Unified Optical Scanning Technology

Chapter 4.10.1.2 - Scanner Devices and Techniques: Resolution of Phased Arrays

4.10.1.2    Resolution of Phased Arrays     Equation 4-48a and its relationship to scanned resolution (Chapter 3) motivate expression of a topic that seldom appears explicitly in the literature. Although the above equation denotes the output beamwidth-and hence the narrowness or breadth of the principal lobe of radiation-it provides no calibration of the number of steered accessible adjacent spots or lobes that may address the full (one- or two dimensional) field of view. In the context of scanned resolution, we seek this number of distinguishable elements of information that may be positioned along a linear track (see Section 3.1.1). In analog form, this scanning process is described by the convolution or shifting function discussed in Section 2.1.3 and illustrated in Figure 2.4. The resolution number is denoted by the letter N, whereas N identifies the number of phase shifting cells in a linear array, as represented relating to Equation 4-48. This is also distinct from typical microwave radar usage of the word resolution, which is defined in that discipline as the smallest increment (finesse) of beam motion.

in which Θ is the full deflected field angle. One-half of this angle is rep-resented by Equation 4-43a as the (positive) first diffracted order of a (blazed) grating (i.e., n = +1 in equation 4-43). When the elements in the array are addressed in complementary phase sequence, the same deflection magnitude results in the opposite (n = -1) direction. Thus, for typically small Θo in Equation 4–43b, and with D = Nd, we form the numerator for Equation 4-49 as

As developed in Section 3.1.1 on Scanned Resolution, the basic Equation 3-3 yields

With Equation 4-48a providing the denominator of Equation 4-49 and adding the one boresight position, the steered resolution reduces to*

independent of the wavelength. Setting aside temporarily the constant 2/a, the ratio of the two variables N and q—that is, the total number of elements divided by the number of elements per phase reset dominates. Thus the number of phase resets in the array is the principal variable that determines the steered resolution (accessible spots) of a one-dimensional phased array. Although the full width of the array D establishes the narrowness of the radiated lobe, the number of such adjacent lobes within the field is expressed more directly dependent on N/q. (Note: N/q = D/Λ‚ where Λ is the array period.) Because q is not constant, this also affects how close the adjacent steering states can be [Wat2].

Equation 4-51 includes the aperture shape factor a, typically a constant of the system. When assumed of value one, it denotes uniform illumination upon a rectangular aperture (in one dimension). This yields the far-field intensity distribution of the sinc2(x) function (Equation 2-2b, Fig. 2.1), having a main lobe within equispaced null intervals. Rayleigh resolution requires this uniform illumination on a rectangular aperture and, further, that the adjacent spots in the far field overlap such that the maximum of each main lobe coincides with the first null of each adjacent spot. Further delimiting Equation 4-51, it is impractical to form a modulo 2π array in which q is less than three cells, in view of the resulting disruption of the ramp wavefronts and the loss in efficiency. Thus, letting a = 1 and setting qmin = 3, steered resolution is sometimes expressed as an assumed Rayleigh resolution with q = 3, forming

.

Depending on aperture illumination, the value of a may differ from 1, and N and q may vary to accommodate spot efficiency and spot steering. A familiar illumination is the Gaussian function, with adjustment of the overfill to make the intensity more uniform across the full aperture width D. The degree of overfill is, however, moderated by the reduction in light throughput due to the loss of illumination beyond the array boundary (vignetting). It is also balanced by the appearance in the far field of fine structure beyond the main lobe, approaching the appearance of the sinc2 function when D is illuminated uniformly.

To quantify this value of a, we consider a different condition than that which generated Table 3.1 in Chapter 3. The left side of that table summarizes the value of a for the Gaussian beam of nominally round cross section falling either completely within the deflecting aperture ('untruncated" column) or when the 1/e2 intensity of the input Gaussian beam occurs at the aperture boundary ('truncated' column). These are the two prominent conditions of illumination of most conventional deflectors. Some, such as the acoustooptic devices, exhibit a rectangular cross section of width >> height, meriting a different evaluation of the shape factor a. The consequences of illumination with a beam that is Gaussian primarily in the D-direction is evaluated and summarized [Bei2], providing data of current interest.

These data are for the variable beam width W (Gaussian at 1/e2 intensity) illuminating the constant width D of a linear phased array. Assigning a parameter ρ = D/W, when ρ = 1 then W = D, whereby the 1/e2 input beamwidth matches the aperture width D. At ρ = 1.5, the array aperture is 1.5 times wider than the 1/e2 beamwidth. In this condition, Figure 2.2 indicates that the aperture delimits the Gaussian function at ±3σ, where its intensity has tapered off to a very small fraction of the maximum value. This represents a practical limit on the narrowness of the illuminating beam. At the other extreme, when ρ → 0, this requires the input beamwidth W >> D, corresponding to the condition of extracting near-uniform illumination from the center of the Gaussian beam and encountering extreme light loss beyond the aperture. This is the case of a = 1. The aperture shape factors for the other one-dimensional gaussian cases are determined [Bei2] and given for ρ = 1, a = 1.15 and for ρ = 1.5, a = 1.35.

Related to the topic of resolution is consideration of the smallest increment of beam position (finesse) addressable by a phased array. Returning to the basic wedge deflector of Figure 4.29a, one can reduce its Wedge angle or its refractive index to provide any small increment of refracted angle, consistent with practical manufacturing tolerances. Figure 4.29c shows, however, the periodic patterning of a phased array requiring a quantized cellular arrangement. Thus minimal beam shifts are limited by integral changes in the phase resets. Equation 4-43a defines the small diffracted angle in one direction as Θo = λ/qd. Thus increasing q (delay elements per phase reset) by one cell narrows the deflection angle incrementally. Adding one cell to each group reduces the angle to (q/q+1) of its initial value. The higher the initial q number, the finer the change in angle. Because it is necessary to form the array with full 2π resets, fractional residues can be filled by approximating uniform spacings with a few alternating (q), (q + 1), (q), (q + 1) periods and programming all to form 2π resets. This minimizes wavefront distortion. Extending this alternating procedure to a larger portion of the array length-not just to equalize a residue-permits achievement of a smaller angular change, especially when the array is composed of groups having a low q number. Not yet evaluated is the possible degradation in efficiency with application of this procedure. Any degradation is likely less significant for groups of high q number.

* Although Equation 4-51 appears to depart from the fundamental Equation 3-5 for scanned resolution N = ΘD/aλ, this Θ is established by the diffractive structure of the array and is not an independent variable. Accordingly, substituting Equation 4-50 into the classic Equation 3-5 and adding one for the boresight forms this Equation 4-51 directly: N= [(2/a)(N/q)] + 1

 

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