a constant quantifying distortion
If we denote the x and y components of the object and image location by xob, yob and xim, yim respectively, then we obtain:
(3.36)
From Eq. (3.35), it is clear that an object represented by straight line that is offset from the optical axis in object space will be presented as a parabolic line in image space. As such, the image is clearly distorted. The sense and character of the distortion is governed by the sign and magnitude of ζ. This is shown in Figures 3.19a,b.
Where ζ and the distortion is positive, the distortion is referred to as pincushion distortion, as suggested by the form shown in Figure 3.19a. On the other hand, if ζ is negative, the resultant image is distended in a form suggested by Figure 3.19b; this is referred to as barrel distortion.
Worked Example 3.1 The distortion of an optical system is given as a WFE by the expression, 4Φ0c3pcosφθ3, where Φ0 is equal to 50 μm and c = 1. The radius of the pupil, r0, is 10 mm. What is the distortion, expressed as a deviation in percent from the paraxial angle, at a field angle of 15°? From Eq. (3.12) and when expressed as an angle, the transverse aberration generated is given by:
The cosφ term expresses the fact that the direction of the transverse aberration is in the same plane as that of the object/axis. The proportional distortion is therefore given by:
(dimensions in mm; angles in radians)
The proportional distortion is therefore 0.13%.
3.6 Summary of Third Order Aberrations
At this stage it will be useful to summarise the five Gauss-Seidel aberrations in terms of the pupil and field dependence of their OPD and ray fans. It should be noted that for all Gauss-Seidel aberrations, the order of the pupil dependence and the order of the field angle dependence sum to four (for the OPD). In particular, it is important for the reader to understand how the different types of aberration vary with both pupil size and field angle. For example, in many optical systems, such as telescopes and microscopes, the range of field angles tend to be significantly smaller than the larger angles subtended to the pupil. Therefore, for such instruments, those aberrations with a higher-order pupil dependence, such as spherical aberration (4) and coma (3), will predominate.
3.6.1 OPD Dependence
The list below sets out the WFE dependence of the five Gauss-Seidel aberrations on pupil function, p, and field angle, θ.
Spherical Aberration: ΦSA ∝ p4
Coma: ΦCO ∝ p3θ
Field Curvature: ΦFC ∝ p2θ2
Astigmatism: ΦAS ∝ p2θ2
Distortion: ΦDI ∝ pθ3
To quantify each aberration, we can define a coefficient, K, which describes the magnitude (in units of length) of the aberration. In addition, as well as normalising the pupil function, we can also normalise the field angle by introducing the quantity, h, which represents the ratio, θ/θ0, the ratio of the field angle to the maximum field angle.
(3.37)
(3.38)
(3.39)
(3.41)
The reader should take particular note of the form of Eq. (3.40). The description of astigmatism here is such that the mean defocus over all orientations of the ray fan is taken to be zero. However, other representations adopt the convention that the defocus is zero for the sagittal ray and the balance of the astigmatism is incorporated into the field curvature. That is to say, in these conventions, the astigmatism is taken to be proportional to cos2φ, rather than cos2φ, as in Eq. (3.40). Of course, in using cos2φ, an average defocus of the same form as field curvature is introduced, hence the reason for adopting the convention used here. If the field curvature and astigmatism were redefined according to that convention, then the following revised description would apply:
(3.42)
(3.43)
3.6.2 Transverse Aberration Dependence
The ray fan or transverse aberration dependence upon pupil function and field angle is such that the order of the two variables sum to three, as opposed to four for OPD. The dependence of transverse aberration is listed below:
Spherical Aberration: tSA ∝ p3
Coma: tCO ∝ p2θ
Field Curvature: tFC ∝ pθ2
Astigmatism: tAS ∝ pθ2
Distortion: ΦAS ∝ θ3
3.6.3 General Representation of Aberration and Seidel Coefficients
The analysis presented in this chapter has demonstrated the power of using the OPD as a way of describing aberrations. More generally, when expressed as a WFE, it can be used to describe the deviation of a specific wavefront from an ideal wavefront that converges on a specific reference point. As such, this deviation can be used to describe defocus, which shows a quadratic dependence on pupil function and tilt, where the WFE is plane surface that is tilted about the x or y axis (the optical axis being the z axis). The standard representation for describing and quantifying generic WFE and aberration behaviour is shown in Eq. (3.44).
