Perturbative analysis of coherent combining efficiency with mismatched lasers

Gregory D. Goodno; Chun-Ching Shih; Joshua E. Rothenberg

doi:10.1364/OE.18.025403

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

η_{B C} = \sum D_{n}^{2},

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

Fig. 1 Power splitting ratios for a 1 x N beamsplitter / combiner.

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η = {| \sum D_{n} E_{n} |}^{2} / \sum {| 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_BC [14].

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_MOexp[iω ₀ t + iψ(t)], where A_MO 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

\begin{matrix} E_{n} (r, t) = A_{n} (r) {[\cos (χ_{n}) u_{x} + \sin (χ_{n}) \exp (i Γ_{n}) u_{y}]}^{} \\ \times \exp [i ω_{0} (t + δ τ_{n}) + i ψ (t + δ τ_{n}) + i ϕ_{n} (r)], \end{matrix}

where u _x and u _y are unit vectors in the two transverse axes r = (x,y); χ_n is a depolarization angle from the desired polarization state (assumed without loss of generality to be linear along u _x); and Γ_n is an a priori random phase shift of the depolarized field component due to uncontrolled birefrigence. δτ_n 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) = 〈 {| \sum D_{n} E_{n} (r, t) |}^{2} 〉 / \sum A_{n} {(r)}^{2},

where the brackets denote time-averaging. The total combining efficiency η is the intensity-weighted average of η′(r) across the BC aperture:

η = \int η^{'} (r) \sum A_{n} {(r)}^{2} d r / \int \sum A_{n} {(r)}^{2} d r,

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 = \frac{1}{P_{t o t}} \int 〈 {| \sum D_{n} E_{n} (r, t) |}^{2} 〉 d r,

where we have identified the denominator of Eq. (5) as the total input power 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:

η = \frac{1}{P_{t o t}} \int 〈 {| \sum D_{n} A_{n} (r) \cos (χ_{n}) \exp [i ω_{0} (t + δ τ_{n}) + i ψ (t + δ τ_{n}) + i ϕ_{n} (r)] |}^{2} 〉 d r .

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:

\begin{matrix} E_{n} (r, t) = [A (r) + δ A_{n} (r)] (1 - δ χ_{n}^{2} / 2) \times \\ \exp {i ω_{0} (t + δ τ_{n}) + i ψ (t) + i Δ ω (t) (t + δ τ_{n}) + i [ϕ (r) + δ ϕ_{n} (r)]} . \end{matrix}

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 ⁻¹ΣA_n(r) and ϕ(r) = N ⁻¹Σϕ_n(r). We have utilized the small-angle approximation for depolarization, cos(χ_n) ≈1 – δχ_n ²/2, where δχ_n are small angular perturbations from the x-axis (χ_n = 0). We have also assumed small path delay mismatches δτ_n to allow substitution of the Taylor expansion ψ(t + δτ_n) ≈ψ(t) + Δω(t)(t + δτ_n), where Δω(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:

D_{n} = \sqrt{η_{B C} / N} + δ D_{n},

where the amplitude split perturbations δD_n << N ^-1/2.

With these approximations, Eq. (7) becomes

η = \frac{1}{P_{t o t}} \int 〈 {| \sum (\sqrt{η_{B C} / N} + δ D_{n}) [A (r) + δ A_{n} (r)] (1 - δ χ_{n}^{2} / 2) \exp [i Δ ω (t) δ τ_{n} + i δ ϕ_{n} (r)] |}^{2} 〉 d r .

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 δτ_n is subsumed into the channel wavefront error δϕ_n. 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):

\begin{matrix} η = η_{B C} - \frac{η_{B C}}{P} \int [\frac{1}{N} \sum δ A_{n} {(r)}^{2} - {(\frac{1}{N} \sum δ A_{n} (r))}^{2}] d r + \frac{2}{P} \sqrt{\frac{η_{B C}}{N}} \int A (r) \sum δ D_{n} δ A_{n} (r) d r \\ - \frac{η_{B C}}{P} \int A {(r)}^{2} [\frac{1}{N} \sum δ ϕ_{n} {(r)}^{2} - {(\frac{1}{N} \sum δ ϕ_{n} (r))}^{2}] d r - N [\frac{1}{N} \sum δ D_{n}^{2} - {(\frac{1}{N} \sum δ D_{n})}^{2}] \\ - η_{B C} 〈 Δ ω {(t)}^{2} 〉 [\frac{1}{N} \sum δ τ_{n}^{2} - {(\frac{1}{N} \sum δ τ_{n})}^{2}] - η_{B C} [\frac{1}{N} \sum δ χ_{n}^{2}], \end{matrix}

where 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:

P_{t o t} \equiv \int \sum A_{n} {(r)}^{2} d r = \int (N A {(r)}^{2} + 2 A (r) \sum δ A_{n} (r) + \sum δ A_{n} {(r)}^{2}) d r,

2 \sum δ D_{n} = - \sqrt{N / η_{B C}} \sum δ D_{n}^{2} .

We can identify each of the square-bracketed terms in Eq. (11) as parameter variances:

σ_{u}^{2} = \frac{1}{N} \sum δ u_{n}^{2} - {(\frac{1}{N} \sum δ u_{n})}^{2},

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 A_n(r) with the corresponding splitting amplitudes D_n:

σ_{A (r), D} = \frac{1}{N} \sum (D_{n} - \sqrt{η_{B C} / N}) [A_{n} (r) - A (r)] = \frac{1}{N} \sum δ D_{n} δ A_{n} (r) .

The efficiency can now be written concisely in terms of statistical nonuniformities between beams:

\begin{matrix} η = η_{B C} [1 - \frac{1}{P} \int (σ_{A (r)}^{2} + A {(r)}^{2} σ_{ϕ (r)}^{2} - 2 \sqrt{N / η_{B C}} A (r) σ_{A (r), D}) d r \\ - 〈 Δ ω {(t)}^{2} 〉 σ_{τ}^{2} - σ_{χ}^{2} - N σ_{D}^{2} / η_{B C}] . \end{matrix}

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 δA_n(r) = ε_nA(r), where ε_n is the fractional change in field amplitude. By direct substitution into Eqs. (14) and (15), one can show that σ_A(r) = A(r)σ_ε and σ_A ₍ _r ₎ _,D = A(r)σ_ε,D. Hence, the integral in Eq. (16) can be factored out:

\frac{1}{P} \int A {(r)}^{2} d r = N \frac{\int A {(r)}^{2} d r}{\int \sum A_{n}^{2} (r) d r} \approx 1.

Here we have approximated 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:

η = η_{B C} (1 - \frac{1}{P} \int A {(r)}^{2} σ_{ϕ (r)}^{2} d r - σ_{ε}^{2} - σ_{χ}^{2} - 〈 Δ ω {(t)}^{2} 〉 σ_{τ}^{2}) + 2 \sqrt{N η_{B C}} σ_{ε, D} - N σ_{D}^{2} .

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 σ_ϕ = σ_χ = σ_τ = 0. If the BC splits power uniformly among channels, then σ_D = σ_A _, _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, σ_ε ² = σ_A ²/A ²:

η = η_{B C} (1 - σ_{A}^{2} / A^{2}) .

Note that since channel powers P_n are proportional to the square of the field amplitudes, then small power fluctuations δP_n∝2AδA_n. Hence fractional power perturbations 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:

η = η_{B C} (1 - σ_{P}^{2} / 4 P^{2}) .

This is equivalent to the results derived in [16] and [17] for a lossless (η_BC = 1), uniform combiner.

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 _, _D = 0, and the BC efficiency is reduced from its limiting value as a splitter: 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_n, where C is a proportionality constant that can be inferred from normalization:

\sum A_{n}^{2} = C^{2} \sum D_{n}^{2} = C^{2} η_{B C}; C = \sqrt{\sum A_{n}^{2} / η_{B C}} \approx \sqrt{N / η_{B C}} A .

Here we approximated

\sum A_{n}^{2} \approx N A^{2}

since it is a coefficient of second order perturbative terms. Hence,

ε_{n} = \sqrt{N / η_{B C}} D_{n}

,

σ_{ε}^{2} = N σ_{D}^{2} / η_{B C}

,

σ_{ε, D} = \sqrt{N / η_{B C}} σ_{D}^{2}

, and the combining efficiency in Eq. (18) reduces to the BC-limited value of η = η_BC. 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, $σ_{D^{'}} \equiv \sqrt{N} σ_{D}$ , the statistical correlation between A and D can be written in terms of the normalized coefficients ε and D’:

cor (A, D) = \frac{σ_{A, D}}{σ_{A} σ_{D}} = \frac{σ_{ε, D^{'}}}{σ_{ε} σ_{D^{'}}} .

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 = η_{B C} (1 - σ_{ε}^{2}) - σ_{D^{'}}^{2} + 2 \sqrt{η_{B C}} σ_{ε, D^{'}} .

For a lossless BC, Fig. 2 displays a plot of efficiency as a function of cor(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.

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.

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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.

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.

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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

η = 1 - \frac{1}{P} \int A {(r)}^{2} [\frac{1}{N} \sum δ ϕ_{n} {(r)}^{2} - {(\frac{1}{N} \sum δ ϕ_{n} (r))}^{2}] d r .

We assume the beams are in-phase, but that their wavefronts are completely uncorrelated, such that the average piston phase

N^{- 1} \sum δ φ_{n} (r) \approx 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 = 1 - \frac{1}{N} \sum \frac{1}{P} \int A {(r)}^{2} δ φ_{n} {(r)}^{2} d r .

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,

σ_{W F E, n}^{2}

:

σ_{W F E, n}^{2} = \frac{1}{P} \int A {(r)}^{2} δ ϕ_{n} {(r)}^{2} d r .

The combining efficiency then becomes

η = 1 - \bar{σ_{W F E}^{2}}

, where

\bar{σ_{W F E}^{2}} = N^{- 1} \sum σ_{W F E, n}^{2}

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 (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 $η = 1 - 〈 Δ ω {(t)}^{2} 〉 σ_{τ}^{2}$ .

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

A (x) = {(2 / π)}^{1 / 4} \sqrt{P / w} \exp [- x^{2} / w^{2}],

such that

\int A {(x)}^{2} d x = 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 (η_BC = 1 and D_n = 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.

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5.1 Beam positioning errors

With small displacement errors δx_n of the n^th channel’s near field position from the array average (defined to be x = 0), the field of the n ^th beam is

A_{n} (x) = {(2 / π)}^{1 / 4} \sqrt{P / w} \exp [- {(x - δ x_{n})}^{2} / w^{2}] .

We can identify the amplitude perturbation:

δ A_{n} (x) = \frac{d A_{n} (x)}{d x} δ x_{n} = - 2 {(2 / π)}^{1 / 4} \sqrt{P / w} \frac{(x - δ x_{n})}{w^{2}} \exp [- {(x - δ x_{n})}^{2} / w^{2}] δ x_{n} .

Hence, the amplitude variance can be written in terms of the beam position variance (neglecting perturbations above second order):

σ_{A}^{2} = \sqrt{2 / π} (4 P / w^{3}) x^{2} \exp [- 2 x^{2} / w^{2}] σ_{x}^{2} / w^{2} .

The combining loss from Eq. (16) is then

\begin{matrix} η = η_{B C} (1 - \frac{1}{P} \int σ_{A}^{2} d x) = η_{B C} (1 - \sqrt{\frac{2}{π}} \frac{4 σ_{x}^{2}}{w^{5}} \int x^{2} \exp [- 2 x^{2} / w^{2}] d x) \\ = η_{B C} (1 - σ_{x}^{2} / w^{2}) . \end{matrix}

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_BC[1 – 2ln(2)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

A_{n} (x) = {(\frac{2}{π})}^{1 / 4} \sqrt{\frac{P}{w + δ w_{n}}} \exp [- x^{2} / {(w + δ w_{n})}^{2}] .

Using the same approach as for beam displacement errors, we find η = η_BC(1 – 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:

A (θ) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} A (x) e^{- i 2 π x θ / λ} d x = {(\frac{2}{π})}^{1 / 4} \sqrt{\frac{P w}{2}} \exp (- θ^{2} / θ_{0}^{2}),

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 = η_BC[1 – σ_θ ²/θ ₀ ²] for array RMS pointing error σ_θ. Written in terms of the FWHM far field divergence Θ = [2ln(2)]^1/2 θ ₀, the efficiency is η = η_BC[1 – 2ln(2)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 Δω_FWHM by <Δω(t)²> = Δω_FWHM ²/[8ln(2)]. In units of Hertz, the RMS frequency spead is Δf_FWHM = Δω_FWHM/2π. Hence <Δω(t)²> = π²Δf_FWHM ²/[2ln(2)], and the loss term in Eq. (16) due to path delay variance σ_τ ² is

η = η_{B C} [1 - 〈 Δ ω {(t)}^{2} 〉 σ_{τ}^{2}] = η_{B C} [1 - \frac{π^{2}}{2 \ln (2)} Δ f_{F W H M}^{2} σ_{τ}^{2}] .

Equation (34) has been shown to agree quantitatively with measurements of fringe visibility (which provide a surrogate for combining efficiency [22]) for a kW fiber amplifier with 21 GHz linewidth [10].

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

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Perturbative analysis of coherent combining efficiency with mismatched lasers

Abstract

Corrections

1. Introduction

2. Coherent combining efficiency

3. Perturbative analysis of combining efficiency