The pupil function for the diffraction-limited imaging system model of Chapter 7 is defined relative to an ideal converging spherical wavefront. A system with aberrations has a wavefront phase surface that deviates from the ideal spherical wave. Aberrations are found in most practical imaging systems, and their effect reduces image quality. If aberrations are significant, then a ray (geometrical) optics approach typically is used for studying image effects. However, if the system is "relatively close" to diffraction limited, then wave optics can be the tool of choice. In this chapter the diffraction-limited imaging theory is extended to include aberrations. The incoherent point spread function (PSF) and modulation transfer function (MTF) are discussed and demonstrated as image quality measures. Aberrated systems tend to cause space-variant imaging, where the impulse response is not the same for each image point. An example of a space-variant image simulation is presented.
8.1 Wavefront Optical Path Difference
Figure 8.1 shows the exit pupil (XP) with an ideal spherical (sp) wavefront and aberrated (ab) wavefront in profile. The wavefront error is described by W(x,y), an optical path difference (OPD) function that represents the difference between the spherical and aberrated wavefront surfaces. x and y are coordinates in the pupil plane.
Aberrated wavefronts arise from various sources. Obvious examples are imperfections in the imaging optics. For systems that peer through the atmosphere, the wavefront disturbances caused by the turbulence can be characterized in terms of aberrated wavefronts. However, even when optical components are made exactly to specification, aberrations will be present. For example, chromatic aberration, where different wavelengths focus at different positions, is caused by the wavelength dependence of the index of refraction of glass. Furthermore, if a system images an extended object scene, light from off-axis points must transit the system at an angle relative to the optical axis, and this generates a departure from a spherical wave.
Wavefront OPD is commonly described by a polynomial series. The Seidel series is used by optical designers because the terms have straightforward mathematical relationships to factors such as lens type and position in the image plane.