Lasers are used in diverse applications, such as free-space optical communication, laser power transmission, and medical systems [1–3]. Power scaling is a major interest in these applications, and fiber lasers have made remarkable progress due to the advantages of efficient heat dissipation, good beam quality, and high conversion efficiency [4]. The single-mode output power of a single fiber laser is currently at the 10-kW level [5–7]. Investigations for a further increase in laser power have suggested several beam-combining techniques, such as spatial beam combining, spectral beam combining (SBC), and coherent beam combining (CBC) [8–11]. Among them, CBC has an extraordinary advantage, which comes from the coherent nature of laser light: In a tiled-aperture CBC system, multiple laser beams are launched into free space, and then overlap at the receiver plane, interfering with each other. Constructive interference after fine adjustment of the phase condition produces a strong central lobe in the interference pattern. The intensity is ideally proportional to the square of the number of combined beams, which is unlike other combining techniques.

However, in a real CBC system, many errors reduce the combining efficiency, and thus predicting the impact of the error factors on the combining efficiency can guide the design of the CBC system. The major error factors in a CBC system are the relative phase error between channels, the tip-tilt error, the polarization-state mismatch, and the beam-quality degradation. The phase of each channel is controlled by algorithms such as stochastic parallel gradient descent (SPGD), locking of optical coherence via single-detector electronic frequency tagging (LOCSET), covariance-matrix adaption-evolution strategy (CMA-ES), or frequency dithering [12–15]. Depending on the sophistication of the algorithms and the system noise, phase errors may occur, reducing the combining efficiency. The laser beam from each channel is directed by the voltage-controlled x-y tip-tilt system in the collimation optics. Depending on the algorithms and the accuracy of the voltages, tip-tilt error may also occur. Although a polarization-maintaining (PM) fiber is used, a CBC system can also suffer from polarization mismatch, due to errors in the fiber’s end-cap angle alignment and system noise. Finally, imperfect beam quality reduces the combining efficiency as well. A higher M ² value means more diffraction at long distances, thereby resulting in low combining efficiency [16]. However, there has been little attempt to clarify the impacts of such errors in a comprehensive and systematic manner, except for a handful of studies covering only a part of them [17].

In this work, the impacts of all major error factors are analyzed and investigated by means of comprehensive numerical simulations, with particular attention paid to the expected combining efficiency and combined beam shapes depending on the levels of the individual errors.

2.1. Coherent-beam-combining System

In Fig. 1, the configuration of a 3-channel CBC system is presented. To maintain mutual coherence among all channels, a single narrowband laser diode is used and its output is split into individual channels, after passing the optical isolators. Inline phase modulators play a crucial role in adjusting the relative phases among the individual channels at high speed. Power amplifiers can be added after the phase modulators, to increase the power of the channel beams. In Fig. 1(b), a 3-channel tip-tilt system is presented. For fine control of the direction of each laser beam, the fiber end cap is equipped with a voltage-controlled x-y alignment component. The relative translation of the fiber end cap in the x-y direction to the fixed collimating lens induces a tilt of the laser beam [12]. The polarization state of each laser is also determined by the rotation of the fiber end cap.

Figure 1. Configuration of a 3-channel CBC system: (a) 3-channel fiber-laser system and (b) 3-channel collimating optics. LD, laser diode; ISO, isolator; SPL, splitter; PM, phase modulator; AMP, amplifier; EC, end-cap; CL, collimating lens; CBC, coherent beam combining.

2.2. Numerical Model of a Multichannel CBC System

The Huygens–Fresnel principle is used to analyze the laser beam’s propagation as follows [18]:

where U₀ and U₁ are the scalar fields at the transmitter plane and the receiver plane respectively, as shown in Fig. 2; $\overrightarrow{r}$ is the displacement vector connecting one position in the transmitter plane to the other in the receiver plane; And r is the corresponding magnitude of the vector. Under the condition that z >> ξ, η, x, y, the above equations are modified to the convolution form with a convolution kernel h(x, y). The numerical calculations can be performed using the fast-Fourier-transform algorithm.

Figure 2. Schematic illustration for the transmitter and receiver planes: (a) The transmitter plane in (ξ, η) and (b) the receiver plane in (x, y).

The numerical analysis of a 3-channel CBC system is performed based on the configuration of Fig. 1(b): Three collimating lenses are in a plane perpendicular to the direction of propagation. It is assumed that the 3-channel laser beams are from single-mode PM fibers, and that the fundamental mode of the single-mode fibers is similar to the fundamental Gaussian mode in free space [19]. The collimated laser beam in the z-direction has the wave vector, k = (0, 0, k). With perfect collimation, each laser beam will only propagate in the z-direction, not to the center of the receiver plane. Therefore, beam steering is required to ensure that the laser beams overlap properly at the center of the receiver plane.

Thus the fundamental Gaussian beam’s field phase distribution for the i^th channel beam in front of the collimation lens is represented by

where (x_i, y_i) is the center of the i^th lens; (x_i,s, y_i,s) is the offset from the center of the i^th lens to steer the direction of the i^th channel beam after transmission through the collimation lens, and thus, (x_i + x_i,s, y_i + y_i,s) becomes the position of the i^th fiber tip; R is the radius of curvature of the Gaussian beam’s wavefront; And f is the focal length of the collimating lens.

As the phase change of a laser beam by the i^th lens is given by

The phase distribution of the channel beam just after the i^th lens becomes

where it is assumed that R = f. As a result, the corresponding wave vector of the transmitted channel beams is given by k_i = (−k x_i,s/f, −k x_i,s/f, k_z). Thus the adjustment of the fiber-tip position determines the tilt angle in the propagation direction of the corresponding laser beam.

For an application in which the distance between the transmitter and the receiver is fixed, a proper steering or offset (x_i,o, y_i,o) exists such that all of the laser beams overlap at the center of the receiver plane. Then the offset x_i,s can be expressed as the sum of two components, x_i,o + x_i,err, where x_i,err is the tip-tilt error. With proper tip-tilt arrangement, all laser beams overlap at the receiver plane. The phase at the center of the i^th beam is denoted as θ_i, and all of the channel beams must be in phase to have constructive interference at the center, which eventually leads to the completion of CBC. To render all channels in-phase, the PMs in Fig. 1(a) control the phases of the individual channel beams, using various methods such as SPGD, LOCSET, CMA-ES, and frequency dithering [12–15]. Despite the high-speed phase control, θ_i may have errors as given by θ_i,err = θ_i − θ_ref + 2n_iπ, where n_i is an integer such that θ_i,err is in the range from −π to π. The parameters used in the simulation are summarized in Table 1.

TABLE 1. Simulation parameters.

Parameter	Value
Wavelength λ (nm)	1,064
1/e2 Beam Radius at the Lens (mm)	10
Lens Diameter (mm)	25.4
Focal Length of Collimating Lens (mm)	100
Distance between Lens Centers (mm)	28
Propagation Distance (m)	1,000
Power of Each Laser Channel (W)	1

With these parameters, the subaperture fill factor, which is defined as the ratio of the beam diameter to the lens diameter, becomes 0.79. Previous research has reported that the subaperture fill factor may be optimized at about 0.85 with an ideal lens. However, the subaperture fill factor in this work is a little smaller than the optimal value, due to a small reduction of the clear aperture of the given lens. The conformal fill factor, which is defined by the ratio of the lens-to-lens distance to the lens diameter, becomes 1.1, slightly larger than unity due to the inclusion of the lens mounts or mechanical structures supporting the lenses [20]. Considering mid- or long-range applications, the propagation distance is set to 1,000 m.

Figure 3 presents the 3-channel CBC outcome with proper phases, tip-tilt, polarization, and beam quality, where it is assumed that the polarization axes of the individual beams are all aligned in the same direction. The 3-channel beams at the transmitter are arranged in a triangular configuration as shown in Fig. 3(a). In Fig. 3(b) the channel phase distributions after the collimating lens are shown. In Fig. 3(c), the intensity profile of the combined channel beams at the receiver plane is shown, which represents a typical CBC beam profile having a strong central lobe and weak surrounding side lobes, obtained under ideal or well-stabilized conditions [21].

Figure 3. Ideal 3-channel coherent beam combining (CBC) system: (a) Intensity profile at the transmitter plane, (b) phase profile after the collimating lenses, and (c) intensity profile of the combined channel beams at the receiver plane.

To quantify the CBC efficiency, the bucket size that encircles a finite area at the center of the receiver plane must be determined [20]. Although the bucket can be of any shape or size, a circle with the radius of the corresponding diffraction-limited beam is normally used. The radius of the corresponding diffraction-limited beam is thus determined as follows [18]:

where D is the aperture diameter of the transmitter, and L is the distance of transmission. In most cases, the bucket size is similar to that of the central lobe of the combined beams.

The power in the bucket (PIB), which is defined by the amount of power received in a specific bucket area, is the figure of merit and is also used to calculate the transmission efficiency, as a PIB over output power delivered by the optical fiber [20].

For example, in Fig. 3 the aperture diameter of the transmitter is given by 28.9 mm, and R_bucket is estimated at 22.5 mm. Thus the PIB is 1.51 W, thereby yielding a transmission efficiency of 50.4%, i.e. (1.51 W) / (3 × 1 W), which is the case without the major error factors. However, the PIB will be degraded if there are error factors. Various cases with error factors are considered in detail in the following sections.

2.3. Impacts of Phase, Tip-tilt, and Polarization-angle Factors on the PIB in a 3-channel CBC System

In this section the effects of the three major error factors (out of four) in terms of the channel phase, tip-tilt, and polarization state are numerically simulated and discussed, being represented by θ_i,err, (x_i,err, y_i,err), and ϕ_i,err (ϕ_i − ϕ_ref), respectively.

First, the phase error θ_i,err is taken into account while the other error factors are set to zero, to determine the effect of phase error separately. The channel-number information is the same as shown in Fig. 3(a). Considering θ₁ as a reference, the values of θ_2,err and θ_3,err are swept from −π to π, and the corresponding PIBs and the combined beam-intensity profiles are calculated. In Figs. 4(a)–4(h), the combined beam-intensity profiles with diverse phase errors are shown. Unlike the ideal situation where the maximum intensity is invariably located at the center of the receiver plane [Fig. 3(c)], the maximum intensity tends to shift away from the center of the receiver plane, depending on the degree of the phase error [Figs. 4(a)–4(d)]. The maximum intensity also tends to be reduced. At an extreme situation of θ_2,err = π shown in Fig. 4(d), even two peaks with the same intensity level are observed. In Figs. 4(e)–4(h) similar trends are observed, except for the direction of the shift.

Figure 4. 3-channel coherent beam combining (CBC) beam profile with phase errors of (a) θ_2,err = 0.25π, (b) θ_2,err = 0.5π, (c) θ_2,err = 0.75π, (d) θ_2,err = π, (e) θ_2,err = 0.25π, θ_3,err = −0.25π, (f) θ_2,err = 0.5π, θ_3,err = −0.5π, (g) θ_2,err = 0.75π, θ_3,err = −0.75π, (h) θ_2,err = π, θ_3,err = −π.

The tip-tilt error is also analyzed in a similar manner, the results of which are shown in Fig. 5. The positional errors in terms of (x_2,err, y_3,err) are swept to determine the corresponding effects in both horizontal and vertical directions. In Figs. 5(a)–5(d), Channel 2 is moved negatively in the x-direction with the increment of the given tip-tilt error, which actually reduces the overlap with the other channel beams in the center of the receiver plane. In Figs. 5(e)–5(h), the tip-tilt errors are applied in both x- and y-directions. Interestingly, in all cases the location of maximum intensity is maintained well at the center, although some asymmetries are observed in its vicinity and surrounding regions, because the phase condition is still kept near the optimal condition.

Figure 5. 3-channel coherent beam combining (CBC) beam profile with tip/tilt errors of (a) x_2,err = 1 μm, (b) x_2,err = 2 μm, (c) x_2,err = 3 μm, (d) x_2,err = 4 μm, (e) x_2,err = 1 μm, x_3,err = −1 μm, (f) x_2,err = 2 μm, x_3,err = −2 μm, (g) x_2,err = 3 μm, x_3,err = −3 μm, (h) x_2,err = 4 μm, x_3,err = −4 μm.

If the polarization axes of the individual channel beams are not exactly aligned, the PIB also tends to diminish. To quantify such behavior, additional numerical simulations are performed. The polarization angle of Channel 1, i.e. ϕ₁, is fixed parallel to the x axis as a reference, and ϕ_2,err and ϕ_3,err are swept from −π to π. In Fig. 6, the combined beam-intensity profiles with diverse polarization angle errors are shown. In Figs. 6(b) and 6(f) peculiar interference patterns are observed, because the orientations of polarization states of the channel beams crucially determine their pattern. Note that only parallel polarizations interfere with each other, while orthogonal polarizations do not interfere at all. In particular, in Fig. 6(d), the combined beam profile is the same as the one shown in Fig. 4(d), which is obvious, because the effect of ϕ_2,err = π is identical to that of θ_2,err = π.

Figure 6. 3-channel coherent beam combining (CBC) beam profile with polarization error of (a) φ_2,err = 0.25π, (b) φ_2,err = 0.5π, (c) φ_2,err = 0.75π, (d) φ_2,err = π, (e) φ_2,err = 0.25π, φ_3,err = −0.25π, (f) φ_2,err = 0.5π, φ_3,err = −0.5π, (g) φ_2,err = 0.75π, φ_3,err = −0.75π, (h) φ_2,err = π, φ_3,err = −π.

In Fig. 7 the overall normalized PIB of the given 3-channel CBC configuration is color-mapped in relation to phase, tip-tilt, and polarization-axis errors. Note that the given PIB is normalized by the maximum PIB under the given condition. In particular, normalized PIBs of 98% and 95% are shown as contours with dashed lines. It is noteworthy that the PIB characteristics in relation to the phase and polarization-axis errors shown a very similar trend, as can be seen in Figs. 7(a) and 7(c) (refer to the Appendix for more theoretical details). This is indeed the result of the extensive study of this work, which would otherwise be difficult to deduce from simply looking at the combined beam profiles shown in Fig. 4 or Fig. 6.

Figure 7. Power in the bucket (PIB) results of 3-channel coherent beam combining (CBC) with (a) phase errors (θ_2,err, θ_3,err), (b) tip/tilt errors (x_2,err, y_3,err), and (c) polarization error (φ_2,err, φ_3,err).

Therefore, to maintain the PIB reduction within 2% or 5% for a single source of error, it is necessary to keep the phase error below 0.65 or 1.06 radians (rad); The tip-tilt error below 0.75 or 1.25 μm; The polarization-angle error below 0.65 or 1.06 rad, respectively. The above error ranges are valid if each error factor exists independently. In a worst-case scenario of composite errors with θ_2,err = 1.06 rad, θ_3,err = −1.06 rad, x_2,err = 1.25 μm, x_3,err = 1.25 μm, ϕ_2,err = 1.06 rad, ϕ_3,err = −1.06 rad, the aggregate effect would reduce the normalized PIB to 83.8%.

2.4. Impacts of Phase, Tip-tilt, and Polarization-angle Errors on the PIB in 7- and 19-channel CBC Systems

In this section, our numerical study is further extended to cases of CBC configurations with even more channel beams, based on a sophisticated statistical approach, assuming that each error factor follows Gaussian statistics with variance σ². In this manner the phase, tip-tilt, and polarization-angle errors can be described by N(0, ${\sigma }_{\theta }^2$), N(0, ${\sigma }_{\Delta x\vee \Delta y}^2$), and N(0, ${\sigma }_{\varphi }^2$) respectively, where N denotes the corresponding Gaussian noise.

Thus the ensemble-averaged PIB is obtained for 100 samples randomly and independently prepared, based on the Gaussian statistics for the individual error factors. The results for 7-channel and 19-channel CBC configurations analyzed under ideal, noise-free conditions are shown in Fig. 8, and one typical case of a 19-channel CBC configuration with random phase errors of N(0, ${\sigma }_{\theta }^2$ = π²) is shown in Fig. 9 for comparison. Unlike the combined beam profile obtained under ideal, noise-free conditions, under noisy conditions we obtain a speckle-like beam profile at the receiver plane. The overall, ensemble-averaged PIB characteristics for the 100 random samples obtained under the random phase error, random tip-tilt error, and random polarization-angle error conditions are shown in Fig. 10, where the ranges of the random errors are considered such that 0 < σ_θ, σ_ϕ < π, and σ_Δx∨Δy < 5 μm. It is noteworthy that gradual degradation with increasing error is observed for all error factors, and that the slight fluctuations in the curves reflect the nature of random sampling and the limited size of the ensemble.

Figure 8. Tiled arrays of (a) 7-channel and (b) 19-channel coherent beam combining (CBC) systems, and the ideal results of (c) 7-channel and (d) 19-channel CBC systems.

Figure 9. Result of 19-channel coherent beam combining (CBC) with random phase errors sampled from N(0, σθ2 = π²) for all laser beams. Dashed circle at the center is the reference bucket.

Figure 10. Ensemble-averaged power in the bucket (PIB) of 7-channel and 19-channel with (a) phase error sampled from N(0, σθ2), (b) tip/tilt error sampled from N(0, σΔx2), and (c) polarization error sampled from N(0, σϕ2), for all laser beams.

2.5. Impact of Beam-quality Factor on the PIB in a Single-channel Laser and in 3-, 7-, and 19-channel CBC Systems

Besides the phase, tip-tilt, and polarization-angle errors, the laser’s beam-quality factor, which in general is quantified by the M ² value, can also affect the PIB of a CBC system. A laser with a larger M ² value will experience greater divergence, resulting in a larger laser beam and reduced efficiency at the receiver plane.

In this section, the impact of the beam-quality factor on the PIB is analyzed in multichannel CBC systems, including 3-, 7-, and 19-channel configurations, which are based on single-mode-based super-Gaussian beams rather than a mixture of multimode beams. This is because most CBC systems normally preclude the use of multimode beams, because the coherent phase control between different channel beams in multimode formats becomes too problematic. However, the use of single-mode-based super-Gaussian beams is often considered to increase the effective fill factor of the aperture of the transmitter [22].

The M ² value for a super-Gaussian-shaped beam can be calculated in an analytical manner as follows [23]:

where E_p is the field of the super-Gaussian beam; Γ is the gamma function; And p is the order of the super-Gaussian beam. In Fig. 11, the evolution of M ² as a function of the order of the super-Gaussian beam is shown, where the M ² value drops to unity if p is set to unity, i.e. if the beam becomes a fundamental Gaussian.

Figure 11. M ² as a function of the order of a super-Gaussian beam.

In Table 2, the simulation results for the PIB and the corresponding degradation ratio are summarized in relation to the M ² value. In the case of a single-channel laser, the PIB degradation ratio in comparison with the case of M ² = 1 is relatively small; However, in multichannel CBC systems the degradation ratio is rather magnified, depending on the channel number. For example, when the M ² value increases from 1 to 1.3, the PIB is reduced by 2.6%, 11.8%, 12.8%, and 13.2% for a single-channel configuration and 3-, 7-, and 19-channel CBC configurations respectively.

TABLE 2. Power in the bucket (PIB) degradation versus super-Gaussian-based M ².

Single-channel Configuration
M 2	PIB (W)	Degradation Ratio (%)
1	0.9582	-
1.1	0.9527	0.57
1.2	0.9433	1.55
1.3	0.9329	2.64

3-channel CBC Configuration
M 2	PIB (W)	Degradation Ratio (%)
1	0.5248	-
1.1	0.4920	6.25
1.2	0.4760	9.30
1.3	0.4630	11.78

7-channel CBC Configuration
M 2	PIB (W)	Degradation Ratio (%)
1	0.5212	-
1.1	0.4853	6.89
1.2	0.4682	10.17
1.3	0.4545	12.80

19-channel CBC Configuration
M 2	PIB (W)	Degradation Ratio (%)
1	0.5109	-
1.1	0.4744	7.14
1.2	0.4570	10.55
1.3	0.4433	13.23

We have performed and analyzed extensive numerical simulations on CBC systems in terms of the four major error factors, including phase, tip-tilt, polarization angle, and beam-quality factor. From a single-channel configuration to 19-channel CBC configurations, the PIB was calculated as a function of each error factor. It has been verified that in the 3-channel CBC configuration, to maintain the PIB within a 5% reduction, the phase error should be kept below 1.06 rad, tip-tilt error below 1.25 μm, and polarization error below 1.06 rad. Note that each of the three parameters is calculated independently, with the other two parameters set to zero. Composite errors of θ_2,err = 1.06 rad, θ_3,err = −1.06 rad, x_2,err = 1.25 μm, x_3,err = 1.25 μm, ϕ_2,err = 1.06 rad, ϕ_3,err = −1.06 rad would reduce the normalized PIB to 83.8%. It has also been verified that if the M ² value increases from 1 to 1.3, the normalized PIB efficiency is reduced by 2.6%, 11.8%, 12.8%, and 13.2% for a single-channel configuration and 3-channel, 7-channel, and 19-channel CBC configurations respectively. It is expected that these numerical results will be useful for the evaluation and design of multichannel CBC systems and other related applications.

Consider the field vector of the l^th channel beam at the center of the receiver, as given by

where a^_ϕl is a unit vector indicating the direction of the polarization axis defined by the polarization angle ϕ_l; E_0l is the amplitude of the field; And θ_l is the phase angle.

Consider the intensity at the center of the receiver plane produced with phase error only, without having any tip-tilt or polarization-axis error (in the same manner as the investigations described in Section 2.3), which is obtained as

Also, consider the intensity at the center of the receiver plane produced with polarization-axis error only, without having any tip-tilt or phase error (in the same manner as the investigations described in Section 2.3), which is obtained as

In fact, the mathematical expressions for I_θ and I_ϕ are identical, except that the phase difference between channels in the former case is simply replaced by the polarization-angle difference between channels in the latter case. As a result, the characteristic change of the PIB depending on the phase and polarization errors must be the same, as can be verified from Figs. 7(a) and 7(c).

Agency for Defense Development of South Korea (UD210019ID); the BK21 FOUR Project.

The authors declare no conflicts of interest.

Data presented in this paper may be obtained from the authors upon reasonable request.