October 1984 / Vol. 9, No. 10 / OPTICS LETTERS 445

Production of short pulses in semiconductor lasers by externallaser excitation through X(3)

D. Haas, J. Wurl, J. McLean, and T. K. Gustafson

Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory, University of California, Berkeley,Berkeley, California 94720

Received May 10, 1984; accepted July 16, 1984

It is shown that parametric mixing through X(3 ) of two injected laser lines can provide index modulation sufficient

for the mode locking of the semiconductor laser. Numerical estimates indicate that picosecond pulses with 100-

GHz repetition rates should be possible with a few watts of external laser power.

During the past several years numerous techniquesfor the generation of short pulses from semiconductorlasers have been demonstrated. These have involvedeither active or passive mode locking by various tech-niques1 or microwave-current-modulated gain switch-ing.2 The gain switching has been limited to less than12 GHz by the electrical properties of the laser.3 All themode-locking techniques have been in conjunction withexternal or integrated optical cavities to provide con-ditions favorable for mode locking. These have, how-ever, limited the repetition rate to the 10-GHz regionand below.

In this Letter we propose parametric refractive-indexmodulation to mode lock a semiconductor laser withoutan external cavity. In particular, we consider utilizingthe nonlinear optical X(3) of GaAs to mix two externallyinjected laser lines to produce the required modula-tion.

It is demonstrated that pulses of the order of 5 psecshould be easily achievable with a 25-psec spacing (i.e.,40-GHz modulation rate). With various techniquessuch as free-electron or multi-quantum-well contribu-tions to enhance the nonlinearity, shorter pulses andmodulation frequencies up to several hundred gigahertzshould be possible.

The technique that we propose provides a relativelysimple means by which to inject the necessary micro-wave-frequency refractive-index modulation withoutthe inconvenience of lossy strip lines and sources. Itfurthermore allows one to consider a distributed-feed-back locking technique that couples the forward-trav-eling component at one frequency with the backward-traveling component at neighboring frequencies.

Since the proposed scheme is distributed, we for-mulate the analysis in terms of the basic pulse-propa-gation equation4

i[O(wo) -k 0 ] + a + ki E(t, z)

= a2E i i(co2 PNL(Z, ( (1)

In this equation

_ 0/3 k2 = 1/2 A|Adws

and E (z, t) is the complex slowly varying opticaltransverse-mode amplitude modulating the plane-wavecarrier at frequency w0 and wave vector ko. We con-sider a single transverse mode specified by invarianceof the transverse profile with propagation in the z di-rection and defined by

VT 2F() - 2 - W2 E (T)jF(T) = 0X (2)

with

VT2 F = -kT 2 F,

for each dielectric layer under consideration. Here VT 2

is the transverse Laplacian. This defines $(w), thedispersion relationship for the particular transverse-mode profile under consideration. ko is then equal toRe [/(coo)], c is the vacuum speed of light, and the totalfield E(0, z, t) = 1/2 E(z, t)F(T) exp[i(wot - koz)] +c.c.

PNL(Z, t) is the nonlinear polarization amplitude ofthe transverse mode induced by guided-wave propa-gation of two laser lines proposed for index modulationand differing in frequency by cin. This is of theform

w02 1 PNL(Z, t)C

2 2ko EO

= am A1 - Cos<t - (31 - / 2)zIE(z, t), (3)where am is proportional to the third-order complexnonlinear susceptibility. AO3 = A1 - 02 is the differencein the modal propagation constants, where we haveassumed that the two modes are copropagating in theGaAs and are undepleted.

The gain medium is assumed to be described by asingle Lorentzian 5 (effective homogeneous line shape).This is an approximation that simplifies the analysis butis not essential. For this situation

3() = con + (Xgi [1 + i(w- o)T2] (4)

0146-9592/84/100445-03$2.00/0 © 1984, Optical Society of America

446 OPTICS LETTERS / Vol. 9, No. 10 / October 1984

where n is the effective refractive index of the mode atfrequency w, ag is the steady-state complex gain coef-ficient of the mode at w = w0, and cIl is the linear losscoefficient, assumed to be frequency independent. InEq. (1) if w is assumed to be on line center,

o = Won, (5a)C

ki = a (n) + agT2, (5b)

k 1= w2(__l 5 _i

where it is understood that n is n(wo). With these, Eq.(1) becomes

(a +k k E( t)az

=gT22C2E -( 2' a) [I -[1 -cos(wot - Aiz)]E(z, t)

+ (cag -al)E(z, t), (6)

where it has been assumed that the dispersion that isdue to the gain profile dominates the modal disper-sion. 6

A simplified analytical solution providing numericalestimates of the parameters required of the injection-laser fields is obtained by assuming that the mode-locked pulses travel near the peak of the microwaveindex modulation5 '7 and have a duration shorter thanthe microwave period; that is,

1-cos(wmt - A/3Z) iz|1[ * )AO (7)

The resultant equation is a generalization of themode-locking equation of Haus for traveling-wavemodulation.5 The steady-state driven field-envelopesolution is obtained by assuming a profile traveling withspeed v = wn/A/3, that is,

E(z, t) = E bt -) = Z 1)

Substituting into Eq. (6), one obtains

= agT22E" + -i am (nto.) 2 + (ag - a,)]E, (8)

where the prime implies d/dc7. (Note that 6TR =2l[Re(k,)-(1/v)] in Haus's theory.} The lowest-ordersolution to Eq. (8) is given by7'8

E(nq) = exp(-1/2 Wp2?i2)exP[- 1/2 -ki) agT2 2 47

with(ag 1/4 )I,1T 2

(9b)

and

1 _ C', - k1 Ij2 = ( )pT2)ag 4(cagT2 ) 2 k-V)

(9C)

wp determines the pulse width and chirp, and Eq. (9c)the complex gain parameter ag. It is seen from this thatthe threshold gain and the pulse width are increased ifvelocity match is not achieved. It should also be em-phasized that this solution is an accurate description aslong as the pulse delay in Eq. (9a) is much less than themicrowave period so that quadratic approximation tothe modulation equation, expression (7), is valid. Al-though this is not strictly valid for the numerical esti-mates that we choose, it provides an order-of-magnitudeestimate of the conditions required for the productionof short pulses.

The usual solutions are obtained by requiring that thevelocity of the index modulation wave be equal to thepulse group velocity (resonant condition),

1 n 1n-Re(k) =- + Re(agT2) 'V C Vg

thereby ensuring that the driven pulse solution has avelocity equal to the linear modal group velocity. Thiscan in principle be established by proper design of thewaveguide for the injected modulating beams. Sincethe two modulating beams share a common waveguide,v is approximately equal to the group velocity of themode.

The discrete longitudinal-mode behavior of the laserenters readily by requiring periodic boundary condi-tions on the pulses in accordance with the Floquettheorem. Thus the total electric field is

( 211E(z,t) = a En-n- ,n=-- V

(10)

where I is the cavity length. In order for each pulse tobe modulated identically each time it passes through theGaAs in the positive z direction, the modulation fre-quency wo must be a harmonic of the round-trip cavityfrequency [(7rs/l)/(1/vg)]. Thus A: = 7rs/l, where s isan integer.

To obtain a numerical estimate of the possibility ofmode locking, we use Eq. (9b) to solve for am, the peakof the modulation in terms of pulse width, gain, andmodulation frequency assuming a perfect velocitymatch. For a GaAs laser 1000 gzm in length, theround-trip cavity time is 24 psec if agT2, which is<<(n/c), is neglected. Thus wm, the modulation fre-quency required, is 2.6 X 1011 rad/sec, 2 ag ; 60 cm-',and wg = 1/T2 - 2 X 1013 rad/sec. 9 Thus if we take apulse width Tp of the order of 4.2 psec where the pulsewidth (FWHM) -rp = 2{(In 2)1/2/[Re(Cwp2)]1/2}, Eq. (9b)gives anm § 0.11 cm-' for a,, assumed real (X(3) real),and Re(wp 2) = Im(Wp2), consistent with Eq. (9b).Equation (9c) shows that, since Iwpi << c)g, taking a realcag is a good approximation for a velocity match.

The electric-field amplitude necessary for locking canbe obtained from an since Eq. (3) using PNL(Z, t) =EoX(3)E

3 gives' 0

6 X 1/2 2n x(3)E2 = a.,cn

October 1984 / Vol. 9, No. 10 / OPTICS LETTERS 447

where the two modulating fields are assumed to be ofequal amplitude, Em. Assuming a mode overlap ofunity for the modulating and lasing transverse modes,and a conservative nonresonant value of X(3) = 1.4 X10-19 (M/V)2 (10-11 e.s.u.),1" one obtains

namX (12Em 2 = (3X 1.3 X 1013 MV

This implies an intensity in the GaAs of 6.7 MW/cm2,which is a reasonable requirement. For a 5-Am-radiusguided beam, one obtains a total power of 5.3 W with anindex change n am X of the order of 3.4 X 10-6. Sinceam scales as rp -4, 10-psec pulses would require -0.3 Wof power with an index change of -2 X 10-7.12

It is anticipated, for various reasons, that the powercan be lowered. With modulating fields havingfrequencies close to the band gap, the X(3) should belarger13 than the above value. In addition, resonantenhancement, 1 4 multi-quantum-well interactions, 15 thefree-electron contributions and nonparabolic bandbending16 provide additional means of increasing X(3).By modulating at extremely high harmonic frequenciesone could expect also to decrease the power considera-bly. Ultimately picosecond pulses with a watt or lessof external power should be obtainable by utilizing acombination of these various techniques. Such powerlevels are available from stabilized quarternary lasers,which would result in an all-solid-state pulse generator.F-center lasers are also interesting possibilities for themodulating fields.

In addition to the traveling-wave case, one could alsohave a standing-wave situation in which frequencycomponents of the forward-traveling pulses are coupledto the neighboring frequency components of back-ward-traveling pulses. In essence, the forward-trav-eling components can Bragg scatter into the back-ward-traveling components because of a standing-waveindex grating set up by the interfering modulating laserbeams. The cross coupling would result because of anoscillation of the induced grating at frequency Wim. Forthis situation, Eq. (3) would be replaced by

Co02 1 PNL(Z, t)

C2 2k0 E0= am[1 - Cos(wmt)cos(AIfz)]E(z, t). (11)

We are experimentally attempting the locking whiledoing further numerical calculations. In particular,both the modal dispersion and the material refractive-index dispersion are expected to be influential since thecavity consists solely of GaAs.6 The gain profile of theGaAs is also being portrayed more accurately than aLorentzian in Eq. (4). Other nonlinear optical inter-actions, such as two-photon absorption in the lasercavity, are also being considered.

In conclusion, we have proposed the use of parametricmixing of two injected laser fields through X(3) for active

mode locking of semiconductor lasers. Since thetechnique does not require an external cavity, intensepulses at a high repetition rate should be possible.More generally, parametrically driven semiconductorlasers and amplifiers offer the possibility of modulationat frequencies at least an order of magnitude higherthan at present and are of potential interest for fiber-optic communication and millimeter-wave transmis-sion.

This research was sponsored in part by NationalScience Foundation grant ECS-8318682 and by the U.S.Air Force Office of Scientific Research under contractno. F49620-79-C-0178.

References

1. P. T. Ho, L. A. Glasser, E. P. Ippen, and H. A. Haus, Appl.Phys. Lett. 33, 241, (1978); H. Yukoyama, H. Ito, and H.Inaba, Appl. Phys. Lett. 40, 105 (1982); C. Harder, J. S.Smith, K. Y. Lau, and A. Yariv, Appl. Phys. Lett. 42,773(1983), and the references therein.

2. R. A. Elliot, Huang DeXiu, R. K. DeFreez, J. M. Hunt, andP. G. Rickman, Appl. Phys. Lett. 42, 1012 (1983).

3. K. Y. Lau, C. Harder, and A. Yariv, Appl. Phys. Lett. 44,273 (1984).

4. T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, andP. L. Kelley, Phys. Rev. 177, 306 (1969).

5. H. Haus, Fields and Waves in Optical Electronics(Prentice-Hall, Englewood Cliffs, N.J., 1984).

6. Background material dispersion in GaAs is estimated tobe -104 psec nm-1 km-' [estimated from data of D. Sellet al., J. Appl. Phys. 45, 2650 (1974)] in contrast to agT22,which gives -105 psec nm-1 km-'. Thus this approxi-mation is expected to be valid.

7. D. J. Kuizenga and A. E. Siegman, IEEE-J. QuantumElectron. QE-6, 694 (1970).

8. E. T. Whittaker and G. N. Watson, A Course In ModernAnalysis (Cambridge U. Press, Cambridge, 1950), Chap.X.

9. H. C. Casey, Jr., and M. B. Panish, Heterostructure La-sers Part B (Academic, New York, 1978), Fig. 7.9-2.

10. M. D. Levenson, Introduction to Nonlinear LaserSpectroscopy (Academic, New York, 1982).

11. C. K. N. Patel, R. E. Slusher, and P. A. Fleurey, Phys. Rev.Lett. 17, 1011 (1966).

12. For a 10-psec pulse approximation (7) is not satisfied, andthus the full cosine modulation should be considered.

13. Y. J. Chen and G. M. Carter, Appl. Phys. Lett. 41, 307(1982).

14. Such as the fundamental lattice resonance at 37.2 ,um,which would also permit group-velocity match withoutguiding [C. J. Johnson, G. H. Sherman, and R. Werl, Appl.Opt. 8, 1667 (1969)].

15. D. S, Chemla, T. C. Daman, D. A. B. Miller, A. C. Gossard,and W. Wiegmann Appl. Phys. Lett. 42, 864 (1983); D. A.B. Miller, D. S. Chemla, D. J. Eilenberger, P. W. Smith,A. C. Gossard, and W. T. Tsang, Appl. Phys. Lett. 41,679(1982).

16. P. A. Wolff and G. A. Pearson, Phys. Rev. Lett. 17,1015(1966).