The influence of the Kramers-Kronig phase is demonstrated in a coherently combined fiber laser where other passive phasing mechanisms such as wavelength tuning have been suppressed. A mathematical model is developed to predict the lasing supermode and is supported by experimental measurements of the gain, phase, and power. The results show that the difference in Kramers-Kronig phase arising from a difference in gain between the two arms partially compensates for an externally applied phase error.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Passive beam combining remains an enticing alternative to active phasing in fiber laser systems. With no external control loop, passive resonators are lightweight and low-complexity, but special efforts must be made to obtain a stable output. A collection of current methods in passive and active beam combining was presented in . For passive resonators based on wavelength tuning, several theoretical treatments have placed an upper limit on the number of fiber cores that can be combined [2–4]. Generally, the branches of a resonator tend to find a common wavelength that cancels their path length difference. As more cores are added, however, the likelihood of finding such a wavelength diminishes, making coherent laser operation unlikely or inefficient beyond roughly ten cores. Experimentally, results have varied widely, and it is uncertain whether the limits of passive coherent beam combining (CBC) have been fully explored. Performance seems to depend on the laser architecture, and an optimal cavity design has not yet become evident.
As demonstrated in a few recent examples, a large number of gain elements () can be combined efficiently in some cases. Up to 20 fibers were phase-locked in a resonator based on phase-contrast imaging, where the gain-dependent phase may have contributed to successful self-phasing . An array of 35 semiconductor lasers was combined in a self-Fourier resonator that utilized regenerative phasing effects due to individual feedback at each laser element . However, a straightforward implementation of passive CBC with a Michelson-type resonator does not appear to provide high combining efficiency , even with a variety of coupling geometries being tested . The most successful passive resonators appear to rely on one or more additional phasing effects besides wavelength tuning, and have a dense coupling architecture where each gain element is coupled to many of the others.
Besides wavelength tuning, a few specific physical effects influence the phase of the beams: thermal effects, the nonlinear Kerr effect, regenerative (Fabry-Perot) phase shifts, and the Kramers-Kronig phase. Thermal effects, including material expansion and temperature-influenced changes in refractive index, produce significant changes to the optical path length (OPL). In a several-meter fiber, the thermal phase noise can be thousands of waves . However, thermal effects are strongly influenced by external conditions like ambient temperature, so they are not well-suited to drive self-phasing. The optical Kerr effect sets in at high power and may affect the phase appreciably in a single-mode CBC system. It was predicted to have a beneficial effect on self-phasing in a system that also had phase effects due to regenerative feedback, where each core was reflective at both facets . Regenerative effects tend to bias the system toward specific phase values, which can increase the combining efficiency within a range of mirror reflectivity values . Finally, the gain-dependent or Kramers-Kronig (K-K) phase shift describes the link between the gain or loss of a medium and its OPL. Measurements of this effect in Yb-doped fiber show phase shifts of π radians with as little as 14 mW of pump power . Although each of these effects are individually well understood, their co-interaction is difficult to engineer. The simultaneous effect of these various physical phenomena has been the subject of numerous theoretical studies (e.g [13, 14].), but it remains difficult to decipher which effects are beneficial to self-phasing due to the complex and interrelated nature of the physics.
In this paper, we consider a two-core fiber laser that is designed to isolate the K-K effect and suppress competing phase mechanisms. In earlier reports, it was observed that the net phase difference between the two beams stays locked to particular values when an externally applied phase difference is adjusted from 0 to 2π . This showed that the resonator mode was adjusting itself to compensate the applied phase difference. Since the only available phasing mechanism was the K-K phase, it was determined that the gain levels of the two cores must be changing to produce the self-phasing effect. This was confirmed in a later experiment where the gain and phase in the active laser were measured and shown to be consistent with the effects due solely to K-K .
Although our previous experiments established that the resonator performed self-phasing through the K-K effect, the dynamics were not fully described. The purpose of the current paper is to reconfirm these results and compare the behavior to a theoretical model. In the following sections, the coupled resonator is written mathematically as a 2×2 matrix that acts on the field in each core. The eigenvectors of this matrix express the supermodes of the resonator. The output power, gain levels, and relative phase of the circulating beams can therefore be deduced as a function of phase difference between the arms, which we adjust externally. A numerical approach to solving the resultant equations is discussed, and finally, new experimental data is presented.
2. Experimental setup
The passive resonator is shown schematically in Fig. 1. A custom-made twin-core double clad polarization-maintaining ytterbium-doped phosphosilicate glass fiber (length: 2.81 meters) is pumped by focusing the light from a laser diode at a wavelength of 975 nm into the inner cladding of the fiber. The left end of the fiber is perpendicularly cleaved and serves as one end of the resonator (4% reflectivity), while the right end of the fiber is angle-polished at 8° to eliminate back reflections. The beams coming out of the cores at the angled facet are collimated by a lens and directed to a common location on a binary phase grating (or “Dammann grating” ), which performs beam combination. A standard diffraction grating, set at the Littrow angle of the desired laser wavelength (1050 nm), serves as the second mirror of the resonator. In the reverse propagation direction, the Dammann grating splits the beam for delivery to both cores. A pinhole spatial filter is included in the cavity to block unwanted orders from the Dammann grating, and a calcite polarizer is used to stabilize the mode polarization.
The resonator is designed to suppress self-phasing effects other than K-K. Thermal phasing effects are eliminated by the fiber design, in which the cores are placed near one another so that they are exposed to the same temperature profile. However, the cores are far enough apart to avoid evanescent coupling (20 µm core spacing; 6 µm mode-field diameter). Kerr effects are avoided by operating at low power (~20 mW CW; nonlinear threshold ~100 W). Phase shifts due to Fabry-Perot feedback are prevented by polishing the rear facet at an 8° angle, producing an estimated reflectivity of −38.7 dB, and eliminating back-reflections so that laser feedback is provided only by the Littrow grating. Although the reflectivity provided by this end of the cavity is quite small (approximately −15.8 dB), the ratio of the internal to external cavity reflectivities is low enough that the regenerative phase contribution should be negligible . Finally, wavelength tuning is restricted by using the Littrow grating to establish a narrow laser bandwidth and carefully coiling the fiber to match the OPLs of the two fiber cores, as described in . The Littrow grating is tilted out of the page in Fig. 1. The wavelength selectivity is set by the grating period (1200 lines/mm), collimating lens (Lens 1: f = 15.3 mm), and the diameter of the fiber modes, yielding a resolution of 0.7 nm. The cold-cavity OPL difference between the fiber cores was measured to be 60 µm, producing a maximum phase difference of 0.23 radians due to wavelength tuning. However, better wavelength stability was observed experimentally (Δλ = 0.1 nm, corresponding to 0.03 rad of phase difference). This is due to mode-matching loss at the fiber cores as a function of wavelength. If the wavelength detunes from Littrow by a small amount, the diffraction angle changes, and this reduces the coupling efficiency back into the fiber cores. This loss shapes the bandwidth into a peak centered at the Littrow wavelength, and therefore slightly off-peak wavelengths are discouraged from lasing.
An external laser is used to probe the fiber medium during laser operation. This allows the gain and relative phase of the actively running fiber cores to be measured while excluding any outside effects that might influence the laser mode. The probe beam is delivered into the cavity by a beam splitter so it co-propagates with the laser mode. The wavelength of the probe (1052 nm) is set near that of the laser mode to ensure that the phasing effects are similar. Its power is kept very low (P/Psat ~0.05) to avoid saturating the gain medium. In the detection arm, the probe and laser mode are spectrally separated, and can thus be measured individually. While viewing just the probe wavelength, the overall phase difference between cores is measured by recording their interference pattern. The position of the fringes is proportional to the phase difference, so the relative phase of the fields E1 and E2 in Fig. 1 can be tracked in response to changes in the cavity. The gain is measured by comparing the transmitted probe power when the laser is oscillating to the probe power when the pump is turned off. A cutback measurement was performed to account for passive absorption of the core mode (–4.9 dB/m). The power of the laser supermode is measured by imaging the cores at the laser wavelength.
Finally, translational movement of the Dammann grating provides an accurate means of adjusting the phase difference between the two laser arms. Since the right-side fiber facet is in the focal plane of a collimating lens, the light fields at the facet and the Dammann grating are related by a spatial Fourier transform. According to the Fourier shift theorem, a spatial shift in the Dammann grating produces a linear phase shift at the fiber facet, and hence translation of the Dammann grating modifies the difference in phase between the two cores. This is analogous to adjusting the arm lengths in a Michelson interferometer.
In a practical system, the phase offset of each beam will be unknown due to manufacturing variations and other conditions such as bending, vibration, and temperature. Complete cancellation of phase errors will lead to a constant output power with respect to phase error. This is possible in principle with active phase adjustment. However, for the passively-adjusted system studied here, some amount of power variation must be tolerated since the phase adjustment mechanism is tied to the laser gain.
A simple thought experiment shows how the power, gain, and relative phase are interrelated. By changing the applied phase, the power on the right-hand side of the cavity changes due to interference of the fields mixed by the Dammann grating. This affects the power coupled back into the cores, which causes the gain values to change according to saturation. Finally, a change in the gain difference leads to a new K-K phase error. This leads to a change in power, and so on, until an equilibrium is reached. Whereas in a single-core resonator, the modal gain and steady-state output power are determined purely by gain saturation, the multi-core resonator is affected by both the gain and phase in each arm. To achieve steady-state lasing, the interrelated gain and phase values adjust themselves to achieve a self-consistent supermode with net loss equal to the net gain.
3. Mathematical model
In this section, we establish mathematical expressions to predict the magnitude and relative phase of the fields in each core. We write the net effect of the resonator in matrix form, producing eigen-equations that express the cavity supermode. Additional equations, representing the physical effects of the Dammann grating, K-K phase, and gain saturation complete the description of the system. Ultimately, a system of nonlinear equations must be solved to obtain the supermode solution. Numerical analysis is described in the next section. Here we will consider the physical effects of each component in the resonator and derive the corresponding mathematics.
3.1 Dammann grating
The Dammann grating consists of a phase-only 50% duty cycle square wave with period T and a controllable translation x0. The phase steps ideally have a depth of π radians to minimize loss to unused grating orders. If the incident field is a plane wave, as is nearly the case for the field traveling leftward from the Littrow grating toward the fiber, then the field at the focal plane of a lens is the spatial Fourier transform of the grating transmittance function. The Fourier coefficients bm of the grating transmittance are Eq. (1) directly expresses the phase applied to each core. Hence, one core receives the phase −∆φ and the other receives +∆φ. Although Eq. (1) implies that the m = 0 order has zero magnitude, this is only the case when the phase depth of the grating is π. The zero order was present to some extent in the current experiment because a π phase depth was not perfectly achieved. It can be ignored without loss of accuracy because it was blocked by the spatial filter to the right of the Dammann grating, and sent into the fiber cladding on the left.
The operation of the Dammann grating is illustrated in Fig. 2. The initial beam traveling right to left in Fig. 2(a) consists of a single collimated Gaussian beam. Upon passing through the Dammann grating, it is split into m=0, ±1, ±2, … orders. The m=±1 orders enter the respective fiber cores, acquiring the phase terms ∓Δφ. In the other direction, beams are combined by the Dammann grating as shown in Fig. 2(b). The beam from each core is split into multiple orders, forming two overlapping sets. The set from the top core is shown with solid shading, and from the bottom, hatched shading. The total power at each discrete angle is the coherent sum of the overlapping orders. The on-axis order consists of the +1 order from the top core and the −1 order from the bottom core, producing an additional phase shift of −∆φ, +∆φ to the fields, respectively. The other orders are blocked by the spatial filter in the rear of the resonator.
3.2 Round-trip matrix
Next we write the effects of the gain fiber and the Dammann grating in matrix form. Our goal is to write a matrix that describes the effect of one round-trip propagation through the cavity. The matrix operates on a column vector E=[E1 E2]T, where E1 and E2 are complex scalars that represent the magnitude and phase of the fields at the left end of the fiber. The matrix for a complete round-trip MRT is written as a sequence of matrices, rM1McM2M1, where M1 represents single-pass propagation in the two cores, M2 represents propagation out of the fiber cores and back, with two passes through the Dammann grating, Mc represents the coupling efficiencies going from free-space back into the fiber, and r is a constant representing scalar field losses.
To write the matrix M1, we consider the effects on the field of traveling through the fiber cores. The gain and phase shift acquired during travel are written together in complex form, as gi exp(jφKK,i), where gi is the field magnitude gain in a single pass, φKK,i is the gain-dependent phase in a single pass, and the subscript denotes the ith core. The propagation for both fields is thus expressed by the matrix
We now write the matrix M2 to describe light propagation in the right-hand side of the cavity. The field exits the cores, propagates through the Dammann grating, passes through the spatial filter, and then reflects off the Littrow grating and passes through the Dammann grating again before returning to the fiber. The effect of propagating through the Dammann grating in each direction is described by the Fourier coefficients in Eq. (1). Hence, by accounting for the grating orders involved and noting that the spatial filter blocks all but the on-axis order, Eq. (1) can be used to determine the effect of a double pass. In matrix form, this is writtenEq. (3) is included in the Appendix (Sect. 8.1).
Losses due to mode-matching and alignment produce imbalanced coupling of the field back into the two cores. The imbalance in coupling efficiency has a strong influence on the self-phasing action, since unequal coupling produces a difference in gain, and thus a difference in K-K phase. Denoting the field coupling efficiency as ai (where 0 ≤ ai ≤ 1), the matrix
Assuming that propagation in the fiber cores from right to left is the same as left to right, the round-trip matrix can be written out completely by multiplying the component matrices. The result isEq. (7), therefore,
3.3 Kramers-Kronig phase shift
The K-K induced phase shift, which appears in Eq. (9) through φT, is related to the gain of the cavity. The K-K effect is responsible for several phenomena in lasers, including frequency pulling close to laser line centers and linewidth broadening in semiconductor lasers. At a given wavelength, it signifies a fixed relationship between the gain or loss of a medium and its refractive index. Modifications to the real and imaginary parts of the refractive index (Δn′ and Δn″, respectively) are related by Δn′=α Δn″, where the constant of proportionality α is called the linewidth enhancement factor or Henry’s alpha parameter .
The change in the real part of the refractive index produces the additional phase delay dφKK=kΔn′ dz during a short propagation step dz, where k is the wavenumber in vacuum. Integration of the delay over the length of the fiber yieldsEqs. (10) and (11) yieldsEq. (2), is given by ΔφKK=−(α/2)ln(G2/G1). From this the total phase can be written
3.4 Gain saturation
The final equation in the system describes how the gain levels are related to the field power. Gain saturation limits the laser power, and in this system, it determines how much the gain levels self-adjust to passively change the K-K phase. A simplistic model of an ideal four-level homogeneously broadened gain medium  yields the saturation expressionEq. (14) is only valid over a short length, however. For long media, the power of the amplified beam varies significantly with distance, producing a gradient in the gain along the propagation axis. To account for this, we use the Rigrod model  which assumes Eq. (14) is valid at each infinitesimal step dz along the length of the fiber, and the sum of the forward and backward propagating beam powers determines the saturation. Further, both ends of the medium are partially reflective, with reflectivity values RL and RR corresponding to left and right. Directly from  (there, Eq. (11)), the power on the right side of the resonator can be written as
An important boundary condition in Rigrod’s analysis is that the gain equals loss at steady state, or RR,iRLGi2=1. This expression is typically seen as the lasing criterion for a single gain element, but it is also valid for each core in the beam-combining system regardless of the interference between them. To see why this is, consider one of the cores as being seeded from the right by power from the combined beam. In this core, the seed is amplified and the gain becomes saturated along the length of the medium, ultimately providing a gain factor Gi in each direction. When the amplified beam exits the right side of the core, it mixes with the other beam in an arbitrary way, and some power returns to seed the core again. However, the returning seed must have the same power as the original, or else the gain will saturate to a different level. Because this seed power must be the same at the beginning and end of a round trip, the steady-state condition must hold.
The effective reflectivity RR,i may change arbitrarily due to the interference between cores. It can be eliminated by substitution of the boundary condition into Eq. (15), yielding
Equation (16) expresses the power on the right-hand side of the fiber, but the power in our system is measured on the left. However, by noting that the power on the right is the power incident on the left after reflecting off of the left-hand mirror and propagating once through the fiber, we may write PR,i=PL,iRLGi. For convenience we will also rewrite the left-side power as PL,i=Pnet|Ei|2, where Pnet is the sum of the power in both cores incident on the left facet. Note that the field vector E is already normalized to unit power (|E1|2+|E2|2=1) according to Eq. (8). As a result of these manipulations, Eq. (16) can be rewritten as
4. Numerical method
We have now established the equations that are needed to find the mode of the two-core resonator. The system is comprised of Eqs. (8), (9), (13), and (17), which has six equations and six unknowns, G1, G2, E1, E2, φT, and Pnet. To solve the system, we use a straightforward but somewhat ad hoc method: the system is first reduced algebraically to two equations and two unknowns, and one of the equations is solved numerically. The solutions are then substituted into the second equation, and those that satisfy it are solutions of the entire system. This method has the advantage that it does not need to rely on optimization to find solutions. Therefore, all solutions are found, regardless of local minima. The method is also easily generalized to an arbitrary number of cores.
The algebraic manipulation that reduces the system from six equations to two eliminates all but the variables G1 and G2. The manipulation is performed rather simply based on the eigenvector in Eq. (8). The norm-square ratio of the eigenvector components (E1 and E2) provides the following relationship to the gain:
This signifies that the powers PL,2 and PL,1 differ only by the effect of single-pass propagation from just after the Dammann grating to the opposite end of the fiber. This is true because the Dammann grating splits the beam power evenly between the ±1 orders, and these beams couple into the cores with efficiency ai2. The beams then propagate through the fiber, picking up a gain factor Gi. Thus, the total change in power is Giai2, and Eq. (18) expresses the ratio of this quantity between cores. Finally, using Eq. (17) yields
This equation directly relates the gain values G1 and G2 to one another, with their balance influenced only by the constants G0, a1, and a2. Remarkably, it implies that one gain value can be found from the other without directly considering the beam-combining dynamics. This is because the effective reflectivity RR, which contains the interference effects, was treated as unknown and eliminated by expressing it in terms of the gain. It can also be seen from Eq. (19) that a1 and a2 must be unequal if there is to be any difference between G1 and G2. Since a K-K phase difference only exists if the gain values are different, the imbalance between core powers is essential to the self-phasing effect.
Numerical solutions to Eq. (19) can be obtained simply by sampling the entire G1G2 space. This approach is possible because the space is bounded in both dimensions. Specifically, 1≤Gi≤ G0. Solutions are found from the sampled space by evaluating both sides of Eq. (19) and looking for intersections. We assume that our sample density is high enough that no solutions are missed. The correspondence between G1 and G2 is one-to-one, and the intersections are stored in a lookup table relating G1 and G2. The table is then passed to the second equation in the system for testing.
The second equation is simply the eigenvalue criterion, Eq. (9). It is related to G1 and G2 via the total phase φT given in Eq. (13). The values in the lookup table are plugged in and used to find crossings of |λ|=1. The bisection method is used to pinpoint each solution after it has been found between points in the lookup table. These solutions satisfy the entire system and they represent the supermode of the resonator.
Multiple solutions may exist at a given applied phase value, meaning that the system is bistable. Optical bistability occurs in resonant cavities where the gain and/or path length depends on the circulating field . A hallmark of bistable systems is hysteresis, whereby the mode will “latch” to a previous lasing state and remain there stably in response to perturbations. There are also unstable solutions, which are distinguished from stable solutions in this work by numerical perturbation tests.
The hysteresis effect can be useful in optical switching applications , but it is mainly an obstacle in the context of power scaling. To maximize power extraction, the system should be made to operate in the high-power state when more than one state is possible. In practice, this could be achieved by seeding the cavity with a high-power source. The additional complication due to hysteresis is not treated in-depth in this paper. For the experiments in the next section, bistability was kept to a minimum by keeping the pump power low. However, multiple solutions can be seen at some values of Δφ.
To fully evaluate the numerical model, we measure experimentally the six unknowns from the previous section (G1, G2, E1, E2, φT, and Pnet) as a function of applied phase. Comparison to theory confirms that K-K is the dominant phasing mechanism and that the calculations account entirely for the behavior of the resonator.
Experimental data were taken in a series of runs, where the Dammann grating was shifted gradually from a fixed start position to a fixed final position. Within each run, images were captured for each Dammann position, showing either the core modes or a fringe pattern in the detection arm (see Fig. 1). The laser output was stable and was replicated consistently between runs. Different a1/a2 ratios could be implemented using the method described in the Appendix (Sect. 8.2). The coupling efficiency ratio was a1/a2 = 0.6 except where noted otherwise.
The relative phase of the fields was measured by observing the fringe pattern of the probe beam on the left-hand side of the resonator. The results are shown in Fig. 3. In absence of any reaction by the resonator, the total phase would equal the applied phase, as shown by the green diagonal line φT=2Δφ. When the a1/a2 ratio is strongly imbalanced (Fig. 3a), the measured phase differs from the applied phase, indicating a strong K-K effect. When a1 and a2 are nearly equal (Fig. 3b), this effect disappears.
For the imbalanced case, the phase difference φT tends to “jump over” odd multiples of π/2, which is a beneficial effect due to K-K. Constructive interference for the combined beam occurs at even multiples of π/2 (marked by solid blue lines), while destructive interference occurs at odd multiples of π/2 (marked by dashed red lines). When the phase mismatch between the arms cancels the phase difference due to K-K, then the beams interfere constructively and the power of the combined beam is maximized. When the phase mismatch is not properly compensated by the K-K phase, then the beams interfere destructively and the combined beam power is low. Hence, the interference conditions can be seen by writing the power of the combined beam as a function of total phase difference. Such an expression is derived in the Appendix (Sect. 8.3).
Associated with the changes in total phase are changes in the power of the laser mode. The combined beam power is shown in Fig. 4a, and the core powers on the left end of the fiber are shown in Fig. 4b. The combined beam occurs on the right-hand side of the cavity at the on-axis order of the Dammann grating. It was measured by sampling the laser mode at the unused port of the intracavity beam splitter. The figure shows that the beam power varies significantly versus applied phase. While this is not ideal for a beam-combining system, it is consistent with a small but beneficial K-K effect, as described below.
To connect the variation in laser power with passive phase adjustment, we refer again to Fig. 3a. The “tread” part of each step on the staircase-shaped curve (the region between the vertical phase jumps) has a rounded shape, with a nonzero slope on the left and a flatter portion on the right. On the left side of each tread, φT tracks the applied phase (plus an offset due to a1≠a2), meaning that the K-K phase difference (ΔφKK) does not adjust itself significantly near the constructive interference states. However, on the right side of each tread region, ΔφKK increasingly counteracts the phase error as the applied phase moves φT toward an odd multiple of π/2.
This is explained by considering how a marginal added phase will affect the gain in each case. Starting at a peak of constructive interference on the left side of a tread region, if a small phase error is introduced, the lasing power will drop a small amount, and the gain will increase in response. This change in gain will be small, however, since the medium is saturated by the high power of the laser mode. Correspondingly, there is little change in ΔφKK. However, at unfavorable phase values, a small shift away from this condition will increase the power via interference. This changes the gain by a relatively large amount, due to the lower saturation from the low power of the mode. The change in ΔφKK will then be relatively large. By this mechanism, the slope of each tread gradually decreases as increasing phase error is applied. This can be viewed as the K-K effect being increasingly stretched to compensate the phase error near the destructive states.
This behavior is supported by measurements of the gain. Figure 5 shows the gain measured by the probe beam during laser operation. In regions of constructive interference (indicated by the locations of the peaks in Fig. 2a, for example), the gain is low due to saturation at high power. The Core 1 and Core 2 curves track one another, indicating that ΔφKK is relatively constant with applied phase. This produces the sloped behavior on the left side of the tread regions in the staircase plot: since the ΔφKK is practically constant, any change to φT is due to the applied phase. In destructive interference regions, the power is low and the gain is relatively high. The distance between the curves decreases as the destructive solutions are approached, indicating a strong pushback by ΔφKK. This coincides with the diminishing slope on the right-hand side of the treads in the staircase.
The theory curves in Figs. 3–5 were calculated without use of fitting parameters. Measured quantities, including the saturation power (14.9 mW), Henry’s alpha (7.3 rad/Neper), and the cold-cavity loss (−29.8 dB round-trip, absent fiber absorption) were used. The small-signal gain G0 was slightly different in the two cores (Core 1: 18.3 dB, Core 2: 18.8 dB) so the mean value was used. The coupling ratio a1/a2 was measured by inserting a pellicle beam splitter in front of the Dammann grating. The gain terms can be canceled after some algebraic manipulations, yielding the ratio a1/a2 from power measurements of the laser mode on both sides of the fiber. No modifications to the curves or data were necessary, aside from arbitrary shifts to the experimental data in Fig. 3 (since the zero-phase position of the fringes is unknown and there is a fixed phase offset between the cores from path-length mismatch), and arbitrary vertical scaling in the theory curve of Fig. 4a (since the cavity losses on either side of the beam splitter are unknown). A constant term was subtracted from the experimental data in Fig. 4a to account for power collected in unwanted back-reflections.
Occasional differences between data sets are noticeable. One such difference is seen in Figs. 4a and 4b near the jump located at 2Δφ=2.3π. In Fig. 4a, the power jumps up to the next peak as Δφ increases, whereas in Fig. 4b the power goes down into the tail of the curve. However, this is a spurious feature caused by the existence of multiple solutions (bistability) and realized by small inadvertent variations in the shift of the Dammann grating run-to-run. It is observed that the jumps are actually simultaneous.
An interesting consequence of the flattening shape of the stair treads in Fig. 3a is that the density of stable solutions increases near odd multiples of π/2. The density of solution states [24, 25] represents the likelihood that a passively phased system will be found operating at a given phase (φT), assuming a random offset (Δφ). It has often been presumed that the more densely-packed states are beneficial for self-phasing, but that is not the case for this system. Although the K-K phase pushes back against applied phase error near the destructively interfering states, the increasing manner in which this occurs results in an increase in the solution density just below the unfavorable odd multiples of φT = π/2. This does not imply that unfavorable states are necessarily higher-density in other systems, but it does suggest that high solution density does not always indicate high-efficiency phase states.
As a separate comment, the ramped side of the tread and its gradual flattening was not observed in our previous reports [15, 16], leaving the impression that K-K was able to perfectly compensate any applied phase difference. It appears that this behavior was due to the presence of off-axis orders from the Dammann grating, which would sustain the lasing mode when the interference loss became large. This behavior can be understood as “beam recycling,” where power from the loss port is fed back into the resonator to compensate interference, which reduces path-length sensitivity . The spurious (but apparently beneficial) behavior in the previous reports could be reproduced in the current system by removing the pinhole in the spatial filter system. This allowed the off-axis Dammann orders to bounce off of the Littrow grating, mix, and eventually couple back into the fiber. The more complex architecture was specifically eliminated in the present work to simplify comparisons with analysis.
A further simplification was achieved by introducing the losses a1 and a2 to achieve an imbalance in gain (and hence K-K phase). However, other methods may be used to imbalance the cores without introducing loss. For instance, if the cores are pumped at two different power levels, then each core will have a different small-signal gain. This will cause each core to follow a different saturation curve, resulting in a similar passive phasing effect even when a1=a2=1.
Finally, in the current setup, the K-K effect was able to partially compensate phase errors, but it came at a large cost to the power and gain. If power/gain could be exchanged more “cheaply” for phase, then the resonator performance could improve significantly. It can be shown that increasing the small signal gain alone does not enhance the K-K phase compensation. However, a greater value of α could significantly improve self-phasing. In the limit of large α, a small change in power produces a small change in gain, which produces a large change in phase. Assuming that a stable mode could be maintained, this could offer a very powerful phasing mechanism. Unfortunately, reported values of α are typically below 10 rad/Neper, so some careful engineering of the gain material may be required. Values for α have been widely reported in semiconductors  but not yet in fiber.
We have taken careful measurements of the laser mode in a beam combining resonator where the Kramers-Kronig effect is responsible for passive phase adjustment. A difference in gain is established between the two arms of the resonator by providing unequal losses at each of the gain elements. This produces a difference in K-K phase. When a phase error is then intentionally introduced, the gain values adjust to partially counteract the phase error. Although the cancellation is not complete, K-K does produce a beneficial effect that results in a redistribution of stable lasing states.
8.1 Coupling matrix M2
The matrix M2 accounts for propagation in the rear of the resonator. It describes the generation and mixing of the Dammann grating orders as the beams exit the fiber and come back. The entire process involves three steps: (1) traveling through the Dammann grating from left to right, (2) passing through the spatial filter, and (3) turning around and passing through the Dammann grating again. These steps can be represented as a sequence of matrices:
Using Eq. (1), a matrix for D can be constructed as:Fig. 2b.
The blocking matrix is easily written as
The product of the matrices yieldsEq. (3).
8.2 Mode-matching efficiencies a1 and a2
The mode matching efficiencies a1 and a2 were changed in the experiment by adjusting the alignment of the Dammann grating and Littrow grating. For “perfect” overlap of the laser mode with the fiber cores, the period T of the Dammann grating corresponds exactly to the spacing d of the fiber cores. This situation is shown in Fig. 6a. In this case, the reflected beams from the rear of the cavity (red shaded circles) overlap exactly with the fiber cores (dashed circles) at the facet. The fields are thus coupled back into the cores with a1=a2=1.
Due to the Fourier-transform action of the lens, the distance between the Dammann orders at the fiber facet is given by D=λf/T, where f is the focal length. When the period of the Dammann grating is perfect, D=d. However, the effective period of the Dammann grating can be easily adjusted by turning the grating for incidence at angle θ. By doing this, the effective grating period becomes Tcosθ, and the spacing between grating orders changes correspondingly. This is shown in Fig. 6b. There, the effective period has been reduced, so the spacing between orders becomes wider. The beams are still centered at the midpoint between the cores, so the mode overlap is equal. Thus, a1=a2<1.
If the rear mirror (Littrow grating) is tilted horizontally in this configuration, the reflected beams are no longer aligned symmetrically. They both move in the direction of the mirror tilt, as shown in Fig. 6c. In this case, the mode overlap is unequal, and the degree of imbalance can be adjusted via the mirror tilt. The figure shows near-perfect overlap at Core 2, but poor overlap at Core 1, producing a1<a2≈1. This was the method for achieving imbalanced mode matching efficiency in the experiment.
It should be mentioned that the pinhole spatial filter must be sized properly to simultaneously block the unwanted Dammann orders in left-to-right propagation, while transmitting the misaligned combined beam as it returns from the Littrow grating. Proper pinhole sizing was fairly easy to achieve with this fiber, since the core separation is much larger than the mode field diameter.
8.3 Constructive/destructive phase conditions
The conditions for constructive and destructive interference are found by writing an expression for the power of the combined beam. A 3×3 matrix is used, as in Sect. 8.1, because the combined beam occupies a different angle than the fields from the cores. The field at the three pertinent angles after leaving the right side of the Dammann grating is given by uR′ = DM1′ψ′ , where ψ is the normalized cavity eigenvector on the left side of the fiber from Eq. (8), the prime again denotes an expansion to a 3×3 matrix size, and D is the Dammann grating matrix given in Eq. (21). Pnet is a scalar that allows the power in each core (PL,1 and PL,2) to be written according to Eq. (17). Writing out the terms yieldsEq. (23) we obtain the power in the combined beam, Pon-axis, from the norm-square value of the middle (on-axis) entry of uR′:
Air Force Office of Scientific Research (AFOSR) (FA9550-14-1-0382 P00001).
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