Theoretical and Experimental Demonstration on Grating Lobes of Liquid Crystal Optical Phased Array

1School of Optoelectronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China 2Science and Technology on Electro-Optical Information Security Control Laboratory, Sanhe 065201, China 3School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China 4Shanghai Aerospace Electronic Technology Institute, Shanghai 201109, China


Introduction
Phase controlled steering effects are most often produced by using an array of unit antennas, the amplitude and initial phase of which are individually controllable.Thereby, the technique of phase array (PA) has been suggested to be a wave director from mechanical wave to electromagnetic wave, from microwave to optical domain [1][2][3].Meanwhile, phased array has two categories: active PA and passive PA [2].Active PA has been more widely deployed on the microwave domain because of its easy control and advanced machining accuracy.However, on the optical domain, the wavelength is almost close to 1 micron.To achieve the feature size of submicron, nanoscale processing has already dramatically developed on the optical domain applications, such as adaptive reshaping [4], optical tweezers [5], and optical Yagi-Uda antenna [6].Even more, an encouraging letter published in Nature in 2013 reported that the first large scale optical phased array (OPA) made by CMOS technology has 64 × 64 units [7].
Comparing with active OPA, passive OPA does not have any other heat deposition, such as quantum loss, nonradiation jump, and Joule heat.Since the promotion on the concept of OPA, there have been at least three methods to realize it [3,8,9]: LiNbO3, PLZT, and liquid crystal (LC).Therein, the OPA using nematic LC usually called liquid crystal optically phased array (LC-OPA) has a great potential to achieve practical OPA system for steering optical laser beam inertia less, nonmechanical, and low SWaP (Size Weight and Power consumption).
LC-OPA has already demonstrated its properties of light weight and high precision on steering laser beam to generate an optical space-time division multiple access network [10].After the survival experiment in the equivalent space environment, modest effects were observed, but none were deemed significant enough to impact the performance of the device for beam steering applications on space communication [11].After the invention of LC-OPA by Dr. McManamon et al. in 1993 [12], he published an all-thing-considered review paper to summarize its nearly 30 years of development [13] including liquid crystal polarization gratings to achieve 99.5% diffraction efficiency [14], simulation on high efficiency improvement [15], volume holographic to amplify steering angle [16], and the scattering free polymer network liquid crystal invented by Dr. Sun et al. to overcome the transparency reduction on the submillisecond liquid crystal devices [17,18].Meanwhile, the concept of LC-OPA has been  suggested into the midinfrared domain [19] and even some alternative methods for steering laser beam such as lenslet array [20] and MEMS-based [21].
Although the performance of LC-OPA has been improved on a variety of features, to date, there has not been a complete analysis on the high order grating lobes.In this paper, it derives a theoretical model on the main lobe and grating lobes and being verified by experimental result.

Theory
Because of the passive feature of LC-OPA, it has a uniform radiative laser source propagating through the LC film shown in Figure 1.Thereby, the wave front of the polarization dependent near field will be modulated by the liquid crystal film with a gradient refractive index distribution.For the device of LC-OPA, modulation on the amplitude of near field could be neglected because the absorption of working medium is very tiny.After long distance propagation, far field could be generated according to the principle of diffraction so as to a steering peak on the given angular position if a suitable near field phase modulation is given.
Steering can be accomplished by a physical prism.If the refractive index of a prism is changeable by loading different voltages, light could be steered as well at different angles.Meanwhile, we can take advantage of the fact that there is no difference for light waves if they have 0, 2, 4, or 2 phase shift.The phase can be made as a 2 subtraction when it exceeds full round 2 in the case of large aperture.According to this property of diffraction, it generates the first scheme, shown in Figure 2(a), to form a deflective beam.and the scheme has already been widely used in the current microwave phased array called variable period grating (VPG), where the steering angle   is governed by sin   = Δ  / 0 , where Δ  is the phase step between two adjacent electrodes,  0 is vacuum wavenumber, and  is width of electrode and gap.Because of the proportional relationship between Δ  and   , it is easy to realize a continuous and full coverage scan on a given domain.Meanwhile, another scheme to steer wave is blazing grating, shown in Figure 2(b).When an in-plane wave is modulated by periodic medium, tens of periodic spots are generated on the far field that is called grating lobes.Therein, the profile of those grating lobes is determined by the modulation function in one period, and the angular position of the th order grating lobe   is determined by the grating equation sin   =  0 /, where  is still the modulation period.When the device is a discrete one, the first order lobe we desire is determined by sin   =  0 /, where  0 is vacuum wavelength and  is the number of electrodes in one period .Comparing with the scheme of VPG, VBG has a constant period .It also always starts at the same initial phase for each period.However, because of the inverse proportional relationship between   and adjustable integer , it is impossible to realize a continuous and full coverage scan on one device.
After long distance propagation through the free space, electric field component of the modulated beam is governed by free space Helmholtz equation.If the distance  meets Fraunhofer's approximation condition, that is, 4 0 ≫  2 , where  is the diameter of transmitting aperture,  field in the far field  far is completely determined by the near field  near according to the Fraunhofer's equation where the constant  is determined by the principle of energy conservation and   is the angular spectrum position.Thereby, the far field distribution can be obtained from a given complicated near field by the numerical method of FFT (Fast Fourier Transformation).
field on the near field is the transmitted beam after the liquid crystal film, where the incident beam  in is usually configured as a standard TEM00 in-plane wave or Gaussian mode with a half beam waist of ,  in = exp(− 2 / 2 ).The transmission function includes two parts: amplitude factor   and phase factor   .The amplitude factor   is only a gate function owing to the limited transmitting aperture with a full width of   ,   = rect(/  ).The phase factor   is determined by the electric controlled liquid crystal film, where () is additional phase retardation.According to the characteristic of Fourier transformation, far field distribution  far can be rewritten as the convolution of two parts that result from gate limited Gaussian beam and phase modulated transmission function, respectively, The shape of main lobe and side lobe is determined by FFT{ in ⋅   }, having the same result as previous theories.
That is, the full width of main lobe is Δ main = 2 0 /  for an in-plane wave through an aperture with a width of   .If the beam waist  of incident Gaussian beam is much smaller than the aperture, the full width of main lobe is Δ main = 4 0 /.Because of the property of transmission function   , a pure phase modulation, if it is a periodic function in the ideal model of LC-OPA, its Fourier transformation must be a group of delta functions, the center position of which is determined by FFT{  }.
The periodic transmission factor   can be expanded on the Fourier series where the integer  is the order number and   is determined by the equation Because   is a phase modulation function on the form of   = exp[()], the coefficient   is rewritten that To evaluate the property clearly, a group of parameters are assumed for simulation: electrode width  = 4 m, electrode gap  = 1 m, electrode length  = 10 mm, and electrode number  = 2000, that is, electrode period  = + = 5 m, and transmitting aperture width   =  = 10 mm.
In the case of VPG, phase modulation  is linear with position  that can be written  = ; then, after a distance of 2/, the phase increase 2, thereby, exp[ ⋅ ( + 2/)] = exp( ⋅ ), so that the factor of phase modulation   is periodic and its period  is 2/, where  is variable.Substituting () into ( 6), the amplitude of each order is coefficient   and it is obtained that  1 = 1 and   = 0 for other orders.So the peak angular position   on the far field can be obtained by sin   = / 0 .
In the case of VBG, a periodic transmission function is that the phase shift is periodic; that is, ( + ) = (), where  is variable.Owing to the periodic property on phase  with a period of   it is not because of wave property of incident beam, the amplitude of each order is coefficient   obtained from (6), and the peak angular position   on the far field can be obtained by sin   =  0 /.
Therefore, no matter VPG or VBG, these two schemes can be unified because they have the same reason to steer the incident beam to the given angle.The phase factor of their transmission function has a common property that it is periodic with a period of  =  0 / sin   .The difference is that on the scheme of VBG, the period  is a user defined variable.On the scheme of VPG, we configure the slope of phase modulation , and the corresponding period  = 2/.
In the practical LC-OPA, fringe effect has already been evaluated in many papers.Owing to the no source boundary condition on the gap between electrodes, there is not only a flyback area for each 2 reset, but also phase dropdown on each electrode gap.Therein, the width of flyback is   .Thereby, the phase shift can be written on the form of () =  ideal ()+ fringe (), where  ideal () is ideal phase modulation, the function of which is linear with position like a saw tooth so as to a periodic transmission function where the subdomain  0 indicates the ordinary domain and   indicates the flyback domain.The width of flyback   is determined by the thickness of liquid crystal device. 1 and   1 are the starting position and ending position of this period, respectively. init and   init are their phase at  1 and   1 , respectively.On the scheme of VBG, the distance | 1 −   1 | = ,  init =   init = 0.However, on the scheme of VPG, the phase on the finite width electrode is loaded step by step, phase shift on the th electrode   is determined by   = ( − 1), when it exceeds 2, and it would make a reset into  init =   − 2 to start another new 2 round where the initial phase is  init , as shown in Figure 2(a).During the range of one quasi period with  electrodes, the fringe effect induced item  fringe () is assumed to be and because of continuous and desirable feature on phase function, the coefficients ,  should be on the condition of  : (1 − ) =  :   .Then, if we assume  = Δ  /2, so  = 1 −   / ⋅ Δ  /2.In the case of VPG, the phase modulation is shown in Figure 3(a); in the case of VBG, it is shown in Figure 3(b).Meanwhile, the values of coefficient and flyback range   are fully governed by the group of EM equations and liquid crystal molecular director equation.
After the numerical FFT operation on the near field  near , if we suppose steering angle   is 1 degree, the far field is as shown in Figure 4. Comparing with the ideal model with only one lobe on the target angle, there are more grating lobes owing to the additional fringe item.These high order grating lobes are on the position of  =   , when  is order number of arbitrary integer.
Therein, the zoom-in figure of Figure 4 is shown in Figure 5; the profile in blue dot line is the FFT of  near in the range of [0, ], where  is calculated by 2/ 0 sin   .The intensity of each grating lobe and main lobe at the position Far field Profile of  =   is governed by the blue dot line.Besides the defined higher order grating lobes, there are also some other neglectable peaks like noise.The intensity of most of them is below 30 dB; those are generated by the overlap of multiside lobes.This property would give us two important issues: first, the phase distribution on the near field can be inversely derived by detecting the intensity and position of each grating lobe; second, by optimizing the phase delay of small account of electrodes in the range of [0, ] can optimize the deflection efficiency.In other words, the phase distribution during one period [0, ] is the major factor to influence the grating distribution or the deflection efficiency.
In the case of VPG, owing to the fringe effect, the transmission factor   () is not perfectly periodic.However, for the most of given value  0 ,   has a property of repetition on  meeting the equation () = ( +   ).Meanwhile, the repetition distance   is 2/ 0 sin   ; that is, the transmission factor   () is a quasi-periodic function of position, where the period   = 2/ 0 sin   .For the most given value of modulated phase, they would generate a group of grating lobes on the diffraction of  =   .And for the phase value which does not have periodic overlap, it would generate the noise lobes.This can be verified in the case of VBG.In contrast, the transmission factor   () is perfectly periodic because of the periodic phase modulation.Then, the far field it generates has a very perfect grating lobes shape without noise as the VPGs as shown in Figure 6 when the steering angle is 1.02 degrees that is the most close to 1 degree by the grating function   = sin −1 ( 0 /).For a given group of steering angle as examples from −3 ∘ to 3 ∘ with an increasing step of 0.5 ∘ , the intensity distributions of far field at those steering angles are captured by CCD and combined together, as shown in Figure 8 from right to left.In order to display the low intensity grating lobe, we adjust the exposure time to the maximum value to detect the grating lobes and side lobes.The main lobes are lined up by a blue dashed line.They all have good accuracy to form good order on one line.And, the zero-order positions are lined up by the yellow dashed line.

Experiment
Meanwhile, in Figure 8, grating lobes of higher order are shown with the same distance between each of them.Its distance is the same as the distance between main lobe and International Journal of Optics original position.It has good agreement with the theoretical analysis that   =  ⋅   .Besides the grating lobes, Figure 8 also illustrates few relatively dark spots between grating lobes.Although they look very clear in this snapped picture, they do not occupy too much energy when we measured using power meter, because of overexposure.

Conclusion
Theoretical and experimental evidence is presented showing that grating lobes and side lobes are generated by fringe effect in the far field.On both working schemes of variable period grating (VPG) and variable blazing grating (VBG), the angular positions of grating lobes are determined by   =  ⋅   ; they have the same physical reason to steer incident beam that their phase factor of transmission function is periodic or quasi-periodic.The normalized intensity of grating lobes is determined by phase modulation in one period or quasi-period.This theoretically and experimentally verified conclusion has been obtained to explain the principle reason on grating lobes.
The principle reason of grating lobes would give us another important issue: optimization on deflection efficiency can be accomplished by optimizing phase delays of small account of electrodes in the range of [0, ], not only VBG scheme but also VPG scheme; higher order and zero order grating lobes can be suppressed as much as possible.Meanwhile, the method of optimization and experimental demonstrations on the steering efficiency improvement is going to be presented in detail in the coming work.

Figure 1 :
Figure 1: The sketch of liquid crystal optical phased array.

Figure 2 :
Figure 2: Ideal phase modulation of LC-OPA on the scheme of VPG (a) and VBG (b).

Figure 3 : 1 Figure 4 :
Figure 3: Fringe effect of LC-OPA on the scheme of (a) VPG and (b) VBG.

Figure 5 :
Figure 5: Zoom-in figure of normalized intensity of far field of VPG scheme.

Figure 6 :Figure 7 :
Figure 6: Zoom-in figure of normalized intensity of far field of VBG scheme.

Figure 7 (
Figure 7(a) depicts the experimental setup for studying the properties of far field.The incident beam is linear polarized Nd:YAG laser.After phase modulation by the LC-OPA phase shifter device, the output laser beam has a coherent far field observed by a high resolution CCD on the focal plane of Fourier lens.Meanwhile, the data loaded on LC-OPA is generated by PC computer and translated by driver module.Meanwhile, the device of LC-OPA we used is developed from grating electrodes fabrication to circuit design and liquid crystal filling.The widths of electrode and gap are 4 m and 1 m, respectively.The thickness of LC cell is maintained by spacers with a diameter of 10 m.The effective optical aperture is 10 mm × 15 mm as shown in Figure7(b).Meanwhile, the LC-OPA module is driven by four parallel ordinary liquid crystal display drivers IC controlled by FPGA.For a given group of steering angle as examples from −3 ∘ to 3 ∘ with an increasing step of 0.5 ∘ , the intensity distributions of far field at those steering angles are captured by CCD and combined together, as shown in Figure8from right to left.In order to display the low intensity grating lobe, we adjust the exposure time to the maximum value to detect the grating

Figure 8 :
Figure 8: Experimental result on the far fields of different steering angles.