Theoretical analysis of nth-order cascadedcontinuous-wave Raman fiber lasers.

II. Optimization and design rules

Bryan Burgoyne, Nicolas Godbout, and Suzanne Lacroix

Centre Optique, Photonique et Laser, Laboratoire des fibres optiques, Department de genie physique, EcolePolytechnique de Montreal, C. P. 6079 Station Centre-Ville, Montreal, Quebec H3C 3A7, Canada

Received October 5, 2004; accepted December 3, 2004

Using the analytical model we developed for an nth-order cascaded Raman laser, we find the design rules forsuch lasers. We determine analytical expressions for the cavity length and the output-coupler reflectivity thatmaximize the output power and minimize threshold power. Simple expressions are obtained in the depleted-pump approximation. Deviations from these expressions when the pump is not completely depleted areshown to be different depending on the parity of the number n of Stokes shifts (cascades). The design rulesshow that the mirror reflectivity is a critical factor in the laser quality and that the ultimate slope efficiency �is found to be gn /g0 . We also find a condition to determine if P-doped fibers are more useful than Ge-dopedfibers in Raman fiber lasers based only on the Raman shift and absorption of the fibers. © 2005 Optical So-ciety of America

OCIS codes: 140.3550, 140.3510.

1. INTRODUCTIONRaman fiber lasers are based on stimulated Raman scat-tering, a third-order, nonlinear interaction between theoptical field and material phonons.1 In a Raman fiber la-ser, a broad Raman gain spectrum is created because ofthe amorphous nature of silica,2 which allows a customi-zable operation wavelength. With the high power han-dling capacity of all-fiber technology, Raman fiber laserscan be tailored to specific applications. Hence, they haveattracted much attention as pumps for networkamplifiers3 or high-power applications.4 Since the gainmedium is nonlinear, it is difficult to predict the behaviorof such lasers. In previous efforts numericalsimulations5,6 were used to determine the optimal param-eters in a specific case but no generalization has beenachieved. We developed an approximate analyticalmodel to predict the behavior of cascaded Raman fiberlasers.7 We now use this model to determine the optimallaser parameters. Our analysis shows which parametersare critical in the laser behavior and yields design rulesfor cascaded, Raman fiber lasers.

2. THEORETICAL MODELUsing the theoretical model we previously developed7 forcascaded Raman lasers as in Fig. 1, it is possible to findthe expressions for the output power Pout , thresholdpower Pthres , and slope efficiency � when the number ofStokes waves n is either even or odd. The model is validas long as the laser spectrum is limited by the Bragg-grating transmission spectrum (i.e., no significant spec-tral broadening is assumed). Depending on the numbern of cascades, those expressions are found to be, for neven

Pout � �ln�Rn

�Rn��

1 � �Rn�

Rn�

�1 � Rn��

�1 � Rn��

� � P infp

4g0L� �j�2

j even

n

� j ��0

2g0� � �

j�1j odd

n�1

� j� , (1)

Pthres � 4g0L� �j�2

j even

n

� j ��0

2g0� � �

j�1j odd

n�1

� j� 1

fp, (2)

�

� �ln�Rn

�Rn��

1 � �Rn�

Rn�

�1 � Rn��

�1 � Rn��

fp

4g0L� �j�2

j even

n

� j ��0

2g0� ,

(3)

where

fp � 1 � R0� exp��4g0L� �

j�2j even

n

� j ��0

2g0� � , (4)

� j �� j

2gj�

ln�Rj�Rj

��

4gjL; (5)

and for n odd

772 J. Opt. Soc. Am. B/Vol. 22, No. 4 /April 2005 Burgoyne et al.

0740-3224/2005/040772-05$15.00 © 2005 Optical Society of America

Pout � �ln�Rn

�Rn��

1 � �Rn�

Rn�

�1 � Rn��

�1 � Rn��

� � P in � Pr

4g0L ��j�1j odd

n

� j

� �j�2

j even

n�1

� j ��0

2g0� , (6)

Pthres � 4g0L� �i�1

i odd

n

� j� � �i�2

i even

n�1

� j ��0

2g0�

� Pr�Pthres�, (7)

� � �ln�Rn

�Rn��

1 � �Rn�

Rn�

�1 � Rn��

�1 � Rn��

1 �dPr

dP in

4g0L �j�1

j odd

n

� j

(8)

In these definitions the subscript j relates to the respec-tive Stokes wave (0 being the pump). P in is the inputpump power, L the cavity length (assumed the same forall the Stokes waves) gj the Raman gain, � j the attenua-tion of the fiber, and Rj

� the Bragg-grating reflectivity atthe output (�) and input (�) ends of the cavity. The ex-pression for the residual pump power Pr is found to be dif-ferent depending whether n is odd or even. For n even

Pr � P inR0� exp��4g0L� �

j�2j even

n

� j ��0

2g0� � ; (9)

for n odd

Pr � P inR0� exp� �

P in � Pr

�j�1

j odd

n

� j � . (10)

3. OPTIMIZATION OF THE LASER:DEPLETED-PUMP APPROXIMATIONA. Optimal Cavity Length and Output-CouplerReflectivityThe aim is to optimize the output power of the laser toachieve maximum efficiency. To do so, different param-eters of the laser can be varied; however, in practice it isusually easier to tailor the cavity length and the Bragg-grating reflectivity than the fiber parameters. We thustry to find the optimal cavity length and output-couplerreflectivity. The optimization of these parameters fromthe analytical expressions of Eqs. (1)–(8) is possible, butdifficult unless some simplifications are made. First, weassume that Rn

� (the high reflector of the last Stokeswave) is sufficiently close to unity to set �Rn

�/Rn�(1

� Rn�)/(1 � Rn

�) � 0. Considering that nowadaysBragg gratings can be manufactured with a reflectivitybetter than 99%, this approximation is justified. We alsoassume that the pump power is completely depleted, i.e.,Pr � 0 and fp � 1 [Eq. (4)]; this is not a restrictive ap-proximation since all the available pump power should beused in an optimized laser. With use of the additionalapproximations, both even and odd cases give almostidentical expressions.

The approximate optimal cavity length Lopt and output-grating reflectivity Ropt are found by differentiating Eq.(1) [or Eq. (6)] with respect to L and Rn

� and setting bothderivatives equal to zero. Both second-order derivativesare verified to be negative so as to insure that Pout is amaximum. We thus obtain in the even case

Lopt,even � �BeBo /AeAo�1/2, (11)

Rn,opt,even� �

1

Rn�

exp� �4gn� Be

Ao� 1/2� � P in

4g0� 1/2

� �AeBo � �Ao � Be� � , (12)

where

� � �ln�Rn

�Rn��

4gn, Ae � �

j�0j even

n� j

2gj, Ao � �

j�1j odd

n�1� j

2gj,

Be � � �j�2

j even

n�2 ln�Rj�Rj

��

4gj, Bo � � �

j�1j odd

n�1 ln�Rj�Rj

��

4gj.

(13)

Ae and Ao are linked to losses and gain while �, Be , andBo are related to the reflection losses at each end of thefiber. We obtained the same expression in the odd caseby switching the e and o subscripts and taking the sum-mations up to n�1 instead of n (and vice versa) in Eqs.(13).

From Eq. (11), we see that the optimized cavity lengthLopt is dictated by the ratio of the reflection losses to thepropagation losses. Thus a longer cavity is preferred tominimize the number of reflections (hence minimizing re-flection losses), while a shorter cavity builds more intra-cavity power and a shorter distance is required to transferthe optical power from one Stokes wave to the next (hence

Fig. 1. Diagram of an nth-order cascaded Raman fiber laser. j is the wavelength of the jth Stokes wave. Only the outputcoupler (OC) is not highly reflective. The vertical lines repre-sent the Bragg gratings used as reflectors with reflectivities Rj

� .

Burgoyne et al. Vol. 22, No. 4 /April 2005 /J. Opt. Soc. Am. B 773

minimizing absorbed power). This can be clearly seen ifwe assume an average absorption (� j � �), average gain( gj � g), and identical intermediate reflectors (Rj

�Rj�

� R2). Eq. (11) then becomes

Lopt,even � �� n � 2

n � 2 � 1/2 ln�R �

�� �0.23� n � 2

n � 2 � 1/2 RdB

�.

(14)

It appears from relation (14) that the optimization de-pends much on the quality of the reflectors. The optimalcavity length changes exponentially with the output-coupler reflectivity, meaning that ultimately it is the qual-ity of the intermediate reflectors that determines thequality of the laser. As a result, to build more compactfiber Raman lasers, one must have access to very-high-reflectivity fiber Bragg gratings. It is notable that in thisapproximation the optimized length Lopt does not dependon the injected pump power. Thus the cavity length re-mains optimized for the different power levels of opera-tion of the laser. When there are only two Stokes waves,Eq. (11) becomes Lopt � 2g1 /�1 and does not depend onthe quality of the reflectors. Finally we see that the op-timal cavity length increases slowly with the number ofcascades.

From Eq. (12) we see that the optimized output-couplerreflectivity Rn,opt

� , as opposed to the optimized length, de-pends on the injected pump power. This implies that thepower level of operation of the laser must be set before de-termining the output-coupler reflectivity. Once again,assuming an average absorption and gain and identicalintermediate reflectors (Rj

�Rj� � R2) provides expres-

sions easier to examine:

Rn,opt� �

1

Rexp� �2 ln�R ���� 1 �

2

n � gP in

� ln�R �

� �n2

4� 1 �

n

2� 1� � . (15)

From relation (15) we see that the output-coupler re-flectivity depends on the fiber used through the absorp-tion and the Raman gain coefficients. Note that the ap-proximate optimal length given by relation (14) dependsonly on �. The quality of the intermediate reflectors rep-resented by R also has a nonnegligible impact on the op-timal output-coupler reflectivity Rn,opt

� , which is alsostrongly dependent on the number n of cascades in the la-ser. This can be explained as follows.

In the stimulated Raman interaction, the effectivenessof the pumping process from one wave to the other is dic-tated by the power of each wave; higher power meansfaster conversion. The power in a resonator is maxi-mized by a short cavity and high-reflectivity mirrors. Sothe conditions for the optimal cavity length implicitly ap-pear here and affect the output reflectivity. However,this only maximizes the conversion efficiency and not theoutput power. A high output-coupler reflectivity does notlet much power out of the cavity, and thus moreroundtrips are made by the last Stokes wave in the cavitybefore it can come out. This means more reflection and

propagation losses. Relations (12) and (15) give the opti-mal output-coupler reflectivity that balances conversionefficiency and losses.

When n is even Eq. (12) is replaced by

R2,opt� �

1

Rn�

exp� �4gn

Ao� � P inAe

4g0Bo� 1/2

� Ae� � . (16)

It is also interesting to see what happens when weminimize the threshold instead of the output power. Thelength that minimizes the threshold power can also becalculated through the derivative:

Lthres,even � �BeBo � ��/AeAo�1/2. (17)

The optimal threshold length [Eq. (16)] is identical to theoptimal output-power length [Eq. (11)] except for the �term. Both terms become identical when � � 0, corre-sponding to Rn

� � 1 and Rn� � 1 (no output coupler). As

can be seen by relation (15) the optimal value of Rn� gets

closer to 1 as R approaches unity. As a result, by usinghigh-quality intermediate reflectors, not only is the out-put power maximized, the threshold power is minimized.

Since we have shown that the quality of the laser is ul-timately determined by the intermediate Bragg-gratingreflectivity, let us define the optimal output power Pout* ,the power at threshold, and the efficiency one can hope toachieve with perfect reflectors (Rj

� � 1 except for the out-put coupler Rn

�) in the even case:

Pout* � �ln�Rn��� P in

g0� 4AeL �ln�Rn

��

gn� � Ao� , (18)

Pthres* � g0Ao�4AeL �ln�Rn

��

gn� , (19)

�* �1

g0� 1

gn�

4AeL

ln�Rn��

��1

. (20)

In this perfect-reflector case the laser quality is limitedby the absorption. It is interesting to note that Ae and Aoplay different roles in these equations. To maximize theoutput power it is preferable to minimize Ae because ithas a greater effect on Pout* than does Ao . However,minimizing Ao has a greater impact on reducing thethreshold power. From Eq. (19) the maximum achiev-able efficiency in a perfect laser is � � gn /g0 , the ratio ofthe Raman gains at the pump and output frequencieswhen L � 0 or Ae � 0. This suggests that the fibershould be designed to minimize absorption at the evenStokes wave frequencies to achieve high efficiency andmaximum output power.

Eq. (19) also shows that the slope efficiency increaseswhen the cavity length or output-coupler reflectivity de-creases. However, when L → 0 and–or Rn

� → 0, we findthat Pthres → �. The limit behaviors of the thresholdpower and slope efficiency are shown in Table 1 (the re-sults for n odd are found by switching the e and o sub-scripts). The data indicate that for very long or veryshort cavities the threshold rises, which is in agreementwith our finding an optimal length in Eq. (16). Thethreshold power is minimized when Rout � 1 because the

774 J. Opt. Soc. Am. B/Vol. 22, No. 4 /April 2005 Burgoyne et al.

losses in the cavity are minimal in this case, but the slopeefficiency is zero because no power is coming out of thecavity.

B. Germanium Ge- versus P-Doped FibersIt has been suggested that P-doped fiber is more efficientin Raman lasers than Ge-doped fiber because the former’sRaman shift is three times larger (40 THz instead of 13.2THz). However, the losses of such fibers are typicallymuch higher, probably due to higher scattering. It isthus interesting to determine whether the increased Ra-man shift offsets the losses in multi-Stokes lasers. Usingthe model described above in the depleted-pump approxi-mation, we define a criterion that delineates the condi-tions under which the use of P-doped fiber is advanta-geous. First we write the optimal output power bysubstituting the optimal cavity length from Eq. (11) andthe optimal output-coupler reflectivity from Eq. (12) inEq. (1):

Pout,opt � 4gn�P in/4g0�1/2 � �AeBo�1/2 � �AoBe�

1/2�2.(21)

The criterion for choosing P-doped over Ge-doped fiber isobviously

Pout,optP Pout,opt

Ge . (22)

When we observe that the Raman gain is roughly thesame for both fibers,8,9 Eq. (21) becomes

�AeBo�1/2 � �AoBe�P

1/2 � �AeBo�1/2 � �AoBe�Ge

1/2.(23)

If we then assume that all intermediate gratings are iden-tical and that the P-doped fiber attenuation differs fromthe Ge-doped one by a factor F� , we now have

� F� �j�0

j even

n�3j

g3j�j�1

j odd

n�1 1

g3j� 1/2

� � F� �j�1

j odd

n�1�3j

g3j�j�2

j even

n�2 1

g3j� 1/2

� �j�0

j even

3n� j

gj�j�1

j odd

3n�1 1

gj� 1/2

� � �j�1

j odd

3n�1� j

gj�j�2

j even

3n�2 1

gj� 1/2

,

(24)

where n is now the number of cascades in the P-doped fi-ber laser and the 3 in the indices comes from the relativeRaman shifts (40 THz/13.2 THz � 3). Relation (24)gives the maximum attenuation a P-doped fiber can haveto be more efficient than a Ge-doped fiber. For the typi-cal values given in Table 2 of Ref. 7 for a laser pumped at

0 � 1.06 �m and emitting at 0 � 1.48 �m, and forn � 2 (corresponding to two cascades in the P-doped fiberand six in the Ge-doped one), relation (24) yields

F� ��P

�Ge� 7.6. (25)

The high absorption coefficient in Table 27 is due to waterabsorption. The P-doped fiber can exhibit up to seventimes more attenuation and still be more efficient in a Ra-man laser than a Ge-doped fiber. It is interesting to notethat the condition set by relation (24) does not depend onthe Bragg-grating reflectivity but only on the gain–absorption ratio. Since the number of cascades requiredto reach the designated wavelength is three times less forthe P-doped fiber, there are much lower propagation andreflection losses.

The optimal parameters for a P-doped fiber are obvi-ously not the same as for a Ge-doped fiber. Since the ab-sorption is higher and there are fewer reflections to reachthe same frequency by using the P-doped fiber than by us-ing the Ge-doped fiber, a shorter cavity length and a low-reflection output coupler are preferred.

4. OPTIMIZATION OF THE LASER:UNDEPLETED PUMPWe now consider the case in which the pump power is notcompletely depleted and see how the optimal parameterscalculated previously are affected. We see from Eqs. (1)–(8) that it is difficult to calculate explicitly the optimalcavity length and output-coupler reflectivity. Instead ofproceeding with the optimization, let us examine thevariation of the residual power. We see from Eqs. (9) and(10) that the expression for the residual pump power Pr isdifferent depending whether the number of cascades n isodd or even. This means that the two cases do not be-have similarly if there is a residual pump power. Onemajor difference is that in the n odd case, the expressionof Pr is transcendental. If we take the derivatives for Prwith respect to L and Rn

� , the minima are found whenPr � 0. From Eqs. (9) and (10) we see that in both n oddand n even cases, increasing the value of L decreases theresidual pump power. Thus the optimal length calcu-lated with Eq. (11) underestimates the actual optimizedlength. This behavior is seen in Fig. 2 where the outputpower of a six-cascade laser calculated by using Eq. (1) isplotted as a function of the cavity length and output-coupler reflectivity for both the depleted and undepletedpump cases with the parameters given in Table 2.7

Those parameters are typical of a Corning PureMode1060™ fiber. We see that the optimal cavity length islonger when we use the undepleted pump equations inboth n odd and n even cases. However, since Poutchanges very slowly around the optimal cavity length, useof the depleted-pump approximation creates an outputpower difference of a few milliwatts from the actual maxi-mal value.

We now examine how the output-coupler optimal reflec-tivity changes when one does not use the depleted-pumpapproximation. In the n even case Eq. (9) shows that alow-reflectivity output coupler minimizes Pr , while in the

Table 1. Parameter Limit Values, n Even Case

Parameter Threshold Efficiency, �

L → 0 Pthres → � gn

g0

�(1 � R0�)

Be � �

L → � Pthres → � 0

Rn� → 0 Pthres → � gn /g0

Rn� → 1 Pthres � 4g0(AeL � Be)(Ao � Bo /L) 0

Burgoyne et al. Vol. 22, No. 4 /April 2005 /J. Opt. Soc. Am. B 775

n odd case it is the opposite: A high-reflectivity outputcoupler is preferred from Eq. (10). Hence, we expect alower value of the optimized output reflectivity Rn,opt

� inthe n even case and a higher one in the n odd case thanthe values calculated with Eq. (12). As seen in Fig. 2, wedo have a lower value of Rout,opt in the n even case but wealso have a lower value for the n odd case. This is ex-plained by the residual pump power Pr diminishing veryrapidly when the cavity length and OC reflectivity are in-creased because of the transcendence. This behavior isclearly seen in Fig. 3 where Pr is plotted as a function of L

and Rout . Hence, Pr is negligible (Pr � 81 mW) whenboth L and Ropt are optimal. Since Pout changes slowlyaround the optimal OC reflectivity, the depleted-pump ap-proximation is still relatively good, miscalculating theoutput power by only a few tens of milliwatts.

5. CONCLUSIONUsing an approximate analytical model describedelsewhere,7 we obtained analytical expressions of the op-timal cavity length and output-coupler reflectivity thatmaximize the laser output power and slope efficiency andminimize threshold power. Our analysis showed that thequality of the laser is critically dependent on the reflec-tivity of the Bragg gratings used in the cascade. Themaximum achievable slope efficiency was shown to be� � gn /g0 . We found a condition that indicates whetherit is preferable to use P-doped fibers instead of standardGe-doped fibers in the laser. This condition depends onthe fiber’s intrinsic gain–loss ratio and on the number ofcascades in the laser. Whenever the pump is not com-pletely depleted, we showed that the laser behaves differ-ent depending whether the number n of cascades waseven or odd. We found that the optimal parameters ob-tained with the depleted-pump approximation are withina few percent of the peak output power calculated nu-merically.

REFERENCES1. R. W. Boyd, ‘‘Stimulated Raman scattering and stimulated

Rayleigh–Wing scattering,’’ in Nonlinear Optics (Academic,1992), pp. 365–388.

2. R. H. Stolen, J. P. Gordon, W. J. Thomlinson, and H. A.Haus, ‘‘Raman response function of silica-core fibers,’’ J.Opt. Soc. Am. B 6, 1159–1165 (1989).

3. M. D. Mermelstein, C. Headley, J. C. Bouteiller, P. Stein-wurzel, C. Horn, K. Feder, and B. J. Eggleton, ‘‘Config-urable three-wavelength Raman fiber laser for Raman am-plification and dynamic gain flattening,’’ IEEE Photon.Technol. Lett. 13, 1286–1288 (2001).

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7. B. Burgoyne, N. Godbout, S. Lacroix, ‘‘Solution of nth-order,cascaded, continuous-wave, Raman fiber lasers. I. Ana-lytical model,’’ J. Opt. Soc. Am. B 22, 764–771 (2005).

8. E. M. Dianov, ‘‘Advances in Raman fibers,’’ J. LightwaveTechnol. 20, 1457–1462 (2002).

9. E. Saulnier, N. Godbout, and S. Lacroix, ‘‘Simultaneousmeasurement of spontaneous and stimulated Raman scat-tering in optical fibers, in Photonics North 2004: OpticalComponents and Devices, S. Fafard, ed., Proc. SPIE 5577,196–203 (2004).

Fig. 2. Output power (in dBm) plotted against the cavity lengthand the output-coupler reflectivity as calculated from Eq. (1).Parameters used for calculation are given in Table 2 of Ref. 7.The left column shows the output power calculated under thedepleted–pump approximation. The top row represents the neven case while the bottom row represents the n odd case. Theopen circle is the optimal cavity length and OC reflectivity ob-tained by using analytical expressions of Eqs. (11) and (12) in thedepleted-pump approximation.

Fig. 3. Residual pump power in dBm plotted against cavitylength and output-coupler reflectivity in the n odd case. The re-sidual pump power diminishes rapidly when the cavity lengthand–or OC reflectivity are–is increased.

776 J. Opt. Soc. Am. B/Vol. 22, No. 4 /April 2005 Burgoyne et al.