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Optical Engineering Science. Stephen RoltЧитать онлайн книгу.

Optical Engineering Science - Stephen Rolt


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      The determinant of the matrix, (AD−BC), is a key parameter. The ratio of the two focal lengths of the system is simply given by the determinant. That is to say the ratio of the two focal lengths is given by:

      (1.28)equation

      (1.29)equation

      This relationship was anticipated in the more generalised discussion in 1.3.9. Looking at the relationships for the principal and nodal points, it is clear when the determinant of the system matrix is unity, i.e. object and image space indices are the same, then the principal and nodal points are co-located.

      In addition to the principal and nodal points, anti-principal points and anti-nodal points are sometimes (rarely) specified. Anti-principal points are conjugate points where the magnification is −1. Similarly, anti-nodal points are conjugate points where the angular magnification is −1.

      1.6.3 Worked Examples

      We can now use the foregoing analysis to see how matrix ray tracing might be used in practice. Here we focus on a number of useful practical examples.

Geometrical illustration of thick lens.

      Worked Example 1.1 Thick Lens

equation equation

      As both object and image space are in the same media, there is a common focal length, f, i.e. f1 = f2 = f. All relevant parameters are calculated from the above matrix using the formulae tabulated in Section 1.6.2.

      The focal length, f, is given by:

equation

      The formula above is similar to the simple, ‘Lensmaker’ formula for a thin lens. In addition there is another term, linear in thickness, t, which accounts for the lens thickness.

      The focal positions are as follows:

equation

      The principal points are as follows:

equation Schematic illustration of Hubble space telescope. equation

      Worked Example 1.2 Hubble Space Telescope

      There are four matrix elements to consider here. First, there is a mirror with a radius of −11.04 m (note sign), followed by a translation of −4.905 m (again note sign). The third matrix element is a mirror (M2) of radius − 1.359 m. Finally, we translate by +4.905 m, so that both the input and output co-ordinates are referenced with respect to the same origin. The matrices are as below:

equation

      The focal positions are:

equation

      The principal points are at:

equation equation

      Although, in terms of overall length, the telescope


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