## Perturbative analysis of coherent combining efficiency with mismatched lasers |

Optics Express, Vol. 18, Issue 24, pp. 25403-25414 (2010)

http://dx.doi.org/10.1364/OE.18.025403

Acrobat PDF (1131 KB)

### Abstract

Coherent combining efficiency is examined analytically for large arrays of non-ideal lasers combined using filled aperture elements with nonuniform splitting ratios. Perturbative expressions are developed for efficiency loss from combiner splitting ratios, power imbalance, spatial misalignments, beam profile nonuniformities, pointing and wavefront errors, depolarization, and temporal dephasing of array elements. It is shown that coupling efficiency of arrays is driven by non-common spatial aberrations, and that common-path aberrations have no impact on coherent combining efficiency. We derive expressions for misalignment losses of Gaussian beams, providing tolerancing metrics for co-alignment and uniformity of arrays of single-mode fiber lasers.

© 2010 OSA

## 1. Introduction

1. T. Y. Fan, “Laser Beam Combining for High-Power, High-Radiance Sources,” IEEE J. Sel. Top. Quantum Electron. **11**(3), 567–577 (2005). [CrossRef]

3. G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. **31**(9), 1247–1249 (2006), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-31-9-1247. [CrossRef] [PubMed]

4. S. J. McNaught, C. P. Asman, H. Injeyan, A. Jankevics, A. M. Johnson, G. C. Jones, H. Komine, J. Machan, J. Marmo, M. McClellan, R. Simpson, J. Sollee, M. M. Valley, M. Weber, and S. B. Weiss, “100-kW Coherently Combined Nd:YAG MOPA Laser Array,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThD2, http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2009-FThD2

5. T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-Channel, High Power, Passively Phase Locked Fiber Array,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2008), paper WA4, http://www.opticsinfobase.org/abstract.cfm?URI=ASSP-2008-WA4

7. D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE **7580**, 75801V (2010). [CrossRef]

8. P. A. Thielen, J. G. Ho, M. Hemmat, G. D. Goodno, R. R. Rice, J. Rothenberg, M. Wickham, J. T. Baker, D. Gallant, C. Robin, C. Vergien, C. Zeringue, T. J. Bronder, T. M. Shay, and A. D. Sanchez, *400-W, High-Efficiency Coherent Combination of Fiber Lasers,” presented at 22nd Annual Solid State and Diode Laser Technology Review* (Newton, MA 2009).

11. H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. **30**(11), 1339–1341 (2005), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-30-11-1339. [CrossRef] [PubMed]

5. T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-Channel, High Power, Passively Phase Locked Fiber Array,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2008), paper WA4, http://www.opticsinfobase.org/abstract.cfm?URI=ASSP-2008-WA4

10. B. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50 W coherently combined passive laser array,” Opt. Lett. **34**(7), 863–865 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-7-863. [CrossRef] [PubMed]

11. H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. **30**(11), 1339–1341 (2005), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-30-11-1339. [CrossRef] [PubMed]

6. T. M. Shay, J. T. Baker, A. D. Sancheza, C. A. Robin, C. L. Vergien, A. Flores, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “Phasing of High Power Fiber Amplifier Arrays,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2010), paper AMA1, http://www.opticsinfobase.org/abstract.cfm?URI=ASSP-2010-AMA1

12. G. D. Goodno, S. J. McNaught, J. E. Rothenberg, T. S. McComb, P. A. Thielen, M. G. Wickham, and M. E. Weber, “Active phase and polarization locking of a 1.4 kW fiber amplifier,” Opt. Lett. **35**(10), 1542–1544 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-10-1542. [CrossRef] [PubMed]

1. T. Y. Fan, “Laser Beam Combining for High-Power, High-Radiance Sources,” IEEE J. Sel. Top. Quantum Electron. **11**(3), 567–577 (2005). [CrossRef]

13. E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. **33**(4), 354–356 (2008), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-33-4-354. [CrossRef] [PubMed]

14. J. R. Leger, G. J. Swanson, and W. B. Veldkamp, “Coherent laser addition using binary phase gratings,” Appl. Opt. **26**(20), 4391–4399 (1987), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-26-20-4391. [CrossRef] [PubMed]

15. C. D. Nabors, “Effects of phase errors on coherent emitter arrays,” Appl. Opt. **33**(12), 2284–2289 (1994), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-33-12-2284. [CrossRef] [PubMed]

16. W. Liang, N. Satyan, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Coherent beam combining with multilevel optical phase-locked loops,” J. Opt. Soc. Am. B **24**(12), 2930–2939 (2007), http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-24-12-2930. [CrossRef]

17. T. Y. Fan, “The effect of amplitude (power) variations on beam combining efficiency for phased arrays,” IEEE J. Sel. Top. Quantum Electron. **15**(2), 291–293 (2009). [CrossRef]

## 2. Coherent combining efficiency

*N*input laser channels that are combined using a 1×

*N*beamsplitter (Fig. 1 ). Owing to the symmetry of propagation, beamsplitters can function in reverse as

*N*× 1 beam combiners (BCs), with power from

*N*properly co-phased input channels combined into a single output channel with good efficiency. A

*N*×1 BC can be a single optical device such as a diffractive optical element [13

13. E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. **33**(4), 354–356 (2008), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-33-4-354. [CrossRef] [PubMed]

14. J. R. Leger, G. J. Swanson, and W. B. Veldkamp, “Coherent laser addition using binary phase gratings,” Appl. Opt. **26**(20), 4391–4399 (1987), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-26-20-4391. [CrossRef] [PubMed]

10. B. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50 W coherently combined passive laser array,” Opt. Lett. **34**(7), 863–865 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-7-863. [CrossRef] [PubMed]

18. S. E. Christensen, and O. Koski, “2-Dimensional Waveguide Coherent Beam Combiner,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper WC1, http://www.opticsinfobase.org/abstract.cfm?URI=ASSP-2007-WC1

*N*×1 BC can represent a cascade of serial splitters whose cumulative effect is to couple

*N*input optical channels into 1 output, e.g., a binary tree or other arrangement of free-space partial reflectors [9, 19

19. J. R. Andrews, “Interferometric power amplifiers,” Opt. Lett. **14**(1), 33–35 (1989), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-14-1-33. [CrossRef] [PubMed]

11. H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. **30**(11), 1339–1341 (2005), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-30-11-1339. [CrossRef] [PubMed]

20. R. Uberna, A. Bratcher, and B. G. Tiemann, “Coherent Polarization Beam Combination,” IEEE J. Quantum Electron. **46**(8), 1191–1196 (2010). [CrossRef]

*N*×1 BC can be described as a 1×

*N*beamsplitter with power splitting fractions

*D*

_{n}^{2}over the desired channels

*n*= 1 to

*N*, where normalization

*n*>

*N*. The BC efficiency as a splitter is thenwhere the summation is over only the

*N*channels of interest. Operated as a

*N*×1 combiner, the coupling efficiency

*η*is the ratio of power in the desired output port to the total input power. It has been shown [14

14. J. R. Leger, G. J. Swanson, and W. B. Veldkamp, “Coherent laser addition using binary phase gratings,” Appl. Opt. **26**(20), 4391–4399 (1987), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-26-20-4391. [CrossRef] [PubMed]

*N*×1 combiner with perfectly aligned plane-wave input beams with powers

*P*~|

_{n}*E*|

_{n}^{2},

*N*channels. If the relative input powers are matched to the splitting fractions, i.e., {|

*E*|

_{n}^{2}}∝{

*D*

_{n}^{2}}, then

*η*reduces to the BC-limited value of

*η*[14

_{BC}**26**(20), 4391–4399 (1987), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-26-20-4391. [CrossRef] [PubMed]

*A*(

_{n}**r**) and wavefronts

*ϕ*(

_{n}**r**) (including piston phase errors), finite spectral content, and depolarization. The beams are assumed to be derived from a cw, single-frequency master oscillator (MO) whose linewidth is broadened using frequency modulation, which is a common method for suppressing stimulated Brillouin scattering in high power fiber amplifiers. The MO field can be written

*A*exp[

_{MO}*iω*

_{0}

*t*+

*iψ*(

*t*)], where

*A*is a constant cw amplitude,

_{MO}*ω*

_{0}is the optical carrier frequency and

*ψ*(

*t*) is a slowly time-varying phase. We assume the spectral content for each channel is unchanged from the MO to the BC. Hence the field of the

*n*

^{th}beam at the BC iswhere

**u**

*and*

_{x}**u**

*are unit vectors in the two transverse axes*

_{y}**r**= (

*x*,

*y*);

*χ*is a depolarization angle from the desired polarization state (assumed without loss of generality to be linear along

_{n}**u**

*); and*

_{x}*Γ*is an

_{n}*a priori*random phase shift of the depolarized field component due to uncontrolled birefrigence.

*δτ*is an optical time delay due to the path length of the

_{n}*n*

^{th}channel. The spatially resolved, time-averaged combining efficiency

*η′*(

**r**) on a

*N*× 1 BC is then:where the brackets denote time-averaging. The total combining efficiency

*η*is the intensity-weighted average of

*η′*(

**r**) across the BC aperture:where

*d*

**r**=

*dxdy*is the differential area element in the transverse BC plane. Substituting Eq. (4) for the point-wise combining efficiency, this reduces towhere we have identified the denominator of Eq. (5) as the total input power

*P*.

_{tot}*N*we can ignore the contibution from depolarized fields since they add incoherently with random phases

*Γ*. Utilizing Eq. (3) for the fields, Eq. (6) reduces to:

_{n}## 3. Perturbative analysis of combining efficiency

*η*analagous to the Marechal approximation [21

21. D. D. Lowenthal, “Maréchal intensity criteria modified for gaussian beams,” Appl. Opt. **13**(9), 2126–2133 (1974), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-13-9-2126. [CrossRef] [PubMed]

*N*input fields have similar profiles, so that after amplification and combination the (

**x**-polarized) electric field of the

*n*

^{th}beam can be written perturbatively:Here

*δA*(

_{n}**r**) and

*δϕ*(

_{n}**r**) are small perturbative deviations of the

*n*

^{th}beam’s amplitude and wavefront distributions from their respective average distributions

*A*(

**r**) =

*N*

^{−1}Σ

*A*(

_{n}**r**) and

*ϕ*(

**r**) =

*N*

^{−1}Σ

*ϕ*(

_{n}**r**). We have utilized the small-angle approximation for depolarization, cos(

*χ*) ≈1 –

_{n}*δχ*

_{n}^{2}/2, where

*δχ*are small angular perturbations from the x-axis (

_{n}*χ*= 0). We have also assumed small path delay mismatches

_{n}*δτ*to allow substitution of the Taylor expansion

_{n}*ψ*(

*t*+

*δτ*) ≈

_{n}*ψ*(

*t*) + Δ

*ω*(

*t*)(

*t*+

*δτ*), where Δ

_{n}*ω*(

*t*) ≡

*dψ*(

*t*)/

*dt*is a time-dependent frequency shift away from the carrier frequency

*ω*

_{0}.

## 4. Discussion of isolated misalignments

*δA*(

_{n}**r**) =

*ε*(

_{n}A**r**), where

*ε*is the fractional change in field amplitude. By direct substitution into Eqs. (14) and (15), one can show that

_{n}*σ*(

_{A}**r**) =

*A*(

**r**)

*σ*and

_{ε}*σ*

_{A}_{(}

_{r}_{)}

*=*

_{,D}*A*(

**r**)

*σ*. Hence, the integral in Eq. (16) can be factored out:Here we have approximated

_{ε,D}*A*(

_{n}**r**) ≈

*A*(

**r**) in the denominator of Eq. (17), which is valid since this factor multiplies terms that are already second-order perturbations. Equation (16) simplifies to:It is illustrative to examine some limiting cases of Eq. (18).

### 4.1 Uniform BC, nonuniform powers

*σ*=

_{ϕ}*σ*=

_{χ}*σ*= 0. If the BC splits power uniformly among channels, then

_{τ}*σ*=

_{D}*σ*

_{A}_{,}

*= 0 and the BC efficiency is reduced from its limiting value as a splitter by an amount proportional to the fractional variance of the field amplitudes,*

_{D}*σ*

_{ε}^{2}=

*σ*

_{A}^{2}/

*A*

^{2}:Note that since channel powers

*P*are proportional to the square of the field amplitudes, then small power fluctuations

_{n}*δP*∝2

_{n}*AδA*. Hence fractional power perturbations

_{n}*δP*/

_{n}*P*are twice the fractional amplitude perturbations

*δA*/

_{n}*A*, and Eq. (19) can be written in terms of fractional RMS power variations

*σ*/

_{P}*P*:This is equivalent to the results derived in [16

16. W. Liang, N. Satyan, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Coherent beam combining with multilevel optical phase-locked loops,” J. Opt. Soc. Am. B **24**(12), 2930–2939 (2007), http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-24-12-2930. [CrossRef]

17. T. Y. Fan, “The effect of amplitude (power) variations on beam combining efficiency for phased arrays,” IEEE J. Sel. Top. Quantum Electron. **15**(2), 291–293 (2009). [CrossRef]

*η*= 1), uniform combiner.

_{BC}### 4.2 Uniform powers, nonuniform BC (σ_{ϕ} = σ_{χ} = σ_{τ} = 0)

*σ*=

_{ε}*σ*

_{ε}_{,}

*= 0, and the BC efficiency is reduced from its limiting value as a splitter:*

_{D}*η*=

*η*–

_{BC}*Nσ*

_{D}^{2}. Hence, for combining arrays of similar lasers, the best combining efficiency arises when using a BC with nearly uniform splitting ratios where

*σ*

_{D}^{2}is small.

### 4.3 Correlation of powers and BC (σ_{ϕ} = σ_{χ} = σ_{τ} = 0)

### 4.4 Piston phase errors (σ_{χ} = σ_{τ} = 0)

*η*= 1) with equal splitting fractions, and allowing for finite phase differences between the input beams, Eq. (18) reduces to the value predicted by Nabors [15

_{BC}15. C. D. Nabors, “Effects of phase errors on coherent emitter arrays,” Appl. Opt. **33**(12), 2284–2289 (1994), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-33-12-2284. [CrossRef] [PubMed]

*η*= 1 –

*σ*

_{ϕ}^{2}.

### 4.5 Uncorrelated wavefront errors (σ_{χ} = σ_{τ} = 0)

### 4.6 Correlated wavefront errors

### 4.7 Polarization errors (σ_{ϕ} = σ_{τ} = 0)

*η*= 1 –

*σ*

_{χ}^{2}.

### 4.8 Path mismatch errors (σ_{χ} = σ_{ϕ} = 0)

## 5. Combining Losses with Gaussian Beams

5. T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-Channel, High Power, Passively Phase Locked Fiber Array,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2008), paper WA4, http://www.opticsinfobase.org/abstract.cfm?URI=ASSP-2008-WA4

13. E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. **33**(4), 354–356 (2008), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-33-4-354. [CrossRef] [PubMed]

*x*. For a Gaussian beam with radius

*w*, the normalized amplitude profile issuch that

### 5.1 Beam positioning errors

*δx*of the

_{n}*n*channel’s near field position from the array average (defined to be

^{th}*x*= 0), the field of the

*n*

^{th}beam isWe can identify the amplitude perturbation:Hence, the amplitude variance can be written in terms of the beam position variance (neglecting perturbations above second order):The combining loss from Eq. (16) is then Equation (31) indicates that combining efficiency is reduced by the fractional variance in beam positioning; e.g., for RMS positioning errors equal to 10% of the Gaussian beam radius, efficiency drops ~1% (see Fig. 4). Written in terms of the Gaussian beam’s full-width at half-maximum

*W*= [2ln(2)]

^{1/2}

*w*, Eq. (31) becomes

*η*=

*η*[1 – 2ln(2)

_{BC}*σ*

_{x}^{2}/

*W*

^{2}].

### 5.2 Beam size errors

### 5.3 Beam pointing errors

*λ*is the optical wavelength and we have identified the 1/e

^{2}far field angular radius,

*θ*

_{0}=

*λ*/

*πw*. By analogy with Eqs. (28) – (31) for near field displacement errors, the consequent drop in combining efficiency is

*η*=

*η*[1 –

_{BC}*σ*

_{θ}^{2}/

*θ*

_{0}

^{2}] for array RMS pointing error

*σ*. Written in terms of the FWHM far field divergence

_{θ}*Θ*= [2ln(2)]

^{1/2}

*θ*

_{0}, the efficiency is

*η*=

*η*[1 – 2ln(2)

_{BC}*σ*

_{θ}^{2}/

*Θ*

^{2}]. We see that pointing errors and near field displacements have identical impact on combining losses when expressed as fractional changes in the relevant near field or far field beam width.

### 5.4 Beam divergence errors

*η*=

*η*(1 –

_{BC}*σ*

_{Θ}^{2}/

*Θ*

^{2}), where

*σ*/

_{Θ}*Θ*is the fractional RMS error in angular FWHM beam divergence.

### 5.5 Path mismatch errors

## 6. Conclusion

- • Nonuniformities in BC splitting fractions and input beam power balance generally result in relatively small impacts to combining efficiency. The impact can be eliminated entirely by matching the beam power fractions to the BC splitting fractions. Conversely, the impact is worsened when the two sets of fractions are anti-correlated.
- • Efficiency is degraded by variations in beam-to-beam intensity profiles.
- • Efficiency is degraded by the intensity-weighted wavefront variance between beams. Hence, only uncorrelated wavefront aberrations impact combining efficiency; correlated aberrations have no efficiency impact and simply “print-through” onto the combined output beam. For uniform plane waves, this reduces to the familiar depdence of combining efficiency on piston phase variance.
- • Path mismatch among beams with finite spectral content reduces efficiency due to dephasing.
- • Depolarization among beams is effectively a direct power loss for the CBC output.

## References and links

1. | T. Y. Fan, “Laser Beam Combining for High-Power, High-Radiance Sources,” IEEE J. Sel. Top. Quantum Electron. |

2. | J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high-power fiber arrays,” Proc. SPIE |

3. | G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. |

4. | S. J. McNaught, C. P. Asman, H. Injeyan, A. Jankevics, A. M. Johnson, G. C. Jones, H. Komine, J. Machan, J. Marmo, M. McClellan, R. Simpson, J. Sollee, M. M. Valley, M. Weber, and S. B. Weiss, “100-kW Coherently Combined Nd:YAG MOPA Laser Array,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThD2, http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2009-FThD2 |

5. | T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-Channel, High Power, Passively Phase Locked Fiber Array,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2008), paper WA4, http://www.opticsinfobase.org/abstract.cfm?URI=ASSP-2008-WA4 |

6. | T. M. Shay, J. T. Baker, A. D. Sancheza, C. A. Robin, C. L. Vergien, A. Flores, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “Phasing of High Power Fiber Amplifier Arrays,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2010), paper AMA1, http://www.opticsinfobase.org/abstract.cfm?URI=ASSP-2010-AMA1 |

7. | D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE |

8. | P. A. Thielen, J. G. Ho, M. Hemmat, G. D. Goodno, R. R. Rice, J. Rothenberg, M. Wickham, J. T. Baker, D. Gallant, C. Robin, C. Vergien, C. Zeringue, T. J. Bronder, T. M. Shay, and A. D. Sanchez, |

9. | C. X. Yu, S. J. Augst, S. Redmond, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Phase Coherence, Phase Noise and Phase Control in High-Power Yb Fiber Amplifiers”, presented at 23rd Annual Solid State and Diode Laser Technology Review, Broomfield, CO (2010). |

10. | B. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50 W coherently combined passive laser array,” Opt. Lett. |

11. | H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. |

12. | G. D. Goodno, S. J. McNaught, J. E. Rothenberg, T. S. McComb, P. A. Thielen, M. G. Wickham, and M. E. Weber, “Active phase and polarization locking of a 1.4 kW fiber amplifier,” Opt. Lett. |

13. | E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. |

14. | J. R. Leger, G. J. Swanson, and W. B. Veldkamp, “Coherent laser addition using binary phase gratings,” Appl. Opt. |

15. | C. D. Nabors, “Effects of phase errors on coherent emitter arrays,” Appl. Opt. |

16. | W. Liang, N. Satyan, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Coherent beam combining with multilevel optical phase-locked loops,” J. Opt. Soc. Am. B |

17. | T. Y. Fan, “The effect of amplitude (power) variations on beam combining efficiency for phased arrays,” IEEE J. Sel. Top. Quantum Electron. |

18. | S. E. Christensen, and O. Koski, “2-Dimensional Waveguide Coherent Beam Combiner,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper WC1, http://www.opticsinfobase.org/abstract.cfm?URI=ASSP-2007-WC1 |

19. | J. R. Andrews, “Interferometric power amplifiers,” Opt. Lett. |

20. | R. Uberna, A. Bratcher, and B. G. Tiemann, “Coherent Polarization Beam Combination,” IEEE J. Quantum Electron. |

21. | D. D. Lowenthal, “Maréchal intensity criteria modified for gaussian beams,” Appl. Opt. |

22. | J. W. Goodman, |

**OCIS Codes**

(140.3290) Lasers and laser optics : Laser arrays

(140.3298) Lasers and laser optics : Laser beam combining

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: October 11, 2010

Manuscript Accepted: November 10, 2010

Published: November 19, 2010

**Citation**

Gregory D. Goodno, Chun-Ching Shih, and Joshua E. Rothenberg, "Perturbative analysis of coherent combining efficiency with mismatched lasers," Opt. Express **18**, 25403-25414 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-25403

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### References

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