Tag Archives: phase

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 →

DOF, intensity, Optics, wavefront

Pupils and Apertures

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We’ve seen in the last article that the job of an ideal photographic lens is simple: to receive photons from a set of directions bounded by a spherical cone with its apex at a point on the object; and to concentrate them in directions bounded  by a spherical cone with its apex at the corresponding point on the image.   In photography both cones are assumed to be in air.

In this article we will distill the photon collecting and distributing function of a complex lens down to its terminal properties, the Entrance and Exit Pupils, allowing us to deal with any lens simply and consistently. Continue reading Pupils and Apertures →

IQ, Optics, Perceptual, Sharpness, Spatial Resolution

DOF and Diffraction: Setup

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The two-thin-lens model for precision Depth Of Field estimates described in the last two articles is almost ready to be deployed.  In this one we will describe the setup that will be used to develop the scenarios that will be outlined in the next one.

The beauty of the hybrid geometrical-Fourier optics approach is that, with an estimate of the field produced at the exit pupil by an on-axis point source, we can generate the image of the resulting Point Spread Function and related Modulation Transfer Function.

Pretend that you are a photon from such a source in front of a f/2.8 lens focused at 10m with about 0.60 microns of third order spherical aberration – and you are about to smash yourself onto the ‘best focus’ observation plane of your camera.  Depending on whether you leave exactly from the in-focus distance of 10 meters or slightly before/after that, the impression you would leave on the sensing plane would look as follows:

Figure 1. PSF of a lens with about 0.6um of third order spherical aberration focused on 10m.

The width of the square above is 30 microns (um), which corresponds to the diameter of the Circle of Confusion used for old-fashioned geometrical DOF calculations with full frame cameras.  The first ring of the in-focus PSF at 10.0m has a diameter of about 2.44\lambda \frac{f}{D} = 3.65 microns.   That’s about the size of the estimated effective square pixel aperture of the Nikon Z7 camera that we are using in these tests.
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Optics, Perceptual, Sharpness, Spatial Resolution

DOF and Diffraction: Image Side

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This investigation of the effect of diffraction on Depth of Field is based on a two-thin-lens model, as suggested by Alan Robinson[1].  We chose this model because it allows us to associate geometrical optics with one lens and Fourier optics with the other, thus simplifying the underlying math and our understanding.

In the last article we discussed how the front element of the model could present at the rear element the wavefront resulting from an on-axis source as a function of distance from the lens.  We accomplished this by using simple geometry in complex notation.  In this one we will take the relative wavefront present at the exit pupil and project it onto the sensing plane, taking diffraction into account numerically.  We already know how to do it since we dealt with this subject in the recent past.

Figure 1. Where is the plane with the Circle of Least Confusion?  Through Focus Line Spread Function Image of a lens at f/2.8 with the indicated third order spherical aberration coefficient, and relative measures of ‘sharpness’ MTF50 and Acutance curves.  Acutance is scaled to the same peak as MTF50 for ease of comparison and refers to my typical pixel peeping conditions: 100% zoom, 16″ away from my 24″ monitor.

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Optics, Perceptual, Sharpness

DOF and Diffraction: Object Side

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In this and the following articles we shall explore the effects of diffraction on Depth of Field through a two-lens model that separates geometrical and Fourier optics in a way that keeps the math simple, though via complex notation.  In the process we will gain a better understanding of how lenses work.

The results of the model are consistent with what can be obtained via classic DOF calculators online but should be more precise in critical situations, like macro photography.  I am not a macro photographer so I would be interested in validation of the results of the explained method by someone who is.

Figure 1. Simple two-thin-lens model for DOF calculations in complex notation.  Adapted under licence.

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Optics, Quantization, Sensors, Spatial Resolution

Wavefront to PSF to MTF: Physical Units

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In the last article we saw that the intensity Point Spread Function and the Modulation Transfer Function of a lens could be easily approximated numerically by applying Discrete Fourier Transforms to its generalized exit pupil function \mathcal{P} twice in sequence.[1]

Numerical Fourier Optics: amplitude Point Spread Function, intensity PSF and MTF

Obtaining the 2D DFTs is easy: simply feed MxN numbers representing the two dimensional complex image of the Exit Pupil function in its uv space to a Fast Fourier Transform routine and, presto, it produces MxN numbers representing the amplitude of the PSF on the xy sensing plane.  Figure 1a shows a simple case where pupil function \mathcal{P} is a uniform disk representing the circular aperture of a perfect lens with MxN = 1024×1024.  Figure 1b is the resulting intensity PSF.

Figure 1a, left: A circular array of ones appearing as a white disk on a black background, representing a circular aperture. Figure 1b, right: Array of numbers representing the PSF of image 1a in the classic shape of an Airy Pattern.
Figure 1. 1a Left: Array of numbers representing a circular aperture (zeros for black and ones for white).  1b Right: Array of numbers representing the PSF of image 1a (contrast slightly boosted).

Simple and fast.  Wonderful.  Below is a slice through the center, the 513th row, zoomed in.  Hmm….  What are the physical units on the axes of displayed data produced by the DFT? Continue reading Wavefront to PSF to MTF: Physical Units →