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Abstract
Self-phasing due to spatial mode selection in a two-element passively coupled fiber laser is studied. We find that the addition of a second supermode in a coupled resonator results in a 90% increase in the average output power and nearly π/2 radians of passive phase adjustment versus applied phase errors between the gain elements. These results require a phase of zero (modulo 2π) between the beams in the external cavity. These findings are supported by an eigenmode analysis of the resonator and show that beam recycling is a useful resonator design feature but must be appropriately implemented to obtain beneficial results.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
There is a limit to the amount of power that can be extracted from a single laser oscillator. Pumping the gain medium at very high levels can induce thermal distortions of the beam and cause damage to the laser. The goal of a coherent beam combining (CBC) system is to overcome these limitations by summing the fields from several low-power gain elements into a single high-power combined output beam. Such a system can increase both the power and radiance emitted by a single laser aperture to levels higher than can be achieved using a single gain medium [1]. Examples of CBC systems based on coupled cavities include the Michelson resonator [2], Dammann grating resonator [3], Talbot cavity [4], and spatially filtered cavity [5]. Rare-earth doped fibers have been used in recent work on CBC because of their resilience to thermally-induced mode distortions and the availability of high gains [6].
CBC systems require accurate phase control of the gain elements. Phase control in an active CBC system is performed using an electronic control system [7–9]. In a passive CBC system, the phases are maintained by the laser array itself, resulting in a simpler system overall. Passive coupling has been successfully applied to arrays with as many as 20 gain fibers [10]. Many physical mechanisms influence the self-phasing of these systems, and because of their complex interaction, it is often unclear why these systems are successful and what influences their behavior. Several mechanisms have been studied in fiber CBC systems, including thermal effects [11], wavelength tuning [12–14], the Kerr nonlinearity [13,15–17], the Kramers-Kronig effect [18,19], and regenerative feedback [20,21].
Spatial mode selection is a self-phasing mechanism that occurs when gain clamping selects the coupled-resonator supermode with the lowest loss for oscillation. We use the term “spatial mode” to refer a transverse supermode of the entire beam combining resonator, not a transverse mode of a conventional laser resonator. This effect has been analyzed in isolation in a Michelson resonator, which found that adding a fourth mirror results in a system whose combined beam power and supermode phase state are less sensitive to phase errors between the gain elements [22]. The resulting generalized Michelson resonator has been described as implementing “beam recycling” since the fourth mirror causes the resonator to recycle energy that would otherwise be lost from the resonator. In other words, spatial mode selection was theoretically shown to be a beneficial effect in CBC systems. A subsequent experiment using a polarization-multiplexed laser validated the theory, but it did not represent a true beam combining system with separate array elements because there was only a single gain medium [23]. More recently, Monte Carlo simulations showed that the average output power from a beam recycling resonator scales with the number of gain elements [24]. However, the scaling occurs only for particular recycling architectures.
In this paper, we present an analysis and experimental demonstration of spatial mode selection in a Dammann grating resonator. We study this effect separate from other self-phasing mechanisms using a beam combining system with two closely spaced gain elements in a single fiber. The Dammann grating allows the phases between the gain elements to be precisely controlled. This provides a way of studying the effects of phase errors that result from thermal and mechanical perturbations, but in a highly controlled environment. In Section 2, we describe the behavior of the Dammann grating and show how it can be used to form a coupled resonator. In addition, we apply eigenmode analysis and a gain saturation model to both a standard and a recycling resonator to compute the supermode phase and combined beam power as a function of applied phase errors between the gain elements. In Section 3, we describe an experimental Dammann grating resonator containing a two-core ytterbium-doped fiber and discuss the various ways we reduce other self-phasing effects. Section 4 describes the results of this work and Section 5 concludes the paper by providing suggestions for future work.
2. Theory
In this section, we analyze the behavior of a Dammann grating resonator containing two gain media. Our goal is to find how the oscillating supermode phase and combined beam power vary with respect to applied phase errors between the gain elements. First, we analyze the Dammann grating and show how it can be used to form a passively coupled resonator. Then, we apply eigenmode analysis with a gain saturation model to both the standard resonator (which does not implement spatial mode selection) and the recycling resonator and compare their properties.
2.1 Dammann grating
A general Dammann grating has a complex periodic structure and couples light into $N$ diffraction orders of equal intensity [25]. Here, we consider the simplest type of Dammann grating ($N$=2), which consists of a 50% duty cycle square wave phase grating with a phase depth of $\pi$ radians, transmittance $t_{A}(x)$, and spatial period $d$ (or fundamental frequency $f_{0}=1/d$). We will show that this optic can couple energy from one beam into two and vice versa, thus performing coherent combination inside a laser resonator.
The angular plane wave spectrum (APWS) of the grating is [26]
To simplify our analysis, we make an approximation to obtain a form that contains only the first and third harmonics of the grating,
where the coefficients areIn the reverse situation, one can show that beams incident on the Dammann grating at angles $\theta =\pm \lambda f_{0}$ result in an output field with components that propagate at $\theta =0$ and $\theta =\pm 2 \lambda f_{0}$. In particular, both incident components contribute equal amplitudes to the $\theta =0$ component of the output field. The $\theta =0$ component represents the combined beam inside the laser resonator.
From this discussion, we see there are five propagation angles of interest in this resonator. The amplitudes and phases of these components can be collected in a 5x1 vector $\mathbf {U}=\left [ \begin {smallmatrix} U_{2} & U_{1} & U_{0} & U_{-1} & U_{-2} \end {smallmatrix} \right ]^{T}$, where $U_{m}$ is the amplitude and phase of the component propagating at the angle $\theta _{m}=m \lambda f_{0}$. In addition, the effect of the Dammann grating on these components can be described using a 5x5 matrix
2.2 Standard resonator
A conceptual diagram of the Dammann grating resonator is shown in Fig. 1. The field at the gain media end mirrors is denoted by $\mathbf {E}= \left [ \begin {smallmatrix} 0 & E_{A} & 0 & E_{B} & 0 \end {smallmatrix} \right ]^{T}$. We begin by forming a matrix that represents one round-trip of propagation through the resonator.
Fig. 1. Model of the Dammann grating resonator. $E_{A}$ and $E_{B}$ denote the field amplitudes at the gain media end mirrors. $\Delta \phi$ is the applied phase error between the gain elements obtained by shifting the Dammann grating. The combined beam is shown as the thick blue solid line, and the uncombined beams as red dashed lines. Recall $\theta _{1}=\lambda f_{0}$ and $\theta _{2}=2 \lambda f_{0}$.
First, the field $\mathbf {E}$ propagates through the gain media represented by the matrix $\mathbf {G}$, given by
After exiting the gain media, the field passes through the Dammann grating $\mathbf {D}$, given in Eq. (7). Next, the field propagates to the resonator end mirror which applies optical feedback of amplitude reflectivity $r$ to only the combined beam propagating along the optical axis. The end mirror matrix $\mathbf {R}$ is given by
The eigenvalues and eigenvectors of the round trip matrix, found from the equation $\mathbf {M_{RT}E}=\lambda \mathbf {E}$, describe the supermodes of the resonator. Since $\mathbf {M_{RT}}$ is rank 1 it has a single nonzero eigenvalue given by
with a corresponding eigenvectorIn addition, we define the relative phase of the supermode as
where $E_{A}$ and $E_{B}$ are the eigenvector components given in Eq. (12). In this case, $\phi _{Rel}= \Delta \phi$, which is simply the applied phase error. This suggests that the resonator cannot self-adjust in response to phase errors in the gain media.Next, we apply the Rigrod gain saturation model [28] to the system in order to find the combined beam power as a function of applied phase error. This method allows us to consider one of the two gain media and represent the rest of the resonator using an effective reflectivity. A gain medium with a single-pass small signal power gain $G_{0}$ and saturated power gain $G_{sat}$ is bounded by mirrors with reflectivities $R_{1}$ and $R_{2}$. The reflectivity $R_{1}$ represents the gain media end mirror and $R_{2}$ represents the effective reflectivity of the remainder of the beam combining resonator. $P_{1}$ and $P_{2}$ are the normalized powers incident on the mirrors $R_{1}$ and $R_{2}$. $P_{1}$ represents the power incident on the gain media end mirror and $P_{2}$ represents the power exiting the gain media into the free-space portion of the beam combining resonator.
First, we compute the saturated gain during oscillation by applying the steady-state condition $\left | \lambda \right |=1$ to Eq. (11). Solving for the single-pass saturated gain $G_{sat}$ gives
From this equation, we see that the saturated power gain depends on the applied phase error $\Delta \phi$. As $\Delta \phi$ increases from zero, the cavity loss increases. Since the gain and loss are equal during oscillation, the saturated gain also increases. When the loss becomes so great that the saturated power gain equals the small signal gain of the medium $G_{0}$, the above equation becomes invalid since oscillations are suppressed. In particular, for $\Delta \phi =n\pi /2$ where $n$ is an odd integer, $G_{sat}$ diverges to infinity, which is not physical.Next, using the gain $G_{sat}$, we can compute $P_{2}$ by using Eq. (11) from the Rigrod analysis [28],
2.3 Recycling resonator
In this section, we analyze the recycling resonator using the same methods developed in the previous section. Recall that the Dammann grating produces uncombined beams propagating at angles $\theta =\pm 2 \lambda f_{0}$ in addition to the on-axis combined beam. This resonator applies feedback to the uncombined beams to achieve beam recycling. In particular, part of the uncombined beam produced by the output from one gain medium is coupled back into the other gain medium, and vice versa. This behavior is shown in Fig. 2. Upon returning to the Dammann grating, the uncombined beams accrue a phase $\phi _{r}$ relative to the combined beam due to the slightly longer propagation length. We call this the recycle phase shift. $\phi _{r}$ can be adjusted by translating the end mirror along the optical axis. The end mirror matrix $\mathbf {R}$ is thus given by
Fig. 2. Illustration of beam recycling. The propagation angle of the uncombined beam from the upper gain medium (dotted line) is inverted after reflection (dashed line) from the end mirror. Similarly, the uncombined beam from the lower gain medium (not shown) couples back into the upper gain medium. The combined beam (not shown) exits the cavity along the optical axis.
The particular value of $\phi _{r}$ has a significant impact on the resonator behavior. One can show that for any $\phi _{r}$ that is not $0$ or $\pi$, two beams that exit the gain media with equal intensities will return with unequal intensities. This means that, during oscillation, the saturated gains of the two media will be unequal and the Kramers-Kronig (KK) effect will be induced in the system [18]. In this situation, spatial mode effects cannot be separated from the KK effect. A proper analysis must include the gain-dependent phase, which is beyond the scope of this work.
If $\phi _{r}=\pi$, one can show that the resonator supports two supermodes, but that their losses are equal for most values of $\Delta \phi$. This means that oscillation in a single-supermode due to gain clamping is difficult to achieve. The losses, shown in Fig. 3(a), are distinct only for a narrow range of phase errors around $\Delta \phi =0$ and $\Delta \phi =\pi$. Even though the losses are equal, the supermodes are distinguishable via their oscillation frequencies. We note that these results are very similar to published results for a spatially filtered cavity [29,30].
Fig. 3. Supermode round trip power loss $L$ versus applied phase error $\Delta \phi$ for the recycling resonator with (a) $\phi _{r}=\pi$ and (b) $\phi _{r}=0$. The end mirror reflectivity is $r=0.048$.
Finally, we show that $\phi _{r}=0$ results in beneficial self-phasing due to spatial mode selection. Substituting $\phi _{r} = 0$ into Eq. (19) above and computing the round trip matrix gives
The cold-cavity losses for these supermodes, defined as $L_{1,2}=1-\left | \lambda _{1,2} \right |^{2}$, are shown below in Fig. 3(b) using an experimentally measured end mirror reflectivity of $r=0.048$. This reflectivity represents the losses from the resonator optics that are used in our experiment, which we discuss in the next section. From the plot, we see that the losses are always less than unity. Since the losses are typically distinct from each other, gain clamping causes the resonator to select the lower-loss mode for oscillation. At $\Delta \phi =\pi /2$, the oscillating supermode switches from $\mathbf {E_{1}}$ to $\mathbf {E_{2}}$, and at $\Delta \phi =3\pi /2$, it switches from $\mathbf {E_{2}}$ to $\mathbf {E_{1}}$. As a result, the loss of the oscillating mode never exceeds roughly 0.984. This demonstrates an improvement over the standard resonator since the oscillating supermode never experiences unity loss (recall that the standard resonator has unity loss for $\Delta \phi =\pi /2$ and $\Delta \phi =3\pi /2$). In addition, Fig. 3(b) suggests that if $\Delta \phi$ varies randomly from 0 to $2\pi$, the oscillating supermode will also vary randomly. In this loss calculation, we have included the loss that is due to the 4% reflectivity that constitutes the end mirror on the left-hand side of the resonator in Fig. 4. The relatively high loss of this experimental resonator can be overcome by the high gain available from the ytterbium-doped two-core fiber, as discussed in the following section. We note that the high loss of the resonator is a consequence of our experimental setup and that in a practical system, the losses can be reduced significantly.
Fig. 4. Diagram of experimental setup. Components inside dotted boxes are for measurement purposes. The spatial filtering method used to adjust the resonator feedback is also shown.
The relative phase of the oscillating mode and the combined beam power versus applied phase errors are computed in the same manner as the standard resonator. These will be shown alongside experimental data in Section 4.
3. Experiment
In this section, we describe the experimental design of the resonator using a two-core ytterbium-doped gain fiber. In addition, we discuss how self-phasing effects other than spatial mode selection are minimized so that we can study spatial mode effects in isolation. We note that the spatial modes studied here are the transverse supermodes of the beam combining resonator and not the transverse modes of the fiber cores. This experimental setup is very similar to one implemented in prior work that studied Kramers-Kronig self phasing [18].
Figure 4 shows a diagram of the experimental setup. A 1-meter length of custom fabricated ytterbium-doped two-core fiber serves as the two gain media. The multimode inner cladding has a diameter of 150 micrometers and facilitates cladding pumping. An aspheric lens with 8mm focal length couples pump light into the inner cladding and collimates light emitted by the fiber cores. The pump source is a 975nm diode laser emitting roughly 0.7 watts that is incident on the fiber cladding. A dichroic mirror (DM) transmits the pump light while reflecting the fiber’s lasing wavelength to a measurement system to assess the laser supermode phase. The fiber cores have a diameter of 3.4 micrometers and a separation of 20 micrometers. This distance is large enough to avoid evanescent coupling between the cores. Two stress rods introduce high birefringence in the fiber cores, resulting in a polarization maintaining fiber. One end of the fiber is prepared with a 0$^{\circ }$ polish. This serves as the gain media end mirror with a 4% reflectivity arising from Fresnel reflections. The other end of the fiber is prepared using a 15$^{\circ }$ polish to suppress any back reflections from this facet. The fiber is coiled on a mandrel to minimize the optical path length (OPL) difference between the fiber cores [31]. Measurements with a probe laser show that the OPL difference is 57 micrometers.
An aspheric lens of focal length $f_{1}=11$ mm collimates light emitted by the 15$^{\circ }$ facet. This field passes through a Dammann grating. The effective period of the grating $d$ is related to the lens focal length by $d=f_{1}\lambda /x$ where $x=10$ micrometers is half of the distance between the fiber cores and $\lambda =1063$ nm, the lasing wavelength. To obtain the correct effective period, we use a grating with period 1.44 mm placed at a 39$^{\circ }$ angle relative to the fiber axis. Thus, $d=1.44\cos {(39^{\circ })}$ mm. The Dammann grating is mounted on a translation stage that allows phase errors to be applied between the fiber cores at a rate of 11.9 rad/mm determined from calibration. A polarizer (PBS) enforces lasing in a polarization state along one of the fiber’s birefringent axes. An afocal system with a spatial filter (ASF) allows for selection of standard feedback or recycling feedback. The width of the spatial filter (slit) is adjusted with a micrometer. The standard resonator is implemented by closing the slit to a size of roughly 400 micrometers which only transmits the combined beam (the zero order of the Dammann grating). The recycling resonator is implemented by opening the slit to a size of roughly 1.2 mm which also transmits the uncombined beams. The spatial filtering method is illustrated in the inset of Fig. 4. Finally, a blazed diffraction grating with 1200 lines/mm aligned at the Littrow angle (LG) serves as a perpendicular high reflectivity end mirror at the lasing wavelength. We use a tunable, fiber-coupled diode laser to measure various properties of this system including the OPL difference, small signal gain, and cold-cavity loss. A beam splitter (BS) together with the Dammann grating couples this probe laser into the fiber cores. The probe power is sufficiently low to avoid saturation of the gain media.
Measurement systems are shown inside dotted boxes in Fig. 4. The spectrally-resolved interference pattern of the supermode emitted by the 0$^{\circ }$ facet is imaged onto a camera CCD-1. The phase of the interference pattern is determined by fitting the data recorded by the camera to a cosine. A second measurement system on the other end of the resonator measures the combined beam power. The Littrow grating produces a zeroth diffraction order (specular reflection) of the incident light. A lens forms the angular plane wave spectrum of this distribution on a second camera CCD-2. The power is measured in the zero order, which is the combined beam.
Several aspects of this design serve to minimize self-phasing mechanisms other than spatial mode selection. The 20 micrometer separation between the fiber cores means the cores are in the same thermal environment. As a result, fields propagating in the two cores will not experience phase shifts with respect to each other due to thermal changes in refractive index. The Kerr nonlinearity is made negligible by operating the laser near threshold so that the circulating power does not exceed several milliwatts. Wavelength tuning is minimized using the combination of the small OPL difference in the fiber and the narrow lasing bandwidth enforced by the Littrow grating (LG). Wavelength tuning is less than 0.1nm in the experiment, which is equivalent to 0.03 rad of wavelength-induced phase shift. This value is negligible, effectively removing wavelength tuning as a self-phasing mechanism. The 15$^{\circ }$ facet’s computed reflectivity is nearly $-40$ dB, which is sufficiently small to avoid regenerative feedback effects [21]. Finally, the KK effect is minimized by achieving near-equal coupling of components into the fiber cores [18].
Several parameters were measured in order to apply the gain saturation model. First, the passive fiber absorption at the laser wavelength found from a cutback measurement was $4.155$ dB/m. The small signal gain at the operating pump power was $G_{0}=19.5$ dB. The cold-cavity loss was $-14.1$ dB, resulting in an end mirror reflectivity of $r=0.048$. Details of the techniques used to measure these quantities are discussed in Appendix A of [32].
4. Results and discussion
4.1 Standard resonator
The supermode relative phase, determined from the interference fringes of the laser mode, is shown in Fig. 5(a). The theoretical predictions from Eq. (14) are plotted alongside for comparison. Overall, the data shows a linear trend, agreeing with predictions. At $\Delta \phi =\pi /2$ and $\Delta \phi =3\pi /2$, the cold-cavity loss is unity, and near these points, a small jump of about $\pi /5$ rad is seen in the data that is not predicted by theory. This may be due to a residual KK effect between the gain media arising from alignment or mismatch between the fiber core sizes.
Fig. 5. Measured and theoretical (a) supermode relative phase $\phi _{Rel}$ and (b) combined beam power for the standard resonator.
The combined beam power is shown in Fig. 5(b). We subtract a constant power that approximately accounts for ASE from all data points and normalize the maximum value of the power to unity. The average power versus phase error is 0.42. Overall, these results agree with theory, and when considered along with the phase data, demonstrate undesirable supermode behavior. In this system, when the phase error is zero, a quarter-wave shift arising from technical noise can quench oscillation. When oscillation is resumed, Q-switching transients can appear that not only affect the output power but could also damage the system due to a dramatic increase in power. These are situations that must be avoided in any practical system. In addition, while the power was stable over time near its maximum values, it varied with time near the high-loss points, likely due to instabilities occurring near threshold.
4.2 Recycling resonator
The supermode relative phase data is shown in Fig. 6(a). The results show a distinct “staircase” shape because the oscillating supermode switches from $\mathbf {E_{1}}$ to $\mathbf {E_{2}}$ and vice versa, as discussed in Section 2.3. This indicates that the supermodes are less sensitive to phase errors compared to the standard resonator. In particular, the modes provide up to 1.41 radians of passive phase adjustment before the modes switch due to spatial mode selection. The data indicates that the resonator oscillates in one of two supermodes depending on the applied phase error.
Fig. 6. Measured and theoretical (a) supermode relative phase $\phi _{Rel}$ and (b) combined beam power for the recycling resonator.
The combined beam power data and theory curve are shown in Fig. 6(b). We subtract the ASE value mentioned previously from all data points and normalize the maximum power to unity. The maximum output power of this system was roughly equal to that of the standard resonator. It is interesting to note that the data shows slightly better performance than theory since the power drops to 0.5 compared to 0.4 near $\Delta \phi =\pi /2$. The power does not fall to zero because the supermode loss never reaches unity. Referring back to Fig. 3(b), when the loss of the first supermode becomes larger than the second supermode, the second supermode will lase. This trade-off between the supermodes prevents the output power from decreasing to zero. Since the power never decreases to zero and the fringe visibility is high throughout, single-supermode lasing occurs for all applied phase errors. The average power versus phase errors is 0.8, which is a 90% increase over the standard resonator. In addition, since the combining efficiency is proportional to the combined beam power, the efficiency of the recycling resonator is always nonzero while the efficiency of the standard resonator is zero near $\Delta \phi =\pi /2$ and $\Delta \phi =3\pi /2$. These results demonstrate the superior performance of the recycling resonator. A practical system implementing beam recycling that is subject to random phase errors from technical noise will be much less sensitive than a system without recycling.
Recall from Section 2.3 that beneficial spatial mode selection only occurs when the recycle phase shift $\phi _{r} = 0$. We conclude this section by showing experimental data for $\phi _{r}=1.93$ and $\phi _{r}=\pi$. In our experiment, we can change the recycle phase by increasing the length of the external cavity at a rate of 0.184 rad/cm found from calibration. A cavity length increase of 10.5 cm results in $\phi _{r} = 1.93$, and an increase of 17 cm results in $\phi _{r} = \pi$. The combined beam power at these two recycle phases $\phi _{r}$ is shown in Fig. 7(a). All data is normalized such that unity represents the maximum output power when $\phi _{r}=0$. In addition, the supermode fringe visibility $V=(I_{max}-I_{min}) / (I_{max}+I_{min})$ is shown in Fig. 7(b). From the data, we see that increasing $\phi _{r}$ causes a decrease in the average output power versus applied phase errors. When $\phi _{r}=1.93$, the average power is 0.25, and when $\phi _{r}=\pi$, the average is 0.5. In addition, the visibility for both cases is a function of the applied phase error. This shows that it is possible for the coherence of the laser to decrease, and the data for $\phi _{r}=\pi$ indicates the laser never oscillates in a single supermode. This was predicted by theory (Fig. 3a), which concluded that for $\phi _{r}=\pi$, there is very little discrimination between supermodes. In contrast, when $\phi _{r}=0$, the theory concluded that the supermodes can be separated at all but two values of $\Delta \phi$. Thus, we see both theoretically and experimentally that a necessary condition for proper beam recycling is to maintain the recycle phase $\phi _{r}$ near zero. We note that this is very easy to do in our experiment, as this relative phase error is extremely tolerant to mechanical errors.
Fig. 7. Measured (a) combined beam power and (b) supermode fringe visibility for recycle phase shifts of $\phi _{r} = 1.93$ and $\phi _{r} = \pi$. Data for $\phi _{r} = 0$ is also shown for reference.
5. Conclusion
In this paper, we have studied spatial mode effects in a two-element passively coupled fiber laser. Experimental data is supported by the results from an eigenmode analysis of the Dammann grating resonator. We have demonstrated beneficial self-phasing in a recycling resonator with matched phases in the recycle arms and contrasted it with (1) a resonator with no recycling feedback and (2) a resonator with mismatched phases in the recycle arms. The results show that spatial mode selection can result in significantly improved performance in passive CBC systems.
Future work on this subject could move in several directions. First, experiments can be performed to determine the power scaling potential of beam recycling resonators when three or more gain elements are combined. In addition, modeling and experiments can be performed to determine the behavior of resonators under the influence of other self-phasing effects such as the Kerr nonlinearity and regenerative feedback.
Funding
Air Force Office of Scientific Research (FA9550-14-1-0382 P00001).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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