IEEE JOURNAL OF QUANTUMELECTRONICS, VOL. QE-5, NO. 9, SEPTEMBER 1969 435 Analysis of Mode Locking and Ultrashort Laser Pulses With a Nonlinear Refractive Index Abstracf-A new method for locking the longitudinal modes of the duration of the pulsing (-1 ps) is larger than the a laser resonator and generating ultrashort pulses of light is de- modulating period [4], [5]. The observed pulsewidths were scribed. The cavity modes are coupled together when a medium X with a refractive index nonlinearity is placed in the cavity. 2 X second forruby and 0.5 lo-* second for A theoretical study is presented which analyzes the mode struc- Nd: glass, while the theoretical values are, respectively, ture of a laser resonator containing a cell filled with an anisotropic lo-'' second and 4 X second, indicating thatthe molecular liquid. It is found thatunder certain conditions the whole linewidth is either not fully mode locked, or that energy exchange between the modes gives rise to a mode-locked the frequency is swept. spectrum and to the attendant generation of ultrashort pulses of An increase in the output power of solid-state lasers light (- 10-11 second for a ruby laser, 10-12 second for a Nd3+: glass laser). has been obtained by the technique of & switching [6]. The output of a non-mode-locked &-switched solid-state INTRODUCTION laser consists typically of a pulse of 10-50 X lo-' second HE OUTPUT electric field of a laser is equal to with a peak power of up to a few hundred megawatts. the sum of the electric fields of the individual modes In these lasers, mode locking has been obtained by in- Tof the cavity that are amplified by the laser medium, serting a saturable absorber inside the cavity [7], [81. i.e., whose frequencies lie within the gain linewidth AvG of A saturable absorber is an element whose optical trans- the amplifying transition. mission is an increasing function of the intensity of the In thenormal mode of oscillation of a laser (no perturba- incident beam. Pulses whose duration is - 10-l' second tion inside the cavity), thephases of the modes are random as short in ruby lasers and - second in Nd:glass and uncorrelated, and the output intensity is fluctuating lasers with peak intensities in excess of lo9 watts have randomly in time around its mean value NI, where N been observed by using this technique. is the number of oscillating cavity modes and is the In this paper we analyze a new method 191, [lo] for average mode intensity. generating high-intensity picosecond pulses in &-switched It has been shown [1]-[3] that if the losses of the laser solid-state lasers. cavity are modulated at a frequency equal to the inter- We show theoretically that the introduction of a re- mode spacing frequency c/2L (L is the length of the fractive index nonlinearity inside a laser resonator gives cavity), mode locking results and the output of the laser rise to a mode-locked spectrum, characteristic of the consists of a continuous train of pulses that have the ultrashort pulsemode of oscillation. The nonlinearities following properties. we consider are provided by liquids consisting of aniso- tropic uniaxial molecules. These molecules, of which 1) The pulsewidth is equal to the reciprocal of the gain nitrobenzene and CS, are two representative examples, linewidth l/Avff. have different polarizabilities along their axis of sym- 2) The pulses are separated in timeby the double metry and along any other axis perpendicular to it. We transit time of the light inside the cavity 2L/c. call these polarizabilities all and aL respectively. A linearly 3) The peak power is equal to N times the average polarized electric field applied to such a liquid induces power of the laser where N is the number of coupled a nonlinear polarization in the medium that is propor- modes. tional to the difference (al1- a*) and to the cube of the Using internal modulators, ultrashort pulses have been electric field and therefore produces a change inthe obtained in continuous-wave gas lasers [a] (with a width dielectric constant of the medium proportional to the of 2.5 X second) and solid-state lasers [3] (8 X lo-" square of the electric field. When a liquid with anisotropic second) with a pulsewidth approaching the theoretical molecules is placed inside a laser resonator where the value l/Avff. Internal modulators have also been used to optical electric fields are large enough to produce an generate ultrashortpulses in pulsed solid-state lasers where appreciable change of the dielectric constant, it couples the longitudinal modes of the resonator together in the followingway. Let us assume that three modes of the Manuscript received January 6, 1969; revised April 22, 1969. This work was supported by the Army Research Office, Durham, resonator (0), (+l),and (- 1) oscillate withtheir re- N. C., and the NASA Electronic Research Center. spective frequencies oo,o,, Q, oo - Q. Q is the radian The authors are with the Division of Engineering and Applied + Science, California Institute of Technology, Pasadena, Cali. intermode frequency Q = m/L. 436 IEEE JOURNAL OF QUANTUM ELECTRONICS, SEPTEMBER 1969 The two modes (0) and (+ l),for example, induce a change in the dielectric constant of the liquid Ae EoEl, 4‘ where E, and E, are the electric fields of the two modes. Thus Ae has a component oscillating at the frequency (coo + €2) - wo = a. The mode (- l), incident upon the liquid, “sees” a modulation of the dielectric constant at frequency €2, which causes the generation of sideband frequencies at (wo - €2) + ma, m = fl, f2, . As an example, the sideband at (coo - a) + Q = coo coincides in frequency with mode (0). The effect of the nonlinearity Ae cc E2is thus seen to be one of coupling modes together, Fig. 1. Orientation of an anisotropic molecule in an electric field. i.e., introducing unique relationships between the ampli- tudes and the phases of the modes. We present below a detailed theoretical analysis of the The quantity s determines the average deviation of the intuitivepicture described inthe last paragraph. The orientation of anisotropic molecules from a purely random results of an experimental investigationhave beenre- orientation. s is the first diagonal element of the anisotropy ported elsewhere [9], [lo]. tensor [ll].From (2) and (3) we obtain The third-order nonlinear polarization induced in the laser cavity bythe anisotropic molecular liquid is expressed E (P.) = ma,,- 4s 3 @I, 2aJ. (4) in termsof the parameters of the liquid as a triple summa- 4- + tion over the cavity modes. Maxwell’s equations with the nonlinear polarization acting as a driving term are then The anisotropy tensor element s can be shown to obey used to find a differential equation obeyed by the modes’ the following differential equation [ll]: amplitudes. A steady-state self-consistent solution is found.This solution applies to practical experimental situations only if sufficient energy exchange takes place between the modes in a time shorter than the duration where E is the linearly polarized electric field. r is the of a &-switchedpulse. The energy exchange time constant timeconstant with which the molecules regain their To is calculated in terms of the parameters of the liquid random orientation afterthe electric field has been turned and of a given laser system. We find that under reasonable off. It is often called the Debye relaxation time [12] or experimental situations Tocan be made short enough so the orientational relaxation time. X can be shown by a that modelocking can take place during theduration simple thermal equilibrium argument [Ill to be given by - 10-30 X lo-’ second of a typical &-switched laser pulse. 1 (.Ill - %). x=-15 kTr = E(q - cyL) COS’ 0 aLE (1) ps + Normal Mode Formalism where 0 is the angle between the direction of the electric In order to describe the mode spectrum of the laser field and theaxis of symmetry of the molecule (see Fig. 1). resonator, we introduce a set of orthonormal electric and The average induced dipole moment of one anisotropic magnetic vector functions EB(r)and H,(r) as defined by molecule is found by replacing cos2 e by its statistical Slater [13], [14]. They are related by the following rela- average (cos2 e} taken over the ensemble of molecules tionships: (PA = ~(a~~- aL)(c0sze> + d. (2) k&,(~) = V X Hn(y), lcJL(r) = V X E,(r) (6) In the absence of any electric field, all the orientations of v -E,(r) = v -H,(r) = 0 (7) the molecular axis of symmetry are equally probable and (cos2 e) = 5. In the presence of a strong electric field, where k, is a constant and n is the index mode number. the molecules tend to align with their axes parallel to the According to (6) and (7), theysatisfy the following field direction and (cos’ e) is different from +. We write differential equations: (cos2 e} = s =-I-5. (3) (VZ+ kW,(Z) = 0, (V2 + lC9)Hn(Z) = 0 (8) LAUSSADEAND YARIV: MODE LOCKING .4ND ULTRASHORT L.4SER PULSES 437 andthey are defined so as to obey the normalization laser medium. Then conditions S, -E&) dr = a,,, 1 K(r)*Hm(r) dr = L,. (9) V .(D$(t)Db(t)exp [i(wo - wb)t] + c.c.). (15) The above integrations are performed over thetotal We look for a solution for s in the following form: volume of the cavity. 8 = SA(t) exp [i(wa - wb)t] + C.C. (16) We express the total electric field E(r, t) (here we go ob over to a scalar notation appropriatefor TEM-like optical resonator modes), and the total magnetic field H(r, t) where sA(t) is a slowly varying function of time compared inside the cavity as to exp [i(wa - ub)t] when a # b.