Here, N is an arbitrary normalization and the time dependence has been suppressed. U(x',y') is the value of the field at the image location (x',y'). (Uniform transmission versions of the formula are derived in Hecht 1987, pp. 461-462; Goodman 1968, pp. 37-41; Born and Wolf 1980, pp. 378-380.) The integral may be performed over the entire surface A, but the integrand is nonzero only inside the aperture pupil.
is called the pupil function and is merely a complex number in circular notation. The transmission coefficient T(x, y) is its modulus2 and the function W(x,y) (in wavelengths) implies the phase. W(x,y) contains aberrations like turbulence, pinched optics, defocusing, astigmatism, etc. The cosine term in brackets is the inclination or obliquity factor. This function will eliminate nonphysical backward propagation. If we place the source far away, then 0S is 0 and cos8S is 1. Set 8r to 180°, as it would be for backward propagation, and the inclination factor vanishes.
In writing Eq. B.l, another approximation has been implicitly made. The aberration function W(x,y) affects the angles in the inclination factor slightly, and the equation doesn't contain this effect anywhere. For all practical purposes, this change in angle is extremely small. Typically, the
2 Modulus is the absolute value of a complex number.
worst wavefront tilts in this book are 30 wavelengths over 100 mm, or about 0.01°. Equation B.l also contains an assumption of linearity.
Key elements in the Fresnel-Kirchhoff formula are the two factors eikr/r and eiks/s (k = 2rc/X). Here, Huygens' principle of re-radiating elemental points is written in mathematical form. Each of these expressions is the time-independent part of a spherical wave function. The denominator will cause the intensity to obey the inverse square law of light (twice as far, one-fourth as bright). The numerator will ensure that the Fresnel zones will be properly painted on the aperture. The "s" spherical wave represents the propagation of the wave from the source to a point on the aperture. It is then reborn as the "r" spherical wave, which propagates to the receiving point.
The intensity is related to the energy, so it cannot be complex-valued. It is calculated from the field above as follows:
I(x',y') = U(x',y') U*(x',y') = U \ (x',y')\2. (B.2)
Once the integral has been performed over the whole open aperture, Eq. B.2 says that the intensity is known for only one point in the image space. To find it at any other location, we must change the values of x' and y' and evaluate the field integral again. To completely map out an entire image this way requires (to say the least) a great deal of time.
Another name for the expression I(x',y') is the point-spread function, or PSF. The PSF determines how diffraction, obstructions, and aberrations degrade a perfectly sharp point source of light into a fuzzy disk. In the case of perfect optics filtered only by a finite circular aperture, the PSF follows the familiar Airy disk pattern.
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