Unified Optical Scanning Technology

Chapter 4.4.1 - Scanner Devices and Techniques: Scanner Configurations and Characteristics

4.4.1 Scanner Configurations and Characteristics

To illustrate the analogous relationship of conventional and holo-graphic devices noted in the introductory paragraph, consider Figure 4.10. It shows a transition of three scanning techniques that yields identical scan geometries. All are radially symmetric, scanning a concentric cylindrical information surface S as for 'internal drum' application. When the cylindrical medium is unwrapped and flattened, the scans appear as parallel straight lines. An alternate information surface S' is flat and mounted perpendicular to the rotating axis. When this medium is indexed 'vertically," the scans form an array of circular arcs. (This can be useful if, for example, an 'original' document is scanned in the same manner to derive a signal for facsimile reproduction.) All three systems yield the same image characteristics. Heuristic Figure 4.10 reresents the transition from (a) a conventional polygon (operating unconventionally) to (b) an uncommon lenticular scanner, to (c) the configuration of a basic form of holographic scanner.

The prismatic polygon at (a) is postobjective. The input beam I is converged by objective lens O initially to axial point Po while overfilling mirror M. Thus operating radially symmetric, the beam reflected from M through angle α scans focal point P in a circular arc on the inside of cylindrical surface S.The device at (b), with marginal lens segments LS, executes the same scan function. Overfilled segment LS redi-rects collimated paraxial input beam I through angle α to focus at P. Upon rotation of the array, it also forms circular arcs on the inside of surface S. At (c) is one basic form of holographic disk scanner. The input beam I is also paraxial, and the overfilled holographic sector HS serves as the lens segment LS to redirect and focus I through angle α to scanned point P on surface S. In this example, holographic segment HS exhibits lenticular power to emulate lens segment LS. It may also be considered a marginal segment of an infinity conjugate zone lens, composed of a concentric array of alternating opaque and transparent circles. The zone lens is developed comprehensively [Bei1] from this infinity conjugate form through the finite conjugate form to the gener-alized form, as relating to the formation of the hologaphic grating and its periodicity.

The choice of hologaphic grating contour is a major design consideration. The two principal options are (per item 6 in 'distinction' listing above) 'linear" and 'lenticular.' The linear form is a plane diffraction grating having equally spaced straight grating elements. It will be seen (Section 4.4.1.1) that this structure is in many respects analogous to a refractive wedge prism. The lenticular form is analogous to a lens element, as represented in (b) and (c) of Figure 4.10 and its zone lens prototype. Although it may appear that the lenticular grating is also to be disposed on a flat surface, this is not a limitation to valid operation (as indicated by item 3 in 'distinctions' listing). In fact, lenticular holograms of notable holographic scanners were formed on curved substrate surfaces [Bei1]. Several such forms are illustrated and discussed below.

With regard to the plane linear transmission grating, Figure 3.8 illustrates its typical application. The equal input and output angles βi and βo (Bragg condition at the center of disk rotation) provide the unique property of reducing significantly the degrading effect of disk wobble Δα. This would otherwise vary the output beam angle (with respect to the facet normal), causing nonuniformity of scan line spacing. Analysis [Kra2, Bei1], starting with the classic grating equation represented above by Equation 3-14 and repeated here

In which d is the grating spacing, reveals a relationship for the tilt error in the vicinity of Bragg operation for both transmission and reflection gratings. This differential in output beam angle dβo for a differential tilt in hologram angle dα within an angular error Δα is given by

in which the upper sign is for transmission and the lower for reflection gratings.

For a typically small Δα error, the reduction of the effect of wobble in transmission gratings is apparent from Equation 4-15, where the fractional term is approximately equal to 1, leaving (for βot ≡ output angle in transmission)

Numerically, for βi = βo = 45°, a substantive angular error of 0.1° (360 arc Sec.) is reduced to 1.3 arc Sec.-a 277x improvement. For the reflec-tive grating, the positive sign before the fraction applies, representing an addition of unity (1) to the existing (1), yielding (for dβor ≡ output angle in reflection)

This is similar to the doubling effect of a polygon (m = 2) in mirror reflection. Thus a transmission grating operating in the Bragg regime ameliorates the effect of such wobble error of a typical reflective facet polygon by a factor of approximately 550 times. Although this indicates an extremely attractive wobble reduction factor compared to that of the polygon, this is achieved only in the center of scan. Furthermore, this condition is also accompanied by a nonminimized scan line bow. However, by unbalancing the nominal Bragg angle condition of equal input and output angles to one having a small differential (Θi - Θo 1.41°) [Kra3], the wobble correction is equalized at the center and ends of scan while minimizing the scan line bow. Under these conditions, the effect of scanner wobble is reduced by a factor of approx-imately 82 times compared to that developed by a typical prismatic polYgon.

With regard to the resolution of holographic scanners, as noted in the introductory paragraph to Section 4.4, this factor is governed by the same equations as for conventional scanners. When the resolution is augmented with finite values of r and f (Equations 3-11 or 3-13), the value of scan magnification m (see Sections 3.4.1 and 3.4.2) is represented, interestingly, by Equations 3-14 or 4-14, the grating equation.

Fig. 4.11 Analogous Bragg diffraetion and prism refraction at minimum deviation. [a] Holographic scanner. Hologram H operating in Bragg regime. [b] Analogous prismatic scanner. Prism P at minimum deviation. From Holographic Scanning, L. Beiser, ©1988 John Wiley & Sons, Inc. Reprinted by permission of Johu Wiley & Sons, Inc.

 

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