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Multi-dimensional and large-sized optical phased array for space laser communication

Ronglei Han, Jianfeng Sun, Peipei Hou, Weijie Ren, Haisheng Cong, Longkun Zhang, Chaoyang Li, and Yuxin Jiang

Open Access Open Access

Abstract

In this study, we propose a novel multi-dimensional and large-sized optical phased array theory for space laser communication that addresses the theoretical limitations of the conventional optical phased array. We theoretically analyzed the principle of this phased array technology. The results of simulation and laboratory experiment clearly showed it can realize the large scanning angle and high optical gain required for communication. The novel optical phased array theory is of great significance to the revolution of miniaturization and networking in the field of space laser communication.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Over the past two decades, space laser communication technology has developed rapidly. A series of experiments were conducted in the form of satellite-ground and intersatellite communication. It is eager to establish laser communication networks to meet the requirements of information globalization. In space laser communication networks, optical phased arrays [1] (OPAs) can realize the weight reduction and miniaturization of laser terminals by replacing the traditional mechanical gimbal, and the link between different terminals can be quickly switched.

The concept of a phased array appeared in the microwave field and was applied to a variety of phased array radars. The optical phased array uses a laser as the carrier and is not subject to radio interference. Several types of OPAs have been developed, such as electro-optical crystal phased arrays, [2] liquid crystal phased arrays, [3] optical waveguide phased arrays [4], and MEMS phased arrays. [5]

In the field of OPA, most studies on phased arrays focus on improving the scanning angle and miniaturization, [6,7] or reaching high transmitting power, [8,9] and there are very few studies on large-size requirement for long-distance laser communication. In the integrated photonics fields, there are many studies on integrated phased array, [10] Jie Sun et al. reported a nanophotonic phased array with 4096 elements on a 576µm×576µm chip. [11] And it also has been reported that the optical phased array has achieved 512 elements. [12,13] These optical phased arrays have high integration and low power consumption, but the photonics platform and micron size of the array means that the emitting aperture is not very large and only a few millimeters at most, which aperture cannot meet the requirement in long-distance laser communication. Moreover, some optical phased arrays use wavelength scanning to realize two-dimensional scanning, [12–15] which has a complicated structure; in addition, the method occupied a large communication band, which has a negative effect on space laser communication. Most conventional OPA theories just control one-dimensional phase for scanning.

There are two important requirements for space laser communication based on an optical phased array: a two-dimensional large scanning angle and high optical gain. Space laser communication needs two optical terminals to keep the link stable during relative movement; therefore, the beam should have a large scanning angle in two dimensions for the needs of pointing and tracking. However, for the laser communication link, the emitting optical system must have some gain, which usually called “emitting antenna gain” or “emitting optical gain” (In the following, it will be referred to “optical gain”), denoting the ratio of the power radiated by the optical antenna to the power of an ideal point light source to omnidirectional radiation. And beams usually needs to transmit at an approximate angle of diffraction limit, thus larger the optical aperture, the smaller the diffraction limit angle, and the higher the optical gain.

For conventional optical phased arrays, the two requirements cannot be met perfectly. Taking a two-dimensional optical phased array with a number of elements N×N as an example, if the radius of a single beam is r, the light frequency is f0, wavelength is λ and the outgoing beam is a plane wave. In the Cartesian coordinate system at the plane of 0 rad beam emission, the optical field of the beams can be expressed as follows:

$$\begin{aligned} U({x,y,t} ) &= \sum\limits_{m = 1}^N \sum\limits_{n = 1}^N \{ {A_{mn}} \cdot circ\left( {\frac{{\sqrt {{{(x - {x_{mn}})}^2} + {{(y - {y_{mn}})}^2}} }}{r}} \right) \\ &\cdot \exp [{i({2\pi {f_0}t + M(t )+ {\phi_{mn}}} )} ]\\ &\cdot \exp \left[ {i\left( {2\pi \frac{{{x_{mn}}\sin {\theta_{xmn}}(t) + {y_{mn}}\sin {\theta_{ymn}}(t)}}{\lambda }} \right)} \right] \} \end{aligned}$$
and
$${\phi _{mn\_piston}} = \frac{{2\pi }}{\lambda }[{{x_{mn}}\sin {\theta_{xmn}}(t) + {y_{mn}}\sin {\theta_{ymn}}(t)} ]$$
where Amn is the amplitude of each beam, x and y are the coordinate values, and the point (x = 0, y = 0) is the center of the array. ${x_{mn}}$ and ${y_{mn}}$ are the components of the distance between the center of the mn-th beam and the center of the whole array in the two-dimensional direction. ${\phi _{mn}}$ is the phase carried by each beam, $M(t)$ is the modulated phase of communication, and m, n = 1, 2, 3, …, N. Equation (2) expresses the piston phase. Piston phase is the phase along the direction of the optical wave-vector, which is the only one-dimensional phase control of a conventional OPA.

We can observe that the single beam of the conventional OPA has a special phase that is related to the position of the beam in the array in Eq. (2). The phase difference between the beams should be sufficient for the combined beam in the far-field scan, and the difference is controlled on only one dimension for the scanning in two directions. The maximum scanning angle in two dimensions is determined by the diffraction factor of a single beam, that is, the diffraction limit, which can be expressed as

$$\sin {\theta _{s\_\max }} < \frac{{1.22\lambda }}{{2r}}$$

For the conventional OPA, increasing the scanning angle will inevitably reduce the size of the array elements; thus, a large number of elements are required to achieve a large aperture and high optical gain for communication; however, numerous elements can make complex systems difficult to control. On the contrary, large-sized elements lead to smaller scanning angles, which means that the two requirements cannot be met perfectly.

In the next parts, we propose a type of multi-dimensional and large-size optical phased array that has large scanning angle and high optical gain at the same time. The “multi-dimensional” here means we add phase control of two more dimensions on the basis of the original controlling piston phase, which called tip phase and tilt phase control. The theory of this OPA will be of great significance to the revolution of miniaturization and networking in the field of space laser communication.

2. Principles of multi-dimensional and large-sized optical phased array

In contrast to conventional OPA, the novel theory we introduce here uses some methods to meet the requirement of multi-dimensional control and large size. The multi-dimensional phase includes tip/tilt phase (the phase controls the wave-plane of single beam) and piston phase. For example, it can achieve the phase control of a transmitted beam or reflective mirrors, as shown in Fig. 1. Beams through the transmission-type phase control device or mirrors will change the tip/tilt phase, which have incline wave plane. However, the important point is that not only the phase of beams has the step distribution, but also the phase plane of each beam has a slope in x or y direction, as shown in Fig. 2, which is taken as an example in the x-direction scanning to express the phase distribution of beams (the same effect in the y-direction). The array is a type of programmable blazed grating, and the peak of the combined beam blazes at the scanning angle.

 figure: Fig. 1.

Fig. 1. Two ways to control the phase of beams (a. transmission type; b. reflection type)

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 figure: Fig. 2.

Fig. 2. Phase distribution of beams (x-direction as example) (a. conventional type; b. proposed phased array)

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It is also assumed that the number of elements is N×N, and the communication system is based on phase modulation. When the scanning angle is 0 rad, the light field of a single beam is as follows:

$${E_{mn}}({x,y,t} )= {A_{mn}} \cdot circ\left( {\frac{{\sqrt {{{(x - {x_{mn}})}^2} + {{(y - {y_{mn}})}^2}} }}{r}} \right) \cdot \exp [{i({2\pi {f_0}t + M(t )+ {\phi_{mn}}} )} ]$$
where the parameters are the same as that in Eq. (1).

When the array is scanned, the emitting light field of a single beam becomes:

$$\begin{aligned} {E_{mn}}^{\prime}({x^{\prime},y^{\prime},t} ) &= {A_{mn}} \cdot circ\left( {\sqrt {{{(\frac{{x - {x_{mn}}}}{{r\cos {\theta_{xmn}}}})}^2} + {{(\frac{{y - {y_{mn}}}}{{r\cos {\theta_{ymn}}}})}^2}} } \right)\\ &\cdot \exp [{i({2\pi {f_0}t + M(t )+ {\phi_{mn}}} )} ]\\ &\cdot \exp \left[ {i\left( {2\pi \frac{{({x - {x_{mn}}} )\sin {\theta_{xmn}}(t) + ({y - {y_{mn}}} )\sin {\theta_{ymn}}(t)}}{\lambda }} \right)} \right] \end{aligned}$$
where λ is the wavelength, ${\theta _{xmn}}(t) = {\theta _x}(t) + \delta {\theta _{xmn}}(t)$ and ${\theta _{ymn}}(t) = {\theta _y}(t) + \delta {\theta _{ymn}}(t)$ is the scan angle of the beams, ${\theta _x}(t)$ and ${\theta _y}(t)$ are the components of the ideal angle value, and $\delta {\theta _{xmn}}(t)$, $\delta {\theta _{ymn}}(t)$ are the components of the angle error value.

From Eq. (5), it can be seen that the phase of the beam can be three dimensional, which can be expressed as

$$\left\{ \begin{array}{l} {\phi_{mn\_piston}} ={-} \frac{{2\pi }}{\lambda }[{{x_{mn}}\sin {\theta_{xmn}}(t) + {y_{mn}}\sin {\theta_{ymn}}(t)} ]\\ \quad{\phi_{mn\_tip}} = \frac{{2\pi }}{\lambda }[{x\sin {\theta_{xmn}}(t)} ]\\ \quad{\phi_{mn\_tilt}} = \frac{{2\pi }}{\lambda }[{y\sin {\theta_{ymn}}(t)} ]\end{array} \right.$$

The tip and tilt phase can make perfect phase slope enough as in Fig. 2(b). The piston part here is opposite to that in Eq. (2), which will result in phase difference between beams and it must be compensated. Each emitting beam path is connected to an optical phase shifter to control the piston phase difference and random phase difference. We assume that the phase of the phase shifter is $\Delta {\phi _{mn}}$, and it should satisfy the following conditions under phase synchronization:

$$\Delta {\phi _{mn}} + {\phi _{mn}} + {\phi _{mn\_piston}} = \Delta {\phi _{11}} + {\phi _{11}} + {\phi _{11\_piston}}$$

Because the phase shifters have maximum phase range, the ${\phi _{mn\_piston}}$ is not completely compensated, the actual compensated piston phase is

$$\phi{_{mn\_piston}} ={-} \bmod [|{\phi _{mn\_piston}}|,\frac{R}{2} - \bmod (\frac{R}{2},2\pi )]$$
where “mod” is the function to get the remainder, R (rad) is the single-direction adjustable range of phase shifters (the value of R can be set freely, usually the half of the maximum range). And Eq. (7) would change into:
$$\Delta {\phi _{mn}} + {\phi _{mn}} + \phi {^{\prime}_{mn\_piston}} = \Delta {\phi _{11}} + {\phi _{11}} + \phi {^{\prime}_{11\_piston}} + 2q\pi \,,\,\,q\,\textrm{is an integer}\textrm{.}$$

The adjustment in Eq. (8) and Eq. (9) can cause the time delay of beams, which has influence on the laser communication since the modulation codes of different time will overlap together. We assume the maximum overlap ratio is 10% of one code as the standard, the communication capacity is B (bit/s) and the light speed is c, there is the condition as follows

$$|{\phi _{mn\_piston}} - \phi {^{\prime}_{mn\_piston}}|\le \frac{1}{{10}}\frac{{2\pi }}{\lambda }\frac{c}{B} = \frac{{\pi c}}{{5\lambda B}}$$

The change in the piston phase value needs to be calculated in real time and quickly compensated for by the phase shifter. In this regard, we use the downlink beams to accomplish phase compensation, using the method of reflective mirrors as an example, as shown in Fig. 3. While emitting beams, the array can also receive the optical signal transmitted by the other terminal. The received light (also called the downlink beam) can be coupled into multi-channel optical fibers after passing through the mirror array and collimators. The receiving beam carries the piston phase to be compensated for according to the characteristics of the emitting and receiving coaxial beams. With phase locking, the phase shifter compensates for the phase difference of the downlink beam, and phase synchronization of the beam emitted from the phased array is also achieved. Finally, the mirrors and phase shifters realize a three-dimensional phase adjustment.

 figure: Fig. 3.

Fig. 3. Phase locking based on the downlink beam

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In Eq. (9), the phase of each beam is consistent with that of the first beam (m = 1, n = 1); hence, the single beam becomes

$$\begin{aligned} {E_{mn}}^{\prime\prime}({x,y,t} ) &= {A_{mn}} \cdot circ\left( {\sqrt {{{(\frac{{x - {x_{mn}}}}{{r\cos {\theta_{xmn}}}})}^2} + {{(\frac{{y - {y_{mn}}}}{{r\cos {\theta_{ymn}}}})}^2}} } \right)\\ &\cdot \exp \left[ {i\left( {2\pi \frac{{x\sin {\theta_{xmn}}(t) + y\sin {\theta_{ymn}}(t)}}{\lambda }} \right)} \right]\\ &\cdot \exp [i(2\pi {f_0}t + M(t) + \Delta {\phi _{11}} + {\phi _{11}} + \phi {^{\prime}_{11\_piston}} + 2q\pi )] \end{aligned}$$

Equation (11) shows that the phase of a single beam is related to x and y and achieves the perfect slope of the phase plane without increasing the number of elements. The emitting beams are like being reflected from a single large plane mirror, which is completely different from that in the conventional OPA and makes a discrete stepped phase. The improvement is equivalent to subdividing one beam into an infinite number of small sub-beams, and the radius ${r_{sub - beam}}$ of the sub-beam is very small; according to Eq. (3), the array can realize a large scanning angle.

The superposition field of multiple beams at the far field at a distance L is

$${I_L}({x_L},{y_L},t) = {\left|{\sum\limits_{m = 1}^N {\sum\limits_{n = 1}^N {\frac{{\exp \left( {\frac{{i2\pi L}}{\lambda }} \right)}}{{i\lambda L}}\int\!\!\!\int {{E_{mn}}^{\prime\prime}(x,y,t)} \exp [i\pi \frac{{{{({x_L} - x)}^2} + {{({y_L} - y)}^2}}}{{\lambda L}}]} } dxdy} \right|^2}$$

${I_L}$ is limited by the amplitude of a single beam whose area decreases with increasing scanning angle.

Under the condition of phase coincidence, the light power of the main peak on the axis in the far field is N2 times as much as that in the incoherent state. The theoretical maximum optical gain of the main peak of the optical phased array in the beams emitting direction depends on N and r, which can be expressed as

$$G = {\left( {\frac{{2\pi Nr}}{\lambda }} \right)^2}\textrm{ }$$

By calculating the optical gain, we can evaluate the main peak power density of the phased array in the far field. The higher the gain, the higher signal light power the receiving terminal can obtain, which could improve the performance of laser communication.

3. Numerical simulation

In the numerical simulation, we assume the distance as 2000 km and the wavelength as 1550nm, which is the common case in the laser communication between satellites. According to the real conditions, the link parameters are assumed as listed in Table 1.

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Table 1. Link Parameters

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From the listed parameters, we can determine that the emitting gain needs to be 101.44 dB, and the equivalent emitting diameter should be larger than 58 mm. Based on this, we set the diameter of a single beam as 8 mm, the array dimensions as 8×8, and the distance between the beams as 16 mm, as shown in Fig. 4. We assume all the emitting beams have the same phase after phase locking, and we simulate the optical field distribution of the far field at an angle of 0 rad, as shown in Fig. 5. The Fig. 5 shows that there are several coherent peaks, with the main peak on the axis having the highest power density. The center of the main peak is the position for the terminal to receive optical signal. When the beams scan, the gain of the main peak will change as in Eq. (14), where ${\left[ {\frac{{2{J_1}(\alpha )}}{\alpha }} \right]^2}$ is the diffraction factor, J1 is the first order Bessel function, $\alpha = \frac{{2\pi r{\theta _s}}}{\lambda }$ and the scanning angle is θs.

$$\left\{ \begin{array}{c} {G_{conventional}} = {\left( {\frac{{2\pi Nr}}{\lambda }} \right)^2}{\left[ {\frac{{2{J_1}(\alpha )}}{\alpha }} \right]^2}\\ {G_{novel}} = {\left( {\frac{{2\pi Nr\cos {\theta_s}}}{\lambda }} \right)^2} \end{array} \right.$$

 figure: Fig. 4.

Fig. 4. 8×8 beam array

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 figure: Fig. 5.

Fig. 5. Optical field distribution of far field

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The theoretical maximum absolute gain of the two type of OPA is the same, which is at the position of θs = 0 rad, and its value is 1.68×1010 (102.25 dB). In Fig. 6 we normalized the gain at different angles by the maximum.

 figure: Fig. 6.

Fig. 6. Optical gain of main peak at different angles

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The gain of the main peak decreases with increasing scanning angle, as shown in Fig. 6, clearly indicating the change in gain of the novel and conventional OPA. We assume the angle range of the gain drops to 50% (−3 dB) as the maximum usable angle for laser communication, denoted as $\Theta $; evidently, the $\Theta $ values of the two types of OPA are as follows:

$$\left\{ {\begin{array}{l} {{\Theta _{conventional}} = 1.992 \times {{10}^{ - 4}}rad}\\ {{\Theta _{novel}} = 0.785 \times 2 = 1.57rad} \end{array}} \right.$$

It is apparent that under the same condition of the 8×8 array, the multi-dimensional phase control can effectively improve the usable scanning angle by approximately 7881.5 times compared to that by the conventional control; moreover, the gain can be maintained at a high level for a large angle range, that means, in the satellite-satellite laser communication, the optical terminal has the larger scanning and angle high optical power for APT (Acquisition, Pointing, Tracking). In contrast, the conventional OPA must adjust the number of elements to balance the scanning angle and optical gain, which means that the size of the elements must be small and make the stepped phase smoother. However, smaller elements should match more quantities and more complex device to obtain a large aperture for optical gain in long-distance transmission and that's not what we want.

4. Novel optical phased array communication experiment based on a 2×2 array

Based on the idea of using the above multi-dimensional large-sized optical phased array and our current condition, we designed a 2×2 array experiment for principle verification in laboratory. The phases of the four emitting beams were controlled by each FSM (Fast Steering Mirror) for two-dimensional scanning of the beam. The structure diagram of the devices is shown in Fig. 7, and the parameters are presented in Table 2.

 figure: Fig. 7.

Fig. 7. Structure diagram of the 2×2 novel OPA experiment

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Table 2. Parameters

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In Fig. 7, terminal A is the emitting part of the OPA, and terminal B is the receiving part. In this experiment, the two terminals use the same laser seed source. The emitting and receiving optical paths were coaxial through the circulator. Terminal B transmits one beam to terminal A as the downlink light, and the closed loop formed by the DSP (Digital Signal Processor) controls the phase shifter to lock the phase of the four received downlink beams at terminal A. Because the emitted signal light and downlink light are in the same path, the phase of the four outgoing beams is also locked, and a clear and stable interference pattern will be generated on the CCD (Charge Coupled Device) at the focal plane of the lens. Furthermore, the fiber coupling point at terminal B is at the maximum light power of the pattern; that is, the phase of the emitting beams is consistent, and therefore, the coupling light power of terminal B will reach a maximum. The signal light coupled at terminal B and the local light were mixed in the optical coherent receiver and demodulated signal. In this experiment, the relative motion of the two terminals was simulated by the complementation of the rotation of the FSM and the rotation stage.

After the closed-loop phase locking by the DSP was turned on, the far-field interference pattern was obtained using the monitoring camera, as shown in Fig. 8. Similar to the simulation above, the pattern has many peaks, and the red circle indicates the calibration point of the optical axis, which is also the fiber coupling position. The monitoring light power at terminals A and B is shown in Fig. 9. It can be seen from the figure that the multi-beam combining power at the two ends both have a step change at the start of the loop, and the value is significantly enhanced, which is the sign of consistent phase. At this time, the phase was locked to the same state.

 figure: Fig. 8.

Fig. 8. Far-field interference pattern

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 figure: Fig. 9.

Fig. 9. Monitoring power at the terminal A (DSP monitoring signal) and terminal B

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We define the phase-locking efficiency as the ratio of the actual beam power to the theoretical value, expressed in Eq. (16), which denoted as γ. The higher the phase-locking efficiency, the better the phase consistency and the stronger the fiber coupling signal at the receiving terminal.

$$\gamma = \frac{P}{{{{(\sqrt {{P_1}} + \sqrt {{P_2}} + \sqrt {{P_3}} + \sqrt {{P_4}} )}^2}}}$$
where P is the actual total optical power after beam combination, and P1, P2, P3, and P4 are the respective optical powers of each beam.

The scanning angles of the beams at ±20 mrad, ±10 mrad, and 0 mrad were selected for data acquisition and analysis. We collected the power of a single beam and the total power by phase-locking the two terminals at different angles to calculate the phase-locked efficiency; the efficiency and the actual optical gain at far field is presented in Table 3.

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Table 3. Phase Locking Efficiency and Optical Gain

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The data in Table 3 indicate that the downlink beam phase was locked at terminal A through phase shifters, and the phase-locking efficiency could reach a maximum of 95.5%. At the same time, the calibration-point at the far-field also reached the peak power, and the phase-locking efficiency was better than 74%. It could be seen that this optical phased array can maintain high power density at different angles. In terms of optical gain, the received light at terminal B could achieve a high gain at each angle, which is higher than 80.40 dB (the theoretical maximum gain is 81.7 dB). The actual data were lower than the theoretical results mainly due to the phase-locking efficiency—higher efficiency leaded to higher gain. Regardless of this, we can calculate that when the angle of the conventional OPA increased to ±177.9 μrad, the gain would decline from maximum (81.7 dB) to 80.40 dB, which demonstrated that our proposed array was improved 112.4 times in terms of the scanning angle comparing to the conventional type.

From the data above, it can be seen that after adding the phase control of two more dimensions through mirrors, the novel array is more suitable for free-space laser communication compared with the conventional-type array.

Finally, we also evaluated the communication performance using an eye diagram through the Q factor and the system bit error rate (BER). the theoretical BER is calculated by Eq. (17). [16]

$$\textrm{BER} = \frac{1}{2}\textrm{erfc}\left( {\frac{Q}{{\sqrt 2 }}} \right)$$
the The receiving eye diagram of terminal B was selected at a scanning angle of 0 mrad, as shown in Fig. 10, from which we could see a clear eye. The received signal power, Q factors and BER at different angles are calculated and presented in Table 4.

 figure: Fig. 10.

Fig. 10. Eye diagram at terminal B

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Table 4. Q Factor and System BER at Different Angles

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From the data in Table 4, we could see that the Q factor was above 9.47, and its maximum value was 10.30. The theoretical system bit error rate of different angles was low, and communication can be implemented without using forward error correction (FEC) at the BER. The results showed that the receiving performance was effective, the 625 Mbps BPSK communication was realized in this experiment, and the novel optical phased array proposed in this paper had basic communication performance.

5. Conclusion

In this paper, we proposed a novel theory of multi-dimensional and large-sized optical phased arrays for space laser communication, which controls the phase in three dimensions to meet the needs of high optical gain and large scanning angle of laser communication. The results showed that the multi-dimensional OPA clearly improved the scanning angle without loss of gain in the case of few elements, which is more suitable for space laser communication. Finally, we designed a 625Mbps BPSK communication experiment based on a 2×2 reflective optical phased array for verification, and the scanning angle reached ± 20 mrad. It was shown that the phase-locking efficiency was higher than 74%, and the optical gain was above 80.40 dB. The scanning angle of the proposed array in the experiment was improved by 112.4 times compared to that of the conventional one, in addition to improving the communication performance. The Q factor was up to 10.30, and receiving performance is effective. The experiment confirmed that our novel optical phased array can achieve basic communication performance. The novel optical phased array theory is of great significance to the revolution of miniaturization and networking in the field of space laser communication. In future work, we will incorporate true time delay communication and improve the scanning angle.

Funding

National Key Research and Development Program of China (2020YFB0408302); Chinese Academy of Sciences Key Project (ZDRW-KT-2019-1-01-0302); National Natural Science Foundation of China (Grant No. 91938302); Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB43030400).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (10)
Fig. 1.
Fig. 1. Two ways to control the phase of beams (a. transmission type; b. reflection type)

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Fig. 2.
Fig. 2. Phase distribution of beams (x-direction as example) (a. conventional type; b. proposed phased array)

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Fig. 3.
Fig. 3. Phase locking based on the downlink beam

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Fig. 4.
Fig. 4. 8×8 beam array

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Fig. 5.
Fig. 5. Optical field distribution of far field

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Fig. 6.
Fig. 6. Optical gain of main peak at different angles

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Fig. 7.
Fig. 7. Structure diagram of the 2×2 novel OPA experiment

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Fig. 8.
Fig. 8. Far-field interference pattern

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Fig. 9.
Fig. 9. Monitoring power at the terminal A (DSP monitoring signal) and terminal B

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Fig. 10.
Fig. 10. Eye diagram at terminal B

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Tables (4)

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Table 1. Link Parameters

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Table 2. Parameters

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Table 3. Phase Locking Efficiency and Optical Gain

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Table 4. Q Factor and System BER at Different Angles

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Equations (17)

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U ( x , y , t ) = ∑ m = 1 N ∑ n = 1 N { A m n ⋅ c i r c ( ( x − x m n ) 2 + ( y − y m n ) 2 r ) ⋅ exp ⁡ [ i ( 2 π f 0 t + M ( t ) + ϕ m n ) ] ⋅ exp ⁡ [ i ( 2 π x m n sin ⁡ θ x m n ( t ) + y m n sin ⁡ θ y m n ( t ) λ ) ] }
ϕ m n _ p i s t o n = 2 π λ [ x m n sin ⁡ θ x m n ( t ) + y m n sin ⁡ θ y m n ( t ) ]
sin ⁡ θ s _ max < 1.22 λ 2 r
E m n ( x , y , t ) = A m n ⋅ c i r c ( ( x − x m n ) 2 + ( y − y m n ) 2 r ) ⋅ exp ⁡ [ i ( 2 π f 0 t + M ( t ) + ϕ m n ) ]
E m n ′ ( x ′ , y ′ , t ) = A m n ⋅ c i r c ( ( x − x m n r cos ⁡ θ x m n ) 2 + ( y − y m n r cos ⁡ θ y m n ) 2 ) ⋅ exp ⁡ [ i ( 2 π f 0 t + M ( t ) + ϕ m n ) ] ⋅ exp ⁡ [ i ( 2 π ( x − x m n ) sin ⁡ θ x m n ( t ) + ( y − y m n ) sin ⁡ θ y m n ( t ) λ ) ]
{ ϕ m n _ p i s t o n = − 2 π λ [ x m n sin ⁡ θ x m n ( t ) + y m n sin ⁡ θ y m n ( t ) ] ϕ m n _ t i p = 2 π λ [ x sin ⁡ θ x m n ( t ) ] ϕ m n _ t i l t = 2 π λ [ y sin ⁡ θ y m n ( t ) ]
Δ ϕ m n + ϕ m n + ϕ m n _ p i s t o n = Δ ϕ 11 + ϕ 11 + ϕ 11 _ p i s t o n
ϕ m n _ p i s t o n = − mod [ | ϕ m n _ p i s t o n | , R 2 − mod ( R 2 , 2 π ) ]
Δ ϕ m n + ϕ m n + ϕ m n _ p i s t o n ′ = Δ ϕ 11 + ϕ 11 + ϕ 11 _ p i s t o n ′ + 2 q π , q is an integer .
| ϕ m n _ p i s t o n − ϕ m n _ p i s t o n ′ | ≤ 1 10 2 π λ c B = π c 5 λ B
E m n ′ ′ ( x , y , t ) = A m n ⋅ c i r c ( ( x − x m n r cos ⁡ θ x m n ) 2 + ( y − y m n r cos ⁡ θ y m n ) 2 ) ⋅ exp ⁡ [ i ( 2 π x sin ⁡ θ x m n ( t ) + y sin ⁡ θ y m n ( t ) λ ) ] ⋅ exp ⁡ [ i ( 2 π f 0 t + M ( t ) + Δ ϕ 11 + ϕ 11 + ϕ 11 _ p i s t o n ′ + 2 q π ) ]
I L ( x L , y L , t ) = | ∑ m = 1 N ∑ n = 1 N exp ⁡ ( i 2 π L λ ) i λ L ∫ ∫ E m n ′ ′ ( x , y , t ) exp ⁡ [ i π ( x L − x ) 2 + ( y L − y ) 2 λ L ] d x d y | 2
G = ( 2 π N r λ ) 2  
{ G c o n v e n t i o n a l = ( 2 π N r λ ) 2 [ 2 J 1 ( α ) α ] 2 G n o v e l = ( 2 π N r cos ⁡ θ s λ ) 2
{ Θ c o n v e n t i o n a l = 1.992 × 10 − 4 r a d Θ n o v e l = 0.785 × 2 = 1.57 r a d
γ = P ( P 1 + P 2 + P 3 + P 4 ) 2
BER = 1 2 erfc ( Q 2 )
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