## 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 Optical Society of America

## Corrections

Gregory D. Goodno, Chun-Ching Shih, and Joshua E. Rothenberg, "Perturbative analysis of coherent combining efficiency with mismatched lasers: errata," Opt. Express**20**, 23587-23518 (2012)

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-20-21-23587

## 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]. Significant progress in servo-controlled phase-locking methods [2] and high power, near-diffraction-limited, narrow-linewidth lasers amenable to phasing [3] has recently culminated in the demonstration of a 100-kW phased array of seven Nd:YAG slab lasers [4].

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, 5–7] and filled-aperture [8–11] implementations, along with both passive [5,10,11] and active [6–9] 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]. For fiber lasers in particular, filled-aperture implementations of CBC promise the best overall efficiency by eliminating near field intensity nonuniformities [1,13].

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.

Previous work has described the efficiency impact of piston phase errors [14,15] or power imbalance [16, 17] between beams in a CBC array. In this paper, we extend this earlier work to the general case of a filled-aperture beam combiner with nonuniform channel couplings. While this analysis could be extended to tiled beam arrays, here we limit our attention to filled aperture combining since this promises higher brightness output due to elimination of the fill-factor impact on beam quality. The analysis treats all nonuniformities that can impact coherent combining efficiency, including wavefront errors, near-field and far-field beam misalignments and mismatches; path mismatch for finite bandwidth lasers; and depolarization. After deriving the analytic form of the combining efficiency, we then develop statistical approximations that are valid for ensembles of phased lasers in the small-error limit. We apply these expressions to limiting cases of isolated aberrations. The specific case of Gaussian beams is analyzed in detail to develop useful tolerancing metrics for a phased array of single-mode fiber lasers.

## 2. Coherent combining efficiency

We consider an array of *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 [13,14], a tapered fiber bundle [10], or a Talbot-imaged waveguide [18]. 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, 19], 3-dB fiber splitters [11], or polarizer/waveplate pairs [20]. Regardless of the specific optical realization, a general *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 ${\sum}_{n=1}^{\infty}{D}_{n}^{2}}=1$ accounts for the possibility of coupling losses intrinsic to the BC into channels

*n*>

*N*. The BC efficiency as a splitter is then

*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] that for a

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

*P*~|

_{n}*E*|

_{n}^{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., {|*E _{n}*|

^{2}}∝{

*D*

_{n}^{2}}, then

*η*reduces to the BC-limited value of

*η*[14].

_{BC}We can generalize Eq. (2) for the case of imperfectly aligned beams with spatially nonuniform amplitudes *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 is

**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:

*η*is the intensity-weighted average of

*η′*(

**r**) across the BC aperture:

*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

*P*.

_{tot}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 *Γ _{n}*. Utilizing Eq. (3) for the fields, Eq. (6) reduces to:

## 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]. 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:

*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}.

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]. Hence, the most useful split ratio is one in which the channels have similar amplitudes:

where the amplitude split perturbations*δD*<<

_{n}*N*

^{-1/2}.

With these approximations, Eq. (7) becomes

*ω*

_{0}

*t*+

*ψ*(

*t*) + Δ

*ω*(

*t*)

*t*+

*ϕ*in Eq. (8) have been factored out, and the delay-dependent piston phase

*ω*

_{0}

*δτ*is subsumed into the channel wavefront error

_{n}*δϕ*. 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):

_{n}*P*=

*P*/

_{tot}*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:

*u*represents any of the parameters {

*A*(

**r**),

*ϕ*(

**r**),

*τ*,

*D*,

*χ*}. We can also identify the covariance of the input field amplitudes

*A*(

_{n}**r**) with the corresponding splitting amplitudes

*D*:

_{n}## 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 *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:

_{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:

#### 4.1 Uniform BC, nonuniform powers

We assume perfect co-phasing, co-polarization, and path matching so that *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}:

*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] and [17] for a lossless (

*η*= 1), uniform combiner.

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

If the input powers are equalized between channels, then *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)

When the input power fractions are perfectly correlated channel-by-channel to the BC power split fractions, *A _{n}* =

*CD*, where

_{n}*C*is a proportionality constant that can be inferred from normalization:

*η*=

*η*. 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).

_{BC}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, ${\sigma}_{{D}^{\prime}}\equiv \sqrt{N}{\sigma}_{D}$, the statistical correlation between *A* and *D* can be written in terms of the normalized coefficients *ε* and *D*’:

*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

*A*,

*D*) with

*σ*=

*σ*=

_{ε}*σ*

_{D}_{’}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
shows the drop in efficiency calculated using both equations for sets of randomly generated input amplitude and splitting coefficients with normalized standard deviations *σ _{A}*/

*A*and

*σ*/

_{D}*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 (σ_{χ} = σ_{τ} = 0)

Assuming flat wavefronts, equal input powers, and a lossless BC (*η _{BC}* = 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],

*η*= 1 –

*σ*

_{ϕ}^{2}.

#### 4.5 Uncorrelated wavefront errors (σ_{χ} = σ_{τ} = 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 *ϕ _{n}*(

**r**) to vary spatially, Eq. (18) becomes

*n*

^{th}beam’s wavefront across the BC aperture, ${\sigma}_{WFE,n}^{2}$:

#### 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 (σ_{ϕ} = σ_{τ} = 0)

Similarly to the case for piston errors, and with the same amplitude assumptions, Eq. (18) reduces to *η* = 1 – *σ _{χ}*

^{2}.

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

Similarly to the case for piston errors, and with the same amplitude assumptions, Eq. (18) reduces to $\eta =1-\u3008\Delta \omega {\left(t\right)}^{2}\u3009{\sigma}_{\tau}^{2}$.

## 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 [2,5–13]. 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

#### 5.1 Beam positioning errors

With small displacement errors *δx _{n}* of the

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

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

*n*

^{th}beam is

*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

For variations *δw _{n}* in Gaussian beam radius of the

*n*

^{th}beam, the field is

*η*=

*η*(1 –

_{BC}*σ*

_{W}^{2}/

*W*

^{2}), where

*σ*/

_{W}*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:

*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

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* = *η _{BC}*(1 –

*σ*

_{Θ}^{2}/

*Θ*

^{2}), where

*σ*/

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

#### 5.5 Path mismatch errors

For a Gaussian spectrum, the frequency variance <Δω(*t*)^{2}> is related to the FWHM linewidth Δ*ω _{FWHM}* by <Δω(

*t*)

^{2}> = Δ

*ω*

_{FWHM}^{2}/[8ln(2)]. In units of Hertz, the RMS frequency spead is Δ

*f*= Δ

_{FWHM}*ω*/2π. Hence <Δω(

_{FWHM}*t*)

^{2}> = π

^{2}Δ

*f*

_{FWHM}^{2}/[2ln(2)], and the loss term in Eq. (16) due to path delay variance

*σ*

_{τ}^{2}is

## 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.

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