Arbitrary multiple beam forming by two cascaded liquid crystal optical phased arrays

Liang Wu; Xiangru Wang; Xiaoxian He; Ziqiang Huang; XinNing Huang; Caidong Xiong

doi:10.1364/OE.26.017066

1. Introduction

Recently, there is growing interest in the use of optical phased arrays (OPAs) for high-precision laser beam controlling in a variety of applications, such as free-space optical (FSO) communications, laser radar (Lidar), and beam combining [1–3]. Unlike mechanical systems, which are burdened with complex mirror objects and rotary gimbals, OPA is a non-mechanical beam forming technology based on phase modulation of light by electronically controlling thousands of individual phase shifters. Thus, it enables fast, high-precision beam steering systems with low size, weight, and power consumption (SWaP). OPA based beam forming systems indeed naturally feature several significant capabilities, such as random-access pointing, multiple beam forming, and dynamic beam generating [4–6]. These advantages have prompted a body of research into OPA schemes, including silicon-on-insulator (SOI) [7,8], nanophotonic phased array (NPA) [9], multiple quantum wells (MQWs) [10], micro-electromechanical mirrors (MEMs) [11], and liquid crystal optical phased array (LC-OPA) [12,13]. Meanwhile, LC-OPA is a particularly promising approach due to the potential to achieve fast speed by various liquid crystal materials [14], large apertures with millions of pixels [15], wide operating optical bandwidth from visible to mid-wave infrared (MWIR) [16], and so on.

One important use case of OPA systems is to generate multiple simultaneous and reconfigurable beams. For OPA beam steerers to be adopted in satellite-based laser communication (Lasercom), one requisite capability is multi-access communication where one terminal may communicate with several terminals so that Lasercom networks can be established [17]. Also, in Lidar applications, beam steering is required to transmit multiple beams from local terminals and/or receive multiple beams reflected from many moving targets, which can enable rapid multiple-object targeting and tracking [18]. Several methods have been proposed to generate multiple beams on LC-OPAs. The most straightforward implementation is cascading a sequence of LC-OPAs directly, however, insertion loss will be accumulated and the number of beams is limited. Another method, the “sub-aperture array” method, can form multiple independent beams using independently programmed sub-apertures. Unfortunately, it suffers from a fundamental tradeoff: the divergence angle of the regenerated beam is inversely proportional to the size of the sub-apertures [19]. A third method is to employ computer generated holograms on liquid crystal spatial light modulators (LC-SLMs) to produce multiple beams. Nevertheless, this method usually requires a comparatively long time to complete the complicated calculations, and hence cannot attain real-time response in practical applications [20–22].

In this work, we propose a novel and potentially useful method of cascaded amplitude and phase (CAP) to achieve fast forming of multi-beam near-field wavefronts based on two custom-fabricated LC-OPAs. By using a 4-f imaging system to relay the optical field from the first LC-OPA to the second LC-OPA, the near-field distribution in terms of the derived amplitude and phase modulation can be obtained. Consequently, arbitrary multiple beams can be realized by a multi-beam forming system based on the CAP method.

2. Principles

Figure 1 shows the schematic of the cascaded amplitude and phase (CAP) method based on two transmissive LC-OPAs and a 4-f imaging system. The OPA-A and OPA-P are identical and responsible for amplitude and phase modulation, respectively. Both LC-OPAs are aligned accurately for pixel-by-pixel generation of the optical near-field of multiple beams. The 4-f system with two achromatic lenses is introduced to relay the optical field from OPA-A onto OPA-P. To satisfy the one-to-one 4-f relay condition, the two lenses must have a same focal length of $f$ , and have a separation of $2 f$ . In the following sections, the principles of LC-OPA and the CAP method will be discussed in detail.

Fig. 1 Schematic of the CAP method to form multiple beams using two transmissive LC-OPAs and a 4-f imaging system. The 4-f imaging system consists of two identical achromatic lenses with a focal length of 300 mm.

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2.1 Liquid crystal optical phased array

The LC-OPA we employed as a key component in our multi-beam forming method is a phase-only transmissive liquid crystal device using a positive nematic liquid crystals (NLC). It is a real-time programmable and addressable phase shifter, which can modify the wavefront by generating an arbitrary phase profile and deflect the beam into a specified direction. Figure 2 illustrates the structure (lower part) of the LC-OPA and the profile of phase modulation (upper part). The linearly polarized beam is incident normally on the transparent, plane-parallel NLC layer of thickness D (confined between two glass-based substrates). The upper substrate is lithographically patterned with transparent and conductive indium-tin-oxide (ITO) grating electrodes, and the lower substrate is the ITO common electrode. The homogeneous alignment LC cells are used, wherein the inside surfaces of the substrates are covered with a polyimide (ZKPI-410) alignment layer and rubbed to achieve a 2-degree pretilt of the aligned LC molecules. According to the physics of electrically controlled birefringence (ECB), the molecules of liquid crystal will rotate parallel to the direction of light propagation when an external electric field applied on the grating electrodes. This rotation will produce an effective refractive index change for the incident beam and consequently translate into a phase modulation, which is expressed by $ϕ = \frac{2 π}{λ} (n_{e} - n_{o}) D$ , where $n_{e}$ and $n_{o}$ are the extraordinary and ordinary refractive index of the LC, respectively [23]. Therefore, the advantage of LC-OPA is that it can flexibly introduce a phase delay across the whole aperture by electronically adjusting the voltage on each electrode unit, enabling agile non-mechanical beam forming and steering.

Fig. 2 Illustration of the structure of phase-only transmissive LC-OPA and the corresponding profile of phase modulation. The light is linearly polarized and incident normally on the LC-OPA, where the thickness of NLC layer is D, and the electrode period is $d$ . The beam is deflected into the direction of $θ_{s}$ , wherein $Δ ϕ$ is the phase shift between two adjacent electrodes.

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Based on the variable period grating (VPG) method, which adopts LC-OPA to approximate a sawtooth shaped phase profile with 2π phase resets, the desired steering angle of $θ_{s}$ for a normally input beam is determined by

\sin θ_{s} = Δ ϕ / (k_{0} d),

where

Δ ϕ

is the phase shift between two adjacent electrodes, as shown in Fig. 2,

k_{0} = \frac{2 π}{λ}

is the vacuum wave number,

λ

is the wavelength of the incident laser beam, and

d

is the electrode period (pitch). Note that the modulo 2π phase shift is utilized to have a unique first diffraction order in the far-field without high-order grating lobes.

Continuous phase distributions cannot be realized owing to the pixelated nature of LC-OPA. Instead, the phase distribution is translated into a discrete form by the digitized drive voltage, resulting in a staircase-like phase profile like the one shown in Fig. 2. As a result, the phase shift on $i$ -th electrode is calculated by

ϕ (i) = r e m [(i - 1) \cdot k_{0} d \sin θ_{s}, 2 π],

where

r e m (*, 2 π)

stands for the remainder operator. During operation, the desired phase shift on each electrode can be calculated by Eq. (2), and the corresponding driving voltage will be calculated using the experimentally measured nonlinear monotone function between phase retardation and driving voltage.

2.2 Cascaded amplitude and phase method

In order to discuss the general operating principles of the CAP method, a complex optical filed can be represented by

E = A (x, y) \exp [j Φ (x, y)],

where

A (x, y)

and

Φ (x, y)

are the amplitude and phase distribution, respectively. For one-dimensional beam forming, we assume that a single laser beam will be split into

n

desired beams in the far-field, and their corresponding angles are

θ_{1}, θ_{2}, \dots, θ_{n}

, respectively. In an ideal case, each of the multiple beams can be considered as a unit impulse function, so the far-field distribution can be represented as the superposition of all the beams

E_{f a r} = δ (θ - θ_{1}) + δ (θ - θ_{2}) + \dots + δ (θ - θ_{n}),

where

δ (*)

is the unit impulse function, and

θ

is the spatial angle between the beam propagation direction

\overset{⇀}{k}

and the z-axis, as shown in Fig. 1. According to diffraction theory, the spatial frequency is defined as

f_{x} = \sin θ / λ

, so the corresponding spectral distribution of the far-field of multiple beams can be calculated as

E_{f a r} = δ (f_{x} - f_{x 1}) + δ (f_{x} - f_{x 2}) + \dots + δ (f_{x} - f_{x n}) .

where

f_{x n} = \sin θ_{n} / λ

is the spatial frequency of the n-th beam. Because of the relationship of Fraunhofer diffraction between the far-field and near-field, the near-field intensity distribution can be calculated by the inverse Fourier transform, so

E_{n e a r} = \exp (j k_{0} \sin θ_{1} \cdot x) + \exp (j k_{0} \sin θ_{2} \cdot x) + \dots + \exp (j k_{0} \sin θ_{n} \cdot x),

where

x

is the coordinate in the near-field plane. Note that

x = i \cdot d

represents the position of the

i

-th electrode. By converting the near-field

E_{n e a r}

of Eq. (6) into the form of Eq. (3), i.e.

E_{n e a r} = A_{x} \exp (j Φ_{x})

, using Euler’s formula, we derive the amplitude term

A_{x}

and phase term

Φ_{x}

as:

A_{x} (i) = \frac{1}{n} \sqrt{n + 2 (\sum_{ξ = 1}^{n - 1} \sum_{η = 2}^{n} \cos (ϕ_{ξ} (i) - ϕ_{η} (i)))}, ξ \neq η,

Φ_{x} (i) = arc \tan \frac{\sum_{m = 1}^{n} \sin ϕ_{m} (i)}{\sum_{m = 1}^{n} \cos ϕ_{m} (i)} .

where

A_{x}

and

Φ_{x}

are the amplitude and phase distribution in x-direction. These distributions will be modulated by the OPA-A and OPA-P, respectively. The subscripts of

ξ

,

η

, and

m

represent the index of the multiple beams, respectively. For any desired number of beams with arbitrary directions, Eqs. (7) and (8) are used to calculate the required modulation for each LC-OPA, and the phase shifts of

ϕ (i)

are calculated by Eq. (2).

Theoretically, as we want multiple beams in the far-field, the near-field should involve an amplitude factor and a phase factor. However, the phase-only OPAs cannot realize amplitude modulation. Moreover, due to the thickness of the two phased array elements, or oblique incidence of light, simply stacking the elements will lead to diffraction effects between the two arrays, and thus errors relative to the above-mentioned near-field. To overcome these problems, the CAP method uses a precise 4-f imaging system to coherently image the amplitude modulation of OPA-A onto OPA-P pixel by pixel.

3. Numerical simulation

To illustrate the validity of the CAP method, two one-dimensional LC-OPAs were used in numerical simulations. The device parameters are listed in Table 1, which are the same as the parameters of our custom-fabricated LC-OPA devices. Thus, the simulation results can be used in the subsequent experiments. Note that with the purpose of simplifying the theoretical model, the following simulations are based on an ideal phase modulation ignoring the fringe effects between electrodes. Based on the theory of Fraunhofer diffraction, the far-field intensity distribution is calculated using the fast Fourier transformation (FFT).

Table 1. Simulation parameters for the CAP method.

View Table

Figure 3 shows the optical intensity distributions when the system is configured to produce 3 beams at ${0.1}^{\circ}, {0.15}^{\circ}, and {0.2}^{\circ}$ . Figure 3(a) is an image of the incident laser source, modeled as a Gaussian beam with a wavelength of 632.8 nm and diameter of 10 mm. It is normally incident on the center of the device so the entire effective apertures of LC-OPAs are illuminated by the input beam. As the OPA-A is addressed with amplitude modulation data calculated by Eq. (7), only the amplitude of the beam will be modulated, and the output intensity distribution after amplitude modulation by OPA-A is shown in Fig. 3(b). This amplitude-modulated beam travels through the OPA-P which is modulated with phase data generated according to Eq. (8). The diffraction patterns are depicted in Fig. 3(c). As we can see, the single beam is split into three quasi-equal beams in the far-field. The intensities are normalized with respect to the maximum intensity, and the corresponding intensity values are inserted beside to show that uniform multi-beam forming is achieved by the CAP method.

Fig. 3 Simulation results of optical intensity distributions when the system is configured to produce 3 beams at $({0.1}^{\circ}, {0.15}^{\circ}, {0.2}^{\circ})$ . (a) Incident laser source, a Gaussian beam with a wavelength of 632.8 nm and diameter of 10 mm. (b) Intensity distribution after amplitude-modulated by OPA-A. (c) Diffraction pattern in the far-field after CAP modulation by two LC-OPAs.

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Several multi-beam forming cases are performed to study the amplitude profiles, phase profiles, and far-field intensities. As shown in Fig. 4, the steering angles of multi-beam forming are assumed to be $({0.1}^{\circ}, {0.2}^{\circ})$ , $({0.1}^{\circ}, {0.2}^{\circ}, {0.3}^{\circ})$ , and $({0.1}^{\circ}, {0.2}^{\circ}, {0.3}^{\circ}, {0.4}^{\circ})$ , respectively. According to Eqs. (7) and (8), when the steering angles are $θ_{1} = {0.1}^{\circ}$ , $θ_{2} = {0.2}^{\circ}$ and $θ_{3} = {0.3}^{\circ}$ , the corresponding phase shift of these three angles are $ϕ_{1}$ , $ϕ_{2}$ and $ϕ_{3}$ , then the amplitude and phase expressions will be calculated as:

Fig. 4 Simulation results of amplitude profiles, phase profiles, and normalized far-field intensity patterns when the steering angles of multiple beams are (a) $({0.1}^{\circ}, {0.2}^{\circ})$ ; (b) ( ${0.1}^{\circ}, {0.2}^{\circ}, {0.3}^{\circ}$ ); and (c) ( ${0.1}^{\circ}, {0.2}^{\circ}, {0.3}^{\circ}, {0.4}^{\circ}$ ). The phase modulations are normalized to $2 π$ . The dashed lines indicate amplitude and phase modulation have the same period.

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A_{x} = \frac{1}{3} \sqrt{3 + 2 [\cos (ϕ_{1} - ϕ_{2}) + \cos (ϕ_{1} - ϕ_{3}) + \cos (ϕ_{2} - ϕ_{3})]},

Φ_{x} = arc \tan \frac{\sin ϕ_{1} + \sin ϕ_{2} + \sin ϕ_{3}}{\cos ϕ_{1} + \cos ϕ_{2} + \cos ϕ_{3}} .

To observe the amplitude and phase profiles clearly, only a small part of the overall profile (from 1mm to 3mm) is shown in the figures. The phase modulations are normalized to $2 π$ . From the upper row, the amplitude modulation profiles are like periodic “sine waves”, and there will be small “sine-waves” coming out between the main “sine-waves” as the number of beams increases. The phase profiles are shown in the middle row, which look like sawtooth curves. It is evident that if the number of beams increase, the oscillations between two adjacent zeros of phase increases. Due to the discrete electrodes and the transverse electric field effects, it is difficult to achieve smooth phase modulations in LC-OPAs. The oscillations will aggravate the effects of the distorted phase modulations, resulting in reduced diffraction efficiency of the CAP system. Meanwhile, phase profiles have the same period with respect to the amplitude profiles, so the fringing field problem may become worse when we cascade two devices since both have the same flyback regions, indicated by the dashed lines in Fig. 4. The bottom row shows that multiple equal-intensity beams are achieved in the desired directions with no grating lobes.

4. Experimental results

The schematic diagram of the experimental setup is illustrated in Fig. 5. A collimated linearly polarized He-Ne laser with a wavelength of 632.8 nm is used as the input light source. An optical attenuator is used to adjust the intensity of the laser, and a beam expander is used to make sure the laser illuminates the whole effective aperture of the LC-OPAs. Because the LC-OPAs we used are phase-only devices, two mutually orthogonal polarizers P1 and P2 are employed to achieve the amplitude modulation required in OPA-A. The polarization axes of P1 and P2 are oriented at $\pm 45^{\circ}$ with respect to the x axis so that maximum amplitude modulation would be generated. Two achromatic lenses L1 and L2 with focal lengths of 300 mm form the 4-f imaging system. A spatial filter SF located in the Fourier plane of the 4-f system is used to suppress the interference caused by reflective surfaces. The input beam is first incident on OPA-A, where amplitude modulation is realized. Then the optical field at the OPA-A surface is relayed onto OPA-P by the accurately calibrated 4-f imaging system. The main experimental challenge is ensuring precise mechanical alignment of the two LC-OPAs. Hence, an adequate calibration procedure of the 4-f system is required to satisfy the 1:1 imaging condition, which will be described below in detail. Finally, the far-field intensity distributions are observed on the CCD camera, which is located in the far-field behind the OPA-P.

Fig. 5 Schematic diagram of the experimental setup for CAP multi-beam forming. BE, beam expander; P, polarizer; L, lens; SF, spatial filter, FL, Fourier lens, Vi, applied voltage.

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4.1 Calibration procedures of the CAP system

The calibration of CAP system involves longitudinal and transverse calibration procedures, which are implemented step by step to obtain imaging with an accuracy necessary for the multi-beam forming condition. In the first step, the distance between the two lenses is set to be two focal lengths. This involves positioning the input laser source and beam expander on an optical track to define the optical axis. A screen is installed on the track at a few meters away from the expander to record the beam size. Then lenses L1 and L2 are placed in the optical path one by one, making sure that the distance between them is 600 mm. The lenses were determined to be aligned when the beam size and position on the screen were the same as the original beam before the two lenses had been placed in the system.

In the second step, the LC-OPAs were accurately placed in the focal planes of their respective lenses for true 4-f imaging, as illustrated in Fig. 5. Longitudinal and transverse alignment was facilitated by imaging the electrode patterns of the LC-OPAs using an objective lens and CCD. The OPA-A is mounted in a position that is close to the back focal plane of L1. A white-light is put in front of the OPA-A, illuminating the center of the device. A blazed grating with pixel-period N (N is chosen to be an integer number) is generated on OPA-A. Then the longitudinal position of OPA-A is aligned by adjusting a micrometer on a translation stage until a sharp periodic electrode pattern is appeared on the CCD. As an example of a 16-pixel-period blazed grating, namely N = 16, the observed electrode pattern of OPA-A is shown in Fig. 6(a). Similarly, the OPA-P is loaded with the same pixel-period of OPA-A, and the position is adjusted by a micrometer on a translation stage until another sharp periodic electrode pattern is observed on the CCD, as shown in Fig. 6(b). Note that Figs. 6(a) and 6(b) are the electrode patterns of OPA-A and OPA-P which are observed individually by changing the field of view of the objective lens. As we can see, both of their electrode patterns are imaged clearly onto the CCD. Through the above two steps, the LC-OPAs and lenses are assembled in the correct planes relative to each another so that unit magnification is obtained in the imaging of OPA-A onto OPA-P.

Fig. 6 Electrode patterns of two LC-OPAs during the calibration procedures of the CAP system. In (a) and (b) Electrode patterns of OPA-A and OPA-P are observed individually (pixel period N = 16). (c)-(f) Show results of non-overlapping and overlapping between LC-OPAs when the pixel-period N are 16 and 32.

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The third step is transverse calibration to ensure the pixels of the two LC-OPAs are entirely parallel and overlapped. Referring to the periodic electrode patterns on the CCD, the angle between the stripes equals the angular difference between the pixels of two devices. The OPA-A is mounted on a goniometer stage such that it can precisely rotate the device. Then only the swing angle of OPA-A is adjusted until the two electrode stripes are parallel. To confirm good pixel-overlap, a number of blazed gratings with different pixel-period N are generated on the two LC-OPAs simultaneously. For pixel-periods of 16 and 32, corresponding results of the non-overlapping and overlapping LC-OPAs are shown in Figs. 6(c)-6(f). Figures 6(c) and 6(e) reveal that the pixels of both LC-OPAs are visible on the CCD but do not overlap with each other. After the adjustment operation, as shown in Figs. 6(d) and 6(f), the pixels are completely overlapped, and the estimated pixel alignment error is 1-3 pixels.

4.2 Multi-beam forming

The LC material we used is characterized by the following parameters: $K_{11} = 11.7 p N$ , $K_{33} = 19.5 p N$ , $ε_{/ /} = 19.5 ε_{0}$ , $ε_{⊥} = 5.1 ε_{0}$ , $n_{o} = 1.52$ , $n_{e} = 1.73$ . The thickness of LC cell is 9 $μ m$ . The measured voltage-dependent phase retardations of the two LC-OPAs are shown in Fig. 7. The voltage is normalized to Freedericksz transition voltage of $V_{t h}$ , which is calculated by $V_{t h} = π \sqrt{K_{11} / Δ ε}$ ( $Δ ε = ε_{/ /} - ε_{⊥}$ ). About 6π phase retardation changes at 632.8 nm light are achieved on both devices, exhibiting the low operating voltage of liquid crystal based devices. Meanwhile, the image inserted in Fig. 7 illustrates the custom-fabricated transmissive-mode, one-dimensional LC-OPA device. The effective optical aperture is 10 mm with 1920 grating electrodes, wherein each pixel has an independently tunable phase shifter that is driven by four chips on glass (COG) drivers. As we demonstrated in section 2, the amplitude and phase modulation are calculated by Eqs. (2), (7) and (8). For the phase-only LC-OPAs, the amplitude modulation must be converted to phase modulation for OPA-A by the equation $Φ_{x}^{O P A - A} = 2 \arcsin (A_{x})$ , where $A_{x}$ is calculated by Eq. (7). Then, the driving voltages that will be loaded on the array of electrodes can be calculated according to the monotone relationship between voltage and phase retardation.

Fig. 7 Measured voltage-dependent phase retardation of the two LC-OPAs at 632.8 nm. The voltage is normalized to the Freedericksz transition voltage $V_{t h}$ , and the phase is normalized to 2π. Insert image, schematic illustration of the custom-fabricated $1 \times 1920$ LC-OPA device, including the grating, driver ICs, I/O ports, and so on.

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As shown in Fig. 8, several multi-beam forming patterns are experimentally obtained using the CAP system: (a) the original single-beam pattern with no voltage applied on the LC-OPAs; (b) two-beam forming; (c) three-beam forming and (d) four-beam forming. All the beam patterns are observed by the same CCD camera, and the original beam center remains unchanged for all cases to be a reference point of 0 degrees. As expected, the original single beam is split and deflected into the target directions in the far-field, so that different multiple beam spots are obtained on the CCD. Examining the beam patterns, the manually-inserted blue lines indicate that a high steering accuracy is achieved by the multi-beam forming system, and the calculated maximum angle deviation is 90 $μ r a d$ , which appears in the four-beam forming case. As shown in Figs. 8(e) and 8(f), we use the beam spots to calculate the normalized intensities for two multi-beam forming cases which are marked in Figs. 8(c) and 8(d) by orange boxes. The steering efficiency (relative to the zero-order beam) of these two cases are 88.18% and 85.69%, respectively, demonstrating that high efficiency is achieved by the CAP multi-beam forming system. The intensities of the beams are almost equal, demonstrating very high uniformity. Figure 9 shows the relationship between the percentage of light and the number of beams, the cyan bars represent the percentage of zero-order, and the magenta bars represent the effective multiple beams. We can see that as the steering angle and the number of beam getting higher, the original zero-order spot intensity grows, so that the diffraction efficiency decreases.

Fig. 8 Experimental multi-beam forming beam patterns. (a) The original single-beam pattern with no voltage applied on the LC-OPAs; (b) Two-beam forming; (c) Three-beam forming and (d) Four-beam forming. (e) and (f) are the normalized intensities corresponding to the multi-beam forming cases marked in the orange boxes, respectively.

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Fig. 9 The relationship between the percentage of light and the number of beams. The cyan bars represent the percentage of zero-order, and the magenta bars represent the effective multiple beams. The corresponding steering angles are shown on the bars.

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5. Conclusions

In this work, we have presented a new and useful multi-beam forming method that is capable of generating arbitrary multiple beams with high efficiency and accuracy. The theoretical formulas for calculating the near-field amplitude and phase factors are deduced analytically through inverse transform from the desired far-field of multiple beams represented as unit impulse functions. Based on a 4-f imaging system which precisely relays the optical field of two LC-OPAs, the derived near-field can be experimentally created. The detailed calibration procedures for the CAP system used to align the experimental multi-beam forming system were also described. Various multi-beam forming cases are investigated numerically and experimentally to show the functionality and flexibility of the proposed CAP method. Our multi-beam forming method delivers high efficiency, accuracy, uniformity, and, with real-time response. The approach can be easily extended to two-dimensional multi-beam forming when two-dimensional devices are used. Moreover, a single CAP system can be used in applications requiring multiple wavelengths (provided the OPAs can achieve 2π phase retardation depth for the corresponding wavelengths) by adopting a time division multiplexing approach to switch the driving voltages between the working wavelengths during different time periods. In all, our CAP method offers a robust and innovative strategy for multi-beam forming and may find extensive applications in areas where multiple beams are required, such as Lasercom networks, Lidar, and optical tweezers, etc.

Funding

National Natural Science Foundation of China (NSFC) (61405029, 91438108, 61775026); China Scholarship Council.

Acknowledgments

L. Wu is grateful to the China Scholarship Council for supporting his study at the University of California, San Diego through the Joint Training PhD Program and Visiting Scholar Program. We all thank Prof. Joseph E. Ford and Dr. William Maxwell Mellette for their constructive grammar and writing revisions on the manuscript.

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Arbitrary multiple beam forming by two cascaded liquid crystal optical phased arrays

Abstract

1. Introduction

2. Principles

2.1 Liquid crystal optical phased array

2.2 Cascaded amplitude and phase method

3. Numerical simulation

4. Experimental results

4.1 Calibration procedures of the CAP system

4.2 Multi-beam forming

5. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (9)

Tables (1)

Equations (10)

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