Читать книгу Optical Engineering Science - Stephen Rolt - Страница 71

The analysis of the aberrations of a single refractive surface is based on the computation of the OPD of a generalised field point to the appropriate order (4th) in terms of field angle, θ and ray height, r, at the pupil. For this analysis, we will assume that the pupil is located at the lens surface. In calculating the OPD, we force all rays to go to the paraxial focus and compute the OPD with respect to the chief. Figure 4.1 shows an object with a field angle, θ, located at a distance, u from a spherical refractive surface of radius R. It must be emphasised, in this instance, that this analysis applies specifically to a spherical surface. In this geometry, it is assumed that the object is displaced from the optical axis in the y direction. The paraxial image is itself located at a distance v from the surface and the position of a ray at the surface (and stop) is described by its components in x and y – h_x and h_y.

The image in this case is the paraxial image and from the paraxial theory, the angle φ may be expressed in terms of θ as θ/n. To compute the optical path of a general ray as it passes from object to paraxial image, we need to define the ray co-ordinates at three points:

The z co-ordinate of the stop position is derived from the binomial expansion for the axial sag of a sphere including terms up to the fourth power. In making this approximation, it is assumed that h is significantly less than R. If we were to adopt the paraxial approximation we would only consider the first r² term in the expansion. In the case of third order aberration, we need to consider the next term. It is then very straightforward to calculate the total optical path, Φ, for a general ray in passing from object to paraxial image:

(4.2)

The two square root terms represent the optical path of two ‘legs’ of the journey, with the path through the glass adding a multiplicative factor of n. The next stage of the process is an extension of the paraxial theory. It is assumed that r_x, r_y, and uθ are all significantly less than u. We can now approximate Φ from Eq. (4.2) using the binomial theorem. In the meantime collecting terms we get:

Before deriving the third order aberration terms, we examine the paraxial contribution which contain terms in h up to order r².

(4.3)

As one would expect, in the paraxial approximation, the optical path length is identical for all rays. However, for third order aberration, terms of up to order h⁴ must be considered. Expanding Eq. (4.2) to consider all relevant terms, we get:

(4.4)

Four of the five Gauss-Seidel terms are present – spherical aberration, coma, astigmatism, and field curvature. However, clearly there is no distortion. In fact, as will be seen later, distortion can only occur where the stop is not at the surface as it is here. Of course, Eq. (4.4) can be simplified if one considers that u, v, and R are dependent variables, as related in Eq. (4.3). Substituting v for u, and R, we can express the OPD in terms of u and R alone. Furthermore, it is useful, at this stage to split the OPD contributions in Eq. (4.4) into Spherical Aberration (SA), Coma (CO), Astigmatism (AS), and Field Curvature (FC). With a little algebraic manipulation this gives:

(4.5a)

(4.5b)

(4.5c)

(4.5d)