Читать книгу Optical Engineering Science - Stephen Rolt - Страница 75

The third order analysis for a spherical mirror proceeds in very much the same way as the single refractive surface. That is to say, a ray is traced from the object location to the mirror and thence to the paraxial focus regardless as to whether the real ray actually terminates there. The general layout is shown in Figure 4.5. The sign convention used here is the same as applied to all previous analyses. That is to say, positive image distance is with the image to the right, and the image distance, as shown in Figure 4.5, is actually negative. However, it must be accepted that, as rays physically converge on this image point, then this image is actually real, despite v being negative. In addition, the same convention is applied to mirror curvature; the mirror depicted in Figure 4.5 has negative curvature.

Figure 4.5 Reflection at spherical mirror.

The analysis proceeds as previously. Firstly, we set out the object and image positions and the ray intercept at the stop.

The optical path is given by:

(4.14)

Rearranging:

In applying the binomial approximation, one needs to be careful with regard to the sign convention. It should be accepted that each of the square root terms in Eq. (4.14) is positive for a real object and real image. That is to say, all rays are physically traced to the appropriate location. In the case of a mirror surface, the definition of a real image corresponds to a negative image distance, u. Once again, we examine the paraxial terms

(4.15)

As for the refractive surface we expand Eq. (4.14) using the binomial theorem to give terms of the fourth order in OPD.

(4.16)

As with the refractive case, four of the five Gauss-Seidel terms are present – spherical aberration, coma, astigmatism, and field curvature. There is also no distortion. As previously, Eq. (4.16) can be simplified considering u, v, and R as dependent variables, as related in Eq. (4.15). We can, once more, express the OPD in terms of u and R alone. Splitting the OPD contributions in Eq. (4.16) into Spherical Aberration (SA), Coma (CO), Astigmatism (AS), and Field Curvature (FC) and with a little algebraic manipulation we have:

(4.17a)

(4.17b)

(4.17c)

(4.17d)

Equations (4.17a)–(4.17c) bear some striking similarities with respect to those for the refractive surface. In fact, if one substitutes n = −1 in the corresponding refractive formulae, one obtains expressions similar to those listed above. Thus, in some ways, a mirror behaves as a refractive surface with a refractive index of minus one. Once again, there are aplanatic points where both spherical aberration and coma are zero. This occurs only where both object and image are co-located at the centre of the spherical surface. The apparent absence of field curvature may appear somewhat surprising. However, the Petzval curvature is non-zero, as will be revealed. We can now cast all terms in the form set out in Chapter 3 and introduce the Lagrange invariant, which is equal to the product of r₀ and θ₀ (the maximum field angle):

(4.18a)

(4.18b)

(4.18c)

(4.18d)

The Petzval curvature is simply given by subtracting twice the K_AS term in Eq. (4.18c) from the field curvature term in Eq. (4.18d). This gives:

(4.19)

Figure 4.6 Petzval curvature for mirror.

In this instance, the Petzval surface has the same sense as that of the mirror itself. However, the radius of the Petzval surface is actually half that of the original surface. This is illustrated in Figure 4.6.

Calculation of the Petzval sum proceeds more or less as the refractive case. However, there is one important distinction in the case of a mirror system. For a system comprising N mirrors, each successive mirror surface inverts the sense of the wavefront error imparted by the previous mirrors.

(4.20)