Unified Optical Scanning Technology

Chapter 4.4.1.1 - Scanner Devices and Techniques: Analogous Relationship of Bragg Diffraction and Wedge

4.4.1.1    Analogous Relationship of Bragg Diffraction and Wedge Prism Refraction    The discussion above of the reduction of the effect of wobble by the linear plane grating identified an analogous rela-tionship to the refractive wedge prism. Pursuit of this characteristic provides a revealing expression of this effect. Consider Figure 4.11 depicting the two configurations. View a is a repeat of Figure 3.8, with notations of the Bragg angles β repaced with the more generic ηota-tions Θ, to be consistent with those of a hypothetical wedge prism scanner of view b.

As may be seen by a comparison of classic optics presentation [J&W] of effective prism tipping and the analysis [Kra1,Bei1] of hologram tipping, a strong parallel appears between the effects of small departures from the Bragg condition and departure from the minimum deviation of a prism. Notably, for a prism (in air) of apex angle 60° and glass index of 1.5, compared with a linear hologram having Θi = Θo = 45°, the plotted curves are almost identical for deviations of ± a few degrees. Keeping in mind that a deviation of ±1º is a 'gigantic' number of 3600 arc seconds for the wobble of a rotating system, which is seldom allowed to exceed 60 arc seconds, the comparison is significant.

For the analysis of the prism system, a ray-trace modification of Figure 4.11b includes identifying the prism with refractive index n and apex angle α, constructing the normals to the input and output faces, and notating the input ray angle to its face normal as Θip, the output ray angle to its face normal as Θop, the input angle of the ray propagating within the prism as αi, and the output angle of the ray propagating within the prism as αo. Then, expressing Snell's law at both faces,

and

The internal angles of the prism can be shown to be related to its apex angle as

By operational symmetry,

yielding,

and

When written in the same form as the grating equation (see Equation 4-14),

whence, for sinα/2 α/2,

Equation 4-22 is seen to be functionally the same as Equation 4-14 with

representing operationally equivalent constants. The product of the refractive index n and the apex angle α of the prism is equivalent to the ratio of the wavelength λ to the grating spacing d of the diffractor. Furthermore, their typical working values each form magnitudes of approximate unity. Thus, following the procedure that yielded Equation 4-15 and its low wobble consequences for the linear holographic scanner, the same inferences may be drawn for a hypothetical scanner utilizing prisms oriented in minimum deviation. Because the operation of such a prism scanner could be burdened by nonuniformities of prism assembly, rotational imbalance, and excessive inertial stress and strain, this analogous configuration is offered for its unifying pedagogic value, rather than for its practical utilization.

 

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