Litton Guidance and Control Systems, MS 19, 5500 Canoga Avenue, Woodland Hills, California 91365
Murray Sargent III
Optical Sciences Center, University of Arizona, Tucson, Arizona 85721
Received January 9, 1984; accepted August 15, 1984
We use the semiclassical strong-signal theory of the laser to predict and explain the onset of side-mode buildup inlasers with one oscillating mode. Two general categories are considered: one for which the side modes and the os-cillating mode all have the same wavelength and the other for which they have different wavelengths. The treat-ments include an arbitrary amount of inhomogeneous broadening. Our approach unifies the treatments of theside-mode instabilities presented earlier and extends them to handle standing waves in addition to the previouslytreated running waves. We write the field and the population matrix elements as Fourier series in the adjacent-mode beat frequency. This approach has been used extensively in both multimode laser theory and saturationspectroscopy. This technique coincides with linear stability analyses used by others, provided that our beat fre-quency includes a contribution that is proportional to the complex side-mode gain. We give a solution that allowsfor detuned operation along with its simpler, centrally tuned special case. The connection with saturation spec-troscopy clearly reveals that the side-mode instabilities require side-mode gain. For the single-wavelength case,nonlinear anomalous dispersion is also required. The side-mode gain and dispersion result from both inhomo-geneous broadening and population pulsations. The lowest instability thresholds occur when both of these mecha-nisms play a role. The approach can also be used to treat instabilities in optical bistability by substituting the ap-propriate equation of state for the strong-mode intensity and by changing the sign of the absorption coefficient.In homogeneously broadened, standing-wave lasers, we show that multiwavelength instabilities depend stronglyon the position of the medium in the cavity. We illustrate the theory by giving numerical results for the output pul-sation frequency and for the instability threshold by using parameters that are appropriate for the He-Xe laser.These results correlate well with experimental observations.
1. INTRODUCTION
A basic question in laser physics is: Given a laser with a singleoscillating mode, can side modes build up? In Doppler-broadened gas lasers, more than one mode can easily oscillate,since, because of spectral hole burning, different modes caninteract with different atoms. Similarly, in standing-wavelasers with homogeneously broadened media, different modescan interact with different atoms because of spatial holeburning. It is not so obvious, however, in a homogeneouslybroadened, unidirectional ring laser, that more than one modecan oscillate. Early laser stability analyses showed that asufficiently intense mode can nevertheless allow side modesto build up. Two general categories were considered: onefor which the side modes and the strong mode have the samewavelength and the other for which each mode has a differentwavelength.
The first category, which occurs in bad cavities, is referredto here as single wavelength, since all the interacting lasermodes are shown to acquire the same wavelength in the me-dium. This case was later shown to be chaotic.' For a reviewof the early development of these instabilities, see Refs. 1 and2.. In 1968, Risken and Nummedal 3 and Graham and Haken 4
extended the above analysis to high-Q, unidirectional cavitiesand discovered an instability for which several modes withdifferent wavelengths can oscillate simultaneously in a ho-mogeneously broadened medium. This instability is referredto in this paper as multiwavelength, since the lasing modeshave different cavity-mode numbers.
More recently, Casperson 5 6 found that side modes couldbuild up with the same wavelength in bad-cavity, inhomo-geneously broadened lasers because of a combination ofside-mode gain and anomalous dispersion generated in thenonlinear polarization by spectral hole burning. The firstconnection between this instability and the earlier instabilitiesthat were predicted in homogeneously broadened media isgiven in Refs. 7 and 8. It was also shown that the early in-stabilities are due to population pulsations (PP's) and that,when combined with the spectral hole-burning mechanism,the threshold for side-mode buildup is reduced appreciably. 7
We also suggested that the chaos predicted for purely homo-geneously broadened systems would persist with the inclusionof inhomogeneous broadening. 8
The reduction of the instability threshold with the increasein homogeneous broadening was discussed in further detailby Minden and Casperson,' 0 who used the same frequency-decomposition techniques suggested by Casperson 6 and usedin Refs. 6-9. This technique was also extended in Ref. 9 tothe detuned operation using a linear stability analysis.Similar solutions for the eigenmodes of the laser were alsogiven by Hillman et al.11 Although Lugiato et al.1 2 showedthat the above solutions apply only to the boundary of theinstability domain, these solutions have the advantage thatthey lend themselves quite well to a physical interpretationof the onset of the instabilities. On the other hand, general-ized linear stability analyses do not have such limitations, aswas shown in the treatments of Refs. 12-16. In this paper,we describe in detail the solutions outlined in Refs. 7-9 and
also describe a simple iterative treatment that generalizes thesolutions in Ref. 9 so as to cover the full domain of the insta-bility. This generalization unifies the above saturation-spectroscopy-type solutions with linear stability analysis.The solutions presented here for the detuned multiwavelengthcase coincide with those of Zorel.13 However, Zorel does nottreat the single-wavelength case and does not show explicitlythe effect of detuning on the pulsation frequency and on thethreshold intensity.
A wealth of experimental information about single-wave-length laser instabilities is provided by Caspersonl,5 and byAbraham and co-workers.17 -23 Such side-mode instabilitieshave been observed in He-Xe lasers5"17-21 and in He-Ne la-sers,21 23 revealing both pulsating and chaotic outputs. Re-cent research in a homogeneously broadened, dye-laser sys-tem2 4 shows that multiwavelength instabilities are also ac-cessible in high-gain, good-cavity laser systems. Other lasersystems also show pulsing behavior, as is discussed in Ref.1.
In this paper, we show that a common thread in all theseinstabilities is fluctuations in the population difference (i.e.,in the PP's) resulting from the presence of correspondingpulsations in the field. They represent the attempt of thepopulation difference to follow any beat frequency that existsbetween laser modes. As is the case in spectral hole burning,the PP's produce a region of anomalous dispersion, in whichseveral frequencies are simultaneously resonant in thecavity.
These pulsations, or, equivalently, the dynamic Stark effect,were first identified by Lamb2 5 26 in his third-order theory ofthe laser. It was later shown27 that PP's cause modes totransfer energy parametrically among one another and tochange their absorption coefficients and indexes of refraction.For example, a strong mode may support a weaker mode orcreate probe gain in an absorbing medium.2 8 Alternatively,side modes that grow from noise can create instability forcoherent light waves propagating in absorbing media.29 -31 Asimilar analysis was applied to optical bistability, and an in-stability was discovered in the upper branch of absorptiveoptical bistability.32 Again, PP's divert energy to sidebands,thereby leading to multimode buildup.3 3
An alternative form of mode splitting for high-gain, inho-mogeneously broadened lasers occurs when the high disper-sion of the medium establishes resonances for frequencieslocated off line center at the regions of anomalous dispersion.3 4
Similarly, homogeneously broadened lasers have similar be-havior, as the unsaturated medium provides the required gainand the anomalous dispersion for side-mode resonances.9
The instability criterion used in this research was firstsuggested by Casperson.6 A single-mode laser is referred toas unstable if that mode acquires sidebands or if other modesappear above threshold. The instability threshold then refersto the intensity of the strong central mode at which this in-stability occurs. To check for single-wavelength instability,the net gain and cavity resonances are examined simulta-neously for a pair of weak side modes symmetrically placedabout a strong central mode. The laser is labeled unstableif both conditions are satisfied.
The strong-signal theory used here is an extension of thatof Hambenne and Sargent,27 35 in which three modes interactwith a possibly inhomogeneously broadened (non-Doppler),two-level system. The central mode remains arbitrarily
strong, whereas the side modes are not allowed to saturate.Although these conditions apply only to threshold (instability)operation, we find that our results correlate well with theobserved fundamental pulsation frequency in the He-Xelaser.1718 20 These experiments also show period-doublingsequences and intermittent and fully developed chaotic out-put. The question of higher-order phase transitions and thethresholds for their appearances is, however, not dealt within this paper. Instead, we address only the question of in-stability threshold occurring in single-mode lasers as a resultof the presence of one pair of weak sidebands.
To establish the role of PP's in the above instabilities, westart with the wave equation (Section 2) and derive laseramplitude and frequency equations. The medium is repre-sented in these equations by the nonlinear polarization, whichwe derive in Sections 3 and 4. Both the standing-wave (SW)case (Fabry-Perot-type cavity) and the running-wave (RW)case (unidirectional ring laser) are treated here. A completederivation is presented here to show the difference betweenour analysis and that of optical-bistability instability. Thepolarization calculation is different from that of Ref. 27 be-cause of the consideration of the SW cavity.
The single-wavelength and the multiwavelength laser in-stabilities are investigated in Sections 5 and 6 for the specialcase of a strong central mode tuned to line center. A gener-alized treatment, which includes detuning effects, is presentedin Section 7. In Section 8, we present some numerical resultsfor homogeneously and inhomogeneously broadened lasersand show that homogeneously broadened lasers have an off-line-center region of multifrequency operation similar to thatshown in inhomogeneously broadened lasers in Ref. 34.
Some numerical correlations and predictions for the He-XeSW laser are shown in Section 9, in which the effects of pres-sure, detuning, cavity Q, and intensity are shown on the in-stability threshold and the pulsation frequency of the laser.
The analysis presented here fills in the details of the deri-vation in earlier papers.7 -9 36 It also establishes the connec-tion among all the above instabilities by the PP's.
2. LASER FIELD EQUATIONS
The equations of motion that determine the dependence ofthe field on space and on time are the field equations. In ouranalysis, the field is treated classically, so a natural startingpoint is the wave equation
vE X 2E 2PV X VXE +Aoauy Af- +A- (1)
where AO, E0, and o are the permeability and permittivity offree space and the conductivity of the medium, respectively;E and P are the electric field and the polarization of the me-dium, respectively. The wave equation can be simplified byassuming that the field is composed of infinite plane waveshaving no transverse variations. Let us decompose the fieldinto its Fourier components:
E(z, t) = 1/2 Z &.(z, t)exp(-ivnt)Un(z) + c.c., (2)n
where 6n = En exp(-i kn), the amplitude coefficient En (z, t),and the phase On (z, t) vary little in an optical frequency periodand vn + n is the oscillation frequency of the mode. Typi-cally, On is complex, whereas En is real. The vector character
S. T. Hendow and M. Sargent III
86 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985
of the field is ignored here, since the medium is composed oftwo-level atoms that contribute to a single polarization com-ponent of the electric field. The spatial character of the fieldis represented by Un (z), which depends on the experimentalconfiguration; for example,
Un (z) = exp (iKnz) for RW's, (3a)
= sin Knz for SW's, (3b)
where K, is the wave number. RW corresponds to the uni-directional ring laser case, whereas SW corresponds to theFabry-Perot-type cavity. The polarization P induced in themedium has the same form as the field, that is,
where Pn(z, t) is the complex, slowly varying component ofthe polarization for the nth mode. The real part of Pn is inphase with the field and represents dispersion resulting fromthe medium; the imaginary part is in quadrature with the fieldand gives rise to gain or loss.
By substituting Eqs. (2) and (4) into Eq. (1) without thecomplex conjugate (valid in the rotating-wave approximation),we get a wave equation for the field components. Since weare concerned primarily with dilute media and slowly varyingfields, we suppose that Vn22n >> Vnn, 9n and that Knacgn/8z>> O26n/OZ2. In these approximations, the wave equationbecomes
[i2Kp d P + (KP 2-OEOVp
2- i.otvp)&p
+ (ou - i2vpI0e0)P]exp(-ivPt)UP(z)
- E ,0vp 2 Pp exp(-ivpt)Up(z), (5)p
where the i associated with op/8z would be absent if SW'swere considered. The dots here denote time derivatives.Equation (5) is a general equation containing all spatial andtemporal variations of the slowly varying field amplitudes.
In high-Q cavities, the mean-field approximation can beused, since the cavity equalizes the field amplitude within it.Consequently, Kn(6n >> d10/z. For bad-cavity lasers, themean-field approximation may still be a good one. This ap-proximation and its limitations for the theory of optical bi-stability are discussed in Ref. 33.
The average round-trip cavity losses are represented by thefictional conductivity a. By setting a = COv/Qn, where v _ nand Qn is the cavity Q for the nth mode, the wave equation forthe nth mode is obtainable from Eq. (5) by projecting ontoUn(z) and by multiplying through by exp(ivnt). By doing sowe get
dn + [,2Q - i(n- Qn)' 6n = i Pn1k IV22 Eo
PEn +- En =-2 IMIPO}
2Qn 2E 2
1 Pn
En 2E0
(7)
(8)
where the passive-cavity-resonance frequency is Qn = cKn,c is the speed of light in vacuum, and Kn is the wave numberfor the nth mode. Equation (7) can be used to define a netgain parameter 3 7 :
v vl1gn =-2Q -2 E IMI (9)
The field amplitude and frequency equations are self-consistent, since the polarization induced in the medium inturn radiates the field that induced it. Equations (6)-(9) areour basic, working laser equations. To investigate the sta-bility of a particular laser mode at line center, we look for sidemodes that satisfy the cavity-resonance condition [Eq. (8)]and that have positive gain [Eq. (9)]. A single-mode laser isstable if no side modes exist, and it is unstable if side modesgrow, causing the total field intensity supported by the me-dium to fluctuate in time. For the more general case of de-tuned operation we use Eq. (6), as is shown in Section 7.
3. POLARIZATION OF THE MEDIUM
Mode-expansion techniques have been highly successful insolving multimode-laser problems. Consequently, the ex-pansion in Eqs. (2) and (3) is used to describe the field. Thefrequencies vn are assumed to be evenly spaced, i.e., v = v1+ (n - 1) A, where, as shown in Fig. 1, A is the intermode beatfrequency. This assumption is self-consistent with the laserinstability problem, since the configuration of interest is thatof a strong mode and two weak but mode-locked sidebands.This mode-locking condition is relaxed in Section 7. Theamplitudes of the sidebands are assumed to be much smallerthan the strong-mode amplitude and the saturation intensityof the medium. The analysis involves the solution for thepolarization of the medium, as observed by one side mode inthe presence of a strong mode and possibly of a second sidemode. The beat frequency A is assumed to be real here; hencethe gain and the dispersion effects of the medium will showup in &n. A generalization of this treatment, however, maybe obtained by allowing A to be complex and including in itall the effects of the medium, as is discussed in Section 7.
Qb
(6)
The amplitude- and frequency-determining equations fora laser mode can then be obtained from the complex waveequation by assuming that &n is real, vn >> &, and Pn = 'Pnexp(-ikn) and by isolating the real and the imaginary partsof Eq. (6). These equations are
EA ).
E2
E3-A -)MI v C) a P
Fig. 1. The amplitude spectrum of three-frequency operation. AM(FM) operation is assumed when the beats v2 - v1 andV3 - v 2 are inphase (180° out of phase).
S. T. Hendow and M. Sargent III
Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 87
The role of the PP's can be intuitively understood by con-sidering the side mode 61 in the presence of the strong mode62. The interference between 61 and 62 creates a beat noteof frequency A = V2- v. Since the medium is interactingwith the field, this beat note is impressed on the populationdifference, thereby creating the PP's. The gain of the me-dium then acquires a sinusoidal time variation that is pro-portional to the PP's amplitude and frequency. Because ofthe interaction of 6 2 with the population inversion, the me-dium now acts as a modulator for 62, producing sidebands onboth sides of the spectrum around v2. The energy of thesesidebands is derived from C2 and is proportional to both 61and &2. If 1 is weak, then only two sidebands are producedwith frequencies ,n A: A. These sidebands interfere with theside modes that produced them, causing appreciable changesin their absorption and dispersion coefficients; for example,side modes propagating in absorbers may experience gain asthe PP's divert energy from the saturator wave.
To solve for the polarization, let us consider an inhomo-geneously broadened medium consisting of two-level atomsinteracting with a near-resonant field expressed by Eq. (2).The medium is treated quantum mechanically and thereforeis described by the elements of the density matrix p. Theseelements have the following equations of motion:
Pab = (y + iCW)Pab + -Yab(Paa - Pbb), (10)h
Paa = Xa -YaPaa - YVabPba + c.C) ()
Pbb = XNb - YbPbb + ( CVabPba + CC (12)
where dk = nak - nbk. We define d-k = d so as to keep Dreal. In the presence of a beat frequency A in the field, thepopulation difference, or dk, tries to follow this beat. How-ever, T1 (the average of the level lifetimes) puts an upper limiton how well the population difference can follow A. There-fore the effect of the PP's is most noticeable when dk varieslittle in time T1 .
N is the unsaturated population difference separable toN(z, w', t) = N(z, t)W(w'). W(w') is the inhomogeneous-broadening function. W(w') = (rlI2Aw)-l exp[-(co -
'01)2/(AC)
2], where A is the inhomogeneous-broadening
width. At the homogeneous limit, W(o') reduces to the deltafunction (w - .
The polarization of the medium [Eq. (4)] is calculated byadding the contribution of all atoms:
P(z, t) = f dw'Pab(Z, ', t) + c-c-_<D
(16)
By using Eq. (13) in Eq. (16) we get
P(z, t) = dw'dpN
X E Pm+i(0', z)exp[-i(vi + mA)t] + c.c. (17)m=-D
The slowly varying complex polarization OPn is defined byequating in Eqs. (4) and (17) the coefficients of exp[-i(v1 +m A)t] and by projecting onto the U,, (z) spatial mode
= 2p IVJ dw'
(18)
where Ya and Yb are the decay rates of the upper and lowerlevels, respectively, and fib = (a + Yb)/2. The dipole decayrate is 'y = Yab + Yph = /T2, where Yph is a dephasing pa-rameter. a and Xb are the pumping rates to the upper andlower levels, respectively; hw is the energy difference betweenthe upper and lower levels. The electric-dipole perturbationenergy is ab = -E, wherep is the electric-dipole matrixelement between levels a and b, and E is the total electricfield.
If one takes a clue from the interaction processes used inperturbation theory, one can see that, in the presence of thefield [Eq. (2)], the polarization element Pab and the populationelement Paa can be written in the following expansion:
= N(z, w', t) E nak(z, w', t)exp(ikAt) (a = a, b). (14)k =--
It is further convenient to define the population differenceD(z, w', t) with the following expansion:
D(z, w', t) = Paa-Pbb = N E dk(z, A', t)exp(ikAt),k=--
where
V = S dzl Un(z)12 = L for RW, (19a)
= L/2 for SW,
N(z, co', t) = W(co')N(z, t),
N(z,t) = a(Z t) _ Xb(Z, t)ta tYb
(19b)
(20a)
(20b)
4. SOLUTION OF THE EQUATIONS OFMOTION
If one uses the expansions of Eqs. (13) and (14) in the equa-tions of motion [Eqs. (10)-(12)] one gets, after equating thecoefficients of exp[-i(vi + mA)t] and exp(ik At),
[y + i' - V1 - A)JPm+i(z)
=-i e End n-m-(z)Un(z), (21)2h ,
(-Ya + ikA)nak = 4 - (6nUnP+n - nUnPn-k)N h
(22)
nbk has an equation of motion that is similar to Eq. (22) exceptthat a - b and - i/h - +i/h. Equations (21) and (22) can
5) be written as follows:
S. T. Hendow and M. Sargent III
X f ' dzN(z, (,', t)p. (z, w', 0 U,-, W,
88 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985 S
3Pm(Z) = -i *2 fZm EZ Gndn-m(z)Un(z), (23)
2h n=1
dk(Z) = kO + i 2 2 (k A)h 'aYYb
X A (-nUnPZ+n -nUnpnk), (24)n
where
.m = 1/[-y + i(CO -v)],
Yr(kA)-=2 (= + A 1 + ikAJ (25)
By specializing Eq. (23) for m = 1-3 and by assuming weakside modes, that is, by assuming that 1, 63 << 62, and thesaturation intensity of the medium, we can drop from Eqs.(23) and (24) all terms that contain 61 and/or 63 twice. Thisapproximation also amounts to limiting the Fourier series [Eq.(14)] to k = 0, t1.
To illustrate this more clearly, we choose to follow a per-turbation-type analysis. If we assume that W = 63 = 0, whichis the single-mode problem, then only P2 and do survive.They are do 1/(1 + 12121 U212) and P2 = -i(pl2h)026 2U2do. By substituting these values into Eq. (24) we getdi , d1, 3; i.e., to generate the term d1 , the probe acts onlyonce (does not saturate). If we carry this approach further,we get da2 ,3, i.e., the probe acts twice. But 61,3 << 62;then da2 leads to a term that is much smaller than do and dai,and, consequently, the term is dropped. A more accuratetreatment is given by Refs. 27 and 35 for the case of two-fre-quency operation with arbitrarily large amplitudes (above-threshold operation).
Equation (23) becomes, for m = 1-3,
Pi = -i 0 1(6 1doUl + 02U2d), (26a)
P2 = -i 0 2 (6 1Uld-l + 02 U2 do + 63 U3 d1 ), (26b)2h
P3 = -i P Z)3(62 U2d-1 + 03U3do). (26c)2h
By specializing Eq. (24) for k = 0, 1 and by using Eq. (26)and the above weak-side-mode approximation, we get
d°~z -1 + I2 L2 U2 U2 (27)
(6 102UlU2fl + 2 U32U3f3) 2 Yab/h'YYaYbdi(z) = -do 1 + I2f2 U2 U2
where
12 = 1/[1 + (C'- V2)2T221,
11(A) = I[l/(Ya + iA)] + [/(Yb + iA)]J/2T1,
Ti = [(/7a) + (1/yb)I/2,
T2 = 1/7,
'2 = JfP 2/hi 2T 1T 2,
A =y1(A)(X2 + D1 )/2,
f2 = y91(A)(3 + . 1 )/2,
f= yf1(A)( J + D 2)/2.
di(z) is called a PP coefficient since it represents the oscilla-tory part of the population difference [Eq. (15)].
The polarization of a laser side mode (n = 1) in the presenceof modes 2 and 3 is given by Eqs. (18) and (26a), namely,
~P1(t) = -i}2 J- dof y d dzN(z, w', t)h -x' JV
X U*01 (6 1doUl + 62dlU2). (30)
This equation shows the dependence of the side-mode po-larization (and field amplitude) on the PP's. The presenceof PP's, therefore, necessitates that these variables containcorresponding variations, without which the system would notresemble any physical situation.
Equation (30) can be broken up into an incoherent part anda coherent part, P1 = lc + 7Pc'h. The incoherent part is as-sociated with do and includes only saturation effects that re-sult from 2, as is shown by Eq. (27). The coherent part,which contains d 1, involves the sum of interferences between2 and its two side modes. This term is dropped whenever
only incoherent effects are to be considered. From Eq. (28)we note that the PP's are dependent on the amplitude and onthe relative phases of the side modes. Setting 3 = 0 leads tothe single-sideband case with its corresponding single-side-band instability. In practice, it is unlikely that this case willbe encountered, since its instability threshold is about fourtimes higher than that of the double-sideband case.
The relative phase between 61 and 3 is an important factorin determining the magnitude of the PP's. Let us considerthe special case of central tuning (v2 = w). The symmetry ofthe situation allows us to set 011 = 131 . When the two side-bands are in phase, i.e., where 1 = 63, the frequency spectrumresembles that of the AM case and is labeled as such. Incontrast, the FM case is achieved when the two side modes arecompletely out of phase (1 =-63). With central tuning, fi= f5; hence by examining Eq. (28) we note that the PP's dropto zero under FM conditions. On the other hand, if 61 and63 are in phase (AM), then the magnitude of the PP's is dou-bled as compared with that of the single-sideband case. Thecancellation (doubling) of d 1 for the FM (AM) case does notoccur, however, for v2 d W.
It is of interest to note that the polarization of the medium[Eq. (30)] is proportional to 6*. In the theory of phase con-jugation, the 6* term leads to the generation of the phase-conjugate signal.
(28)
We now carry out the integration over z in Eq. (30). Firstwe start with Dflnc. Using Eqs. (27) and (30), we get
upinc = -i f2 So dw'W('):ZJ 1 1
X1 L NUUl (31X-f dz- 1+( 2 (31)
Let us consider first the single-wavelength instability of thelaser. Since all the frequencies have the same wavelength inthe medium we can replace all K, by K in Eq. (31). We canalso isolate the slow-varying N(z, t) from the fast-varying
(29) function cost 2Kz by averaging the latter over a wavelength.
.
S. T. Hendow and M. Sargent III
Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 89
Equation (31) becomes
tpinc = -i -- N X dV'W(c')J1°- h E_ co'Wc'&D
L fx dZI 2 (32)
Jv X JO 1 + I2 X21 U" 2 (2
where N is the average unsaturated population differencedefined by
1- Ndz. (33)L
The solution of the incoherent part of the polarization [Eq.(32), single-wavelength case] is shown in Appendix A. It hasthe following form:
inc = -i N Xh _ doWc)00
the modes involved have the same number of wavelengths perround trip.
Let us proceed now to treat the multiwavelength instabilityfor the SW laser. In this case, we cannot assume that all Kvectors are equal. Instead, the spatial projection in Eqs. (31)and (36) is performed keeping K& as is. In multimode lasertheory this leads to mode coupling, which depends on theposition of the medium inside the cavity.26 A similar de-pendence is present in optical bistability and instability in anSW cavity, as was first pointed out by Sargent. 36 In our case,the threshold of the multiwavelength instability shows acorresponding dependence on the position of the mediuminside the cavity, as is discussed in Section 6.
I 1/(1 + I2-L2 ) for RW, (34)
2/(1 + I212 + 1+ 1212) for SW (35)
These results [Eqs. (34) and (35)] apply to the following twocases: (1) the undirectional ring (RW) and the SW laser withsingle-wavelength instability, and (2) the undirectional ring(RW) laser with multiwavelength instability. The coherentpart of the polarization is
pcoh = -i 2 X ddW(w')JJ 1< 2h d-'W006
X-g dzNUlU2d,(z). (36)
By using Eq. (28) and the argument preceding Eq. (32), wecan rewrite Eq. (36) as
Pcoh i 4 YabNhryyy JY
+ 62 ) (J X So1 1 +I21 2 1 +I 2f21 2)
As shown in Appendix A, the solution of Eq. (37) is
1pcoh = i N X dw'W(cv')D1rI2(if1 + ifm)h f_
(37)
1
1 + I22 122 h
i2 22f2 1-
As shown in Appendix A, the incoherent and the coherentparts of the polarization take the following form:
Note that, by examining Eq. (37), 1oh = 0 at I2 = 0. Theseresults [Eqs. (38) and (39)] apply to the same two cases listedafter Eq. (35).
The total polarization for mode 1 is the sum of the coherentand incoherent parts as given by Eqs. (34), (35), (38), and (39).As can be seen from these expressions, the polarization is in-dependent of the position of the medium in the cavity.Consequently, the single-wavelength instability has no suchdependence, in contrast to the multiwavelength instability,which does. Note that the expression single wavelength isapplicable only to the case in which the medium completelyoverfills the cavity. There, all wavelengths are exactly phasematched. This degeneracy in the wavelength of the modesinvolved is, however, removed for media that do not overfillthe cavity. In this case, it becomes more accurate to say that
where i = 1, 2, qi = I212, and q2 = 12f2-Note that the approach presented in this paper and the
corresponding physical interpretation for the onset of theseinstabilities also apply to the instabilities in absorptive opticalbistability.3 3 For that case, the expressions for the polar-izations are identical to the ones derived above except that theaverage unsaturated population difference N is replaced by-N. In addition, the laser equations for the amplitude andthe frequency are replaced by the amplitude-determiningequation for the field inside the cavity.
S. T. Hendow and M. Sargent III
90 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985
5. SINGLE-MODE LASER INSTABILITIES:SPECIAL CASE OF THE CENTRALLY TUNEDSTRONG MODE
As pointed out in Section 3, the amplitude of the PP's reachesa maximum when two in-phase and mode-locked side modesare symmetrically placed about the strong mode. Thereforethe initial assumption of the mode structure shown in Fig. 1is self-consistent with the presence of instability. A set thatis not mode locked would cause PP's to wash out if the rate ofchange of PP's is more than 1T 1. A mode-locked set, on theother hand, would provide gain to the side modes, which inturn produce a periodic envelope for the laser output. Insidea laser cavity, these side modes must satisfy the cavityboundary condition. They could either have the samewavelength as the main mode (single-wavelength instability)or belong to different passive-cavity modes (multiwavelengthinstability). Let us examine the instability conditions forthese cases.
The multiwavelength instability is one in which severalmodes with different wavelengths can oscillate simulta-neously. In this case, the side modes belong to differentlongitudinal modes, and hence the resonance conditions arealready established. These side modes may also belong toother transverse modes. The instability threshold is ex-pected, however, to be much higher because of the decreaseof spatial overlap leading to the decrease of the PP's and ofthe cross saturation between the modes. In addition to spatialeffects, there are averaging effects of the transverse-fieldvariations that lead to the reduction or washing out of thepp'S.18 The instability requirement for the multiwavelengthcase is that these side modes have positive net gain. To in-vestigate that, let us consider the gain parameter [Eq. (9)] forthe laser. Since the case of interest in this section is centraltuning ( 2 = ), the phase angle 0 (Section 2) can be set to zeroby symmetry arguments. The gain parameter is
v vi1gn= -- I IMm' } (44)
2Q 2o En.
Let us define [only for v2 = w and 61 = 63; see Eq. (54)]
a1 + iK = i. (45)2coE1
By specializing Eq. (44) for n = 1 and using Eq. (45), we get
= [ 1i + (O-2z Z )Re(al + iK)], (46)
where ao = vp2N/2hye0 is homogeneous-broadening linearabsorption coefficient. 91 is the relative excitation for thestrong mode and is defined in Appendix C. Note that (v/2Q)(9T/ao)[Aw/yZi (y)] = 1 and that ao, has units of inverseseconds. This factor is introduced into Eq. (46) as a conve-nient way of separating the cavity Q from the relative exci-tation 91. It also introduces the normalization factor a 0 , whichfacilitates the numerical work. Note that, in the limit of Aw,= 0, the inhomogeneous-broadening factor AwlyZi (-y) = 1.Re(al + iK') is the net effective gain coefficient for the sidemode, and a and K are the complex absorption and thecoupling coefficients, respectively, usually defined in problemsof phase conjugation.
From Eq. (46), the condition for the multiwavelength in-stability is g, > 0 or
91 A,- Re(al + iK*) > 1.
ao azi () (47)
To solve relation (47) graphically, we plot the left-hand sideof relation (47) versus the side-mode detuning (i.e., vi - v2).The effect of the PP's is to introduce gain to side modes de-tuned from line center by roughly the Rabi flopping frequency.If the PP's are strong enough, relation (47) would be satisfiedfor these side modes, and the laser would be unstable if apassive-cavity mode exists at that detuned value.
The single-wavelength instability, unlike the multiwave-length type, occurs in bad cavities. To show how the PP's leadto this instability, let us examine the frequency-determiningequation for the laser cavity [Eq. (8)]. By specializing Eq. (8)for n = 1, by taking into account that the passive-cavity res-onance frequencies for the side modes are Q = Q2 = Q3 = V2,
and by using Eq. (45), we get
-(v 1 - 2) - =- Im(a1 + iKl). (48)v ao yZi (y)
The right-hand side of Eq. (48) represents the normalizeddispersion introduced by the medium and amplified by therelative excitation 91. The left-hand side of Eq. (48) holds theproperties of the passive cavity and represents the equationof a straight line having a slope of -2Q/v. This equationrepresents the resonance condition for side modes in thepresence of a saturator wave. Both sides are plotted versusthe side-mode detuning, i.e., versus v - v 2. Intersectionsbetween the dispersion curve and the straight line representresonances for side modes and a possible instability if thoseside modes have positive net gain, i.e., if they satisfy relation(47). Hence there are two conditions for the onset of thesingle-wavelength instability: (1) The medium should beanomalously dispersive to the point at which the index re-versals cross the cavity line, creating resonances for sidemodes, and (2) those side modes have positive net gain.
The physical significance of the intersections between thecavity line and the dispersion curve of the medium is that theside-mode frequencies corresponding to these intersectionsand the frequency of the main mode are all phase matched.The wavelength of these side modes in the medium dependson the index of refraction, which is given by 1 + Im(al + iK')/v.
At these intersections, the anomalous dispersion of the me-dium is large enough so that reversals in the index of refractionshorten (lengthen) the round-trip cavity length for thosesidebands having frequencies higher (lower) than the mainmode. This has the effect of keeping the sidebands resonantin the cavity. Index reversals resulting from spectral holeburning in inhomogeneously broadened media have the sameeffect as the PP's, and together they play a major role in de-termining if a side mode is resonant in the cavity.
Note that the frequency-determining equation [Eq. (8)]does not contain the cavity Q. It appears, then, that theresonance condition is independent of the cavity Q. However,the amplitude- and frequency-determining equations of thelaser are coupled; therefore a change in the Q factor also af-fects the frequency of operation of the laser. The introductionof the relative excitation in Eq. (48), on the other hand, ef-fectively decouples the graphic solutions of the gain and theresonance equations, namely, it allows one to change the cavityQ without having to adjust the level pumping rates to achievethe same mode intensity. This step will prove useful in de-
S. T. Hendow and M. Sargent III
Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 91
fining a necessary condition for the single-wavelength insta-bility in homogeneously broadened lasers, namely, the bad-cavity limit of v/2Q > (1T,) + (1/T2).
We proceed now to show that the condition for the onset ofinstabilities in the homogeneously broadened limit and forunidirectional lasers is, in fact, the same as that given in Refs.2 and 3. Specifically, we write the steady-state version for thewave equation [Eq. (6)]:
v 1Pn- £2n + K = - - - n
2cE En(49)
2
1
F,-I ~s0
0
-
a rwhere K v/2Q, which is not to be confused with the couplingcoefficient defined above. By specializing Eqs. (34), (38), and(C4) (see below) for homogeneously broadened media andinserting them into Eq. (49) and by taking Q2 = = V2, A =
2- v 1 , 6 = Q2- Qa, and ya = Yb, we find that for n = 1
This is the same as Eq. (3.7) of Ref. 3, provided that one sub-stitutes A - -i, 6 - -a, and I2 - X\ into Eq. (50). Thisshows the equivalence (in the limits chosen) of the Fouriermethods used to derive the polarization [Eq. (30)] and thestability approach of Ref. 3. Nonzero 's lead to the multi-wavelength instability.
The side-mode dispersion relation for absorptive opticalbistability is also given by Eq. (49), although Pn has the op-posite sign and the relation for I2 is determined by the tran-scendental steady-state optical bistability equation. Theeigenvalue equation that we get from Eq. (49) is the same asthat derived in Ref. 33.
Setting 3 = 0 in Eq. (50) is equivalent to asking that all threefrequencies correspond to the same Q2, i.e., to the same pas-sive-cavity wavelength. This leads to the following instabilityconditionsl 3:
K > 'Y + Ya (bad-cavity limit),
I2 >(y + 7a + K)(y + K).
'(K - Y - Ya)
-2 L I I I I 1-15 -10 - 5 0 5 10 15
( ,- )/ (a)
2
;__ 0-I-
_~ I
(51a) -2(51b)
Although the SW laser has a dispersion equation that issimilar to Eq. (49), no attempt is made here to analyze its in-stability conditions because of the complexity of the expres-sion for the polarization. Relations (51) are for the three-wavecase, in which two sidebands exist simultaneously. Setting63 to zero leads to corresponding instability when only onesideband is present.3 9 We find that the threshold for this caseis approximately four times higher than that of the double-sideband case.
The instability conditions [relation (47) and Eq. (48)] aregiven in terms of the effective gain and the dispersion coeffi-cients for the side mode. We proceed now to consider a fewexamples by presenting the gain and the dispersion of the sidemode, as the side-mode frequency is detuned away fromatomic resonance.3 9 Let us start first with the single-wave-length instability. Figures 2-4 are the normalized absorptionand dispersion coefficients for the unidirectional ring (or RW)laser, whereas Figs. 5 and 6 are for the SW laser. In all thesefigures, the frequency of the strong mode, v2, is tuned to theatomic resonance frequency . The symmetry of the problemin frequency space allows us to equate the amplitudes of thesidebands and also sets the phase angle ( 3 + 1 - 2v2)t + (03
-15 -10 -5 0(v - c) / r
(b)
5 10 15
Fig. 2. Normalized sideband (a) absorption and (b) dispersioncoefficients for the two-RW configuration and for a homogeneouslybroadened medium. The incoherent (dashed) and coherent (dot-ted-dashed) contributions are shown separately along with their sum(solid lines). Equal lifetimes are assumed (T, = T 2 ), and the coeffi-cients are multiplied by the relative excitation % = 1 + 12 appropriatefor laser operation. Here, I2 = 70 and v2 = a, i.e., the strong mode istuned to line center. HB stands for homogeneously broadened. Notethat (a) shows relation (47) satisfied for A/ 9,y, whereas (b) satisfiesEq. (48). The straight line in (b) is the passive-cavity line of Eq.(48).
+ 01 - 202) to zero, i.e., the mode spectrum is AM.2 7 Thisassumption is indicated in the figures by "3 wave am." Thesimpler case of a single sideband with 63 = 0, i.e., two-wave,is treated first, since the added complication of the phase angledoes not occur, and thus the restriction to central tuning (v2= w) is lifted. Figure 3 shows the detuning curves for thenormalized absorption [left-hand side of relation (47)] and thenormalized dispersion [right-hand side of Eq. (48)] for a sat-urator wave intensity of I2 = 70. The dashed and the dot-
S. T. Hendow and M. Sargent III
92 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985
1.5
1.0
0
-1.0
-l ,5 -6
2
0
*1
-2 L
-3 0 3 6( - C)/r
(a)
-6 -3 0 3 6(V - ca) /h
(b)
Fig. 3. The normalized sideband (a) absorption and (b) dispersioncoefficients for the three-RW case and for various intensities. Otherparameters are as in Fig. 2.
ted-dashed curves are the contributions of the incoherent [Eq.(34)] and the coherent fEq. (38)] parts plotted separately. InFig. 2(b), the passive-cavity line [left-hand side of equation(48)] is also plotted. As shown, there are four intersectionsleading to four resonances of sidebands. Only the two side-bands farthest from line center have gain; this means that oneof them could oscillate, leading to a single-sideband (or atwo-wave) instability. The intensity of 12 at which this in-stability occurs is about four times higher than that of thethree-wave AM case.
The three-wave AM case (actually phase-matched three-wave phase conjugation) is presented by Figs. 3 and 4. ForT = T2, instability occurs at I2 15, i.e., 15 times the satu-ration intensity of the medium, as shown in Fig. 1 of Ref. 8.Figure 5 shows the effect of the PP's in inhomogeneously
*_Z
a#Ct
broadened media; the figure illustrates that the role of thePP's in these instabilities and in saturation spectroscopycannot be ignored even if the strong-mode intensity is belowthe saturation intensity of the medium. Casperson6 utilizedthe index reversals resulting from spectral hole burning topredict the onset of instability in inhomogeneously broadenedmedia. When the PP's are included in the analysis, we ob-serve a substantial decrease of the saturator intensity at whichthis instability occurs in inhomogeneously broadened sys-tems. 7 -10,12,14
Figure 5 shows the above coefficients using parameters thatare appropriate for the He-Xe laser at 12 = 1. The minimumfundamental pulsation frequency for this case is predicted tobe 12.69 MHz. The frequency is lower for cavities havinghigher Q factors. Note that no index reversals are present in
1.5
1.0
0
-1.0
-1.5
2
1tZ_.-_
+ 0
I
6 3 0 3
( aI-w)/t(a)
-2L
6
-6 -3 0 3 6( Vl -J) / t
(b)
Fig. 4. The normalized sideband (a) absorption and (b) dispersioncoefficients for the three-RW case and for various values of T1/T 2.Other parameters are as in Fig. 2.
l
._.
U,
__
4-
+
S. T. Hendow and M. Sargent III
I
I
Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 93
quency. Dropping the PP's, on the other hand, reverses theemphasis from the gain to the resonance condition and wouldlead one to conclude that, near the instability threshold, thepulsation frequency is close to zero; this does not agree withexperimental observations.18
The cooperative effect of the PP's and spectral hole burningbecomes clear when one considers the dominant instabilitycondition for each case. In homogeneously broadened lasers,side-mode resonances are established at a point in operatingintensity far below that at which side-mode gain becomesavailable. Clearly, the dominant condition is the gain con-dition. The situation is reversed for the hypothetical case ofan inhomogeneously broadened laser without PP's. There-fore the presence of inhomogeneous broadening in a laser
8-2 0 2 4
( v- o) /Y(a)
-4 -2 0 2 4(Y.- co)y
(b)
Fig. 5. Normalized (a) absorption and (b) dispersion coefficientsversus sideband detuning for the three-RW case and for an inhomo-geneously broadened medium having Ac/y = 2.05. Other parametersare I2 = 1, 'a = Yb, T1/T2 = 11.685, and 9 = 1.659. The fundamentalpulsation frequency cop here is 12.69 MHz. The dotted and dot-ted-dashed curves are for the incoherent and coherent contributions,respectively.
* 4i
._
as
-2 1 1 1
-12 -9 -6 -,
4
2
% 0
6-
4 -2.0
-4the dashed curve of Fig. 5(b); this shows that spectral holeburning is insufficient to explain this instability at these valuesof I2 and at the given homogeneous- and inhomogeneous-broadening widths. These reversals in the index and in thecorresponding side-mode resonances are seen to exist evenwith intensities as low as I2 = 0.1; this indicates that thedominant condition for instability is the side-mode gaincondition. Hence, as the relative excitation is gradually in-creased from 1.0, the laser abruptly transits from a stable cwstate to an unstable state with a megahertz pulsation fre-
3 0 3 6 9 12
( - ) (a)
-12 -9 -6 -3 0 3 6 9 12
( Y. - ) )/ (b)
Fig. 6. Dependence of (a) the absorption coefficients and (b) thedispersion coefficients on the position of the medium in the SW cavityas given by Eqs. (40)-(43) for the multiwavelength case. The mediumis homogeneously broadened (HB). N 2 = 1 (-1) corresponds toplacing the medium at the end (in the middle) of the cavity. N2 =0 occurs when the medium fills the cavity.
1.5
1.0
0.5
0-
-0.5
- 1.0- 4
a |7^ 0.5
% 0+
I- 0.5
L
I L
S. T. Hendow and M. Sargent III
94 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985
decreases the threshold by compensating for the unavailabilityof side-mode gain. Figure 5 shows that side-mode resonancesare established by the PP's, whereas side-mode gain is pro-vided by groups of atoms that are not fully saturated by thestrong lasing mode.
6. MULTIWAVELENGTH INSTABILITY:CENTRAL TUNING
The instability threshold for the central tuning case is givenby relation (47), whereas the expressions for the polarizationsare given by Eqs. (34) and (38) for the RW case and by Eqs.(40)-(43) for the SW case. The RW or unidirectional-ring-laser case is illustrated in Figs. 2(a), 3(a), and 4(a), in whichthe laser is unstable if a passive-cavity resonance frequencyfalls within the region of positive net gain, i.e., if relation (47)is satisfied. The single-sideband case is shown in Fig. 2(a),whereas Fig. 3(a) shows the corresponding double-sidebandor three-wave AM case. The instability threshold (value ofI2 at which instability occurs) for the multiwavelength caseis the same as that for the single-wavelength case. Experi-mentally, the difference between the two is in cavity length,cavity Q, and relative excitation of the laser.
The SW cavity with its characteristic spatial hole burninghas an additional property, namely, the dependence of theinstability threshold on the position of the medium in thecavity. The presence of two or more modes with differentwavelengths creates a spatial interference pattern that in-fluences the cross saturation between these modes.2 6 Nearthe ends, the spatial holes corresponding to the differentmodes are in phase (overlap), and therefore cross-saturationeffects and N2 [Eq. (41)] are maximized. Therefore, for V2= 1, one expects the sideband absorption coefficient for themultiwavelength case to be identical to that of the single-wavelength case. Figure 6 shows the dependence of thesecoefficients on N2. Note that Fig. 6 with N2 = 1 is similar toFig. 1 of Ref. 8. For an instability to occur, cavity resonancesare required at frequency detunings of approximately the Rabiflopping frequency. Figure 6 (N2 = 1) shows that, at thesefrequencies, the gain condition [relation (47)] is satisfied.
If the medium is placed in the middle of the cavity (N2 =-1), the spatial holes corresponding to the different modesdo not overlap. This causes the cross-saturation effects to-gether with the PP's to vanish. Under these conditions, theabsorption coefficient approaches that of an unperturbedmedium, i.e., it approaches the Lorentzian function.
A mixture of saturation occurs when the medium overfillsthe cavity (N2 = 0) and thus the absorption coefficient takesa shape midway between the above two cases. This increasesthe instability threshold for side modes located at about theRabi flopping frequency. However, side modes that haveresonances close to the atomic resonance frequency may sat-isfy the gain condition [relation (47)] and thus can promoteinstability. Again, these predictions apply only to thresholdconditions, and a steady-state solution is required to deter-mine if continuous multiwavelength operation of the SW laseris possible.
Such dependencies of the properties of SW devices on theposition of the medium inside the cavity also occur in SW la-sers26 and in SW optical bistable devices.3 6 In the laser, crosssaturation limits multimode operation, whereas it enhancesbistability and instability in bistable devices.
7. SINGLE-MODE INSTABILITY: THEDETUNED CASE
As the strong laser mode is tuned to line center, a homoge-neously broadened medium is completely saturated, andtherefore the gain of the side modes is primarily composed ofdiverted energy from the central mode. However, detuningleads to partial saturation of the medium. The side modesthen can promote instability by relying on the medium forincoherent gain. The result is a lower instability thresholdand a substantial reduction in the role of the PP's in causinginstability off line center. Other effects of detuning are thenoticeable change of the central mode intensity and thechange of the relative phase angle between the side modes andtheir relative amplitudes.' 1 The decrease in the intensity withdetuning is a major factor that influences the pulsation fre-quency of the laser.
To investigate the onset of instability in a detuned system,we need to calculate the eigenvalues for the dispersion equa-tions of the side modes. For simplicity, let us consider thespecial case of the RW laser. The dispersion equation is givenby the wave equation for the laser [Eq. (6)]. By incorporatingthe gain and the coupling coefficients of Eq. (45) into Eq. (6),we get for n = 1 and n = 3
6g = [al - + i(v ) 61 + iK'6&,
[3 -- ( 3 Q3 iK3 1.[s 2Q3
(52)
(53)
These are the coupled mode equations for the two side modesin the presence of a strong mode. The parameters ai and K,
i = 1, 3, are the same complex absorption and coupling coef-ficients introduced above in Eq. (46). The full expression forthe single-wavelength case and for the RW laser is given by
a, = ao S .dwo'W(w') Il (1- I2/f _ 1 + I2X2 1+If2
for RW,
(54)
-iK'= ao C dw'W(w') -+2i, 12f3J- 1 +1I2L 2 1 +2f2
for RW.
(55)
The expressions for a 3 and K3 are the same as those for a, andK, except that the indices 3 and 1 are interchanged and A isreplaced by -A. The corresponding expressions for SW la-sers may be obtained by using Eqs. (35), (39), (49), and(52).
To check the stability of the laser we choose to calculate theeigenvalues X1,2 of Eqs. (52) and (53). The laser is unstableif either of the two eigenvalues has a positive real part. Thiscondition is sufficient for instability, provided that the sidemodes are resonant in the cavity, such as is the case in themultiwavelength instability. These conditions do not existin the single-wavelength instability, and so all side modes areforced to acquire the same wavelength as the main mode.This extra condition appears in the form of a vanishingimaginary part of the eigenvalues; this indicates that anom-alous dispersion of the nonlinear medium has compensated
S. T. Hendow and M. Sargent III
Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 95
for the side-mode frequency offsets and that the side-modefield amplitude now grows exponentially. These conditionscan be modified to the following form:
Re[(Xj + i)/KI > 0, (56)
Im[(Xj + iA)/K] = -K-'(Vl - 2), (57)
where A = V2 - v1. The left-hand sides of expressions (56)and (57) represent the normalized side-mode absorption anddispersion coefficients, respectively, whereas the right-handside of Eq. (57) represents the familiar equation of a straightline with slope -2Q/v. Intersections of the curves locateresonances, and instability occurs if relation (56) is satisfied.Note that in working with expressions (56) and (57), thenormalization factors after Eq. (46) are used.
The solution for the eigenvalues for Eqs. (52) and (53) startswith the coefficient matrix
a, 1 - -V + i(v 1 - 1)2Qi
-iK 3
iWl 1 = 0.
3 Q3 i(P3-Q3)
(58)
By setting PI - 3 = -2A, 3 = 2 - 1 = Q3 - 2, and (vI +v3)/2 = v2, and by using Eq. (8) with n = 2 and Eq. (C4), we getthe eigenvalues
X1,2 = -K-iA + i + 2
± {K*K3 + F(a1 - a3) + i(V2 -Q2)1211/
4(a) 9.0
0
-4-15 0 15
15I -41
where
W- V2 AwC 9SL2V2 - 2 = K
Y YZ (7) 1 + I 2L 2(60)
Note that, for 6 = 0 (single wavelength), (X1,2 + i)/K is
independent of K and is symmetric in A. Equations (59) area general equation. They include the changes in side-modeamplitudes as well as the changes in the relative phase anglebetween them. They also need not be restricted to themode-locked case, since any deviations of the frequencies ofthe modes outside the mode-locked configuration can be in-cluded in 6, This approximation is valid, provided thatthese frequency offsets are small compared with 1/T2. Theresults obtained here are similar to those of Ref. 11, which alsoshows the effects of detuning on the relative phase angle andon the side-mode amplitudes.
At the limit of a homogeneously broadened medium witha centrally tuned strong mode and v2 = and 3 = 0, we get forEqs. (59)
(X1 + iA)/K = -1 + RyO 1do,
(X2 + i)/K = -1 + RhZ0do (1 - 12 I2f.
(61)
(62)
Note that Eq. (61) contains only the incoherent term, whereasEq. (62) has the sum of the coherent and incoherent contri-butions to the polarization. The factor of 2 in the right-handparentheses in Eq. (62) represents the doubling of the PP'sthat results from the three-wave AM operation.
Equations (59) show that the real parts of the eigenvaluesare independent of 6, and therefore both the multiwavelengthand the single-wavelength instabilities can be addressed si-multaneously (for the RW case). Here, A and 6 are taken asreal quantities. As discussed in Ref. 12, for a strong enough
(g) 9 =3
I _ -\M ___-IC 1 fle ,
4(d
0
I 5
- 15 0 15
4 (h) 9=3
0
- I
Fig. 7. Effects of detuning the strong mode on the eigenvalues of the sideband for a homogeneously broadened, unidirectional-ring-laser case.The relative excitation 9% is 16, and the cavity detuning is given by q = (c-v2)/y. X1 and X2 are shown separately on the left-hand and on theright-hand sides, respectively. The even function is the real part, whereas the odd function is the imaginary part. The intensities of the strongmode are (a), (b) I2 = 15; (c), (d) I2 = 14.001; (e), (f) I2 = 11.003; and (g), (h) I2 = 5.996. The vertical axis is Re[Xl,2 + iA/K and Im[XI,2 + iA]/K;
the horizontal axis is (vi - V2)/1y.
ol A ,A
-15 0 15
S. T. Hendow and M. Sargent III
_-1,.4I-1
I _.
-in I Z lb1 U 10Uu1
96 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985
relative excitation both eigenvalues in Eqs. (59) are abovethreshold, leading to two pulsation frequencies. The eigen-value with the lowest threshold sets the instability thresholdof the laser, whereas the pulsation frequency of the secondeigenvalue may correspond to a secondary pulsation frequencyin the laser.'2,"9
The above approach is valid only at the instability bound-ary.12 However, a generalization that extends our treatmentbeyond threshold could be performed by permitting the beatfrequency A to be complex and by including in it iterativelythe effects of the medium. The initial assumption that A isreal led to the eigenvalues given in Eqs. (59). These eigen-values can be self-consistently eliminated by substitutingthem into A. Replacing iA by iA + (a, + a5)/2 i / anditerating the calculation in exactly the same fashion resultsin the second-order eigenvalues of XN = -K -i A. This alsomodifies all the complex denominators (O,'s) and Lorent-zians (, 2's) in a similar way, since each iA is replaced by itsnew value. The imaginary part of i gives the pulsationfrequency, whereas the real part determines if the system isstable. This iterative treatment, which converges exactly inone iteration, is an improvement to the slowly varying enve-lope approximation in Lamb's theory.2 6 It also connects thedual-side-mode approach of Casperson6 to the generalizedlinear stability analysisl2 -'4' 33 and to saturation-spectros-copy-type solutions. 7 -9
8. OFF- LINE-CENTER OPERATION INHOMOGENEOUSLY BROADENED LASERS:SINGLE-WAVELENGTH CASE
Let us examine the effects of the detuning on the side-modecoefficients for a homogeneously broadened medium. 3 9 InFig. 7 we show a series of plots for the real and imaginary partsof the normalized eigenvalues (X1,2 + iA)/K versus side-modedetuning as the strong-mode detuning (q) is increased, where
0.80
0.140
0
- 0. 40
-0.80
0.50
0.25
-
E
IHB, SW 11 - 1.59
14 -2 0 2 14('l- '2)I1
(a)
IHB, Sw-2.0
AI
q = ( - V2)/'y. Note that, at zero detuning [q = 0; Figs. 7(a)and 7(b)], the real and imaginary parts of the eigenvalues areidentical to the absorption and dispersion coefficients for theAM and FM special cases. Also compare Figs. 7(a) and 7(b)with Fig. 3 and with Fig. 14 of Ref. 33.
On line center, Fig. 7(a) shows one eigenvalue having apositive real part, indicating the possibility for instabilitydepending on the side-mode resonances. However, instabilityconditions are quickly destroyed as the laser is tuned off linecenter; this leads to the high localization of pulsing about linecenter in homogeneously broadened lasers. This instability,however, occurs over a broad range of detunings in inhomo-geneously broadened lasers.
As cavity detuning q increases, the medium gets only par-tially saturated, and so the side modes acquire incoherent gainthat approaches a Lorentzian shape as I2 drops to zero. InFig. 7(c), we see one eigenvalue acquiring a positive real part.However, no single-wavelength instability occurs here, sincethe eigenvalue that has a positive real part does not have re-versals in its imaginary part, and thus Eq. (57) cannot besatisfied. This, however, does occur with higher central modedetuning values, as is shown in Figs. 7(e) and 7(g).
Therefore, for high values of relative excitations, a homo-geneously broadened laser has two regions of single-wave-length instability: about line center and off line center.These two regions are analogous to those predicted and ob-served in inhomogeneously broadened lasers. The first isanalogous to that of Casperson,6 whereas the second is anal-ogous to that of Casperson and Yariv.3 4 As shown in Fig. 7(b),PP's are responsible for the line-center instability, whereasthe off-line-center instability relies on the unsaturated me-dium for gain and anomalous dispersion. The two side peaksin Fig. 7(g) represent the incoherent gain available to the sidemode as the main mode is detuned off line center.
To simulate an experimental configuration, we chose tokeep the relative excitation constant for all these plots. The
0.80
0.140
0
-0. 40 F
-0-. 80-4 -2
0.50
- 0.25.1
_ 0
-0.25
- 0. 500 2
(b )
0 2 4(v, -' 2)/r(c)
-4 -2 0 2 40 d-)/f
(d)
Fig. 8. The real [(a), (c)] and imaginary [(b), (d)] parts of the normalized eigenvalues versus probe detuning for an inhomogeneously broadened(IHB) medium with a 1/e width of Ax = 10y. At detunings of X - V2 = 0 and 8-y, the saturator wave intensities are I2 = 2.0 and I2 = 1.375, re-spectively. All curves have the same y. The straight lines in (b) and (d) are the cavity lines.
IHB,SW ] 1 1.59
IHB,SW - .81
o:S X Q's4
S. T. Hendow and M. Sargent III
-0.25 -
-0.50-4 -2
A
Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 97
intensity of the strong mode as it is detuned ( 2 d c) wouldthen be adjusted in accordance with Eq. (C4).
It is evident that the PP's have a less important role in es-tablishing the off-line-center single-wavelength instability.Reversals in the index are still a necessary condition for in-stability. However, the incoherent anomalous dispersion ofthe medium contributes more to these reversals than the PP'sdo [see Fig. 7(g)]. It is not necessary that both side modes beabove threshold. For sufficiently large detuning values, it isexpected that only a single sideband will exist.
Figure 7(g) shows clearly that intersections may occur be-tween a cavity line (of slope -2Q/v) and the dispersion curve.Hence side modes will certainly grow above threshold. Onenotes, however, that the resonance conditions are highlysusceptible to saturation effects, and therefore there existsa point in side-mode intensity at which the resonances of theside modes in the cavity may be ruined by self-saturation.
Finally, Fig. 8 shows the effects of detuning on the eigen-values for an inhomogeneously broadened medium. One ofthe eigenvalues shows the expected hole burning and indexreversal encountered in the three-wave AM type of operation.The other eigenvalue shows a small and shallow central bump,where the positions of the minima are at approximately theRabi flopping frequency. This fine structure is real and canbe shown to occur in FM-type saturation spectroscopy. Thiscentral bump, however, washes out for T 1 >> T 2 and for de-tunings far from line center.
Although the figures in this section are presented for thespecial case of 6 = 0 (single wavelength), it should be notedthat the real parts of the eigenvalues also apply to the multi-wavelength case. Of course, steady-state solutions are re-quired to determine if continuous multimode operation ispossible in homogeneously broadened lasers.
9. THE HE-XE LASER
In this section, we utilize Fig. 8 to predict threshold and pul-sation characteristics of the He-Xe laser. First, the absorp-tion and dispersion coefficients are plotted versus the de-tuning for a parameter set that is suitable for the He-Xe laser.The cavity line is then drawn with a slope equal to twice thecavity lifetime. The fundamental pulsation frequency of thelaser is then given by | - v1 1 if an intersection occurs betweenthe curve and the line and if the gain is positive.
Linewidth characteristics of the He-Xe laser are4O41 aninhomogeneous-broadening width of 100 MHz (full width atlie points) and a natural homogeneous linewidth (FWHM)of 4.6 ± 0.2 MHz. Added pressure broadening is 10.9 i 1.3MHz/Torr for the Xe gas and 18.6 i 0.7 MHz/Torr for the He.The laser operates on the 3.51-um transition in Xe.
Figure 9 shows quite good correlation between our modeland the experimental results as given in Fig. 8 of Ref. 18. Thedashed line in Fig. 9, which represents the minimum pulsationfrequency, comes to within 1 MHz of the observed pulsationfrequencies.
The detuning characteristics of the laser are shown in Fig.10 together with the mode intensity. This figure shows thatthe most dominant factor influencing wp is the intensity ofthe strong mode. Since the theory is for a non-Doppler me-dium, it follows that the Lamb dip does not exist here. This,however, is shown clearly in the experiment, as is shown in Fig.4 of Ref. 18 (see also Fig. 2 of Ref. 20). Nevertheless, the
MHz
p
6
2
4 12 20MHz
Fig. 9. The fundamental pulsation frequency wp versus homoge-neous-broadening width for various values of intensity for an SWlaser. The dashed line represents the boundary between stable andunstable operation of the laser. The parameters used are Aw = 60MHz, V2 = , Ya = 0.2 MHz, Yb = 5 MHz, and t, = 0.5 nsec.
1.6
'2
0.8
0
MHz
6
cp
2
l I I I I
-8 0 8
detuning ( C)W-V2)
Fig. 10. The detuning characteristics of (a) the lasing intensity I2and (b) the pulsation frequency wp for an SW cavity laser for relativeexcitations of 9Z = 1.2 and 91 = 1.43. Parameters used are ya = Yb
= 3.5 MHz, y = 2.84 MHz, Aw = 60 MHz, and cavity lifetime t, = 0.5nsec.
-~~~~~ 9
.0 \ >
O\~~~~~~~ I,~~~~~~~~,
,', , . .I
S. T. Hendow and M. Sargent III
10
98 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985
MHz
I0
p
6
2
1.0 1.2 1.4 1.6
Relative Excitation (n )Fig. 11. Fundamental pulsation frequency for a SW laser versusrelative excitation for various values of homogeneous-broadeningwidths (HWHM). The dashed curve represents the boundary be-tween stable and unstable laser operation. Parameter values are ya= 0.2 MHz, b = 5 MHz. Other parameters are the same as in Fig.10.
agreement between this theory and the experiment is evident.The decrease in the observed cop near line center1 8 may beexplained by the drop in intensity resulting from the Lambdip. Note that, although instability persists over a broaddetuning range, it stops at a small but nonzero intensityvalue.20 This detuning range in which instability occurs isseen to shrink to a small range about line center for near-threshold operation.
The influence of inhomogeneous broadening is of particularinterest, since instability threshold decreases by about anorder of magnitude with the presence of inhomogeneousbroadening. Figure 9 shows a mapping of wp versus homo-geneous-broadening widths y for various values of strong-mode intensity I2. The dashed line here divides the space intostable and unstable laser operation. The laser is unstable ifthe intensity I2 is higher than the dashed line for a laser op-erating with a certain y. The decrease of cop with y for acertain I2 is observed only at these high ratios of YalYb. For'Ya = Yb these curves show a continuous increase of wp with,y and consequently lead to the conclusion that the lowestinstability threshold occurs when ya = Yb.' 0 The effect ofincreasing the buffer-gas pressure (or y) on the instability ofthe He-Xe laser is also shown in Figs. 11 and 12.
At high excitation rates, Fig. 11 shows wp to increase firstand then to decrease as y is increased. This behavior is at-tributed to the broadening effects of pressure on the lasinglinewidth, whereas the decrease of op is primarily due to thedrop in intensity as y is increased. Figure 12 is similar to Fig.10 in that the space is separated into stable and unstable op-eration. Given the gas pressure, the laser is then stable (un-stable) if the operating point falls below (above) the =constant curve. Alternatively, the vertical axis in this figuremay be labeled as the threshold intensity required for insta-bility. Note the gradual increase of the instability thresholdwith cavity lifetime; this occurs in low-pressure lasers andsuggests that cavities with higher lifetimes may be usedwithout incurring a substantial increase in the instability
threshold. However, cavities with higher lifetimes have thedisadvantage of a more rapid increase of instability thresholdwith the increase in pressure. The change of the pulsationfrequency with cavity lifetime may also be plotted. In general,the frequency drops with an increase in t. The space is alsodivided into stable and unstable operation.
The effects of intensity change on p are shown explicitlyin Fig. 13 for line-center operation and for various values ofcavity lifetimes t. The scaling of p with 12 correlates wellwith the experimental result of Figs. 2 and 5 of Ref. 20. Notealso that these curves are discontinuous at low intensities.
1.6
N
'-4
1.2
0.8
0.4
00 40 80 nsec
cavity lifetime ( tC )Fig. 12. Threshold intensity versus cavity lifetime t for variousvalues of y. The threshold intensity is the minimum strong-modeintensity required for instability. Laser parameters are the same asin Fig. 11. Each curve separates the space into stable laser operation(below the curve) and unstable operation (above). The cavity lifetimeis t = Q/V.
MHz
8
p
4
01.0 1.3
Relative Excitation1.6
(3i)
Fig. 13. The predicted pulsation frequency wp versus the relativeexcitation for various values of cavity lifetimes. y = 5 MHz; otherlaser parameters are the same as in Fig. 6. The relationship betweenthe strong-mode intensity 2 and 91 is close to linear. At = 1.6, I2= 2.
S. T. Hendow and M. Sargent III
Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 99
The low instability thresholds calculated here for theHe-Xe laser (down to 10% of the saturation intensity of themedium) suggests that even the single-sideband case of Sec-tion 5 may be present here. Figures 9-13 are all for the SWlaser. Except for some numerical differences, the unidirec-tional ring laser has pulsing characteristics that are highlysimilar to those of the SW case, and therefore these charac-teristics are not given here.
10. SUMMARY
We have presented an examination of the onset of single-modelaser instabilities. The analysis involved the derivation ofexpressions for the nonlinear polarization of the medium forthe two cases of the unidirectional ring and the SW cavitiesand also for a medium having an arbitrary amount of inho-mogeneous broadening. The field, the population difference,and the polarization of the medium are Fourier analyzed in
for the He-Xe laser. This examination shows the generalgood agreement between theory and experiment.
APPENDIX A: SOLUTION OF THE SPATIALPROJECTION FOR THE POLARIZATIONS
The incoherent contribution to the polarization [Eq. (32)] forthe laser case can be solved easily by using the following in-tegral4 2:
S A+Bcosx B Ab -aB 2f a+bcosx b b /a2- 2
X an- 1 N~a tan (x/2)
I a+b(Al)
where a 2 > b2. Expression (Al) is needed only for the SWlaser case, since the spatial projection for the RW laser istrivial. Equation (32) becomes
for RW, (A2)
for SW. (A3)
the side-mode beat frequency. Because the population dif-ference follows the changes in the field envelope, it breaksdown into a dc component and a PP component. The po-larization of the medium, on the other hand, follows the fieldin form and is shown to break down into incoherent and co-herent parts that correspond to the dc and ac parts of thepopulation difference, respectively. In inhomogeneouslybroadened media, the incoherent part of the polarizationcontains effects of spectral hole burning, whereas the coherentpart contains the modulation effects of the PP's on the fieldand on the polarization.
To examine the onset of laser instabilities, two approachesare described. The first is a beat-frequency saturation-spectroscopy-type solution for line-center operation, and thesecond is a linear stability analysis of the mode structure.Both treatments are included here to show the equivalencebetween the two and also to emphasize the physical pictureof the onset of these instabilities.
Single-mode laser instabilities are divided into the twogeneral categories of single wavelength and multiwavelength.For the latter case, the laser modes belong to different cavitymodes, whereas, in the former case, all the modes have thesame number of wavelengths per round trip in the cavity. Forthe multiwavelength instability to occur, the side mode hasto satisfy the side-mode gain condition. The single-wave-length instability, on the other hand, has an added side-modecavity resonance condition that limits this instability to lasershaving bad cavities. The multiwavelength instability is alsodifferent from the single-wavelength case in that there is adependence of the instability threshold on the position of themedium in the SW cavity. In this case, minimum thresholdoccurs when the medium is placed closest to the mirrors.
In Section 7, a generalization of the linear stability analysiswas discussed, showing the equivalence between saturation-spectroscopy-type solutions, linear stability analysis of theside-mode amplitudes, and standard linear stability analysis.Finally, the dependence of the instability threshold and thefundamental pulsation frequency of the laser on operatingconditions was discussed using parameters that were suitable
The coherent part of the polarization is given by Eq. (37). Tosolve that projection, let us define the following parame-ters:
q = I2C2, q2 = I 2f 2. (A4)
Equation (45) contains the following two functions:
1 1 1 b B
a+blJQ2A+BIU12 Ab-aBta+bJl2 A+BljU2J(A5)
and
qI 4 22 2
l+q1U]2 1+qjU]2(A6)
By using Eqs. (A5) and (A6) and by setting 62 = 6 , we findthat the polarization [Eq. (37)] breaks down into two parts:
t1ph = i2 I2 5 dw'W(o')lJ6( f, + & f)(F11 + F12),
(A7)
where
Fij L (_ _ )i 1 I 2 (J2 \
.A q2 - q X JoI + jq1I ]2) (A8)
where j = 1, 2. By making use of expression (Al), the inte-gration [Eq. (A8)] becomes
(-l)i qj/(l + qj)Flj= X 2 1q2 - l1 --
qj
for RW,
for SW.
(A9)
By using Eqs. (A4) and (A9) in Eq. (A7), we get the coherentpart of the polarization:
pcoh =i2N I +2 -p1oh h J dw'W(w'X'1iI2(6J1fi + 6 '212/2
The case of the multiwavelength instability has to betreated separately. It requires the solution of the spatialprojection in Eqs. (32) and (37) without assuming identicalwavelengths for all the modes. Let us consider first the in-coherent contribution given by Eq. (32). By setting K =QnIC = (Q 2 /C) + (n - 2)A/c and expanding the sin Knz func-tion, we get
2 sin2 Knz = 1 - cos (Q2 ) cos (n - 2)A
-sin (2 -) sin (n - 2)A] (A12)
The second term in the brackets is an odd function; therefore
where P2 is given by Eq. (26b). By setting 61 = 63 = 0, weobtain
P2 =-i 0262doU22h
(B2)
It follows from substituting Eq. (B2) into Eq. (Bl) and iso-lating the fast-varying spatial function that
P2 =-i PN dC0'0 2&2W(w')
X 1 A) dzU2U2
X o 1 + I2L 2U2U2
By using expression (Al), we get
(B3)
dco'D2e2W(w')1/(1 + I2L32)
I2/(1 + I2-C2 +
for RW, (B4)
for SW. (B5)
it can be dropped from Eq. (A12) since the denominator of Eq.(32) is an even function. The remainder of the expression canbe integrated easily after separating the fast-varying spatialfunction from the slow one. By using expression (Al), weget
pinc =-i 1 3 N f dw'W(') i Vi 6
where V2(1+1212/2+ v1+I2L2)1 (A13)where
N11 =L 2ANV2 = dz N cos -Z.
NL fo ~~~C(14)
The coherent part of the polarization [Eq. (37)] can be treatedsimilarly. The following expansion is helpful:
sin Kiz sin K3z sin2 K2z = (` 2A - 24 c c
X cos 2K 2 z - cos 2 K2z + cos2 2K 2z) (A15)
If one uses expression (Al) and Eqs. (A4)-(A6), (A14), and(A15), the projection in Eq. (37) is solved. The expressionsfor p)oh are given in Eqs. (42) and (43).
APPENDIX B: POLARIZATION OF THESTRONG MODE
The polarization of the strong mode is given by Eq. (18) withn = 2:
Note that, for single-mode operation, there are no PP's(dko = 0); therefore the appropriate complex polarizationis identical to that obtained using the rate-equation approx-imation solution. As expected, Eq. (B4) is identical to Eq.(70) of Ref. 35, provided that Eq. (70) is modified to take intoaccount the proper definition of the intensity given in Eq.(29).
APPENDIX C: RELATIVE EXCITATION OFTHE STRONG MODE
Let us specialize the gain parameter [Eq. (9)] for n = 2:
g2 = -2 - , Im- 2 ,2Q 26'2
(Cl)
where P 2 is given by Eqs. (B4) and (B5). At threshold I2 =0, N = Nth, andg 2 = 0. By setting w - = 0, we get from Eq.(Cl)
Nt =EQ h AcoNth-=Q 2Zi-) (C2)
where Zi (y) is the imaginary part of the plasma-dispersionfunction 2 6:
Z(,Y)=7fdw ) + i(w -dw) (C3)
Atthehomogeneouslimit, Aw/Zi(y) = y. ByusingEqs. (B4),(B5), and (C2) in Eq. (7) with n = 2 and by assuming steady-state conditions, we get
1/(1 + 12L2)dw'W(')13 2
2/(1 + I2132 + 12132
for RW, (C4)
for SW, (C5)
where % is the relative excitation defined by
9 = NINth- (C6)
2= -i Nh _
1 Aw r_ = Ac
T Yzi (Y) J
P2 = f dco' dzN(z,co',t)p 2 U2(z), (Bi).N X
S. T. Hendow and M. Sargent III
Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 101
ACKNOWLEDGMENTS
We wish to thank N. B. Abraham, R. Gioggia, J. Maloney, M.Minden, L. Narducci, C. Stroud, and S. Stuut for stimulatingdiscussions. S. T. Hendow wishes to express his appreciationto the Institute for Modern Optics, University of New Mexico,for its hospitality during the early stages of this research.
This paper is based in part on material submitted by S. T.Hendow in partial fulfillment of the requirements for thePh.D. degree at the University of Arizona, Tucson, Arizona85721.
This research is supported partially by the U.S. Air ForceOffice of Scientific Research and the U.S. Office of NavalResearch.
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