where R is the radius of curvature of the reference sphere. The integration here is taken over infinite limits although in (3.439) it was taken over the area of the pupil. This change is made to simplify the application of Fourier transform theory and is validated, for example, in Marechal-Francon [3.26] and Welford [3.6]. If £/AR and n/AR are considered as single variables as in Eq. (3.434), then Eq. (3.489) expresses U(£,n) as the two-dimensional
Fourier transform of F(£,n) with the appropriate re-scaling (normalization) of £, n. This statement is generally true for any shape or size of pupil and any aberration function W(x, y). If the transmission over the pupil is not uniform (apodisation), this can also be introduced into the function F(£, n). The constant factors omitted are given by Born-Wolf [3.120(e)] or Marechal-Francon [3.26] and disappear in the subsequent normalization process. The intensity point spread function is given by
Again following the treatments of Welford [3.6] and Marechal-Francon [3.26], we now generalize from the expression (3.487) for two dimensions to give the OTF as the normalized inverse Fourier transform of the intensity PSF as
where s and t are the spatial frequencies in the £ and n directions respectively. They are expressed as lines/mm (or line-pairs/mm in television usage) if linear dimensions are in mm. L(s) in (3.487) is simply the one-dimensional equivalent and is equal to L(s,0). From (3.491), the OTF in the ^-direction only is
The function il(£) is the line spread function (LSF), the image of an infinitely narrow line source illuminated incoherently. Eq. (3.492) states that the one-dimensional OTF is the Fourier transform of the LSF. The denominator in (3.492) is the normalizing factor which gives L(0,0) = 1, a condition expressed in (3.485) above giving C0 = 1 for the object with no loss of contrast.
It is shown by Marechal-Francon [3.26(g)] that the expression (3.491) for the OTF can be converted into an alternative form which has an elegant physical interpretation. This form uses the autocorrelation function of the pupil function:
f+° fF(x + ARs, y + ARt)F* (x, y)dx dy L(s,t) = J - ° J- ° +-^-^-—---(3.494)
As before, the denominator normalizes to make L(0,0) = 1. Applying the theorem of Parseval, it is shown in Marechal-Francon [3.26(h)] that the numerator of (3.494) can be transformed into the symmetrically equivalent form
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