Abstract

The effect of stimulated Raman and Brillouin scattering on the power handling capacity of optical fibers is considered and found to be important especially when low loss optical fibers are used. A critical power below which stimulated effects may be neglected is defined for forward and backward Raman scattering and for backward Brillouin scattering. This critical power is determined by the effective core area A, the small signal attenuation constant of the fiber α, and the gain coefficient for the stimulated scattering process γ0, by the approximate relation Pcrit ≈ 20/γ0. For a fiber with 20-dB/km attenuation and an area of 10−7 cm2Pcrit ≈ 35 mW for stimulated Brillouin scattering. For stimulated Raman scattering Pcrit is approximately two orders of magnitude higher. It is concluded that these effects must be considered in the design of optical communication systems using low loss fibers.

© 1972 Optical Society of America

Introduction

Low loss optical fibers are currently being considered as transmission media for optical communication systems. At low power densities the losses of an optical fiber will be determined by spontaneous Raman, Brillouin, and Rayleigh scattering, absorption losses in the bulk material, scattering at the core-cladding interface, and mode conversion from low-loss trapped modes to high loss cladding modes. On the other hand at high power densities within the fiber, Raman and Brillouin scattering processes can become stimulated, introducing power dependent loss mechanisms.[1][3] In the case of forward stimulated Raman scattering the principal effect is a frequency shift of the radiation transmitted by the fiber to lower frequencies. Sufficiently large frequency shifts could produce amplitude distortion at the receiver if the detector is intrinsically frequency sensitive or if narrow band filters are used. On the other hand backward-wave stimulated scattering processes (either Raman or Brillouin) will result in a severe attenuation of the forward traveling, information carrying wave, due to the transfer of energy to the stimulated backward wave.

In this paper both forward and backward stimulated scattering processes are discussed. Raman scattering may be either forward or backward-wave in character whereas stimulated Brillouin scattering is exclusively a backward-wave process in amorphous media and in the geometry considered. The differential equation governing the stimulated processes is solved under the assumption that the primary radiation—termed the pump—follows the natural exponential attenuation of the fiber, i.e., pump depletion due to the stimulated process is neglected. In this approximation the equation for the forward and backward stimulated radiation fields may be solved exactly. In both cases the criterion that the stimulated wave be everywhere less than the forward traveling injected wave yields a critical input power above which nonlinear effects play an important role.

Using available data for the magnitudes of the nonlinear scattering cross sections it is found that stimulated Brillouin scattering should determine the maximum power handling capacity of a fiber. The maximum power that can be launched is given to a good approximation by Pcrit.Brillouin20(αA/γ0), where α is the attenuation constant and A the area of the fiber and γ0 is the gain coefficient of the nonlinear process. Using this relation and the value of γ0 for Brillouin scattering a fiber with a core area of 10−7 cm2 and an attenuation of 20 dB/km gives Pcrit. = 35 mW for either cw or long pulse radiation. For short pulses of radiation this critical power is greater due to the backward-wave nature of the interaction and the finite line width of the Brillouin scattering.

Analysis

In this section we evaluate the input power density to an optical fiber for which the nonlinear process, either forward or backward wave stimulated scattering becomes important. The criterion is arbitrarily interpreted to be that set of conditions for which the new radiation field generated by the nonlinear interaction is comparable to the incident field. For forward Raman scattering we require that the magnitude of the Stokes shifted wave be less than the injected wave at all points along the fiber. For backward scattering, either Raman or Brillouin, we require that at the entrance face the backward traveling wave be less than the injected wave. All calculations are made assuming single pass amplification of spontaneous emission. For Raman scattering feedback due to scattering could be a problem due to the isotropy of the Raman gain but is not specifically considered here.

Forward Wave Interaction

Consider a uniform transmission medium of effective cross sectional area A and length L with an attenuation constant α. For convenience we label the injected radiation with the subscript p, refer to it as the pump, and denote its frequency by ωp. Let the pump be injected at z = 0, traveling in the +z direction with a power Pp and an effective power density SpPp/A. In the absence of any nonlinear interaction the pump propagates as

Pp(z)=Pp(0)exp(-αpz),

with an identical dependence for the power density. Assuming that the guiding medium is Raman active, the differential equation for a forward traveling wave at the Stokes frequency is given by

[(d/dz)+αs]Ps(z)=γSp(z)Ps(z),

where the subscript s refers to the Stokes component and Ps is the Stokes power at any point. The gain constant γ depends upon the Raman scattering cross section and is frequency dependent.

A similar equation can be written for the pump wave where the nonlinear interaction is explicitly taken into account. Since we are looking for an upper bound to the input power where nonlinear effects become important we make the simplifying assumption that the pump is not attenuated by the nonlinear interaction. In this case Eq. (2) becomes

[(d/dz)+αs]Ps(z)=γPs(z)Sp(0)exp(-αpz),

where Sp(0) is the pump intensity at the input. The solution to Eq. (3) is

Ps(z)=Ps(0)exp{-αsz+γSp(0)αp[1-exp(-αpz)]}.

At the output of the fiber, z = L, assuming that αpL ≫ 1, Eq. (4) becomes

Ps(L)Ps(0)exp[-αsL+γSp(0)αp].

Thus under the assumption that the pump attenuates according to Eq. (1), Eq. (5) says that the electronic gain at the Stokes frequency is [γSp(0)/αp], equivalent to the gain produced by the incident pump intensity for a distance leff = 1/αp.

Equation (5) assumes a Stokes wave injected at (z = 0). In practice no Stokes wave would be injected; any Stokes power appearing at the output would be due to amplified spontaneous Stokes scattering occurring throughout the length of the fiber. It is shown in Appendix A that the summation over the length of all spontaneous emission weighted by its net gain is equivalent to assuming an input flux at z = 0 of one photon per mode (longitudinal and transverse) of the fiber. In this case the total Stokes flux at z = L is given by

Ps(L)=transversemodesdν(hν)exp[-αsL+Sp(0)αpγ(ν),],

where we now take into account the frequency dependence of the gain coefficient. Assuming a Lorentzian gain profile with full width at half maximum Δνfwhm and assuming the peak gain to be large, Eq. (6) becomes

Ps(L)={transversemodes(hνs)exp[-αsL+Sp(0)γ0αp]}Beff,

where γ0 is the peak gain coefficient. The effective bandwidth or number of longitudinal modes is thus

Beff=π2Δνfwhm[Sp(0)γ0/αp]12,

and the effective input Stokes power is

Ps(0)eff=(hνs)(Beff)(numberoftransversemodes.)

In order that the nonlinear process not be of importance we demand that the Stokes power computed from Eq. (7) be less than the signal power at z = L, i.e.,

Ps(0)effexp[-αsL+Sp(0)γ0αp]<Pp(0)exp(-αpL).

An absolute upper limit to the input pump power at which point nonlinear effects must be considered would be when Eq. (10) is satisfied as an equality. We define this pump power as the critical power, Pcrit. For a single transverse mode fiber and assuming α = αp the relation from which the critical power is determined is

π2(hνs)(γ0Aαp)Δνfwhm=(γ0PcritAαp)/23exp(-γ0PcritAαp).

As an example consider the case of Raman scattering in crystalline quartz at a pump wavelength of 1 μ. The peak Raman gain coefficient is γ0 ≈ 5 × 10−10 cm/W.[4] Assuming a single mode fiber of cross sectional area 10−7 cm2 (diameter ≈3 μ), an attenuation coefficient of α ≈ 5 × 10−5 cm−1 (20 dB/km) for both the Stokes and pump waves, and taking Δνfwhm ≈ 10 cm−1, the relation for the critical power becomes

exp(100Pcrit)=2×108Pcrit/23,

where Pcrit is expressed in watts. The critical power determined from this relation is Pcrit = 160 mW. The value of the critical power is reasonably insensitive to the choice of Δνfwhm. For example changing the line width of a factor of 10 changes the critical power by approximately 15%. On the other hand the critical power is extremely sensitive to α, γ0, and A. For example with αp = 1 × 10−5 (4.3 dB/km), Pcrit = 30 mW and for αp = 5 × 10−6 (2 dB/km), Pcrit = 12 mW, assuming A = 10−7 cm2 and γ0 = 5 × 10−10 cm/W. For the range of parameters involved here the critical power in watts is given to a very good approximation by the relation

Pcrit16(Aαp/γ0),

where A is given in cm2, α in cm−1, and γ0 in cm/W.

The gain coefficient, used in the above example, γ0 = 5 × 10−10 cm/W, applies to Raman scattering in α-quartz.[4] Stolen et al.[2] report that the line width for Raman scattering in amorphous glasses is of the order of several hundred cm−1, and the peak gain constant is smaller than the value quoted above by roughly an order of magnitude. It is thus concluded that a fiber with an area of 10−7 cm2 and an attenuation constant of 20 dB/km should be capable of handling an optical power of the order of 1 W without problems from stimulated forward Raman scattering. For lower loss fibers the power handling capacity will be correspondingly reduced.

Backward Wave Interaction

Because of the flat character of the dispersion relation for optical phonons near the center of the Brillouin zone, phase matching of the Raman process may be achieved for both forward and backward scattering; phase matching requirements for Brillouin scattering in isotropic media prohibit forward scattering and so SBS will be a backward wave interaction in the fiber geometry. The analysis for the backward-wave interaction below applies to either Raman or Brillouin scattering, whereas the preceding analysis applies only to Raman scattering.

For the backward-wave interaction we apply the same approach as for the case of the forward-wave interaction except that we consider a Stokes wave, either Brillouin or Raman, traveling in a direction opposite to the pump. Taking z = 0 as the point at which the pump is injected and assuming the pump to propagate as exp(−αpz) the gain experienced by a Stokes wave injected at some point z0, defined as the ratio Ps(0)/Ps(z0) is, Appendix B,

G=exp{-αsz0+[γSp(0)/αp][1-exp(-αpz0)]}.

The total backward-traveling, stimulated Stokes power reaching the entrance face is then found by summing all contributions from spontaneous emission along the fiber multiplied by the gain, Eq. (13). In Appendix B it is shown that for purely spontaneous scattering this summation is approximately equivalent to the injection of a single Stokes photon per mode at the point along the fiber where the nonlinear gain exactly equals the natural loss of the fiber, i.e., where γSp(0) exp(−αpz) = αs. When αs = αp the net effective gain for this single fictitious photon is [see Eq. (B6)]

Geff(ν)=exp{[Sp(0)γ(ν)]/αp}[Sp(0)γ(ν)/αp].

The total backward Stokes radiation is thus

Ps(0)=transversemodesdν(hν)·Geff(ν).

Neglecting the frequency dependence of the denominator in Eq. (14) and assuming a Lorentzian line shape, the effective bandwidth is given by Eq. (8) and the effective input Stokes power by Eq. (9). For the case of Brillouin scattering where the phonons are thermally activated the effective input Stokes power must be multiplied by (kT/a + 1) ≈ kT/a, where k is Boltzmann’s constant, T the temperature, and νa the frequency of the acoustic phonon.[5] This factor, equal to the average phonon occupation number, will be typically of the order of 100 to 200 depending upon the pump wavelength and the velocity of sound in the medium.

The critical power for the backward wave process is arbitrarily defined to be that input power for which the backward stimulated Stokes power equals the input pump power at z = 0. For a single transverse mode fiber the critical powers for stimulated Raman and Brillouin scattering are found from the following relations:

π2(hνs)(γ0αpA)ΔνfwhmRaman=(γ0PcritAαp)/25exp(-γ0PcritAαp),(Raman)
π2(νsνa)(kT)(γ0αpA)ΔνfwhmBrill.=(γ0PcritAαp)/25exp(-γ0PcritAαp).(Brillouin)

The relation for Pcrit′ for backward Raman scattering gives a greater critical power than forward scattering by roughly 25%, the relation being approximately

Pcrit=20(Aαp/γ0).

For typical values associated with Brillouin scattering and low loss single-mode fibers the critical power is approximately given by

Pcrit21(Aαp/γ0).

Equations (12), (18), and (19) all give roughly the same relation, the numerical factor in each case being approximately the natural logarithm of the gain required to bring the spontaneous emission to the level of the input pump power.

The expression for the peak gain coefficient for stimulated Brillouin scattering is[5]

γ0=(2π2νsνaM2)/c2αa,

where νs and νa are the frequencies of the Stokes wave and the acoustic phonon, respectively, αa is the attenuation constant of the acoustic intensity, c is the velocity of light, and M2 is the elastooptic figure of merit given by[6]

M2=n6p2/ρVa3.

In Eq. (21) n is the index of refraction, ρ the elastooptic constant, ρ the density, and Va the velocity of sound. For fused silica M2 = 1.51 × 10−18 sec3/g.[6]

Pine[7] has measured the room temperature line width for spontaneous Brillouin scattering in fused silica at λ = 6328 Å, obtaining a value of approximately 80 MHz. The corresponding attenuation constant is αa = 2πΔνfwhm/Va ≈ 800 cm−1. Walder and Tang[8] estimate an attenuation constant of 560 cm−1 at λ = 6943 Å from stimulated Brillouin measurements in the same material. In estimating the acoustic attenuation constant appropriate for λp = 1.064 μ and a corresponding acoustic frequency of 16.4 GHz, we take Pine’s directly measured value and assume a linear dependence on acoustic frequency giving αa ≃ 500 cm−1p = 1.06 μ). The peak gain coefficient thus becomes γ0 ≈ 3 × 10−9 cm/W. Taking αp = 5 × 10−5 cm−1 (≈20 dB/km) and A = 10−7 cm2, the critical power for stimulated Brillouin scattering, Eq. (19), is Pcrit = 35 mW. This value will be correspondingly lower if fibers of smaller cross sectional areas or lower attenuation constants are used. For example Rich and Pinnow[9] have reported bulk losses at 1.064 μ of less than 3 dB/km.

The critical powers derived above for stimulated forward and backward scattering processes implicitly assume a single frequency pump source. For a pump source of finite line width Eqs. (11), (16), and (17) approximately hold so long as the line width of the pump is less than the line width of the spontaneous scattering, Δνfwhm. For Raman scattering Δνfwhm is sufficiently large that Eqs. (11) and (16) are expected to be valid for most laser sources. On the other hand for Brillouin scattering at λp = 1.06 μ, Δνfwhm ≈ 50 MHz. Thus in Eq. (17) Pcrit refers to the power within a 50-MHz bandwidth and hence under most circumstances to the power in a single laser mode. In terms of the power handling capacity of a fiber, Eq. (17) is interpreted to imply that stimulated Brillouin scattering will not be of importance provided the power in any individual laser mode, or more specifically the power within any bandwidth of size Δνfwhm, is less than Pcrit.

The critical powers have also been derived assuming steady-state or cw operation. Consider the effects of using a pulsed source as would apply when a digital format is used. For the forward-wave interaction Eq. (12) would still apply to the peak pump power unless the pulse width of the pump were so short that either the bandwidth of its envelope exceeded the line width of the spontaneous emission in which case the transient scattering problem must be treated, or the difference in group velocities between the Stokes wave and the pump times the pulse width exceeds 1/αp in which case the interaction length is effectively shortened.

For backward wave interactions where the pump is pulsed the interaction length is limited to a value leffvgΔt/2, where vg is the group velocity and Δt is the pulse duration, due to the opposite direction of propagation of the waves. For a single pulse the critical pump power will exceed the values determined by Eqs. (18) and (19) by the factor ≈1/(leffαp). In the limit of very short optical pulses where the inverse of the pulse width exceeds the line width of the scattering process the transient nature of the scattering problem must be taken into account. This effect is most pronounced for stimulated Brillouin scattering and will further raise the threshold for stimulated scattering. There remains, however, the possibility of a succession of pump pulses each amplifying the backward traveling Stokes wave. For a consecutive string of ones, each with a duty cycle of D the critical power exceeds the value given by Eqs. (18) or (19) by the factor 1/D. Such an occurrence represents a worst case and is probably relatively unlikely to occur.

In the above calculations the modal character of the various fields have been neglected. For stimulated Raman scattering the principal modification caused by considering the detailed spatial variation of the fields is to introduce a filling factor or overlap integral into the expressions for the critical powers. For most cases this integral will be of the order of unity, and only a slight modification of the critical powers is expected. The same filling factor arguments apply to the case of stimulated Brillouin scattering in a single mode fiber. On the other hand for multimode fibers the Brillouin scattering threshold will be further modified by the fact that the different transverse modes within the fiber propagate with different phase velocities. In this case a particular backward traveling Stokes mode will interact with a given forward traveling pump wave with a corresponding phonon frequency given by

νa=νp(Va/c)(nm+nm),

where nm and nm′ are the effective refractive indices for the forward and backward modes. A given Stokes Brillouin wave will effectively interact with only those pump modes that give an acoustic frequency within the acoustic line width. This limitation on the range of modes which can effectively interact further raises the critical power for stimulated Brillouin scattering in multimode guides.

Discussion

The effect of stimulated Raman and Brillouin scattering on the optical power handling capacity of low loss optical fibers has been evaluated. The results are presented for each scattering process in terms of a critical power, defined as the input power level to the fiber for which the particular nonlinear scattering process is clearly of importance. Approximate expressions for the critical powers for forward and backward Raman scattering and backward Brillouin scattering are found in Eqs. (12), (18), and (19), respectively. Because of the exponential dependence of these stimulated processes on pump power level, a reduction of the input power from the critical value by approximately 1 dB reduces the level of the stimulated scattering by approximately 20 dB. Thus the power that may be sent down a fiber without serious interference from nonlinear effects will be typically of the order of 1 dB below Pcrit, and hence the critical power so defined serves as a useful benchmark.

The ultimate power handling capacity of a fiber will depend upon the fiber parameters A, α, γ0 as well as the spectral characteristics of the optical pump. When the pump source is a laser operating in a single longitudinal mode and hence with a narrow line width, stimulated Brillouin scattering will have the lowest critical power. The estimated critical power for a fused silica fiber with A = 10−7 cm2 and α = 5 × 10−5 cm−1 (≈20 dB/km) is 35 mW for a cw pump. The critical power for stimulated Raman scattering is one to two orders of magnitude greater than this value. When the pump source is multimode, stimulated Brillouin scattering will not be of importance so long as the pump power within any frequency interval Δνfwhm is less than the critical value. For an extremely broadband pump source stimulated Raman scattering will ultimately set the upper limit to the power handling capabilities of the fiber.

Appendix A: Forward Wave Interaction

The differential equation for the photon occupation number of a given Stokes mode is

[(d/dz)+αs]Ns=γSp(z)(Ns+1),

where the gain constant on the right-hand side is chosen to agree with the classical limit, and the additional factor of unity takes into account the spontaneous emission whose rate is

(d/dz)Nsspontaneous=γSp(z).

Neglecting spontaneous emission, the solution to Eq. (A1) is

Ns(z2)Ns(z1)=exp[αs(z1-z2)+z1z2γSp(z)dz].

We next argue huristically that the spontaneous emission from each incremental length—γSp(z)dz—will be amplified according to Eq. (A3). At any point z the photon occupation number of a given mode due to spontaneous emission, in addition to any injected photons Ns(0), is given by

Ns(z)=0zdξγSp(ξ)exp[αs(ξ-z)+ξzγSp(η)dη]+Ns(0)exp[-αsz+0zγSp(η)dη],

which is easily verified to be the general solution to Eq. (A1). Under the assumption of a pump intensity of the form Sp(z) = Sp(0) exp (−αpz), Eq. (A4) becomes

Ns(z)=0zdξγSp(0)exp{-αpξ+αs(ξ-z)+γSp(0)αp×[exp(-αpξ)-exp(-αpz)]}+Ns(0)×exp{-αsz+γSp(0)αp[1-exp(-αpz)]}.

Consider the case where there is no injected Stokes wave, Ns(0) = 0 and for simplicity assume αs = αp. At the exit plane, z = L Eq. (A5) becomes

Ns(L)=exp(-αsL)0LγSp(0)dξexp{γSp(0)αp×[exp(-αpξ)-exp(-αpL)]}.

For fiber systems of interest αpL ≫ 1 and the second exponential factor in braces is approximately zero. When γSp(0)/αp ≫ 1, as will be the case when stimulated emission is important, the major contribution to the integral will occur while αpξ < 1. Expanding exp (−αpξ) ≃ 1 − αpξ Eq. (A6) becomes

Ns(L)exp[-αsL˙+γSp(0)αp]0LγSp(0)dξexp[-γSp(0)ξ]=exp[-αsL+γSp(0)αp][1-exp(-γSpL)][-αsL+γSp(0)αp],

where γSpL > γSp/αp ≫ 1. From Eq. (A5) this is seen to be identical to the result where Ns(0) = 1 and spontaneous emission is neglected. Hence the net result of the amplification of all spontaneous emission is equivalent to the fictitious injection of a single photon at the input plane, z = 0.

Appendix B: Backward Wave Interaction

Let the incident pump wave travel in the +z direction from z = 0 to z = L with the direction of propagation of the Stokes wave in the −z direction. The differential equation for the photon occupation number of the backward wave is

[(d/dz)-αs]Ns=-γSp(z)(Ns+1).

Neglecting the spontaneous emission term the solution to Eq. (B1) is

Ns(z2)Ns(z1)=exp[αs(z2-z1)+z2z1γSp(z)dz].

The solution to Eq. (B1) including spontaneous emission as well as Ns(L) photons injected at z = L is

Ns(z)=zLdξγSp(ξ)exp[αs(z-ξ)+zξγSp(η)dη]+Ns(L)exp[αs(z-L)+zLγSp(η)dη],

which satisfies Eq. (B1). For a pump taking the form Sp(z) = Sp(0) exp (−αpz) Eq. (B3) becomes

Ns(z)=zLdξγSp(0)exp{αsz-(αs+αp)ξ+γSp(0)αp×[exp(-αpz)-exp(-αpξ)]}+Ns(L)exp{αs(z-L)+γSp(0)αp[exp(-αpz)-exp(-αpL)]}.

With no injected Stokes wave the total amplified spontaneous emission at the input plane, z = 0 is

Ns(0)=0LdξγSp(0)exp{-(αs+αp)ξ+γSp(0)αp[1-exp(-αpξ)]}.

For simplicity assume αs = αp = α and consider the limit αL ≫, then

Ns(0)=exp[γSp(0)/αp][γSp(0)/αp],

the result used in Eq. (15). Consider now Eq. (B1). The net gain for a real photon is gnet = γSp(z) − αs. For an exponentially decaying pump gnet = 0 when

γSp(0)exp(-αpz0)=αs

or

αpz0=ln[γSp(0)/αs].

Assume that a single photon is injected at L = z0 with no spontaneous emission and take αs = αp, Eq. (B4) then gives

Ns(0)=1/[γSp(0)/αp]exp{[γSp(0)/αp]-1}.

In the limit γSp(0)/αp ≫ 1, Eq. (B8) reduces to Eq. (B6), and hence the net effect of amplified spontaneous emission from the full interaction length is equivalent to the injection of a single photon per mode at z0 given by Eq. (B7).

References

1. E. P. Ippen, Appl. Phys. Lett. 16, 303 (1970) [CrossRef]  .

2. R. H. Stolen, E. P. Ippen, and A. R. Tynes, Appl. Phys. Lett. 20, 62 (1972) [CrossRef]  .

3. E. P. Ippen and R. H. Stolen, Paper F9, 7th International Quantum Electronics Conference, Montreal, May 1972.

4. W. D. Johnston Jr., I. P. Kaminow, and J. G. Bergman Jr., Appl. Phys. Lett. 13, 190 (1968) [CrossRef]  .

5. C. L. Tang, J. Appl. Phys. 37, 2945 (1966) [CrossRef]  .

6. D. A. Pinnow, in Handbook of Lasers, R. J. Pressley, Ed. (Chemical Rubber Co., Cleveland, 1972).

7. A. S. Pine, Phys. Rev. 185, 1187 (1969) [CrossRef]  .

8. J. Walder and C. L. Tang, Phys. Rev. Lett. 19, 623 (1967) [CrossRef]  .

9. T. C. Rich and D. A. Pinnow, Appl. Phys. Lett. 20, 264 (1972) [CrossRef]  .

References

  • View by:

  1. E. P. Ippen, Appl. Phys. Lett. 16, 303 (1970).
    [Crossref]
  2. R. H. Stolen, E. P. Ippen, A. R. Tynes, Appl. Phys. Lett. 20, 62 (1972).
    [Crossref]
  3. E. P. Ippen, R. H. Stolen, Paper F9, 7th International Quantum Electronics Conference, Montreal, May 1972.
  4. W. D. Johnston, I. P. Kaminow, J. G. Bergman, Appl. Phys. Lett. 13, 190 (1968).
    [Crossref]
  5. C. L. Tang, J. Appl. Phys. 37, 2945 (1966).
    [Crossref]
  6. D. A. Pinnow, in Handbook of Lasers, R. J. Pressley, Ed. (Chemical Rubber Co., Cleveland, 1972).
  7. A. S. Pine, Phys. Rev. 185, 1187 (1969).
    [Crossref]
  8. J. Walder, C. L. Tang, Phys. Rev. Lett. 19, 623 (1967).
    [Crossref]
  9. T. C. Rich, D. A. Pinnow, Appl. Phys. Lett. 20, 264 (1972).
    [Crossref]

1972 (2)

R. H. Stolen, E. P. Ippen, A. R. Tynes, Appl. Phys. Lett. 20, 62 (1972).
[Crossref]

T. C. Rich, D. A. Pinnow, Appl. Phys. Lett. 20, 264 (1972).
[Crossref]

1970 (1)

E. P. Ippen, Appl. Phys. Lett. 16, 303 (1970).
[Crossref]

1969 (1)

A. S. Pine, Phys. Rev. 185, 1187 (1969).
[Crossref]

1968 (1)

W. D. Johnston, I. P. Kaminow, J. G. Bergman, Appl. Phys. Lett. 13, 190 (1968).
[Crossref]

1967 (1)

J. Walder, C. L. Tang, Phys. Rev. Lett. 19, 623 (1967).
[Crossref]

1966 (1)

C. L. Tang, J. Appl. Phys. 37, 2945 (1966).
[Crossref]

Bergman, J. G.

W. D. Johnston, I. P. Kaminow, J. G. Bergman, Appl. Phys. Lett. 13, 190 (1968).
[Crossref]

Ippen, E. P.

R. H. Stolen, E. P. Ippen, A. R. Tynes, Appl. Phys. Lett. 20, 62 (1972).
[Crossref]

E. P. Ippen, Appl. Phys. Lett. 16, 303 (1970).
[Crossref]

E. P. Ippen, R. H. Stolen, Paper F9, 7th International Quantum Electronics Conference, Montreal, May 1972.

Johnston, W. D.

W. D. Johnston, I. P. Kaminow, J. G. Bergman, Appl. Phys. Lett. 13, 190 (1968).
[Crossref]

Kaminow, I. P.

W. D. Johnston, I. P. Kaminow, J. G. Bergman, Appl. Phys. Lett. 13, 190 (1968).
[Crossref]

Pine, A. S.

A. S. Pine, Phys. Rev. 185, 1187 (1969).
[Crossref]

Pinnow, D. A.

T. C. Rich, D. A. Pinnow, Appl. Phys. Lett. 20, 264 (1972).
[Crossref]

D. A. Pinnow, in Handbook of Lasers, R. J. Pressley, Ed. (Chemical Rubber Co., Cleveland, 1972).

Rich, T. C.

T. C. Rich, D. A. Pinnow, Appl. Phys. Lett. 20, 264 (1972).
[Crossref]

Stolen, R. H.

R. H. Stolen, E. P. Ippen, A. R. Tynes, Appl. Phys. Lett. 20, 62 (1972).
[Crossref]

E. P. Ippen, R. H. Stolen, Paper F9, 7th International Quantum Electronics Conference, Montreal, May 1972.

Tang, C. L.

J. Walder, C. L. Tang, Phys. Rev. Lett. 19, 623 (1967).
[Crossref]

C. L. Tang, J. Appl. Phys. 37, 2945 (1966).
[Crossref]

Tynes, A. R.

R. H. Stolen, E. P. Ippen, A. R. Tynes, Appl. Phys. Lett. 20, 62 (1972).
[Crossref]

Walder, J.

J. Walder, C. L. Tang, Phys. Rev. Lett. 19, 623 (1967).
[Crossref]

Appl. Phys. Lett. (4)

E. P. Ippen, Appl. Phys. Lett. 16, 303 (1970).
[Crossref]

R. H. Stolen, E. P. Ippen, A. R. Tynes, Appl. Phys. Lett. 20, 62 (1972).
[Crossref]

W. D. Johnston, I. P. Kaminow, J. G. Bergman, Appl. Phys. Lett. 13, 190 (1968).
[Crossref]

T. C. Rich, D. A. Pinnow, Appl. Phys. Lett. 20, 264 (1972).
[Crossref]

J. Appl. Phys. (1)

C. L. Tang, J. Appl. Phys. 37, 2945 (1966).
[Crossref]

Phys. Rev. (1)

A. S. Pine, Phys. Rev. 185, 1187 (1969).
[Crossref]

Phys. Rev. Lett. (1)

J. Walder, C. L. Tang, Phys. Rev. Lett. 19, 623 (1967).
[Crossref]

Other (2)

D. A. Pinnow, in Handbook of Lasers, R. J. Pressley, Ed. (Chemical Rubber Co., Cleveland, 1972).

E. P. Ippen, R. H. Stolen, Paper F9, 7th International Quantum Electronics Conference, Montreal, May 1972.

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Equations (39)

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P p ( z ) = P p ( 0 ) exp ( - α p z ) ,
[ ( d / d z ) + α s ] P s ( z ) = γ S p ( z ) P s ( z ) ,
[ ( d / d z ) + α s ] P s ( z ) = γ P s ( z ) S p ( 0 ) exp ( - α p z ) ,
P s ( z ) = P s ( 0 ) exp { - α s z + γ S p ( 0 ) α p [ 1 - exp ( - α p z ) ] } .
P s ( L ) P s ( 0 ) exp [ - α s L + γ S p ( 0 ) α p ] .
P s ( L ) = transverse modes d ν ( h ν ) exp [ - α s L + S p ( 0 ) α p γ ( ν ) , ] ,
P s ( L ) = { transverse modes ( h ν s ) exp [ - α s L + S p ( 0 ) γ 0 α p ] } B eff ,
B eff = π 2 Δ ν f w h m [ S p ( 0 ) γ 0 / α p ] 1 2 ,
P s ( 0 ) eff = ( h ν s ) ( B eff ) ( number of transverse modes . )
P s ( 0 ) eff exp [ - α s L + S p ( 0 ) γ 0 α p ] < P p ( 0 ) exp ( - α p L ) .
π 2 ( h ν s ) ( γ 0 A α p ) Δ ν f w h m = ( γ 0 P crit A α p ) / 2 3 exp ( - γ 0 P crit A α p ) .
exp ( 100 P crit ) = 2 × 10 8 P crit / 2 3 ,
P crit 16 ( A α p / γ 0 ) ,
G = exp { - α s z 0 + [ γ S p ( 0 ) / α p ] [ 1 - exp ( - α p z 0 ) ] } .
G eff ( ν ) = exp { [ S p ( 0 ) γ ( ν ) ] / α p } [ S p ( 0 ) γ ( ν ) / α p ] .
P s ( 0 ) = transverse modes d ν ( h ν ) · G eff ( ν ) .
π 2 ( h ν s ) ( γ 0 α p A ) Δ ν f w h m Raman = ( γ 0 P crit A α p ) / 2 5 exp ( - γ 0 P crit A α p ) , ( Raman )
π 2 ( ν s ν a ) ( k T ) ( γ 0 α p A ) Δ ν f w h m Brill . = ( γ 0 P crit A α p ) / 2 5 exp ( - γ 0 P crit A α p ) . ( Brillouin )
P crit = 20 ( A α p / γ 0 ) .
P crit 21 ( A α p / γ 0 ) .
γ 0 = ( 2 π 2 ν s ν a M 2 ) / c 2 α a ,
M 2 = n 6 p 2 / ρ V a 3 .
ν a = ν p ( V a / c ) ( n m + n m ) ,
[ ( d / d z ) + α s ] N s = γ S p ( z ) ( N s + 1 ) ,
( d / d z ) N s spontaneous = γ S p ( z ) .
N s ( z 2 ) N s ( z 1 ) = exp [ α s ( z 1 - z 2 ) + z 1 z 2 γ S p ( z ) d z ] .
N s ( z ) = 0 z d ξ γ S p ( ξ ) exp [ α s ( ξ - z ) + ξ z γ S p ( η ) d η ] + N s ( 0 ) exp [ - α s z + 0 z γ S p ( η ) d η ] ,
N s ( z ) = 0 z d ξ γ S p ( 0 ) exp { - α p ξ + α s ( ξ - z ) + γ S p ( 0 ) α p × [ exp ( - α p ξ ) - exp ( - α p z ) ] } + N s ( 0 ) × exp { - α s z + γ S p ( 0 ) α p [ 1 - exp ( - α p z ) ] } .
N s ( L ) = exp ( - α s L ) 0 L γ S p ( 0 ) d ξ exp { γ S p ( 0 ) α p × [ exp ( - α p ξ ) - exp ( - α p L ) ] } .
N s ( L ) exp [ - α s L ˙ + γ S p ( 0 ) α p ] 0 L γ S p ( 0 ) d ξ exp [ - γ S p ( 0 ) ξ ] = exp [ - α s L + γ S p ( 0 ) α p ] [ 1 - exp ( - γ S p L ) ] [ - α s L + γ S p ( 0 ) α p ] ,
[ ( d / d z ) - α s ] N s = - γ S p ( z ) ( N s + 1 ) .
N s ( z 2 ) N s ( z 1 ) = exp [ α s ( z 2 - z 1 ) + z 2 z 1 γ S p ( z ) d z ] .
N s ( z ) = z L d ξ γ S p ( ξ ) exp [ α s ( z - ξ ) + z ξ γ S p ( η ) d η ] + N s ( L ) exp [ α s ( z - L ) + z L γ S p ( η ) d η ] ,
N s ( z ) = z L d ξ γ S p ( 0 ) exp { α s z - ( α s + α p ) ξ + γ S p ( 0 ) α p × [ exp ( - α p z ) - exp ( - α p ξ ) ] } + N s ( L ) exp { α s ( z - L ) + γ S p ( 0 ) α p [ exp ( - α p z ) - exp ( - α p L ) ] } .
N s ( 0 ) = 0 L d ξ γ S p ( 0 ) exp { - ( α s + α p ) ξ + γ S p ( 0 ) α p [ 1 - exp ( - α p ξ ) ] } .
N s ( 0 ) = exp [ γ S p ( 0 ) / α p ] [ γ S p ( 0 ) / α p ] ,
γ S p ( 0 ) exp ( - α p z 0 ) = α s
α p z 0 = ln [ γ S p ( 0 ) / α s ] .
N s ( 0 ) = 1 / [ γ S p ( 0 ) / α p ] exp { [ γ S p ( 0 ) / α p ] - 1 } .

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