## Abstract

In this paper, a novel phase modulation technique based on a corner-cube reflector (CCR) array is proposed and demonstrated experimentally. The piezoceramics are linked behind each CCR. When the beams irradiate on the CCR array, the phase modulation can be realized by applying a voltage to piezoceramics to control the spatial location of each CCR. The piston phase errors of the device itself are compensated by employing the stochastic parallel gradient descent (SPGD) algorithm. Then, the piezoceramics are loaded with preset voltages to obtain the expected phase, and the anticipative optical field is generated. In the experiment, the piston phase errors of the 7-way and 19-way CCR array are corrected well. In order to further verify the phase control capability of the device, a vortex beam carrying orbital angular momentum (OAM) of 1 is generated by utilizing the 6-way CCR array. The experimental results confirm the feasibility of the concept.

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

## 1. Introduction

The corner-cube reflector (CCR) is a beam retroreflective element, the incident beam can return along the original path after being reflected by three mutually perpendicular reflective surfaces. In addition, as the CCR rotates along the normal to the center of the incident surface, the reflection ability is not changed. Owing to these special properties of the CCR, it is widely used in optical communication [1,2], lasers [3,4], target homing [5] and ranging [6–9]. In these application, the large diameter CCR is often required, unfortunately, it is difficult to machine. Consequently, the CCR arrays are generally employed to replace the large diameter CCR. For example, The LRRR (Laser Ranging Retro Reflector), which was placed on the Moon by Apollo 15, consisted of 300 CRRs, is used to measure the distance between the Earth and the Moon [6]. Until 2018, the researchers from Yunnan Observatories, Chinese Academy of Sciences can still receive signals returned by the LRRR [9].

Due to manufacturing errors and installation errors, there are co-phasing errors in a CCR array, which reduce the peak power of the reflected beam in the far-field, and weaken the performance of the CCR array. Therefore, the correction of the piston phase errors for the CCR array is of great importance, but relatively little research has been done in this area. As early as in 1982, Takuso Sato et al designed a coherent optical adaptive technique based on 2×2 CCRs to generate phase-conjugate wavefront, the electrostrictive material Pb(Mg_{1/3}-Nb_{2/3})O_{3}(PMN) was employed to control piston phase of the CCR [10]. In our previous work, the piston phase errors of three CCRs were corrected manually by observing the far-field distribution, which proved that phase modulation can be achieved by controlling the position of each CCR [11]. Additionally, the piezoceramics are often used in deformable mirrors of adaptive optical systems to correct aberrations in real time [12,13]. Based on these principles, a multi-aperture phase modulation technique based on the CCR array is proposed. The spatial position of each CCR is controlled by employed piezoceramics, and utilizing the stochastic parallel gradient descent (SPGD) algorithm, the piston phase errors are compensated. Then, the piezoceramics are loaded with preset voltages, and accurately control the spatial position of each CCR to achieve phase modulation.

In recent years, coherent beam combining (CBC) technology has been used in optical field modulation [14–16]. However, in CBC systems, in addition to piston aberrations, there are also tilted aberrations and dynamic phase noise, which complicates phase modulation [17]. In the system we designed, due to the characteristics of the CCR, there are only piston phase errors. In addition, phase modulation is realized by controlling the spatial position of the CCR, which can generate the phase stably. In the experiment, the piston phase errors of 7-way and 19-way CCR arrays are well corrected. Simultaneously, compared with spatial light modulators, this device can modulate large-aperture beams without a telescopic beam shrinking system. In order to further verify the phase control capability of the device, a vortex beam is generated. Due to carrying the orbital angular momentum (OAM), the vortex beam is widely applied [18–21]. Meanwhile, its optical field distribution is quite sensitive to phase, when the phase change around the optical axis is integer multiple of 2π, the optical field is a ring. Otherwise, the optical field has an obvious low-intensity gap in the intensity ring [22,23]. Therefore, the vortex beam carrying OAM of 1 is generated, the experimental result is in good agreement with the simulation result, demonstrating that the device designed has excellent phase-modulation capability.

## 2. Theoretical analysis and simulation

Figure 1(a) shows the ideal CCR array, and the phase difference of each CCR is an integer multiple of 2π, and can be considered as co-phasing. Nonetheless, manual installation is difficult to ensure sub-wavelength accuracy, and each mirror is not machined with the same accuracy, resulting in the piston phase errors in the CCR array, as shown in Fig. 1(b). Figure 1(c) displays the schematic of a CCR array with *n _{j}* CCRs on the

*j*th circle

*.*Thus, the parallel light is reflected by the CCR array can be expressed as a collection of some sub-beams, as follow.

*j*represents the circle number of the CCR array, and

*g*denotes the CCR number on

*j*th circle. Here each sub-beam should be linearly polarized with identical polarization directions. In this paper, we assume that 19 CCRs form an array unit, and several array units are spliced together to form a larger array, as shown in Fig. 3(c). In Eq. (1),

*s*is the array unit number, $A_{j,g}^s$ and $\phi _{j,g}^s$ are the amplitude and relative piston phase of each sub-beam respectively. Here, $\vec{\rho }_{j,g}^s = {\vec{r}^s} - \vec{r}_{j,g}^s$ is the radius vector with respect to $\vec{r}_{j,g}^s$, which represents the center coordinate of the

*s*th array unit, and ${\vec{r}^s}$ is the radius vector of the

*s*th array unit in the input plane. The complex amplitude on the far-field can be obtained by the Fourier transform of the near field

*E*with some additional phase factors.

_{near}*z*is transmission distance, ${\vec{r}_{far}}$ represents the radius vector in the observation plane.

*k*= 2π/λ is the wave number with wavelength

*λ*. The intensity in the far-field can be calculated by the following formula. According to Eq. (1) ∼ (3), the far-field of the beam reflected by the 6-way CCR array with piston phase error is shown in Fig. 2(a), which displays random interference without any specificity. Figure 2(b) depicts the far-field image of the beam reflected by the 6-way ideal CCR array, it is an ideal Airy disk. Figure 2(c) displays the normalized transverse intensity profiles comparison of (a) and (b). From Fig. 2(c), when there is no piston phase error, the far-field distribution is more concentrated and the sidelobe is lower in power. In contrast, when piston phase errors exist, the far-field light intensity drops significantly. Thus, it is possible to determine whether the piston phase errors exist in the CCR array according to the intensity distribution in the far-field.

Next, we analyzed the impact of the piston phase errors on the far-field as the CCR increases. Figure 3 (a)-(c) display the CCR distribution of 7-way, 19-way, and 57-way CCR array respectively. Figure 3 (d)-(f) show the statistical results of the Strehl Ratio (SR) on the far-field for the corresponding CCR array with random piston aberrations and 10000 simulations. The statistical interval is 0.01. The abscissa is the value of the SR and the ordinate is the proportion of different SR values.

Table 1 illustrates the mode, mean, and variance of the statistical data. As the number of the CCR increases, the mode and mean of SR decrease, and the variance become also smaller. The above data indicate that the piston phase errors greatly affect the performance of the CCR array, particularly large-scale CCR arrays. For example, when detecting the signal reflected by the CCR array with the piston phase errors, it will increase the difficulty of receiving the signal. In terms of the mean, if we correct the piston aberration, the SR of the CCR array with 7-way, 19-way, and 57-way will increase by 2.3, 4.7, 10.8 times, respectively. As a consequence, it is necessary to use phase correction methods for the CCR array, which can greatly improve the performance of the CR arrays.

The methods of phase-locked control based on far-field information are similar to the active phase control optimization algorithm methods in CBC systems, most of which employ the SPDG algorithm. Thus, the SPDG algorithm can also be utilized to correct the piston phase errors of the CCR array. After correcting the piston phase errors of the CCR array, it can be considered that no co-phasing errors in the CCR array. Then, preset voltage is applied to the piezoceramics to control the spatial position of each CCR, and generate the expected phase. In our work, a vortex beam is generated to verify the performance of device designed, and the phase of each sub-beam should be as follows.

Here,*n*is the total number of the CCR on

_{j}*j*th circle,

*l*is the topological charge of the vortex beam. According to Eq. (4), the voltage of the

*g*th CCR on

*j*th circle should be:

*V*

_{2π}is the voltage required for piezoelectric ceramics to extend one wavelength. Then, the optical field reflected by the CCR array can be expressed as

## 3. Experimental device

Figure 4 displays the designed 19-way CCR array. Figure 4(a) shows the designed CCR unit, composed of the CCR, cylindrical sleeve, and piezoceramics. The CCR is placed in the sleeve, and the sleeve is connected to the piezoceramics. Since the beam reflected by the CCR can return in the original path, the optical axes of the sub-beams reflected are parallel, meanwhile, rotating the CCR can not affect its performance. Thus, in the designed CCR array, only piston phase errors exist. And the front and back displacement can be controlled via using the piezoceramics actuators, which can meet the precision of the nanometer. Accordingly, the spatial position of the CCR can be precisely controlled theoretically to modulate the piston phase of the reflected sub-beam. Figure 4(b) is the manufactured pedestal for mounting the CCR. Figure 4(c) displays the installed 19-way CCR array. The depth and diameter of the CCR are 22mm and 25.4mm, respectively, and the distance of adjacent CCR is 31mm. In addition, the effective diameter of the CCR array is 149.4mm, and the duty cycle is 55%. As the CCR unit can be unloaded and installed at will, and several pedestals designed can be assembled together, as consequence, different arrangements and larger-scale CCR arrays can be obtained, and different phases can be generated.

The experimental setup as shown in Fig. 5. The laser is emitted from the optical fiber as a point source, and becomes parallel light after passing through the lens L_{1}. Then, the parallel light is reflected by the CCR array placed behind the lens L_{1}, and the reflected beam is composed of a series of sub-beams. The reflected beam passes through the lens L_{1} again and is split into two beams by the beam splitter mirror BS. One beam is focused on the CCD to observe the far-field, and the other beam is focused into the optical fiber in the original way, which collected by the photodetector (PD) through the optical fiber circulator, which obtains the magnitude of the total light intensity in bucket. Then, the PD converts the light signal into the electrical signal, transmits it to the control computer, and runs the SPGD algorithm to correct the piston phase errors of the CCR array, forming a closed-loop system.

Here, we take advantage of the characteristics of the CCR, the beam emitted from the optical fiber is reflected by the CCR array and then returns to the light source following the original path. There is no need to align the PD with the lens when building the optical path, which greatly simplifies the optical system, reduces environmental interference, and improves the robustness of the system.

## 4. Experimental results

The experiment results in this paper demonstrate that the designed 19-way CCR array has excellent phase modulation capability. The laser wavelength in the experiment is 1064 nm, the diameter and focal length of the lens L_{1} are 180mm and 1000mm respectively. The results in the following experiments were normalized. As shown in Fig. 6, we firstly selected 7 CCRs at the center for the experiment of piston phase error correction. Figure 6(a) displays the far-field before correction, which is diffuse due to the piston phase errors, and the peak power is 0.46. The corrected optical field is shown in Fig. 6(b), the peak power is 1. It is obvious that the intensity is much improved and the peak power is more than doubled. Figure 6(d) is the variation curve of the power in bucket (PIB) during the correction process. The PIB is at a relatively low level before correction, after the closed-loop, the PI B increased steadily and converged at 13.5s. At 21.2s, the control of the SPGD algorithm is withdrawn, the system is in the open-loop, and the PIB remains still relatively stable. Figure 6 (c) shows the far-field after 20 minutes in the open-loop, the peak power drops to 0.98, almost unchanged, proving that the system has superior stabilized. Figure 6(e) is the far-field simulation results of the 7-way CCR array without piston phase errors, the corrected result is almost the same as the simulation result. Figure 6(f) indicates the normalized transverse intensity profiles comparison of simulation and experiment results. The main lobe curves are almost identical, and the side lobe power of the experiment is slightly larger than the simulation result, demonstrating that the piston phase errors of the CCR array are well compensated.

Subsequently, the piston phase errors correction of the 19-way CCR array is carried out, as shown in Fig. 7. Figure 7(a) displays the far-field before correction, the peak power is 0.22. Compared with Fig. 6(a), due to the increased number of the CCR, the far-field is more diffuse and peak power is lower, which is consistent with the analysis in Fig. 3 above. Figure 7(b) shows the far-field after correction, the peak power is 1, which is increased by 4.5 times. The variation of the PIB in the far-field is shown in Fig. 7(d). At t = 22.4s, the algorithm control is cancelled, and the PIB remains stable in the open-loop. Figure 7(c) is the far-field after 20 minutes in the open-loop, the peak energy drops to 0.98, and the light field is nearly unchanged. Figure 7(e) displays the far-field simulation result of the 19-way CCR array, the experimental results are almost the same as the simulation results, demonstrating that the piston phase error of the 19-way CCR array can also be corrected well. Figure 7(f) depicts the normalized transverse intensity profiles comparison of simulation and experiment results. The two curves are slightly different, and the overall trend is uniform. The main lobe of the experiment results is wider, and the power of the side lobes is larger, which may be related to the numerical aperture of the lens L_{1}. In the experiment, the effective diameter of the 19-way CCR array is slightly smaller than the lens L_{1}, which may introduce additional aberrations to the sub-beam reflected by the outer ring CCR, such as spherical aberration.

The above experiments illustrate that the piston phase errors of the CCR array can be corrected well. Meanwhile, after correction, the expected optical field can be obtained by applying preset voltage to the piezoceramics. In the next experiment, in order to further verify the phase control capability of the device, the 6 CCRs on the first ring are utilized to generate a vortex beam with *l *= 1. The 6-way CCR array distribution as shown in Fig. 8(a). According to Eq. (4), the phase loaded of each sub-beam can be calculated, Fig. 8(b) displays the spiral phase with *l *= 1 loaded of the 6 sub-beams, and its far-field simulation results as shown in Fig. 8(c), whose center is a hexagonal ring. Figure 8(d) depicts the initial far-field of the 6-way CCR array. After correction, the far-field is closed to the ideal far-field, as shown in Fig. 8(e), which demonstrate that the piston phase errors of the CCR array are corrected. Then, the piezoceramics behind each CCR are loaded with preset voltages to generate a fragmented spiral phase. Here, the voltage can be obtained according to Eq. (5) and the measured V_{2π}=11V(λ=1064nm). The voltages loaded of the 6 CCRs are 1.83V, 3.67V, 5.5V, 7.33V, 9.17V, 11V, as shown in Fig. 8(f). The experimental result is shown in Fig. 8(g), which is almost the same as the simulation result, a hexagonal intensity ring without low-intensity gap exists at the center. According to the description of Ref. 15, the topological charge of the fragmented spiral wavefront at the input plane is consistent with the far-field optical field. From Fig. 8(f), the voltages are spirally distributed, and displacement variations of the piezoceramics correspond to a wavelength, namely 2π (*l *= 1), therefore, we have reason to believe that the experimentally far field is an optical vortex with *l *= 1. The above experiments demonstrate that this method can precisely modulate the phase and generate the desired optical field.

## 5. Discussion

In the above experiments, 6 elements on a ring are employed to generate a vortex beam, and the same is done for coherent beam combining technology. However, can 7-way array with the central element be used to generate a vortex beam? Fig. 9 displays the experimental and simulation results of the vortex beam with l = 1 by utilizing the 7-way CCR array. The 7-way CCR array with the central CCR distribution as shown in Fig. 9(a). In Fig. 9(b), the fragmented spiral phase with *l *= 1 is loaded on the ring, and the phase is 0 at the center. However, the experiment and simulation results in a crescent-shaped optical field, which is obviously not an integer-order vortex beam. Since a vortex beam can already be generated by employed 6-way CCR array, this optical field can be regarded as the result of the coherent combination of a vortex beam and a plane wave with phase = 0 at the center.

Furthermore, in simulation, the vortex beams with *l *= 1 are generated by utilizing 18-way CCRs without the central CCR and 19-way CCRs, as shown in Fig. 10. From Fig. 10(c) and (f), the optical field generated by 19-way CCRs is close to the vortex beam obtained by 18-way CCRs. Therefore, we infer that the effect of the central CCR on the generated vortex beam is related to the number of CCRs, namely spatial resolution, and the effect of the central CCR can be neglected when there are enough CCRs. For example, when using a spatial light modulator to generate a vortex beam, the center of the loaded spiral phase is not a singularity, but some pixels whose phase is 0 or 2π, however, this does not affect the generation of the vortex beam. As a result, in the input of low spatial resolution, removing the central element, the vortex beam can be better generated.

In the experiment, the voltage frequency is set to 50 Hz. In actuality, the changing frequency of piezoceramics can reach kHz, thus allowing for faster modulation. In addition, the piezoceramics used in this paper can supply 9.82um displacement at a voltage of 150V. Therefore, for 1064nm wavelength, we only need to consider ±10V voltage control, without the use of high-voltage amplifiers, which reduces system complexity and control difficulties. And under this control voltage, the effect of the hysteresis nonlinear characteristics of the piezoceramics is relatively small. At present, many researchers have investigated this area [24–26]. And in practical applications, we will consider employing these methods to compensate for it to enable more accurate phase control. Moreover, we merely employ 19 CCR in this paper, in the near future, we will comprehensively investigate the CCR array with hundreds of elements, which can modulate larger aperture beams and can also be employed to correct for atmospheric turbulence. Furthermore, the polarization adjustment mask and intensity adjustment mask can also be installed to each unit of the CCR array to generate vector vortex beam or other structured beams, such as dark hollow beam and flat top beam [27], which deserves further study.

## 6. Conclusion

In conclusion, a new phase modulation technique based on the CCR array is proposed and experimentally demonstrated. During the closed-loop, the piston phase errors of the CCR array can be compensated, and based on the correction results, an anticipative optical field can be obtained by applying pre-set voltage to the piezoceramics. In the letter, the piston phase errors of 7-way and 19-way CCR array are corrected, and a vortex beam with *l *= 1 is generated by utilizing 6 CCRs, which demonstrate that the designed device can accurately control the piston phase. This method improves the performance of the CCR array, and greatly expands its application scenarios. Meanwhile, it has a good application prospect in the optical field manipulation for large-aperture beams.

## Funding

National Natural Science Foundation of China (62005286, 62175241); Frontier Research Fund of Institute of Optics and Electronics, China Academy of Sciences (C21K006); Equipment Pre-research Key Laboratory Fund (6142A04190212).

## 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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