Abstract

We show that phase matching for the four-photon mixing process in a single-mode fiber depends only on the propagation constant total dispersion βII(ω). Frequencies ωs and ωa for the Stokes and anti-Stokes waves must satisfy ωs < ωzd < ωa, where ωzd is the frequency for which dispersion is zero; βII(ωzd) = 0. Variations in the frequency shift Ω(ωp) are described for pump frequency ωp around ωzd, i.e., in the region where delicate balances of material and waveguide dispersion effects are used in fiber design. We show that no new waves are created when the pump and zero-dispersion frequencies coincide, i.e., Ω(ωzd) = 0. Since the creation of Stokes and anti-Stokes waves is intimately related to the βII(ω) versus ω curve, some interesting results are predicted for advanced design fibers.

© 1986 Optical Society of America

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References

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  1. R. H. Stolen, in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979), Chap. 5, p. 125.
  2. R. H. Stolen, J. E. Bjorkholm, IEEE J. Quantum Electron. QE-18, 1062 (1982).
    [CrossRef]
  3. K. O. Hill, D. C. Johnson, B. S. Kawasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1974).
    [CrossRef]
  4. R. H. Stolen, IEEE J. Quantum Electron. QE-11, 100 (1975).
    [CrossRef]
  5. R. H. Stolen, M. A. Bosch, C. Lin, Opt. Lett. 6, 213 (1981).
    [CrossRef] [PubMed]
  6. C. Lin, W. A. Reed, A. D. Pearson, H.-T. Shang, Opt. Lett. 6, 493 (1981).
    [CrossRef] [PubMed]
  7. K. Washio, K. Inque, T. Tanigawa, Electron. Lett. 16, 333 (1980).
  8. W. A. Gambling, H. Matsumura, C. M. Ragdale, Microwaves Opt. Acoust. 3, 239 (1979).
    [CrossRef]
  9. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1969), p. 95.
  10. S. Kobayashi, S. Shibata, N. Shibata, T. Izawa, in Proceedings of First International Conference on Integrated Optics and Optical Fiber Communication (Institute of Electrical Engineers of Japan, Tokyo, 1977).
  11. M. Monerie, IEEE J. Quantum Electron. QE-18, 535 (1982).
    [CrossRef]
  12. C. M. Lemrow, V. A. Bhagavatula, Laser Focus 21(3), 82 (1985).

1985 (1)

C. M. Lemrow, V. A. Bhagavatula, Laser Focus 21(3), 82 (1985).

1982 (2)

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

M. Monerie, IEEE J. Quantum Electron. QE-18, 535 (1982).
[CrossRef]

1981 (2)

1980 (1)

K. Washio, K. Inque, T. Tanigawa, Electron. Lett. 16, 333 (1980).

1979 (1)

W. A. Gambling, H. Matsumura, C. M. Ragdale, Microwaves Opt. Acoust. 3, 239 (1979).
[CrossRef]

1975 (1)

R. H. Stolen, IEEE J. Quantum Electron. QE-11, 100 (1975).
[CrossRef]

1974 (1)

K. O. Hill, D. C. Johnson, B. S. Kawasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1974).
[CrossRef]

Bhagavatula, V. A.

C. M. Lemrow, V. A. Bhagavatula, Laser Focus 21(3), 82 (1985).

Bjorkholm, J. E.

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

Bosch, M. A.

Gambling, W. A.

W. A. Gambling, H. Matsumura, C. M. Ragdale, Microwaves Opt. Acoust. 3, 239 (1979).
[CrossRef]

Hill, K. O.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1974).
[CrossRef]

Inque, K.

K. Washio, K. Inque, T. Tanigawa, Electron. Lett. 16, 333 (1980).

Izawa, T.

S. Kobayashi, S. Shibata, N. Shibata, T. Izawa, in Proceedings of First International Conference on Integrated Optics and Optical Fiber Communication (Institute of Electrical Engineers of Japan, Tokyo, 1977).

Johnson, D. C.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1974).
[CrossRef]

Kawasaki, B. S.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1974).
[CrossRef]

Kobayashi, S.

S. Kobayashi, S. Shibata, N. Shibata, T. Izawa, in Proceedings of First International Conference on Integrated Optics and Optical Fiber Communication (Institute of Electrical Engineers of Japan, Tokyo, 1977).

Lemrow, C. M.

C. M. Lemrow, V. A. Bhagavatula, Laser Focus 21(3), 82 (1985).

Lin, C.

MacDonald, R. I.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1974).
[CrossRef]

Matsumura, H.

W. A. Gambling, H. Matsumura, C. M. Ragdale, Microwaves Opt. Acoust. 3, 239 (1979).
[CrossRef]

Monerie, M.

M. Monerie, IEEE J. Quantum Electron. QE-18, 535 (1982).
[CrossRef]

Pearson, A. D.

Ragdale, C. M.

W. A. Gambling, H. Matsumura, C. M. Ragdale, Microwaves Opt. Acoust. 3, 239 (1979).
[CrossRef]

Reed, W. A.

Shang, H.-T.

Shibata, N.

S. Kobayashi, S. Shibata, N. Shibata, T. Izawa, in Proceedings of First International Conference on Integrated Optics and Optical Fiber Communication (Institute of Electrical Engineers of Japan, Tokyo, 1977).

Shibata, S.

S. Kobayashi, S. Shibata, N. Shibata, T. Izawa, in Proceedings of First International Conference on Integrated Optics and Optical Fiber Communication (Institute of Electrical Engineers of Japan, Tokyo, 1977).

Stolen, R. H.

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

R. H. Stolen, M. A. Bosch, C. Lin, Opt. Lett. 6, 213 (1981).
[CrossRef] [PubMed]

R. H. Stolen, IEEE J. Quantum Electron. QE-11, 100 (1975).
[CrossRef]

R. H. Stolen, in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979), Chap. 5, p. 125.

Tanigawa, T.

K. Washio, K. Inque, T. Tanigawa, Electron. Lett. 16, 333 (1980).

Washio, K.

K. Washio, K. Inque, T. Tanigawa, Electron. Lett. 16, 333 (1980).

Watson, G. N.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1969), p. 95.

Whittaker, E. T.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1969), p. 95.

Electron. Lett. (1)

K. Washio, K. Inque, T. Tanigawa, Electron. Lett. 16, 333 (1980).

IEEE J. Quantum Electron. (3)

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

R. H. Stolen, IEEE J. Quantum Electron. QE-11, 100 (1975).
[CrossRef]

M. Monerie, IEEE J. Quantum Electron. QE-18, 535 (1982).
[CrossRef]

J. Appl. Phys. (1)

K. O. Hill, D. C. Johnson, B. S. Kawasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1974).
[CrossRef]

Laser Focus (1)

C. M. Lemrow, V. A. Bhagavatula, Laser Focus 21(3), 82 (1985).

Microwaves Opt. Acoust. (1)

W. A. Gambling, H. Matsumura, C. M. Ragdale, Microwaves Opt. Acoust. 3, 239 (1979).
[CrossRef]

Opt. Lett. (2)

Other (3)

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1969), p. 95.

S. Kobayashi, S. Shibata, N. Shibata, T. Izawa, in Proceedings of First International Conference on Integrated Optics and Optical Fiber Communication (Institute of Electrical Engineers of Japan, Tokyo, 1977).

R. H. Stolen, in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979), Chap. 5, p. 125.

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Figures (4)

Fig. 1
Fig. 1

Normalized pump and output frequencies for two different GeO2-doped step-index fibers with ρ = 3.5 μm. Profile heights are Δ1 ≈ 0.0035 (3.1 mol % GeO2) and Δ2 ≈ 0.0060 (5.8 mol % GeO2).

Fig. 2
Fig. 2

Frequency shift Ω versus pump frequency. The solid curve is the exact result, while the broken curve is the approximation [expression (7)]. The fiber has a 5.8-mol % GeO2-doped step-index core (Δ ≈ 0.0060) with ρ = 3.5 μm. The dotted curve is a suitably scaled βII.

Fig. 3
Fig. 3

Frequency shift against pump frequency for a clad power-law profile fiber with fixed ρ = 3.5 μm. The core is 5.8-mol % GeO2 doped (Δ ≈ 0.0060), and curves are labeled by the power-law index α. Step-index profile has α = ∞.

Fig. 4
Fig. 4

Frequency shift Ω for a parabolic dispersion model. βII, normalized by an arbitrary scale factor, is also indicated. Ωmax = (3/2)1/2(ω2ω1).

Equations (9)

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Ω = ( ω p - ω s ) = ( ω a - ω p ) .
Δ k ( Ω ) β ( ω a ) + β ( ω s ) - 2 β ( ω p ) = 0.
β = k n 0 ( 1 - 2 Δ U 2 / V 2 ) 1 / 2 ,
β ( ω a ) = β ( ω p ) + Ω β ( ω p ) + Ω 2 0 1 ( 1 - t ) β II ( ω p + t Ω ) d t ,
ω s = ω p - Ω ω a = ω p + Ω { 1 - | ω p - f Ω | } β II ( f ) d f = 0 .
ω s < ω zd < ω a
Ω [ - 12 β II ( ω p ) / β IV ( ω p ) ] 1 / 2 ,
β II ( ω ) = A ( ω - ω 1 ) ( ω 2 - ω ) ,
{ ω s < ω 1 ω p < ω a < ω 2 }             for ω 1 < ω p < ω mid , { ω 1 < ω s < ω p ω a > ω 2 }             for ω mid < ω p < ω 2 .

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