## Abstract

The first theory for two novel coherent beam combination architectures that are the first electronic beam combination architectures that completely eliminate the need for a separate reference beam are presented. Detailed theoretical models are developed and presented for the first time.

©2006 Optical Society of America

## 1. Introduction

To achieve the high brightness required for many laser applications it is necessary to phase lock multiple element optical arrays. Recently, IPG Photonics has reported 2.5-kW of power out of a single mode fiber with a near diffraction limited optical beam [1]. The intensity and hence the power available from a single-mode optical fiber is limited either by optical surface damage or nonlinear optical effects. These limitations can be overcome by coherent beam combining of the power from multiple optical fibers. The novel coherent beam combining system presented in this paper offers not only highly accurate and robust phase locking but, in addition, is readily scalable to more than 100 elements. Furthermore, this is the first phased array locking system that doesn’t require an external reference beam. Here the first theory for two new active coherent beam combining techniques, the self-referenced LOCSET and the self-synchronous LOCSET techniques, are presented.

Accurate control of the optical phase is required for any phase locked multi-fiber approach. In a master oscillator power amplifier configuration, the optical paths of each of the fibers must be locked to within a fraction of the wavelength in order to coherently combine the individual outputs into a single, high-power beam. As a result of time varying thermal loads and other disturbances, active feedback is required in order to provide for stable coherent addition. There have been a number of experimental and theoretical research efforts addressing the need the for very high brightness fiber laser sources. The technical approaches that have been attempted include the optical self-organized approaches [2–8] and RF phase locking methods [9–11]. Active phase locking has demonstrated high fringe visibility for both passive [9–14] and amplified systems [10–12] and powers of 470 watts [12] have been demonstrated. In the previous active phase locked fiber arrays, an external reference beam was phase modulated at an RF frequency [9–12]. In those systems, the light emerging from each element is then interfered with the light from a reference element because the same RF frequency is used to modulate each array element the light from each element must be sent to a spatially isolated photodetector. Good fringe visibilities of > 94% and hence very low phase errors have been consistently achieved using the active phase locking methods.

In the self-referenced and self-synchronous LOCSET techniques [15] the array elements are phase modulated at unique RF frequencies in an identical manner to the first LOCSET technique [13, 14]. The phase error signal for an individual phase modulated element originates from the RF beat note generated by the interference between the overlapping fields of the individual array elements. Therefore, the fields of all of the array elements must overlap on the photodetector to obtain the error signal. The optical phase shift between the optical wave in the unmodulated element (for self-referenced LOCSET) or the array mean phase (for selfsynchronous LOCSET) and in the RF modulated elements are measured separately in the electronic domain and the phase error signal is fed back to the LiNbO_{3} phase modulator for each element individually to minimize the phase error for that element. The general theories for both the self-referenced and self-synchronous LOCSET are derived below.

## 2. Phase locking model

In self-synchronous LOCSET all of the array elements are phase modulated, while in the selfreferenced LOCSET configuration one array element is unmodulated while all of the remaining array elements are phase modulated. The equations that model the self-synchronous LOCSET technique can be obtained from the equations for the self-referenced LOCSET by simply setting the amplitude of the unmodulated field to zero. Therefore, the theoretical model for the self-referenced LOCSET will be derived first, since mathematically self-synchronous LOCSET is a special case of the self-referenced LOCSET analysis.

Assuming that the unmodulated and phase modulated fields are plane waves and are identically polarized, then the unmodulated element optical field, E_{u}(t) and the i^{th} array element optical fields, E_{i}(t) are,

where E_{u0} and E_{i0} represent the field amplitudes for the unmodulated element and i^{th} phase modulated element, respectively. ω_{L} represents the laser frequency. ϕ_{u} and ϕ_{i} represent the optical phases of the unmodulated and the i^{th} array elements, respectively. β_{i} represents the phase modulation amplitude for the i^{th} array element. ω_{i} represents the RF modulation frequency for the i^{th} array element.

Substituting the trigonometric identity for the cosine of the sum of two angles into Eq. (2) we obtain,

where the sinusoids of the optical and RF frequencies have been separated. Approaching the solution from this point of view provides physical insight into the phase locking techniques.

The optical fields from the unmodulated array element and all of the phase modulated array elements are superimposed on the photodetector. Therefore the photodetector current is,

where N represents the number of phase modulated elements in the optical array, l and j represent the summation indices for the phase modulated elements, μ_{o} and ε_{o} represent the magnetic and electric permeabilities of free space and R_{PD} represents the responsivity of the photodetector, A represents the photodetector area.

For clarity the photodetector currents are separated into 3 terms

represents the photocurrent due to the unmodulated field squared. The photocurrent due to the beating of the unmodulated beam with the phase modulated phase modulated elements is,

Finally the ac photocurrent due to the beat notes of the squared sum of the phase modulated phase modulated elements is,

Substituting Eq. (1) and Eq. (2) into Eq. (5) and neglecting the optical frequency terms that the photodetector cannot follow the optical frequency terms greatly simplifies the right hand side and Eq. (5) becomes,

This photocurrent does not depend upon the optical phase and will be rejected by ac coupling.

Substituting, Eq. (1) and Eq. (2) into Eq. (7), neglecting the terms oscillating at optical frequencies, and substituting the Fourier series expansion for the Cos(β_{i} Sin(ω_{i} t)) and Sin(β_{i} Sin(ω_{i} t)) terms, the photocurrent for the beating of the phase modulated fields with each other is,

$$\sum _{l=1}^{N}\sqrt{{P}_{l}}\xb7\sum _{\begin{array}{c}j=1\\ j\ne l\end{array}}^{N}\sqrt{{P}_{j}}\left[\begin{array}{c}\left\{\begin{array}{c}\mathrm{Cos}\left({\varphi}_{l}-{\varphi}_{j}\right)\left({J}_{0}\left({\beta}_{l}\right)+2\xb7\sum _{{n}_{l}=1}^{\infty}{J}_{2{n}_{l}}\left({\beta}_{l}\right)\xb7\mathit{Cos}(2\xb7{n}_{l}\xb7{\omega}_{l}\xb7t)\right)\\ \left({J}_{0}\left({\mathit{\beta}}_{j}\right)+2\xb7\sum _{{n}_{j}=1}^{\infty}{J}_{2{n}_{j}}\left({\mathit{\beta}}_{j}\right)\xb7\mathit{Cos}(2\xb7{n}_{j}\xb7{\mathit{\omega}}_{j}\xb7t)\right)\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\end{array}\right\}\\ -\left\{\begin{array}{c}\mathit{Sin}\left({\varphi}_{l}-{\varphi}_{j}\right)\xb72\xb7\sum _{{n}_{l}=1}^{\infty}{J}_{2{n}_{l}-1}\left({\beta}_{l}\right)\xb7\mathit{Sin}\left(\right(2\xb7{n}_{l}-1)\xb7{\omega}_{l}\xb7t))\\ \left({J}_{0}\left({\mathit{\beta}}_{j}\right)+2\xb7\sum _{{n}_{j}=1}^{\infty}{J}_{2{n}_{j}}\left({\mathit{\beta}}_{j}\right)\xb7\mathit{Cos}(2\xb7{n}_{j}\xb7{\mathit{\omega}}_{j}\xb7t)\right)\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\end{array}\right\}\\ +\left\{\begin{array}{c}\mathit{Sin}\left({\varphi}_{l}-{\varphi}_{j}\right)\left({J}_{0}\left({\beta}_{l}\right)+2\xb7\sum _{{n}_{j}=1}^{\infty}{J}_{2{n}_{j}}\left({\beta}_{j}\right)\xb7\mathit{Cos}(2\xb7{n}_{j}\xb7{\omega}_{j}\xb7t)\right)\\ 2\xb7\sum _{{n}_{j}=1}^{\infty}{J}_{{2n}_{j}-1}\left({\mathit{\beta}}_{j}\right)\xb7\mathit{Sin}\left(\right(2\xb7{n}_{j}-1)\xb7{\omega}_{j}\xb7t)\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\end{array}\right\}\\ +\left\{\begin{array}{c}\mathit{Cos}\left({\varphi}_{l}-{\varphi}_{j}\right)\left(2\xb7\sum _{{n}_{j}=1}^{\infty}{J}_{2{n}_{j}-1}\left({\beta}_{j}\right)\mathit{\xb7Sin}\left(\right(2\xb7{n}_{j}-1)\xb7{\omega}_{j}\xb7t)\right)\\ 2\xb7\sum _{{n}_{l}=1}^{\infty}{J}_{2{n}_{l}-1}\left({\mathit{\beta}}_{l}\right)\mathit{\xb7Sin}\left(\right(2\xb7{n}_{l}-1)\xb7{\omega}_{l}\xb7t)\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\end{array}\right\}\end{array}\right],$$

this equation explicitly shows the RF frequency components of the photodetector current due to this term.

Substituting Eq. (1) and Eq. (2) into Eq. (7), neglecting the photocurrent terms oscillating at optical frequencies and substituting the Fourier series expansions for the Cosine and Sine of β Sin(ω t), the photocurrent due to the beating of the unmodulated field with the phase modulated fields is,

$$\sum _{j=1}^{N}\sqrt{{P}_{j}}\xb7\left[\begin{array}{c}\left\{\mathit{Cos}({\varphi}_{j}-{\varphi}_{u})\left({J}_{0}\left({\beta}_{j}\right)+2\xb7\sum _{{n}_{j}=1}^{\infty}{J}_{{2n}_{j}}\left({\beta}_{j}\right)\xb7\mathit{Cos}(2\xb7{n}_{j}\xb7{\omega}_{j}\xb7t)\right)\right\}\\ +\left\{\mathit{Sin}({\varphi}_{j}-{\varphi}_{u})\xb72\xb7\sum _{{n}_{j}=1}^{\infty}{J}_{{2n}_{j}-1}\left({\beta}_{j}\right)\xb7\mathit{Sin}\left(\left(2\xb7{n}_{j}-1\right)\xb7{\omega}_{j}\xb7t\right)\right\}\end{array}\right],$$

where J represents a Bessel function of the first kind and n is an integer. Eq. (10) shows explicitly the spectral content of the photocurrent signal due to the beating between the unmodulated element and the phase modulated elements. The second term in the square brackets of Eq. (10) is proportional to Sin(ϕ_{u}-ϕ_{j}) and has the correct characteristics for an error signal. This term is used as one portion of the error signal for the control loop. Electronic signal processing will be used to isolate the phase error signals for each phase modulated element in the frequency domain.

Next, the photocurrent signal due to the beating of the unmodulated field with the phase modulated field is signal processed to extract a portion of the phase control signal. The error signal will be extracted using a coherent demodulation in the RF domain. The RF demodulation is implemented by multiplying the photodetector current by sin(ω_{c} t) [16] and integrated over a time, τ. When ω_{c} is the phase modulation frequency of one of the phase modulated elements and the integration time τ, is selected to simultaneously isolate the individual phase control signals of the phase modulated elements and short enough so that the phase control loop can effectively cancel the phase disturbances of the system, then this can be used for phase error cancellation. The signal processing of the two photocurrent terms that depend upon the optical phases will be evaluated individually. The portion of the phase control signal due to the beating of the unmodulated beam with the phase modulated fields is,

$$\frac{1}{\tau}\xb7\underset{0}{\overset{\tau}{\int}}\left\{\begin{array}{c}2\xb7{R}_{\mathit{PD}}\xb7\sqrt{{P}_{u}}\xb7\mathit{Sin}({\omega}_{c}\xb7t)\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\\ \sum _{i=1}^{N}\sqrt{{P}_{i}}\xb7\left[\begin{array}{c}\left\{\mathit{Cos}({\varphi}_{i}-{\varphi}_{u})\left({J}_{0}\left({\beta}_{i}\right)+2\xb7\sum _{{n}_{i}=1}^{\infty}{J}_{{2n}_{i}}\left({\beta}_{i}\right)\xb7\mathit{Cos}(2\xb7{n}_{i}\xb7{\omega}_{i}\xb7t)\right)\right\}\\ +\left\{\mathit{Sin}({\varphi}_{u}-{\varphi}_{i})\xb72\xb7\sum _{{n}_{i}=1}^{\infty}{J}_{{2n}_{i}-1}\left({\beta}_{i}\right)\xb7\mathit{Sin}\left(\left(2\xb7{n}_{i}-1\right)\xb7{\omega}_{i}\xb7t\right)\right\}\end{array}\right]\end{array}\right\}\xb7\mathit{dt}.$$

If ω_{i}=ω_{c} and if the integration time, τ≫2 π/ω_{i} and τ≫2 π/|(ω_{i}-ω_{j})| for all i and j when j≠i then Eq. (11) is to an excellent approximation,

the error signal changes sign as the phase difference, (ϕ_{u} - ϕ_{i}), changes sign and is zero when the phase difference is zero. The strength of the demodulated signal is proportional to the square root of the product of the optical powers in the unmodulated and i^{th} phase modulated element of the array and J_{1}(β_{i}). The demodulation process is equivalent to extracting the amplitude of the Fourier Sine series component of the current at frequency, ω_{i}. This portion of the phase control signal for each phase modulated element is extracted by simply demodulating the error signal with the modulation frequency for that element.

The photocurrent due to the unmodulated field squared makes no contribution to the phase control signal because it has no frequency components at any of the phase modulation frequencies, ω_{i}. However, the phase modulated self beating photocurrent does have frequency components at frequency, ω_{i}. If ω_{i}=ω_{c} and the integration, τ≫2 π/ω_{i} and 2 π/|(ω_{i} -ω_{j})| then the demodulation of the phase modulated element self-beating terms is,

$${S}_{\mathit{iji}}={R}_{\mathit{PD}}\xb7\sqrt{{p}_{i}}\xb7{J}_{1}({\beta}_{i})\xb7\sum _{j=1}^{N}\sqrt{{p}_{j}}\xb7{J}_{0}\left({\beta}_{j}\right)\xb7\mathit{Sin}\left({\varphi}_{j}-{\varphi}_{i}\right)\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}},$$

where ϕ_{j} and ϕ_{i} represent the phases of the j^{th} phase modulated element and the i^{th} phase modulated elements, respectively. Eq. (13) is an additional phase correction signal and serves to further stabilize the phase of the phase modulated elements to each other. Therefore the total phase control signal is the sum of Eq. (12) and Eq. (13),

$${S}_{\mathit{SRi}}={R}_{\mathit{PD}}\xb7\sqrt{{P}_{i}}\xb7{J}_{1}\left({\beta}_{i}\right)\left(\begin{array}{c}\sqrt{{P}_{u}}\xb7\mathit{Sin}\left({\varphi}_{u}-{\varphi}_{i}\right)\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\\ +\sum _{j=1}^{N}\sqrt{{P}_{j}}\xb7{J}_{0}\left({\beta}_{j}\right)\xb7\mathit{Sin}\left({\varphi}_{j}-{\varphi}_{i}\right)\end{array}\right)\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}},$$

where S_{SRi} represents the phase error control signal for the self-referenced LOCSET configuration. In Eq. (14) the first term in the bracket has the same form as the phase error signals used by previous phase locked arrays and the second term in the bracket is the selfsynchronous term. By using the error signal in Eq. (14) in the feedback control loop the phase is locked. Eq. (14) is the phase error signal that is used for the self-referenced LOCSET configuration. In this mode the phases of the phase modulated array elements are adjusted by the feedback loop to track the phase of the unmodulated element. Therefore, the relative phases between the unmodulated array element and each of the phase modulated array elements at the sensing photodetector are preserved when any or all of the array elements are disturbed.

In the self-referenced LOCSET configuration one array element is not phase modulated whereas, in the self-synchronous LOCSET configuration all of the array elements are phase modulated. The phase error control signal for the self-synchronous LOCSET configuration is obtained from Eq. (13) by simply setting Pu equal to zero,

where S_{SSi} represents the phase error control signal for the self-synchronous configuration of the system. When the error signal in Eq. (15) is used in the feedback control loop the phases of the array elements are locked together. In the self-synchronous configuration the phases of all of the elements adjust themselves to minimize the phase difference between the array elements and the relative phases remain locked when any or all of the array elements are disturbed. Hence the control loop for element strives to null ϕi relative to the mean phase of the remaining array elements.

Assuming that the loop integrator time constant, τ≫1/|ω_{i}- ω_{j}| for any pair of phase modulation frequencies and that a portion of the central lobe of the far-field pattern of the array be imaged on the photodetector active area, then the phase error signal for the selfreferenced LOCSET and self-synchronous LOCSET configurations are derived and are presented in Eqs. (14) and (15), respectively.

## 3. Control loop analysis

The closed loop performance of the control loop is analyzed in this section. The expressions for the control loop signal-to-noise ratio will be derived and discussed. The phases for j^{th} array element are,

where ϕ_{jo} represents mean phase for the j^{th} array element and Δ*ϕ _{j}* (

*t*) represents the time varying component of the jth array element phase when the control is opened. Substituting Eq. (16) into Eq. (15),

where δϕ_{je} represents the closed loop phase error for the other array elements. Assuming that the control set point is adjusted so that mean phases for all the control loops are equal, i. e. ϕ_{jo}=ϕ_{io} and that the control loop gain is sufficiently high so that the phase fluctuations are small then,

In Eq. (18) it is evident that the responses of the i^{th} control signal due to a disturbance the phase of the i^{th} element is N times stronger than the response of the i^{th} control signal to an equal magnitude disturbance in one of the other array elements. This is understood by considering that in Eq. (18) the first term in summation corresponds to a weighted ensemble average of the phase fluctuations in the array that reduces the impact of the fluctuations of other array elements on the stability of the i^{th} array elements phase while the second term in the summation increases the control loop signal for fluctuations in the element that is being controlled, the i^{th} element. This results in an increase in error signal strength as the number of elements increase and a simultaneous decrease in the influence of fluctuation in any of the other array elements on the control loop for the i^{th} array element. Both of those features result in a control loop that scales gracefully as the number of elements are increased.

Figure 1 is a linear systems model block diagram for the self-synchronous LOCSET control loop. In the control loop for the phase of the i^{th} array element, the phase of the i^{th} element is subtracted from the mean phase of the remaining array elements and as the number of elements increase the strength of this signal increases. In practice, all of these functions are performed in the optical photodetector. The electronic signal processing and phase modulators functions are represented in the block on the far right-hand-side of Fig. 1. In the control loop block diagram the coupling of the j^{th} elements phase to the i^{th} elements phase control, K_{ij}, are

The closed loop phase error for the the i^{th} element, δϕ_{i} can be derived. The closed loop phase error for the i^{th} element, δϕ_{ie_i}, due to an external phase disturbance in the i^{th} element, Δϕ_{i} is,

where s represents the LaPlace variable jω, A_{e} represents the RF amplifier gain, K_{PM} represents the phase modulator’s voltage to phase conversion factor and τ represents the integrator time constant. Increasing the gain of the feedback loop results in a decrease in the amplitude of the phase fluctuation and for high gains the induced phase error is very stable. Each element in the array has a similar control loop that accurately controls the phase for that element. When the system is properly designed this results in accurate phase control with very small phase errors for disturbances whose frequency spectrum is within the control loop bandwidth. The control loop bandwidth in radians/sec is the second term in the denominator of Eq. (20).

The signal-to-noise ratio for the phase control signal of the i^{th} element is,

where δϕ_{ie_rms} represents the root-mean-phase error in closed loop operation, q represents the charge of an electron, and B represents the electronic bandwidth of the system. The signal-tonoise ratio clearly depends upon the number of array elements. The dependence of the signalto-noise ratio is elucidated by assuming that β_{j} is small so that J_{0}(β_{j})=1, all of the array elements have equal powers, that the coupling elements, K_{ij} are all equal, N is much greater than 1 and substituting for the control loop bandwidth from Eq. (20) then,

as the number of elements are increased the signal-to-noise ratio remains constant. A significant practical consequence of Eq. (22) is that the self-synchronous LOCSET system remains stable as the numbers of elements are increased.

## 4. Summary

The first theory for two novel coherent beam combination techniques that are the first electronic beam combination techniques that completely eliminate the need for a separate reference beam are presented. The self-referenced LOCSET and self-synchronous LOCSET techniques are less complex electronic phase locking techniques than previous electronic phase locking systems. Both of these techniques eliminate the external reference beam and use a single photodetector, while previous electronic phase locking techniques required one photodetector for each array element and an external reference beam. In addition, the phase control signal for the LOCSET techniques is obtained by imaging the central portion of the far-field onto the photodetector used in the control loop. The elimination of the external reference beam from the electronic phase locking systems greatly simplifies the optical alignment as well as providing for use of a single photodetector per array instead of N detectors for an N element array.

The basic operating principles are illuminated and the theory necessary to design and model the basic control loop performance are developed in this paper. The conditions necessary to design the control loop are analyzed, a linear model of the control loop is developed and it is shown that the signal-to-noise ratio of this system remains constant as the numbers of array elements are increased.

## Acknowledgments

The author wishes to thank Dr. Fassil Ghebremichael, Dr. Erik Bochove, Capt. Benjamin Ward, and Dr. Theodore Salvi for helpful discussions.

## References and links

**1. **V. P. Gapontsev, “New milestones in the development of super high power fiber lasers,” presented at Photonics West, OE/LASE 2006, San Jose, CA, Jan 21–26, 2006.

**2. **P. K Cheo, A. Liu, and G. G. King, “A high brightness laser beam from a Phase-Locked Multicore Yb-Doped Fiber Laser Array,” IEEE Photon. Technol. Lett. **13**, 439–441 (2001). [CrossRef]

**3. **E. J. Bochove, P. K. Cheo, and G. G. King, “Self-organization in a multicore fiber laser array,” Opt. Lett. **28**, 1200–1202 (2003). [CrossRef] [PubMed]

**4. **H. Bruesselbach, D. C. Jones, M. S. Mangir, M. I. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. **30**, 1339–1341 (2003). [CrossRef]

**5. **R. J. Beach, M. D. Feit, S. C. Mitchell, K. P. Culter, J. W. Dawson, S. A. Payne, R. W. Mead, J. S. Hayden, D. Krashkevich, and D. A. Alunni, “Ribbon fiber with multiple phase-locked gain cores,” in *Advances in Fiber Lasers*, L. N. Durvasula, ed., Proc. SPIE4974, 7–16 (2003). [CrossRef]

**6. **R. A. Beach, M. D. Feit, R. H. Page, L. D. Brasure, R. Wilcox, and S. A. Payne, “Scalable antiguided ribbon laser,” J. Opt. Soc. Am. B **19**, 1521–1534 (2002). [CrossRef]

**7. **C. J. Corcoran, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. **86**, 201118–201121 (2005). [CrossRef]

**8. **B. W. Grimes, W. B. Roh, and T. G. Alley, “Beam phasing multiple fiber amplifiers using a fiber phase conjugate mirror,” in *Fiber Lasers III: Technology, Systems, and Applications*, A. J. W. Brown, J. Nilsson, D. J. Harter, and A. Tunnermann, eds., Proc. SPIE6102, 61021C-1 to 61021C-8 (2006).

**9. **R. R. Rice et, J. A. Davis, J. S. Whitely, J. H. Hollister, and N. F. Ruggieri, “Coherent Fiber MOPA,” Presented at 14^{th} Annual Solid State and Diode Laser Technology Review, Sean Ross, ed., Albuquerque, NM (2001).

**10. **J. Abderegg, S. J. Brosnan, M. E. Weber, H. Komine, and M. G. Wickham, “8-watt coherently-phased 4-element fiber array,” in *Advances in Fiber Lasers*, L. N. Durvasula, ed., Proc. SPIE4974, 1–6 (2003). [CrossRef]

**11. **S. J. Augst, T. Y. Fan, and A. Sanchez, “Coherent beam combining and phase noise measurements of Yt fiber amplifiers,” Opt. Lett. **29**, 474–476 (2004). [CrossRef] [PubMed]

**12. **M. Wickham, “Coherently coupled high power fiber arrays,” in Fiber Lasers III: Technology, Systems, and Applications, A. J. W. Brown, J. Nilsson, D. J. Harter, and A. Tunnermann, eds., Proc. SPIE6102, 61020U-1 to 61020U-5 (2006).

**13. **
“A novel technique for phase locking Optical Fiber Arrays,” T. M. Shay and V. Benham, in *Free-Space Laser Communications IV*, J. C. Ricklin and D. G. Voelz, eds., Proc. SPIE5550, 313–319 (2004). [CrossRef]

**14. **
“First experimental demonstration of fiber array phase locking by RF phase modulation,” T. M. Shay and V. Benham, *Proceedings of the 17th Solid State and Diode Laser Technology Review*, S. Ross, ed., pg. BEAM-7 (2004).

**15. **
“Self-synchronous locking of optical coherence by single-detector electronic-frequency tagging,” T. M. Shay, US Patent 7,058,098, June 2006.

**16. **
Note that in principle any odd harmonic of the modulation frequency, ω_{i}, can be used to demodulate the phase error signal. However, the fundamental frequency generally produces the highest signal-to-noise ratio. Therefore, in this analysis the demodulation the fundamental frequency is always used for demodulation of the phase error signals.