Unified Optical Scanning Technology

Chapter 4.3.5.2 - Scanner Devices and Techniques: Relationships Between Scanner and Lens

4.3.5.2    Relationships Between Scanner and Lens    The scanner size and relationship to the flat-field lens may now be determined. Figure 4.2 illustrates a typical prismatic polygon operating with all beams in the plane of the polygon substrate (the plane of the paper). One of n facets of width Da is shown (solid lines) in three positions: undeflecting (neutral) and its two limit positions. The reflected beams are shown in corresponding positions. A lens housing edge denotes the input surface of a flat-field lens. Angle γ provides clear separation of the input beam and the down-deflected beam or lens housing. The pupil relief distance P (perpendicular to the lens housing at axis point a) and its slant distance Peestablish angle α as

Angle α is the off-axis illumination on the polygon that broadens the input beam width on the facet from D to Dm. (Note that α approximates the angle between the input beam and the normal to the up-deflecting facet.) An additional safety factor t (1 ≤ t ≤ 1.4) limits one-sided truncation of the beam by the edge of the facet at the end of scan, yielding

Replacing D in Equation 4-2 with Dm and solving for the full facet width Da,

which allows for maximum off-axis landing of the input beam and also avoids truncation of the input beam during the end of the active portion of scan.

As developed earlier [Bei5], the polygon outer (circumscribed) diameter DP = Da/sin(π/n) may now be expressed with operational parameters of Figure 4.2,

Solution of Equation 4-9 or one of similar form [Kes] entails determination of α, the angle of off-axis illumination on the facet, usually requiring a detailed layout similar to that of Figure 4.2. Aiding system perception, Equation 4-6 and Figure 4.2 reveal that

Substituting this into Equation 4-9 yields,

still requiring a determination of Pe for a given pupil relief distance P. After some accurate small-angle approximations and series expansion of cosα, the relationship for Dp may be expressed [Bei5] with Pe implicit in terms of the principal design parameters,

The numerator of the second term, represented originally by 1 + ΘDs/2P [Bei5], was simplified here by letting the beam spacing factor s take a practical value of 2 for good beam clearance. One may observe also in the numerator of the second term the reappearance of the resolution invariant ΘD, discussed particularly in Section 3.3. The terms in Equation 4-12 are sufficiently basic as initial design parameters to allow reasonable estimate of the polygon diameter. Evaluation of the rigorous Equation 4-9 vs. this equation in typical operational conditions reveals a reduced value by Equation 4-12 of approximately 1.3%.

Proper orientation of the polygon also requires the height h (Fig. 4.2), the distance from the polygon rotational axis o normal to the lens axis. Assume first a typical polygon of relatively high facet count (n ≥ 12) to which the input beam is directed to the center of the undeflected facet, encountering the inscribed circle of radius R' where R' = R cos(π/n). Because sinβ = h/R', then

However, the intersection point of the input beam at the polygon facet moves during scan as the facet rotates from this center position. As the facet shifts under the beam, the intersection point moves toward the source of the beam, "lifting up" on the corners of the facet [Kle, L&K, Mar1]. Also, because the polygon is illuminated asymmetrically by the input beam (which is displaced from the facet normal by angle β), this intersection of the beam with the facet also becomes more asymmetric at lower facet count (where the arc of the superscribed circle may no longer be assumed equal in length to its chord, the length of the facet.) The consequence of this is analyzed in a recent publication [Mar3].

 

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