Tag Archives: SA

Fourier Optics, Geometrical Optics, MTF, PSF, wavefront

Fourier Optics and the Complex Pupil Function

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In the last article we learned that a complex lens can be modeled as just an entrance pupil, an exit pupil and a geometrical optics black-box in between.  Goodman[1] suggests that all optical path errors for a given Gaussian point on the image plane can be thought of as being introduced by a custom phase plate at the pupil plane, delaying or advancing the light wavefront locally according to aberration function \Delta W(u,v) as earlier described.

The phase plate distorts the forming wavefront, introducing diffraction and aberrations, while otherwise allowing us to treat the rest of the lens as if it followed geometrical optics rules.  It can be associated with either the entrance or the exit pupil.  Photographers are usually concerned with the effects of the lens on the image plane so we will associate it with the adjacent Exit Pupil.

aberrations coded as phase plate in exit pupil generalized complex pupil function
Figure 1.  Aberrations can be fully described by distortions introduced by a fictitious phase plate inserted at the uv exit pupil plane.  The phase error distribution is the same as the path length error described by wavefront aberration function ΔW(u,v), introduced in the previous article.

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Fourier Optics, Geometrical Optics, intensity, MTF, Optics, PSF, wavefront

An Introduction to Pupil Aberrations

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As discussed in the previous article, so far we have assumed ideal optics, with spherical wavefronts propagating into and out of the lens’ Entrance and Exit pupils respectively.  That would only be true if there were no aberrations. In that case the photon distribution within the pupils would be uniform and such an optical system would be said to be diffraction limited.

Figure 1.   Optics as a black box, fully described for our purposes by its terminal properties at the Entrance and Exit pupils.  A horrible attempt at perspective by your correspondent: the Object, Pupils and Image planes should all be parallel and share the optical axis z.

On the other hand if lens imperfections, aka aberrations, were present the photon distribution in the Exit Pupil would be distorted, thus unable to form a perfectly  spherical wavefront out of it, with consequences for the intensity distribution of photons reaching the image.

Either pupil can be used to fully describe the light collection and concentration characteristics of a lens.  In imaging we are typically interested in what happens after the lens so we will choose to associate the performance of the optics with the Exit Pupil. Continue reading An Introduction to Pupil Aberrations →

Optics, Sharpness, Spatial Resolution

A Simple Model for Sharpness in Digital Cameras – Polychromatic Light

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We now know how to calculate the two dimensional Modulation Transfer Function of a perfect lens affected by diffraction, defocus and third order Spherical Aberration  – under monochromatic light at the given wavelength and f-number.  In digital photography however we almost never deal with light of a single wavelength.  So what effect does an illuminant with a wide spectral power distribution, going through the color filter of a typical digital camera CFA  before the sensor have on the spatial frequency responses discussed thus far?

Monochrome vs Polychromatic Light

Not much, it turns out. Continue reading A Simple Model for Sharpness in Digital Cameras – Polychromatic Light →

Optics, Sharpness, Spatial Resolution

A Simple Model for Sharpness in Digital Cameras – Spherical Aberrations

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Spherical Aberration (SA) is one key component missing from our MTF toolkit for modeling an ideal imaging system’s ‘sharpness’ in the center of the field of view in the frequency domain.  In this article formulas will be presented to compute the two dimensional Point Spread and Modulation Transfer Functions of the combination of diffraction, defocus and third order Spherical Aberration for an otherwise perfect lens with a circular aperture.

Spherical Aberrations result because most photographic lenses are designed with quasi spherical surfaces that do not necessarily behave ideally in all situations.  For instance, they may focus light on systematically different planes depending on whether the respective ray goes through the exit pupil closer or farther from the optical axis, as shown below:

371px-spherical_aberration_2
Figure 1. Top: an ideal spherical lens focuses all rays on the same focal point. Bottom: a practical lens with Spherical Aberration focuses rays that go through the exit pupil based on their radial distance from the optical axis. Image courtesy Andrei Stroe.

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