1. Introduction

Fiber lasers in general, and ultrashort pulse fiber lasers in particular, have demonstrated a remarkable increase in average power performance over the past decade [1, 2]. This is due to the fiber geometry, since a large surface area to volume ratio facilitates rapid heat dissipation and consequently allows for scalability to high average powers. But the tradeoff with the fiber geometry is that the optical signal is tightly confined to a relatively small transverse area over relatively long lengths. This sets limits on achieving high pulse energies in fibers because of saturation-fluence, optical damage, and nonlinear effects such as stimulated Raman scattering (SRS), stimulated Brillouin Scattering (SBS), four-wave-mixing (FWM), or self-phase modulation (SPM). Limitations on pulse energy are particularly severe for chirped-pulse amplification (CPA) of ultrashort pulses in fibers [3], where recompressed-pulse distortions caused by SPM occur at relatively low pulse energies, in the ~mJ range [4].

A general approach to overcoming single-laser energy and power limitations is to combine the outputs from an array of lasers [5]. Active coherent phasing appears best suited for combining large numbers of individual laser channels [6], and has been demonstrated with cw [7], pulsed [8], and ultrashort pulse [9, 10] fiber lasers, with combined powers ranging from hundreds of Watts to kW and with up to millijoule energies for long and short pulses. The two key technical challenges associated with active coherent combining are (i) how to spatially combine multiple output beams, and (ii) how to temporally combine multiple beams, i.e. track and correct phasing errors in each individual channel. Multiple beams can be tiled spatially [11], thus combining only in the far-field, or can be combined into a single beam using a binary-tree type of arrangement based on either interferometric 50:50 beam splitters/combiners [12] or polarization beam splitters (PBS) [13]. A single diffraction-limited beam can also be obtained from a coherently-phased spatially-tiled beam array using diffractive-optics [14] or multi-mode interference effects in hollow-waveguides [15].

There are different strategies that can be used to track phasing errors in each individual channel. Applicability of a particular phasing approach, however, does depend on the beam combining method used. For example, one strategy is based on the spatial recognition of each channel phase in a tiled array output by using a detector array and heterodyne phase detection with respect to a reference channel [12, 16], which requires spatial monitoring of a tiled-array output. Another scheme, Hänsch-Couillaud detection, measuring deviation from a linear polarization can be used to track relative phases between pairs of channels [9], but this strategy is only applicable to PBS combiners in a binary-tree type arrangement and requires a matching tree of detectors. Alternatively, each channel can be “tagged” by individual-frequency modulation which allows tracking relative phases of all the channels with a single detector [17, 18], the so-called LOCSET technique (Locking of Optical Coherence by Single-detector Electronic-frequency Tagging). This strategy appears to be the most general approach, applicable to all spatial beam combining methods, and therefore might be the best path for phasing a large number of channels.

In this paper we report on coherent combining of four parallel femtosecond pulse fiber amplifiers using the LOCSET phasing scheme and a binary-tree type of beam combining, and we explore the combining efficiency of such a system as a function of the number of parallel channels. Understanding the array-size scalability is crucial for the development of high power and high energy ultrashort pulse fiber laser arrays. Fundamentally, scalability of a combined-array size is determined by the effect of phase and amplitude noise on the combining efficiency. General statistical analysis indicates [19, 20] that if the phase-noise average is zero, then efficiency should converge to a fixed value at very large number of channels, but if this average is different from zero then the combining efficiency continuously degrades with increasing channel number [19]. Here we study the extrapolated performance of coherently combined system at a very large number of parallel channels phased using LOCSET, by first developing a theoretical model, and then validating its accuracy through comparison of its predictions with the experimentally characterized performance of our combining system. Finally, we use this “calibrated” model to predict the combining performance with increasing number of channels in the presence of temporal amplitude and phase variations in each of the parallel-channel signal paths. We show that at very large number of combined channels using the LOCSET phasing arrangement, combining efficiency converges to a fixed value, determined only by the magnitude of the phase and amplitude errors.

Section 2 of the paper describes the experimental system and the details of its operation. Measured performance of this coherently-combined fiber CPA array system is presented in Section 3. The theoretical model and its experimental validation by comparing it to the measured characteristics of a coherently combined array is given in Section 4. In Section 5 we analyze the coherent-combining efficiency of such an array with increasing number of channels and explore its dependence on the magnitude of phase and amplitude errors. Conclusions are given in Section 6.

2. Experiment

2.1 Fiber chirped pulse amplifier array

To explore the coherent phasing of multiple parallel fiber CPA channels, we built an experimental coherently combined system based on an all-fiber, four-channel amplifier array. All fibers and fiber-optic components in the system were polarization preserving—resulting in the measured polarization extinction ratio at the output of the system to be ~20.5dB. The system layout is shown in Fig. 1 . It consists of a mode-locked femtosecond pulse fiber oscillator, a pulse stretcher, a parallel fiber amplifier array, a beam combiner and a pulse compressor. It also contains the control electronics for coherent phasing of the parallel fiber amplifiers.

 

Fig. 1 Experimental setup for four channel monolithic fiber pulse combining.

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The femtosecond oscillator is an All Normal Dispersion (ANDi) femtosecond fiber oscillator [21] producing 7 nm bandwidth pulses at 1050.5 nm central wavelength and 47 MHz repetition rate with an average output power of 30 mW. Since this is a stretched-pulse oscillator, the generated pulses are positively chirped with a pulse duration of ~15 psec. After de-chirping, the duration of these pulses is reduced to ~300 fsec. The seed pulses from the oscillator are stretched in a standard Martinez-type diffraction-grating pulse stretcher to ~900 psec. The pulse stretcher is arranged in a folded configuration (to reduce its length) and contains a single 10-cm wide grating with 1800 lines/mm groove density. The stretched pulses are coupled into single-mode polarization maintaining (PM) fiber, and then split with 50:50 single-mode fiber splitters, shown in Fig. 1, into four separate channels. Parallel amplification channels were built using standard single-mode PM fiber components (based on PM980 fiber for passive components), and all four channels consisted of identical components with identical fiber lengths to ensure that each optical path is of equal length and with equal amount of linear and higher-order dispersion. To achieve accurate optical-path matching each channel includes a compact adjustable delay line, described in more detail further in the text. For correcting the phase drift between the channels, three of the channels include fiber piezo-stretcher (PZT) based phase modulators. The one channel without the modulator had an equivalent length of identical passive fiber spliced into its path, to match that of a PZT stretcher. Amplification in each of the channels was implemented using standard in-core pumped Yb-doped single-mode PM fibers (PM-YSF-HI from Nufern), WDM components for combining pump and signal paths and standard telecom-grade single-mode pump diodes. The total length of fiber in each individual channel was about 30 m.

Four parallel amplified signals at the output of the fiber array are beam combined using a binary-tree type of arrangement. In a series of experiments, we interchangeably used both bulk-component and all-fiber based beam combiners. For the bulk combiner, we used a PBS-tree arrangement to implement polarization combining [13]. The all-fiber beam combiner used a 50:50 single-mode PM fiber arrangement, which is essentially a “reverse” of the signal splitter at the array input. It is clear, however, that an all-fiber beam combiner is incompatible with high power combining or high pulse energies. The reason this monolithic arrangement was used was to achieve an accurate measurement of array-phasing performance. Indeed, in this all-fiber beam combiner, complete beam overlap between all combined signals is automatically ensured. Therefore, all the effects on the combining efficiency associated with a non-perfect spatial beam overlap [22] are eliminated, and the combining efficiency is determined solely by interferometric addition of errors between different channels, allowing very accurate measurement of the phasing effects.

A combined beam of stretched and amplified pulses was launched into a standard Treacy-type diffraction grating compressor. Just like the stretcher, the compressor was arranged in a folded configuration, and therefore, uses only a single diffraction grating, identical in specifications to the one used in the stretcher. Compressor throughput efficiency was approximately 65%. A small fraction of the output power is “sampled” by either a glass wedge placed prior to the compressor, or alternatively, by zero-order reflection from the compressor grating. The “sampled” beam is directed onto a single detector to provide a feedback signal for active phase locking. Without phase locking (i.e. free running operation / open control loop) the relative phases between channels drift, resulting in random output power fluctuations after the combiner. This is of course expected, since without phase control, time-varying random constructive or destructive interference occurs between the channels. Using a feedback loop set to maximize the output power forces the channels to interfere constructively. The feedback signal contains the phase error information for each of the three of the channels with PZT modulators (the phase error is with respect to the fourth, i.e. reference, channel). The phase error signals are individually extracted with three separate feedback electronic signal processing units and the appropriate error canceling signals sent to the phase controllers, using the so-called LOCSET scheme, described in more detail further in the text.

2.2 Equalization of parallel-channel optical paths

Coherent combining of ultrashort optical pulses requires not only robust phasing, but also accurate matching of the group delays between parallel channels, so that the combined pulses are exactly overlapped in time. Errors in timing cause both pulse distortions and a loss in combining efficiency. In practice, acceptable group-delay errors should be much smaller than the pulse duration, and for femtosecond pulses should be on the order of few micrometers. Achieving such fiber-length accuracy by simply cutting the fiber is not practical. Instead, an arrangement for adjustable fiber length control should be used. Implementing adjustable length control in an array consisting of a large number of parallel channels has to be compact and cost-effective to manufacture. With this practical constraint in mind, we demonstrate an adjustable and compact delay line built using standard single-mode fiber based micro-optical components. The schematic of this arrangement and its 3D rendering is shown in Fig. 2 . The adjustable delay line exploits the non-reciprocal nature of a fiber circulator. An input pulse is sent into port 1 of a fiber circulator, it then travels out of port 2 with a spliced-on fiber collimator, and in free space propagates over a variable length (i.e. the delay) before being retro-reflected back into port 2 with a micro-optic mirror. Adjustment of the double-pass delay in this arrangement has been achieved with a compact micro-optical linear translation stage. After coupling back through the collimator into the single-mode fiber of port 2 the delayed pulse passes the circulator the second time and due to the circulator non-reciprocity is directed out of port 3. This configuration was selected since it minimizes the number of free-space degrees of freedom need for signal back-coupling adjustment—only two angularadjustments of only the reflecting mirror are needed. The insertion loss of this delay line is approximately 6 dB.

 

Fig. 2 Schematic and 3D rendering of the micro-optic delay line.

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During combining experiments, we observed that it was very important to match dispersions of the parallel channels. A four-channel array needs only three adjustable delay lines to achieve complete equalization between all four channels. Therefore, in an early implementation of our four channel combining setup we did not include a fiber circulator delay line in one of the channels (reference channel four). But each circulator introduces significant dispersion, which led to the unbalanced dispersion in the reference channel, and subsequently, inefficient combining of this channel with the other three. This inefficiency was completely eliminated after an identical circulator-based delay line was incorporated in the reference channel to equalize the dispersions between all four channels.

2.3 Channel active-phasing control system

As stated earlier, the four-channel fiber CPA array setup was actively phased using a LOCSET approach. The optical phase in three of the channels was controlled using fiber piezo-stretchers. The fourth channel was left as a reference for the phase controlled channels to match. The piezo-stretcher gives a π phase shift per 2.6 V of drive voltage. The maximum voltage range of the device is ± 500 V and therefore supports up to 384π of continuous phase control. The feedback-control electronics was limited to an output of ± 5 V and therefore high-voltage amplifier chips were used to reach at least ± 100 V (76π) driving voltages. Using only the limited ± 5 V (3.8π) range would result in temporary, on average once every ~30 sec, instances of unlocked operation lasting less than ~200 msec before the system relocked to a modulo 2π phase. The extended phase control range with the 100 V high voltage amplifiers eliminated these temporary unlocks and resulted in robust locking that we tested from several minutes to one hour.

In the self-referenced LOCSET technique for active coherent beam combining, a single detector is used to supply a feedback signal to each of the feedback electronic units [18]. The principle of this technique is to modulate each phase controlled channel with a unique radio frequency (RF) value. One channel remains unmodulated. The modulated channels are then phase controlled to track the phase of the unmodulated channel. A single detector measures a photocurrent that includes a superposition of all the interference effects between the channels. Demodulation of this photocurrent by feedback electronic signal processing units is performed for each modulated channel. Since the channels have different RF modulations the demodulation can be tuned to isolate the error signal for just a particular channel. The result of the LOCSET scheme is an RF phase locked loop that stabilizes the phases of the modulated channels with respect to the unmodulated channel.

The choice of values for RF modulation depends on how quickly the phase of a channel fluctuates. To determine the appropriate RF values, we first monitored the interferometric power fluctuations from the channels without feedback control (i.e. in the free running state, open control loop). In this case, the interference varies randomly over time and gives an indication of the phase noise. Figure 3 shows measurements of the fluctuating power in both the time (a) and frequency (b) domains. The frequency domain data was obtained by taking a ten second interval of data at 1 kHz sampling rate, subtracting out the mean, and then performing a Fast Fourier Transform. The DET data shows the measured detector noise when there is no optical power on the detector.

 

Fig. 3 Power noise in the unlocked state for one, two, three, and four channels: (a) time domain, (b) frequency domain. The DET data indicates our detector noise floor.

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Figure 3(a) shows that without phase-locking parallel-channel outputs interfere randomly, producing random power drifts and fluctuations, corresponding to the sources of noise associated with acoustic vibrations and temperature drifts. Experiments with cw fiber laser combining also confirm that noise is dominated by frequencies below 100 Hz [23, 24]. We determined that RF modulation frequencies in the kilohertz range are sufficiently in the low-noise regime of our environment. In our multi-channel experiments, the modulation frequencies were set to 5 kHz, 6 kHz, and 7.5 kHz. The integration time of the feedback electronics was set to 50 msec. This allowed the phase control loop to cancel phase disturbances up to 20 Hz in frequency.

3. Measured performance of the coherently-combined fiber CPA array

3.1 Combined pulses

An important performance metric for a multi-channel ultrashort-pulse combining system is the combined-pulse quality, compared to individual-channel pulses. The measured spectra and autocorrelation traces of the combined and individual channel pulses are shown in Fig. 4 . In this figure both spectral and autocorrelation traces from individual channels and of the combined signal overlap with each other very accurately, clearly indicating the complete absence of spectral and temporal distortions due to the coherent phasing. For reference, a calculated bandwidth-limited pulse autocorrelation trace is shown as a dashed line in Fig. 4(b) (it was calculated by taking Fourier-transform of the spectra in Fig. 4(a)). Autocorrelation-trace wings of the measured pulses that are appearing relative to the “ideal” bandwidth-limited trace, indicate the presence of some residual third-order dispersion in the system, most likely due to imperfect alignment between the diffraction-grating stretcher and compressor. Deconvolving the measured autocorrelation traces of the four channel combined pulse with a calculated pulse shape factor gives a pulse duration of 524 fsec. This is slightly longer than the bandwidth-limited 410 fsec duration. A deconvolution factor of ~1.4 for relating the measured autocorrelation trace to the actual pulse duration was estimated by calculating the bandwidth-limited pulse and its autocorrelation from the measured pulse spectrum. We should remark that when the system is unlocked the autocorrelation trace of the combined signal fluctuates in magnitude but not in shape.

 

Fig. 4 Pulse quality results: (a) normalized spectrum of individual channels and all four channels combined, (b) normalized autocorrelation traces of individual channels and all four channels combined – the dashed line shows the calculated (from the spectral measurement) bandwidth limited autocorrelation of the combined pulse.

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3.2 Combining efficiency

Since the primary reason for using laser beam combining is power scaling, combining efficiency is one of the key performance metrics when characterizing coherently combined laser arrays. Our objective here was to perform very accurate efficiency measurements with the experimental 4-channel fiber CPA array and then to use these results to extrapolate performance for very large arrays. In this regard it is very useful to distinguish between “internal” effects on combining efficiency associated with a variety of laser-array signal errors such as phase errors, power fluctuations, incorrect balance between different-channel output powers, etc., and the “external” effect directly associated with the beam-combiner loss. The latter depends on a specific type, design and quality of a beam-combining element used in the system and does not depend on the performance of the fiber-laser array itself. Here we are primarily concerned with the “intrinsic” effects that characterize fundamental limitations onarray combining efficiency. In order to distinguish between these “internal” and “external” effects experimentally, we used the notion of absolute and relative combining efficiencies, η_abs and η_relative respectively, as well as the combiner efficiency η_combiner. Absolute combining efficiency is simply a ratio between the combined output power and the sum of all the individual powers from all the channels at the input of the combining element. Combiner efficiency characterizes cumulative power transmission/loss from all the channels in the combining element, and “internal” effects are characterized by the relative combining efficiency η_relative. All three different efficiencies are related through η_abs = η_combiner·η_relative.

While it is straightforward to measure absolute combining efficiency, the question arises of how to measure the relative efficiency, particularly in an all-integrated system. As it turns out a convenient and rigorous measurement procedure exists to determine the magnitude of η_relative accurately. Mathematical derivation of this procedure for a general case of N signals combined with a binary-tree beam type combiner is given in Appendix A. This procedure requires measuring the power only at the output of the system, and does not require doing any measurements before the beam combiner. The procedure consists of measuring the coherently-combined power P^comb when all the channels are seeded and pumped and phase-locking is turned-on, and then measuring each power P_k^out transmitted through the combiner from each k-th channel individually, when signals from all other channels m ≠□k are blocked (e.g. by turning-off the pump diodes and blocking signal paths in the corresponding delay lines for all channels m ≠ k). Then a relative combining efficiency of N coherently-phased channels is given by (Appendix A)

Figure 5(a) shows the relative combining efficiency for two, three, and four channels with monolithic 50:50 PM fiber coupler combiners over a time period of five minutes. The mean efficiency is 96.4%, 94.0%, and 93.9% for two, three, and four channel combining respectively. Even though different combinations of channels can be used for two and three channel combining, the results are within 1% of each other.

 

Fig. 5 Combining efficiency and power noise: (a) combining efficiency for two, three, and four channel locking over a five minute time period, (b) four channel locked and unlocked noise.

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Measured output powers for two, three, and four channel combining were 15.2 mW, 33.2 mW, and 58.6 mW respectively. We note that the output power from the combining system was kept safely below the 300 mW damage threshold of the 50:50 PM fiber couplers. We also note that each fiber coupler has an insertion loss of ˗1.2 dB, resulting in ˗1.2 dB combiner loss for the two-channel system and ˗2.4 dB loss in the three-channel and four-channel systems (two stages of 50:50 couplers). The combiner loss contributes to reducing the absolute combining efficiency. In separate experiments with free-space PBS based combiners instead of monolithic combiners, absolute efficiency in the four-channel system was 85%. Observing the electronic error signals on an oscilloscope indicated that the channel phase errors drifted over a range of less than 20π over time. Figure 5(b) compares the power noise of the locked four channel result with the previous, Fig. 3(b), unlocked four channel measurement. We see that at the frequencies measured, the locked result is within the measurement floor of our detector. Figure 6 shows that the four channel combining is robust over a longer period of time, even when we occasionally block the signal to the feedback detector. Without feedback the system drifts, but immediately recovers when the feedback signal is restored. These intentional feedback ‘blocks’ lasted for less than one second and show that the combining efficiency re-stabilizes due to electronic feedback system.

 

Fig. 6 Four channel combining efficiency with feedback blocks. The momentary drop-outs in the combined signal are due to intentional interruptions to demonstrate system robustness and its reaction to abrupt external interruptions in system operation.

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4. Experimental validation of the theoretical combining-efficiency model

In the LOCSET method of active phase control, a small phase modulation is purposely applied to each of the phase controlled channels. The electric field of a phase controlled channel varies as$\cos \left({\omega }_{L}t+\varphi +\beta \sin ({\omega }_{RF}t)\right)$, where ω_L is the laser frequency, ϕ is an initial phase, ω_RF is an RF modulation frequency, and β is the phase modulation amplitude. From this expression it is apparent that this phase modulation should affect combining efficiency similarly to the random variation of the signal phase $\varphi$. Consequently, this allows validating experimentally the theoretical prediction of the combining efficiency as a function of the phase-variation magnitude, by measuring combining efficiency as a function of the phase modulation amplitude β. Additionally, this analysis allows to determine the effect of the finite amplitude β in the LOCSET scheme on the overall combining efficiency at large channel numbers.

Analytically, the LOCSET scheme combining efficiency as a function of modulation amplitude β is (see Appendix B)

 

Fig. 7 Effect of phase modulation on the combining efficiency: (a) experimental and theoretical combining efficiency as a function of phase modulation amplitude, (b) theoretical combining efficiency as a function of number of channels for different values of phase modulation amplitude.

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For a fixed β value, the theoretical combining efficiency given by Eq. (2) is plotted in Fig. 7(b) as a function of the number of channels under the assumption that all individual channel powers are equal. Figure 7(b) shows the dependence on channel number for three different β values. What is noteworthy is that the efficiency converges to a stable value as the number of channels is increased. In fact, by taking the limit of Eq. (2) as N goes to ∞ we get ${\eta }_{relative}(\beta )=J_0{}^2(\beta )$. At β = 0.25 the theoretical efficiency converges to 96.9%. This result indicates that the effect of the LOCSET phase modulation on the overall combining efficiency can be negligibly small.

5. Coherent combining of large fiber arrays

Since the experimental validation given above shows a good agreement between the measured and calculated combining efficiency as a function of phase-modulation magnitude, one can use this to extrapolate this result to larger numbers of combined channels and to evaluate achievable combining efficiencies at large combined array sizes. For this it is necessary to generalize Eq. (2) to the case of unequal phases between different channels. The combining efficiency then is given by

To analyze Eq. (3) statistically, we make the assumption that the channel powers $P_i^{out}$ are independent Gaussian random variables with mean $P_{avg}^{out}$ and standard deviation σ_P, and that the phases ϕ_i are independent Gaussian random variables with zero mean and standard deviation σ_ϕ. For these random variables, the following expressions for expected value are useful in simplifying Eq. (3): $E\{\cos ({\varphi }_i-{\varphi }_j)\}=\exp (-{\sigma }_{\varphi }{}^2)$ and $E\left\{\sqrt{P_i^{out}{P}_j^{out}}\right\}\cong P_{avg}^{out}{[1-\frac{1}{8}{({\sigma }_P/P_{avg}^{out})}^2-\frac{3}{16}{({\sigma }_P/P_{avg}^{out})}^4]}^2$. The expected value of Eq. (3) is

With Eq. (4) we have a formula for combining efficiency in terms of the variables N, σ_P/$P_{avg}^{out}$, σ_ϕ, and β. For calculations, we assume that locking occurs at β = 0.25 (the value we currently are able to achieve stable locking). Figure 8 plots Eq. (4) versus number of channels for different power variations σ_P and different phase errors σ_ϕ. The important result is that in both cases the efficiency converges to a fixed value for very large number of channels, thus indicating that LOCSET based coherently combined systems should scale gracefully with array size. The efficiency value towards which the large array converges depends on the magnitude of the phase and amplitude noise amplitudes. Results in Fig. 8(b) indicate that for phase-errors smaller than ~λ/20 the combining efficiency should exceed 90%. Figure 8(a) indicates that combining efficiency is quite insensitive to amplitude noise in the channels. Indeed, even for large amplitude noise of ~20% the predicted combining efficiency exceeds 95%. Explanation of such convergence of combining efficiency at large channel numbers should be generally associated with the fact that combined power increases linearly with the number of channels, while the total noise increases only proportionally to the square-root of the number of channels, thus leading to decrease in single-to-noise ratio with increasing array size.

 

Fig. 8 Scalability of a multi-channel combining system with LOCSET locking at β = 0.25 and different magnitudes of errors: (a) only power variation errors, (b) only temporal phase errors.

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We also explored how this convergence-efficiency depends on the magnitude of the phase and amplitude errors assuming very large combined array sizes (i.e. when N → ∞). Results are plotted in Fig. 9 . Figure 9(a) indicates again that efficiency dependence on the amplitude noise is relatively weak, compared to the phase noise effect, which is plotted in Fig. 9(b). From the latter figure one can conclude that, remarkably, achieving combining efficiency of >90% requires relatively modest phasing accuracy of > λ/20. Note also that when combining efficiency degrades to zero for large phasing errors in Fig. 9(b), the array-output power simply is dissipated through all the intermediate binary-tree output ports, not the single combined-output port.

 

Fig. 9 Combining efficiency for very large arrays as a function of power and phase noise magnitude: (a) combining efficiency as a function of amplitude noise only, and (b) of phase noise only.

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It is also important to note that this scalability analysis is limited to zeroth-order phase errors in the time domain (recall that the electric field in Eq. (31) is written only with time as a variable). For our monolithically integrated combining system it is appropriate to neglect spatial errors since the waveguide geometry automatically overlaps Gaussian spatial modes of different channels. But high power systems must use free-space combining arrangements. For a filled-aperture architecture, the possibilities include using polarization beam splitters [13], a diffractive optical element [14], or a hollow-core waveguide [15]. In these cases, spatial errors, particularly wavefront errors, might have a dominant effect on reducing the combining efficiency [22].

6. Conclusion

We have demonstrated a multi-channel fiber femtosecond pulse combining system with 96.4%, 94.0%, and 93.9% relative combining efficiencies for two, three, and four channels respectively. The combined and compressed ~500 fs pulses have identical shape to the compressed pulses from individual channels, indicating that pulse quality is preserved in a multi-channel pulse combining system. Furthermore, we established convenient experimental and theoretical metrics for characterizing combined system performance due to phase and amplitude errors in the parallel-channel array. It is based on a notion of relative combining efficiency, which can be experimentally determined using a straightforward measurement procedure. Inherent advantage of this “figure of merit” is that it directly relates to the efficiency of the combined array, and can be easily calculated and measured. Although this metric was rigorously developed for a binary-tree type of combiner, it should be directly applicable to other types of beam combiners (e.g. holographic beam combiners).

Our analysis of combining efficiency dependence on amplitude and phase errors shows that LOCSET feedback based combining systems should scale gracefully to very large numbers of channels. Although our interest here was primarily associated with ultrashort-pulse combining, these conclusions are of general nature and could be equally well applied to cw and pulsed systems. Of course, a full description of ultrashort-pulse combining scalability should also include consideration of pulse-dispersion effects. However, our experiment indicates if all the channels are identical these dispersion effects can be cancelled out, at least in the low-nonlinearity case. For analysis quantifying specific short pulse-related effects on coherent combining efficiency see [25] for a theoretical approach and [26] for an experimental approach. Coherent combining of femtosecond pulses in multi-channel parallel fiber CPA systems offers a possible path towards simultaneously generating high energy and high average power ultrashort laser pulses.

Appendix A: Coherent-combining efficiency using binary-tree type combiners

First let’s consider combining efficiency with a single two-port combiner. Combining efficiency in general depends on

• 1) Relationship between combiner splitting ratio X and the ratio between input beam powers (or powers) X’
• 2) Combiner loss for each of the two inputs η₁ (for input P₁ⁱⁿ) and η₂ (for input P₂ⁱⁿ)
• 3) Phase difference Δφ between the two beams

If there is only a single input beam into the combiner then each input beam P₁ⁱⁿ and P₂ⁱⁿ will be partially transmitted and partially reflected, as shown in Fig. 10, i .e. beam combiner acts as a beam splitter. We can always choose one direction as a combiner output, for example as selected in the Fig. 10. Then $P_1^{out}$and $P_2^{out}$denotes beam splitter output powers when only either $P_1^{in}$or $P_2^{in}$input beam is present, respectively.

 

Fig. 10 Beam splitter: (a) when only beam P₁ is present, (b) when only beam P₂ is present.

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When both input beams $P_1^{in}$and $P_2^{in}$are present and their relative phases are fixed (i.e. they are phased with each other) the same device acts as an interferometric beam-combiner, as shown in Fig. 11 . Combined power· P^comb is found by direct calculation:

 

Fig. 11 Beam combiner.

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(5)$$\begin{array}{l}{P}^{comb}=c{\epsilon }_{0}nA\cdot 〈{\left(E_1^{out}\cdot \cos \left(\omega t+{\varphi }_1\right)+E_2^{out}\cdot \cos \left(\omega t+{\varphi }_2\right)\right)}^2〉\\ =P_1^{out}+P_2^{out}+2\sqrt{P_1^{out}\cdot P_2^{out}}\cdot \cos \Delta \varphi .\end{array}$$

Here brackets $〈\mathrm{...}〉$denotes time average, and output power is related to the corresponding field amplitude as$P_i^{out}=c{\epsilon }_{0}nA\cdot {\left(E_i^{out}\right)}^2/2$, where A is beam area. The maximum constructive interference occurs when both beams are in phase, i.e. $\Delta \varphi =\left({\varphi }_1-{\varphi }_2\right)=0$, producing the in-phase combined power:

(6)$$P^{comb\_in-phase}=P_1^{out}+P_2^{out}+2\sqrt{P_1^{out}\cdot P_2^{out}}={\left(\sqrt{P_1^{out}}+\sqrt{P_2^{out}}\right)}^2.$$

Let’s denote$P_1^{/}={\eta }_1\cdot P_1^{in}$,$P_2^{/}={\eta }_2\cdot P_2^{in}$, and $P_{tot}^{/}=P_1^{/}+P_2^{/}$. It can be shown that the maximum combining efficiency is achieved when

(7)$$P_1^{/}=X_1\cdot P_{tot}^{/}.$$

It is straightforward to show that in this case it is also$P_2^{/}=X_2\cdot P_{tot}^{/}=\left(1-X_1\right)\cdot P_{tot}^{/}$. It is more convenient to express this condition for achieving the maximum combining efficiency as

(8)$$\frac{P_1^{in}}{P_2^{in}}=\frac{X_1\cdot {\eta }_2}{X_2\cdot {\eta }_1}=\frac{X_1\cdot {\eta }_2}{\left(1-X_1\right)\cdot {\eta }_1}.$$

Since $P_1^{out}=X_1\cdot P_1^{/}$and $P_2^{out}=X_2\cdot P_2^{/}=\left(1-X_1\right)\cdot P_2^{/}$then substituting the condition for the maximum combining efficiency into Eq. (6) leads to the expression for the maximum achievable combined power:

(9)$$\begin{array}{l}{P}^{comb\_MAX}={\left(\sqrt{P_1^{out}}+\sqrt{P_2^{out}}\right)}^2\\ ={\left(X_1\cdot \sqrt{P_{tot}^{/}}+\left(1-X_1\right)\cdot \sqrt{P_{tot}^{/}}\right)}^2\\ =P_{tot}^{/}={\eta }_1\cdot P_1^{in}+{\eta }_2\cdot P_2^{in}.\end{array}$$

This combining efficiency is achieved when both inputs are perfectly in-phase with respect to each other, are in the correct ratio (described by Eq. (8)) with respect to each other, and the only factor reducing the combined efficiency is each-beam losses in the combiner.

Based on this analysis we can define absolute and relative combining efficiencies denoted by ${\eta }_{abs}$ and ${\eta }_{relative}$ respectively, as well as the combiner efficiency${\eta }_{combiner}$:

(10)$${\eta }_{abs}=\frac{P^{comb}}{P_1^{in}+P_2^{in}};{\eta }_{relative}=\frac{P^{comb}}{P^{comb\_MAX}};{\eta }_{combiner}=\frac{P^{comb\_MAX}}{P_1^{in}+P_2^{in}}=\frac{{\eta }_1{P}_1^{in}+{\eta }_2{P}_2^{in}}{P_1^{in}+P_2^{in}}.$$

Then

(11)$${\eta }_{abs}={\eta }_{combiner}\cdot {\eta }_{relative}.$$

Now let’s generalize this to the case of arbitrarily-large binary-tree combiner, which combines N inputs into one output. Figure 12 depicts a general binary-tree configuration consisting of m stages. The 1-st stage (output stage) consists of one combiner with 2¹ = 2 internal inputs, the 2-nd stage consists of two combiners and 2² = 4 internal inputs, and so on until the last m-th stage (actual input stage) which consists of m combiners with N = 2^m inputs, which are also the actual external inputs into this binary-tree combining arrangement. Total number of individual combining elements in this arrangement is

 

Fig. 12 Schematic of a general N→1 binary-tree beam combiner.

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(12)$$K=2^{m-1}+2^{m-2}+\mathrm{...}+2^0=2^m-1=N-1.$$

This figure also introduces a system for labeling all the internal-stage inputs $P_{k_i}^i$ into each combining element by the stage number i and the corresponding internal-input number k_i, here k_i = 1, 2, ..., 2ⁱ, as well as combining-element efficiencies ${\eta }_{k_i}^i$ and splitting ratios $X_{k_i}^i$for each corresponding input at each combining element in each binary-tree stage. The N = 2^m inputs $P_{k_m}^m$ into the m-th stage can also be denoted as external inputs $P_{k_m}^{\text{in}}$ into the complete binary-tree arrangement. Each internal or external input field is characterized by complex amplitude${\tilde{E}}_{k_i}^i=E_{k_i}^i\cdot e^{i{\varphi }_{ki}^i}$, and power is related to field amplitude through$P_{k_i}^i=c{\epsilon }_{0}nA\cdot {\left(E_{k_i}^i\right)}^2/2.$

Each combining element in the i-th stage of this binary tree combines two internal input signals $E_{k_i}^i$and $E_{k_i+1}^i$, and k_i in this pair should be an odd-integer number and k_i +1 should be an even-integer number. Based on that splitting ratio for each of the two input beams into a combining element is $X_{k_i}^i$if k_i is an odd-integer, and $X_{k_i+1}^i=1-X_{k_i}^i$ if k_i +1 is an even-integer.

Each individual binary-tree input $I_{k_m}^{\text{in}}{}^{/}$ travels an individual path from the tree input stage to the common combined output. This path can be identified by a particular sequence of combining-element inputs the signal passes, expressed as a sequence of the corresponding stage-label values

(13)$$k_m^{/}\Rightarrow k_m^{/},k_{m-1}^{/},\mathrm{...},k_i^{/},\mathrm{...},k_2^{/},k_1^{/}.$$

where symbol $\Rightarrow$ indicates that each individual path corresponds uniquely to an external input with a label $k_m^{/}$. Obviously, there are N different individual paths.

Using this notation we can express the output of this binary-tree combiner$P_{k_m}^{out}$ when only one input signal $P_{k_m}^{in}$ present at the input of the binary tree (i.e. no coherent combining) as

(14)$$P_{k_m}^{out}={\eta }_{k_m}^m\cdot X_{k_m}^m\cdot \mathrm{...}\cdot {\eta }_{k_i}^i\cdot X_{k_i}^i\cdot \mathrm{...}\cdot {\eta }_{k_1}^1\cdot X_{k_1}^1\cdot P_{k_m}^{in},$$

where$k_m=1,\mathrm{...},N$.

Here all the efficiencies and splitting ratios correspond to a particular and unique path that the input signal $P_{k_m}^{in}$ takes when propagating from the tree input to its common output.

In order to describe the coherently-combined output from the tree with all input signals present let’s first consider the combining in the 1-st stage (the output stage), expressed in terms of amplitudes$E_{1,2}^1$ and phases ${\varphi }_{1,2}^1$ of the signals at this stage input:

(15)$${\tilde{E}}^{comb}=\left(\sqrt{{\eta }_1^1{X}_1^1}{E}_1^1\cdot e^{i{\varphi }_1^1}+\sqrt{{\eta }_2^1{X}_2^1}{E}_2^1\cdot e^{i{\varphi }_2^1}\right).$$

Each of the two input amplitudes in this expression can be expressed through individual amplitudes $E_{1,2,3,4}^2$and phases ${\phi }_{1,2,3,4}^2$of inputs into the second stage:

(16)$$\begin{array}{l}{E}_1^1=\left(\sqrt{{\eta }_1^2{X}_1^2}{E}_1^2\cdot e^{i{\varphi }_1^2}+\sqrt{{\eta }_2^2{X}_2^2}{E}_2^2\cdot e^{i{\varphi }_2^2}\right),\\ E_2^1=\left(\sqrt{{\eta }_3^2{X}_3^2}{E}_3^2\cdot e^{i{\phi }_3^2}+\sqrt{{\eta }_4^2{X}_4^2}{E}_4^2\cdot e^{i{\phi }_4^2}\right).\end{array}$$

This should be continued all the way until the m-th stage (input). It is straightforward to see that the overall result could be expressed as follows:

(17)$${\tilde{E}}^{comb}={\displaystyle \sum _{k_m=1,N}\sqrt{{\eta }_{k_m}^m\cdot X_{k_m}^m\cdot \mathrm{...}\cdot {\eta }_{k_1}^1\cdot X_{k_1}^1}\cdot }{E}_{k_m}^{in}\cdot e^{i\left(\omega t+{\varphi }_{km}^m+\mathrm{...}+{\varphi }_{k1}^1\right)}.$$

Here summation over $k_m=1,N$ implies the convention of Eq. (13), where each particular value of $k_m$ identifies a particular sequence of all other indices of all the $\eta$-s, X-s and φ-s in each term of the summation sequence, thus identifying the path this input signal passes in the binary tree.

Since the binary tree consists of two-port combiners described earlier, it is easy to see that coherent combining efficiency in general depends on relationships between each combining-element splitting ratio X_i and the ratio between powers X_i’ of the signals at the input of this combining element, combining element losses corresponding to each of the two inputs into each combiner η^/_i and η^/_i+1, relative phase difference Δφ between the inputs into each combining element.

This means that in order to achieve maximum combining efficiency for a given binary-combiner tree (with fixed losses in each of the combining elements) it is necessary to have K degrees of freedom in controlling all internal-input amplitudes $E_{k_i}^i$ and K degrees of freedom in controlling all internal-input phases ${\varphi }_{k_i}^i$ in the binary-tree combiner. It is straightforward to show that since there are N = K + 1 external-input signals, this control can be achieved at any total power by controlling N external-input signal amplitudes and N external-input signal phases. With this consideration Eq. (17) could be rewritten slightly differently:

(18)$${\tilde{E}}^{comb}={\displaystyle \sum _{k_m=1,N}\sqrt{{\eta }_{k_m}^m\cdot X_{k_m}^m\cdot \mathrm{...}\cdot {\eta }_{k_1}^1\cdot X_{k_1}^1}\cdot }{E}_{k_m}^{in}\cdot e^{i\left(\omega t+{\varphi }_{km}^{out}\right)}.$$

Here we replaced the sum of all the inter-stage phases${\varphi }_{km}^m+\mathrm{...}+{\varphi }_{k1}^1$ by a “cumulative” phase ${\phi }_{km}^{out}$at the tree output, corresponding to an external input labeled by $k_m$. Note that implicitly this also includes any phases that signals might acquire when propagating between individual combining elements, since all these additional phase contributions can also be compensated by controlling the phases of the N external-input signals.

Combined output power can be calculated from Eq. (18):

(19)$$\begin{array}{l}{P}^{comb}=c{\epsilon }_{0}nA\cdot 〈{\left(\frac{1}{2}{\displaystyle \sum _{k_m=1,N}\sqrt{{\eta }_{k_m}^m\cdot X_{k_m}^m\cdot \mathrm{...}\cdot {\eta }_{k_1}^1\cdot X_{k_1}^1}\cdot }{E}_{k_m}^{in}\cdot e^{i\left(\omega t+{\varphi }_{km}^{out}\right)}+\frac{1}{2}c.c.\right)}^2〉\\ =\frac{c{\epsilon }_{0}nA}{2}{\displaystyle \sum _{k_n=1,N}{\displaystyle \sum _{k_m=1,N}\sqrt{{\eta }_{k_n}^n\cdot X_{k_n}^n\cdot \mathrm{...}\cdot {\eta }_{k_1}^1\cdot X_{k_1}^1}\sqrt{{\eta }_{k_m}^m\cdot X_{k_m}^m\cdot \mathrm{...}\cdot {\eta }_{k_1}^1\cdot X_{k_1}^1}\cdot }}\\ \times E_{k_n}^{in}\cdot E_{k_m}^{in}\cdot \cos ({\varphi }_{km}^{out}-{\varphi }_{kn}^{out})\\ ={\displaystyle \sum _{i=1,N}{P}_i^{out}}+{\displaystyle \sum _{\begin{array}{l}j=1,N\\ j\ne i\end{array}}{\displaystyle \sum _{i=1,N}\sqrt{P_j^{out}\cdot P_i^{out}}\cdot \cos \Delta {\varphi }_{ji}}}.\end{array}$$

Here brackets $〈\mathrm{...}〉$denote time average, $\Delta {\varphi }_{ij}=\left({\varphi }_j-{\varphi }_i\right)$is the phase difference between a pair of corresponding output signals i and j, and each output power is defined by Eq. (14).

The maximum constructive interference occurs when all $\Delta {\varphi }_{ij}=\left({\varphi }_j-{\varphi }_i\right)=0$, i.e. there are no phase errors, producing the maximum achievable combined power:

(20)$$P^{comb\_MAX}={\left({\displaystyle \sum _{i=1,N}\sqrt{P_i^{out}}}\right)}^2.$$

(21)$$\text{Let}�\text{s}\text{denote}{P}_{k_m}^{/}={\eta }_{k_m}^m\cdot \mathrm{...}\cdot {\eta }_{k_i}^i\cdot \mathrm{...}\cdot {\eta }_{k_1}^1\cdot P_{k_m}^{in}and{P}_{tot}^{/}={\displaystyle \sum _{k_m=1,N}{P}_{k_m}^{/}}$$.

According to Eq. (14)

(22)$$P_{k_m}^{out}={\eta }_{k_m}^m\cdot X_{k_m}^m\cdot \mathrm{...}\cdot {\eta }_{k_i}^i\cdot X_{k_i}^i\cdot \mathrm{...}\cdot {\eta }_{k_1}^1\cdot X_{k_1}^1\cdot P_{k_m}^{in}=X_{k_m}^m\cdot \mathrm{...}\cdot X_{k_i}^i\cdot \mathrm{...}\cdot X_{k_1}^1\cdot P_{k_m}^{/}.$$

It can be shown that the maximum in-phase combining efficiency occurs when external-input powers are:

(23)$$P_{k_m}^{/}=X_{k_m}^m\cdot \mathrm{...}\cdot X_{k_i}^i\cdot \mathrm{...}\cdot X_{k_1}^1\cdot P_{tot}^{/}.$$

In this case, substitute Eq. (23) and Eq. (22) into Eq. (20), and we get:

(24)$$P^{comb\_MAX}=P_{tot}^{/}{\left({\displaystyle \sum _{k_m=1,N}\left(X_{k_m}^m\cdot \mathrm{...}\cdot X_{k_i}^i\cdot \mathrm{...}\cdot X_{k_1}^1\right)}\right)}^2.$$

Here the same summation convention as in Eq. (17) applies. Since as it was noted earlier, for each binary-tree combining component splitting ratios between two inputs are related by $X_{k_i+1}^i=1-X_{k_i}^i$if ki is an odd-integer, it is straightforward to show that

(25)$${\displaystyle \sum _{k_m=1,N}\left(X_{k_m}^m\cdot \mathrm{...}\cdot X_{k_i}^i\cdot \mathrm{...}\cdot X_{k_1}^1\right)}=1.$$

Consequently we can obtain an alternative expression to the Eq. (20) of the maximum achievable combining power

(26)$$P^{comb\_MAX}=P_{tot}^{/}={\displaystyle \sum _{k_m=1,N}{P}_{k_m}^{/}}={\displaystyle \sum _{k_m=1,N}\left({\eta }_{k_m}^m\cdot \mathrm{...}\cdot {\eta }_{k_i}^i\cdot \mathrm{...}\cdot {\eta }_{k_1}^1\cdot P_{k_m}^{in}\right)}.$$

Based on this analysis we can define absolute and relative combining efficiencies denoted by ${\eta }_{abs}$ and ${\eta }_{relative}$ respectively, as well as the combiner efficiency${\eta }_{combiner}$:

(27)$${\eta }_{abs}=\frac{P^{comb}}{{\displaystyle \sum _{k_m=1,N}{P}_{k_m}^{in}}},$$

(28)$${\eta }_{relative}=\frac{P^{comb}}{P^{comb\_MAX}}=\frac{P^{comb}}{{\left({\displaystyle \sum _{k_m=1,N}\sqrt{P_{k_m}^{out}}}\right)}^2}=\frac{P^{comb}}{{\displaystyle \sum _{k_m=1,N}\left({\eta }_{k_m}^m\cdot \mathrm{...}\cdot {\eta }_{k_i}^i\cdot \mathrm{...}\cdot {\eta }_{k_1}^1\cdot P_{k_m}^{in}\right)}},$$

and

(29)$${\eta }_{combiner}=\frac{P^{comb\_MAX}}{{\displaystyle \sum _{k_m=1,N}{P}_{k_m}^{in}}}=\frac{{\left({\displaystyle \sum _{k_m=1,N}\sqrt{P_{k_m}^{out}}}\right)}^2}{{\displaystyle \sum _{k_m=1,N}{P}_{k_m}^{in}}}=\frac{{\displaystyle \sum _{k_m=1,N}\left({\eta }_{k_m}^m\cdot \mathrm{...}\cdot {\eta }_{k_i}^i\cdot \mathrm{...}\cdot {\eta }_{k_1}^1\cdot P_{k_m}^{in}\right)}}{{\displaystyle \sum _{k_m=1,N}{P}_{k_m}^{in}}}.$$

Then

(30)$${\eta }_{abs}={\eta }_{combiner}\cdot {\eta }_{relative}.$$

Appendix B: LOCSET combining efficiency as a function of phase-modulation amplitude

Here we show how the analytical expression for combining efficiency in the LOCSET scheme as a function of phase modulation amplitude, Eq. (2), is derived. The combined electric field in the time domain from a system employing the self-referenced LOCSET technique [17] can be written as

(31)$$E_{}^{comb}(t)=E_1^{out}\cos \left({\omega }_{L}t+{\varphi }_1\right)+{\displaystyle \sum _{k=2}^N{E}_k^{out}\cos \left({\omega }_{L}t+{\varphi }_k+{\beta }_k\sin \left({\omega }_{k}t\right)\right)},$$

where N is the total number of elements, $E_1^{out}$and $E_k^{out}$are the field amplitudes for the unmodulated and k^th phase modulated elements, ϕ₁ and ϕ_k are optical phases, ω_L is the laser frequency, ω_k is an RF modulation frequency, and β_k is a phase modulation amplitude.

For calculations we assume the phase locked condition and set ϕ₁ and ϕ_k equal to zero. The phase modulation amplitudes are all set to the same value β. The combined power P^comb is equal to the total field squared (to within a proportionality constant) and when averaged over a time T that is larger than the optical period (2π/ω_L) but smaller than any of the modulation periods (2π/ω_k) gives

(32)$$\begin{array}{l}{P}^{comb}(t,\beta )=\frac{c{\epsilon }_{0}nA}{T}{\displaystyle {\int }_{t-\frac{1}{2}T}^{t+\frac{1}{2}T}{\left|E_{}^{comb}(\tau )\right|}^{2}d\tau }\\ =c{\epsilon }_{0}nA\cdot [\frac{1}{2}{\displaystyle \sum _{k=1}^N{E}_k^{out}{E}_k^{out}}+{\displaystyle \sum _{k=2}^N{E}_1^{out}{E}_k^{out}}\cos (\beta \sin ({\omega }_{k}t))\\ +\frac{1}{2}{\displaystyle \sum _{k=2}^N{\displaystyle \sum _{\begin{array}{l}m=2,\\ m\ne k\end{array}}^N{E}_m^{out}{E}_k^{out}[\cos (\beta \sin ({\omega }_{k}t))}}\cos (\beta \sin ({\omega }_{m}t))\\ +\sin (\beta \sin ({\omega }_{k}t))\sin (\beta \sin ({\omega }_{m}t))]].\end{array}$$

As seen in Eq. (32) the combined power is still time dependent and depends on the phase modulation amplitude β. Our feedback detector does indeed measure the time dependent combined power (this is necessary because the feedback electronics have to demodulate the individual RF modulations), but when measuring combining efficiencies we used a slower detector that did not respond to the modulation time dependence. Therefore the measured time averaged power is given by

(33)$$P^{comb}(\beta )=\frac{1}{T_2}{\displaystyle \underset{0}{\overset{T_2}{\int }}{P}^{comb}(t,\beta )dt},$$

where T₂ (the response time of the slow detector) is several times greater than 2π/|ω_k−ω_m| for any possible k and m values. The theoretical maximum combined power occurs in the limit of β equal to zero and is given by

(34)$$P_{\max }^{comb}=\frac{c{\epsilon }_{0}nA}{2}{\left({\displaystyle \sum _{k=1}^N{E}_k^{out}}\right)}^2.$$

The combining efficiency as a function of β becomes

(35)$${\eta }_{relative}(\beta )=P^{comb}(\beta )/P_{\max }^{comb}.$$

Using the Fourier series expansions

(36)$$\begin{array}{l}\cos (\beta \sin ({\omega }_{k}t))=J_0(\beta )+2{\displaystyle \sum _{m=1}^{\infty }{J}_{2m}(}\beta )\cos (2m{\omega }_{k}t),\\ \sin (\beta \sin ({\omega }_{k}t))=2{\displaystyle \sum _{n=1}^{\infty }{J}_{2n-1}(}\beta )\sin ((2n-1){\omega }_{k}t),\end{array}$$

where J_m is a Bessel function of order m of the first kind to rewrite the terms in Eq. (32), we substitute Eq. (32) into Eq. (33), take the asymptotic limit for large T₂, and calculate the combining efficiency from Eq. (35) to give

(37)$${\eta }_{relative}(\beta )=\frac{{\displaystyle \sum _{k=1}^N{E}_k^{out}{}^2}+{\displaystyle \sum _{k=2}^{N}2E_1^{out}{E}_k^{out}{J}_0(\beta )}+{\displaystyle \sum _{k=2}^N{\displaystyle \sum _{\begin{array}{l}m=2,\\ m\ne k\end{array}}^N{E}_m^{out}{E}_k^{out}{J}_0{}^2(\beta )}}}{{\left({\displaystyle \sum _{k=1}^N{E}_k^{out}}\right)}^2}.$$

Equation (37) is equivalent to the desired result, Eq. (2), when written in terms of powers rather than electric field amplitudes.

Acknowledgments

The authors acknowledge the financial support from Office of Naval Research (Grant No. N00014-07-1-1155).

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