Generation of dark hollow beam via coherent combination based on adaptive optics

Yi Zheng; Xiaohua Wang; Feng Shen; Xinyang Li

doi:10.1364/OE.18.026946

1. Introduction

In recent years, coherent beam combination (CBC) techniques of laser arrays have attracted much attention because of their success in saving beam quality where dynamic phase noise exists, such as in high-power systems with thermal-effect aberrations [1,2] or in long-distance propagation with atmospheric turbulence [3,4]. CBC techniques with an adaptive optics (AO) servo, such as a master oscillator power amplifier (MOPA) [1,5] and an optical phased array [6], have a powerful ability to correct phase noise in unit beams. Thus, an output laser could achieve high coherent efficiency and keep good beam quality.

The propagation properties of a laser array have been investigated comprehensively. The results indicate that some special beams, such as a dark hollow beam (DHB) [7], a flat-topped beam (FTB) [8], and a vortex beam [9,10] can be obtained by using a laser array with a suitable initial phase and amplitude distributions. Beams with different intensity profiles are useful in various fields––for example, in optical trapping and manipulations, particularly in biophysical science for trapping living cells and organelles [11,12], and in atomic physics for manipulating neutral atoms [13,14]. For such applications, Gaussian beams and FTBs are suited to trap micron-sized particles with a refraction index higher than that of the ambient index, whereas doughnut-shaped beams, such as DHBs and Laguerre–Gaussian beams, have to be adopted to trap particles with a refraction index lower than that of the ambient index. A potential application of DHBs carrying optical vortices propagating through atmospheric turbulence would be in optical communication, where the topological charge could be used as the information carrier [15]. Actually, methods for generating different types of beams and beam shaping have already been proposed and studied extensively in areas such as holograms [16], diffractive optics [17], and specific waveguides [18]. Most of these methods are often implemented with a single beam and/or fixed phase plate. The motivation for using AO and a laser array scheme is mainly because of the significant advantages in solving the problem of beam quality in a dynamic phase noise environment, which is unavoidable in some applications such as optical communications [10,15] and high-power systems [19,20]. A laser array can be regarded as multiple sub-apertures divided from a large single beam, and the correction of phase noise, including not only piston and small-tilt errors but also high-order phase aberrations, can be accomplished via the piston error correction of each unit beam by the AO servo [3,4].

In this paper, a novel method for generating a DHB is proposed. The concept of using a laser array and an AO servo is studied theoretically and experimentally in detail. In Section 2, the theoretical formulation is presented. The experimental configuration and results are given in Section 3 and Section 4, respectively. Finally, Section 5 summarizes the main results obtained in this paper.

2. Theoretical formulation

Consider the laser array shown in Fig. 1 . The unit beams are located symmetrically on rings, and the radius difference between two adjacent rings is equal to R. Each beam is a single-mode Gaussian laser. The complex amplitude of N unit beams on input plane $(x, y; Z = 0)$ is

E (x, y; Z = 0) = \sum_{j = 0}^{N - 1} E_{j} (x, y; Z = 0) = \sum_{j = 0}^{N - 1} a_{j} \cdot \exp (i ϕ_{j}) \cdot \exp [- \frac{{(x - x_{j})}^{2} + {(y - y_{j})}^{2}}{ω_{j}^{2}}],

where

a_{j}

,

ϕ_{j}

,

ω_{j}^{}

, and

(x_{j}, y_{j})

are the amplitude, relative phase, waist radius, and the center coordinates of the j ^th unit, respectively.

Fig. 1 Schematic diagram of the laser array on input plane. The red part indicates the actual laser array in our experiment. The dashed lines represent the annular area where the beams are located symmetrically.

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The propagation of the beam passing through a first-order optical system parameterized by an ABCD transfer matrix is characterized by the Huygens–Fresnel diffraction integral [21]. The field distribution of the unit beam on the output plane $(u, v; Z = z)$ could be expressed as

\begin{array}{l} E_{j} (u, v; Z = z) = - \frac{i k}{2 π B} \exp (i k L) \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} E_{j} (x, y; Z = 0) \\ \times \exp {\frac{i k}{2 B} [A (x^{2} + y^{2}) + D (u^{2} + v^{2}) - 2 (x u + y v)]} d x d y . \end{array}

Here,

k = 2 π / λ

is the wave number with wavelength λ, and L is the axial optical path length. Substituting Eq. (1) into Eq. (2) and after tedious integral calculations, the field distribution could be described as

\begin{array}{l} E_{j} (u, v; Z = z) = - a_{j} (\frac{i z_{R}}{α}) \exp (i k L + i ϕ_{j}) \exp [\frac{i k A (x_{j}^{2} + y_{j}^{2})}{2 α}] \\ \times \exp [- \frac{i k (x_{j} u + y_{j} v)}{α}] \exp [\frac{i k β (u^{2} + v^{2})}{2}], \end{array}

where

z_{R} = k ω_{0}^{2} / 2

is the Rayleigh distance,

ω_{0}

is the uniform waist radius of the unit beams,

α = B - i A z_{R}

, and

β = (D - i C z_{R}) / α

. It should be noticed that in free space the transfer matrix satisfies

A D - B C = 1

. Equation (3) indicates that the Gaussian property of the unit beam will not be changed after propagating through an ABCD system without an aperture. The combined field is

\begin{array}{l} E (u, v; Z = z) = \sum_{j = 0}^{N - 1} E_{j} (u, v; Z = z) \\ = - (\frac{i z_{R}}{α}) \exp (i k L) \exp [\frac{i k β (u^{2} + v^{2})}{2}] \\ \times \sum_{j = 0}^{N - 1} a_{j} \exp [\frac{i k A (x_{j}^{2} + y_{j}^{2}) - 2 i k (x_{j} u + y_{j} v)}{2 α} + i ϕ_{j}], \end{array}

and the corresponding intensity distribution reads as

I (u, v; Z = z) = E (u, v; Z = z) E^{*} (u, v; Z = z) .

The transfer matrix of the beam path shown in Fig. 2 could be calculated as

Fig. 2 Beam path of the laser array passing through a thin lens.

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\begin{array}{l} (\begin{matrix} A & B \\ C & D \end{matrix}) = (\begin{matrix} 1 & z \\ 0 & 1 \end{matrix}) (\begin{matrix} 1 & 0 \\ - 1 / f & 1 \end{matrix}) (\begin{matrix} 1 & l \\ 0 & 1 \end{matrix}) \\ = (\begin{matrix} 1 - \frac{z}{f} & z + (1 - \frac{z}{f}) l \\ - \frac{1}{f} & 1 - \frac{l}{f} \end{matrix}) . \end{array}

Consider the Fraunhofer diffraction in which the length parameters become $l = 0, z = f$ . Equation (6) could be simplified to

(\begin{matrix} A & B \\ C & D \end{matrix}) = (\begin{matrix} 0 & f \\ - \frac{1}{f} & 1 \end{matrix}) .

If the field on the output plane is the DHB, the intensity of the center spot should be zero:

I (0, 0; Z = z) = 0.

From Eqs. (4), (7), and (8), we can obtain the formula of the DHB field:

(\sum_{j = 0}^{N - 1} a_{j} \exp (i ϕ_{j})) {(\sum_{j = 0}^{N - 1} a_{j} \exp (i ϕ_{j}))}^{*} = 0.

There are infinite solutions to this formula. Here we consider a special case in which seven unit beams are hexagonally distributed, as in the red part in Fig. 1. According to the symmetrical characteristics, the amplitude and phase of the six surrounding beams should be equal and are assumed to be $a_{1}$ and $ϕ_{1}$ . Then Eq. (9) is simplified to

a_{0}^{2} + 36 a_{1}^{2} + 12 a_{0} a_{1} \cos (ϕ_{1} - ϕ_{0}) = 0.

One solution is

a_{1} = 1 / 6 a_{0} (\approx 0.17 a_{0}), (ϕ_{1} - ϕ_{0}) = π .

It should be clarified that we only are concerned with one period, owing to the $2 π$ periodicity of phase. Now we can substitute Eq. (11) and (7) into Eq. (4) and compute the far field via numerical simulation. The related parameters refer to the experimental values: wavelength $λ = 1064 nm$ , focus length $f = 1000 mm$ , waist diameter $d = 10 mm$ , and space between two rings $R = 16 mm$ . Figure 3 shows the simulation results:

Fig. 3 Numerical simulation of DHB generation. (a) The phase distribution of the near field. The center and surroundings obey $(ϕ_{1} - ϕ_{0}) = π$ . (b) The amplitude distribution of the near field. The center and surroundings obey $a_{1} = 1 / 6 a_{0}$ . (c) The far field of the DHB.

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In the far field the DHB appears, and the obvious dark area exists in the center. Furthermore, the influences of the amplitude and phase distributions are analyzed. Figure 4 shows the far fields deriving from different near fields. Respectively, Fig. 4(a) is in different amplitude distributions while phases keep $(ϕ_{1} - ϕ_{0}) = π$ , and Fig. 4(b) is in different phase distributions while amplitudes keep $a_{1} = 0.17 a_{0}$ . The curves are $(u, v = 0)$ profiles of the corresponding patterns.

Fig. 4 Far fields deriving from different near fields. (a) Different amplitude distributions. The phases keep $(ϕ_{1} - ϕ_{0}) = π$ . (b) Different phase distributions. The amplitudes keep $a_{1} = 0.17 a_{0}$ .

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From Fig. 4(a) we can conclude that the amplitude distribution would influence the intensity distribution of the center area. The center becomes darker while the ratio of $a_{1}$ and $a_{0}$ increases to the proper extent. A special case of $a_{1} = 0.016 a_{0}$ should be noticed, in which the center is flat-topped. It indicates that the FTB could also be generated via the same scheme. Besides, Fig. 4(b) indicates that the phase distribution would influence the form of the far field. Particularly, according to [9,10], the vortex beams could be obtained with a suitable phase distribution. Discussion about this is beyond our topic, so we will not go into that detail. Here, the phase shift is adverse because it will destroy the dark-hollow character. Therefore, when the phase noise is dynamic, an AO servo is necessary for phase-locking. Experimentally, the amplitudes of laser arrays are correspondingly stable; thus, the stabilization of $(ϕ_{1} - ϕ_{0}) = π$ is the main purpose and challenge. We design the zero-order interference detection method to extract real-time piston errors by using a reference beam. When the reference beam superposes the unit beam coaxially without tilt error, a thorough combination will be achieved and a zero-order interference fringe will appear. The complex amplitude of j ^th fringe is

Z_{j} = u_{j} + u_{r e f} = a_{j} \exp (i ϕ_{j}) + a_{r e f} \exp (i ϕ_{r e f}) .

Then the intensity is

I = Z_{j} Z_{j}^{*} = a_{j}^{2} + a_{r e f}^{2} + 2 a_{j} a_{r e f} \cos (ϕ_{r e f} - ϕ_{j}) .

Equation (13) indicates that the intensity fluctuation of the fringe corresponds with the piston error between two beams. Especially, when the piston error is zero the fringe is brightest, and when it is π the fringe is darkest. Therefore, to achieve the phase-locking of $(ϕ_{1} - ϕ_{0}) = π$ , the AO servo should stabilize the center fringe in the brightest status and the surrounding sixes in the darkest. Furthermore, the taking place of the zero-order interference phenomenon requires constraint of the tilt angle, which is

θ_{t i l t} << \arcsin \frac{λ}{π 2 ω_{0}} .

In our experiment, this value should be $θ_{t i l t} << 3
.39 \times 10^{-8} rad$ , which means that the tilt error is hardly acceptable. Precise adjustment of the beam path is required to eliminate the tilt errors.

3. Experimental configuration

Figure 5 shows a block diagram of the experimental configuration. All of the fiber-connected devices are polarization-maintaining so that interference efficiency could be guaranteed. The master oscillator is a semiconductor laser generator protected by an isolator. It provides the single-mode laser with a 1064 nm wavelength and a linewidth less than 1 MHz. The output power is adjustable from 0 to 150 mW. The beam is split by splitter1. One is for producing the seven unit beams via splitter2, and another is for producing the reference beam. The intensity adjuster in each channel is employed to control the amplitude distribution. Then the seven fiber lasers are linked to the AO servo with a beam-arraying structure (BAS) and an active segmented mirror (ASM). These two parts are adopted to array the unit beams into a hexagonal distribution and to provide proper phase modulation. The output beam is divided into two parts by BS1. One is for the sample system. The beam is focused by Lens1 ( $f = 1000 mm$ ), and a high-speed CMOS is employed to record the far field. A 4× micro-objective lens is employed to expand the pattern size to be suitable for the detecting area. The second part is for the zero-order interference detection system, which can extract the piston errors with the help of the custom-made photo-detectors (PDs) array. The last part behind BS2, with a camera and monitor, helps to solve the difficulty in the manual adjustment of the beam expander and BAS. The tilt errors will be eliminated once the reference beam and the unit beams are confocal on the focus plane of Lens2.

Fig. 5 Experimental configuration of seven-laser CBC system based on AO. The thin line represents the fiber link and the thick represents the cable link.

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3.1 The experimental AO servo

Figure 6 shows the main devices of the AO servo. The ASM, shown in Fig. 6(a), is composed of seven sub-mirrors with hexagonal distribution. Each sub-mirror is round in shape with a 16 mm diameter and is supported by three piezoceramic actuators. They are driven by a high-voltage amplifier (HVA) and can supply $\pm 2 μm$ displacement. The sub-mirror can modulate the reflected beam with 3 degrees of freedom, including the piston and tilts. In our current experiment, three actuators are controlled by the same voltage, so the mirror moves vertically and only the piston modulation is implemented. Figure 6(b) shows a picture of the BAS. It is composed mainly of the collimators, mounting brackets, and a hexagonal pyramid. The waist diameter of the Gaussian beam emitted from the collimator is 10 mm. The collimators are held by the mounting brackets with 4 degrees of freedom. The glass hexagonal pyramid is axis-drilled and polished, with a wavelength reflective coating of 1064 nm (>99%). Six beams are reflected by the pyramid, and the 7^th beam goes through the axis in the fashion of Fig. 6(c). All of the parts are assembled on an aluminum supporting plate and adjusted very carefully to assure that the seven beams are horizontal and coaxial.

Fig. 6 Key devices of AO servo. (a) The ASM and the HVA. (b) The BAS. (c) Schematic diagram of the hexagonal pyramid.

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3.2 The feedback control loop of AO servo

The feedback control loop of an AO servo is mainly composed of a piston extracting system and a PC-based algorithm platform. The piston extracting system is designed according to a zero-order interference phenomenon. Figure 7 shows the main devices of it. An adjustable varifocal telescope is employed to be the beam expander, as shown in Fig. 7(a). It can expand the collimated beam into the reference beam with a large diameter (D> = 50 mm) and can be precisely adjusted via the multi-degrees-of-freedom mounting bracket. A LiNbO₃ phase modulator (PM) and function generator are equipped in this channel to import phase noise. Although the unit beams are still in a stable condition, the dynamic piston error in the reference channel could completely represent the actual case in which unit beams are dynamic, whereas the reference is stable. For an experimental test of the performance of the AO servo, there is no difference between these two cases, thanks to the relativity of the piston phase. The seven zero-order fringes then irradiate on the PDs array, whose photosensitive areas are arranged to correspond with the distribution of the laser array, shown in Fig. 7(b). The size of the sub-area is $5 \times 5 mm$ . The intensity of each fringe could be detected individually and quantified in the form of voltage signals by the gain-adjustable amplifying circuit. These signals are acquired by an AD card on the PC platform. The feedback control signals are calculated by an extremum approach (EA) algorithm and exported to the HVA by a DA card.

Fig. 7 Key devices of piston extracting system. (a) The beam expander. (b) The PDs array.

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The EA algorithm works as in the following steps:

(1) Generate a sine wave and export it to the ASM to sweep the phase. The intensities of the fringes will fluctuate. Record the corresponding voltage curves. After adequate periods, the maximum and minimum voltage of each channel can be identified.
(2) Set the voltage standards to $V_{j}^{\max}$ and $V_{j}^{\min}$ . The values are 90% of the maximum voltage and 110% of minimum voltage of j ^th channel, respectively.
(3) Acquire the real-time signal from the PD arrays at the k ^th iteration, $V_{j}^{(k)}$ , which reflects the piston error between j ^th beam and the reference.
(4) The feedback control voltages are calculated by:
$V_{j}^{(k + 1)} = V_{j}^{(k)} + γ_{j} (V_{j}^{s t d} - V_{j}^{(k)}) .$

Particularly, while $j = 0$ , $V_{j}^{s t d} = V_{j}^{\max}$ , else $V_{j}^{s t d} = V_{j}^{\min}$ ; that means the algorithm implements the maximization in the center channel and the minimization in the surrounding six channels. Therefore, once the algorithm is enabled, the phase-locking of $(ϕ_{1} - ϕ_{0}) = π$ could be accomplished. $γ_{j}$ is the proportion factor that influences the control effect.

4. Experimental investigation

The first step of our experiment is the calibration of the PD arrays. We close the unit channels so the device is irradiated by the reference beam only. Then we adjust the gain knobs of the amplifying circuit to make the voltage signal from each sub-PD be equal. Then we close the reference and open the unit beams to adjust the amplitudes. As the required distribution is $A_{1} = 0.17 A_{0}$ , the intensity of the six surrounding beams should be 2.9% of the center. Finally, we stop the intensity adjustment, open all the channels, and adjust the gain knobs once again to make the fluctuation of voltage signals be suited to the measurement range of the AD card (1~4v). This is beneficial in order to improve the precision and stability of the AO servo.

The second step is to simulate the dynamic environment via LiNbO₃ PM. The function generator exports a sine wave to the PM, as shown in Fig. 8(a) . The amplitude and frequency are $\pm 0.25$ V and 170 Hz. As the half-wavelength voltage of the PM is 2 V, the amplitude of the piston error is $π / 4$ .

Fig. 8 Voltage signals recorded by oscilloscope. (a) The sine wave provided by function generator. (b) The PD output signal of center channel in open loop and (c) in closed loop. The Y-axis (voltage) scales are respectively 200 mV, 2 V,2 V and the x-axis (time) scales are both 2 ms.

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The AO correcting effect can be recognized from Figs. 8(b) and 8(c), which record the voltage signal of the center channel ( $V_{0}$ ) in open and closed loops, respectively. One figure is adequate because the other channels obey the same rule, thanks to the parallel character of the feedback. The figure indicates that the AO servo successfully stabilized the intensity of the zero-order fringe in the maximum (or minimum) phase. To extract the piston error from the voltage signal, we can review Eq. (13) and find

I_{j}^{\max} = a_{j}^{2} + a_{r e f}^{2} + 2 a_{j} a_{r e f}, I_{j}^{\min} = a_{j}^{2} + a_{r e f}^{2} - 2 a_{j} a_{r e f} .

So the intensity of zero-order fringe can be expressed as

I = (I_{j}^{\max} + I_{j}^{\min}) / 2 + \cos (ϕ_{r e f} - ϕ_{j}) \cdot (I_{j}^{\max} - I_{j}^{\min}) / 2.

As the voltage signal is identical to the intensity, the piston extracting formula is

ϕ_{r e f} - ϕ_{j} = \arccos [\frac{2 V - (V_{j}^{\max} + V_{j}^{\min})}{V_{j}^{\max} - V_{j}^{\min}}] .

Depending on Eq. (18) we can obtain the evolution curves of piston errors in a π period, shown in Fig. 9 . When the control loop is open, the piston errors between the reference beam and the unit beams dither rapidly and randomly. When the loop is closed, the piston phase of the center channel is locked to be synchronized with the reference channel [Fig. 9(a)]; in the meanwhile, the six surrounding channels are locked with the π piston shift [Fig. 9(b)]. The AO servo works effectively against dynamic phase noise.

Fig. 9 Evolution curves of piston errors between reference channel and unit channels. The upper is the center channel, and the lower is the representative one of six surrounding channels.

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To demonstrate the capability of the AO servo, we calculate the phase spectral density (PSD), shown in Fig. 10(a) . The curves are obtained by applying the Fourier transform to the piston errors. The peak on 170 Hz is owing to the imported phase noise. The second peak on 30 Hz, which could be also detected in the other non-presented PSD curves, is ascribed to the circuit noise in the HVA. In addition, the error rejection transfer function (ERTF) in Fig. 10(b) reveals the control bandwidth. The curve is obtained by

Fig. 10 AO capability analysis. (a) The phase spectral density in two stages. (b) The error rejection transfer function.

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E R T F = \log_{10} (\frac{P S D_{c l o s e}}{P S D_{o p e n}}) .

The zero passage frequency, which is approximately 100 Hz, presents the valid correction bandwidth of the AO servo. Furthermore, the characteristic of the frequency component is similar to that of systems with power amplifiers [5,22] or atmospheric turbulence [23–25], which means that our AO servo is also competent for those usages.

Figure 11 shows the 10 s long-exposure patterns of the far field in open and closed loops. Each pattern is the average of more than 20000 real-time frames recorded by the CMOS. When the loop is open, the far field displays random interference without any particularity. When the loop is closed, the DHB appears, and the intensity distribution is similar to the theoretical simulation. The beam quality is rather good. Nevertheless, the weakened symmetry is owing to the residual errors caused by the reasons such as the limited precision of intensity adjusters and detectors, maladjustments of high-frequency components, errors in polarization maintaining, and so on.

Fig. 11 10 s long-exposure patterns of far-field. (a) Open loop. (b) Closed loop. (c)Theoretical simulation. The intensity-scale bar is distinguished.

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5. Conclusion

In this paper, a novel method for generating a DHB from a laser array has been proposed and studied, both theoretically and experimentally, for the first time to our knowledge. It also has been demonstrated that a CBC with an AO servo is a powerful technique for beam shaping to eliminate the influence of dynamic phase noise in some applications. A variety of beam profiles such as FTB and vortex beams could also be realized by using such a concept, which deserves further study.

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Generation of dark hollow beam via coherent combination based on adaptive optics

Abstract

1. Introduction

2. Theoretical formulation

3. Experimental configuration

3.1 The experimental AO servo

3.2 The feedback control loop of AO servo

4. Experimental investigation

5. Conclusion

References and links

Cited By

Figures (11)

Equations (19)

Optics Express