Unified Optical Scanning Technology

Chapter 4.10.1.1 - Scanner Devices and Techniques: Basis of Phased Array Beam Steering

4.10.1.1    Basis of Phased Array Beam Steering    The alteration of an incident optical wavefront by phase variation is characterized by considering the effect of a refraction prism. Figure 4.29a illustrates a plane wavefront in air, incident parallel to the plane surface of a dielectric wedge. Within the material of refractive index >1, the wavelength is compressed proportionately while the fronts remain parallel to the incident wavefront. A linearly increasing local phase delay results from the progressive retardation of the wavefronts across the enlarging wedge thickness. Upon encountering the tilted boundary, the wavelengths are reexpanded and their angle of propagation is altered as illustrated. This familiar refractive process is exemplified in the prismatic electrooptic gradient deflector of Figure 4.27b.

To provide a wide aperture when composed of material having a moderate refractive index, this configuration can exhibit substantive bulk. To relieve this burden, the long wedge profile is divided into an array of smaller wedge periods, each forming a linear phase delay of from 0 to 2π. As illustrated in Figure 4.29b, when implemented as described below, this technique provides the same optical deflection as the continuous single wedge. Along with the need to accommodate the combination of slope and refractive index of the material, one must dimension the periods of the wedges such that they form 2π phase differentials (or multiples thereof, i.e., modulo 2π) at the operating wavelength, to assemble continuous nonstaircased wavefronts in the near field. This is functionally analogous to the refractive or the more familiar reflective blazed grating, in which high efficiency is achieved when the angle of specular reflection (from the sawtooth slopes of the grating surfaces) coincides with the angle of diffraction at the selected wavelength. Accordingly, an array providing (single or multiple) 2π phase differentials normally exhibits a throughput efficiency having a wavelength sensitivity, thus limiting one composed of fixed full-phase segments to near-monochromatic operation at the selected diffraction order.

The above examples provide continuous phase retardation by virtue of their linear surface slopes, either continuous or in increments. Incremental phase retardation can be achieved by other means, in transmission by an array of small electrooptically controlled cells and in reflection by precise actuation of individual mirrored pistons. A functional illustration of an array of refractive phase retarders and the formation of the radiated wavelets into contiguous wavefronts is provided in Figure 4.29c. Operation is similar with pistons, except that the reflective piston requires displacement of only ½ of the 2π phase retardation distance. A thin electrooptic retarder having a reflective ground plane not only requires ½- thickness but, if composed of a relatively slow liquid crystal, attains a fourfold increase in switching speed. This figure illus-

Fig. 4.29 Evolution of optical phased array beam steering in transmission; analogous in reflection. [a] Prototype prismatic wedge illustrating familiar refraction of incident plane (constant phase) wavefront. [b] Synthesis of [a] with array of wedges; each width imparts 2π phase delay over the array period. [c] Synthesis of [b] with multiple delay elements (4 per 2π period). Superposition of output wavelets forms wavefronts at idealized efficiency of 81%. greater multiplicity provides higher efficiency. After A.S. Keys, R.L. Fork, T.R. Nelson, Jr., and J.P Loehr, "Resonant transmissive modulator construction for use in beam steering array," in Optical Scanning:Design and Application, SPIE, Vοl. 3787 (1995). Reproduced by permission of the publisher.

trates another option: division of the full 2π phase change into a number of substeps, four steps per full cycle in this case, providing 81% diffracted into the first order. The more steps, the higher the efficiency. Eight steps per cycle attains an otherwise lossless efficiency of 95%. Figure 4.29 is a pedagogic unification of three illustrations of [Keys], providing an introduction to the technology. Informative reviews appear in [McM, Mc&W] and in the Air Force report [Dor], Chapter 2.

Some of the heuristic observations expressed above are affirmed by the considerations that follow. The angular relationships of the phased array are expressed by the diffraction grating equation, presented earlier as Equation 4-25 for beam deflection of the acoustooptic grating and represented here for this application,

where Θiand Θoare the input and output beam angles with respect to the grating normal (boresight), n is the diffraction order, λ is the free space wavelength, and Λ is the grating (array) period, per Figure 4.29, b or c. As in Figure 4.29c showing four delay elements per array period, for q delay elements, each separated by a fixed distance d, Λ = qd. Because the number of elements in each period is q = 2π/Φ‚ where Φ is the phase shift between elements, then Λ = (2π/Φ)d, the distance required to assemble a one-wave phase difference. When, as is typical, Θi= 0 and we seek the angle of first-order wave propagation (per Fig. 4.29, b or c), then

The normalized intensity of the radiation pattern follows the analogy of the one-dimensional microwave phased array [Sko], expressed compactly as

in which

where Θ is the angle with respect to the grating normal at which the field in free space wavelength λ is measured, N is the number of phase shifters in the array, and, as stated above, the uniform elemental spacings d are assumed to provide uniform phase difference Φ between elements.

A direct analog to Equation 2-2b is that for the efficiency ηqof a linear array having the nominal (blazed) 2π phase resets illustrated in Figure 4.29, b and c; expressed by

in which q = the number of elements per 2π delay period. This may be recognized as the [Fourier transform]2 of a uniformly illuminated linear aperture, as described at the end of Section 2.4.1, expressed by Equation 2-2, and illustrated in Figure 2.1. Inserting values of q = 4 and 8, Equation 4-46 yields η4 = 0.81 and η8 = 0.95, respectively, as indicated above. Lower efficiency due to reduced q represents depletion of the main lobe to the sidelobes caused by disruptive wavefront staircasing.

Typical liquid crystal phase retarder elements exhibit a unique loss factor established by the minimum space required to relax its orientation from a 2π phase shift to zero. This 'flyback' transition is analogous to the flyback time τ of many conventional scanners (including acoustooptic), discussed in Section 4.3.2, Duty Cycle, and expressed in Equation 4-1 as η = 1 - τ/T, where T is the full scan period. Note that this represents a time loss, affecting among other things the bandwidth for a required overall transfer time. In the case of liquid crystal elements, the time terms are replaced with small A representing the flyback width and Λ the full 2π width, yielding the duty cycle

squared to denote the radiated intensity rather than the time. The topic of efficiency merits attention for at least two other parameters, the fill factor and the overall vignetting factor. The fill factor accounts for the practical burden of creating a refractive or reflective cell having an operating portion that occupies its full allotted area. The vignetting factor considers the typical loss of input illumination beyond the boundary of the array. In context of power throughput, the final efficiency is the product of the individual factors.

The far field angular beamwidth ΘB is expressed as a minor variation to the diffraction relation of Equation 2-16, which accounts for the distribution of the input illumination upon the full aperture width Nd and is given by

in which a is the aperture shape factor modifying the beamwidth, as discussed in Sections 2.1.2 and 3.2 regarding spot size and scanned resolution. Because the output beam angle Θo seldom exceeds 10°, its cosine renders a negligible effect on the beamwidth, yielding a more concise expression, as is that of Equation 2-16

in which D = Nd is the is the full aperture width.

 

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