We explore the physics of excitations of a small number of quanta in microresonators. In particular, we examine this physics as it relates to the dynamics of nonlinearly coupled microlaser oscillators used to generate time resolved coherent optical wavefronts. We seek wave fronts that can be both stabilized and also rapidly reconfigured by purely electro-optic means. Novel opportunities are offered by reductions in the number of quanta needed for laser, or laser-like, action; advances in microcavity nonlinear optics; densely packed arrays of microlasers; adjustable micro-optical delay lines; synchronization of pulse envelopes in physically distinct lasers; and locking of optical fields in physically distinct lasers. Quantum statistical issues could become important, but are not emphasized here. Strategies for realizing an optical analog of high repetition rate agile microwave phased array radar with true time delay are examined.

Keywords: microresonators, microlasers, modelocking, optical phased array, nonlinear optics, VCSELS, non-mechanical optical beam steering, microcavity physics, short optical pulses, agile

 We examine the physics of an array of nonlinearly coupled microresonators excited at relatively low quantum number per resonator. The concern here is more with the novel dynamics that can be accessed by combining close packing density with very specifically controlled nonlinear behavior in three dimensional periodically structured arrays, than with quantum statistical issues per se. The latter could, however, become relevant with advances in low quantum number nonlinear optical microcavity physics.

We use as an illustration the task of achieving an optical analog of agile phased array microwave radar. A particular concern is pulse forming and true time delay capabilities for an array of modelocked nonlinearly coupled microlaser oscillators packed at interelement spacings of the order of an optical wavelength. In essence this requires phaselocking the entire set of modes of an array of physically distinct microlaser oscillators into a quasi-stable state and then rapidly reconfiguring that phaselocking along a well controlled trajectory to a different, and also quasi-stable, state.

This model of an array of synchronized and phaselocked microlaser oscillators has strong similarities to phased array microwave radar, especially as regards the radiated reconfigurable electromagnetic wavefront. The array of microlaser oscillators, however, differs significantly in the pulse forming elements, the means of locking the relative phases of those elements, and the means of reconfiguring the relationship of those elements. In essence the pulse forming elements become optoelectronically controlled microlaser oscillators as opposed to electronic circuits. The differences between the optical and the microwave phased arrays tend to occur, as one would expect, largely because of the substantial differences in the wavelength and the period of the radiation and the differences between optical oscillators and microwave oscillators.

Mastery of techniques for generating and shaping pulses of a few optical cycles duration [1], reductions in the number of quanta required for laser[2,3] and laser-like action[4], fabrication of densely packed arrays of microlaser oscillators [5]; demonstration of a micro-optical delay line susceptible of integration with a microresonator [6]; demonstration of synchronization of pulse envelopes in physically distinct modelocked laser oscillators[7]; and demonstration of phase locking of the optical carrier fields in physically distinct modelocked laser oscillators [8]; suggest that the demanding task of constructing an agile optical phased array technology can be seriously addressed [9].

1.1 Definition of terminology

A definition of terminology is important since the terminology needed to discuss this technology has been used in many applications, sometimes with varied meaning. By spatially extended modelocking we mean a phaselocking of the entire set of longitudinal modes for two or more physically distinct laser oscillators. In particular, we confine attention to the case where the two or more physically distinct laser oscillators are themselves each modelocked to produce a repetitive train of short optical pulses from each laser. The long term concern is with large two dimensional arrays; however, to illustrate the essential physics we will confine attention to the case of two nonlinearly coupled microlaser oscillators and the required controls for those two oscillators.

By agile we mean that the array of pulses produced in an array of microlasers, such as indicated in Fig. 2, would be capable of being formed, configured into a given quasi-stable state, and then reconfigured to a new, also again quasi-stable state, within a short time, e.g. order of a microsecond or less. The main point of this feature is to gain improved agility as compared to current methods of beam scanning and to also provide non-mechanical beam steering. Non-mechanical scanning tends to be particularly useful for small spacecraft, e.g., where inertial concerns favor non-mechanical agile controls.

    Fig. 1:  Schematic diagram of two nonlinearly coupled, but otherwise physically distinct, modelocked laser oscillators.  The adjustable parameters needed to produce and maintain synchronization of the pulses and phaselocking of the optical fields are shown.  The mechanisms required for conventional laser modelocking (CML) are indicated in the unshaded boxes.  The mechanisms that must typically be added to conventional modelocking to achieve soliton-like modelocking (SML) are indicated in the lightly-shaded boxes.  The mechanisms that we suggest must necessarily be added to CML and SML to achieve spatially extended agile modelocking (SEAM) are indicated in the most deeply shaded boxes bordered by the dotted box.  Higher order connections will probably also be needed.

We show in Fig. I a schematic diagram of two nonlinearly coupled physically distinct modelocked lasers. The controls required to modelock the individual lasers, synchronize the pulse envelopes in the two different lasers, and also lock the relative phase difference of the two different optical fields in the two different lasers are indicated. This task has been demonstrated, however, to the best of our knowledge, only in a few specific cases, and then only for macroscopic laser oscillators [7,8].

For very short pulses the frequency spectrum can be relatively broad. The question can be raised as to the meaning of the phase of the optical fields constituting the pulses in this case since there are a large number of different frequency components. In practice soliton-like pulses formed in a well adjusted pulse shaping environment and propagating on nonlinearly coupled guides can be described by the coupled nonlinear Schrodinger equation using functions of the form [11-13].

E_k(z,t) = A_k(z,t) exp[-iw_kt+iF_k(z)] (1)

Here E_k(z,t) denotes the electric field of the pulse in the kth laser oscillator. We assume, somewhat arbitrarily for this discussion, that any given microlaser oscillator contains only one pulse. Then A_k(z,t) is the complex amplitude envelope of that pulse and the carrier frequency w_k is defined as the center of gravity of the frequency spectrum of the pulse defined by E_k(z,t).

If t is measured in the frame of reference retarded by z/v_gk, where v_gk is the group velocity of the kth pulse, F_k(z) in equation (1) determines the position of the carrier wave with respect to the envelope A_k(z,t) identified in equation (1) and also, of course, the position of the carrier wave with respect to the carrier wave in other pulses. Denote the electric field of that other pulse as E_l(z,t)=A_l(z,t) exp[-iw₁t+iF₁(z)]. This phase relation of optical carrier fields F_k(z) - F ₁(z) evaluated at a common location z is critical to the application we discuss. Stable values of F_k(z)- F₁(z) are relatively rarely achieved for two laser oscillators and even more rarely achieved for two modelocked oscillators. We also note that an additional requirement in the laser oscillators we discuss is that the round trip phase delay in the medium must be an integral multiple of 2p , i.e.

ò F _k(z)dz = m2p , (2)

where m is an integer and the integral is over a complete round trip in the resonator. The integration path includes all mechanisms so that the task of satisfying (2) can sensitively depend on any linear or nonlinear change in the laser k, including changes due to the evanescent fields of neighboring lasers.

In general the synchronization of pulse envelopes and the establishment of a well defined phase relation of the carrier phase in two different optical pulses can be easily achieved by the simple method of dividing a pulse by a beam splitter. A beamsplitter is simple and reliable, but does not provide the capacity for agile reconfiguration that we seek here. An alternative is to use the beam splitter to produce two similar pulses and then shift one of the pulses in time using a delay line; however this latter approach is limited in the magnitude of the phase and group delay that can be agilely and reliably introduced.

Our concern is with introducing large phase and group delays and also introducing those phase and group delays in a rapid and reliable manner, consistent with equation (2) for a large array of nonlinearly coupled optical beams. We also note in agreement with reference 12 that by exerting control over the pulse generation process one also gains control, at least in principle, over the relation of the carrier phase to the pulse envelope of any given pulse. This becomes particularly important as the pulse durations approach a few cycles of the optical field [1, 12]. We do not expect this result in microlasers in the short term; however, the possibility does not appear to be ruled out by the physics of microlasers.

By synchronized pulses we thus mean pulse envelopes synchronized so the random variation in relative position of any two envelopes A_k(z,t) and A_l(z,t) is small compared to the pulse duration. By phaselocked pulses we mean that the random variation in the relative phase F_k(z) - F ₁(z) of the mean optical fields (as defined above) of two different pulses in two different microlasers is small compared to an optical period. Phaselocking of the optical fields in two physically distinct laser oscillators is thus typically much more demanding than synchronization of the pulse envelopes. The precision required for phase locking optical fields is a small fraction of an optical period. The precision required for synchronizing pulse envelopes is only a small fraction of the pulse duration.

By microresonator we mean an optical resonator with transverse dimensions of the order of an optical wavelength permitting packing densities approximating interelement spacings of the order of an optical wavelength. We show a schematic diagram of an array of microlasers in Fig. 2.

    Figure 2:  Diagram of an array of microresonators including micro-optical delay lines and elements to produce laser action, laser oscillator modelocking, and nonlinear coupling of the microlaser oscillators.  The arrays of principal interest would also include rows of microlasers extending normal to the plane of the figure so as to form a two dimensional array.  The length of the microlaser would typically be long compared to an optical wavelength so that the entire structure would take on a three dimensional character with all three dimensions large compared to an optical wavelength.

 We use the term microcontrol to refer to control elements that are incorporated within the microlaser structure. The micro-optical delay line indicated in Fig. 2 and discussed in reference 6 is such a micro-control. A practical constraint of significance is a need to avoid locating this micro-optical delay line within a distance of another periodic structure where that distance is less than the spatial extent (in the medium) of the pulse. The reason is that the two periodic structures interact. The coupled structure then restricts the bandwidth of the transmission resonance. This latter coupling may be acceptable under some circumstances, but needs to be handled with care.

When we refer to a two-dimensional array we mean an array of microlasers extending along, e.g., both the x and y axes of an xy plane of a Cartesian coordinate system (call these dimensions W_x and W_y) with the microresonator axes extending in the z direction (call this dimension L_z). Given the need for a longer microresonator path than available in current microlasers, e.g. VCSELs as they are currently fabricated, this implies an extension of the microresonators in the z direction that could be long compared to optical wavelengths. The entire structure would then be a two dimensional array with W_x, W_y and L_z all large compared to the optical wavelength. The structure would take on a three dimensional character given the need for length and transverse dimensions large compared to an optical wavelength.

A primary distinguishing feature of this technique is the ability to make stable arrays of phaselocked short optical pulses and then rapidly reconfigure that particular array to a different, but also stable phaselocked array. We use the term quasi-stable to refer to these states that are stable for as long as needed, but that are also susceptible of a rapid transition along a well defined trajectory to a new and also quasi- stable state.

The wavefronts of primary initial interest would be time resolved, continuous, and susceptible of orientation in a variety of directions. In general, the cases most easily realized, and presumably also those of primary practical interest, would be those where pulse envelopes overlap to some degree, both in the dimensions transverse to, and also along, the direction of propagation. We do not restrict attention to this condition, however, the simple cases discussed here as examples approximate the situation where the pulse envelopes are separated by distances that are of the order of, or less than, the spatial extent of the pulse envelope (in both the direction of propagation and also the direction transverse to the direction of propagation) as measured within the microlaser medium. More complex wavefronts could, of course, be considered and will presumably be the focus of further analysis.

The orientation of the wavefront within the microlaser array would be constrained to some degree by the dimensions of the array. In the simplest case the range of angles available would be determined by the aspect ratio of array. If W_x is the width of the array (measured normal to the resonator axes) and L_z is the length of the microresonator (measured along the resonator axes), the angles of the wavefront with respect to the normal to the array that could be simply accessed would be Q = tan^-1(L_z/W_x). With programmed output coupling (time dependent Q-switching of individual resonators) angles well in excess of this limit could, in principle, be realized. This would be very demanding, but does not appear to be excluded by the physics of the system.

By modelocking of a laser oscillator we mean spontaneous formation of short, approximately transform limited optical pulses as a quasi-stable state of the laser oscillator without requiring the introduction of other optical pulses that have been formed external to the microlaser. Our meaning here is to exclude modelocking caused solely by introducing externally generated pulses. This does not exclude the use of external optical pulses for reference purposes.

Conventional modelocking (CML) is taken to mean the active or passive locking in phase and frequency of the longitudinal modes of each physically distinct laser oscillator so as to produce approximately transform limited short optical pulses within the individual rnicrolaser. The pulses are assumed to occur at the reciprocal of the round trip delay in the resonator. In general the cases of interest will be those where the individual laser oscillators each operate on the lowest order transverse mode of their particular laser resonator.

By soliton-like modelocking (SML) we mean the use of the mechanisms of self phase modulation and group velocity dispersion in a manner similar to that used to form solitons in optical fiber, however, within a laser oscillator, where conventional active or passive modelocking already makes significant contribution to the pulse formation. Attention to these soliton-like modelocking mechanisms is typically required to obtain the shortest pulses and the most stable modelocking [15,16]. A related consequence of soliton-like modelocking is the presence of quasi-forces that occur between pulses that propagate on coupled optical guides and that also exhibit soliton-like properties. When we refer to quasi-forces we mean those forces between optical pulses as described in reference 11.

1.2 Prior work

A considerable amount of analytical work has been done on phase locking via evanescent field overlap in semiconductor laser arrays. In general, however, such phase locked microlaser oscillators are not modelocked so as to produce short optical pulses within the laser itself and the phaselocking that does occur is unstable and not particularly adjustable [17]. A recent review article on microlaser arrays provides a good description of the state of the art for microlasers [18]. A common problem for arrays of semiconductor lasers coupled by evanescent field coupling is that the lasers lock so that nearest neighbor lasers are 180 degrees out of phase. This does not produce an output that lends itself to short optical pulse generation and is usually not helpful in producing a uniform wavefront.

Optical pulses generated external to the laser oscillator have been introduced within a microlaser oscillator to produce a type of modelocking of a microlaser [19]. That strategy tends to beg the question posed here, namely how one generates reconfigurable arrays of optical pulses spontaneously within a microlaser oscillator array, see above.

Colliding pulse modelocking of lasers where two counterpropagating pulses are produced and synchronized within the same macroscopic laser [16,20] has been widely demonstrated. Colliding pulse modelocking is similar to the spatially extended modelooking we discuss in at least two respects: (1) there are at least two distinct optical pulses that are free to lowest order to adjust their relative position within a laser resonator; (2) the two pulse envelopes can be synchronized in time by a shared nonlinear susceptibility.

Colliding pulse modelocking is different from the spatially extended modelocking we discuss here in other respects: (1) the pulses, in the simplest case, propagate in opposite, rather than in the same direction; (2) the optical phases F _k(z) - F₁(z) of the two pulses propagating in opposite directions are not necessarily locked for a significant time period (although this is possible); and (3) the two lasers are not physically distinct, since the two optical paths are simply counterpropagating paths in the same physical resonator. A consequence of the latter property is that one lacks the freedom to easily adjust the relative phase and relative envelope synchronization of the two countexpropagating pulses. We note that colliding pulse operation in a microlaser, and hence in a microlaser array, is allowed by the physics of the system for a sufficiently fast saturable absorber response relative to the round trip time [16,20] and could under some circumstance be useful.

Synchronization of pulse envelopes and phase locking have been experimentally demonstrated for two nonlinearly coupled, but otherwise physically distinct colliding pulse modelocked lasers [7,8]. Note that there were four pulses in that particular system. In that case two bistable configurations of the four pulse envelopes were demonstrated, and also under some circumstances, a phase locking of the optical fields for a given pair of co-propagating pulses, one in each of the two physically distinct lasers.

Such a synchronization of pulse envelopes and phase locking of optical carrier fields for two or more physically distinct modelocked lasers had, to our knowledge, not been previously demonstrated. The observed phase locking, which was only obtained for longer pulses, appears understandable in terms of the advantage gained by superimposed nonlinearly induced pairs of gratings in the shared absorber produced by four phase locked pulses. In that case the two induced superimposed gratings produced by the two pairs of counterpropagating pulses would produce a minimum loss for each of the four pulses. Multiple pairs of counterpropagating pulses may, or may not, be a preferred configuration in the microlasers we discuss here.

In this prior experimental demonstration of rudimentary spatially extended agile modelocking in two macroscopic modelocked lasers, all the adjustable mechanisms indicated in Fig. I were available and used. Adjustment of the position of the saturable absorber provided an adjustable self phase modulation, the prism sequence provided an adjustable group velocity dispersion, a mechanically translated mirror and means of translating a prism normal to its base provided adjustable group delay and an adjustable phase delay, although these two delays did not have a simple independent character. The shared saturable absorber provided an adjustable complex nonlinear shared susceptibility. Both a fast (Kerr effect in the absorber medium) and slow component (bleaching of the absorber) were present. The presence of phase locking of the optical fields could be monitored by interfering the optical fields produced by the two physically distinct laser oscillators on a screen. The pulse positions were monitored by electronic means and also by use of an autocorrelator.

The phase locking of the optical fields of two single mode ew macroscopic laser oscillators by means of a closed feedback loop has also been demonstrated[ 10]. That work did not produce modelocked lasers or short pulses. seat work also required an unusually stable environment for the original implementation. That work does demonstrate, however, the feasibility of maintaining the round trip optical path of two physically distinct laser resonators at the same value to within a small fraction of an optical wavelength using only opto-electronic sensing of the interference pattern and electronically controlled feedback.

 As mentioned above destabilizing quasi-forces can occur due to the nonlinear coupling of the pulses propagating on adjacent optical guides. Even when the pulse envelopes are stable and synchronized, and the optical carrier fields set to a particular optimal phase relationship, and all external perturbations rendered unimportant, the pulses can still be distorted and the relative positions of the pulse envelope drift in time due to the quasi-forces that are exerted between the pulses. These quasi-forces are due to the nonlinear coupling on adjacent optical guides via the evanescent fields [11]. The desired pulse relationships can consequently be lost even given stable pulses, optimal phase stability, optimal relative phase relationships and absence of disruptive external perturbations. One goal here is to examine means of using intentionally introduced stabilizing mechanisms to dominate these destabilizing mechanisms, see Fig. 3.

A separate topic, but of potential long term interest, is recent advances demonstrating cooperative excitations in microresonators. These excitations suggest the possibility of obtaining nonlinear processes in microresonator arrays at extremely low quantum number per resonator. In particular, where excitonic resonances of a semiconductor occur and can be made coincident with an optical resonance of the microresonator one obtains polariton states that might be used to implement the phenomena we discuss at very low excitation levels [4].

Aceves and coworkers [21] have extensively analyzed states of nonlinearly coupled optical fiber guides where the packing densities are characterized by interelement spacings having optical wavelength dimensions. This is interesting work; however, those workers do not address modelocking or microlaser oscillators per se, and do not include the detailed controls or combination of pulse formation and agile reconfiguration of a coherent highly articulated wavefronts that we address here.

1.3 Spatially extended agile modelocking

We examine, in effect, laser oscillator modelocking that is more spatially extended and more agilely reconfigurable than conventional laser modelocking. We will refer to this here as spatially extended agile modelocking (SEAM). We focus on the case where the phases of the modes of each individual laser oscillator are locked so as to form short pulses in each individual laser oscillator, and also locked in a relationship to the modes in all the other laser oscillators so that the pulses produced by the set of microlaser oscillators form a well defined array, both as regards the positions of the pulse envelopes and also as regards the relative phases F_k(z) - F ₁(z). This has been achieved for the simple case of two modelocked macroscopic laser oscillators [7,8], but has not, to the best of our knowledge, been achieved for microlaser oscillators.

There is a need to transfer the learning from the demonstration of this mechanism in two macroscopic oscillators to a densely packed array of microscopic laser oscillators. The task is to introduce the particular niicro-controls that provide the adjustments indicated in Fig. I within an array of microlaser oscillators where the overall dimensions of the array are large compared to an optical wavelength, but the interelement spacing is of the order of an optical wavelength. This includes an obvious need to lengthen the optical path in the microresonator and to shorten the pulse duration so that the pulse spatial extent along, the direction of propagation in the microresonator medium is short compared to the microresonator optical length. Maintaining an interelement spacing of the order of an optical wavelength, while introducing the needed sensing and controls constitutes a major design and fabrication challenge, but does not appear to be precluded by the physics of the problem.

1.4 Micro-optical delay line

A device that appears important in integrating micro-controls within the microlaser oscillator is a recently demonstrated micro-optical delay line[6]. The work to date indicates that this delay line has a potential for introducing adjustable group delay, adjustable phase delay, and also adjustable group velocity dispersion within the microresonator environment. The particular niicro-optical delay line that was demonstrated is a series of alternating layers of GaAs and AlAs that form a photonic band edge resonance that matches the spectrum of the optical pulse to be delayed. The delay line consisted of 30 alternating layers of GaAs and AlAs that formed a 6 micron long structure, as measured normal to the plane of the layers. This micro-optical delay line has the desirable property that the group velocity of the pulse is reduced by a large factor, e.g., to 1/18 of the velocity in free space.

The pulses in the reported work were approximately 2 psec in duration and centered in wavelength near 1.55 microns. This group delay was varied over a large range (e.g., for 118 micron to 20 microns of effective optical path) in the experimental work by tilting the sample. Analysis of the delay line indicates that a similar variation could be obtained by applying electric fields selectively to the alternating layers in a structure via electrodes incorporated within the delay line.

This delay line has a high transmission, e.g., ~95%, for the short optical pulses and also does not significantly distort the pulse envelope despite the large reduction in group velocity. Also the structure is compatible with the planar fabrication techniques that are used to form microlasers. We characterize the delay line in Fig. 2 as a series of alternating layers similar to the layers of material that are used to form the mirrors for the microlasers. In practice the delay line is very similar to the Bragg reflectors of the microlaser. The principal difference is a slightly different spacing of the periodic layers so that the pulse spectrum coincides with the first transmission resonance at the one dimensional photonic band edge. In considering use of this delay line in the microresonator environment one should bear in mind that the physics of the transmission process requires that the delay line be short compared to the pulse length as measured along the direction of propagation within the laser medium.

 The task of realizing spatially extended agile modelocking is one of introducing a relatively large number of detailed, well managed, and rapidly adjusted, micro-controls into a densely packed microlaser environment. We do not wish to minimize the very imposing fabrication issues; however, we focus here on what are essentially the physics issues. The question addressed here is whether the physics can be identified that might make this task possible. The main issue is whether identifiable correction mechanisms can dominate the known destabilizing mechanisms. In general, the electronically controlled devices such as the micro-optical delay line can introduce relatively large corrections, but only at rates typical of integrated electronics. The shared nonlinear complex optical susceptibility can introduce corrections that are small, but nevertheless executed are rates that are optical in character, e.g., having a response time of a few femtoseconds. Our experience with the experimental work and the numerical simulations suggests that this task can be accomplished. A detailed answer will depend on a combination of numerical simulations and experimental work. We discuss the issues more fully below.

2.1 Reconfiguration trajectories

The consequence of physically distinct laser oscillators is that the pulse in any given laser oscillator is free to lowest order to be positioned anywhere in time relative to the other pulses of the array, i.e., within the constraint imposed by the laser resonator length. One thus needs controls that not only produce and maintain any particular quasi-stable array of short pulses, but also controls that provide means of causing a given particular array of pulses to follow a well controlled trajectory to a new, and also intentionally maintained, quasi-stable state.

We have used numerical simulation to explore some of these trajectory issues for the simple case of two coupled modelocked oscillators that include the controls indicated in Fig. I. We illustrate pulse trajectories for such a simple two pulse array in a numerical simulation shown in Fig. 3. The criteria are that the controlled changes be such as could be produced by externally adjusting the properties of the micro-optical delay line, as, e.g,. by applied electric fields. In particular, we seek to show that the controlled changes introduced by the micro-optical delay line would be adequate to dominate the destabilizing mechanisms caused by mechanisms internal to the individual microlaser. Our principal concern here is the unavoidable nonlinear interactions as discussed in reference 11, or other externally caused random perturbations.

The case illustrated in Fig. 3 is somewhat simplistic, but serves the purpose of illustrating the tasks that must be addressed in producing and reconfiguring a given array of phaselocked pulses. We have used our numerical simulation capability to explore at one time or another in some degree all the mechanisms illustrated in Fig. 1. In this particular simulation, aside from assuming a given pulse shape established by the conventional modelocking, we only examine the interplay of the nonlinear interaction of a pulse pair arising from the soliton-like shaping mechanisms (the mechanisms indicated by SML in Fig. 1) and corrections to the group and phase delay. such as could be introduced via the micro-optical delay line [6].
 
 

View Figure 3 (100k)
 
 

Within the relatively short propagation distance shown in Fig. 3 we find the micro-optical delay line is more than able to both move the pulses by distances large compared to the pulse duration and also to stabilize the pulse pair against the, in this case attractive force, that arises from the soliton-like interactions of the pulse pair[ I I]. These latter forces tend to become important only at the extreme end of the trajectory where the evanescent field overlap becomes significant.

We do not prove here that some combination of the shared complex nonlinear optical susceptibility and the corrections introduced by the micro-optical delay line would make spatially extended agile modelocking feasible, the positive results from the experiment [7,8] and this simulation, as well as the many advances occurring in the relevant technologies lead us to believe that a successful realization of this techniques is feasible via the micro-optical delay line. In particular, given that the experimental case of macroscopic oscillators did shown long term stable pulse synchronization and, under some conditions, phase locking of the optical carrier fields we expect that proper combining of the multiple shaping mechanisms indicated in Fig. I will provide stable behavior over, essentially arbitrarily long, propagation distances.

A principal conclusion to be derived from Fig. 3 is that the destabilizing forces, that we know about from the work in reference 11 e.g., appear to be relatively small compared to the stabilizing forces that can be realized. Note that equation 2 must be satisfied so that the corrections that can be introduced are restricted to corrections that satisfy that condition, namely that the round trip phase delay in the microresonator be an integral multiple of 2p . In effect the path shown in Fig. 3 is the folded path corresponding to many transits through the microresonator where the destabilizing mechanisms are present and the corrections are introduced so that phase delay is maintained, modulo 2p .

In the experimental demonstration of synchronization of two pulses and phase locking [7,8] the array could be reconfigured between two bistable states (with one or the other pulse leading), but only two bistable states. This reconfiguration could be carried out as rapidly as permitted by the mechanical adjustments available for those lasers. In that case the shared nonlinear susceptibility played a major role. We expect that a similar nonlinear element would be advantageous, and probably necessary, in maintaining the pulse relationships in an optimal microlaser system given the complexities of the nonlinear environment.

2.2 Technical requirements for spaCLAy extended agile modelocking

We summarize what we believe to be the features that are required to achieve this spatially extended agile modelocking: (1) nonlinear gain in each microlaser oscillator adequate to produce laser, or laser-like, nonlinear behavior; (2) a nonlinear susceptibility that phase locks the modes of each oscillator to form short pulses; (3) an optical pulse length in the microlaser medium short compared to the optical length of the microresonator; (4) means of producing and adjusting the group velocity dispersion so as to produce a minimally short, minimally chirped, optical pulse; (5) self phase modulation that is optimally matched to the group velocity; (6) means for adjusting the group delay in each microresonator; (7) means for adjusting the phase delay independently of the group delay in each resonator; (8) means for adjusting higher order corrections in each microresonator; (9) a shared complex nonlinear optical susceptibility that synchronizes the pulse envelopes of at least adjacent microlasers; (10) a shared complex nonlinear optical susceptibility that phaselocks adjacent microlasers; (11) a structure for the microlasers that localizes the optical energy in the dimensions transverse to the direction of propagation so that the evanescent coupling is not too strong or too weak; (11) packing densities where the interelement spacings are of the order of optical wavelengths; and (12) means of sensing departures of the system from the desired condition, given the typical perturbations and the nature of the coupled states, and of introducing the needed corrections into the various components via closed loop feedback, see Fig.2.

2-3 Agile scanning of wavefronts

While the case of two pulses shown in Fig. 3 is a simple example it demonstrates a strategy for moving pulses relative to each other within the local time frame of the propagating array and also for stabilizing the relative position of the pulses. The rate of variation of the pulse position for a change in group delay of 10 microns per transit, which is a small fraction of the variation in delay produced by the 6 micron long experimentally demonstrated delay line [6], the intraresonator optical path and a round trip time of 10 psec, the rate of scanning of pulse position is 10⁶ meters per sec. The pulse envelopes could presumably be moved to a new configuration in a relatively short elapsed time.

2.4 Principal differences from conventional modelocking

A basic difference between conventional modelocking and this more spatially extended agile modelocking is the phaselocking of the modes of many physically distinct, but coupled laser oscillators. An important distinguishing feature is the additional degree of freedom of moving individual pulses relative to each other in local time within a given laser oscillator and hence within the array. A conventional modelocked laser produces a single train of pulses where the interval between the pulses is relatively difficult to change without major changes in the laser resonator. Because the pulse in a given laser in the microlaser array is free to lowest order free to assume any position relative to a pulse in another resonator one can, in principle, obtain a wide variety of arrays of pulses. The capability discussed here does not become especially meaningful until many microlasers are synchronized and phase locked in arrays. The physics of the pulse formation and phase locking have, however, been demonstrated with two modelocked, synchronized, and phase locked macroscopic lasers. The extension of this strategy to many densely packed and rapidly switched microlasers appears to be allowed by the relevant physics and recent advances.

2.5 Limitations

There are a number of limiting mechanisms that need to be addressed. An obvious constraint, of course, is that the different parts of the array can only communicate at best at optical velocities. Another concern is heat dissipation. For an element having a cross sectional area of a square optical wavelength, an optical wavelength of one micron, and a lifetime for the resonator excitations of 100 psec the heat generated per centimeter squared for optical wavelength interelement spacing per quantum dissipated in each element is -0.2 Watts/cm². This is a simplistic calculation, but implies that one can reasonably pursue systems of this kind given the relatively low number of quanta now required to produce laser, and laser-like, action in microresonators, provided a relatively small fraction of the quanta are dissipated in the resonator.

Two other important difficulties are the very short length of the resonator and the very close interelernent spacing of the laser oscillators. The short length provides an important capacity to translate a relatively small adjustment of the pulse position per transit into a relatively large displacement in actual relative position of pulses in a short time. The time available for recovery of a saturable absorber is, however, also very short. The microdelay line is helpful in this regard as it can increase the optical path significantly without necessarily increasing the actual physical length of the resonator. We note that absorber recovery times as short as 150 fs have been reported [22]. The fast Kerr effect also provides a mechanism that functions as a saturable absorber, but that also has a very fast (a few femtoseconds) recovery. The lengthening of the microresonator makes this Kerr effect strategy of greater interest because of the longer optical path and the greater opportunity for self focusing to play a role.

Some of the physical mechanisms that can be a problem are the Raman gain, self steepening, third order dispersion, and linear coupling between neighboring guides. One of the additional problems of producing very short modelocked pulses within a semiconductor rnicroresonator is the chirp on the pulses. The micro-optical delay line could help in this regard as it can be adjusted to introduce a controlled group velocity dispersion. A further need is to properly include in numerical simulations all the relevant mechanisms that could occur and explore stability over very long propagation distances. In addition the modeled microstructure should approximate as closely as possible the materials and properties that could realistically be fabricated.

Our conclusion is that the physics of excitations having a small number of quanta in a microresonator permit strategies that could lead to successful realization of a type of modelocking we refer to as spatially extended agile modelocking. The principal issue is whether the extensive and complex set of agile controls that are required can be incorporated in a densely packed microlaser environment. The demonstration of the desired physical behavior in macroscopic laser oscillators, the demonstration of a micro-optical control line that provides a number of key capabilities, the marked reductions in the number of quanta required for laser, or laser-like action, numerical simulations, and other recent relevant advances support this conclusion. We list below the set of relevant factors.

This possibility of an optical analog of microwave phased array radar occurs because of a number of recent advances: (1) reduced threshold for laser oscillator action in vertical cavity surface emitting lasers; (2) achievement of high packing densities for vertical cavity surface emitfing lasers; (3) demonstration of a micro-optical delay line that is susceptible of incorporation in vertical cavity microresonators; (4) apparent capability of the micro-optical delay line to control not only group delay, but also phase delay and group velocity dispersion; (5) demonstration (in macroscopic laser oscillators) of pulse synchronization in physically distinct laser oscillators by a shared complex nonlinear optical susceptibility; (6) demonstration (also in macroscopic laser oscillators) of phase locking of the optical fields of physically distinct laser oscillators (also by a shared complex nonlinear optical susceptibility); (7) simulations that provide a detailed description of the nonlinear interaction of short pulses propagating on coupled optical guides; and (8) evidence that the magnitude of the corrections by the micro-optical delay line can dominate the expected disruptive quasi-forces experienced by short optical pulses on coupled optical guides.

We recognize that the tasks that must be addressed are formidable and will require substantial time. The combination of the advances, and the guidance of the numerical simulations are, however, encouraging.

 We thank Joseph Haus, Dennis Deppe, and P.P. Banedee for valuable discussions. We are also indebted to Joseph Haus and Gary Shaulov for an adaptation of code written by Ben Luce that was used for the simulation shown in Fig. 3. We also thank Clifford Pate for progranu-ning. This work was supported in part by AFOSR Grant F49620-96-1-0206 and AFOSR Grant F49620-93-1-0410

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3. See, e.g., R.L. Fork, "Physics of optical switching", Phys. Rev. A26, 2049-2064 (1982). For reported threshold currents and typical microresonator photon lifetimes the mean number of quanta in the resonator for the lowest reported thresholds for conventional laser action is order of a few hundred.

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