Experimental Demonstration of Central-Lobe Energy Enhancement Based on Amplitude Modulation of Beamlets in 19 Elements Fiber Laser Phased Array

Coherent beam combination of fiber laser phased array is a promising technique to achieve high-brightness laser output. In this paper, we focus on central-lobe energy enhancement of the combined beam in the far-field by theoretically and experimentally investigating the relationship between the power-in-the-bucket (PIB) of the combined beam in the far-field and the beamlets amplitude modulation in the near-field. We employ a 19-elements fiber laser phased array integrating adaptive fiber-optic collimators with a beamlet filling factor of 0.903 and achieve phase-locking and beamlet pointing correction among fiber lasers using the SPGD algorithm. The amplitude modulation applied to the tiled sub-apertures is used to control the shape of the emissive array's amplitude envelope to further change the PIB of the combined beam under the same total output power. According to theoretical analysis, the fiber laser phased array achieves PIB = 0.51 for a zero truncation ratio (Tr), which is increased to PIB = 0.57 for Tr = 1.3. Experimentally, we observe PIB = 0.49 for Tr = 0 and PIB= 0.56 for Tr = 1.3. The experimental results are in good agreement with the theoretical analysis, suggesting that the amplitude modulation is helpful for further improving the energy density of central-lobe.


Introduction
Coherent beam combination (CBC) of fiber laser phased array (FLPA) provides an approach for achieving laser output with high energy density and high beam quality [1]- [5]. A nearly diffractionlimited equivalent large-aperture output laser can be obtained by correcting the phase and tip/tilt errors in each sub-aperture of the FLPA. At present, both large-scale FLPA and high-power fiber laser coherent combining technologies have achieved breakthroughs. Currently, the maximum scaling and maximum combined power of the FLPA are over 100 elements and over 16kW [2], [3]. Although the tip/tilt error control has not been included in these reports, the FPLA with only phase-locking control has shown great potential in improving the brightness of the combined beam. Due to these advantages, several related research projects have been launched in recent years [6]- [9]. For example, in 2006, Mourou et al. proposed the International Coherent Amplification Network (ICAN) project in the European Union, aiming to achieve a femtosecond laser with high repetition rate, high average power, and high peak power through coherent combining technology of FLPA [4]. This plan looks forward to the application prospects of the ICAN system in particle acceleration and space debris cleaning.
In FLPA applications, the input electrical power provided to the laser system is sometimes limited. For example, in 2014, Soulard et al. proposed to carry the ICAN system in a space vehicle, and the solar panels on the carrying platform provide electrical energy to ICAN to remove space debris [7]. Therefore, it is vital to obtain a higher energy density central lobe in the far-field when the total electrical input power is equal. In fact, the structure of the FLPA is similar to an optical phased array, where amplitude modulation is often used to improve the energy in central lobes [10]- [12]. Recently, Wang et al. found that the central-lobe energy of a FLPA system can be enhanced by amplitude modulation in the near-field, to make the amplitude envelope of the beamlets following a nonuniform distribution [13]. However, their work is purely based on numerical calculations, without a mathematical model and experimental demonstration.
In this paper, we propose a central-lobe energy enhancement method via tuning the amplitude envelope of the beamlets in the near-field to further increase the central-lobe energy in the combined beam after the phase and tip/tilt errors are both corrected. Both theoretical model and experimental results of a 19 elements FLPA system are presented to demonstrate the feasibility of the proposed method.

Basic Theory
The amplitude modulation is employed to adjust the characteristics of the central-lobe in far-field in phased array technique and the amplitude coefficients of array wavelets can be obtained indirectly by solving the radiation field of the equivalent single large-aperture electromagnetic wave [14], [15]. According to the feature that the array is formed by splicing multiple wavelets, the amplitude coefficients of the array wavelets can be obtained by sampling the amplitude of the equivalent single large-aperture wave-front. Until now, the effects of uniform and non-uniform amplitude distribution (Chebyshev distribution, Taylor distribution, and Gaussian distribution) on the central-lobe enhancement has been compared in microwave phased array techniques. The results confirm that a non-uniform distribution ratio in the emissive beam provides better performance than the uniform distribution and the Gaussian distribution allows one to achieve a better suppression effect than the other two distributions. Similar to microwaves, light waves also belong to the electromagnetic wave, and the physical characteristics of the two are usually consistent [16]. Therefore, we can directly analyze the radiation characteristics of the beamlet array in the far-field by analyzing the radiation characteristics of the Gaussian beam with the same diameter as the circumscribed circle of the beamlet array in the far-field, and then use stochastic parallel gradient descent (SPGD) algorithm to directly optimize the amplitudes of the outgoing beamlets in CBC system so that the effect of beamlet amplitude modulation can be verified [17], [18]. Fig. 1 shows a schematic diagram of the plane of a hexagonal array. The parameters describing the emissive array of 19 elements FLPA are as follows: the 19 sub-apertures are arranged in a regular hexagonal array in a circle of radius R = 76 mm. The sub-aperture diameter d = 28 mm, and the distance between the centers of adjacent sub-apertures s = 31 mm. The operating wavelength λ = 1064 nm.

Diffraction Model of Single Gaussian Beam
The complex amplitude distribution of a Gaussian beam in the Fraunhofer diffraction region is analyzed [19]- [21]. According to the Fresnel-Kirchhoff diffraction principle, the distribution of the light field at any point P(x, y ) in the far-field is given by Where E near (x, y ) is the complex amplitude, Q(x, y, 0) and P(x, y, z) stand for the points of the near-field coordinate plane and of the far-field coordinate plane, respectively.
To simplify the analysis, we assume the Gaussian beam's waist 0 is in the pupil plane and the pupil position is at z 0 is the distance from the Gaussian beam waist to the diffraction screen. A is the amplitude coefficient, k is the wave number, and z r is the Rayleigh distance.
Moving to polar coordinates Q(ρ, ϕ, 0) and P(ρ , ϕ , z) we may rewrite Eq. (1) as follows Where the pupil aperture is a circular hole with radius R = 76mm, J 0 is a zero-order Bessel function and According to the definition of beam intensity (I far (P) = E * (P) · E (P)), the beam intensity at point P in the far-field is given by (4), one may see that the beam intensity distribution in the Fraunhofer diffraction region is regulated by the pupil's size (R) and the Gaussian beam waist 0 . The truncation ratio (Tr = R/ 0 ) is used to describe the relationship between R and 0 . n is the order of the Bessel function J n and when Tr = 0, higher-order terms with n = 2 and above can be ignored.
Equation (5) is the formula of far-field beam intensity distribution of flat-top beam. By looking at Eqs. (4) and (5), one may appreciate the difference between a Gaussian beam and a flat-top beam in the far-field: the Gaussian beam's diffraction field is formed by superimposing Bessel functions of various orders, which have no common zeros. Because of this, the peak power of the first-order diffraction ring decreases and its zero-point position expands outward. The energy in the first-order diffraction ring thus increases and the ring size increases. At the same time, the energy of the secondary diffraction ring is reduced.
An index to evaluate the quality of laser transmission is the average power density in the farfield, which is generally characterized by power-in-the-bucket (PIB). It is defined as the ratio of the combined beam's power in the bucket to the total power of the combined beam. The bucket size Fig. 2(a) shows the behavior of PIB for different Tr values. When Tr = 0, it means that 0 is infinity, and the amplitude shape of the single beam is flat-top type, which is widely used in high power laser system. In this state, we have PIB = 0.8378. The maximum PIB 0.9691 occurs at Tr = 1.3, which is 0.1313 higher than that at Tr = 0. In Fig. 2(b), when Tr = 1.3, it can be seen that the far-field spot becomes wider and the peak energy decreases compared to the case of Tr = 0. This effect is more obvious as Tr further increases. When Tr = 2.5, diffraction is very weak, which indicates that the circular hole has almost no limit to the incident beam and that the intensity distribution curve is the result of the direct projection of the incident Gaussian beam.

Mathematical Model of FLPA System
The CBC of the FLPA system can be viewed as a multi-beam interference [22]- [24]. To specifically study the effect of central-lobe energy enhancement based on amplitude modulation of beamlets, we assume that there are no phase errors and tip/tilt errors in the system, and the beamlets are derived from the same seed laser source. As it is shown in Fig. 1, the amplitude of each beamlet can be adjusted to match the intensity of a specific envelope at different Tr. The near-field complex amplitude of the nth beamlet U n can be expressed as The far-field intensity distribution of the coherent combined beam is Where and (x n , y n , z n ) are the coordinates of the nth beam at the emissive surface. z is the distance traveled by the laser beam and the z-plane is the coherent focal plane. ω 0 = 11 mm, which is the waist radius of beamlet in the emissive plane. A n is the amplitude of the nth sub-aperture beam, which can be used to adjust the intensity to a specific distribution (Tr). The centers of the whole sub-apertures are arranged in circles of four different diameters from the center sub-aperture to the outside, such that there are only four levels of amplitude index A n that need to be adjusted, as shown in Fig. 3. Fig. 4 shows the PIB of the 19 elements FLPA and the variation of the central-lobe widths of the combined spots. When Tr = 0, the amplitudes of the outgoing beamlets are equal and we have PIB = 0.51. Different Tr represents different amplitude envelope shape of outgoing beamlets and by changing the value of A n , the amplitude of each beamlet can be adjusted to a specific distribution. Keeping the total output power constant, and when Tr = 1.3 we have PIB = 0.57, which is the maximum value. The difference between the two cases is that PIB of the FLPA system in Fig. 4(a) is generally smaller than the PIB value of the equivalent beam, which is caused by the cut effect and the arrangement of sub-apertures of the array. To improve the combining efficiency,  the Gaussian beamlet ω 0 is cut to obtain a more uniform intensity distribution wave-front in each sub-aperture. However, the energy of outgoing beamlets will be decreased by cut operation. The energy transmission efficiency η (defined as the ratio of transmission power to the total power) reflects the power loss of the fiber array and is related to the diameter of the sub-aperture d in which It is contradictory to increase the PIB and η at the same time and when the sub-aperture diameter d is 28 mm, the system has a higher PIB value and lower power loss. Moreover, there are some gaps between the sub-apertures, this also causes PIB of the array beamlets to be lower than PIB of the equivalent single beam. However, the changing trend of the spot width in the far-field is consistent.
In order to check whether the Gaussian distribution is the best, we directly optimize the beamlet amplitude of the CBC system using SPGD algorithm [25]. It can be seen that the optimized result is very close to PIB value at Tr = 1.3, as shown in Fig. 5. The energy of the central-lobes of the combined spots is indeed increased under the same total emissive power.

Phenomenon in Different Scale Arrays
For the regular hexagonal FLPA system with a filling factor of 0.903 (defined as the ratio of sub aperture diameter to center distance of adjacent aperture), the relationship between the scale of the array and PIB is analyzed. By applying the same method mentioned above, PIB values corresponding to CBC systems with 7, 19, 37, and 61 fiber lasers are obtained. The behavior of PIB, together with the combined spot radius and the peak intensity of the emissive array are shown in Fig. 6. The near-field intensity distribution and amplitude ratio of the array are shown in Fig. 7. It can be seen that independent of the array's scale, PIB always is maximum when Tr = 1.3, as shown in Fig. 6(a). Figs. 6(b) and 6(c) show the central lobe radius and normalized peak intensity at different array scales. It can be seen that when Tr = 1.3, the peak power of the central-lobe in the fiber arrays of different sizes decreases slightly, and the width becomes wider (to be analyzed in Section III).

Experimental Results and Discussion
The experimental setup of 19-element FLPA system is illustrated in Fig. 8. The key components for phase lock control and tip/tilt control are homemade [26]. The piezoelectric ceramic ring fiber optic   phase modulator (PM) has a half-wave voltage of approximately 2.5 V and first-order resonance frequency of approximately 32 kHz. The maximum optical-length displacement of the PM is about 4 λ with a control voltage from -10 V to 10 V, and the maximum optical power handling is about W-level, with a PM-980 fiber. It is produced by twisting the polarization-maintaining fiber around a piezoelectric ceramic ring. The adaptive fiber optics collimator (AFOC) has an offset range of optical fiber end face ±30 μm and first-order resonance frequency about 2 kHz. The control signal is amplified 100 times by a high-voltage amplifier (HVA) to move the position of the fiber end face at the focal plane of the lens to change the direction of the outgoing beam. The source laser is a linearly polarized single-mode fiber laser at 1064 nm wavelength with a maximum output power of 110 mW and line-width of 20 kHz. The source is equally divided into 19 channels by a fiber splitter, and each one is connected to a PM, which is later employed to correct the phase-type errors of the array. We have also 19 beam amplitude modulators (AM), each one connected to the output of a PM to adjust the final emitted beam intensity of each beamlets. Each AM is connected to the AFOC's tail fiber and then emitted into free space. The 19 beamlets have identical polarization direction, a feature obtained by rotating the fiber tip of each AFOC during assembling. One part is focused by a transform lens with a focal length of 2 m and then the focused beamlets are sent to a beam splitter (BS). After passing through the BS, a part of the beamlet is sent to a 10× micro-objective (MO) and detected by a high-speed CCD camera to observe and calculate the actual PIB. Another part is sent to a photo detector (PD) with a pin hole of 10 μm diameter on the focal plane of the transform lens, used to assess performance index and to produce the cost function for the SPGD controller, which can optimize the phase error and tip/tilt error. Moreover, because the PD has a high sampling rate, the cost function (voltage acquired by PD) can also accurately reflect the control bandwidth (iteration rate), the enhancement capacity of PIB (average value), and the stability of the convergence stage (MSE) during the experiment.
The behavior of a 15 s normalized cost function collected by the PD of the 19-element FLPA is shown in Fig. 9. When the phase error and tip/tilt errors are not corrected, the system is in an open loop state: the average (Ave) of the cost function is 0.0182 and the mean square error (MSE) is 0.0162, where Ave and MSE are dimensionless results. After 36 ms, the system enters the phaselocked state in which the iteration rate is 9.8 kHz (the convergence time of the SPGD algorithm is defined as the time taken for the normalized cost function to reach 90% of the steady state value). When the phase error is corrected and the system is in a closed loop state (PL), the Ave of cost function rises to 0.335 and the MSE decreases from 0.0162 to 0.0021. After 670 ms, the phase errors and tip/tilt errors are both corrected (PL&TT) in which the iteration rate of tip/tilt controlling is around 1 kHz. The Ave of the cost function reaches 0.994, and the MSE rises slightly to 0.0034 due to the fluctuations of the fiber end face in AFOC during the tip/tilt control. Comparing the cost function curves in the three states, it can be seen that the energy concentration in the pinhole may reach the highest level only when the phase and tip/tilt errors are corrected simultaneously.  A 5 s long exposure pattern of the combined beam in the far-field is shown in Fig. 10. PIB is obtained by calculating the ratio of the pixel intensity in the bucket to the whole far-field image's pixel intensity, which reflects the relationship between the central lobe power of the combined beam and the total transmitting power. Here, the bucket's size is the pixel area occupied by the Airy Disk when the beam amplitude is evenly distributed. Panel (a) shows the situation when the phase-lock and tip/tilt controls are not performed. It is seen that there is nearly no interference pattern at the camera's target surface, i.e. the system is in an incoherent state, and the energy concentration of the combined beams is very weak. The corresponding mean value of PIB is 0.08. Panel (b) shows the far-field spot long exposure image when the phase-lock control is performed without tip/tilt control. A bright spot appears on the target surface of the camera, which is the central lobe of the combined beam, together with four side lobes around the central lobe, which means that after the phase-locked control, the combined beam is characterized by some coherent combining effects. At this stage, the combined beam has a mean PIB of 0.26, which is 0.18 higher than that of the open loop state. Panel (c) is a far-field long exposure image in which the phase and tip/tilt errors are both corrected (PL&TT). When the closed loop state is stabilized, there are 7 clear spots in the far-field. At this stage, the intensity of the central lobe is significantly higher than that of the other two states, the strength of the side lobe is weakened, and the distribution is more uniform. The average PIB of the combined beam is 0.49, reaching the highest level during the entire experiment.
Using the same transmit power, the amplitude of the transmit array may be redistributed by adjusting AM to obtain a Gaussian distribution of the envelopes with different Tr values. The comparison between the simulation results and the experimental results is shown in Fig. 11 (a) and the normalized long-exposure intensity patterns of the combined beams in far-field are shown in Fig. 12. Experimental results show that as Tr increases, the amplitude envelope of the transmitting array changes from a uniformly distributed flat-topped type to a discretely distributed Gaussian type. PIB also gradually increases from 0.49 (Tr = 0) to 0.56 (Tr = 1.3), and PIB is increased by 14.3% compared to the maximum value during the entire experiment. Then, as Tr continues to increase, PIB decreases and even go below the initial value 0.49. Moreover, we perform two experiments with different total transmit power values. During each experiment, the total transmit power remains unchanged, and Tr obtained by numerical simulation is used to adjust the amplitude distribution of the beamlets. As shown in Fig. 11(a), when P total is 1.52 mW and 0.76 mW, PIB shows the same trend. This means that the average power density of the combined beam can be improved by amplitude modulation and the best experimental results are obtained only when Tr is appropriate. Fig. 11(b) shows the normalized one-dimensional beam intensity distribution of the combined beams at different Tr. As Tr increases from 0 to 1.3, the peak power of the side lobes decreases because the power is transferred to the central-lobe. This change is also seen directly in Fig. 12. Experimental results show that central-lobe energy can be further improved via tuning the amplitude envelope of the beamlets in the near-field after the phase and tip/tilt errors are both corrected. This method is suitable for applications where the total input power of the FLPA is limited. For example, when the FLPA is used for space debris cleaning.

Summary
The traditional approach to improve PIB of FLPA is increasing the filling factor of the transmitting plane and introducing phase lock and tip/tilt controls into the system. To further enhance PIB of FLPA, we employed a method based on amplitude weighting, which allowed us to increase the central lobe energy. Numerical simulations showed that fiber phased array with Gaussian amplitude envelope shows a considerable enhancement effect on far-field central lobe energy compared to FLPA systems using uniform amplitude distribution of aperture laser emission. We found the same behavior with 7, 19, 37, and 61 FLPA systems at the same filling factor. Optimal performance was obtained with Tr = 1.3. Experimental results showed that PIB gradually increased from 0.49 (Tr = 0) to 0.56 (Tr = 1.3), after correcting the phase and tip/tilt errors. The experimental results were in agreement with the theoretical analysis and confirmed that amplitude modulation is a promising way to achieve high energy density fiber laser output. Our results demonstrated a new method to optimize PIB of FLPA besides the tip/tilt control and phase-locked controls under limited total power output.