Abstract

The growth of the Raman Stokes spectrum in single-mode optical fibers is studied in circumstances where there is a significant variation of loss with wavelength. It is shown that the dependence of pump and total Stokes power with fiber length can be calculated from only two equations when an effective Stokes loss is used. The effective Stokes loss is calculated both analytically and numerically for a linear increase of loss with wavelength. Good agreement is found for the two methods of computation.

© 1986 Optical Society of America

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References

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  1. R. H. Stolen, “Fiber Raman Lasers,” in Fiber and Integrated Optics, D. B. Ostrowsky, Ed. (Plenum, New York, 1979), pp. 157–182.
    [CrossRef]
  2. R. H. Stolen, E. P. Ippen, A. R. Tynes, “Raman Oscillation in Glass Optical Waveguides,” Appl. Phys. Lett. 20, 62 (1972).
    [CrossRef]
  3. W. P. Urquhart, P. J. R. Laybourn, “Stimulated Raman Scattering in Optical Fibers with Nonconstant Losses: a Two-Wavelength Model,” Appl. Opt. 25, 1746 (1986).
    [CrossRef] [PubMed]
  4. J. AuYeung, A. Yariv, “Spontaneous and Stimulated Raman Scattering in Long Low Loss Fibers,” IEEE J. Quantum Electron. QE-14, 347 (1978).
    [CrossRef]
  5. W. P. Urquhart, P. J. R. Laybourn, “The Effective Core Area for Stimulated Raman Scattering in Single-Mode Optical Fibres,” Proc. IEE (J) Optoelectron. 132, 201 (1985).
    [CrossRef]
  6. W. P. Urquhart, Ph.D. Thesis, U. Glasgow (1984).
  7. D. Von der Linde, M. Maier, W. Kaiser. “Quantitative Investigation of the Stimulated Raman Effect Using Subnanosecond Light Pulses,” Phys. Rev. 178, 11 (1969).
    [CrossRef]

1986

1985

W. P. Urquhart, P. J. R. Laybourn, “The Effective Core Area for Stimulated Raman Scattering in Single-Mode Optical Fibres,” Proc. IEE (J) Optoelectron. 132, 201 (1985).
[CrossRef]

1978

J. AuYeung, A. Yariv, “Spontaneous and Stimulated Raman Scattering in Long Low Loss Fibers,” IEEE J. Quantum Electron. QE-14, 347 (1978).
[CrossRef]

1972

R. H. Stolen, E. P. Ippen, A. R. Tynes, “Raman Oscillation in Glass Optical Waveguides,” Appl. Phys. Lett. 20, 62 (1972).
[CrossRef]

1969

D. Von der Linde, M. Maier, W. Kaiser. “Quantitative Investigation of the Stimulated Raman Effect Using Subnanosecond Light Pulses,” Phys. Rev. 178, 11 (1969).
[CrossRef]

AuYeung, J.

J. AuYeung, A. Yariv, “Spontaneous and Stimulated Raman Scattering in Long Low Loss Fibers,” IEEE J. Quantum Electron. QE-14, 347 (1978).
[CrossRef]

Ippen, E. P.

R. H. Stolen, E. P. Ippen, A. R. Tynes, “Raman Oscillation in Glass Optical Waveguides,” Appl. Phys. Lett. 20, 62 (1972).
[CrossRef]

Kaiser, W.

D. Von der Linde, M. Maier, W. Kaiser. “Quantitative Investigation of the Stimulated Raman Effect Using Subnanosecond Light Pulses,” Phys. Rev. 178, 11 (1969).
[CrossRef]

Laybourn, P. J. R.

W. P. Urquhart, P. J. R. Laybourn, “Stimulated Raman Scattering in Optical Fibers with Nonconstant Losses: a Two-Wavelength Model,” Appl. Opt. 25, 1746 (1986).
[CrossRef] [PubMed]

W. P. Urquhart, P. J. R. Laybourn, “The Effective Core Area for Stimulated Raman Scattering in Single-Mode Optical Fibres,” Proc. IEE (J) Optoelectron. 132, 201 (1985).
[CrossRef]

Maier, M.

D. Von der Linde, M. Maier, W. Kaiser. “Quantitative Investigation of the Stimulated Raman Effect Using Subnanosecond Light Pulses,” Phys. Rev. 178, 11 (1969).
[CrossRef]

Stolen, R. H.

R. H. Stolen, E. P. Ippen, A. R. Tynes, “Raman Oscillation in Glass Optical Waveguides,” Appl. Phys. Lett. 20, 62 (1972).
[CrossRef]

R. H. Stolen, “Fiber Raman Lasers,” in Fiber and Integrated Optics, D. B. Ostrowsky, Ed. (Plenum, New York, 1979), pp. 157–182.
[CrossRef]

Tynes, A. R.

R. H. Stolen, E. P. Ippen, A. R. Tynes, “Raman Oscillation in Glass Optical Waveguides,” Appl. Phys. Lett. 20, 62 (1972).
[CrossRef]

Urquhart, W. P.

W. P. Urquhart, P. J. R. Laybourn, “Stimulated Raman Scattering in Optical Fibers with Nonconstant Losses: a Two-Wavelength Model,” Appl. Opt. 25, 1746 (1986).
[CrossRef] [PubMed]

W. P. Urquhart, P. J. R. Laybourn, “The Effective Core Area for Stimulated Raman Scattering in Single-Mode Optical Fibres,” Proc. IEE (J) Optoelectron. 132, 201 (1985).
[CrossRef]

W. P. Urquhart, Ph.D. Thesis, U. Glasgow (1984).

Von der Linde, D.

D. Von der Linde, M. Maier, W. Kaiser. “Quantitative Investigation of the Stimulated Raman Effect Using Subnanosecond Light Pulses,” Phys. Rev. 178, 11 (1969).
[CrossRef]

Yariv, A.

J. AuYeung, A. Yariv, “Spontaneous and Stimulated Raman Scattering in Long Low Loss Fibers,” IEEE J. Quantum Electron. QE-14, 347 (1978).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

R. H. Stolen, E. P. Ippen, A. R. Tynes, “Raman Oscillation in Glass Optical Waveguides,” Appl. Phys. Lett. 20, 62 (1972).
[CrossRef]

IEEE J. Quantum Electron.

J. AuYeung, A. Yariv, “Spontaneous and Stimulated Raman Scattering in Long Low Loss Fibers,” IEEE J. Quantum Electron. QE-14, 347 (1978).
[CrossRef]

Phys. Rev.

D. Von der Linde, M. Maier, W. Kaiser. “Quantitative Investigation of the Stimulated Raman Effect Using Subnanosecond Light Pulses,” Phys. Rev. 178, 11 (1969).
[CrossRef]

Proc. IEE (J) Optoelectron.

W. P. Urquhart, P. J. R. Laybourn, “The Effective Core Area for Stimulated Raman Scattering in Single-Mode Optical Fibres,” Proc. IEE (J) Optoelectron. 132, 201 (1985).
[CrossRef]

Other

W. P. Urquhart, Ph.D. Thesis, U. Glasgow (1984).

R. H. Stolen, “Fiber Raman Lasers,” in Fiber and Integrated Optics, D. B. Ostrowsky, Ed. (Plenum, New York, 1979), pp. 157–182.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Top: six Stokes spectra computed using Eqs. (1) and (2). Bottom: corresponding loss profiles. The two horizontal scales are directly equivalent. See Table I for data used in computation.

Fig. 2
Fig. 2

Isometric projection representing the evolution of the power spectrum during propagation along the length of an optical fiber. Input power = 8.0 W. The loss spectrum is (A) of Fig. 1 (bottom). Other data used in computations are in Table I.

Fig. 3
Fig. 3

Depletion of the pump power with the corresponding growth of the Stokes power. The latter is computed by splitting the Stokes spectrum into 91 elements and summing the power contributions from each. Loss at pump wavelength, β l = 8.7. The loss spectra for the stokes wavelengths are lines (A), (C), and (E) of the lower graph of Fig. 1. Other data used in computations are in Table I.

Fig. 4
Fig. 4

(top) and (middle) are the variation of the normalized effective Stokes loss with respect to wavelength, where (top) twenty-one and (middle) ninety-one simultaneous equations are used to represent the Stokes power evolution. (bottom) is the loss profile. See Table I for data used in computations.

Fig. 5
Fig. 5

(A), (B), and (C) variation of the normalized effective Stokes loss with the maximum Stokes loss. The pump loss is held constant at β l = 8.73 × 10−3 (0.2 dB km−1), and the loss profile is linear. Fiber lengths: Z = (A) 6.3, (B) 31.6, (C) 63.3. Upper graph: direct numerical calculation using Eq. (23), lower analytical result, using Eq. (28).

Fig. 6
Fig. 6

Variation of the normalized effective Stokes loss with the number of simultaneous equations representing the Stokes power. Values of the effective Stokes loss with m = 91: Z = 6.3: (A) 0.2963, (B) 0.4890, (C) 0.4888; Z = 31.6: (A) 0.2976, (B) 0.4925, (C) 0.4957; Δ = (A); □ = (B); * = (C).

Fig. 7
Fig. 7

Comparison of two methods of solutions. Solid lines: solution by splitting the Stokes spectrum into ninety-one elements and summing the power contributions from each. Dotted lines: by the use of the effective Stokes loss and only one equation to represent the Stokes power evolution. The loss spectra for the Stokes wavelengths are lines (A), (B), and (C) of Fig. 4 (bottom). Normalized effective Stokes losses: (A) 0.2873; (B) 0.4823; (C) 0.4823. Other data used in computation are in Table I.

Tables (2)

Tables Icon

Table I Values of Constants for the Numerical Examples used Throughouta

Tables Icon

Table II Calculation of Diff [from Eq. (16)] Where the Data of Fig. 1 are Compared with Gaussian and Lorentzian Functions of FWHM = 55.2 cm−1

Equations (37)

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d R l d Z + β l R l = - R l i = 1 q L i ( R s i + Q / q ) .
d R s i d Z + β s i R s i = Λ i L i R l ( R s i + Q / q ) .
R s = i = 1 q R s i .
R l = P l / P 0 ,
R s i = P s i / P 0 .
Z = z P 0 g 0 A ,
β l = α l A g 0 P 0 ,
β s i = α s i A g 0 P 0 ,
Λ i = λ l / λ s i .
g i / g 0 = L i ,
Q = p q P 0 ,
g ( ν ) = { g 0 L ( ν ) = g 0 [ 1 + ( ν - ν s δ ν / 2 ) 2 ] - 1 ; ν 0 - < ν < ν 0 + , 0 elsewhere ,
L = ( δ ν / 2 ) 2 ( δ ν / 2 ) 2 + [ ( i - m + 1 2 m ) ( ν s m - ν l ) ] 2 .
R l = 1 ,
R s i = Q ( δ ν / 2 π ) · ( ν s m - ν l m ) ( δ ν 2 ) 2 + { [ i - ( m + 1 2 ) m ] ( ν s m - ν l ) } 2 .
diff = i = 1 91 { R s ( ν i ) - R s ( ν 46 ) exp [ - ( ν i - ν 46 Δ ν / 2 ) 2 ] } .
β s = i = 1 q β s i R s i i = 1 q R s i .
d d Z i = 1 q R s i + β s i = 1 q R s i = R l i = 1 q Λ i L i ( R s i + Q / q )
d R l d Z + β l R l = - R l ( R s + Q ) ;
d R s d Z + β s R s = Λ R l ( R s + Q ) .
β s ( min ) β s β s ( max ) .
[ β s ( max ) i = 1 q β s i R s i β s ( max ) ] / i = 1 q R s i β s ( max ) .
i = 1 q β s i R s i β s ( max ) i = 1 q R s i ,
[ β s ( min ) i = 1 q β s i R s i β s ( min ) ] / i = 1 q R s i β s ( min ) .
β s = ν l ν s max R s ( ν ) · β ( ν ) d ν ν l ν s max R s ( ν ) d ν ;
β s N = β s - β s ( min ) β s ( max ) - β s ( min ) ;
R s ( ν ) exp [ - ( ν - ν s Δ ν / 2 ) 2 ] .
β N ( ν ) = ν - ν l ν s m - ν l .
β s N = ν s - ν l ν s m - ν l .
β s N = ν l ν s m ( ν - ν l ν s m - ν l ) exp [ - ( ν - ν s Δ ν / 2 ) 2 ] d ν ν l ν s m exp [ - ( ν - ν s Δ ν / 2 ) 2 ] d ν .
ν = ν - ν s Δ ν / 2 .
Δ ν π 4 [ erf ( ν s m - ν s Δ ν / 2 ) + erf ( ν s - ν l Δ ν / 2 ) ] ;
( - ν l ν s m - ν l ) ( Δ ν π 2 ) .
ν = ( ν - ν s ν / 2 )
Δ ν 2 π ( ν s ν s m - ν l ) - Δ ν 2 ( ν s m - ν s ) / ( Δ ν / 2 ) ν exp ( - ν 2 ) d ν ( ν l - ν s ) / ( Δ ν / 2 ) .
β s N = ( ν s - ν l ν s m - ν l ) - 1 π { exp [ - ( ν s m - ν s Δ ν / 2 ) 2 ] + exp [ - ( ν l - ν s Δ ν / 2 ) 2 ] } .
β s N = ( ν s - ν l ν s m - ν l ) .

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