulated optical fiber systems,” IEEE J. Lightwave Technol. LT-2, 6–10 (1984).
[CrossRef]

Ippen, E. P.

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

Jain, R. K.

Kitayama, K.

K. Kitayama, M. Ohashi, and T. K. Gustafson, “Sequence-frequency three-wave mixing emissions in a birefringent optical fiber,” IEEE J. Quantum Electron. QE-21, 1689–1700 (1985).
[CrossRef]

Lee, C.

Leibolt, W. N.

Lin, C.

Manassam, J. T.

Marcuse, D.

A. R. Chraplyvy, D. Marcuse, and P. S. Henry, “Carrier-induced phase noise in angle-modulated optical fiber systems,” IEEE J. Lightwave Technol. LT-2, 6–10 (1984).
[CrossRef]

Nakashima, T.

Nakazawa, M.

Ohashi, M.

K. Kitayama, M. Ohashi, and T. K. Gustafson, “Sequence-frequency three-wave mixing emissions in a birefringent optical fiber,” IEEE J. Quantum Electron. QE-21, 1689–1700 (1985).
[CrossRef]

Pask, C.

Pearson, A. D.

Reed, W. A.

Seikai, S.

Shang, H. T.

Smith, R. G.

Snyder, A. W.

Stolen, R. H.

R. H. Stolen, C. Lee, and R. K. Jain, “Development of the stimulated Raman spectrum in single-mode silica fibers,” J. Opt. Soc. Am. B 1, 652–657 (1984).
[CrossRef]

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

R. H. Stolen, J. Botineau, and A. Ashkin, “Intensity discrimination of an optical pulse with birefringent fibers,” Opt. Lett. 7, 512–514 (1982).
[CrossRef] [PubMed]

J. M. Dziedzic, R. H. Stolen, and A. Ashkin, “Optical Kerr effect in long fibers,” Appl. Opt. 20, 1403–1406 (1981).
[CrossRef] [PubMed]

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
[CrossRef]

R. H. Stolen and W. N. Leibolt, “Optical fiber modes using stimulated four photon mixing,” Appl. Opt. 15, 239–243 (1976).
[CrossRef] [PubMed]

R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. QE-11, 100–103 (1975).
[CrossRef]

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

Vatarescu, A.

A. Vatarescu, “Light conversion in nonlinear monomode optical fiber,” IEEE J. Lightwave Technol. LT-5, 1652–1659 (1987).
[CrossRef]

Yariv, A.

J. Anyeung and A. Yariv, “Spontaneous and stimulated Raman scattering in long low loss fibers,” IEEE J. Quantum Electron. QE-14, 347–352 (1978).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

IEEE J. Lightwave Technol. (2)

A. Vatarescu, “Light conversion in nonlinear monomode optical fiber,” IEEE J. Lightwave Technol. LT-5, 1652–1659 (1987).
[CrossRef]

A. R. Chraplyvy, D. Marcuse, and P. S. Henry, “Carrier-induced phase noise in angle-modulated optical fiber systems,” IEEE J. Lightwave Technol. LT-2, 6–10 (1984).
[CrossRef]

IEEE J. Quantum Electron. (4)

K. Kitayama, M. Ohashi, and T. K. Gustafson, “Sequence-frequency three-wave mixing emissions in a birefringent optical fiber,” IEEE J. Quantum Electron. QE-21, 1689–1700 (1985).
[CrossRef]

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

J. Anyeung and A. Yariv, “Spontaneous and stimulated Raman scattering in long low loss fibers,” IEEE J. Quantum Electron. QE-14, 347–352 (1978).
[CrossRef]

R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. QE-11, 100–103 (1975).
[CrossRef]

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

Opt. Lett. (5)

Phys. Rev. A (1)

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
[CrossRef]

Other (1)

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

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Figures (7)

Fig. 1
Fig. 1

Normalized gain coefficients versus normalized index mismatch Δβ/g0 for various ratios χ3′/χ3″ = 15, 20, and 25, corresponding to frequency shifts of Δf = 450, 350, and 300 cm−1, shown by solid, dashed, and dotted-dashed curves, respectively, with one pump mode; the long-short-dashed line is for χ3′/χ3″ = 0 and Δf = 450 cm−1. (a)δβ ≠ 0, (b)δβ = 0.

Fig. 2
Fig. 2

Normalized gain coefficients versus normalized index mismatch Δβ/g0 for various ratios χ3′/χ3″ = 6, 8, and 12, corresponding to frequency shifts Δf = 450, 350, and 250 cm−1, shown by solid, dashed, and dotted-dashed lines, respectively, with two pump modes P1 = P2. (a)δβ ≠ 0, (b)δβ = 0.

Fig. 3
Fig. 3

Normalized gain coefficients versus normalized index mismatch for various ratios P1/P2 with χ3′/χ3″ = 6, Δf = 450 cm−1, and the other parameters the same as in Fig. 2.

Fig. 4
Fig. 4

Normalized power distributions P ¯ i = P i / P p ( 0 ) and relative phase θ versus normalized distance Γ for P ¯ s ( 0 ) = 10 9, P ¯ 3 ( 0 ) = 10 3, and P ¯ 4 ( 0 ) = 10 9. (a) ΔS = 0.489 (phase-matching point), (b) ΔS = 0.3, (c) ΔS = 0, (d) ΔS = 0.489.

Fig. 5
Fig. 5

Normalized powers P ¯ i = P i / P p ( 0 ) versus normalized distance Γ for P ¯ s ( 0 ) = 10 3 and P ¯ 3 ( 0 ) = P ¯ 4 ( 0 ) = 10 9. (a) ΔS = 0.489 (phase-matching point), (b) ΔS = 0.3, (c) ΔS = 0.

Fig. 6
Fig. 6

Normalized powers P ¯ i = P i / P p ( 0 ) versus normalized distance Γ for P ¯ s ( 0 ) = P ¯ 3 ( 0 ) = P 4 ( 0 ) = 10 9. (a) ΔS = 0.489 (phase-matching point), (b) ΔS = 0.3, (c) ΔS = 0.

Fig. 7
Fig. 7

Normalized powers P ¯ i = P i / P p ( 0 ) versus normalized distance Γ for P ¯ s ( 0 ) = P ¯ 3 ( 0 ) = P ¯ 4 ( 0 ) = 10 3. (a) ΔS = 0.489 (phase-matching point), (b) ΔS = 0.3, (c) ΔS = 0.

Equations (45)

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E t = i A i ( z ) e i ( r , θ ) exp [ j ( β i z ω i t ) ] ,
1 2 e i × h i * d A = 1 ,
d A s d z + α s A s = j 3 2 0 I χ 3 ω s A 1 A 2 A a * exp ( j Δ β z ) + j 3 2 0 I s 1 χ 3 ω s A 1 A 1 * A s + j 3 2 0 I s 2 χ 3 ω s A 2 A 2 * A s ,
d A a d z + α a A a = j 3 2 0 I χ 3 * ω a A 1 A 2 A s * exp ( j Δ β z ) + j 3 2 0 I a 1 χ 3 * ω a A 1 A 1 * A a + j 3 2 0 I a 2 χ 3 * ω a A 2 A 2 * A a ,
A 1 = P 1 exp [ j ( δ β 1 z + ϕ 1 ) ] ,
A 2 = P 2 exp [ j ( δ β 2 z + ϕ 2 ) ] ,
δ β 1 = 3 / 2 0 χ 3 ω 1 ( I 11 P 1 + I 12 P 2 ) ,
δ β 2 = 3 / 2 0 χ 3 ω 2 ( I 12 P 1 + I 22 P 2 ) ,
Δ β = β 1 + β 2 β s β a ,
I i i = 1 2 e i 4 d A ,
I i m = e i 2 e m 2 d A ( i m ) ,
I = e 1 e 2 e s e a d A ,
χ 3 = χ 3 ( e ) + χ 3 ( R ) ,
A s ( z ) = exp [ ( j Δ β + δ β 2 α ) z ] { A s ( 0 ) ( cosh C z C 1 C sinh C z ) + A a * ( 0 ) j C 2 C ω s χ 3 sinh C z exp [ j ( ϕ 1 + ϕ 2 ) ] } ,
A a * ( z ) = exp [ ( j Δ β + δ β 2 + α ) z ] { A a * ( 0 ) ( cosh C z + C 1 C × sinh C z ) A s ( 0 ) j C 2 C ω a χ 3 sinh C z exp [ j ( ϕ 1 + ϕ 2 ) ] } ,
C = [ C 1 2 + C 2 2 ω s ω a χ 3 2 ] 1 / 2 , δ β = δ β 1 + δ β 2 , α = ½ [ α s + α a + j ( C 4 C 3 ) χ 3 ] , C 1 = ½ [ α s α a + j ( Δ β + δ β ) j ( C 3 + C 4 ) χ 3 ] , C 2 = 3 / 2 0 ( P 1 P 2 ) 1 / 2 I , C 3 = 3 / 2 0 ω s ( I s 1 P 1 + I s 2 P 2 ) , C 4 = 3 / 2 0 ω a ( I a 1 P 1 + I a 2 P 2 ) .
P s ( z ) = { B 1 P s ( 0 ) + B 2 P a ( 0 ) + B 3 [ P s ( 0 ) P a ( 0 ) ] 1 / 2 } × exp [ ( 2 g α s α a ) z ] ,
P a ( z ) = { B 4 P a ( 0 ) + B 5 P s ( 0 ) + B 6 [ P s ( 0 ) P a ( 0 ) ] 1 / 2 } × exp [ ( 2 g α a + α s ) z ] ,
g = 1 2 ( C 3 C 4 ) χ 3 + 1 2 [ ( T 1 2 + T 2 2 ) 1 / 2 + T 1 ] 1 / 2 ,
T 1 = ¼ { [ ( C 3 + C 4 ) χ 3 α s + α a ] 2 [ Δ β + δ β ( C 3 + C 4 ) χ 3 ] 2 + 4 ω s ω a C 2 2 ( χ 3 2 χ 3 2 ) } ,
T 2 = ½ [ ( C 3 + C 4 ) χ 3 a s + a a ] [ Δ β + δ β ( C 3 + C 4 ) χ 3 ] + 2 ω s ω a C 2 2 χ 3 χ 3 .
T 1 = ¼ { [ ( c 3 + c 4 ) 2 4 ω s ω a C 2 2 ] χ 3 2 Δ β 2 } ,
T 2 = ½ ( C 3 + C 4 ) χ 3 Δ β ,
g = g 0 + [ ( C 4 + C 3 ) 2 4 ω s ω a C 2 2 ] 2 8 ( C 3 + C 4 ) Δ β 2 χ 3 2 g 0 ,
Δ β 0 = ( C 3 + C 4 ) χ 3 δ β + [ ( C 3 + C 4 ) χ 3 α s + α a ] × χ 3 χ 3 = Δ β p 0 + [ ( C 3 + C 4 ) χ 3 α s + α a ] χ 3 χ 3 ,
g m = 1 2 ( C 3 C 4 ) χ 3 + 1 2 [ ( T 1 m 2 + T 2 m 2 ) 1 / 2 + T 1 m ] 1 / 2 ,
T 1 m = ¼ { [ ( C 3 + C 4 ) χ 3 α s + α a ] 2 [ 1 ( χ 3 χ 3 ) 2 ] + 4 ω s ω a C 2 2 ( χ 3 2 χ 3 2 ) } ,
T 2 m = ½ [ ( C 3 + C 4 ) χ 3 α s + α a ] 2 χ 3 / χ 3 + 2 ω s ω a χ 3 χ 3 ,
g = g 0 + 2 ω s ω a C 2 2 χ 3 χ 3 Δ β Δ β 0 ,
g max g 0 = χ 3 χ 3 ( P 1 P 2 ) 1 / 2 I I s 1 P 1 + I s 2 P 2 ,
d A s d z = O 2 ω s χ 3 I p s A p A p * A s ,
d A p d z = j O 1 ω p A p * A 3 A 4 exp ( j Δ β z ) + j O 2 χ 3 ω p A p i I p i | A i | 2 O 2 ω p χ 3 I p s A s * A p ,
d A 3 d z = j O 1 2 ω 3 A p 2 A 4 * exp ( j Δ β z ) + j O 2 χ 3 ω s A 3 i I 3 i | A i | 2 ,
d A 4 d z = j O 1 2 ω 4 A p 2 A 3 * exp ( j Δ β z ) + j O 2 χ 3 ω 4 A 4 i I 4 i | A i | 2 ,
O 1 = χ 3 O 2 e P 2 e 3 e 4 d A ,
d ζ s d Γ = O 3 ζ p 2 ζ s ,
d ζ p d Γ = 2 ζ p ζ 3 ζ 4 sin θ O 3 ζ p ζ s 2 ,
d ζ 3 d Γ = ζ p 2 ζ 4 sin θ ,
d ζ 4 d Γ = ζ p 2 ζ 3 sin θ ,
d θ d Γ = Δ S + ( 4 ζ 3 ζ 4 ζ p 2 ζ 4 ζ 3 ζ p 2 ζ 3 ζ 4 ) × cos θ + Q s ζ s 2 + Q p ζ p 2 + Q 3 ζ 3 2 + Q 4 ζ 4 2 ,
P = P s ( 0 ) ω s + P p ( 0 ) ω p + P 3 ( 0 ) ω 3 + P 4 ( 0 ) ω 4 ,
O 3 = χ 3 2 ω s ( ω 3 ω 4 ) 1 / 2 I p s O 2 O 1 ,
Q n = 2 i i ω i I i n ω n ω p ( ω 3 ω 4 ) 1 / 2 O 1 χ 3 O 2 ,
i = { 2 i = P 1 i = 3 , 4 ,
B 1 = ¼ [ ( 1 C 1 r C r + C 1 i C i ) 2 + ( C 1 r C i + C r C 1 i ) 2 ] , B 2 = 1 4 C 2 2 ω s 2 | χ 3 | 2 ( T 1 2 + T 2 2 ) 1 / 2 , B 3 = 2 ( B 1 B 2 ) 1 / 2 cos ( ϕ s + ϕ a ϕ 1 ϕ 2 π 2 tan 1 C 1 r C i + C r C 1 i 1 C 1 r C r + C 1 i C i + tan 1 C r χ 3 C i χ 3 C r χ 3 + C i χ 3 ) , B 4 = ¼ [ ( 1 + C 1 r C r C 1 i C i ) 2 + ( C 1 r C i + C r C 1 i ) 2 ] , B 5 = ω a 2 ω s 2 B 2 , B 6 = 2 ( B 4 B 5 ) 1 / 2 cos ( ϕ s + ϕ a ϕ 1 ϕ 2 π 2 tan 1 C 1 r C i + C r C 1 i 1 + C 1 r C r C 1 i C i tan 1 C r χ 3 C i χ 3 C r χ 3 + C i χ 3 ) , C 1 r = ½ [ ( α s α a ) ( C 3 + C 4 ) χ 3 ] , C 1 i = ½ [ Δ β + δ β ( C 3 + C 4 ) χ 3 ] , C r = [ ( T 1 2 + T 2 2 ) 1 / 2 + T 1 ] 1 / 2 / [ 2 ( T 1 2 + T 2 2 ) ] 2 , C i = [ ( T 1 2 + T 2 2 ) 1 / 2 T 1 ) 1 / 2 / [ 2 ( T 1 2 + T 2 2 ) ] 1 / 2 ,

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