1. INTRODUCTIONFrom a naive point of view, as far as only longitudinalmodes are involved, multimode emission in lasers re-quires the existence of some kind of inhomogeneity in theamplifying medium, since in a perfectly homogeneous me-dium all modes interact with all atoms and competitionbetween modes ensures that only one of them survives.Inhomogeneities can be of a spatial or a spectral nature.Fabry–Perot-type resonators are inhomogeneous, as dif-ferent cavity modes can interact with molecules located atdifferent points in the cavity given the standing-waveform of the intracavity field, and thus mode–mode compe-tition is weakened (spatial hole burning). In unidirec-tional ring resonators this kind of inhomogeneity does notexist. Spectral inhomogeneity arises when differentgroups of atoms in the amplifying medium have differentresonances (e.g., in a gas laser this is due to the Dopplereffect). In this case, different groups of atoms differing intheir transition frequency can interact with different cav-ity modes and, again, mode–mode competition is weak-ened (spectral hole burning).
Thus at a first sight, multimode emission is not ex-pected to occur in a homogeneously broadened ring laser.Nevertheless, for large intracavity field intensity, nonlin-earities provide an alternative mechanism for multimodeemission even in a homogeneously broadened mediumthrough the Rabi splitting of the lasing transition. Thisis the so-called Risken–Nummedal–Graham–Haken(RNGH) instability.1–3 The existence of this instability iswell known since the late sixties, but it has been experi-mentally identified in lasers only three years ago in a ringerbium-doped fiber laser4 (EDFL) (there is a previous ob-
servation in optical bistability in a microwaveresonator5,6).
The above reasons lead to the conclusion that in inho-mogeneously broadened ring lasers two different mecha-nisms for multimode emission coexist since Rabi splittingoccurs also in inhomogeneously broadened media. Thisis the main concern of the present study; i.e., we are in-terested in determining the conditions for multimodeemission in inhomogeneously broadened ring-cavity la-sers and in clarifying whether the two mechanisms areseparable from each other.
Our main motivation for revisiting this basic problemis the understanding of what is really being observed inexperiments with ring EDFLs because these systems arenot homogeneously broadened, the inhomogeneous line-width being around two or three times the homogeneouswidth.7 Thus we find it important to clarify how theRNGH mechanism for multimode emission is modified bythe simultaneous presence of inhomogeneous broadeningor, in other words, which is the influence of the RNGH in-stability on the multimode emission in inhomogeneouslybroadened lasers.
Quite surprisingly, this subject has not been studied indetail to our knowledge. The only relevant previous re-search known by the authors is the derivation of the char-acteristic polynomial governing the instability made inRefs. 8 and 9. In Ref. 8 the analysis of multimode emis-sion is restricted to the limit of equal relaxation rates forpolarization (g') and population inversion (g i) and thelimit of large inhomogeneous broadening, and in Ref. 9,only the good cavity limit k ! g' , g i , k being the cavitylinewidth, is summarily considered. A systematic study
2001 Optical Society of America
1602 J. Opt. Soc. Am. B/Vol. 18, No. 11 /November 2001 Roldan et al.
of the problem is still lacking, and in particular the classB laser limit, defined as g i ! k ! g' , has never beenanalyzed to our knowledge.
We show below that there is not a sharp distinction be-tween the two mechanisms responsible for the instability,in the sense that there is always a single instability do-main, which reduces to that of the RNGH instability asthe inhomogeneous linewidth goes to zero. Yet we dem-onstrate that the relative importance of the two mecha-nisms depends strongly on the length of the laser cavity,in the sense that although coherent effects can be safelyneglected in short cavities, they become important for la-sers such as EDFLs with a cavity so long that the freespectral range is much smaller than the homogeneouslinewidth of the transition.
The rest of the paper is organized as follows. In Sec-tion 2 we introduce the model equations and derive thesingle-mode steady solutions and the condition for multi-mode emission when only spectral hole burning is consid-ered. In Section 3 we derive the characteristic equationsgoverning the linear stability of the single-mode solutionand derive the multimode-emission threshold in somespecial cases. The instability in class A (k ! g' , g i) andclass C (k ; g' ; g i) lasers is numerically addressed inSection 4. Class B lasers are considered in Section 5,where analytical expressions for the instability thresholdare derived for long and short cavities, and a quantitativecriterion for the cavity length above which coherent ef-fects play an essential role is given. These results areapplied to the case of EDFLs in Section 6. Finally, Sec-tion 7 contains the main conclusions.
2. MODEL AND STATIONARY SOLUTIONSWe consider an incoherently pumped and inhomoge-neously broadened two-level active medium of length Lm ,contained in a ring cavity of length Lc , interacting with aunidirectional plane-wave laser field. We assume thatthe cavity is resonant with the center of the atomic-transition frequencies distribution and that the cavitymirrors’ reflectivities are close to unity so that theuniform-field limit holds.
A. Maxwell–Bloch EquationsThe Maxwell–Bloch equations describing such a laser canbe written in the form8
~]t 1 ]z!F~z, t! 5 sF2F 1 AE2`
1`
dvL~v!PG , (1)
]t P~v, z, t! 5 g21@2~1 1 iv!P 1 FD#, (2)
]t D~v, z, t! 5 g@1 2 D 2 ~FP* 1 F* P !/2#. (3)
In these equations F(z, t) is the normalized slowly vary-ing envelope of the laser field, and P(v, z, t) andD(v, z, t) are the normalized slowly varying envelope ofthe medium polarization and the population inversion, re-spectively, for molecules detuned by v with respect to thecavity resonance. We use the adimensional time t andlongitudinal coordinate z, which are related with time tand space z through
t 5 Ag ig't, z 5 2pz/~ aLm!, (4)
where g i and g' are the decay rates of D and P, respec-tively, and
a 5 2pc/~LcAg ig'! (5)
is the adimensional free spectral range of the cavity, c be-ing the light velocity in the host medium. The decayrates s and g are defined as
s 5 k/Ag ig', g 5 Ag i /g', (6)
with k the cavity linewidth. With this definition for g theactual free spectral range of the cavity, FSR, reads
FSR 5 2pc/Lc 5 ~ga!g' ; (7)
hence ga represents the FSR measured in units of the ho-mogeneous linewidth. A is the incoherent pump param-eter. L(v) represents the spectral distribution of atomicresonances, which, in order to deal with analytical expres-sions, is taken as a Lorentzian distribution of width(HWHM) u,
L~v! 51
p
u
u2 1 v2 , (8)
where both v and u are frequencies scaled to g' . Withour notation the unscaled total gain linewidth (HWHM) ofthe medium is g'(1 1 u). Finally the boundary condi-tion for the electric field reads
F~0, t! 5 F~2p/a, t!, (9)
which means that F must be a periodic function of z;hence it can be expressed as a superposition of planewaves with a spatial wave number a equal to an integermultiple of a.
A remark on the influence of cavity detuning is in order.Here we are assuming that the gain line center is in reso-nance with a cavity longitudinal mode. The existence ofcavity mistuning would be important only for cavitieswhose free spectral range is larger than, or of the sameorder as, the homogeneous gain linewidth (see Ref. 10 forthe influence of detuning in the homogeneously broad-ened case). Clearly for long cavities, cavity detuning isirrelevant, since there is always a mode in resonance insuch a case. Thus the results derived below are valid forshort cavities in resonance and always valid, irrespec-tively, of the existence of cavity detuning, for long cavities.
B. Single-Mode SolutionsEquations (1)–(3) admit two types of steady solutions, thelaser-off solution (F 5 P 5 0, D 5 1) and a family ofsingle-mode lasing solutions of the form
F 5 AIa exp@i~az 2 Vt!#, (10)
P~v! 5 pa~v!exp@i~az 2 Vt!#, (11)
D~v! 5 da~v!, (12)
parameterized by the spatial wave number a. We obtain
pa~v! 51 2 i~v 2 gV!
1 1 Ia 1 ~v 2 gV!2AIa, (13)
da~v! 51 1 ~v 2 gV!2
1 1 Ia 1 ~v 2 gV!2 , (14)
Roldan et al. Vol. 18, No. 11 /November 2001 /J. Opt. Soc. Am. B 1603
A 5 AIa 1 1~AIa 1 1 1 u !
3 F1 1 S ga
AIa 1 1~1 1 gs! 1 uD 2G , (15)
together with the dispersion relation
V 5 aAIa 1 1 1 u
AIa 1 1~1 1 gs! 1 u, (16)
where Ia [ uFu2 is the steady-field intensity. The thresh-old for this single-mode solution reads [Ia 5 0 in Eq. (15)]
Ath~a! 5 ~1 1 u !@1 1 ~ga!2/~1 1 gs 1 u !2#. (17)
This threshold is minimum for the resonant mode a 5 0[for which V 5 0, according to Eq. (16)] and takes thevalue
A0 [ Ath~0 ! 5 1 1 u. (18)
A useful auxiliary variable is defined as
R 5 A1 1 I0, (19)
where I0 is the intensity of the resonant mode, in terms ofwhich we can write the ratio of the pump parameter to itslasing threshold value as
r 5 A/A0 5 R~R 1 u !/~1 1 u !, (20)
so that, at threshold, R 5 r 5 1.
C. Estimation of the Multimode-Emission Threshold inthe Pure Spectral-Hole-Burning LimitA simple way to estimate the multimode-emission thresh-old is by assuming that the threshold for amplification ofa detuned mode is not affected by the already existingresonant mode. This assumption is valid if (i) the inho-mogeneous width u is large, because in that case the in-tensity I0 of the resonant mode at threshold for multi-mode emission will be small, and (ii) if the frequency V (ora) of the detuned mode is sufficiently larger than the nor-malized homogeneous width g21, which is a measure ofthe width of the spectral hole. If these conditions are ful-filled, multimode emission is caused by a pure spectral-hole-burning mechanism.
Under these conditions a side mode of spatial frequencya will start oscillating at the pump value A 5 Ath(a)given by Eq. (17). In terms of the pump parameter r de-fined by Eq. (20) the threshold for multimode emissionreads
rth~a! 5 1 1 ~ga!2/~gs 1 1 1 u !2. (21)
Multimode emission is predicted for r . rth . The termga/(gs 1 1 1 u) represents the side-mode frequency off-set divided by the sum of the cavity linewidth and the to-tal gain linewidth.
3. STABILITY ANALYSISWe are interested in the conditions under which the reso-nant mode, which is the one with the lowest threshold,loses stability versus nonresonant perturbations. Wethus analyze the stability of the single-mode solution Eqs.(13)–(16), with a 5 V 5 0, versus perturbations of the
form dF(z, t) 5 df exp(iaz 1 lt), dP(v, z, t)5 dp(v)exp(iaz 1 lt), and dD(v, z, t) 5 dd(v)exp(iaz1 lt), where, in order to satisfy boundary condition (9), amust be an integer multiple of the free spectral range a.
A. Characteristic EquationsAfter some algebra, one obtains the following characteris-tic equations:
l 1 s 1 ia 5 s~1 1 gl!FJ2 1 S 1 2 gR2 2 1
g 1 lDJ0G ,
(22)1 1 ia/l 5 gs@J2 2 GJ0 /~1 1 gl!#, (23)
with
G 5 ~1 1 gl!@1 1 gl 1 g ~R2 2 1 !/~g 1 l!#, (24)
Jn 5 AE2`
1`
dvL~v!vn
~R2 1 v2!~G 1 v2!. (25)
The integrals J0 and J2 can be worked out explicitly,yielding
J0 5R 1 u 1 AG
AG~R 1 AG!~u 1 AG!, (26)
J2 5Ru
~R 1 AG!~u 1 AG!. (27)
Equation (22) is associated with an amplitude instability,and Eq. (23) is associated with a phase instability.1,2
They generalize those derived for g51 by Mandel in hisbook.8
Equations (22) and (23) cannot be solved analytically inthe general case. By a squaring operation they can bewritten, respectively, as a tenth- and a ninth-order poly-nomial, whose coefficients are too long to be given here.The stability boundaries shown in the Subsection 3.3 cor-respond to the points where the root of each of these poly-nomials with the largest real part changes its sign. Indoing that one must be careful, since the squaring opera-tion introduces spurious solutions.
Next we consider some special limits where analyticalinformation can be obtained from the characteristic equa-tions. The analysis of the class B laser limit, in whichgeneral analytical results can be derived, is in Section 5.
B. Some Special Cases
1. Homogeneously Broadened Transition (u50)In this limit A0 5 1, R2 5 r 5 A, J0 5 1/G, and J2 5 0,and the characteristic equations (22) and (23) reduce to
0 5 gl3 1 ~1 1 gs 1 g2 1 iag!l2 1 @g ~r 1 gs!
1 ia~1 1 g2!#l 1 g@2s~r 2 1 ! 1 ira#, (28)
0 5 gl2 1 g ~1 1 gs 1 iag!l 1 ia, (29)
respectively, which coincide with those obtained byRisken and Nummedal.1 Let us just remember that theminimum instability threshold occurs at the pump value1
rc 5 5 1 3g2 1 2A2~g2 1 1 !~g2 1 2 ! (30)
1604 J. Opt. Soc. Am. B/Vol. 18, No. 11 /November 2001 Roldan et al.
and at the frequency
ac 5 A3~rc 2 1 ! 2 g2
2 F1 2sg
1 1 g2
rc 2 1 1 g2
3~rc 2 1 ! 2 g2G .
(31)The instability threshold takes its minimum value, rc5 9, in the class B laser limit g ! 1, and correspond-ingly ac 5 A12. For g 5 1 we have rc 5 14.928 and ac5 4.515 2 0.826s. In the following subsections we onlyconsider the amplitude instability for reasons that areclarified below.
2. Small Inhomogeneous Broadening (u ! 1)In this limit the tenth-order characteristic polynomial as-sociated with Eq. (22) reduces to a sixth-order polynomialif one retains only up to terms linear in u. In general, nosimple analysis is possible. In the special case g 5 1,however, Eq. (22) can be simplified to a quartic,
and this is the case we consider. The boundaries of theinstability domain can be found setting l 5 2iv. Byequating separately to zero the real and imaginary partsof Eq. (32), we obtain
v4 2 ~3R2 1 2uR 2 4 !v2 1 2R~R3 1 uR2 2 R 2 2u !
5 0, (33)
a 5 @~2 1 s!v2 2 2s~R2 2 uR 2 1 !#/~2v!. (34)
At R 5 Rc , with Rc the solution of the equation
Rc4 1 4uRc
3 2 16Rc2 1 16 5 0, (35)
the two roots v62 of Eq. (33) coalesce and are equal to
vc2 5 3Rc
2/2 1 uRc 2 2. (36)
The corresponding critical value ac for a can be obtainedsetting R 5 Rc and v 5 vc in Eq. (34). This approxi-mate result is strictly valid for u ! 1, but we havechecked numerically that the results provided by theanalysis are remarkably good for values of u as large as0.76. Equation (35) can be solved perturbatively in pow-ers of u. Making use of Eqs. (20), (34), and (36), we ob-tain the first-order correction to the RNGH values for rcand ac :
Even very small values of u can lower dramatically the in-stability threshold, whereas ac changes little, especially ifs is of order 1.
3. Inhomogeneous Limit (u @ 1)We look for an asymptotic solution of Eq. (22) in the formR 5 R0 1 R1 /u 1 R2 /u2 1 ... and l 5 l0 1 l1 /u1 l2 /u2 1 ... . At order 0 and order 1 we obtain, re-spectively, R0 5 1, l0 5 2ia, and R1 5 0, l1 5 iags.
At order 2 we obtain the first real correction to l that setsthe instability condition, which reads R2 5 a2f(a, g),with
f~a, g!
5g4a6 1 g2~5 1 g4!a4 1 ~4 1 5g4!a2 1 4g2
g2a6 1 ~1 1 3g2 1 g4!a4 1 ~6 2 4g2!a2 2 8. (39)
Taking into account Eq. (20), the instability is predictedto occur for
r > 1 1 ~a/u !2f~a, g!. (40)
This result represents the first correction to the instabil-ity threshold (r 5 1) obtained by Mandel8 for the specialcase g 5 1. For g 5 1, Eq. (39) reduces to
f~a, 1! 5 ~a2 1 1 !2/~a4 1 a2 2 2 !. (41)
The threshold is the minimum at ac2 5 2.177, and it takes
the value rc 5 1 1 4.469/u2. For g ! 1 and a 5 O(1),Eq. (39) can be approximated with
f~a, 0! 5 4a2/~a4 1 6a2 2 8 !. (42)
In this case the threshold is the minimum at ac2 5 8/3,
and it is given by rc 5 1 1 32/17u2.
4. Short-Cavity Limit (ga @ 1)In this case [see Eq. (7)] the free spectral range of the cav-ity is much larger than the homogeneous linewidth.Since g < 1 by definition, the condition ga @ 1 impliesa @ 1. The physically allowed values for a 5 na arethus accordingly large. As the imaginary part of the un-stable eigenvalue l is of order a, the eigenvalue itself is oforder a, and then both l and gl are, in absolute value,much larger than one. This circumstance allows for a no-ticeable simplification of the problem. In fact, assumingR2 2 1( 5 I0) 5 O(1), we can neglect the term g (R2
2 1)/(g 1 l) in Eqs. (22) and (24) so that AG ; 1 1 gl; gl, and the right-hand member of the characteristicequation reduces to sAG(J0 1 J2). Moreover, we canneglect R with respect to AG in J0 and J2 and write theapproximated characteristic equation
l 5 2ia 1 sFu 1 gl~Ru 1 1 !
gl~u 1 gl!2 1G . (43)
We are considering the uniform-field limit in which, bydefinition, the cavity linewidth s ! a( < a). ThereforeEq. (43) tells us that the leading term in l is 2ia, i.e., itis purely imaginary, and the stability condition can be ob-tained evaluating the real part of the next-order correc-tion, which is of order s.
Up to now we have made no assumption on the order ofmagnitude of u. If u 5 O(1), the first term inside thesquare brackets is of order 1/gl ; 1/ga, and thus it canbe neglected with respect to 1. The approximated solu-tion is l 5 2ia 2 s, which means that, in this limit, theresonant mode never loses stability against a side mode.This is expected since, in units of g' , the frequency ga ofthe considered side mode is much larger than the inhomo-geneous width u, which is of order 1. Notice that thisdoes not mean that there is no multimode instability in ashort cavity with moderate inhomogeneous broadening,
Roldan et al. Vol. 18, No. 11 /November 2001 /J. Opt. Soc. Am. B 1605
but that this instability occurs for large values of thepump, for which the approximation R 5 O(1) does nothold.
If u ; ga(@1) the numerator inside the square brack-ets can be approximated by glRu (since Ru @ 1 and gl@ 1) and Eq. (43) can be approximated by
l 5 2ia 1 s@21 1 Ru/~u 1 gl!#. (44)
Since a @ s, we can look for an approximated solution ofthe form l 5 2ia 1 l0 , with l0 given by
l0 5 s@21 1 Ru/~u 1 gl!#l52ia
5 sS Ru2
u2 1 g2a2 2 1 1 iRuga
u2 1 g2a2D . (45)
Taking into account that r 5 R in the limit u @ 1, the in-stability condition to the leading order is
r > 1 1 ~ga/u !2, (46)
which exactly coincides with our estimate, Eq. (21), in thelimit of large u. We have demonstrated that Eq. (21)gives the threshold for multimode operation with a goodapproximation in the limiting conditions where it wassupposed to be valid, i.e., for ga @ 1 (short cavity), andu @ 1 (large inhomogeneous broadening).
We finally note that Eq. (46) coincides with Eq. (40) fora @ 1. Thus this last equation can be regarded as valid,whenever u @ 1, independently of the order of magnitudeof a.
4. MULTIMODE INSTABILITY IN CLASS AAND CLASS C LASERSAs there are two characteristic equations, one for the am-plitude, Eq. (22), and another for the phase, Eq. (23), onecould expect the existence of two different instabilitymechanisms. But this is not the case because the ampli-tude instability always precedes the phase instability.We cannot demonstrate this analytically, but we have ex-haustively tested this conclusion numerically. Moreover,since Eq. (22) gives rise to a single instability boundary,there is a single instability leading to multimode emis-sion, and consequently there is not an analytical distinc-tion between the RNGH instability and multimode emis-sion due to inhomogeneous broadening.
In this section we address these questions in class Aand class C lasers, defined by the relations k ! g i ,g' ( s ! g, g21) and k ; g i ; g' ( s ; g ; g21), respec-tively. Class B lasers are analyzed in Section 5.
In Fig. 1 the instability boundary is represented in theplane (r, a) for the special case g 5 1, s 5 0.01 (class Alaser) and for the values of the inhomogeneous width u in-dicated in the figure. We have checked that the samequalitative results are obtained with s 5 1 (class C la-ser). The first remarkable feature is the continuous tran-sition observed in the curves from homogeneous broaden-ing (RNGH instability) to inhomogeneous broadeningcases. The multimode instability threshold lowers dra-matically as the inhomogeneous width is increased. Forexample, for homogeneous broadening u 5 0 the (RNGH)instability occurs at the minimum pump rc 5 14.928given by Eq. (30); for u 5 1 this threshold has lowered to
approximately 3, and for u 5 10 it is very close to unity.For small u this behavior is in good agreement with theanalysis of Subsection 3.B.2. This means that the insta-bility is essentially of the RNGH type, but inhomogeneousbroadening, even in a small amount, produces strong ef-fects. This is more clearly appreciated in Fig. 2(a), wherethe minimum instability threshold rc is plotted as a func-tion of u for s 5 0.01 and two values of g.
Fig. 1. Multimode-emission threshold for g 5 1, s 5 0.01, andfor the indicated values of the inhomogeneous gain linewidth u.The single-mode solution is unstable to the right of each curve.a is the scaled frequency of a side mode, which must be an inte-ger multiple of the free spectral range a, and r is the pump pa-rameter scaled to the laser threshold.
Fig. 2. (a) Minimum instability threshold rc and (b) instabilitythreshold at the spatial frequency a 5 10 as functions of the in-homogeneous gain linewidth u for two values of g.
1606 J. Opt. Soc. Am. B/Vol. 18, No. 11 /November 2001 Roldan et al.
Figure 1 also shows that the curvature of the instabil-ity boundary decreases rapidly, especially for the largestvalues of u. In other words, the decreasing of the insta-bility threshold is even more dramatic for large values ofa. In Fig. 2(b) we represent the instability threshold r asa function of u for a 5 10, i.e., around three times thevalue of a at which the instability threshold is minimumfor u 5 0, for two values of g and s 5 0.01.
This behavior of the instability threshold has a clearphysical meaning if we remember that for a cavity of fixedlength, the wave numbers that can be amplified are thoseexceeding a certain minimum given by the free spectralrange a of the cavity. For a homogeneously or a slightlyinhomogeneously broadened laser line a long cavity(small a) is necessary to obtain multimode emission; forlasers with large inhomogeneous broadening, multimodeemission is easy to achieve even for short cavities (withlarge a). Moreover, for a given cavity length the numberof unstable modes will be much larger than in the absenceof inhomogeneous broadening, a circumstance favorablefor self-mode locking if mode coupling is strong enough.
Let us compare now the exact curves obtained herewith those predicted in the pure spectral-hole-burninglimit. Figure 3 is similar to Fig. 1, but we have alsoadded (dashed curves) the multimode-emission thresholdpredicted by Eq. (21). It can be seen that in the limit ofvalidity of Eq. (21), that is, ga @ 1 and u @ 1, the ap-proximation is indeed very good. Equation (21) fails forsmall u, even if ga is large, because in this case thethreshold for multimode emission is relatively high,which means that at threshold the resonant mode isstrong and nonlinear effects cannot be neglected. Thefailure of Eq. (21) for small ga and any value of u, beyondthe above reason, is because, when the frequency separa-tion is comparable with the homogeneous linewidth, it be-comes also comparable with the Rabi splitting induced bythe resonant mode; hence a full description of mode cou-
Fig. 3. Comparison between the exact instability threshold(solid curves) and the instability threshold predicted by Eq. (21)in the pure spectral-hole-burning limit (dashed curves), for thesame g and s of Fig. 1, and for the indicated values of the inho-mogeneous gain linewidth u.
pling (including coherent effects) is necessary to deter-mine correctly the multimode instability.
Before proceeding to the analysis of the class B limit, inFig. 4 we display the two instability thresholds for ampli-tude and phase instability in the (r, a) plane for two val-ues of g, s 5 0.01, and u 5 2. As we advanced at the be-ginning of this section, the instability threshold providedby the phase equation (dashed curve) is always precededby the one provided by the amplitude equation (solidcurve). Nevertheless, it is to be remarked that the twoboundaries approach each other for large a. This factshould have consequences on the dynamics of the laserbeyond the instability threshold, which should be differ-ent for large and short cavities since in the latter case thephase instability should manifest itself together with theamplitude instability.
5. MULTIMODE INSTABILITY IN CLASS BLASERSA class B laser is defined by the conditions g i ! k! g' (g ! 1, g ! s ! g21). For this kind of laser it isconvenient to consider separately the limits (a) a @ 1 and(b) a 5 O(1), i.e., ga ! 1. Condition (a) includes theshort-cavity limit ga @ 1 already discussed in Subsection3.B.4 but also the case of a cavity of intermediate lengthwhose free spectral range is of the same order as the ho-mogeneous linewidth, i.e., ga 5 O(1). Limit (b) is thelong-cavity limit.
A. Limit a š 1In a class B laser the condition a @ 1 is enough to writeEq. (22) in a form similar to that of Eq. (43) even if R isnot of order 1, but its value is bounded by the much lessrestrictive condition R2 ! a/g. In fact, this condition isenough to neglect the term g (R2 2 1)/(l 1 g) in Eqs.(22) and (24). The only differences with respect to Eq.(43) is that now, in general, AG ; 1 1 gl (since gl maybe of order 1), and R cannot be neglected in J0 and J2 (be-
Fig. 4. Instability threshold associated with the amplitude in-stability (solid curves) and the phase instability (dashed curves)for the indicated values of g. s 5 0.01, u 5 2.
Roldan et al. Vol. 18, No. 11 /November 2001 /J. Opt. Soc. Am. B 1607
cause it may be of the same order as AG or even larger).The approximated characteristic equation reads
l 5 2ia 1 sFR 1 u 1 ~1 1 gl!~Ru 1 1 !
~R 1 1 1 gl!~u 1 1 1 gl!2 1G , (47)
and the approximated solution is l 5 2ia 1 l0 , with l0given by
l0 5 sFR 1 u 1 ~1 1 gl!~Ru 1 1 !
~R 1 1 1 gl!~u 1 1 1 gl!2 1G
l52ia
. (48)
After some calculation, one finds
Re l0 5 2sg2a2D21@g2a2 1 H~R, u !#, (49)
with D 5 @(R 1 1)(u 1 1) 2 g2a2#2 1 g2a2(R 1 u1 2)2 . 0, and
H~R, u ! 5 R2~1 2 u ! 1 R~1 2 u !~u 1 2 ! 1 ~u 1 1 !2.
(50)
Thus the instability condition (Re l0 . 0) is
g2a2 1 H~R, u ! , 0. (51)
This inequality coincides with the instability conditionderived by Khanin in his book [Eq. (4.154) of Ref. 11], us-ing a rate-equations model where coherent effects (popu-lation pulsations) are neglected. Thus we can say thatEq. (51) gives the instability threshold when coherentmode–mode coupling is unimportant and only gain satu-ration is relevant. An important property of Eq. (51) isthat it sets a necessary condition for the instability to oc-cur in the considered limit, given by the inequality u. 1.
Let us now consider the condition R2 ! a/g we haveassumed and check whether it is consistent with Eq. (51).Clearly R at the instability threshold is larger the largera is and the smaller u is. Assuming u 5 O(1) andga @ 1, it follows from Eq. (51) that R ; ga, and pre-cisely
R 5 ga/Au 2 1 1 O~1 !. (52)
Then R2 ; g2a2, and the condition R2 ! a/g requires a! g23. Thus Eq. (51) seems to be valid for 1 ! a! g23. We show that, however, Eq. (52) remains validwhen a 5 O(g23) or larger; thus Eq. (51) is correct forany a @ 1. We then consider the scaling gR2 ; a ; l5 O(g2n) with n > 3. In this limit it is easy to showthat the characteristic equation Eq. (22) has theasymptotic form
l 5 2ia 1 s@21 1 Ru/~R 1 gl!# (53)
for any n. The approximated solution again has the forml 5 2ia 1 l0 , with l0 given by
l0 5 s@21 1 Ru/~R 1 gl!#l52ia ; (54)
thus
Re l0 5 2@g2a2 1 R2~1 2 u !#/~g2a2 1 R2!, (55)
and the instability is produced at the same value of Rgiven by Eq. (52).
Let us finally note that, using Eq. (20), we can write Eq.(51) in the form
r > 1 1 ~ga!2/~1 1 u !2 1 2R~R 1 1 !/~1 1 u !2
> rth~a!ug!1 , (56)
where rth(a) is the approximate instability threshold inthe pure spectral-hole limit given in Eq. (21). The insta-bility threshold given by Eq. (51) is thus always largerthan the one predicted by Eq. (21), which was admittedlyvalid for a @ 1. Both boundaries tend to coincide in thelimits of validity of Eq. (21), i.e., u @ 1 and R 5 O(1).Equation (51), with respect to Eq. (21), contains one in-gredient more, namely, the saturation of the sidebandgain due to the resonant mode, and this explains why itgives a larger value for the instability threshold.
B. Long-Cavity LimitWhen a 5 O(1), Eq. (51) loses its validity because coher-ent effects become important. However, unlike in thegeneral case, in the case of a class B laser, analytical re-sults can be derived in this case also for any value of u.We start from the exact equations (22) and (23) and lookfor a perturbative solution setting l 5 l0 1 gl1 1 g2l21 ... (with g ! 1). We then solve the characteristicequations at the different orders in g. The amplitudeequation (22) gives l0 5 2ia,
l1 5 isa/~1 1 u !
3 F1 2 uR 2 1
R 1 12 2
~R 2 1 !~R 1 u 1 1 !
a2 G ,
(57)
which is purely imaginary, and
Re l2 5 2sPamp /@a~R 1 1 !~u 1 1 !#2, (58)
where
Pamp 5 H~R, u !a4 2 3~R2 2 1 !
3 @~R 1 u 1 1 !2 2 Ru#a2
1 R~R2 2 1 !~R 1 1 !2~R 1 u !~u 1 2 !. (59)
Thus the amplitude-instability condition (Re l2 . 0) is
Pamp , 0, (60)
which does not depend on s. With respect to the phaseequation (23) we obtained l0 5 2ia, and
l1 5 2isa@u~R 2 1 !/~R 1 1 ! 2 1#/~1 1 u !, (61)
Re l2 5 2sPphase /@~R 1 1 !~u 1 1 !#2, (62)
where
Pphase 5 H~R, u !a2 1 Ru~R2 2 1 !~R 1 u 1 2 !, (63)
the phase-instability condition being
Pphase , 0, (64)
which again does not depend on s. Notice that this con-dition, as well as condition (51), can be satisfied only ifu . 1.
Equation Pamp 5 0 is a quadratic equation in a2, as itoccurs in the RNGH instability.1,3 Like in the homoge-neous case, the minimum instability threshold is given by
1608 J. Opt. Soc. Am. B/Vol. 18, No. 11 /November 2001 Roldan et al.
the value R 5 Rc for which the discriminant is equal tozero. Hence Rc is the solution of the equation
~2u 1 1 !2R5 1 ~2u 1 1 !2~2u 1 3 !R4
1 ~4u4 1 16u3 1 27u2 1 10u 2 6 !R3
1 ~10u3 1 9u2 2 26u 2 26!R2
1 ~u 1 1 !2~5u2 2 8u 2 27!R 2 9~u 1 1 !4 5 0,
(65)
and the corresponding critical value for a is
ac2 5 3~Rc
2 2 1 !@~Rc 1 u 1 1 !2 2 Rcu#/2H~Rc , u !.(66)
In the limit of small inhomogeneous broadening u ! 1 weobtain the approximated expressions
rc 5 9 2 36u 1 O~u2!, (67)
ac2 5 12 2 33.75u 1 O~u2!, (68)
which show that the effects of a small amount of inhomo-geneity is even more relevant than in the case g 5 1. Inthe opposite limit u @ 1 one gets
rc 5 1 1 32/~17u2! 1 O~u23!, (69)
ac2 5 8/3 1 80/~17u2! 1 O~u23!. (70)
These expressions agree with those derived in Subsection3.B.3. We note that, to determine the 1/u2 correction toac
2, it was necessary to calculate rc up to the term of order1/u4 [not shown in Eq. (69)]. This explains why in theanalysis of Subsection 3.B.3, which was limited to the1/u2 correction to rc , we simply found ac
2 5 8/3.
C. Comparison of the Different LimitsWe want to establish a connection among the different in-stability conditions for class B lasers, namely, condition(51), valid for short cavities and cavities of intermediatelength, and conditions (60) and (64), valid for long cavi-ties.
The coefficients of the leading terms in a in Eqs. (59)and (63) coincide and are equal to H(R, u). Thus forlarge a both the amplitude and phase instabilities occur,to the leading order, at the same value of R 5 R` givenby H(R` , u) 5 0:
R` 5u 1 2
2FA1 1
4~u 1 1 !2
~u 2 1 !~u 1 2 !2 2 1G , (71)
which requires u . 1 (for u , 1, R` does not exist, whichmeans that in the limit a → ` the value of R does nottend to a finite constant). Thus the amplitude- andphase-instability boundaries display a vertical asymptoteon the plane (r, a) at r 5 r` 5 R`(R` 1 1)/(u 1 1).Two important limiting cases are R` , r` → ` as u → 1,and R` , r` → 1 1 4/u2 as u → `. We also note that theinstability threshold given by condition (51) also readsH(R, u) 5 0 when ga is small; thus this boundary tendsto r 5 r` for ga → 0.
As in the general case, the phase-instability thresholdgiven by condition (64) is always larger than theamplitude-instability threshold given by condition (60).Therefore in the following we focus on the amplitude in-stabilities. We can say that r` is the value to which the
instability threshold tends in the upper limit of the valid-ity range of condition (60) and in the lower limit of the va-lidity range of condition (51).
Let us consider a specific value of g, namely, g5 1023. Condition (51) is valid, in principle, for a5 O(g21) or larger, i.e., a ; 103 or larger; condition (60)is valid for ga ! 1, i.e., a ! 103. Thus at a ; 10 –100and r ; r` we expect to observe a smooth transition fromone instability threshold to the other.
These considerations are summarized in Fig. 5, wherewe represent for g 5 1023 and u 5 4 the exact instabilitythreshold obtained from Eq. (22) (diamonds) and the ap-proximated results given by conditions (51) (dotted curve)and (60) (solid curve). Note the logarithmic scale in the aaxis. The two approximated curves work well in their re-spective domains of validity. Moreover, for a relativelylarge value of u such as u 5 4, even Eq. (40) gives a verygood approximation of the exact curve over the entirerange of a considered (not shown). For the calculation ofthe exact curve of Fig. 5 we set s 5 0.1 (note that thevalue of s does not appear in the approximated expres-sions, as commented).
D. When Are Coherent Effects Important?We can estimate which is the value of a 5 acoh that sepa-rates the domains of validity of conditions (60) and (51).Physically acoh separates the region where coherent ef-fects are essential (a , acoh) from that where mode–mode coupling is basically given by gain saturation (a. acoh). This transition can be defined to occur whenthe horizontal distance between the two curves in Fig. 5 isat minimum. The value of acoh cannot be calculated ex-actly, but we find numerically that it is between 2.3g21/2
for u * 5 and 4g21/2 for u 5 1.5 [remember that condi-tion (51) predicts no multimode emission for u < 1].With this hint an approximated expression for acoh can beobtained by looking at asymptotic solutions to conditions(60) and (51) for a 5 O(g21/2) in the class B limit g→ 0. In this way we obtain R 5 R1 5 R` 2 f1(u)/a2
Fig. 5. Comparison between the exact instability threshold in aclass B laser (diamonds) and the approximated curves valid inthe limit a @ 1 (dotted curve) and a 5 O(1) (solid curve). Theparameters are g 5 0.001, s 5 0.1, and u 5 4.
Roldan et al. Vol. 18, No. 11 /November 2001 /J. Opt. Soc. Am. B 1609
1 O(1/a4) from condition (60) and R 5 R2 5 R`
1 f2(u)g2a2 1 O(g2a4) from condition (51), where
f1~u ! 5 3R`
2 2 1
u 2 1
R`2 1 ~u 1 1 !2 1 ~u 1 2 !R`
2R` 1 u 1 2, (72)
f2~u ! 5 @u~u 2 1 !~8 1 7u 1 u2!#21/2. (73)
From these expressions we compute the squared distances 5 (r1 2 r2)2 between both thresholds by use of Eq. (20),which, to the leading order, reads
s 5 ~2R` 1 u !2@ f1~u ! 1 g2a4f2~u !#2/@a2~1 1 u !#2.
(74)Finally, we minimize s with respect to a with the result
g2acoh4 5 f1~u !/f2~u !. (75)
Since (for u . 1) f1(u), f2(u) . 0, the value acoh exists,and this justifies a posteriori the correctness of the scalinga 5 O(g21/2) used in the derivation. From Eq. (75) wefind that acoh is a decreasing function of u that tends to241/4g21/2 5 2.21g21/2 for u → ` and diverges as u → 1.By way of example, Eq. (75) yields acoh 5 2.36g21/2 for u5 5 and acoh 5 3.83g21/2 for u 5 1.5, which comparewell with the numerics. From Eq. (5), the cavity lengththat separates the domains of validity of conditions (60)and (51), i.e., the cavity length above which coherent ef-fects must be taken into account, can be estimated to be ofthe order of (taking a 5 2g21/2)
Lcoh 5 pc~g' /g i!1/4g'
21. (76)
For example, for CO2 lasers Lcoh ' 100 m, for 632.8-nmHe–Ne lasers Lcoh ' 20 m, for Er31-doped fiber lasersLcoh ' 5 cm, and for Nd–glass lasers Lcoh ' 3 mm.Thus the necessity of using the full Maxwell–Bloch de-scription of the laser to describe multimode emission de-pends strongly on the particular laser system under con-sideration: it is necessary for Nd–glass and Er31-dopedfiber lasers and unnecessary for CO2 and He–Ne lasers.Finally, notice that the expression for Lcoh gives a good es-timate for the critical length for values of u > 2; forsmaller values of u, Lcoh can be considerably smaller(tending to 0 for u → 1) so that the necessity of consider-ing coherent effects is still more important in this limit.
6. APPLICATION TO Er31-DOPED FIBERLASERSThe class B limit analyzed in Section 5 is of particular rel-evance since it applies to the case of a ring EDFL [g. 1026, s 5 O(1)],12 which is the only laser system inwhich the RNGH instability has been identified so far.4,13
These lasers are, however, inhomogeneously broadenedwith an inhomogeneous width that is two or three timesthe homogeneous width,7 i.e., 2 & u & 3. Following ourprevious analysis, and given that the typical length ofthese lasers (;10 m) is orders of magnitude larger thanLcoh ('5 cm), coherent effects are essential for correctlydescribing multimode emission in EDFLs.
To apply the previous analyses to a fiber laser, it mustbe taken into account that in such a laser the amplifyingmedium must be described as a three-level rather than a
two-level system.14 In Ref. 12 it was demonstrated thatthe two-level model equations apply to such a laser bysuitably modifying the pump parameter and the relax-ation of the population inversion.
The pump parameter A of the equivalent two-levelmodel is linked to the optical pumping rate W by
A 5 G~W 2 g i!/~W 1 g i!, (77)
where G is the radiation–matter coupling constant, whichis commonly a quantity much larger than unity in ED-FLs. From Eq. (77) it follows that
W 5 g i~G 1 A !/~G 2 A !
5 @G 1 R~R 1 u !#/@G 2 R~R 1 u !#, (78)
where Eq. (20) is used. The optical pumping at laserthreshold (A 5 A0 5 1 1 u) is
W0 5 g i~G 1 1 1 u !/~G 2 1 2 u !, (79)
and the ratio W/W0 can be written as
W
W05
G 1 R~R 1 u !
G 2 R~R 1 u !
G 2 1 2 u
G 1 1 1 u. (80)
As for the relaxation rate for population inversion (g i), inthe equivalent two-level model it must be multiplied bythe coefficient r, defined as12
r 5 2G/~G 2 A ! 5 2G/@G 2 R~R 1 u !#. (81)
Consequently, the frequency a in Eq. (59) must be substi-tuted with a/Ar because of the normalization used inwriting Eqs. (1)–(3). With this change for a, equationPamp 5 0 provides the instability boundary for a three-level laser as a function of R, which is translated in termsof W/W0 by using Eq. (80). Let us remark that, as al-ready observed in Ref. 12, the large value of G character-izing EDFLs already produces a relevant reduction of theratio between instability and laser threshold if this ratiois measured in terms of the optical pump parameter W,which is the actual control parameter in the experiments,and not in terms of the rescaled pump parameter A.
In Fig. 6 we represent the instability boundary for G5 60 (Ref. 12) and u 5 0, 1, 2, and 3. Notice that inho-mogeneous broadening largely reduces the instabilitythreshold but not as dramatically as in the two-level la-ser. For Er31-doped fiber lasers the most important ef-fect of inhomogeneous broadening is the enhancement ofthe number of modes that become unstable for a givenpump value and becomes virtually infinite for W largerthan its value at the vertical asymptotes in Fig. 6. Theseasymptotes are given by
W`
W05
G 1 R`~R` 1 u !
G 2 R`~R` 1 u !
G 2 1 2 u
G 1 1 1 u(82)
and take the value 1.10 for u 5 2 and 1.05 for u 5 3 withour choice of the parameter G. Thus in a three-level la-ser pumped relatively close to threshold the theory pre-dicts that a very large number of modes are unstable.Note in the figure that the value of a 5 amin for which thethreshold is minimum is ;2 both for u 5 2 and u 5 3. Ifthis minimum threshold is to be detected, amin must be aninteger multiple of a, which implies [through Eq. (5)] that
1610 J. Opt. Soc. Am. B/Vol. 18, No. 11 /November 2001 Roldan et al.
the cavity length must be an integer multiple of the quan-tity Lc,min 5 pc/Ag ig', which is 26 m for the parametersof EDFLs.12 This value is to be compared with the cor-responding prediction (Lc,min 5 11 m) from the homoge-neous broadening model of Ref. 12.
The analysis of Ref. 12 had demonstrated that thethreshold for multimode emission can be much lower thanthe one predicted by the standard RNGH equations if thethree-level structure of the active material is taken intoaccount. This represented a first step in the comprehen-sion of the experimental results reported in Ref. 4. How-ever, one problem remained open, namely, the fact that,with the typical parameters of EDFLs, a 5 O(1) andtherefore only a few modes turned out to be unstable if in-homogeneous broadening is neglected. This was in con-trast with the experimental evidence of self-mode lockingwith a large number of active modes.
Now we have shown that inhomogeneous broadeningstrongly modifies the unstable domain, in such a way thata very large number of modes can be unstable even forsmall values of the pump. Therefore self-mode locking isjustified theoretically.
Let us remark that for a full comparison with the ex-periment the present analysis should be extended to thecase in which the mirror’s reflectivity is so low that theuniform-field approximation does not apply,4,13 but this isbeyond the scope of the present paper and will be the sub-ject of future research.
7. CONCLUSIONSWe have derived the multimode-emission threshold for aring-cavity laser with Lorentzian inhomogeneous broad-ening from a full Maxwell–Bloch description of the sys-tem. Our analysis shows that even a small amount of in-homogeneous broadening strongly reduces the thresholdfor multimode emission. Moreover, when the inhomoge-neous broadening is large, the shape of the unstable do-main changes considerably, and modes highly out of reso-
Fig. 6. Multimode-emission threshold in an erbium-doped fiberlaser with G 5 60 for the indicated values of the inhomogeneousgain linewidth u.
nance can become unstable. We have also analyzed howthe relative importance of the two physical mechanisms,spectral hole burning and coherent effects, that producethe instability varies as the cavity length changes.
In a short cavity, defined as a cavity whose free spectralrange is much larger than the homogeneous linewidth,not only coherent effects but even gain saturation at thefrequency of the side mode can be safely neglected, pro-vided the inhomogeneous broadening is large. Thismeans that multimode emission is due in this case to apure spectral-hole burning mechanism.
When the free spectral range becomes comparablewith, or smaller than, the homogeneous linewidth, coher-ent effects and gain saturation can no longer be neglected.In the case of class B lasers we were able to derive ap-proximated analytical expressions for the instabilitythreshold for any value of the cavity length. We haveshown that for a free spectral range of the same order asthe homogeneous linewidth the threshold for multimodeemission can be correctly estimated taking into accountonly gain saturation. In this case the multimode-emission threshold is given by Eq. (51), an equation al-ready derived by Khanin11 by neglecting coherent effects(population pulsations). However, for longer cavities theexact result can only be obtained by including coherent ef-fects. In this case the multimode-emission threshold isgiven by Eq. (60). This equation is one of the main re-sults of this paper together with the finding of a quanti-tative criterion for the minimum cavity length, Lcoh [Eq.(76)], above which coherent effects (and thus a fullMaxwell–Bloch description) must be considered.
These considerations are particularly important forEDFLs, which are typically characterized by a free spec-tral range much smaller than the homogenous linewidth.Our analysis shows that, even in the presence of inhomo-geneous broadening, the rate-equations approach is inad-equate to describe multimode emission in such lasers, anda full Maxwell–Bloch description is needed, which takesinto account coherent effects such as Rabi splitting. Inthis case the most outstanding effect of the inhomoge-neous broadening is a dramatic flattening of the instabil-ity threshold with respect to the side-mode frequency,leading to the amplification of a huge number of sidemodes for low pumps. This may pave the way for the un-derstanding of the self-mode locking behavior experimen-tally observed even very close to the first laser threshold.
ACKNOWLEDGMENTSWe gratefully acknowledge interesting discussions withFedor Mitschke (Universitat Rostock, Germany). Thisresearch has been supported by the Spanish Government(Direccion General de Ensenanza Superior e Investiga-cion Cientıfica) through Project PB98-0935-C03-02.
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