Coherent combining efficiency in tiled and filled aperture approaches

V. E. Leshchenko

doi:10.1364/OE.23.015944

1. Introduction

In the last decade the sufficient increase of the interest to the coherent combining technique is observed. The interest is stimulated by nearly achieving technological limits of different single channel amplification schemes caused by various thermo-optical and nonlinear effects and damage threshold. Therefore the development of new approaches to the creation of high power, high energy and high intensity laser facilities is required, and coherent beam combining is the most promising one. Thereby the projects aimed at the development of fundamental high field physics including nonlinear quantum electrodynamics and the achievement of near Exawatt peak power and about 10²⁵W/cm² peak intensity [1, 2] are based on coherent beam combining.

To date the coherent combining of continuous wave (CW) [3, 4] and pulsed [5, 6] radiations amplified in bulk, fiber and semiconductor amplifies is realized. Both chirped-pulse amplifiers (CPAs) [5, 6] and optical parametric chirped-pulse amplifiers (OPCPAs) [7–9] are coherently combined. The most impressive results have been obtained in CW regime. A 100-kW Nd:YAG CW solid-state laser system was developed by combining of seven 15 kW lasers [3]. Moreover the record 64 number of fiber amplified beams (but with low 0.06 mW power) [4] and 218 semiconductor amplifiers [10] were coherently combined. In pulsed regime achievements are more modest which caused by the larger number of parameters to be unified. In pulsed regime the largest number of combined beams is 4 [5, 6], the highest pulse energy is 100 mJ (peak power is 4 TW) [8], the highest peak intensity is 10²⁰W/cm² [9], and the shortest pulse duration is 23 fs [8, 9]. Note that about 10 channels will be enough [1, 2, 9] to achieve ultra-relativistic intensity (>10²³W/cm²) in the facilities based on bulk amplifiers. Whereas in fiber systems it will require combination of about 10⁴ fiber amplifiers [11] but currently only 4 femtosecond fiber amplifiers have been coherently combined [5, 6]. Therefore the considerable increase of the peak power and the peak intensity of individual beams and its number as well as the development of the techniques to stabilize and combine such unprecedented number of amplifiers (in the case of fiber facilities) are required. Thus coherent combining of powerful femtosecond pulses is only in the very beginning currently and it is important to obtain analytical expressions and to determine major parameters and effects which drastically affect combining efficiency with the scaling of intensity and power as well as the number of combining beams. Such an analysis provides guidelines for the designing of laser systems based on coherent beam combining and enables the extrapolation of obtained experimental results.

2. Approaches to coherent beam combining and combining efficiency definitions

There are two general coherent combining techniques. Different combining elements are used in different methods which cause the use of slightly or rather different definitions of coherent combining efficiency. This discrepancy is rather important and need to be clarified. The first and the most widely used approach is the so called filled aperture combining (see Fig. 1(a)) according to which beam splitters or polarization beam splitters are used. The second approach is the tiled aperture combining (see Fig. 1(b)) according to which all beams are placed side by side and focused (generally by a parabolic mirror) to form high-intensity combined beam in the focal plane.

Fig. 1 General schemes of filled (a) and tiled (b) aperture combining.

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In experiments on filled aperture combining the efficiency (η) is generally defined as the ratio of the energy (W_Σ) or the power in the combined beam to the sum of the energies (powers) of the beams before combining

η_{f i l l e d}^{W} = \frac{W_{Σ}}{W_{Σ}^{\max}} = \frac{W_{Σ}}{\sum_{n = 1}^{N} W_{n}},

where W_n is the energy in the n-th channel and N is the number of combining beams. Therefore the efficiency decreases due to losses on combiners which appear because of not absolute destructive interference in the secondary arms. For filled aperture combining there is one more quantity to characterize the performance of the coherent beam combining. It is so called figure of merit (FOM) [12] which is defined as

F O M = \frac{W_{\sum} - \sum_{n = 1}^{N} W_{secondary n}}{W_{\sum} + \sum_{n = 1}^{N} W_{secondary n}},

and it is related to η^W according to

F O M = 2 η_{f i l l e d}^{W} - 1.

This is because assuming no absorption in combining elements:

\sum_{n = 1}^{N} W_{n} = W_{\sum} + \sum_{n = 1}^{N} W_{secondary n} .

In tiled aperture combining there are intrinsically almost no losses in combiners because there are no secondary arms. This causes η^W always to be 1 (in real situation this efficiency is slightly less than unity because of some losses due to nonperfect reflectivity of optical elements). Therefore a different definition of the coherent combining efficiency is required. In tiled aperture combining the main goal generally is to increase peak intensity so the natural value to estimate quality of combining is the ratio of the focused peak intensity of the combined beam (I_Σ) to the maximum value (I_Σ^max which generally equals to $I_{Σ}^{\max} = {(\sum_{n = 1}^{N} \sqrt{I_{n}})}^{2}$ ) achievable under the combining of perfectly coherent beams:

η_{}^{I} = \frac{I_{Σ}}{I_{Σ}^{\max}} .

However direct registration of peak intensity is a challengeable experimental task [13]. Generally focused peak fluency (F_Σ) is registered which is simply done by CCD-cameras. The efficiency in terms of peak fluency (η^F) is defined as

η_{}^{F} = \frac{F_{Σ}}{F_{Σ}^{\max}} .

where F_Σ^max is the maximum value of the focused peak fluency of the combined beam achievable under combining of perfectly coherent beams and generally equaling to

F_{Σ}^{\max} = {(\sum_{n = 1}^{N} \sqrt{F_{n}})}^{2}

. Also the interference figure contrast (IFC) is used to determine the combining performance in tiled aperture combining:

I F C = \frac{F_{\max} - F_{\min}}{F_{\max} + F_{\min}},

where F_max and F_min are the maximum fluency and the adjacent minimum in the combined beam pattern, respectively. IFC is similar to FOM and has the same relation with efficiency:

I F C = 2 η_{}^{F} - 1.

Note that η^F for filled aperture combining (η^F_filled) coincides with η^F for tiled aperture combining (η^F_tiled) and the same situation is for η^I_filled and η^I_tiled. As will be discussed in details later (section 3.1) there is some difference between η^F_filled and η^F_tiled, and η^I_filled and η^I_tiled only under the presence of large phase misalignment and for small number of channels (N<10).

Therefore the most versatile value to characterize combining performance is the efficiency in terms of focused peak fluency (η^F). However η^W or FOM inevitably will be used in experiments on filled aperture combining so it is important to find a correspondence between η^F, η^W and η^I. Also pulse stabilization techniques are generally similar or even identical in different coherent combining techniques. There are a plenty of phase and delay stabilization techniques such as heterodyne technique [5, 7], optical cross-correlators [14], LOCSET and synchronous multi-dither LOCSET [6, 15], SPGD [16], single frequency dithering technique [17, 18] and some others, but their direct comparison is difficult when different definitions of coherent combining efficiency are used. However in the view of the necessity of the development of stabilization systems one should have the ability to compare results obtained in all approaches. Thus some unification and relation between terms used in different approaches is required which will be one of the subjects of the paper.

It should be noted that the emphasis will be placed on the coherent combining of femtosecond pulses in this paper, which are required to achieve ultra-high intensity. However, of course, all obtained results can be simply applied to longer pulses and CW radiation by inserting corresponding pulse duration, wavelength and other parameters required for an equation.

3. Influence of different instabilities and mismatches on coherent combining efficiency

3.1 Optical path mismatch and phase difference

According to experimental data [5–9], phase and time overlap of combining beams are the most critical effects for the coherent beam combining so the analysis of coherent combining efficiency is begun with these parameters. In the time domain the laser electric field in n-th channel assuming Gaussian spectrum can be expressed as

E (t, δ T_{n}, δ φ_{n}) = \sqrt{I_{0}} \exp [- 2 l n (2) \frac{{(t - δ T_{n})}^{2}}{τ^{2}} - i ω_{0} t + i δ φ_{n}],

and equivalently in the spectral domain

E (ω, δ T_{n}, δ φ_{n}) = \sqrt{I_{0}} \frac{τ_{}}{\sqrt{4 l n (2)}} \exp [- \frac{{(ω - ω_{0})}^{2} τ_{}^{2}}{8 l n (2)} + i δ T_{n} (ω - ω_{0}) + i δ φ_{n}],

where I₀ is the peak intensity, δT is the time delay related to optical path length error according to δL = cδT, τ is the full width at a half maximum (FWHM) pulse duration, ω₀ is the carrier frequency, δφ_n = δφ_{ceo_n} + ω₀δT_n is the full phase, δφ_ceo is the carrier-envelope offset phase.

Assuming δT_n and δφ_n to be independent standard normal random variables with mean 0, averaged on δT and δφ peak intensity of a combined beam is determined by

\begin{array}{l} {\bar{I}}_{Σ}^{p e a k} (N, σ_{T}^{}, σ_{φ}^{}) = \int {| \sum_{n = 1}^{N} E (0, δ T_{n}, δ φ_{n}) |}^{2} \frac{e^{- \frac{\sum_{n = 1}^{N} δ T_{n}^{2}}{2 σ_{T}^{2}}}}{{(\sqrt{2 π} σ_{T}^{})}^{N}} d δ T_{1} \times ... \times d δ T_{N} \frac{e^{- \frac{\sum_{n = 1}^{N} δ φ_{n}^{2}}{2 σ_{φ}^{2}}}}{{(\sqrt{2 π} σ_{φ}^{})}^{N}} d δ φ_{1} \times ... \times d δ φ_{N} = \\ = I_{0} (\frac{N}{\sqrt{1 + \frac{8 \ln (2) σ_{T}^{2}}{τ^{2}}}} + \frac{N (N - 1)}{1 + \frac{4 \ln (2) σ_{T}^{2}}{τ^{2}}} \exp [- σ_{φ}^{2}]) . \end{array}

Here and further σ_X is the root mean squared (rms) instability of parameter “X”. Taking into account the relation between delay and path length errors: σ_L = cσ_T, Eq. (11) becomes

{\bar{I}}_{Σ}^{p e a k} (N, σ_{L}^{}, σ_{φ}^{}) = I_{0} (\frac{N}{\sqrt{1 + \frac{8 \ln (2) σ_{L}^{2}}{c^{2} τ^{2}}}} + \frac{N (N - 1)}{1 + \frac{4 \ln (2) σ_{L}^{2}}{c^{2} τ^{2}}} \exp [- σ_{φ}^{2}]) .

In the above expression the maximum of the combined intensity to be at t = 0 is assumed because of zero mean value of δT. The validity of the assumption will be examined later during comparison with exact numerical results. The maximum value of peak intensity achievable under the absence of distortions (when σ_L = 0 and σ_φ = 0) equals to

I_{Σ_\max}^{p e a k} = N^{2} I_{0} .

According to Eq. (5) efficiency of coherent combining in terms of peak intensity is

η_{}^{I} (N, σ_{L}^{}, σ_{φ}^{}) = \frac{1}{N^{2}} (\frac{N}{\sqrt{1 + \frac{8 \ln (2) σ_{L}^{2}}{c^{2} τ^{2}}}} + \frac{N (N - 1)}{1 + \frac{4 \ln (2) σ_{L}^{2}}{c^{2} τ^{2}}} \exp [- σ_{φ}^{2}]) .

The dependence of the combining efficiency on optical path mismatch and phase difference given by Eq. (14) is presented in Fig. 2. The white and black lines in Fig. 2(b-d) define the boundary where the efficiency is above 95% and 90%, respectively. As follow from Eq. (14) and clearly seen in Fig. 2(a), for small number of channels the first term in Eq. (14) is rather important and results in higher combining efficiency for smaller N especially on large phase instabilities. For N>~10 there is almost no efficiency changes with further increase of N because the impact of the first term decreases as 1/N.

Fig. 2 Analytically calculated coherent combining efficiency in terms of peak intensity (η^I): (a) as a function of only phase instability (σ_φ); (b) of only optical path lengths mismatch (σ_L); and of both parameters for 2 (c), 8 (d) and 100 (e) channels. The white and black lines define η = 0.95 and η = 0.9, respectively.

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The useful limit of Eq. (14) for the large number of channels and small instabilities is

η_{N > > 1, \frac{σ_{L}^{}}{c τ} < < 1, σ_{φ}^{} < < 1}^{I} (σ_{L}^{}, σ_{φ}^{}) = 1 - σ_{φ}^{2} - \frac{4 \ln (2) σ_{L}^{2}}{c^{2} τ^{2}} .

The above calculations have been performed under the assumption that the maximum of the combined pulse is at t = 0. Moreover spatial distribution has been neglected. Note that in filled aperture combining phase difference results in energy losses in secondary arms and there are no profile changes. Whereas in tiled aperture combining phase difference results in combined beam profile changes as shown in Fig. 3. Thus spatial distribution is important for the analysis of time and phase misalignments in only tiled aperture combining.

Fig. 3 The profile of a combined beam in tiled aperture combining of two beams with phase difference equal to 0 (a) and π (b).

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To determine the validity of the assumption and to obtain the exact solution, direct numerical computation was performed by Monte-Carlo technique. Averaged on realizations peak intensity of the combined pulse is

{\bar{I}}_{Σ}^{p e a k} = \frac{1}{M} \sum_{m = 1}^{M} M a x [{| \sum_{n = 1}^{N} E (x, t, δ T_{n}^{m}, δ φ_{n}^{m}) |}^{2}],

M is adjusted so that the computation error to be negligible (<0.5%). Exact numerical value of η^I_tiled obtained under the assumption of the normal distribution of variations and Gaussian profiles of the combining beams is shown in Fig. 4. Note that η^I_filled dependence on phase coincides with Fig. 2(a) and on optical path mismatch with Fig. 4(b). Comparing exact numerical results presented in Fig. 4 and analytical results shown in Fig. 2, one can find that there is almost no difference between them for more than 8 channels whereas for smaller number of channels there is a slight discrepancy especially on large instabilities. The difference between analytical and numerical results achieves about 15% for 2-4 channels and 1.5 rad phase mismatch, whereas for more than 8 channels the discrepancy don’t exceed 2% on small (σ_φ<0.5rad, σ_L/cτ<0.5) and 5% on large misalignments. Thus the assumption of t = 0 used in Eq. (11) is valid for the large number of channels (N>8) and for N<8 but small misalignments (σ_φ <0.5rad, σ_L/cτ <0.5).

Fig. 4 Exact (numerically calculated) coherent combining efficiency in terms of peak intensity (η^I): (a) as a function of only phase instability (σ_φ); (b) of only optical path lengths mismatch (σ_L); and of both parameters for 2 (c), 8 (d) and 100 (e) channels. The white and black lines define η = 0.95 and η = 0.9, respectively.

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To obtain efficiency in terms of peak fluency and energy (for filed aperture combining), the energy of a combined pulse (W_Σ) which coincides in the discussed case with fluency was calculated using Eq. (9):

\begin{array}{l} W_{Σ}^{} (N, δ T_{1}, ..., δ T_{N}, δ φ_{1}, ..., δ φ_{N}) \propto F_{Σ}^{} (x = 0) = \int_{- \infty}^{\infty} \sum_{n = 1}^{N} \sum_{m = 1}^{N} E (t, δ T_{n}, δ φ_{n}) E^{*} (t, δ T_{m}, δ φ_{m}) d t = \\ = I_{0} \sqrt{\frac{π}{4 l n (2)}} τ \sum_{n = 1}^{N} \sum_{m = 1}^{N} \exp [i (δ φ_{n} - δ φ_{m})] \exp [- 4 l n (2) \frac{{(δ T_{n} - δ T_{m})}^{2}}{4 τ^{2}}] . \end{array}

The maximum value of W_Σ obtained under the absence of distortions is

W_{Σ}^{\max} = N^{2} I_{0} \sqrt{\frac{π}{4 l n (2)}} τ .

The averaged over δT and δφ value of W_Σ is

\begin{array}{l} {\bar{W}}_{Σ}^{} (N, σ_{T}^{}, σ_{φ}^{}) = \int W_{Σ} (δ T_{1}, ..., δ T_{N}, δ φ_{1}, ..., δ φ_{N}) \frac{e^{- \frac{\sum_{n = 1}^{N} δ T_{n}^{2}}{2 σ_{t}^{2}}}}{{(\sqrt{2 π} σ_{T}^{})}^{N}} d δ T_{1} \times ... \times d δ T_{N} \times \\ \times \frac{e^{- \frac{\sum_{n = 1}^{N} δ φ_{n}^{2}}{2 σ_{φ}^{2}}}}{{(\sqrt{2 π} σ_{φ}^{})}^{N}} d δ φ_{1} \times ... \times d δ φ_{N} = I_{0} \sqrt{\frac{π}{4 l n (2)}} τ_{} (N + \frac{N (N - 1)}{\sqrt{1 + 4 l n (2) \frac{σ_{T}^{2}}{τ_{}^{2}}}} \exp [- σ_{φ}^{2}]) \end{array}

Taking into account that σ_L = cσ_T and according to Eq. (1) coherent combing efficiency is

η_{f i l l e d}^{W} (N, σ_{L}^{}, σ_{φ}^{}) = η_{}^{F} (N, σ_{L}^{}, σ_{φ}^{}) = \frac{1}{N^{2}} (N + \frac{N (N - 1)}{\sqrt{1 + 4 l n (2) \frac{σ_{L}^{2}}{c^{2} τ_{0}^{2}}}} \exp [- σ_{φ}^{2}]) .

Examples of Eq. (20) are presented in Fig. 5. Comparing these results with the efficiency in terms of peak intensity shown in Fig. 4, one can see that under only phase instability (see Fig. 4(a) and 5(a)) they coincide for the large number of channels and η^I is slightly higher η^W for small number of channels. Whereas under optical path mismatch (see Fig. 4(b) and 5(b)) η^I is lower η^W.

Fig. 5 Coherent combining efficiency in terms of peak fluency (η^F) and energy (η^W): (a) as a function of only phase instability (σ_φ), (b) of only optical path lengths mismatch (σ_L), and of both parameters for 2 (c), 8 (d) and 100 (e) channels. The white and black lines define η = 0.95 and η = 0.9, respectively.

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Spatial structure of the combined pulse important for tiled aperture combining has not been taken into account in Eq. (20). Taking into consideration this aspect, the efficiency will depend on phase misalignment the same way as in the case of peak intensity (see Fig. 4(a)). Whereas the dependence on optical path mismatch coincides with analytically calculated one presented in Fig. 5(b).

With the large number of channels and small instabilities Eq. (20) reduces to

η_{f i l l e d}^{W}_{N > > 1, \frac{σ_{L}^{}}{c τ_{0}} < < 1, σ_{φ}^{} < < 1}^{} (σ_{L}^{}, σ_{φ}^{}) = η_{N > > 1, \frac{σ_{L}^{}}{c τ_{0}} < < 1, σ_{φ}^{} < < 1}^{F} (σ_{L}^{}, σ_{φ}^{}) = 1 - σ_{φ}^{2} - \frac{2 \ln (2) σ_{L}^{2}}{c^{2} τ_{0}^{2}} .

Note that the efficiency limit for large number of channels and small instabilities (Eq. (21)) coincides with the result obtained in [19] (taking into account that Δf_FWHM = 2ln(2)/(πτ₀)).

As can be seen from the obtained results, the shorter pulse the higher requirements on the accuracy of the optical path length adjustment. For example to combine 10 fs pulses with 95% efficiency (η^F) path length mismatch should be less than 0.6 μm, whereas for 5 fs pulses it should be less than 0.3 μm.

Generally only the difference of full phases is stabilized by slight adjustment of optical path lengths but carrier-envelope offset (ceo) phases remains unstabilized. So requirements on the ceo-phase difference stability under stabilized full phase difference should be analyzed. Assuming δφ to be perfectly stabilized so that δφ_n = 0 = δφ_{ceo_n} + ω₀δL_n/c, the ceo-phase error translates into the path length error δL_n__ceo = cδφ_{ceo_n}/ω₀ and σ_L__ceo = cσ_ceo /ω₀. Therefore in the above equations (under stabilized full phase difference) optical path length variation should be replaced with

σ_{L}^{2} \to σ_{L}^{2} + σ_{L_c e o}^{2} .

To combine 10 fs pulses (800 nm central wavelength) with 95% efficiency (η^F) the instability of the ceo-phase should be less than 4.7 rad and for longer pulses requirements are even lower. Whereas in for example OPCPA systems the value of the instability of the ceo-phase induced by amplification and compression generally does not exceed 1 rad [20] even without active stabilization. Thus under stabilized differences of full phases requirements on the stability of the difference of ceo-phases are not high and generally can be satisfied without active stabilization. However the obtaining of not only high combining efficiency but also stable zero ceo-phase of the combined pulse is the most interesting from experimental point of view [21]. A principal scheme of a stabilization setup which can be used to obtain zero ceo-phase in the combined beam is presented in Fig. 6. Optical path length errors are measured by optical balanced cross-correlators and stabilized by adjusting channel lengths by for example piezoelectric actuator delay lines. In the reference channel ceo-phase is stabilized and adjusted to absolute zero value through the detection system based on for example above threshold ionization (ATI) [21]. Difference of full phases is measured through one of the numerous techniques [5–7, 14–18] and stabilized by ceo-phase adjustment in slave channels.

Fig. 6 Principal scheme of the stabilization system to achieve zero ceo-phase in the combined beam.

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3.2 Dispersion compensation accuracy

Assuming Gaussian pulse spectrum and the presence of only second order dispersion peak intensity in the n-th channel with δk_2n dispersion compensation error equals to

I^{p e a k} (δ k_{2_n}) = \frac{I_{0}}{\sqrt{1 + {(4 \ln (2) \frac{δ k_{2_n}}{τ_{0}^{2}})}^{2}}},

where I₀ is the peak intensity of a transform limited pulse, τ₀ is the FWHM transform limited pulse duration. Assuming δk_2n to be independent normally distributed random variables with 0 means, the peak intensity of a combined pulse is

\begin{array}{l} {\bar{I}}_{Σ}^{p e a k} (N, σ_{k_{2}}^{}) = \int {| \sum_{n = 1}^{N} \sqrt{I^{p e a k} (δ k_{2_n})} |}^{2} \frac{e^{- \frac{\sum_{n = 1}^{N} δ k_{2_n}^{2}}{2 σ_{k_{2}}^{2}}}}{{(\sqrt{2 π} σ_{k_{2}}^{})}^{N}} d δ k_{2_1} \times ... \times d δ k_{2_N} = \\ = I_{0} (\begin{array}{l} N \frac{τ_{0}^{2}}{\sqrt{2 π} 4 \ln (2) σ_{k_{2}}^{}} \exp [\frac{τ_{0}^{4}}{64 \ln^{2} (2) σ_{k_{2}}^{2}}] K_{0} [\frac{τ_{0}^{4}}{64 \ln^{2} (2) σ_{k_{2}}^{2}}] + \\ + N (N - 1) \frac{τ_{0}^{4}}{32 \ln^{2} (2) σ_{k_{2}}^{2}} U^{2} [\frac{1}{2}, \frac{5}{4}, \frac{τ_{0}^{4}}{32 \ln^{2} (2) σ_{k_{2}}^{2}}] \end{array}), \end{array}

where K₀ is the modified Bessel function of the second kind (Macdonald function), U is the confluent hypergeometric function of the second kind with the integral representation

U [a, b, z] = \frac{1}{Γ (a)} \int_{0}^{\infty} e^{- z t} t^{a - 1} {(1 + t)}^{b - a - 1} d t .

According to Eq. (5) the efficiency in terms of peak intensity equals to

η_{}^{I} (N, σ_{k_{2}}^{}) = \frac{1}{N^{2}} (\begin{array}{l} N \frac{τ_{0}^{2}}{\sqrt{2 π} 4 \ln (2) σ_{k_{2}}^{}} \exp [\frac{τ_{0}^{4}}{64 \ln^{2} (2) σ_{k_{2}}^{2}}] K_{0} [\frac{τ_{0}^{4}}{64 \ln^{2} (2) σ_{k_{2}}^{2}}] + \\ + N (N - 1) \frac{τ_{0}^{4}}{32 \ln^{2} (2) σ_{k_{2}}^{2}} U^{2} [\frac{1}{2}, \frac{5}{4}, \frac{τ_{0}^{4}}{32 \ln^{2} (2) σ_{k_{2}}^{2}}] \end{array}) .

Examples of Eq. (26) are depicted in Fig. 7(a, blue lines). According to Eq. (26) and Fig. 7(a, blue lines) to achieve 95% efficiency in terms of peak intensity the accuracy of the second order dispersion compensation should not be worse than 0.125 × τ₀².

Fig. 7 The dependence of the coherent combining efficiency in terms of peak intensity (η^I, blue lines), and in terms of peak fluency and energy (η^F and η^W, green lines) on dispersion compensation accuracy for second (a), third (b), fourth (c) and fifth (d) dispersion orders. (blue lines in (a) are analytical results (Eq. (26), the other results are obtained numerically)

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With the large number of channels and small instabilities Eq. (26) reduces to

η_{N > > 1, \frac{σ_{k_{2}}^{}}{τ_{0}^{2}} < < 1}^{I} (σ_{k_{2}}^{}) = 1 - \frac{1}{2} {(\frac{4 \ln (2) σ_{k_{2}}^{}}{τ_{0}^{2}})}^{2} .

The influence of higher order dispersions proved to be difficult to calculate analytically so it is calculated numerically. Numerical computation is performed by Monte-Carlo technique and similar to the one describe in the previous section. The obtained η^I dependence on third, fourth and fifth orders dispersion compensation accuracy is shown in Fig. 7 (blue lines).

Then the efficiency in terms of peak fluency and energy is discussed. In this case the energy of a sum pulse is proportional to the peak fluency:

\begin{array}{l} W_{Σ}^{} \propto F_{Σ}^{} (x = 0) = \int_{- \infty}^{\infty} I_{Σ}^{} (t) d t = \int_{- \infty}^{\infty} I_{Σ}^{} (ω) d ω = \\ = I_{0} {(\frac{τ_{0}}{\sqrt{4 l n (2)}})}^{2} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \int_{- \infty}^{\infty} \exp [- \frac{{(ω - ω_{0})}^{2} τ_{0}^{2}}{4 l n (2)} + i (δ Φ_{n} [ω - ω_{0}] - δ Φ_{m} [ω - ω_{0}])] d ω, \end{array}

where

δ Φ_{n} [ω - ω_{0}] = δ k_{2_n} \frac{{(ω - ω_{0})}^{2}}{2} + δ k_{3_n} \frac{{(ω - ω_{0})}^{3}}{3!} + ...

is the spectral phase in the n-th channel. The averaged combined pulse energy is

{\bar{W}}_{Σ}^{} = \int W_{Σ} \frac{e^{- \frac{\sum_{n = 1}^{N} δ Φ_{n}^{2}}{2 σ_{Φ}^{2}}}}{{(\sqrt{2 π} σ_{Φ}^{})}^{N}} d δ Φ_{1} \times ... \times d δ Φ_{N},

where dδΦ_n = dδk_{2_n} × dδk_{3_n} × … . For example allowing only δk₂ errors Eq. (29) is

\begin{array}{l} {\bar{W}}_{Σ}^{} = I_{0} {(\frac{τ_{0}}{\sqrt{4 l n (2)}})}^{2} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \int \int \int_{- \infty}^{\infty} \exp [- \frac{{(ω - ω_{0})}^{2} τ_{0}^{2}}{4 l n (2)} + i (δ k_{2_n} - δ k_{2_m}) \frac{{(ω - ω_{0})}^{2}}{2}] \times \\ \times \frac{e^{- \frac{δ k_{2_n}^{2} + δ k_{2_m}^{2}}{2 σ_{k 2}^{2}}}}{{(\sqrt{2 π} σ_{k 2}^{})}^{2}} d ω \times d δ k_{2_n} \times d δ k_{2_m} . \end{array}

It should be noted that only differences of dispersions (δk_{2_n}-δk_{2_m}) but not their absolute values are important according to Eq. (30). According to Eq. (1) and (6) the coherent combining efficiency in terms of energy and peak fluency is

η_{f i l l e d}^{W} = η_{}^{F} = \frac{{\bar{W}}_{Σ}^{}}{N^{2} I_{0} {(\frac{\sqrt{π} τ_{0}}{\sqrt{4 l n (2)}})}^{}} .

The efficiency given by Eq. (31) was calculated numerically, the obtained results for second, third, fourth and fifth order dispersions are presented in Fig. 7 (green lines).

Note that according to the obtained results presented in Fig. 8, requirements on dispersion compensation increase as (τ₁/τ₂)^k with pulse duration decrease from τ₁ to τ₂, where k is the dispersion order. For example to realize η^I = 0.95 for τ₀ = 20 fs, σ_k2 should be less than 50 fs², σ_k3 less than 2240 fs³ and σ_k4 less than 18600 fs⁴, whereas for τ₀ = 10 fs it should be σ_k2<12.5 fs², σ_k3<280 fs³, σ_k4<1160 fs⁴.

Fig. 8 The dependence of the coherent combining efficiency in terms of peak intensity (η^I, blue lines), and in terms of peak fluency and energy (η^F and η^W, green lines) on dispersion compensation accuracy for second (a), third (b) and fourth (c) dispersion orders.

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3.3 Pointing instability

To examine the influence of pointing instability, electrical field in the far field can be expressed as

E (ψ, δ ψ_{n}) = \sqrt{I_{0}} \exp [- 2 \ln (2) \frac{{(ψ - δ ψ_{n})}^{2}}{Ψ^{2}} + i α_{n} ψ],

where Ψ is the FWHM beam divergence, α_n is the phase tilt due to the displacement of a beam (dX) from the axis of a focusing element: α_n = kdX, and Gaussian beam profile is assumed. Intensity of the combined pulse is

I_{Σ}^{} (N, ψ, δ ψ_{1}, ..., δ ψ_{N}) \propto {| \sum_{n = 1}^{N} E (ψ, δ ψ_{n}) |}^{2} .

Assuming pointing errors to be independent normally distributed random variables with 0 means, the averaged peak intensity of a combined pulse equals to

{\bar{I}}_{Σ}^{p e a k} (N, σ_{ψ}^{}) = \int I_{Σ}^{} (N, 0, δ ψ_{1}, ..., δ ψ_{N}) \frac{e^{- \frac{\sum_{n = 1}^{N} δ ψ_{n}^{2}}{2 σ_{ψ}^{2}}}}{{(\sqrt{2 π} σ_{ψ}^{})}^{N}} d δ ψ_{1} \times ... \times d δ ψ_{N},

where the maximum of the intensity pattern of the combined pulse to be at ψ = 0 is assumed. The validity of this assumption will be examined later during comparison with exact numerical results. In the discussed case peak fluency within a constant factor coincides with peak intensity

{\bar{F}}_{Σ}^{p e a k} (N, σ_{ψ}^{}) = \sqrt{\frac{π}{4 \ln (2) τ^{2}}} {\bar{I}}_{Σ}^{p e a k} (N, σ_{ψ}^{}) .

Therefore efficiencies in terms of peak intensity and peak fluency coincide: η^I(σ_ψ) = η^F(σ_ψ) under the presence of pointing instability. Substituting Eq. (32) and (33) into Eq. (34), integrating and dividing it by I₀N² and taking into account that for a Gaussian pulse Ψ = 4ln(2)/(k₀D₀) where D₀ is the FWHM beam diameter on a focusing mirror, the efficiency is

η_{}^{I} (N, σ_{ψ}^{}) = η_{}^{F} (N, σ_{ψ}^{}) = \frac{1}{N^{2}} (\frac{N}{{(1 + \frac{{(k_{0} D_{0} σ_{ψ}^{})}^{2}}{2 \ln (2)})}^{1 / 2}} + \frac{N (N - 1)}{(1 + \frac{{(k_{0} D_{0} σ_{ψ}^{})}^{2}}{4 \ln (2)})}) .

Analytically calculated (Eq. (36)) efficiency dependence on pointing instability is shown in Fig. 9(a, blue lines).

Fig. 9 The dependence of the coherent combining efficiency on beam pointing instability. (a) In terms of peak intensity (η^I) and peak fluency (η^F). (b) In terms of energy (η^W).

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In the large number of channels and small instabilities limit combining efficiency (Eq. (36)) reduces to

η_{}^{I}_{N > > 1, k_{0} D_{0} σ_{ψ}^{} < < 1} (σ_{ψ}^{}) = η_{}^{F}_{N > > 1, k_{0} D_{0} σ_{ψ}^{} < < 1} (σ_{ψ}^{}) = 1 - \frac{{(k_{0} D_{0} σ_{ψ}^{})}^{2}}{4 \ln (2)} .

Above the assumption that the maximum of a combined pulse intensity distribution to be at ψ = 0 has been used. To determine its validity exact numerical (Monte Carlo) computation without this assumption was performed. The procedure is the same as used in the section 3.1. The comparison of approximate (analytical) and exact (numerical) results is presented in Fig. 9(a). It is clearly seen that for small beam pointing instability (σ_ψ) and any N and for N>8 and any σ_ψ there is very good agreement because difference doesn’t exceed 5%. For small number of channels (N<8) there is a discrepancy (up to 20%) between analytical and numerical results because of rather high possibility for the shift of the location of the peak on intensity pattern from ψ = 0 which results in the error in analytical computations for small number of channels. Thus the assumption of ψ = 0 in Eq. (34) is valid for small σ_ψ and any N, and for N>8 and any σ_ψ.

For other beam profiles solving the integral in Eq. (34) proves to be difficult so numerical calculation is required. Results for super-Gaussian profile ( $E (r) = \sqrt{I_{0}} \exp [- r^{4} / w^{4}]$ ) and top-hat profile ( $E (r) = \sqrt{I_{0}} П [r / w]$ , where П[a] is the rectangle function which is 1 for |a|<1 and 0 outside this range) are depicted in Fig. 10(a). From experimental point of view beam aperture is more important than beam diameter because it is the aperture which specifies the requirements on the size of optical elements. Therefore data in Fig. 10 are for the same diameter of the beam aperture of 10 mm. The diameter of the aperture in the paper is defined as the diameter of the cycle containing 99% of the pulse energy. As can be seen from the presented results the requirements for different beam profiles almost coincide.

Fig. 10 The dependence of the coherent combining efficiency on pointing instability for different beam profiles with the aperture diameter of 10 mm, N = 100. (a) In terms of peak intensity (η^I) and peak fluency (η^F). (b) In terms of energy (η^W).

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Further filled aperture combining efficiency in terms of energy is discussed. For filled aperture combining pointing errors result in phase difference changes across combiners and consequently in energy losses. In this case the electric field (for the n-th channel at the output of the final combiner) can be expressed as

E (x, δ ψ_{n}) = \sqrt{I_{0}} \exp [- \frac{2 \ln (2) x^{2}}{D_{0}^{2}} + i k_{0} x δ ψ_{n}],

where Gaussian beam profile is assumed. The energy of the combined pulse is

W_{Σ}^{} (N, δ ψ_{1}, ..., δ ψ_{N}) \propto \int_{- \infty}^{\infty} {| \sum_{n = 1}^{N} E (x, δ ψ_{n}) |}^{2} d x = I_{0} \sqrt{\frac{π}{4 l n (2)}} D_{0}^{} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \exp [- \frac{D_{0}^{2} k_{0}^{2}}{4 \ln (2)} \frac{{(δ ψ_{n} - δ ψ_{m})}^{2}}{4}] .

Assuming δψ_n to be independent normally distributed random variables with 0 means, averaged over δψ combined pulse energy is

\begin{array}{l} {\bar{W}}_{Σ}^{} (N, σ_{ψ}^{}) = \int W_{Σ} (δ ψ_{1}, ..., δ ψ_{N}) \frac{e^{- \frac{\sum_{n = 1}^{N} δ ψ_{n}^{2}}{2 σ_{ψ}^{2}}}}{{(\sqrt{2 π} σ_{ψ}^{})}^{N}} d δ ψ_{1} \times ... \times d δ ψ_{N} = \\ = I_{0} \sqrt{\frac{π}{4 l n (2)}} D_{0}^{} (N + N (N - 1) \frac{1}{\sqrt{1 + \frac{{(k_{0} D_{0} σ_{ψ}^{})}^{2}}{4 \ln (2)}}}) . \end{array}

According to Eq. (6) and using Eq. (40) the efficiency of coherent combining for filled aperture combining in terms of energy is

η_{f i l l e d}^{W} (N, σ_{ψ}^{}) = \frac{1}{N^{2}} (N + N (N - 1) \frac{1}{\sqrt{1 + \frac{{(k_{0} D_{0} σ_{ψ}^{})}^{2}}{4 \ln (2)}}}) .

Examples of Eq. (41) are presented in Fig. 9(b). The useful limit of Eq. (41) for the large number of channels and small beam pointing instability is

η_{f i l l e d}^{W}_{N > > 1, k_{0} D_{0} σ_{ψ}^{} < < 1} (σ_{ψ}^{}) = 1 - \frac{{(k_{0} D_{0} σ_{ψ}^{})}^{2}}{8 \ln (2)} .

Note that the efficiency limit for large number of channels and small instabilities (Eq. (42)) coincides with the result obtained in [19] (taking into account that slightly different definitions are used: θ = Ψ = 4ln(2)/k₀D₀ and σ_θ = σ_ψ).

Numerical results for super-Gaussian and top-hat profiles obtained by numerical integration of Eq. (40-41) with correspondent E(x) are shown in Fig. 10(b). The results for different beam profiles almost coincide, in spite of larger differences comparing with the efficiency in terms of peak intensity.

The obtained dependence of the pointing stability required to achieve 95% efficiency on beam size is shown in Fig. 11. The results for different beam profiles almost coincide. As follows from Eq. (36) and (41) and clearly seen in Fig. 11, the larger beam diameter the higher requirements on pointing stability (smaller instability is required). Therefore beam pointing stability is one of the most critical parameters for peak power and intensity scaling because the higher peak power the larger beam size and consequently the smaller pointing instability is required.

Fig. 11 The dependence of the combining efficiency in terms of peak intensity and fluency (a), and energy (b) on beam aperture diameter for different beam profiles.

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3.4 Angular chirp

One another parameter important for the combining of femtosecond pulses is the angular chirp and the pulse front tilt directly related to the angular chirp [22]. Angular chirp results in peak intensity decrease due to the increase of the focal spot size and effective pulse duration. Assuming Gaussian spectral and space pulse shapes, both effects result in focused peak intensity to be [23]

I_{n}^{p e a k} (С а) = \frac{I_{0}}{1 + {(\frac{С а \times λ_{F W H M} π D_{0}}{2 \ln (2) λ_{0}})}^{2}} = \frac{I_{0}}{1 + {(\frac{δ θ_{n} k_{0} D_{0}}{4 \ln (2)})}^{2}} = \frac{I_{0}}{1 + {(\frac{δ θ_{n}}{Ψ})}^{2}},

where Ca = dθ/dλ = δθ/λ_FWHM is the angular chirp, δθ is the angular spread of spectral components on FWHM level, λ_FWHM is the FWHM spectral width. The peak intensity of a combined pulse is

I_{Σ} = {(\sum_{n = 1}^{N} \sqrt{I_{n}})}^{2} .

Assuming normal distribution of angular chirp errors (δθ_n) and calculating averaged over δθ peak intensity by the same way as in the previous sections, the efficiency of coherent beam combining in terms of peak intensity is

η_{}^{I} (N, σ_{θ}^{}) = \frac{1}{N^{2}} (\begin{array}{l} N \frac{4 π \ln (2)}{k_{0} D_{0} \sqrt{2 π} σ_{θ}^{}} \exp [\frac{1}{2} {(\frac{4 \ln (2)}{k_{0} D_{0} σ_{θ}^{}})}^{2}] (1 - E r f [\frac{4 \ln (2)}{\sqrt{2} k_{0} D_{0} σ_{θ}^{}}]) + \\ + N (N - 1) {(\frac{4 \ln (2)}{k_{0} D_{0} \sqrt{2 π} σ_{θ}^{}})}^{2} \exp [2 {(\frac{2 \ln (2)}{k_{0} D_{0} σ_{θ}^{}})}^{2}] K_{0}^{2} [{(\frac{2 \ln (2)}{k_{0} D_{0} σ_{θ}^{}})}^{2}] \end{array}),

where Erf is the error function. The dependence of the coherent combining efficiency in terms of peak intensity on angular chirp given by Eq. (45) is shown in Fig. 12(a). It follows from the obtained results that angular chirp should not exceed 0.24Ψ so as to achieve 95% combining efficiency in terms of peak intensity.

Fig. 12 The dependence of the coherent combining efficiency on angular chirp compensation accuracy (σ_θ). (a) In terms of peak intensity (η^I,). (b) In terms of peak fluency (η^F). (c) In terms of energy (η^W). (Ψ is the FWHM beam divergence which for a Gaussian beam equal to Ψ = 4ln(2)/(k₀D₀)).

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By taking the large number of channels and small instability limit of Eq. (45), it is simplified to

η^{I}_{N > > 1, k_{0} D_{0} σ_{θ}^{} < < 1} (σ_{θ}^{}) = 1 - {(\frac{k_{0} D_{0} σ_{θ}^{}}{4 \ln (2)})}^{2} .

Unlike the case of beam pointing in the case of angular chirp the efficiency in terms of peak fluency differs from the efficiency in terms of peak intensity. The electric field in the far field under the presence of angular chirp is [22, 23]

E (x = 0, t, δ θ_{n}) = \frac{\sqrt{I_{0}}}{\sqrt{1 + {(\frac{δ θ_{n} k_{0} D_{0}}{4 \ln (2)})}^{2}}} \exp [- \frac{2 \ln (2) t^{2}}{τ_{0} (1 + {(\frac{δ θ_{n} k_{0} D_{0}}{4 \ln (2)})}^{2})} - i ω_{0} t],

where δθ = Ca/λ_FWHM as it was defined earlier. Peak fluency is

F_{Σ}^{p e a k} (N, δ θ_{1}, ..., δ θ_{N}) \propto \int_{- \infty}^{\infty} {| \sum_{n = 1}^{N} E (0, t, δ θ_{n}) |}^{2} d t = \int_{- \infty}^{\infty} \sum_{n = 1}^{N} \sum_{m = 1}^{N} E (0, t, δ θ_{n}) E^{*} (0, t, δ θ_{m}) d t .

Peak fluency averaged over δθ is

{\bar{F}}_{Σ}^{p e a k} (N, σ_{θ}^{}) = \sum_{n = 1}^{N} \sum_{m = 1}^{N} ∭ E (0, t, δ θ_{n}) E^{*} (0, t, δ θ_{m}) \frac{e^{- \frac{δ θ_{n}^{2} + δ θ_{m}^{2}}{2 σ_{θ}^{2}}}}{{(\sqrt{2 π} σ_{θ}^{})}^{2}} d t d δ θ_{n} d δ θ_{m} .

This integral was calculated numerically. Obtained dependence of η^F on angular chirp is shown in Fig. 12(b). To achieve 95% combining efficiency in terms of peak fluency angular chirp should not exceed 0.34Ψ.

Further η^W_filled is considered. Under the presence of the angular chirp and assuming Gaussian time and space profiles, the electric field on a combining element equal to [22]

E (x, t, δ θ_{n}) = \sqrt{I_{0}} \exp [- 2 \ln (2) \frac{x^{2}}{D_{0}^{2}} - 2 \ln (2) \frac{{(t - \frac{x δ θ_{n} k_{0} τ}{4 \ln (2)})}^{2}}{τ^{2}} + i k_{0} x - i ω_{0} t] .

As can be seen from the second summand in the exponent there is the pulse front tilt which result in imperfect pulse overlap on combining elements and energy loses. The energy of the combined pulse is

W_{Σ}^{} (N, δ θ_{1}, ..., δ θ_{N}) \propto \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {| \sum_{n = 1}^{N} E (x, t, δ θ_{n}) |}^{2} d x d t .

Under the same assumptions as earlier the efficiency is computed to be

η_{f i l l e d}^{W} (N, σ_{θ}^{}) = \frac{1}{N^{2}} (N + N (N - 1) \frac{4 \ln {(2)}^{}}{\sqrt{π} k_{0} D_{0}^{} σ_{θ}^{}} e^{\frac{1}{2} {(\frac{4 \ln {(2)}^{}}{k_{0} D_{0}^{} σ_{θ}^{}})}^{2}} K_{0} (\frac{1}{2} {(\frac{4 \ln {(2)}^{}}{k_{0} D_{0}^{} σ_{θ}^{}})}^{2})) .

Examples of Eq. (52) are shown in Fig. 12(c). To achieve η^W = 0.95 angular chirp should not exceed 0.5Ψ. With the large number of channels and small angular chirp misalignments Eq. (52) reduces to

η_{f i l l e d}^{W}_{N > > 1, k_{0} D_{0} σ_{Δ θ}^{} < < 1} (σ_{θ}^{}) = 1 - {(\frac{k_{0} D_{0} σ_{θ}^{}}{8 \ln (2)})}^{2} .

According to the obtained equations, requirements on angular chirp compensation increase (smaller value of instability is required) with beam diameter increasing. For instance to achieve 95% efficiency (η^I) under D₀ = 10 mm σ_θ should be less than 8.4 µrad. Whereas for combining of 20 mm beams with 95% efficiency σ_θ should be less than 4.2 µrad. Therefore the higher peak power the larger beam size and the smaller angular chirp is required.

3.5 Aberrations

To analyze the influence of aberrations on combining efficiency the decomposition of wavefront distortions on Zernike polynomials [24, 25] is used in the paper. Note that three first polynomials corresponded to the piston phase and angular errors in two directions (tip and tilt) have been examined earlier (sections 3.1 and 3.3) so they are assumed to absent in this section. The influence of aberrations is analyzed numerically by Monte-Carlo technique. In each particular experimental case aberrations can considerably differ from each other so each Zernike term is examined separately and phase in each channel is

Φ_{n} (r, φ) = δ A_{n} Z_{m}^{k} (r, φ) .

where Z is a Zernike polynomial, δA is the rms wavefront distortion. Normalization of Zernike polynomial on unity rms value is used [24, 25]. For example for coma

Z_{c o m a} (r, φ) = \frac{1}{\sqrt{8}} (3 {(\frac{r}{a})}^{3} - 2 (\frac{r}{a})) \cos (φ),

where a is the radius of the beam aperture. δA_n is assumed to be independent normally distributed random variables with 0 means. According to Eq. (5) and (6) and taking into account that in this case peak fluency is proportional to peak intensity, the efficiencies in terms of peak intensity and peak fluency are

η_{}^{F} (σ_{Φ}) = η_{}^{I} (σ_{Φ}) = \frac{{\bar{I}}_{Σ}^{p e a k} (σ_{Φ})}{{\bar{I}}_{Σ}^{p e a k} (σ_{Φ} = 0)} .

where σ_Φ is the standard deviation of rms wave front error in beam aperture. The averaged on realizations focused peak intensity (

{\bar{I}}_{Σ}^{p e a k}

) of a combined pulse was calculated numerically

{\bar{I}}_{Σ}^{p e a k} = \frac{1}{M} \sum_{m = 1}^{M} I_{Σ}^{p e a k_m} .

M was adjusted so that the computation error to be negligible (<0.5%). The computed efficiency dependence on aberrations for different number of channels and Gaussian profile is shown in Fig. 13. It is well seen that there is almost no discrepancy for different number of channels so the calculation of the efficiency dependence on aberrations for three different beam profiles was performed for the large number of channels (N = 100). Obtained results are shown in Fig. 14.

Fig. 13 The dependence of the coherent combining efficiency in terms of peak intensity (η^I) and peak fluency (η^F) on rms value of wave front distortions for Gaussian profile for spherical aberration (a) and astigmatism (b).

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Fig. 14 The dependence of the coherent combining efficiency in terms of peak intensity (η^I) and peak fluency (η^F) on rms aberrations in beam aperture for Gaussian (a), super-Gaussian (b) and top-hat (c) profiles.

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There is no considerable discrepancy in efficiency dependence on wavefront distortions for different profiles and to achieve more than 95% absolute efficiency of coherent combining rms value of aberrations should be less than 0.1-0.4 rad as follow from the data in Fig. 14. In the previous sentence absolute efficiency has been mentioned because absolute and relative definitions of the coherent combining efficiency can be defined and give different results for some misalignments. It is important because only absolute efficiency is affected by aberrations. The absolute efficiency is the ratio of achieved peak intensity (fluency) of the combined pulse to the maximum value achievable under the absence of aberrations so it coincide with Eq. (5) and with a so called Strehl ratio. Whereas relative efficiency (η_rel) is

η_{r e l}^{I} = {\bar{I}}_{Σ}^{p e a k} / {(\sum_{n = 1}^{N} \sqrt{I_{n}})}^{2} .

It should be noted that the absolute efficiency doesn’t exceed relative one because

{(\sum_{n = 1}^{N} \sqrt{I_{n}})}^{2} \leq I_{Σ}^{\max}

. Moreover under the presence of only aberrations

{\bar{I}}_{Σ}^{p e a k} = {(\sum_{n = 1}^{N} \sqrt{I_{n}})}^{2}

so the relative efficiency always equals to unity. Note also that for the coherent combining efficiency in terms of energy there is no difference between relative and absolute values because of the validity of

W_{\max}^{} = \sum_{n = 1}^{N} W_{n} .

Considering filled aperture combining in terms of energy and assuming identical aberration type (Z) in each channel, the energy of the combined pulse is

W_{Σ}^{} (N, δ A_{1}, ..., δ A_{N}) \propto \int {| \sum_{n = 1}^{N} E_{n} (x, y) |}^{2} d x d y = I_{0} \int_{0}^{\infty} {| \Pr (x, y) |}^{2} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \exp [i (δ A_{n} - δ A_{m}) Z (x, y)] d x d y .

where Pr is the pulse profile. The efficiency is

\begin{array}{l} η_{f i l l e d}^{W} (N, σ_{A}^{}) = \frac{1}{I_{0} N^{2}} \int W_{Σ} (δ A_{1}, ..., δ A_{N}) \frac{e^{- \frac{\sum_{n = 1}^{N} δ A_{n}^{2}}{2 σ_{A}^{2}}}}{{(\sqrt{2 π} σ_{A}^{})}^{N}} d δ A_{1} \times ... \times d δ A_{N} = \\ = \frac{1}{N^{2}} (N + N (N - 1) \int_{}^{} {| \Pr (x, y) |}^{2} \exp [- Z^{2} (x, y) σ_{A}^{2}] d x d y) . \end{array}

For the large number of channels Eq. (61) reduces to

η_{f i l l e d_N > > 1}^{W} (σ_{A}^{}) = \int_{}^{} {| \Pr (x, y) |}^{2} \exp [- Z^{2} (x, y) σ_{A}^{2}] d x d y .

This integral was calculated numerically for Gaussian, super-Gaussian (E(r)~exp[-r⁴/w⁴]) and top-hat profiles and different Zernike terms. Obtained results are presented in Fig. 15.

Fig. 15 The dependence of the coherent combining efficiency in terms of energy (η^W) on rms aberration in beam aperture for Gaussian (a), super-Gaussian (b) and top-hat (c) profiles.

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To achieve more than 95% efficiency of coherent combining (η^W) rms value of aberrations should be less than 0.2-0.5 rad according to the obtained results. The rather large spread of requirements is defined by the discrepancy between coherent combining efficiency dependences on rms wavefront distortion for different types of aberrations (Zernike terms) which is clearly seen in Fig. 14 and 15. Moreover comparing results for different definitions (see Fig. 14 and 15) one can see that requirements on wavefront distortions are close.

3.6 Polarization

Generally combined radiations have linear polarization, so in this section the case of linear polarization is considered. Of course the obtained result can be extended to any polarization because it can be expressed as a sum of two orthogonal polarizations with appropriate amplitudes and phases. To examine the influence of polarization, the electric field in the n-th channel can be expressed as

\vec{E} (δ χ_{n}) = \sqrt{I_{0}} (\cos (δ χ_{n}) \vec{e} {}_{x}+ \sin (δ χ_{n}) \vec{e} {}_{y}) .

As earlier assuming δχ_n to be independent normally distributed random variables with 0 means and, the combining efficiency calculated by the same way as in the previous sections is

η^{I} (N, σ_{χ}^{}) = η^{F} (N, σ_{χ}^{}) = η^{W} (N, σ_{χ}^{}) = \frac{1}{N^{2}} (N + N (N - 1) \exp [- σ_{χ}^{2}]) .

Note that efficiencies in all terms coincide under the presence of polarization errors because

{\bar{W}}_{Σ} (σ_{χ}^{}) \propto {\bar{P}}_{Σ}^{p e a k} (σ_{χ}^{}) \propto {\bar{I}}_{Σ}^{p e a k} (σ_{χ}^{})

. The dependence of the coherent beam combining efficiency on polarization error (Eq. (64)) is presented in Fig. 16.

Fig. 16 Coherent combining efficiency dependence on: (a) polarization error under fixed number of channels; (b) the number of channels under fixed polarization error.

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The limit of Eq. (64) for the large number of channels and small polarization errors is $1 - σ_{χ}^{2}$ which coincides with the result obtained in [19].

As follow from the data to achieve 95% combining efficiency polarization error should be less than 0.22 rad (12.6°) which in many cases especially for parametric amplifiers is satisfied automatically without any stabilization.

3.7 Spatial chirp and other parameters important only for filled aperture combining

There are some parameters which have no influence on combining efficiency in tiled aperture combining but affect efficiency in filled aperture combing. They are spatial chirp, energy instability and inequality, beam position and diameter on combining elements, beam splitter coefficients inequality. This is due to the fact that in tiled aperture approach all radiation reaches the focal plane (there are no secondary arms) in spite of the differences in parameters under discussion. Therefore there is ability to effectively combine beams with different diameters, profiles and energies. Whereas in filled aperture combining the discrepancy in any of the parameters results in energy loses on combiners because some part of radiation leaks into the secondary arms. Analysis of energy instability and inequality, beam position and diameter on combining element and beam splitter coefficients inequality for filled aperture combining can be found elsewhere [19, 26] and is not discussed in the paper. Coherent combining efficiency under the spatial chirp has not been discussed elsewhere to the best of my knowledge so it is analyzed further.

Under the presence of the spatial chirp (Sc = 2πdx/dω = 2πδx/ω_FWHM), the electric field on a combining element is [22]

E (x, t, δ x_{n}) = \frac{\sqrt{I_{0}}}{\sqrt{1 + {(\frac{δ x_{n}}{D_{0}})}^{2}}} e x p [- \frac{2 \ln (2) {(\frac{x^{}}{D_{0}^{}} - \frac{i t}{τ_{0}} \frac{δ x_{n}}{D_{0}})}^{2}}{(1 + {(\frac{δ x_{n}}{D_{0}})}^{2})} - \frac{2 \ln (2) t^{2}}{τ_{0}^{2}}] .

Assuming normal distribution of δx_n and according to Eq. (1) the efficiency (in all terms) is

η_{f i l l e d}^{} (N, σ_{x}^{}) = \frac{1}{N^{2}} (N + N (N - 1) \frac{D_{0}^{}}{\sqrt{π} σ_{x}^{}} e^{\frac{1}{2} {(\frac{D_{0}^{}}{σ_{x}^{}})}^{2}} K_{0} (\frac{1}{2} {(\frac{D_{0}^{}}{σ_{x}^{}})}^{2})) .

Graphs given by Eq. (66) are presented in Fig. 17. For the large number of channels and small instability Eq. (66) reduces to

Fig. 17 The dependence of the coherent combining efficiency in filled aperture combining (η_filled) on spatial chirp.

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η_{f i l l e d}^{}_{N > > 1, σ_{x}^{} / D_{0} < < 1}^{} (σ_{x}^{}) = 1 - {(\frac{σ_{x}^{}}{2 D_{0}^{}})}^{2} .

According to the obtained results, requirements on spatial chirp are rather moderate. For example to achieve 95% coherent combining efficiency spatial chirp should be less than σ_x/D₀ = 0.5. Moreover the larger beam diameter the lower requirements on spatial chirp compensation (larger value of instability is required). Nearly the same situation is for the other parameters under discussion in the section which causes them to be not very important for coherent beam combining.

3. Total efficiency

In the previous sections the influence of each parameter has been discussed separately. It was done to avoid huge expressions which are difficult to analyze and interpret. Thus the question of the total efficiency under the presence of all misalignments should be discussed. Low level of errors is the most interesting since the aim is to achieve high efficiency. Assuming small misalignments electric field in the n-th channel becomes a product of the impacts of the instabilities presented in the previous sections.

\begin{array}{l} \vec{E} (t, ψ, δ T_{n}, δ φ_{n}, ...) = \sqrt{I_{0}} (\cos (δ χ_{n}) \vec{e} {}_{x}+ \sin (δ χ_{n}) \vec{e} {}_{y}) \exp [- 2 l n (2) \frac{{(t - δ T_{n})}^{2}}{τ_{0}^{2}} - i ω_{0} t] \exp [i δ φ_{n}] \times \\ \times \frac{\exp [- 2 \ln (2) \frac{{(ψ - δ ψ_{n})}^{2}}{Ψ_{}^{2}} + i α_{n} ψ]}{{(1 + {(4 \ln (2) \frac{δ k_{2_n}}{τ_{0}^{2}})}^{2})}^{1 / 4} {(1 + {(\frac{δ x_{n}}{D_{0}})}^{2})}^{1 / 4} \sqrt{1 + {(\frac{δ θ_{n} k_{0} D_{0}}{4 \ln (2)})}^{2}}} = \sqrt{I_{0}} \prod_{i} E_{i} (δ X_{n_i}) . \end{array}

where X_{n_i} is the i-th parameter in the n-th chanel (for example X_{1_2} = φ₁).

According to [27], when E is factorized, the expression for the efficiency becomes the product of the impacts of the misalignments calculated above

η_{t o t a l} = \prod_{i} η (σ_{i}) .

Thus the total efficiency under the presence of all misalignments is the product of the individual impacts of the instabilities (at least for small errors).

4. Discussion

Performed analysis allows the determination of the requirements on different misalignments with the number of channels taken into account. Table 1 shows the requirements on the discussed parameters including the example of the particular and interesting case of 10 fs pulses combining.

Table 1. Coherent combining loss for each misalignments discussed in the paper and tolerances for each effect individually under 10 fs pulses combining (τ is the FWHM pulse duration; Ψ is the FWHM beam divergence).

View Table

Preformed analysis allows the determination of the parameters critical for peak intensity scaling. With the enhancement of pulse peak power and energy, beam aperture generally increase too which causes the increase of the requirements on beam pointing stability and angular chirp correction. For example a petawatt beam has an aperture of several tens of centimeters which according to the performed analysis (section 3.3) requires pointing stability on 1-3 µrad level but currently such a pointing stability has not been achieved in a petawatt or even subpetawatt laser facility. Thus beam pointing and angular chirp are very important parameters for combining of high power pulses and the development of pointing stabilization systems will be an important and challengeable task in the nearest future.

Whereas requirements on the other parameters presented in the Table 1 don’t depend on beam size so don’t change with peak power scaling. The experimental ability to achieve the stability of these parameters required for the coherent combining of 23fs, 100 mJ, 4TW, relativistic intensity laser pulses with more than 90% efficiency was demonstrated for example in [9]. Moreover offered and realized [9, 28] stabilization technique exploiting additional unamplified radiation to improve amplified pulses stability should preserve achieved in [8, 9] parameters stability with peak power and the number of channels scaling and enable pointing stabilization on the required 1 µrad level.

Furthermore obtained results of efficiency dependence on the number of combining channels (see Fig. 18) allows to conclude that with the scaling of the number of beams from currently realized 2-4 and 90-95% efficiency to the larger value (N~100-10000), the efficiency will drop by not larger than 5-10% under conditions of preserving achieved accuracy of misalignments correction in each channel. This should encourage such projects as ELI-ultra-high-field and International Coherent Amplification Network but of course many challenges will have to be overcome to create such facilities including stabilization systems development.

Fig. 18 The relation between the efficiency of the coherent combining of two (N = 2) and large (N>>1) number of beams for tiled (a) and filled (b) aperture combining. Gray filled aria shows the range of possible values.

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Combining efficiency definition in terms of peak fluency is the most versatile and measurable but as it was discussed earlier in filled aperture combining the definition in terms of energy is the most frequently used. Therefore the relation between the definitions has been found. According to the performed analysis for the one group of parameters such as optical path mismatch, dispersion compensation accuracy, aberrations and polarization η^F and η^W_filled coincide. Whereas for the other group of parameters, namely phase difference, beam pointing, angular and spatial chirp, the decrease of η^F is slightly higher that the decrease of η^W_filled. Thus η^W_filled should not be less than η^F. If it is possible to determine the impact of all discussed parameters, the exact relation between different definitions can be found using the formulas calculated in the paper. If the determination of the parameters impacts or of the most critical parameter has not or cannot be done, the range of possible efficiencies can be found on the basis of the performed analysis. The obtained relation is presented in Fig. 19(a).

Fig. 19 (a) The relation between the efficiency of the coherent beam combining in terms of peak fluency (η^F) and energy(η^W). (b) The relation between the efficiency of the coherent beam combining in terms of peak fluency and peak intensity (η^I). Gray filled aria shows the range of possible values.

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For many projects the final goal is the increase of peak intensity but not peak fluency so the relation between them was found. For phase difference, beam pointing, spatial chirp, aberrations and polarization η^F and η^I coincide, for optical path mismatch, dispersion compensation accuracy and angular chirp the decrease of η^I is not more than twice the decrease of η^F. The range of possible values is shown in Fig. 19(b). To calculate the exact relation between the definitions one should determine the impacts of all discussed misalignments and use the formulas presented in the paper.

5. Conclusions

In the paper the guidelines to compare results obtained in different combining approaches and evaluated using different combining efficiency definitions are obtained. Different approaches to qualify combining performance in tiled and filled aperture combining experiments are discussed. The dependence of the combining efficiency on the number of combining pulses and different effects for all approaches has been investigated and analytical equations for its evaluation have been obtained. They enable the determination of the most important parameters and effects which strongly affect combining efficiency with intensity scaling. The analysis provide design guidelines for laser systems based on coherent beam combining and enables extrapolation of experimentally achieved results. It is determined that under preserving the levels of misalignments, combining efficiency decrease will not exceed 5-10% with the number of channels increase from currently realized 2-4 (and efficiency 90-95%) to 10-10000. This proves good prospects to scale peak intensity.

Appendix: Efficiency calculation on the example of the detailed derivation of Eq. (24, 26)

Assuming δk_2n to be independent normally distributed random variables with 0 means, the peak intensity of a combined pulse is

{\bar{I}}_{Σ}^{p e a k} (N, σ_{k_{2}}^{}) = \int {| \sum_{n = 1}^{N} \sqrt{I^{p e a k} (δ k_{2_n})} |}^{2} \frac{e^{- \frac{\sum_{n = 1}^{N} δ k_{2_n}^{2}}{2 σ_{k_{2}}^{2}}}}{{(\sqrt{2 π} σ_{k_{2}}^{})}^{N}} d δ k_{2_1} \times ... \times d δ k_{2_N} .

Taking into account Eq. (23) one obtains:

\begin{array}{l} {| \sum_{n = 1}^{N} \sqrt{I^{p e a k} (δ k_{2_n})} |}^{2} = {| \sum_{n = 1}^{N} \frac{1}{\sqrt[4]{1 + {(4 \ln (2) \frac{δ k_{2_n}}{τ_{0}^{2}})}^{2}}} |}^{2} = \sum_{n = 1}^{N} \sum_{m = 1}^{N} \frac{1}{\sqrt[4]{1 + {(4 \ln (2) \frac{δ k_{2_n}}{τ_{0}^{2}})}^{2}} \sqrt[4]{1 + {(4 \ln (2) \frac{δ k_{2_m}}{τ_{0}^{2}})}^{2}}} = \\ = \sum_{n = 1}^{N} \frac{1}{\sqrt[]{1 + {(4 \ln (2) \frac{δ k_{2_n}}{τ_{0}^{2}})}^{2}}} + \sum_{n = 1}^{N} \sum_{m \neq n}^{N} \frac{1}{\sqrt[4]{1 + {(4 \ln (2) \frac{δ k_{2_n}}{τ_{0}^{2}})}^{2}} \sqrt[4]{1 + {(4 \ln (2) \frac{δ k_{2_m}}{τ_{0}^{2}})}^{2}}} . \end{array}

So the peak intensity of a combined pulse is

\begin{array}{l} {\bar{I}}_{Σ}^{p e a k} (N, σ_{k_{2}}^{}) = I_{0} (\begin{array}{l} \sum_{n = 1}^{N} [\int_{- \infty}^{\infty} \frac{1}{\sqrt[]{1 + {(4 \ln (2) \frac{δ k_{2_n}}{τ_{0}^{2}})}^{2}}} \frac{e^{- \frac{δ k_{2_n}^{2}}{2 σ_{k_{2}}^{2}}}}{{(\sqrt{2 π} σ_{k_{2}}^{})}^{}} d δ k_{2_n}] + \\ + \sum_{n = 1}^{N} \sum_{m \neq n}^{N} [\int_{- \infty}^{\infty} \frac{1}{\sqrt[4]{1 + {(4 \ln (2) \frac{δ k_{2_n}}{τ_{0}^{2}})}^{2}}} \frac{e^{- \frac{δ k_{2_n}^{2}}{2 σ_{k_{2}}^{2}}}}{{(\sqrt{2 π} σ_{k_{2}}^{})}^{}} d δ k_{2_n} \times \int_{- \infty}^{\infty} \frac{1}{\sqrt[4]{1 + {(4 \ln (2) \frac{δ k_{2_m}}{τ_{0}^{2}})}^{2}}} \frac{e^{- \frac{δ k_{2_m}^{2}}{2 σ_{k_{2}}^{2}}}}{{(\sqrt{2 π} σ_{k_{2}}^{})}^{}} d δ k_{2_m}] \end{array}) = \\ = I_{0} (\sum_{n = 1}^{N} [\int_{- \infty}^{\infty} \frac{1}{\sqrt[]{1 + {(4 \ln (2) \frac{δ k_{2_n}}{τ_{0}^{2}})}^{2}}} \frac{e^{- \frac{δ k_{2_n}^{2}}{2 σ_{k_{2}}^{2}}}}{{(\sqrt{2 π} σ_{k_{2}}^{})}^{}} d δ k_{2_n}] + \sum_{n = 1}^{N} \sum_{m \neq n}^{N} {[\int_{- \infty}^{\infty} \frac{1}{\sqrt[4]{1 + {(4 \ln (2) \frac{δ k_{2_n}}{τ_{0}^{2}})}^{2}}} \frac{e^{- \frac{δ k_{2_n}^{2}}{2 σ_{k_{2}}^{2}}}}{{(\sqrt{2 π} σ_{k_{2}}^{})}^{}} d δ k_{2_n}]}^{2}) = \\ = I_{0} (2 N \int_{0}^{\infty} \frac{1}{\sqrt[]{1 + {(4 \ln (2) \frac{δ k_{2_n}}{τ_{0}^{2}})}^{2}}} \frac{e^{- \frac{δ k_{2_n}^{2}}{2 σ_{k_{2}}^{2}}}}{{(\sqrt{2 π} σ_{k_{2}}^{})}^{}} d δ k_{2_n} + N (N - 1) {[2 \int_{0}^{\infty} \frac{1}{\sqrt[4]{1 + {(4 \ln (2) \frac{δ k_{2_n}}{τ_{0}^{2}})}^{2}}} \frac{e^{- \frac{δ k_{2_n}^{2}}{2 σ_{k_{2}}^{2}}}}{{(\sqrt{2 π} σ_{k_{2}}^{})}^{}} d δ k_{2_n}]}^{2}) . \end{array}

Replacing the variable in the first integral with q defined as $(\cos h (q) - 1) / 2 = {(4 \ln (2) δ k_{2_n} / τ_{0}^{2})}^{2}$ and in the second one with $q^{'} = {(4 \ln (2) δ k_{2_n} / τ_{0}^{2})}^{2}$ , Eq. (72) becomes

{\bar{I}}_{Σ}^{p e a k} (N, σ_{k_{2}}^{}) = I_{0} (\begin{array}{l} N \frac{τ_{0}^{2}}{4 \ln (2) \sqrt{2 π} σ_{k_{2}}^{}} e^{\frac{τ_{0}^{4}}{64 \ln^{2} (2) σ_{k_{2}}^{2}}} \int_{0}^{\infty} e^{- \frac{τ_{0}^{4} \cosh (q)}{64 \ln^{2} (2) σ_{k_{2}}^{2}}} d q + \\ + N (N - 1) {[\frac{τ_{0}^{2}}{\sqrt{32} \ln (2) σ_{k_{2}}^{} \sqrt{π}} \int_{0}^{\infty} e^{- \frac{τ_{0}^{4} q^{'}}{32 \ln^{2} (2) σ_{k_{2}}^{2}}} \frac{1}{\sqrt{q^{'}}} \frac{1}{\sqrt[4]{1 + q^{'}}} d q^{'}]}^{2} \end{array}) .

Using the integral representation of the modified Bessel function of the second kind $K_{0} [x] = \int_{0}^{\infty} \exp [- x \cos h (q)] d q$ and of the confluent hypergeometric function of the second kind $U [a, b, z] = \frac{1}{Γ (a)} \int_{0}^{\infty} e^{- z t} t^{a - 1} {(1 + t)}^{b - a - 1} d t$ and taking into account that $Γ (\frac{1}{2}) = \sqrt{π}$ , Eq. (24) is obtained. Eq. (26) is obtained by dividing Eq. (24) by I₀N².

Acknowledgments

This work has been partly supported by the Russian Academy of Sciences under the Program of Basic Research “Extreme Light Fields and Their Applications”. The author acknowledges V.I. Trunov and E.V. Pestryakov for stimulating discussions of some results.

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parameter	Equations for combining loss estimation for N>>1 and small instabilities, 1-η			Exact tolerances for 5% loss (τ₀ = 10 fs, λ₀ = 800 nm, D₀ = 10 mm, Gaussian beam profile, N>>1)
	η^I	η^F	η^W_filled	η^I	η^F	η^W_filled
phase difference (σ_φ)	σ_φ²	σ_φ²	σ_φ²	220 mrad	220 mrad	220 mrad
optical path mismatch (σ_L)	$\frac{4 \ln (2) σ_{L}^{2}}{c^{2} τ^{2}}$	$\frac{2 \ln (2) σ_{L}^{2}}{c^{2} τ_{0}^{2}}$	$\frac{2 \ln (2) σ_{L}^{2}}{c^{2} τ_{0}^{2}}$	0.4 µm	0.59 µm	0.59 µm
dispersion (σ_k2)	$\frac{1}{2} {(\frac{4 \ln (2) σ_{k_{2}}^{}}{τ_{0}^{2}})}^{2}$	$3 {(\frac{\ln (2) σ_{k_{2}}^{}}{τ_{0}^{2}})}^{2}$	$3 {(\frac{\ln (2) σ_{k_{2}}^{}}{τ_{0}^{2}})}^{2}$	12.5 fs²	22 fs²	22 fs²
beam pointing (σ_ψ)	$4 \ln (2) {(\frac{σ_{ψ}^{}}{Ψ})}^{2}$	$4 \ln (2) {(\frac{σ_{ψ}^{}}{Ψ})}^{2}$	$2 \ln (2) {(\frac{σ_{ψ}^{}}{Ψ})}^{2}$	4.8 µrad	4.8 µrad	6.9 µrad
angular chirp (σ_θ)	${(\frac{σ_{θ}^{}}{Ψ})}^{2}$	$\frac{1}{2} {(\frac{σ_{θ}^{}}{Ψ})}^{2}$	$\frac{1}{4} {(\frac{σ_{θ}^{}}{Ψ})}^{2}$	8.3 µrad / 100 nm	12 µrad / 100 nm	17.5 µrad / 100 nm
aberrations (σ_A)	σ_A²	σ_A²	σ_A²	0.1-0.4 rad	0.1-0.4 rad	0.2-0.5 rad
polarization (σ_χ)	σ_χ²	σ_χ²	σ_χ²	0.22 rad	0.22 rad	0.22 rad

Coherent combining efficiency in tiled and filled aperture approaches

Abstract

1. Introduction

2. Approaches to coherent beam combining and combining efficiency definitions

3. Influence of different instabilities and mismatches on coherent combining efficiency

3.1 Optical path mismatch and phase difference

3.2 Dispersion compensation accuracy

3.3 Pointing instability

3.4 Angular chirp

3.5 Aberrations

3.6 Polarization

3.7 Spatial chirp and other parameters important only for filled aperture combining

3. Total efficiency

4. Discussion

5. Conclusions

Appendix: Efficiency calculation on the example of the detailed derivation of Eq. (24, 26)

Acknowledgments

References and links

Cited By

Figures (19)

Tables (1)

Equations (73)

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