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Optical Engineering Science. Stephen RoltЧитать онлайн книгу.

Optical Engineering Science - Stephen Rolt


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rel="nofollow" href="#ulink_13fd97f0-00a8-5dd5-b989-c0547a4b1792">Figure 5.1 is that the aggregate distance F1P + PF2 is always constant. By virtue of Fermat's principle, this inevitably implies that, since the optical path is the same in all cases, F1 and F2, from an optical perspective, represent perfect focal points with no aberration whatsoever generated by reflection from the ellipsoidal surface. In describing the ellipsoid above, it is useful to express it in terms of polar coordinates defined with respect to the focal points. If we label the distance F1P as d, then this distance may be expressed in the following way in terms of the polar angle, θ:

Geometrical illustration of ellipsoid of revolution.

      (5.4)equation

      From the above, it is easy to calculate the conjugate parameter for this conjugate pair:

equation

      Thus, it is straightforward to demonstrate that for a conic section, there exists one pair of conjugates for which perfect image formation is possible. Of course, the most well known of these is where k = −1, which defines the paraboloidal shape. From Eq. (5.5), the corresponding conjugate parameter is −1 and relates to the infinite conjugate. This forms the basis of the paraboloidal mirror used widely (at the infinite conjugate) in reflecting telescopes and other imaging systems.

      As for the spherical mirror, the effective focal length of the mirror remains the same as for the paraxial relationship:

      More generally, the spherical aberration produced by a conic mirror is of a similar form as for the spherical mirror but with an offset:

      (5.7)equation

      Worked Example 5.1 Simple Mirror-Based Magnifier

      We wish to construct a simple magnification system with a simple conic mirror. The system magnification is to be two and the object distance 100 mm. There is to be no on axis aberration. What is the prescription of the mirror, i.e. base radius and conic constant?

      It is assumed that object and image are located the same side of the mirror, so that, in this context, the image distance is −200 mm. The overall set up is illustrated in the diagram:

Geometrical illustration of an object and image located the same side of the mirror. The image distance is –200 mm. equation

      This gives R = −133 mm.

      We now need to calculate the conjugate parameter, t:

equation

      From Eq. (5.5) it is straightforward to see that k = −(1/t)2 and thus k = −0.1111. The shape is that of a slightly prolate ellipsoid.

      The practical significance of a perfect on axis set up described in this example, is that it forms the basis of an ideal manufacturing test for such a conic surface. This will be described in more detail later in this text.

      5.2.3 Conic Refracting Surfaces

      If the refractive index of the surface is n, assuming that the object is in air/vacuum, then the conic constant of the ideal surface is –n2. In fact, the shape is that of a hyperboloid. The abscissa of the hyperboloid effectively produce grazing incidence for rays originating from the object. By definition, therefore, the angle that the surface normal makes with the optical axis at the abscissa is equal to the critical angle. This restricts the maximum numerical aperture that can be collected by the system. With this constraint, it is clear that the maximum numerical aperture is equal to 1/n. In summary therefore:

      (5.8)equation

      Unfortunately, no other general condition for perfect image formation results for a conic surface. However, for perfect image correction, all orders of (on axis) aberration are corrected. Thus, although no condition for perfect image formation is possible, it is still possible, nevertheless, to correct for third order spherical aberration with a single refractive surface.

Geometrical illustration of single refractive surface at infinite conjugate.
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