Pobreza

It is convenient to have analytical expressions for the wavefront deformations caused by aberrations. In a general manner, without assuming any symmetries, the shape of a wavefront as measured at the exit pupil of the optical system may be represented by a polynomial: which includes high-order aberration terms, where N is the degree of this polyno-mial. In polar coordinates, we define

x = S sin θ (7.8)

and

y = S cos θ, (7.9)

where S is the radial distance at the exit pupil and the angle θ is measured with respect to the y axis, as shown in Figure 7.2. Then, the wavefront shape may be written in polar coordinates, expressing it as a series of cosⁿθ plus a series of sinⁿθ as follows:

162 Handbook of Optical Design

where the cos θ and sin θ terms describe the symmetrical and antisymmetrical com-ponents of the wavefront, respectively. However, not all possible values of n and l are permitted. To have a single valued function, we must satisfy the condition

W(S,θ) = W(−S,θ + π). (7.11)

Then, it is easy to see that to satisfy this condition, n and l must be both odd or both even. On the other hand, if this expression for the wavefront is converted into Cartesian coordinates W(x,y), it becomes an infinite series, unless l ≤ n. Thus, if we want Equation 7.10 to be equivalent to the finite series in Equation 7.7, we have to impose this condition, which is almost always satisfied, except in some very rare cases, for example, in the representation of some rotational shearing interferograms, as pointed out by Malacara and DeVore (1992).

Before proceeding, it is important to remember that these wavefront deformations W(s,θ) are measured with respect to a reference sphere whose vertex is at the center of the exit pupil and that its shape is close to that of the wavefront. The vertex of the reference sphere and the vertex of the wavefront are in contact and crossed by the principal ray. The center of curvature of this reference wavefront, which, for simplic-ity, we will call the point C, can be placed at many different locations. For a given image height h, the point C is at some point along the principal ray. The curvatures and tilts of the small circular region close to the vertex of these surfaces, called the parabasal region, define the location of the point C. Thus, the location of the point C is only defined by the linear and quadratic terms. The linear terms define the travel-ing direction of the principal ray and the quadratic terms define the position of the point C along the principal ray. On the other hand, the quadratic terms are only the defocusing and the primary astigmatism terms.

We see that if both the defocusing and the astigmatism are zero, the reference sphere becomes the osculating sphere, defined as a sphere with the same curvature as the wavefront at their common vertex. The presence of a defocusing term would be an indication of a difference between the parabasal curvature of the wavefront and the curvature of the reference sphere.

y

S x θ

FIGURE 7.2 Polar coordinates for the ray on the entrance pupil of an optical system.

7.3.1 h. h. hoPkins waVefront aberration

Polynomialfor centered systems

Now, if we restrict ourselves to the case of a wavefront produced by a centered, also called axially symmetrical optical system, with a point object displaced along the y axis, the wavefront is symmetrical about the tangential or meridional plane, obtaining

W S a S_nl ^{n l} where the angle θ is measured in the entrance pupil, from the y axis, by convenience, owing to the symmetry of centered systems about the meridional plane. The wave-front deformation changes for different image heights h. Thus, the coefficients anl

are functions of the image height that can also be written in a polynomial form as

anl w hknl k

where K is the maximum power for the image height h. As shown by Hopkins (1950), if we include in Equation 7.12 the image height h and impose the conditions

W(S,θ,h) = W(S,−θ,h) (7.14)

because of the symmetry about the meridional plane and

W(S,θ,h) = W(S,θ + π,−h) (7.15)

because of the rotational symmetry of the lens system about the optical axis, the wavefront expression may be shown to have only terms of the form

F h

and their products. An interesting consequence is that the sum of the powers of S and h is always an even number, whose value minus one is known as the aberration order.

In other words, the order of these aberrations is given by the sum of the power of S and the power of h, as (n + k)/−1, as in Table 7.1. The wavefront aberration W(S,θ,h) may be represented by a linear combination of these aberrations, with terms w_knl Sⁿ h^k cos^lθ, where k is the power of the image height h, n is the power of the semiaper-ture S, and l is the power of cos θ, with l ≤ n.

Each of these terms has a name, but not all are higher-order terms. Unfortunately, there is no universal agreement on the notation for aberration coefficients, but some people use this representation for the wavefront aberrations (Geary 2002; Hopkins 1950; Sasian 2006). The following general names had been proposed:

164 Handbook of Optical Design

1. Spherical aberrations: terms independent of θ (k = 0) 2. Comatic aberrations: terms with odd powers of cos θ (k odd) 3. Astigmatic aberrations: terms with even powers of cos θ (k even)

As an example, the fifth-order aberration w₃₃₃ S³ h³ cos³θ is named an ellipti-cal coma because, as illustrated later in Figure 7.6, it transforms the circles at the exit pupil into ellipses at the image. Sometimes, this aberration is called triangular astigmatism by telescope makers, because it appears on-axis with a slightly different form, with a triangular deformation, due to a mirror deformation and not as a result of an off-axis displacement of the image.

A wavefront aberration polynomial up to fifth-order aberrations for a centered system (Kidger 2004) can be expressed with a linear combination of the Hopkins aberrations in Table 7.1, as follows:

W S h w w S w h w Sh