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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 24 — Nov. 22, 2010
  • pp: 25403–25414
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Perturbative analysis of coherent combining efficiency with mismatched lasers

Gregory D. Goodno, Chun-Ching Shih, and Joshua E. Rothenberg  »View Author Affiliations


Optics Express, Vol. 18, Issue 24, pp. 25403-25414 (2010)
http://dx.doi.org/10.1364/OE.18.025403


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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

Coherent beam combination (CBC) is a method for parallel brightness scaling, in which outputs from multiple laser apertures are combined into a single beam while conserving beam quality [1

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]

]. Significant progress in servo-controlled phase-locking methods [2

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 6102, 61020U1 (2006).

] and high power, near-diffraction-limited, narrow-linewidth lasers amenable to phasing [3

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]

] has recently culminated in the demonstration of a 100-kW phased array of seven Nd:YAG slab lasers [4

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

].

It is of great interest to extend CBC methods from free space solid-state lasers to fiber lasers, due to their promise of higher efficiency, packageability, and single-mode beam quality. Recent high-power fiber CBC work has demonstrated both tiled-aperture [2

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 6102, 61020U1 (2006).

, 5

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

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]

] and filled-aperture [8

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

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]

] implementations, along with both passive [5

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

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

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]

] and active [6

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

9

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).

] phasing approaches. Fiber lasers naturally generate near-Gaussian beams whose output even above the kW level has been shown to be coherently combinable with good efficiency [12

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]

]. For fiber lasers in particular, filled-aperture implementations of CBC promise the best overall efficiency by eliminating near field intensity nonuniformities [1

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

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]

].

The primary requirement for high efficiency CBC is that the individual lasers must be virtually identical to allow complete constructive interference. This means the lasers must be mutually coherent, spatially mode-matched and co-aligned, co-polarized, and locked in phase with high precision. When these requirements aren’t perfectly met, combining efficiency suffers.

2. Coherent combining efficiency

We consider an array of N input laser channels that are combined using a 1×N beamsplitter (Fig. 1
Fig. 1 Power splitting ratios for a 1 x N beamsplitter / combiner.
). 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

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]

], a tapered fiber bundle [10

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]

], or a Talbot-imaged waveguide [18

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

]. Equivalently, a 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

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).

, 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]

], 3-dB fiber splitters [11

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]

], or polarizer/waveplate pairs [20

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

]. Regardless of the specific optical realization, a general N×1 BC can be described as a 1×N beamsplitter with power splitting fractions Dn 2 over the desired channels n = 1 to N, where normalization n=1Dn2=1 accounts for the possibility of coupling losses intrinsic to the BC into channels n > N. The BC efficiency as a splitter is then
ηBC=Dn2,
(1)
where 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]

] that for a N×1 combiner with perfectly aligned plane-wave input beams with powers Pn ~|En|2,

η=|DnEn|2/|En|2.
(2)

Equation (2) indicates that an array of equal power input beams will exhibit the highest combining efficiency for a BC that has uniform power splitting among the N channels. If the relative input powers are matched to the splitting fractions, i.e., {|En|2}∝{Dn 2}, then η reduces to the BC-limited value of ηBC [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]

].

We can generalize Eq. (2) for the case of imperfectly aligned beams with spatially nonuniform amplitudes An(r) and wavefronts ϕn(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 AMOexp[ 0 t + (t)], where AMO is a constant cw amplitude, 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 is
En(r,t)=An(r)[cos(χn)ux+sin(χn)exp(iΓn)uy]×exp[iω0(t+δτn)+iψ(t+δτn)+iϕn(r)],
(3)
where u x and u y are unit vectors in the two transverse axes r = (x,y); χn is a depolarization angle from the desired polarization state (assumed without loss of generality to be linear along u x); and Γn is an a priori random phase shift of the depolarized field component due to uncontrolled birefrigence. δτn is an optical time delay due to the path length of the n th channel. The spatially resolved, time-averaged combining efficiency η′(r) on a N × 1 BC is then:
η(r)=|DnEn(r,t)|2/An(r)2,
(4)
where the brackets denote time-averaging. The total combining efficiency η is the intensity-weighted average of η′(r) across the BC aperture:
η=η(r)An(r)2dr/An(r)2dr,
(5)
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 to
η=1Ptot|DnEn(r,t)|2dr,
(6)
where we have identified the denominator of Eq. (5) as the total input power Ptot.

While in general both the desired polarization and depolarized field components will contribute to the combining efficiency, in the limit of large N we can ignore the contibution from depolarized fields since they add incoherently with random phases Γn. Utilizing Eq. (3) for the fields, Eq. (6) reduces to:

η=1Ptot|DnAn(r)cos(χn)exp[iω0(t+δτn)+iψ(t+δτn)+iϕn(r)]|2dr.
(7)

3. Perturbative analysis of combining efficiency

While Eq. (7) is an accurate expression for combining efficiency, it provides little direct insight into the sensitivity of the BC to small input beam misalignments or aberrations. The impact of small, non-common errors in beam intensity profiles, wavefronts, polarization angles, and path length can be determined by developing an expression for 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]

]. We assume each of the 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:
En(r,t)=[A(r)+δAn(r)](1δχn2/2)×exp{iω0(t+δτn)+iψ(t)+iΔω(t)(t+δτn)+i[ϕ(r)+δϕn(r)]}.
(8)
Here δAn(r) and δϕn(r) are small perturbative deviations of the n th beam’s amplitude and wavefront distributions from their respective average distributions A(r) = N −1ΣAn(r) and ϕ(r) = N −1Σϕn(r). We have utilized the small-angle approximation for depolarization, cos(χn) ≈1 – δχn 2/2, where δχn are small angular perturbations from the x-axis (χn = 0). We have also assumed small path delay mismatches δτn to allow substitution of the Taylor expansion ψ(t + δτn) ≈ψ(t) + Δω(t)(t + δτn), where Δω(t) ≡ (t)/dt is a time-dependent frequency shift away from the carrier frequency ω 0.

The final approximation is to restrict attention to “quasi-uniform” BCs with near-equal splitting ratios. The justification is that the highest combining efficiency [cf. Eq. (2)] is obtained by matching the input channel power balance to the corresponding splitting ratio for each channel. For practical manufacturing purposes, most beam combining architectures involve lasers that share a common design to provide an economy of scale [4

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

]. Hence, the most useful split ratio is one in which the channels have similar amplitudes:
Dn=ηBC/N+δDn,
(9)
where the amplitude split perturbations δDn << N -1/2.

With these approximations, Eq. (7) becomes
η=1Ptot|(ηBC/N+δDn)[A(r)+δAn(r)](1δχn2/2)exp[iΔω(t)δτn+iδϕn(r)]|2dr.
(10)
Here the common path (channel-independent) global phasor terms ω 0 t + ψ(t) + Δω(t)t + ϕ in Eq. (8) have been factored out, and the delay-dependent piston phase ω 0 δτn is subsumed into the channel wavefront error δϕn. Expanding the exponential in Eq. (10) to second order, evaluating the modulus-square, and neglecting all perturbative terms higher than second order yields (after some algebra):
η=ηBCηBCP[1NδAn(r)2(1NδAn(r))2]dr+2PηBCNA(r)δDnδAn(r)drηBCPA(r)2[1Nδϕn(r)2(1Nδϕn(r))2]drN[1NδDn2(1NδDn)2]ηBCΔω(t)2[1Nδτn2(1Nδτn)2]ηBC[1Nδχn2],
(11)
where P = Ptot/N is the average input beam power (to within a constant). Here we have moved all time-invariant terms outside the time-average brackets and have utilized the two normalization relations:
PtotAn(r)2dr=(NA(r)2+2A(r)δAn(r)+δAn(r)2)dr,
(12)
2δDn=N/ηBCδDn2.
(13)
We can identify each of the square-bracketed terms in Eq. (11) as parameter variances:
σu2=1Nδun2(1Nδun)2,
(14)
where u represents any of the parameters {A(r), ϕ(r), τ, D, χ}. We can also identify the covariance of the input field amplitudes An(r) with the corresponding splitting amplitudes Dn:
σA(r),D=1N(DnηBC/N)[An(r)A(r)]=1NδDnδAn(r).
(15)
The efficiency can now be written concisely in terms of statistical nonuniformities between beams:
η=ηBC[11P(σA(r)2+A(r)2σϕ(r)22N/ηBCA(r)σA(r),D)drΔω(t)2στ2σχ2NσD2/ηBC].
(16)
Note that the covariance term will be positive when the splitting fractions are positively correlated with the input power fractions, hence increasing the combining efficiency. When the sets of splitting fractions and power fractions are anti-correlated, the covariance is negative and the combinining efficiency is decreased. When the two sets of amplitude coefficients are identical, then the covariance term reduces to the DOE splitting variance, cancelling the effect of split nonuniformity as expected from the exact expression Eq. (2).

4. Discussion of isolated misalignments

To illustrate the isolated effects of power balancing, nonuniform BC channel splitting, wavefront and phase error, path mismatch, and depolarization, it is useful to consider the limiting cases in which all parameters but the one in question are perfectly aligned or matched. These isolated limits are generally valid even for multiple simultaneous misalignments due to the second order perturbative approximation used to arrive at Eq. (16).

If all beams have identical near field shapes, but are not necessarily matched in power, we can write δAn(r) = εnA(r), where εn is the fractional change in field amplitude. By direct substitution into Eqs. (14) and (15), one can show that σA(r) = A(r)σε and σA ( r ) ,D = A(r)σε,D. Hence, the integral in Eq. (16) can be factored out:
1PA(r)2dr=NA(r)2drAn2(r)dr1.
(17)
Here we have approximated An(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:
η=ηBC(11PA(r)2σϕ(r)2drσε2σχ2Δω(t)2στ2)+2NηBCσε,DNσD2.
(18)
It is illustrative to examine some limiting cases of Eq. (18).

4.1 Uniform BC, nonuniform powers

We assume perfect co-phasing, co-polarization, and path matching so that σϕ = σχ = στ = 0. If the BC splits power uniformly among channels, then σD = σA , D = 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, σε 2 = σA 2/A 2:
η=ηBC(1σA2/A2).
(19)
Note that since channel powers Pn are proportional to the square of the field amplitudes, then small power fluctuations δPn∝2AδAn. Hence fractional power perturbations δPn/P are twice the fractional amplitude perturbations δAn/A, and Eq. (19) can be written in terms of fractional RMS power variations σP/P:
η=ηBC(1σP2/4P2).
(20)
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]

] and [17

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]

] for a lossless (ηBC = 1), uniform combiner.

4.2 Uniform powers, nonuniform BC (σϕ = σχ = στ = 0)

If the input powers are equalized between channels, then σε = σε , D = 0, and the BC efficiency is reduced from its limiting value as a splitter: η = ηBCD 2. Hence, for combining arrays of similar lasers, the best combining efficiency arises when using a BC with nearly uniform splitting ratios where σD 2 is small.

4.3 Correlation of powers and BC (σϕ = σχ = στ = 0)

When the input power fractions are perfectly correlated channel-by-channel to the BC power split fractions, An = CDn, where C is a proportionality constant that can be inferred from normalization:
An2=C2Dn2=C2ηBC;C=An2/ηBCN/ηBCA.
(21)
Here we approximatedAn2NA2 since it is a coefficient of second order perturbative terms. Hence, εn=N/ηBCDn, σε2=NσD2/ηBC, σε,D=N/ηBCσD2, and the combining efficiency in Eq. (18) reduces to the BC-limited value of η = ηBC. This means the BC has the highest possible efficiency when the input channel powers are matched to the channel splitting ratios, as expected from the exact expression in Eq. (2).

In a more general case, we can examine the effect of correlations between the input powers and BC splitting fractions. Defining the fractional standard deviation for BC split amplitudes, σDNσD, the statistical correlation between A and D can be written in terms of the normalized coefficients ε and D’:
cor(A,D)=σA,DσAσD=σε,DσεσD.
(22)
When cor(A,D) = 1 the two sets of fractions are perfectly correlated between channels; when cor(A,D) = –1 they are anti-correlated; and when cor(A,D) = 0 they are uncorrelated. Expressed in terms of the normalized variances, the efficiency is
η=ηBC(1σε2)σD2+2ηBCσε,D.
(23)
For a lossless BC, Fig. 2
Fig. 2 Effect of channel-by-channel correlations between input powers and BC splitting fractions on the combining efficiency for a lossless BC and perfectly aligned plane waves. The parameter σ refers to the fractional (normalized) standard deviations of A and D, which are assumed to be equal.
displays a plot of efficiency as a function of cor(A,D) with σ = σε = σD as a parameter, i.e., with the assumption that the normalized input amplitude and BC split amplitude coefficients have equal spread. As the correlation between A and D shifts from negative to positive, the combining efficiency approaches 100% regardless of the amplitude nonuniformity between channels.

It is also useful to examine the region over which the perturbative efficiency shown in Eq. (23) represents a valid approximation for the exact expression of Eq. (2). Figure 3
Fig. 3 Comparison of the exact (Monte Carlo) and approximate solutions for combining efficiency of large arrays (N = 103) with nonuniform and uncorrelated input powers and BC splitting ratios, assuming a lossless splitter and perfectly aligned plane waves.
shows the drop in efficiency calculated using both equations for sets of randomly generated input amplitude and splitting coefficients with normalized standard deviations σA/A and σD/D. The error in the efficiency calculation is < 1% for normalized standard deviations < 20% of the average across the array.

4.4 Piston phase errors (σχ = στ = 0)

Assuming flat wavefronts, equal input powers, and a lossless BC (ηBC = 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

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 – σϕ 2.

4.5 Uncorrelated wavefront errors (σχ = στ = 0)

Wavefront errors (WFE) across each beam that are uncorrelated from beam-to-beam affect η in a manner similar to piston errors. With the same assumptions as for piston errors, but now allowing ϕn(r) to vary spatially, Eq. (18) becomes
η=11PA(r)2[1Nδϕn(r)2(1Nδϕn(r))2]dr.
(24)
We assume the beams are in-phase, but that their wavefronts are completely uncorrelated, such that the average piston phase N1δφn(r)0 at all points. This is often the case for WFE arising from uncontrolled optic or amplifier imperfections. Since the remaining integration and summation in Eq. (24) are linear operations, we are free to exchange their order to obtain
η=11N1PA(r)2δφn(r)2dr.
(25)
We can identify the summand of Eq. (25) as the intensity-weighted spatial variance of the n th beam’s wavefront across the BC aperture, σWFE,n2:
σWFE,n2=1PA(r)2δϕn(r)2dr.
(26)
The combining efficiency then becomes η=1σWFE2¯, where σWFE2¯=N1σWFE,n2 is the array average of the uncorrelated, intensity-weighted WFE spatial variance across each beam.

4.6 Correlated wavefront errors

It is apparent from inspection of Eq. (14) that addition of an identical common-path wavefront to each beam has no impact on the beam-to-beam wavefront variance at any point. Hence, combining efficiency is unaffected by wavefront aberrations that are common to all beams. One should note, however, that the combined beam quality can depend quite strongly on common path input wavefront aberrations, which effectively “print through” the BC onto the combined output beam. Also, depending on the specific combiner architecture, common wavefront error can impose spatial filtering losses unrelated to coherent coupling (e.g., for single-mode fiber delivery of each channel to the BC).

4.7 Polarization errors (σϕ = στ = 0)

Similarly to the case for piston errors, and with the same amplitude assumptions, Eq. (18) reduces to η = 1 – σχ 2.

4.8 Path mismatch errors (σχ = σϕ = 0)

Similarly to the case for piston errors, and with the same amplitude assumptions, Eq. (18) reduces to η=1Δω(t)2στ2.

5. Combining Losses with Gaussian Beams

Fiber lasers are natural candidates for beam combining, since they can provide near-diffraction-limited output beams at multi-kW output powers. Accordingly, much of the current activity in beam combined laser systems is devoted to fiber lasers [2

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 6102, 61020U1 (2006).

,5

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

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]

]. Single-mode fibers typically emit beams that are, to a very good approximation, Gaussian. Hence, it is practically useful to develop analytic expressions for combining losses from Eq. (16) that are specific to the case of Gaussian beams. These expressions can be used to determine engineering tolerances for array co-alignments and BC uniformity as trades against efficiency.

Due to symmetry, we can confine our attention to a single transverse coordinate x. For a Gaussian beam with radius w, the normalized amplitude profile is
A(x)=(2/π)1/4P/wexp[x2/w2],
(27)
such that A(x)2dx=P. Starting from Eq. (27), we can develop analytic expressions for common beam misalignments. In the second order perturbative limit, these misalignments do not interact with each other and can be considered separately. The results of the below analysis are summarized in Fig. 4
Fig. 4 Co-alignment and uniformity tolerances for spatially and spectrally Gaussian beams with 1% allowance for combining loss for each effect. The BC is assumed to be lossless and uniform (ηBC = 1 and Dn = N -1/2). RMS parameter variation refers to beam-to-beam mismatch. Fractional parameters and values are relative to the array average FWHM parameter, colored black in each sketch.
.

5.1 Beam positioning errors

With small displacement errors δxn of the nth channel’s near field position from the array average (defined to be x = 0), the field of the n th beam is
An(x)=(2/π)1/4P/wexp[(xδxn)2/w2].
(28)
We can identify the amplitude perturbation:
δAn(x)=dAn(x)dxδxn=2(2/π)1/4P/w(xδxn)w2exp[(xδxn)2/w2]δxn.
(29)
Hence, the amplitude variance can be written in terms of the beam position variance (neglecting perturbations above second order):
σA2=2/π(4P/w3)x2exp[2x2/w2]σx2/w2.
(30)
The combining loss from Eq. (16) is then
η=ηBC(11PσA2dx)=ηBC(12π4σx2w5x2exp[2x2/w2]dx)=ηBC(1σx2/w2).
(31)
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 = ηBC[1 – 2ln(2)σx 2/W 2].

5.2 Beam size errors

For variations δwn in Gaussian beam radius of the n th beam, the field is
An(x)=(2π)1/4Pw+δwnexp[x2/(w+δwn)2].
(32)
Using the same approach as for beam displacement errors, we find η = ηBC(1 – σW 2/W 2), where σW/W is the fractional RMS error in beam FWHM.

5.3 Beam pointing errors

Pointing errors (far field displacements) can be treated similarly to near field displacements since the beam remains Gaussian upon being Fourier transformed to the far field:
A(θ)=12πA(x)ei2πxθ/λdx=(2π)1/4Pw2exp(θ2/θ02),
(33)
where λ is the optical wavelength and we have identified the 1/e2 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 = ηBC[1 – σθ 2/θ 0 2] for array RMS pointing error σθ. Written in terms of the FWHM far field divergence Θ = [2ln(2)]1/2 θ 0, the efficiency is η = ηBC[1 – 2ln(2)σθ 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

Variations in beam divergence among array elements is equivalent to changes in the far field beam size, and by analogy with changes in the near field beam size [Eq. (32)] the combining efficiency is η = ηBC(1 – σΘ 2/Θ 2), where σΘ/Θ is the fractional RMS error in angular FWHM beam divergence.

5.5 Path mismatch errors

6. Conclusion

We have examined the loss of coherent combining efficiency for arrays of imperfectly aligned laser beams using non-ideal, filled-aperture beam combiners. In the perturbative limit of small deviations from perfect alignment, the combining loss is proportional to the normalized variances of each beam parameter. The main results of this analysis can be summarized qualitatively as follows:

  • 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.

We have applied this analysis to derive expressions for combining losses due to misalignment of coherent arrays of Gaussian beams. Typically, for ~1% impact to combining efficiency, beams must be mode-matched to within ~10% of the diffraction-limited beam waist, position, pointing angle, divergence, and coherence length. These results can be used to guide the design of large phased arrays of coherently combined fiber lasers emitting single, high-brightness beams.

References and links

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]

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 6102, 61020U1 (2006).

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

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 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).

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. 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]

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]

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]

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. 14(1), 33–35 (1989), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-14-1-33. [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]

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]

22.

J. W. Goodman, Statistical Optics (Wiley, 2000), pp. 163–165.

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

  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]
  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 6102, 61020U1 (2006).
  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
  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 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).
  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. 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]
  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]
  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]
  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. 14(1), 33–35 (1989), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-14-1-33 . [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]
  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]
  22. J. W. Goodman, Statistical Optics (Wiley, 2000), pp. 163–165.

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