Supermodes of high-repetition-rate passively mode-locked semiconductor lasers

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 32, NO. 6, JUNE 1996 94I Supermodes of High-Repetition-Rate Passively Mode-Locked Semiconductor Lasers Randal A. Salvatore, Steve Sanders, Thomas Schrans, and Amnon Yariv, L~ Fellow, IEEE Abstract-We present a steady-state analysis of high-repetition- between neighboring modes can be significant, and typically rate passively mode-locked semiconductor lasers. The analysis only a small number of rnodes (around 3-10) dominate. includes effects of amplitude-to-phase coupling in both gain and Active modelocking, on the other hand, has been analyzed absorber sections. A many-mode eigenvalue approach is prethoroughly in both the time domain and the frequency domain sented to obtain supermode solutions. Using a nearest-neighbor mode coupling approximation, chirp-free pulse generation and [4]-[6]. It has been suggested that passive mode-locking electrically chirp-controlled operation are explained for the first should be analyzed in the time domain since simple products in time. The presence of a nonzero alpha parameter is found to the time-domain analysis result in cumbersome convolutions change the symmetry of the supermode and significantly reduce in the frequency domain analysis 171, however, in the case of the mode-locking range over which the lowest order supermode remains the minimum gain solution. An increase in absorber high-repetition-rate passive modelocking, where few modes strength tends to lead to downchirped pulses. The effects of are involved and the induced carrier modulation is much individual laser parameters are considered, and agreement with closer to a sinusoid [8], the frequency domain approach recent experimental results is discussed. becomes more appropriate. In this paper, we present a steady- I. INTRODUCTION P REVIOUSLY, the theory of passive modelocking has been analyzed thoroughly in the time domain [ 11. Haus’ analysis has provided a clear picture of the evolution of pulses through gain, absorptive, and bandwidth-limiting elements within a cavity. A steady-state solution was found when these effects are included. Certain approximations were deemed necessary in order to present an analytic solution. For example, in the steady-state solution, a symmetric and unchirped pulse envelope is assumed as limited by the approximation of all time-domain effects only up to the quadratic term. The model has been extended to include chirped pulses due to selfphase modulation (SPM) yet only for a fast absorber [2], [3], and still restricts the analysis to exponents quadratic in time and achieves symmetric pulses. No recovery is assumed to occur during pulses. Additionally, both models include an approximation of the discrete-mode spectrum by a continuous spectrum. Although the latter approximation works well for mode-locked lasers having many closely-spaced modes, and a slightly-varying gain with frequency, it, along with the assumption of no material recovery during the pulse, is not adequate for the case of high-repetition-rate passively modelocked lasers (250 GHz). In this case, the difference in gain Manuscript received August I , 199.5; revised January 30, 1996. This work was supported by the National Science Foundation under Grant ECS-9001272 and by ARPA and the Office of Naval Research under Grant N00014-91-J119.5. R. A. Salvatore is with the Electrical and Computer Engineering Department, University of California, Santa Barbara, Santa Barbara, CA 93106 USA. S. Sandcrs is with SDL Inc., 80 Rose Orchard Way, San Jose, CA 95134 USA. T. Schrans is with the Ortel Corp., 201.5 West Chestnut Street. Alhambra, CA 91803 USA. A. Yariv is with the Department of Applied Physics 128-95, California Institute of Technology, Pasadena, CA 91125 USA. Publisher ltem Identifier S 0018-9197(96)04131-0. state analysis of passive modelocking directed toward highrepetition-rate semiconductor lasers. The analysis is done in the frequency domain extending that presented in [8]. For the first time, passive mode-locking supermodes are found while amplitude-to-phase coupling from slow saturation is permitted. Section I1 describes the model and arrives at an equation for each mode in the supermode. It incorporates dispersive effects through the common semiconductor laser parameters and unlike previous frequency domain calculations, does not force all modes beyond (the minimum) three modes to contribute zero coupling. Section I11 describes the eigenvalue formulation used to arrive at a self-consistent solution of the coupled nonlinear equations. Section IV presents an approximate analytical expression based on (the minimum) three modes in order to reduce the complexity and allow one to build physical intuition about the gain requirements and amplitudes and phases of the supermode spectrum. Section V presents results for the full calculation. Section VI compares the results with experiments for high-repetition-rate passively mode-locked lasers. Finally, Section VI1 includes conclusions. 11. THE MODEL High-repetition-rate modelocking ( 2 5 0 GHz) was first demonstrated by Vasil’ev [9] and by Sanders et al. [lo]. To date, semiconductor lasers are the only mode-locked lasers that have been able to generate repetition rates of hundreds of GHz. Due to their large material gain coefficients, fast recovery times, and the ability to be made into short monolithic cavities, high-repetition-rate pulse trains can be generated easily. Typically, high-repetition-rate lasers involve a monolithic semiconductor laser structure, meaning no external cavity is used. The model presented is intended to analyze the monolithic multisection laser, and no intention of including an external cavity is made here although one could easily modify 0018-9197/96$05.00 0 1996 IEEE IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 32, NO. 6, JUNE 1996 942 were disallowed [see Fig. 2(b)]. One can write an equation for the net gain of each mode including the coupling effects due to each of its neighboring modes. Also there are phase effects, and for stable mode-locking one requires that all the modes will be equally spaced in frequency. The rest of this section will be devoted to deriving an equation for each of these coupled modes which will subsequently be solved to find the supermode for the high-repetition-rate passively mode-locked laser. The net optical field inside the laser can be written as a sum over individual modes, w,t ) =p , ( t ) W , (1) n Fig. 1. Schematic for two-section monolithic passively mode-locked laser. lm 1 Single Mode (a) where E, ( t ) represents the time dependence of mode n, and G,(?) represents the nth spatial eigenmode of the cold cavity and satisfies v J 2 G n ( F )+ ~ ~ E , R ~ G , ( ? )= 0. Here, is the magnetic permeability of free space, E, is the electric permittivity, and 0, is the resonant frequency of the nth mode of the cold cavity. Assuming we have some uniform guiding (through index or gain-guiding) structure longitudinally throughout the laser, we can write Gn(F)= Jzqz, y) cos (P,Z). AM Mode-locking (b) Fig. 2. Without any mode coupling, a homogeneously broadened laser will lase in the single mode at which the gain and loss are (a) equal. Allowing mode coupling, amplitude modulated (AM) passive modelocking may permit a cooperative saturation of the absorber during some part of the repetition cycle and allow supermode lasing to occur (b) with a lower threshold gain than if mode coupling were disallowed. the modeled cavity to include a reflection-free facet and some length of free space to account for an external cavity. Passive modelocking- requires a minimum of two sections such that one section is pumped above transparency and one remains below. A standard two-section monolithic passively mode-locked laser structure is shown in Fig. 1. More complex structures have been made to achieve Bragg filtering [ I l l , incorporate additional sections [ 121, [ 131, change recombination rates [14], or develop transient gratings to increase the effectiveness of absorber saturation [ 121. The steady-state effects of each of these can be taken into account by adjusting the appropriate parameters of the model in Fig. 1. Physically, one may model the average net gain of a semiconductor laser as having an approximately parabolic spectrum near its peak. Typically, if one pumps the gain strongly enough, it will reach a point at which the gain equals the loss as shown in the left side of Fig. 2. If the gain equals the loss for some mode, this mode will start lasing, the homogeneous gain will become clamped, and further pumping will go into generating light in the lasing mode. On the other hand, if one allows a coupling to exist between the modes, the presence of light in mode n, under some conditions, can make it easier for the light in mode n+ 1 (and vice versa) to saturate through the absorber at certain times during the repetition cycle. Thus, the laser may lase at a lower average carrier density than it could if mode coupling (2) These modes of the cold cavity may be delta-function nor- J' G,(q . Gm(?) dV = V,S,,. (3) Similar to (1), the net electronic polarization can be written as a sum of terms separable in space and time. Upon writing the wave equation for the net field polarization projecting onto Cn(F): (4) where P,(t) = (l/Vc) $(?, t ) . G,(F')dV is the projection of the polarization on mode n.Here rp, represents the photon lifetime for the nth mode. With the optical frequency much greater than the repetition rate, we may write E,(t) as the slowly varying complex envelope of &,(t) such that &,(t)= $E,(t)e- + (5) C.C., where w, is the optical angular frequency of the nth lasing mode (w, # R, for nonzero detuning), and correspondingly P, ( t )may be written as the slowly varying complex envelope of the polarization. Thus, ~- dt i(0, 1 - - wn)E,(t) + -E,(t) 27pn ~ SALVATOREet al.: SUPERMODES OF HIGH-REPETITION-RATE PASSIVELY MODE-LOCKED SEMICONDUCTOR LASERS where pn(t) will contain coupling terms to electric fields spaced at harmonics of the repetition rate, A = w, - wn-l> since the net polarization is given by where may possess optical-pulse-induced oscillations in the carrier density [SI, and f(w,) takes into account the frequency dependent gain or loss of the material. Although in general the material's loss spectrum has somewhat different center and shape than that of the material's gain spectrum, we shall not attempt to model that in this paper. Because lasers tend to operate at their gain peak and semiconductor lasers have a significant contribution of gaindependent phase shift at their gain peak, xk(F) = xk(F) i x i ( F ) presents not only a gain, but a change in refractive index as well. The mode-locked laser is in fact no better in this respect. It tends to operate at an even longer wavelength than a continuous wave (CW) laser (due to the presence of the absorber) [SI and is expected to produce even a slightly larger amplitude-to-phase coupling factor 01 in its gain section [15], where a = - & ( f l / x ; ( F ) . Since pn(t)is computed from a projection of @(F, t ) onto Cn(F) over the whole length of the laser, there is a contribution from both the gain and absorbing sections 943 and this term is proportional to thie average single pass gain where r is the confinement factor, c is the speed of light, ti is the material gain coefficient of the absorber section (ti < 0), I , is the total laser length, and h, and h, are the ratios of the gain section and absorber section lengths, respectively, to the full laser length. The term & ~ , e - ~ v ~ 9will be determined from the carrier dynamics by using a linear approximation for the change in optical gain (loss) versus carrier density for the gain (absorber) section with g = G[n,(t)-1201. Here, G is the differential gain, n, ( t ) is the time dependent carrier density, and 110 is the carrier density at transparency. Correspondingly, ti = A[n,(t) - no] for the absorber. Gain and absorber dynamics result from the photon intensity, which is proportional to S ( X ,t ) = S O + s ~ ( x COS ) (kat) (13) k with + and where ( ) A represents a spatial average over a wavelength. Permitting this form, one notices from the carrier rate equation that a modulation in the light intensity will induce a modulation in the carrier density at tlhe same fundamental and harmonics of that frequency. However, the effect of both the small number of modes and the shorter in-phase overlaps of quickly beating pairs of modes causes the coupling of higher harmonics to drop off. Ignoring the terms responsible for second nearest neighbor and higher coupling terms to simplify the problem and still keep it suitable for high-repetition rate modelocking, from the carrier rate equation, where and (The notation ( g / a ) indicates quantities pertaining to the gain or absorber region, respectively.) We will use x'' = g p L r c / w ~ to relate the material gain coefficient, g, to the imaginary part of the susceptibility with pT being the cold cavity refractive index. The imaginary part of (10) yields (16) we find a saturated material gain g for the gain section, dependent on the gain recombination time r, and the injection pumping Rp(,/a)and correspondingly 6 for the absorber section dependent on the absorber recombination time r, , Here, g' and 6'represent the unsaturated gain and unsaturated Loss. Additionally, the carrier density is written n ( , l a ) ( z , t ) = IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 32, NO. 6, JUNE 1996 944 n O ( g l a ) + n l ( g l COS , , ( ~[At+$(,/,)]+. ) . . , and terms showing modulation at the first harmonic in the rate equation lead to: [accomplished through a(g/a),],giving the single pass net gain and phase effects that are not due to coupling as LL For steady state, we can ignore all time derivatives and using (6), (9), (23), (25), (26), (28), and (29), the equation for mode n becomes, {2i.r,,(Qn - U,) - 1 + f ( % ) [ ( l ia,,)io + + (1 + iaan)60]}E, da2 ($ + + + -S1( v n - G - 1 + l?n+E,+l) = 0 , (30) SO where we have defined coupling coefficients for the nearest neighbor modes, As,)' 'fn- = f ( W n ) [ K g ( l and + iCYgn)f?ilLG v,+ = f ( w , ) [ n , ( ~ + icvgn)e-'+g So the carrier modulation becomes small and it lags the optical pulses by nearly 7r/2 radians since the repetition rate is well beyond the recombination rate or saturation rate. Computing the spatial integrals in (1 l), we find that + 1 + zaa,)e-4a] -.SO (32) 2 These two terms are completely determined by the structure of the laser and the average photon intensity. Let a single detuning in the separation of modes be defined, 6 = w, - R, - (wn-l - Rn-l), since for stable modelocking the detuning of the repetition rate, 6,must equal the detuning in the separation between all neighboring modes. The detuning, S,, of mode n with respect to R, is then the detuning of the zeroth mode plus n times the repetition rate detuning, 6, = 60 nS. The general equation then for the nth mode with nearest neighbor coupling, for a parameters incorporated for the gain and absorber, and with geometric overlap factors included is Ka( + 1 1 I + nS) + (1+ iagn)in+ (1+ iaa,)E, . E , + sl('fn-En-l + ?j,+B,+l) = 0. sin (27rhg) [-2zrPn(S0 (33) Here 51 = s1/sO, and the material gain bandwidth is taken into account with g, = f ( w n ) g o , and 6, = f(w,)&. and is the normalized gain. Likewise for the part of the integral over the absorber, where K, -A& = 111. THE SOLUTION The coupled nonlinear equations (33) can be solved systematically. Also, one should solve the problem for a large enough number of equations such that the result does not depend strongly on the fact that the modes beyond those considered have been forced to have an electric field of zero. To reduce the number of parameters for the calculation, it will be helpful to transform to dimensionless parameters, s=- 1 1 I sin (27rha) - 11 A G' (34) (35) (27) and ti0 CTP = riih, -. Pr (28) One can write the single pass gain from (12) along with its corresponding phase contribution. Also, for generality, one should allow the inclusion of a frequency dependence 1161 of ' -3 - Gr, ' and (39) SALVATORE et ul.: SUPERMODES OF HIGH-REPETITION-RATE PASSIVELY MODE-LOCKED SEMICONDUCTOR LASERS One may subtract out the detuning of mode zero from the set of equations (33). Defining a constant, R = (ijo-E-1 - 51 + ijO+E+l)T, EO Choose initial values of Gnand 6 (40) fFindnew -7 Im (R) is the component of detuning of the center mode due to mode coupling and Re ( R )is the reduction in required average gain for the center mode due to mode coupling, similar to that discussed in [17]. Taking the imaginary part of the n = 0 equation and subtracting it from the general mode n equation leads to [-2iTp,,n6 go T- Update %atrix, recalculate E, and test i(cyyojo + + G")(l bn2) + a , , ~ , ~-) 1 i Im (R)]E, - + S I ( % - L + ijn+En+l)= 0. (50 - ,ibn2(crg,jo --t Update q, R, and test (41) The net gain spectrum of the semiconductor material is concave downward and may be represented by the form f ( u T 1=) 1/[1 +(U, - w ~ ) ~ / ( A w )Since ~ ] . to second order, one may write f(w,,) = 1 -bn2, substituting this, and since b << 1 and the coupling term is of the same order, we may ignore their product which goes like b 2 . Now the general equation for mode n with center mode detuning subtracted finally becomes [-2iTp,7L6 \ + (1 + i(Vgrl)jn+ (1+ ia,,,)a, + a , o i i o ) - i Im ( R )- 1]E, + &(ijn-G--l+ ijn+En+l) = 0. - 945 Recalculate 1 6 - Fig. 3. - (42) Considering a set of 2q+ 1 modal equations (all are complex except for the .n = 0 equation), there are 4qf 1 real equations and a list of 4q+ 3 unknowns including 4q+ 1 unknowns to specify the fields [we may take arg(E0) = 0 to define an absolute optical phase] and two other unknowns, go and 6.The phase of the repetition rate is also a degree of freedom and one may specify arg (91) = 0. Then, the modulation response of the laser sections can be referenced relative to the phase of the optical pulses. Since physically one considers a laser operating with a specific dc pumping (or more appropriately here, a constant average output) power, one may specify a particular average cavity photon intensity for 50.The latter two conditions, without loss of generality, reduce the number of unknowns in the field vector to 4q- 1, making the problem completely determined. Due to the nonlinear dependence of the parameters go, ?io, 6, Im ( R ) ,G's, and SXl on the vector + E , the problem remains challenging. However, the solution is vastly simplified by viewing it as an eigenvalue problem. For example, one may directly write the problem in a matrix form as (43) found at the bottom of the page. Through multiplications of the rows by the appropriate complex factors Flowchart of calculation for self-consistent supermode solution. one may also show that the problem can always be written, having a single complex eigenvalue, i, in the form i)- Ij,]zm= 0, [A"(go, 6, (44) where A" (go, 6,E ) is a modified complex matrix and E m is a modified eigenvector. The problem is more easily solved by keeping it in the form of (43), however. For a nontrivial eigensolution, we require that the real and imaginary parts of the determinant of the matrix in (43) equal zero. This gives two conditions from which one may find a best estimate for ,GO and b, and this was done simply through Newton's method. With this better estimate of the eigenvalue we proceed to update the relevant parameters and find a new estimate of the eigenvalue. The process is repeated as shown in Fig. 3 until convergence is reached. The computation gives tlhe supennode solutions of the high-repetition-rate laser for the chosen average operating power go. Iv. RESULTS FROM AN APPROXIMATE THREE-MODE SOLUTION The full numerical solution is complicated, involving a large number of interrelated parameters, and it does not quickly IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 32, NO. 6, JUNE 1996 946 lead to a simple intuitive picture of the effects of the device parameters. To supplement the full numerical solution, an approximate analytical description involving only three modes and an approximation of the supermode symmetry is pursued. One may show ,that if the Q parameters of the gain and absorber sections are ignored, and the gain bandwidth is symmetric relative to the cavity modes, a totally symmetric (odd symmetry) supermode solution for any number of modes will result. The form of the supermode solution will be En = El,, (45) TABLE I PARAMETER VALUES USED IN THE CALCULATIONS Variable Symbol Kumbei of Modes Considered Zq+l 15 Center Wavelength X 0.85 Effective Index of Refraction p. 36 Diffcrcntial Gain G 1 x 10-15 .9 2.2 / Diff. Ratio of Diff. Abs. since phase effects resulting from the Q parameter greatly outweigh the effects present when the Q'S were zero. Since a simple, analytic, and reasonably accurate result can be obtained assuming (46) when some nonzero Q is present, we derive a solution for three mode-locked modes using this even symmetric assumption. The term - i b [ ~ , ( * ~ ) g o ~ , ( * ~ 1 i i o ] is found to have little effect on the net gain, amplitudes, or phases of the supermode and will, for this reason, be ignored in this three-mode approximation. From the n = 1 and n = - 1 equations of (42), the expressions + and can be obtained. Combining this with the n = 0 equation, we can find the reduction in required gain for the center mode, Re ( R )= -(GO + Zo - 1) From (48), we will find that a chirp-free solution will exist if k g a g sin ?Lg = -&aa sin $,. (50) In this case, a soliton-like compensation effect occurs in the monolithic laser cavity. This condition implies that the selfphase modulation (SPM) of the absorber section may exactly oppose the SPM from the gain section [ 181, [ 191. For a larger rJ Gdin Spction n c c o v ~ r yTime Rdtio of Abs. Krcov. Time and one can always find a three-mode solution having all three modes exactly in phase. However, as soon as ag # 0 or a , # 0 is chosen, the symmetry is broken and one finds that now a chirpfree supermode solution of this form will not generally exist. Thus, no passively mode-locked supermode will exist having the form of (45) when the amplitude-to-phase coupling is taken into account. The relative phases of the modes in the supermode depend strongly on the amplitude-to-phase coupling. One finds, for numerous solutions of the full numerical analysis that once a nonzero Q parameter is chosen, the solutions are of the even symmetric form Gam 1 Gain Recov. Time FunddrnriiI.al RepPt,ition Rate Vdhir I 0.3 A/2* 80 C a n Section a parameter 0" 4 a, 2I Pholon Cavit.y Lifetime r, 10 Confincnient Factor r 0.05 / Total Laser Length pni cmz x 10-9 r Absorber Scction a pararnekr Katio of Absorber Length Units h, 0.25 Uormalized TTnsdturatrri Absorptiun a; -2.0 Gain Bandwidth Aw/Zn 10 Coefficient for 0's dependencc on intensity ai 0.25 GHz ps THa ratio of cyg : a,, a net upchirp (optical frequency rising with time during the pulse) due to SPM will occur. In the frequenc? domain picture this corresponds to a phase term. eza("-'"o) , multiplying the optical spectrum, where a is negative. For a smaller ratio of ag : a,, a net downchirp due to SPM is found to occur. A plot of chirp verses the ratio of a g :a , for a specific laser operating point will be shown in the next section, using the full calculation. Evidence of both these regimes has recently been demonstrated [20]. V. THE FULLSUPERMODE CALCULATION As formulated in Section 111, the high-repetition-rate laser supermode can be found numerically. This may be accomplished even while eliminating all assumptions on the modal phase and removing any restrictions on the number of participating Fabry-Perot modes. One finds that if a large enough number of modes is allowed such that the outermost modes have powers of < loF6 compared to the strongest modes, there is little further change in the result if additional modes are included. Given reasonable parameters for laser material and structure, such as those shown in Table I, one can find the supermode solution. In general, one would not expect the a parameter from the gain and absorber regions to be equal. Previously [ 151, the dependence of the interband transition component of this parameter has been calculated. One would expect a smaller Q parameter for laser sections pumped to lower carrier densities. This, in fact, is found to be an important consideration in finding a stable supermode solution. Lau [8] has calculated supermode solutions for three modes with cy = 0 for both sections. We find reasonably good qualitative agreement with these results even as the number of modes considered is increased. The plots resulting from Q = 0, SALVATORE et al.: SUPERMODES OF HIGH-UEPETITION-KATE PASSIVELY MODE-LOCKED SEMICONDUCTOR LASERS - % 2 6 941 d 2 'I 3 3 2.5 c) 0.8 2c d 0.6 .4 a -3 3 -8 -6 - 4 -2 0 2 4 6 8 3.5 d 2 3 -I4 2 1.5 1 0.5 -0 5 Mode Number, n Fig. 4. Calculated mode structure of supermode assuming no amplitude-to-phase coupling, a y = 0, clcl = 0, .go = 2.5, and using the other parameter values as given in Table 1. s 2 -1.5 t' -8 10 15 20 25 Avg. Photon Intensity, yo 30 Fig. 6. Calculated plot of R e ( X ) , the reduction in threshold gain due to mode coupling, versus average cavity intensity is plotted for a , = 0, u a = 0, ai = 0. The corresponding detuning, 6, of the repetition rate is shown on the scale at right. tb -0.53 v I ' -6 - 4 I I -2 0 ' I 2 4 Mode Number, n l l 6 8 Fig. 5 . Calculated phase of the supcrmode assuming no amplitudc-to-phase coupling, a!, = 0, a, = 0, .So = 2.5, and using Table I to define all other parameters. -1 0 5 10 15 20 25 -1.5 30 \ h) h G a Avg. Photon Intensity, Fo Fig. 7. The calculated modulation depth €or the signal at the first harmonic of the repetition rate is plotted for a q = 0, a , = 0, 0 1 = 0. <?o = 2.5, a 15-mode calculation, and the parameters in Table I are shown in Figs. 4 and 5. From here on, the frequency a point far beyond the knee of the nonlinearity, the minimum mode coupling again cannot be obtained. This explains why dependence of the cold cavity loss is neglected so rpn= rp. mode-locking may only be obtained over a finite range in Fig. 4 shows the calculated field strengths for the 15-mode Fig. 6. The right scale in Fig. 7 shows in this case where a g = supermode. Fig. 5 shows the corresponding modal phases, 0 and cy, = 0, one does not expect SPM to generate any pulse where 4% is defined as the optical phase in Ene2(wnt+d'n). chirp effects and the quadratic phase (41 4-1 - 2 4 0 ) / 2 = Clearly, the symmetry of (45) is present here. Fig. 6 shows the 0 indicates that, to first order, no linear chirp is present in threshold gain difference, Re ( R ) ,as a function of 50,defined this case. previously and is displayed in units of IOW4 times the cold As discussed in the previous section, the a parameter cavity loss (from rp).The right side scale of this plot shows the can have a large effect on the phase of each optical mode. expected detuning, 6, of the cavity repetition rate. Fig. 7 shows Assuming an N parameter of ag = 4 for the gain section, the modulation depth at the first harmonic as a function of only a limited range of values for cy,, the N parameter for average intensity. The threshold gain for single mode operation the absorber section, was found to give stable self-consistent must be greater than the mode-locking threshold gain, meaning solutions. A calculation of the approximately linear chirp Re ( R )> 0 for stable modelocking to be realized [8]. In ideal (quadratic phase) at the center of the optical spectrum, (41 amplitude modulated (AM) passive modelocking, a minimum 4-1 - 2 4 0 ) / 2 , versus a,/ag is plotted in Fig. 8 for the range mode coupling is required in order to obtain simultaneous of stable mode-locked solutions. The range is quite narrow and lasing from 3 or more modes of a homogeneously broadened corresponds to a region where the SPM effects from absorber laser. This requires a minimum nonlinearity to be present. and gain nearly cancel as discussed in [20]. The dependence of Hence, if the average cavity intensity, 50, is too low, an the a's on frequency is ignored in this and subsequent plots. inadequate amount of mode coupling is generated, and modeThe same plots as shown in Figs. 4-7 can be shown for locked operation cannot be obtained. Additionally, if the cavity the case including effects of reasonable nonzero a's. The intensity is large such that the absorber is strongly saturated to new calculated field strength for the 15-mode supermode with + + IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 32, NO. 6, JUNE 1996 94s 0.3 I I I I I A a 0.2 v 0 0 - 0 r 2 o.l ?! 0 - 0- 91 0 2 -0.1 - - v 0 G -0.2 .e 6 -0.3 - - 0o I I I I -1.5 ’ -8 I I I I I I I I -6 -4 -2 0 2 4 6 8 %lag Mode Number, n Fig. 8. A calculation of the linear chirp, the quadratic phase around the center of the optical spectrum, for difl‘erent values of Fig. 10. Calculated phase of supermode when allowing amplitude-to-phase coupling, a y = 4, u a = 2.1. Parameters are exactly as shown in Table 1. I ti .C I -1.2 I -1.4 g -1.6 5‘ -1.8 2 1.35 &a 0 a a 0.9 .- c-’ d -2 ’c) 3 0 -6 -4 -2 0 2 4 6 8 -2.2 v $ -2.4 0 5 0.5 1 Fig. 9. Calculated mode structure of supermode when allowing amplitude-lo-phase coupling, a s = 4, a a = 2.1, and So = 2.5. Parameters are exactly as shown in Table I. ag = 4 and a, = 2.1 is shown in Fig. 9. Fig. 10 shows the corresponding modal phases, 4%. The previously discussed change in supermode symmetry is mainly shown in this plot of dn. Before discussing the other three plots, it should be mentioned that physically as the gain current in the laser is increased to raise the average intensity, go, one weakens the rss”~), absorber section through the relation ko = ii;/(l where kb is the section’s normalized unsaturated absorption. The strength of the gain is also weakened since we require 0. Thus, the two sections both operate that go i i o - 1 closer to transparency as 50 is increased. This implies a change in each section’s a parameter also occurs and their dependence on 50 will be approximated to first order here by Aa, = -alA50 and Aa, = alASo, where a1 takes into account a linear decrease (increase) in ag (0,) as the cavity intensity is increased. Here, Q Z is taken as 0.25 around the point sXo = 2.5. Fig. 11 shows the plot of required gain reduction, Re ( R ) , and the expected detuning in the repetition rate as a function of 50. Fig. 12 shows the modulation depth and an estimate of the mode-locked laser’s linear chirp (41 4-1- 249)/2. One can see that the expected mode-locking range over which the coupled equations can be simultaneously satisfied is severely limited when the phase condition including the a parameter is + + 1.5 2 2.5 3 3.5 4 Avg. Photon Intensity, To Mode Number, n + 1 a \ 0.45 e! -8 0, Fig. 11. Calculated plot of Re (I?), the reduction in threshold gain due to mode coupling, versus average cavity intensity is plotted for a s = 4, aa = 2.1. The corresponding detuning of the repetition rate is shown on the scale at right. 1.8 10.2 121‘ 1.6 10.15 I ti e 1.4 a $ 1.2 . I 5 0.8 zzzE 0.6 0.4 - 0.4 e i; it n it 0.2 0.5 1 ae + 1 !ipd.i I n 0.05 --. h) -0.05 I I I I I 1.5 2 2.5 3 3.5 i: 4 Avg. Photon Intensity, Fig. 12. The calculated modulation depth for the signal at the first harmonic of the repetition rate is plotted for a g = 4, cua I 2.1. The corresponding linear chirp, ($1 4-1- 240)/2 is shown on the scale at right. + considered (although other parameters remain identical). This is a direct result of the presence of the a parameters in the coupling terms and occurs consistently regardless of whether or not one includes more allowed modes in the calculation. It is expected that the mode-locked laser’s operation will change if one modifies the structure or bias parameters. These - SALVATORE et al.: SUPERMODES OF HIGH-REPETiTlON-RATE PASSlVELY MODE-LOCKED SEMICONDUCTOR LASERS s 1.2 2d 1 h I 1 I I 0.2 2 .s d-d .a F- . c e 4 g 1.2 1 0.4 I 0.1 ;f; O F+ -0.1 8 -0.2 3 0.8 0.6 3 0.4 s* h I 0.2 I I I -0.3 2 a v 0.4 II_T___~ 2 0.3 g 1 .- 0.2 U U 0 C .I 0.6 -0.1 .U 2.g 8 3. 0.1 0.8 p: 2 g -0.2 0.4 g h -0.3 v -0 - . A. I 949 I 2 1.5 1./5 I L z.zs -0.4 c.3 Unsaturated Absorption Strength, & Fig. 15. Calculated value of R e ( R ) , the reduction in threshold gain due to mode coupling, plotted against (ib, unsaturated absorption strength. 5 6 - 1.2 - 0.3 51 - 0.2 - 0.1 3 s+ 2 . I 1 c-) 5 0.8 - -0.1 - -0.2 d a 0.4 8 - -0.3 0.2 I I 4 5 0.4 - 0 2.c .B 0.6 9 0.4 1.4 0 1 0.2 I I I 0.25 0.3 0.35 I 0.4 0.45 -0.4 0.5 0.3 1.05 g 5 Ratio of Absorber/Gain Recovery Times, r 3‘ 0.1 3 1- 0 g 5 0.2 h h -0.1 r 2 25 2 $ 5 -0.3 5-0.2 0.45 0.3 h d -0.4 0.1 0.15 0.2 0.25 0.3 0.35 0.05 Relative Length of Absorber, ha Fig. 14. The calculated variation of Re (R), the reduction in threshold gain due to mode coupling, plotted against P , the ratio of absorber recovery time to gain recovery time. Fig. 16. Calculated value of Re ( R ) ,the reduction in threshold gain due to mode coupling, plotted against h a , the physical length of the absorber as a fraction of the laser’s full length. effects are important if one intends to understand or optimize the laser’s operation. We have calculated results one would expect from modifying key laser parameters and using the nearest-neighbor mode coupling approximation for the range of supermode solutions that exist around the case considered in Fig. 9. One finds that if s , the ratio of the differential absorption to differential gain is increased, a larger mode coupling is obtained. This leads to a larger value of Re ( R ) ,the reduction in the mode-locking threshold relative to the single-mode threshold, as shown in Fig. 13, which is expected to lead to a more stable mode-locked supermode. Fig. 13 also shows that a decreased upchirp or increased downchirp is expected to occur if a larger s is present and all other parameters are unchanged. The effect of T , the ratio of absorber recovery time to gain recovery time, is expected to be nearly the opposite. Shown in Fig. 14. an increased r leads to a decrease in the mode-locked gain reduction and ultimately a loss of a stable mode-locked solution altogether as the ratio is increased above T = 0.46 in this case. Simultaneously the increased value of T will lead to an increased upchirp as shown in Fig. 14. It i s known that one can reduce the value of r through stronger reverse bias or ion implantation into the absorber section. An increase in tib, the unsaturated absorption strength of the saturable absorber, is shown to lead (as shown in Fig. 15) to an increase in the mode-locked gain reduction, Re ( R ) . The strength of the unsaturated absorption is proportional to this section’s length and absorption coefficient. An increase in either of these is expected to lead to a more strongly downchirped pulse as shown in Fig. 15. This agrees with expectations described in [20] where a stronger saturable absorber is cited as the reason for a significant downchirp being obtained over most of the experimental chirp-versuscurrent curve. Unlike the time-domain analyses [ 11-[3], the frequency domain analysis allows one to account for the spatial geometry of the laser. Recently, this has been noted by Martins-Filho et al. [13]. The ratio of the physical length of the absorber to the total laser length, ha, is expected to change the effectiveness of mode coupling. If the same absorber strength can be incorporated into a smaller segment of the laser, one can achieve a more effective mode coupling and obtain a larger mode-locked gain reduction, Re (12). This is consistent with results determined in [21]. Fig. 16 also shows the effects on pulse chirp when the parameter h, is varied. The mode-locked laser’s round-trip frequency is determined by the laser’s cavity length. A larger cavity round-trip fre- IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 32, NO. 6, JUNE 1996 950 0.2 0.1 aQ ,i 0.2 / + / o t s / I -0.1 / $y / h, n -z a -0.2 0 Cavity Round-Trip Frequency, A (GHz) Fig. 17. Calculated value of Re ( R ) ,the reduction in threshold gain due to mode coupling, plotted againat A, the rrprtitiori ratr. - ?5 2 .s 1.4 1.2 8' 1 0.05 0 % ' 9 ,- .z 0.6 8 0.4 N Y . :: 0.2 c! % 5 o + + 0.8 p: 13 " " " " " ' 853.2 853.4 853.6 853.8 Wavelength (nm) measured in frequency domain, in terms Of the Fig. 19. phase of optical spectrum versus the center wavelength of a 0.5 nm rectangular spectral filter. The measurement is done with 30 mA into gain section and with absorber grounded and shows an essentially linear downchirp of 1.7 pshm. 0.1 d 3 -0.05 " 853 n -0.05 6 8 10 12 14 16 18 z e 20 Material Bandwidth, A w / 2 (THz) ~ (n), the reduction in threshold gain due to Fig. 18. Calculated value of Re mode coupling, plotted against A w , material bandwidth. quency is expected to result in reduced mode coupling due to a reduction in ng and n,. This will eventually lead to a point where the minimum mode coupling cannot be obtained and no stable mode-locked supermode exists. Although the point is ~ 1 0 GHz 5 in this case (Fig. 17), using larger values of s (=5), we have obtained stable supermode solutions slightly beyond 200 GHz. This agrees well with the theoretical results presented by Lau [SI. In this case, larger mode coupling effects resulted in a reduced downchirp. As intuitively expected, lasers having a larger gain and absorber bandwidth will obtain a greater mode-locked gain reduction. Fig. 18 shows the expected increase in Re(R) as one solves the supermode equations allowing successively larger material bandwidths. Even larger advantages are found to occur if one assumes a gain bandwidth wider than the laser's absorption bandwidth. An expected decreased pulse chirp for larger material bandwidth is also shown in Fig. 18. VI. EXPERIMENTAL RESULTS AND DISCUSSION Experimental measurements of the spectrum, pulse chirp, and the variation of pulse chirp with injection current have previously been published [20]. The laser used was a monolithic two-section quadruple quantum well GaAs laser having a repetition rate of 73 GHz. It showed qualitatively the same characteristics as the calculation for 80 GHz in the previous section. A broader spectrum and longer pulses as found from streak camera results can typically be obtained at higher bias conditions. An optical spectrum for the laser operating at 30 mA has previously been shown in [20]. The chirp of this spectrum has been measured through cross-correlation techniques [20], and integration of these results leads to phase values, 4(X), of the optical spectrum plotted in Fig. 19. The figure shows a phase of the optical spectrum corresponding to a train of pulses with a 1.7 pshm downchirp and a timebandwidth product, &-Ai/,which is 18% larger than the compressed pulse time-bandwidth product achieved in the experiment. This regime Of Operation qualitatively corresponds to the calculated optical phase in Fig. 10. Additionally, the experimental measurement of 1.7 pshm downchirp for this laser is equivalent to (41 4-1- 2 4 0 ) / 2 = 0.07 rad. Previously presented experimental results have demonstrated the effect of changes in the dc gain section injection current from the preceding case. The experimental results are shown in Fig. 20. While the laser is above threshold, the changes in dc injection current are nearly linearly related to the average photon intensity inside the cavity, go. Hence, we expect Fig. 20 to show agreement with the calculated pulse chirp in Fig. 12. Both show a sequence of upchirped, chirp-free, and downchirped operation as the photon intensity inside the cavity is increased. Although good agreement between theory and experiment is obtained, we do not intend to imply that we have found the actual parameters of the mode-locked laser. However, we believe that the chosen parameters place the calculation in qualitatively the same regime of operation and that the calculated effects of the Q parameter, the laser structure parameters, and the bias parameters will show a good correspondence with further experimental results. Additionally, all results presented in this paper are believed to be for the lowest-order supermode-the one which possesses a minimum threshold gain. We have found some relatively small regions in the parameter space in which a second supermode solution could be found as a self-consistent solution. For a set of reasonable parameters and an arbitrary go, + SALVATORE et al.: SUPERMODES OF HIGH-REPETITION-RATE PASSIVELY MODE-LOCKED SEMICONDUCTOR LASERS 95 1 and downchirped pulses all from a single laser under different gain section bias. Results of the full supermode calculation (with nearest-neighbor-only coupling) were presented. The supermode magnitude and phase were plotted in the case where no amplitude-to-phase coupling exists in either laser section. Here, supermode solutions could be obtained over a broad range of cavity intensities. In this case, parameter values 1 and mode-loclung ranges show good agreement with previous calculations by Lau [8]. Other characteristics of the supermode solution were plotted as a function of cavity intensity also. When reasonable amplitude-to-phase coupling factors were chosen for both laser sections, the supermode symmetry was 15 20 25 30 35 40 45 Current (mA) severely changed. The phase was found to take on a predominantly quadratic shape in the region of the spectrum where the Fig. 20. Measured values of chirp as a function of gain section currents. An increasing downchirp is seen as the gain current is increased. mode strengths are significant. This indicated the presence of essentially linearly chirped pulses. The presence of a nonzero a parameter was found to drastically limit the range (in terms the second supermode was always found to exist for a slightly the variation of cavity intensity) over which stable modedifferent repetition rate detuning, 6, and a higher required locked solutions could be found owing to the added phase gain, ,GO. The supermode solutions were not orthogonal. This effects. Near regions where the phase effects from gain and is contrary to some assumptions in a recent publication on absorber nearly compensated each other, the effect of the a passively mode-locked laser noise [22]. One would expect parameters on the reduction in gain due to coupling were actively mode-locked supermode solutions to be orthogonal. neither very advantageous nor very harmful. They typically However, passively mode-locked lasers are inherently nonled to a slight weakening in Re ( R ) ,the reduction in threshlinear, i.e., mode coupling is a direct consequence of the old gain due to mode coupling. To facilitate understanding saturation effects resulting from the beating of the Fabry-Perot and optimization of high-repetition-rate passively mode-locked laser modes. Only in a linear coupled mode problem would lasers, calculations of the reduction in gain provided through one expect the eigenvectors to be orthogonal. In passive mode coupling and of the expected linear chirp were presented mode-locking, however, the presence of one supermode will for variations in parameters of the laser structure and bias. modify the system (the laser) such that conditions will not Comparisons were made lo expectations and to results from permit a second supermode to exist simultaneously. Thus a other models. superposition of supermodes is not a valid solution to the set Next, experimental results from a high-repetition-rate pasof coupled nonlinear equations. The characteristic shape of sively mode-locked laser at 73 GWz were compared to the the second supermode we have found has essentially the same supermode calculations in this paper. A good qualitative shape for its supermode envelope but was offset by half of agreement for the spectral shape, chirp, and variation in chirp one mode spacing from the usual spectrum center. Two of the with changing injection current was found. The calculated sulasing modes in its supermode thus possessed nearly equal field permodes analyzed were typically not as broad as the measured strengths, the nearly quadratic phase was essentially centered supermode. The reason for choosing narrower supermodes is about this offset point and the necessary gain was always found that in this case the higher-order coupling effects (e.g., second to be higher than that required for the supermode which was nearest neighbor, third nearest neighbor coupling, etc.) are not offset. expected to be smaller. Thus, in this case, the nearest-neighborcoupling approximation is expected to be more accurate. VII. CONCLUSION However, by including second-nearest-neighbor coupling and In conclusion, we presented a steady-state analysis for high- higher-order coupling in the matrix for the supermode solution, repetition-rate passively mode-locked semiconductor lasers. one may more accurately model lower repetition rate modeWe derived an equation for an arbitrary mode that exists locked lasers down to lower repetition rates ( ~ 5 5GHz) which in the supermode of the laser. The equation requires gain are viable for data rates in communication systems which are to balance loss and incorporates phase effects that result practical today. from amplitude-to-phase coupling in each section of the laser. REFERENCES Additionally, mode coupling enters through the nonlinearity H. A. Haus, “Theory of mode locking with a slow saturable absorber,” of both the saturable gain and saturable absorption sections. IEEE J. Quantum Electron., vol. QE-11, pp. 136-146, Sept. 1975. A nonlinear eigenvalue problem approach was presented to 0. E. Martinez, R. L. Fork, and J. P. Gordon, “Theory of passively mode-locked lasers for the case of a nonlinear complex-propagation numerically solve for the passively mode-locked laser’s sucoefficient,” J. Opt. Soc. Amer. B, vol. 2, pp. 753-760, May 1985. permode. An approximation of nearest-neighbor-only coupling J . C. Chen, H. A. Haus, and E. P. Ippen, “Stability of lasers modewas used in this paper. 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Mirtchev, “Chirping effects accompanying light pulse shortening by nonlinear amplification and saturable absorption,” Optical Quantum Electron., vol. 23, pp. 1161-1168, Dec. 1991. [19] G. P. Agrawal and N.A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers.” I€€€ J. Quantum Electron., vol. 25, pp. 2297-2306, Nov. 1989. [20] R. A. Salvatore and A. Yariv, “Demonstration of down-chirped and chirpfree pulses from high-repetition-rate passively mode-locked lasers,” f E E E Photon. Technol. Left., vol. 7, pp. 1151-1153. Oct. 1995. [21] S. Sanders, “Passive mode-locking and millimeter-wave modulation of quantum well lasers,” Ph.D. dissertation, California Inst. of Technol.. 1991. [22] I. Kim and K. Y. Lau, “Frequency and timing stability of mode-locked semiconductor lasers-Passive and active mode locking up to millimeter wave frequencies,” IEEE J. Quanrum Electron., vol. 29, pp. 1081-1090, Apr. 1993. Randal A. Salvatore was born in Dearborn. MI, on January 2, 1967 He received the B S (sznnmi cum luude) from the University of Michigan Ann Arbor, in 1990, and the M S and Ph D from the California Institute of Technology in 1991 and 1995, respectively, all in electrical engineenng He is presently an Assistant Research Engineer at the University of California, Santa Barbara His current research interests are in semiconductor lasers, pulse shaping, and devices and systems for optical communications. Dr Salvatore is a member of the Optical Society of America, IEEELEOS, Phi Beta Kappa, and Tau Beta Pi View publication stats IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 32, NO. 6, JUNE 1996 Steve Sanders was born in New York, NY on Apr. 18, 1964. He received the B.S. degree in physics and applied physics from Yale University in 1986, and the M.S. and Ph.D. degrees in applied physics from the California Institute of Technology, Pasadena, CA, in 1988 and 1991, respectively. His graduate work was supported by a National Science Foundation Graduate Fellowship. His doctoral research involved the development of passively mode-locked quantum well lasers, resulting in the demonstration of monolithic devices emitting pulse trains at greater than 100 GHz repetition rates After completing his Ph.D , he uorked as a Research Fellow at the California Institute of Technology until 1993. on rare-earth doped fiber laser noise reduction In 1993, he joined the research and development group at SDL, Inc where he has been working on phasing o f laser diode arrays to develop high power spatially coherent sources and on application of high power spatially coherent laser diodes to frequency conversion for the generation of visible and mid-IR radiation Thomas Schrans was born in Gent, Belgium, in 1964 He received the engineenng degree in Electronics Engineering from the University of Gent, Belgium, in 1987 He received the M S. and Ph D degrees from the California Institute of Technology in 1988, and 1994 His doctoral research was on modelocking of semiconductor lasers, and modeling of DFB lasers In 1987, he joined the California Institute of Technology on a one year fellowship from the Belgian American Educational Foundation In 1994, he joined the IBM T J Watson Research Center, in Hawthorne, NY, as a Post-Doctoral Fellow, where he performed research on components and subsystems for WDM networks Since June 1995, he is a Staff Scienlist at Ortel Corporation, Alhambra, CA His research interests are semiconductor lasers and optical communications Amnon Yariv (S’56-M’59-F’70-LF’95) was born in Tel Aviv, Israel. He received the B.S. (1954), M.S. (1956), and Ph.D. (1958) degrees in electrical engineering from the University of California in Berkeley. In 1959. he went to Bell Telephone Laboratories, Murray Hill, NJ, joining the early stages of the laser effort. He came to the California Institute of Technology in 1964 as an Associate Professor of Electrical Engineering, and became a Professor in 1966. In 1980, he became the Thomas G. Myers Professor of Electrical Engineering and Applied Physics. On the technical side, he took part (with various co-workers) in the discovery of a number of early solid-state laser systems, in proposing and demonstrating the field of semiconductor integrated optics, the invention of the semiconductor distributed feedback laser and in pioneering the field of phase conjugate optics. His present research efforts are in the areas of nonlinear optics, semiconductor lasers and integrated optics with emphasis on communication and computation. He is a founder and chairman of the board of the ORTEL Corporation. He has published widely in the laser and optics fields, and has written a number of basic texts in quantum electronics, optics and quantum mechanics. Dr. Yariv is a member of the American Physical Society, Phi Beta Kappa, the American Academies of Arts and Sciences, the National Academies of Engineering and Sciences, and a Fellow of the Optical Society of America. He was the recipient of the 1980 Quantum Electronics Award of the IEEE, the 1985 University of Pennsylvania Pender Award, and the 1986 Optical Society of America Ives Medal, and the 1992 Harvey Prize.