A detailed analysis of mode-locking is presented in which the nonlinear mode-coupling behavior in a waveguide array, dual-core fiber, and/or fiber array is used to achieve stable and robust passive mode-locking. By using the discrete, nearest-neighbor spatial coupling of these nonlinear mode-coupling devices, low-intensity light can be transferred to the neighboring waveguides and ejected (attenuated) from the laser cavity. In contrast, higher intensity light is self-focused in the launch waveguide and remains largely unaffected. This nonlinear effect, which is a discrete Kerr lens effect, leads to the temporal intensity discrimination required in the laser cavity for mode-locking. Numerical studies of this pulse shaping mechanism show that using current waveguide arrays, fiber-arrays, or dual-core fibers in conjunction with standard optical fiber technology, stable and robust mode-locked soliton-like pulses are produced.
©2005 Optical Society of America
Mode-locked lasers are one of the most successful commercial devices which are based fundamentally on nonlinear optics. Commercially available pulsed mode-locked lasers are compact, cheap, reliable and robust and have a wide range of applications from optical communications to medical devices. The mode-locking in these packaged lasers are commonly based upon nonlinear polarization rotation, saturable absorbers, nonlinear interferometry, or acouto-optic modulators. In this paper, we present a new mode-locking technique which is based upon the nonlinear mode-coupling (NLMC) produced by waveguide arrays, dual-core fibers, or fiber arrays. From a theoretical and simulation standpoint, the resulting mode-locking is highly robust and self-starting, making it a viable candidate for the ever-maturing technology of mode-locked lasers.
Physically, the successful operation of a mode-locked laser [1, 2] is achieved by utilizing an intensity discrimination perturbation in the laser cavity in conjunction with bandwidth limited gain . In its simplest form, intensity discrimination preferentially attenuates weaker intensity portions of individual pulses or electromagnetic energy. This attenuation is compensated by the gain medium (e.g. Erbium-doped fiber) which acts to preserve the total cavity energy. Thus pulse shaping occurs since the peak of a pulse, for instance, experiences a higher net gain per round trip than its lower intensity wings. This time-domain narrowing (compression) of a propagating pulse is limited, however, by the bandwidth of the gain medium (typically ≈ 20– 40 nm [1, 2]). This mode-locking mechanism, which is only a small perturbation to the governing laser cavity equations, has been successfully achieved in a wide variety of experimental configurations including: a fiber ring laser with a linear polarizer [3, 4, 5, 6], the figure-eight laser with nonlinear interferometry [7, 8, 9, 10], a linear-cavity configuration with a semi-conductor saturable absorber [11, 12, 13], and a laser cavity with an acousto-optic modulator [14, 15].
Mode-locking using NLMC has been proposed previously using a waveguide array [16, 17], long-period fiber grating [18, 19], or dual-core fiber . In the current manuscript, three key concepts and extensions are explored. First, the more subtle aspects of the waveguide array dynamics are explored. Specifically, the temporal evolution of electromagnetic energy in neighboring waveguides is considered along with the energy transfered and lost to these waveguides. Further, the evolution of electromagnetic energy for large cavity energy is considered and shown to produce unstable mode-locking. Both of these issues are ignored in previous work [16, 17]. Second, for the dual-core fiber based mode-locking, a new model is proposed which is not based upon coupled nonlinear Schrödinger equations . Rather, the discrete components of the fiber propagation and dual-core propagation are separated. This model is necessary to achieve the intensity discrimination required for mode-locking to occur. Further, it shows that dual-core mode-locking can only occur if a high n 2 value is achieved in the dual-core fiber segment, something not addressed in previous work . Third, we propose and demonstrate for the first time mode-locking using a fiber array. Although it is a natural extension of the NLMC concept, it provides the first theoretical proof of concept of the feasibility of using a fiber array for generating stable mode-locked pulses. Thus the NLMC concept for mode-locking is not new. However, its practical implementation, operational requirements, and engineering feasibility are developed fully here for three different mode-locked laser systems.
In this paper, NLMC is proposed as the mechanism responsible for generating the intensity discrimination required for mode-locking. The intensity discrimination property of NLMC has been studied extensively in the context of a directional coupler [21, 22, 23, 24, 25] since the late 1980s. Here, the aim is to use this property for temporal pulse shaping. Figure 1 illustrates two possible fiber laser cavity configurations based upon a NLMC element. The underlying concept is as follows: a resonant and linear mode-coupling interaction transfers energy periodically between two or more propagating modes in neighboring waveguides. Nonlinearity, however, can be used to detune the resonant interaction by shifting the propagation constant of each mode via the self-phase and cross-phase modulation. Thus the NLMC can be arranged so that a resonant linear interaction occurs between the primary propagation mode and a second neighboring propagation mode. Then the low-intensity parts of a pulse which propagate through the nonlinear mode-coupling device are efficiently coupled to the neighboring mode and attenuated. In contrast, for sufficiently high intensities the peak of the pulse is detuned from the resonant coupling and is transmitted more efficiently in the primary mode. Thus, the wings of a pulse, or more generally the noise floor or low intensity dispersive radiation, are attenuated slightly more than its peak. This gives the necessary pulse shaping required to form soliton pulses by bringing into balance the chromatic dispersion and self-phase modulation in the cavity, thus achieving stable mode-locked operation after many round trips of the fiber laser cavity .
The NLMC concept has been applied previously to mode-locking with a long-period fiber grating (LPFG) [18, 19] and a dual-core fiber configuration . In each of these laser cavities, the application of the theoretical principles to the optical engineering has met with difficulties: in the long-period fiber grating because of the need of a high n 2 LPFG (which has not yet been manufactured), and in the dual-core fiber because of the lack of fidelity of the resonant linear coupling between core modes over the length of the laser cavity considered . However, with the emergence of new technologies, i.e. waveguide arrays, fiber arrays and chacogenide glass, the technical issues which prevented mode-locking via NLMC can be circumvented. Indeed, using these current technologies holds great promise for such commercially important applications as mode-locking.
To be more precise about the NLMC technology available, we consider in Fig. 2 three possible NLMC configurations in the laser cavities of Fig. 1. For instance, the spatial self-focusing and NLMC properties of a waveguide array have been well established in theory and experiment [26, 27, 28, 29, 30]. Waveguide arrays have been studied extensively in the context of all-optical signal processing (routing and switching) in fiber optic networks and devices [26, 27, 28, 29, 30]. A discrete form of Kerr lens mode-locking [31, 32] occurs since high intensity light injected into a waveguide array self-focuses while low intensity light is spatially coupled to neighboring waveguides through discrete diffraction . Since only light in the launch waveguide is retained, intensity discrimination is achieved in the laser cavity. Thus, high intensity light launched into the waveguide array passes through virtually unattenuated whereas low intensity light is diffracted to the neighboring waveguides and lost, i.e. intensity discrimination is achieved. Thus the discrete waveguide coupling, when conjoined to the optical fiber cavity (see Fig. 1), gives the necessary intensity-discrimination mechanism allowing for mode-locking . In a similar fashion, the dual-core fiber in Fig. 2(b) or a fiber array such as that depicted in Fig. 2(c) [33, 34] also discriminate low intensity light by resonantly coupling it to nearby waveguides for which the energy is discarded. Thus, this new NLCM technology in conjunction with mature fiber optics technology presents a promising candidate for mode-locked lasers.
The paper is outlined as follows: The governing optical fiber equations are presented in Sec. II. This is followed in Sec. III by an in-depth discussion of the mode-coupling equations for three possible NLMC components: a waveguide array, a dual-core fiber, and a fiber array structure. Section IV conjoins the optical fiber and NLMC element to explore the dynamics of mode-locking. Special attention is given to the robustness of the mode-locking under large perturbations in the cavity parameters. A brief summary and discussion of the results is given in the concluding Sec. V.
2. Laser Cavity Equations
The theoretical model for the dynamic evolution of electromagnetic energy in the laser cavity is composed of two components: the optical fiber and the NLMC element. The pulse propagation in a laser cavity is governed by the interaction of chromatic dispersion, self-phase modulation, linear attenuation, and bandwidth limited gain. For convenience, we consider an optical fiber laser such as depicted in Fig. 1. The propagation is given by
and Q represents the electric field envelope normalized by the peak field power |Q 0|2. Here the variable T represents the physical time in the rest frame of the pulse normalized by T 0/1.76 where T 0=200 fs is the typical full-width at half-maximum of the pulse. The variable Z is scaled on the dispersion length Z 0 = (2πc) /( D̄)(T 0/1.76)2 corresponding to an average cavity dispersion D̄ 12 ps/km-nm. This gives the one-soliton peak field power |Q 0|2 = λ 0 A eff/(4πn 2 Z 0). Further, n 2 = 2.6 × 10-16 cm2/W is the nonlinear coefficient in the fiber, A eff = 60 μm2 is the effective cross-sectional area, λ 0 = 155 μm is the free-space wavelength, c is the speed of light, and γ = ΓZ 0 (Γ = 0.2 dB/km) is the fiber loss. The bandwidth limited gain in the fiber is incorporated through the dimensionless parameters g and τ = (1/Ω2) (1.76/T 0)2. For a gain bandwidth which can vary from Δλ=20–40 nm, Ω = (2πc/)Δλ so that τ≈ 0.08 – 0.32. The parameter τ controls the spectral gain bandwidth of the mode-locking process, limiting the pulse width.
For illustrative purposes, a small number of simulations considered in Sec. IV will propagate according to Eq. Eq. (1) with a fixed gain, i.e. g(Z) = g 0 instead of the saturated gain Eq. (2). This is for convenience only. The mode-locking is highly robust and insensitive to changes in the gain model. Indeed, the NLMC acts as an ideal saturable absorber and even large perturbations in the cavity parameters (e.g. dispersion-management, attenuation, polarization rotation, higher-order dispersion, etc.) do not destabilize the mode-locking.
Finally, it should be noted that a solid-state configuration can also be used to construct the laser cavity. As with optical fibers, the solid state components of the laser can be engineered to control the various physical effects associated with Eq. (1). Given the robustness of the mode-locking observed, the theoretical and computational predictions considered here are expected to hold for the solid-state setup.
3. Nonlinear Mode-Coupling
In addition to the cavity propagation Eq. (1) and Eq. (2), theoretical models are required to describe the NLMC element. This NLMC section is divided into three subsections, each of which provides a quantitative model for a physically realizable NLMC element. Of importance in the three models is the ability of each model to provide quantitatively accurate results. Where possible, the models considered are based on experimentally measured parameters. The NLMC models are fundamentally the same, the only real difference being in the number of modes coupled together. In all cases, robust mode-locking is possible. It should be noted that the NLMC theory presented here is an idealization of the dynamics of the full Maxwell’s equations. For very short temporal pulses (i.e. tens of femtoseconds or less), modifications and corrections to the theory may be necessary.
3.1. Waveguide Array
where An represents the normalized electric field amplitude in the nth waveguide (n = -N, …,-1,0,1,…,N and there are 2N + 1 waveguides). The peak field power is again normalized by |Q 0|2 as in Eq. Eq. (1). Here, the variable ξ is scaled by the typical waveguide array length  of =6 mm. This gives C = and β = (γ * /γZ 0). To make connection with a physically realizable waveguide array , we take the linear coupling coefficient to be c = 0.82 mm-1 and the nonlinear self-phase modulation parameter to be γ* = 3.6 m-1W-1. Note that for the fiber parameters considered, the nonlinear fiber parameter is γ=2πn 2/(λ 0 A eff)=0.0017m-1W-1. These physical values give C = 4.92 and β = 15.1. The periodic waveguide spacing is fixed so that the nearest-neighbor linear coupling dominates the interaction between waveguides. Over the distances of propagation considered here (e.g. = 6 mm), chromatic dispersion and linear attenuation can be ignored in the waveguide array.
Equation (3) is simulated with 41 (N = 20) waveguides  for two different launch powers. Figure 3 demonstrates the well-known, self-focusing behavior as a function of increased launch power [26, 27, 28, 29, 30]. For this simulation, light was launched in the center waveguide with initial amplitude A 0(0) = 1 (top)and A 0(0) = 3 (bottom). Lower intensities are clearly diffracted via nearest-neighbor coupling whereas the higher intensities remain spatially localized due to self-focusing. The spatial self-focusing can be understood as a consequence of Eq. (3) being a second-order accurate, finite-difference discretization of the focusing nonlinear Schrödinger equation . This fundamental behavior has been extensively verified ex-periementally [27, 28, 29, 30]. Note that this model, and those that follow, allow for secondary coupling back into the central waveguide.
In a laser cavity configuration, the spatial self-focusing of the waveguide array which arises from NLMC can be used as the mode-locking element in the laser cavity. Specifically, we can consider the propagation of temporal pulses through the waveguide array Eq. (3). Figure 4 shows the evolution of a hyperbolic secant pulse through the waveguide array with different peak powers. The input and output of the temporal pulse in the center waveguide shows the pulse shaping which occurs due to NLMC. Note that the low intensity tails of the higher intensity pulses (b) and (c) are strongly attenuated due to coupling to neighboring waveguides. For low intensities (a), much of the pulse energy is coupled out to the neighboring waveguides. For this model, the temporal dispersion in the waveguide array is neglected because the typical dispersion length is approximately 1 meter while the waveguide array is only 6 millimeters. Thus the contribution from the dispersion is expected to be perturbatively small. This temporal pulse narrowing is the fundamental intensity discrimination element required for mode-locking.
3.2. Dual-Core Fiber
Yet another example of mode-locking induced by NLMC arises from dual-core optical fibers. In principle, the mode-locking mechanism is identical to that of the waveguide array, i.e. high intensity portions of a pulse are almost completely transmitted in one core of the fiber whereas low intensity portions are coupled to the second core and lost. The concept is to then engineer the core-to-core spacing and fiber length so that in the low intensity regime electromagnetic energy is almost completely coupled from one core to its neighbor. This coupling, as in the waveguide array, is an evanescent mode-coupling. Thus low intensity portions of a pulse will be effectively attenuated and the appropriate intensity discrimation is established. For longer fibers, energy from the second core begins to couple back to the original core mode. This should be avoided for the sake of this mode-locking application. Note that for the dual-core fiber based mode-locking, a new model is proposed which is not based upon coupled nonlinear Schrödinger equations . Rather, the discrete components of the fiber propagation and dual-core propagation are separated. This model is necessary to achieve the intensity discrimination required for mode-locking to occur.
The normalized equations which govern the nearest neighbor coupling of the waveguides are given by 
where An (n = 1,2) represents the electric field amplitude in the nth core. The peak field power is again normalized by |Q 0|2 as in Eq. Eq. (1). Here, the variable ξ is scaled by the assumed dual-core fiber length of = 10 cm. This gives C = and β = (γ * /γZ 0). To make connection with a physically realizable system, we take the linear coupling coefficient to be c = 0.010 mm-1. Note that this value is substantially smaller than that of the waveguide array since the dual cores are typically separated by 10–30 microns. In order to enhance the nonlinear self-phase modulation parameter, we assume the dual-core fiber is made of chalcogenide glass [36, 37, 38, 39, 40] which can have an n 2 value up to 3–4 orders of magnitude larger than standard glass. Thus we assume γ * = 200γ where γ is the nonlinear fiber coefficient given in the previous subsection. These physical values give C=1 and β = 23.72. Over the distances of propagation considered here (e.g. = 10 cm), chromatic dispersion and linear attenuation can be ignored in the dual-core fiber as with the waveguide array. Specifically, the dual-core fiber length is only ≈ 10% of the dispersion length.
A simulation of Eq. (4) serves to illustrate the dynamics of the dual-core coupling. Initially, electromagnetic radiation initially enters only into a single core A 1. The initial energy in A 1 will couple through evanescent wave interaction to the neighboring core mode A 2. Figure 5 demonstrates the effective spatial exchange of energy in the dual-core fiber for two different initial conditions. In the first (top), the initial field A 1 (0) = 1 so that the nonlinearity has little effect on the linear coupling dynamics. For higher initial intensities (bottom), i.e. A 1(0) = 3, the nonlinearity provides self-focusing which helps to confine the initial energy to the launch core-mode. Only a small amount of energy is coupled to the neighboring core-mode in this case.
As with the waveguide array, the temporal pulse shaping aspects of the dual-core coupling are what determine the suitability of the dual-core fiber for mode-locking. To illustrate the analog of Fig. 4 of the waveguide array, a temporal initial condition is launched into the dual-core fiber subject to Eq. (4). Figure 6 illustrates the pulse shaping which occurs for a series of initial hyperbolic secant pulses at different power levels. The light solid lines are the initial data while the heavy lines represent the output of the launch waveguide. Note that the low amplitude pulse is coupled out to the neighboring waveguides and lost, whereas the high amplitude pulse is subject to self-phase modulation and remains localized in the launch waveguide. Note further that the higher intensity pulse is narrowed in the time-domain as is expected of a mode-locking mechanism.
3.3. Fiber Arrays
The final example considered here for mode-locking induced by NLMC arises from fiber arrays. In principle, the mode-locking mechanism is identical to that of the waveguide array or dual-core fiber, i.e. high intensity portions of a pulse are almost completely transmitted in one core of the fiber whereas low intensity portions are coupled to neighboring core modes and lost. The concept is to then engineer the fiber array spacing and length so that in the linear regime electromagnetic energy is almost completely coupled from one core to its neighbor. This coupling, as in the waveguide array and dual-core fiber, is an evanescent mode-coupling. Thus low intensity portions of a pulse will be effectively attenuated and the appropriate intensity discrimation is established.
The normalized equations which govern the nearest neighbor coupling of the fiber array are given by 
where Am,n represents the electric field amplitude in the m and nth core (see Fig. 7). The peak field power is again normalized by |Q 0|2 as in Eq. Eq. (1). Here, the variable ξ is scaled by the assumed fiber array length of = 10 cm. This gives Cm,n = cm,n and β = (γ * /γZ 0). To make connection with a physically realizable system, we take the average linear coupling coefficients to be Ci,j = 0.046 mm-1  which correspondes to a core-to-core spacing of ≈12.5 microns. Further, just as with the dual-core fiber, the nonlinear self-phase modulation parameter is enhanced by using chalcogenide glass [36, 37, 38, 39, 40] . Again assuming γ * = 200γ gives the parameter values C = 4.6 and β = 23.72. Over the distances of propagation considered here (e.g. = 10 cm), chromatic dispersion and linear attenuation can be ignored in the dual-core fiber as with the waveguide array. Specifically, the fiber array length is only ≈ 10% of the dispersion length.
Simulations of Eq. (5) serve to illustrate the NLMC dynamics of the fiber array. Initially, electromagnetic radiation enters only into a single core A 0,0. The initial energy in A 0,0 will couple through evanescent wave interaction to the six neighboring core modes (A ±1,0,A 0,±1,A 1,1 and A -1, -1) illustrated in Fig. 7. Figure 8 demonstrates the effective spatial diffraction in the fiber array for two different initial conditions. In the first (Fig. 8(a)), the initial field A 0,0 (0) = 1 so that the nonlinearity has little effect on the linear coupling dynamics. For higher initial intensities (Fig. 8(b)), i.e. A 0,0(0) = 3, the nonlinearity provides self-focusing which helps to confine the initial energy to the launch core-mode. Only a small amount of energy is coupled to the neighboring core-modes over the 4 centimeters of propagation.
In a laser cavity configuration, the spatial self-focusing of the fiber array which arises from NLMC can be used as the mode-locking element in the laser cavity. Specifically, we can consider the propagation of temporal pulses through the waveguide array Eq. (3). Figure 9 shows the evolution of a hyperbolic secant pulse through the waveguide array with different peak powers. The input and output of the temporal pulse in the center waveguide shows the pulse shaping which occurs due to NLMC. Note that the low intensity tails of the higher intensity pulses (b) and (c) are strongly attenuated due to coupling to neighboring waveguides. These results for the intensity discrimination are qualitatively identical to those of the waveguide array (Fig. 4) and dual-core fiber (Fig. 6). As before, the contribution from the dispersion is expected to be perturbatively small.
4. Mode-Locking Dynamics
In the preceding section, the temporal intensity discrimination necessary for mode-locking was demonstrated with a waveguide array, dual-core fiber and fiber array. Thus each of these basic components has the potential to be a passive NLMC element in the laser cavities depicted in Fig. 1. In this section, we simulate the laser cavity and mode-locking dynamics associated with each of the NLMC elements of the last section. An in-depth study of the mode-locking dynamics will be given for the waveguide array. The dual-core fiber and fiber array lasers will only be given a cursory discussion as their mode-locking dynamics are qualitatively identical to the waveguide array mode-locking. Note that although the waveguide array and dual-core fiber mode-locking have been proposed elsewhere [16, 17], the practical feasibility of such mode-locking and the detailed dynamics are considered here in detail. Further, previous work only considered mode-locking with a constant gain g(Z) = g 0 whereas this manuscript considers the more physically realistic gain Eq. (2).
4.1. Waveguide Array
When placed within an optical fiber cavity (see Fig. 1), the pulse shaping associated with Figs. 4, 6, and 9 leads to robust and stable mode-locking behavior. The computational model considered in this subsection evolves Eq. Eq. (1) while periodically applying Eq. Eq. (3) every round trip of the laser cavity. The simulations assume a cavity length of 5 m and a gain bandwidth of 25 nm (τ≈0.1). The loss parameter is taken to be γ= 0.1 which accounts for losses due to the output coupler and fiber attenuation. To account for the significant butt-coupling losses between the waveguide array and the optical fiber, an additional loss is taken at the beginning and end of the waveguide array. Note that the waveguide parameters are exactly those used in Figs. 3 and 4.
The gain parameter g 0 is critical in determining the mode-locking. Specifically, for g 0 below a critical value, the cavity losses dominate and no mode-locking occurs. As g 0 is increased, stable mode-locking occurs with exactly one pulse per round trip. Increasing g 0 further leads to the formation of multiple pulses in the cavity, i.e. harmonic mode-locking. Two gain models will be considered here: a constant gain g(Z) = g 0 and the saturated gain model of Eq. Eq. (2). In either case, stable mode-locking is observed. This, in part, demonstrates the robust nature of the NLMC mode-locking.
Figure 10(a) demonstrates the stable mode-locked pulse formation over 40 round trips of the laser cavity starting from noisy initial conditions with a coupling loss in and out of the waveguide array of 20% and with a constant gain g(Z) = g 0. Due to the excellent intensity discrimination properties of the waveguide array, the mode-locked laser converges extremely rapidly to the steady-state mode-locked solution. It is typical that most theoretical mode-locking models take significantly longer to converge to the mode-locked solution, i.e. 100–1000 round trips. The gain level g 0 has been chosen so that only a single pulse per round trip is supported. Note that in Fig. 10, the initial condition is chosen for convenience only. Figure 10(b) demonstrates the qualitatively similar stable mode-locking which occurs for the saturable gain model Eq. Eq. (2). The mode-locking is robust and persists even with substantial changes in the cavity parameters. Further, the mode-locking is self-starting in an anomalous dispersion cavity since noisy initial CW radiation undergoes a modulational instability which generates local peaks in the electric field leading to soliton-like pulse formation as demonstrated in Fig. 10. The power levels required to achieve stable mode-locking are comparable to other laser cavities .
To further characterize the NLMC behavior, we consider the energy transfered to the neighboring waveguides once stable mode-locking operation has been achieved as depicted in Fig. 10(a). Figure 11 shows the temporal output of intensity for the launch waveguide A 0 and its two neighboring wave-guides (A ±2 and A ±1). The launch waveguide also depicts the initial input temporal pulse shape (dotted line). Note the clear transfer of energy from the low intensity tails of the pulse. Once mode-locked, Fig. 11 also predicts that 94% of the energy remains in the central waveguide A 0. The energy in each of A ±1 and A ±2 is 1.5% and 2.7% respectively. Thus only a ≈6% loss of energy is required to achieve the NLMC pulse shaping.
The gain parameter g 0, which is easily modified in an experiment, can alter the basic one pulse per round trip scenario. As g 0 is increased, the laser cavity is harmonically mode-locked with multiple pulses per round trip. A typical example of this dynamics is shown in Fig. 12. For this simulation, the gain parameter is increased to g(Z) = g 0 = 0.475. Additionally, a 40% coupling loss is assumed at the input and output of the waveguide array. Note the formation and persistence of multi-pulses per round trip. To accurately model the ensuing pulse-to-pulse interactions, more comprehensive details of the laser cavity are required. For instance, gain saturation and depletion dynamics can lead to equally spaced pulses . Such augmentations to the basic laser cavity model are not considered here. Further, to characterize the behavior of an over-driven system at high peak powers, modification to the governing equations need to be considered.
4.2. Dual-Core Fiber
We now consider the pulse shaping associated with the dual-core fiber in Fig. 6. As before, the computational model considered in this subsection evolves Eq. Eq. (1) while periodically applying Eq. Eq. (4) every round trip of the laser cavity. The simulations assume a cavity length of 5 m and a gain bandwidth of 25 nm (τ ≈0.1). The loss parameter is taken to be γ= 0.1 which accounts for losses due to the output coupler and fiber attenuation. To account for butt-coupling losses between the dual-core fiber and the optical fiber, an additional 20% loss is taken at the beginning and end of the dual-core fiber. Note that the dual-core fiber parameters are exactly those used in Figs. 5 and 6.
Figure 13(a) demonstrates the stable mode-locked pulse formation over 50 round trips of the laser cavity starting from noisy initial conditions and with g 0 = 0.39 in Eq. Eq. (2). The gain level g 0 has been chosen so that only a single pulse per round trip is supported. As with the waveguide array mode-locking, the initial condition is chosen for convenience only. Note the similarity between Fig. 13(a) and 10. The mode-locking is robust and persists even with substantial changes in the cavity parameters.
4.3. Fiber Arrays
Finally, we consider the pulse shaping associated with Fig. 9. As before, the computational model considered in this subsection evolves Eq. Eq. (1) while periodically applying Eq. Eq. (5) every round trip of the laser cavity. The simulations assume a cavity length of 5 m and a gain bandwidth of 25 nm (τ≈0.1). The loss parameter is taken to be γ= 0.1 which accounts for losses due to the output coupler and fiber attenuation. To account for butt-coupling losses between the dual-core fiber and the optical fiber, an additional 20% loss is taken at the beginning and end of the fiber array. Note that the fiber array parameters are exactly those used in Figs. 8 and 9.
Figure 13(b) demonstrates the stable mode-locked pulse formation over 50 round trips of the laser cavity starting from noisy initial conditions and with g 0 = 0.5 in Eq. Eq. (2). The gain level g 0 has been chosen so that only a single pulse per round trip is supported. As with the waveguide array mode-locking, the initial condition is chosen for convenience only. Note the similarity between Fig. 13(b), 13(a), and 10. The mode-locking is robust and persists even with substantial changes in the cavity parameters.
In conclusion, robust and stable mode-locking has been achieved by NLMC in a novel optical fiber laser cavity using a waveguide array, dual-core fiber, or fiber array as the mode-locking mechanism. The spatial self-focusing behavior which arises from the nonlinear mode-coupling of these mode-locking elements gives the ideal intensity discrimination (or saturable absorption) required for temporal pulse shaping and mode-locking. Although only a limited number of illustrative numerical experiments are presented here, extensive numerical simulations of the laser cavity with a waveguide shows a remarkably robust mode-locking behavior. Specifically, the cavity parameters can be altered significantly, the coupling losses can be increased, and the gain model altered, and yet the mode-locking persists for a sufficiently high value of g 0. This demonstrates, in theory, the promising technological implementation of this device in an experiment.
In practice, the technology and components are currently available  to construct a NLMC mode-locked laser based upon a waveguide array. An advantage of this technology is the short interaction region and robust intensity-discrimination (saturable absorption) provided by the NLMC. The only drawback of the waveguide array is the perhaps large coupling losses which can occur due to core size mismatch between the fiber and waveguide array, thus ruling out high-Q cavities. Simulations show, however, that even with large coupling losses (50%) stable mode-locking can be achieved. In practice, fiber tapering or free-space optics may be helpful to circumvent the losses incurred from coupling. For the fiber array or dual-core fiber, a large nonlinear index of refraction is necessary in order to keep the NLMC device short (e.g. 10 centimeters). Such large nonlinear index of refraction values may be achieved with emerging chalcogenide glass technologies [36, 37, 38, 39, 40]. Here the coupling losses with the fiber laser cavity are not expected to be as large as those experienced by the waveguide array due to comparable core radii. Finally, it is expected that the waveguide array and fiber array might be more effective NLMC devices in comparison with the dual-core fiber since the low-intensity energy continues to couple out and away from the launch waveguide whereas the dual-core fiber couples low-intensity energy back into the launch waveguide, thus defeating the purpose of the NLMC for mode-locking.In fact, Fig. 13(a) demonstrates the slower settling to the mode-locked state using the dual-core fiber in comparison to the waveguide array (Fig. 10) and fiber array (Fig. 13(b)).
In any of the three laser cavities proposed, index-matching materials, tapered couplers, polarization controllers and isolators may be useful and necessary to help stabilize the theoretically idealized dynamics presented in the model here. Thus a mode-locked laser cavity operating by NLMC is an excellent candidate for a compact, cheap, and reliable pulse source based upon the union of the emerging technology of waveguide arrays, dual-core fibers and fiber arrays with traditional fiber optical engineering.
We are especially indebted to D. Christodoulides and R. Morandotti for discussions concerning aspects of the waveguide arrays and their implementation. J. N. Kutz acknowledges support from the National Science Foundation (DMS-0092682).
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