Читать книгу Optical Engineering Science - Stephen Rolt - Страница 64

The fifth and final Gauss-Seidel aberration term is distortion. The WFE associated with this aberration is third order in field angle, but linear in pupil function. In fact, a linear variation of WFE with pupil function implies a flat, but tilted wavefront surface. Therefore, distortion merely produces a tilted wavefront but without any apparent blurring of the spot. The WFE variation is set out in Eq. (3.34).

(3.34)

Thus, the only effect produced by distortion is a shift (in the y direction) in the geometric spot centroid; this shift is proportional to the cube of the field angle. However, this shift is global across the entire pupil, so the image remains entirely sharp. The shift is radial in direction, in the sense that the centroid shift is in the same plane (tangential) as the field offset. So, the OPD fan for the tangential fan is linear in pupil function and zero for the sagittal fan. The ray fan is zero for both tangential and sagittal fans, emphasising the lack of blurring.

Taken together with the linear (paraxial) magnification produced by a perfect Gaussian imaging system, distortion introduces another cubic term. That is to say, the relationship between the transverse image and object locations is no longer a linear one; magnification varies with field angle. If the height of the object is h_ob and that of the image is h_im, then the two quantities are related as follows:

(3.35)

Figure 3.19 (a) Pincushion (positive) distortion. (b) Barrel (negative) distortion.

M₀ is the paraxial magnification; ζ is a constant quantifying distortion

If we denote the x and y components of the object and image location by x_ob, y_ob and x_im, y_im 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%.