Abstract

The analytical solutions to differential equations governing nonlinear four-photon mixing in optical fibers are presented for the general case of depleted pump power. The expressions reduce to those derived earlier with the assumption of nondepleted pump power. We find that the maximum power available for conversion from the pump waves to the signal waves resulting from a third-order nonlinearity depends on the index mismatch and that the index mismatch required for achieving maximum possible power transfer is a function of the frequency shift.

© 1989 Optical Society of America

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References

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962).
  2. K. O. Hill, D. C. Johnson, B. S. Kowasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1978).
  3. R. H. Stolen, J. E. Bjorkholm, IEEE J. Quantum Electron. QE-18, 1062 (1982).
  4. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions, Applied Mathematics Series 55 (National Bureau of Standards, Washington, D.C., 1964).
  5. Here the term that is not large is a relative description comparing it with the case of the large frequency shift discussed below. For instance, for usual fiber parameters (core radius ρ = 3 μm, relative refractive-index difference Δ = 0.0066) at the long-wavelength range λp = 1.32 μm, C02 will be greater than zero when the frequency shift is smaller than ~2600 cm−1; for ρ = 3.5 μm and Δ = 0.006, at the short-wavelength range λp = 0.532 μm, C02 will be greater than zero at a frequency shift less than ~9000 cm−1. These critical frequency shifts (2600 and 9000 cm−1) are much larger than those within the Raman frequency band.
  6. D. F. Walls, Phys. Lett. 32A, 476 (1970).
  7. C. Lin, W. A. Reed, A. D. Pearson, H. T. Shang, Opt. Lett. 6, 493 (1981).

1982 (1)

R. H. Stolen, J. E. Bjorkholm, IEEE J. Quantum Electron. QE-18, 1062 (1982).

1981 (1)

1978 (1)

K. O. Hill, D. C. Johnson, B. S. Kowasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1978).

1970 (1)

D. F. Walls, Phys. Lett. 32A, 476 (1970).

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962).

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962).

Bjorkholm, J. E.

R. H. Stolen, J. E. Bjorkholm, IEEE J. Quantum Electron. QE-18, 1062 (1982).

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962).

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962).

Hill, K. O.

K. O. Hill, D. C. Johnson, B. S. Kowasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1978).

Johnson, D. C.

K. O. Hill, D. C. Johnson, B. S. Kowasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1978).

Kowasaki, B. S.

K. O. Hill, D. C. Johnson, B. S. Kowasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1978).

Lin, C.

MacDonald, R. I.

K. O. Hill, D. C. Johnson, B. S. Kowasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1978).

Pearson, A. D.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962).

Reed, W. A.

Shang, H. T.

Stolen, R. H.

R. H. Stolen, J. E. Bjorkholm, IEEE J. Quantum Electron. QE-18, 1062 (1982).

Walls, D. F.

D. F. Walls, Phys. Lett. 32A, 476 (1970).

IEEE J. Quantum Electron. (1)

R. H. Stolen, J. E. Bjorkholm, IEEE J. Quantum Electron. QE-18, 1062 (1982).

J. Appl. Phys. (1)

K. O. Hill, D. C. Johnson, B. S. Kowasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1978).

Opt. Lett. (1)

Phys. Lett. (1)

D. F. Walls, Phys. Lett. 32A, 476 (1970).

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962).

Other (2)

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions, Applied Mathematics Series 55 (National Bureau of Standards, Washington, D.C., 1964).

Here the term that is not large is a relative description comparing it with the case of the large frequency shift discussed below. For instance, for usual fiber parameters (core radius ρ = 3 μm, relative refractive-index difference Δ = 0.0066) at the long-wavelength range λp = 1.32 μm, C02 will be greater than zero when the frequency shift is smaller than ~2600 cm−1; for ρ = 3.5 μm and Δ = 0.006, at the short-wavelength range λp = 0.532 μm, C02 will be greater than zero at a frequency shift less than ~9000 cm−1. These critical frequency shifts (2600 and 9000 cm−1) are much larger than those within the Raman frequency band.

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Figures (2)

Fig. 1
Fig. 1

Optimum normalized index mismatch ΔSmax (at which ηslp is maximum) versus frequency shift Δf for initial value P4(0) = 0, P3(0) = 0.001P2(0), P2(0) = 0.5Pp(0) (the three-wave case), fiber core radius ρ = 3 μm, and a relative refractive index Δ = 0.0066.

Fig. 2
Fig. 2

Normalized power P ¯ i = P i / P versus normalized distance Γ for a frequency shift Δf = 1600 cm−1 and the other parameters the same as in Fig. 1. (a) ΔS = ΔSmax = 0.2445, (b) ΔS = 0.3, (c) ΔS = 0.15.

Equations (18)

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d A m d z = j 3 2 0 R e ( χ 3 ) ω m I γ m exp ( j m Δ β z ) + j 3 2 0 χ 3 ( 0 ) ω m A m i I m i | A i | 2 .
P 4 ( z ) = ω 4 P { η 2 η η 1 s n 2 [ ( z + z 0 ) / z c ] | k } / { 1 η s n 2 [ ( z + z 0 ) / z c ] | k ] } ,
P i ( z ) = P i ( 0 ) + i [ P 4 ( 0 ) P 4 ( z ) ] ω i / ω 4 ,
z c 1 = O 1 P | C 0 | [ ( η 3 η 1 ) ( η 4 η 2 ) ω 1 ω 2 ω 3 ω 4 ] 1 / 2 ,
k = [ ( η 3 η 2 ) ( η 4 η 1 ) / ( η 3 η 1 ) ( η 4 η 2 ) ] 1 / 2 ,
η = ( η 3 η 2 ) / ( η 3 η 1 ) ,
z 0 = z c F ( sin 1 { [ P 4 ( 0 ) P ω 4 η 2 ] / η [ P 4 ( 0 ) P ω 4 η 1 ] } 1 / 2 | k ) ,
C 0 = [ 1 1 16 ( Q 1 + Q 2 Q 3 Q 4 ) 2 ] 1 / 2 ,
Q m = ω m χ 3 ( 0 ) i j ω i I i m / I R e ( χ 3 ) ( ω 1 ω 2 ω 3 ω 4 ) 1 / 2 .
x ( x 1 x ) ( x 2 x ) ( x 3 + x ) [ C + Δ S 2 x Q 1 4 ( x 1 x ) 2 Q 2 4 ( x 2 x ) 2 + Q 3 4 ( x 3 + x ) 2 + Q 4 4 x 2 ] 2 = 0 ,
P 4 ( z ) = ω 4 P { η 3 η η 2 s n 2 [ ( z + z 0 ) / z c ] | k } / { 1 η s n 2 [ ( z + z 0 ) / z c ] | k } ,
k = [ ( η 4 η 3 ) ( η 2 η 1 ) / ( η 3 η 1 ) ( η 4 η 2 ) ] 1 / 2 ,
η = ( η 4 η 3 ) ( η 4 η 2 ) ,
z 0 = z c F ( sin 1 { [ P 4 ( 0 ) ω 4 P η 3 ] / η [ P 4 ( 0 ) ω 4 P η 2 ] } 1 / 2 | k ) .
P 3 ( z ) P 3 ( 0 ) cosh 2 z / z c ,
P 4 ( z ) ( ω 4 / ω 3 ) P 3 ( 0 ) sinh 2 z / z c ,
P 4 ( Z max ) P 4 ( 0 ) = ω 4 P η slp P 4 ( 0 ) ,
P 3 ( Z max ) P 3 ( 0 ) = ω 3 P η slp ω 3 ω 4 P 4 ( 0 ) ,

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