13 Wavefronts
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Last updated: October 15, 2025
13.1 Wavefronts overview
Modeling light fields as the sum of waves is less intuitive than imaging rays. One purpose of this chapter is to provide some intuitive explanations of wave properties. In particular, I would like to connect the concepts of complex number representations of the wavefronts with basic hardware devices.
A second reason is that one often hears people describe optical imperfections using the concept of wavefront aberrations. They will use the orthogonal Zernike polynomials and the Seidel Aberrations (non-orthogonal) to characterize these imperfections. Their widespread use in commercial applications makes it important to understand the logical basis for their use.
Finally, I want to explain how wavefront calculations with complex numbers fit into ISETCam calculations. ISETCam uses complex wavefront representations to characterize and model shift-invariant lenses. I would like you to understand what we are doing, and I hope this section clearly explains why we made this choice.
13.2 Wave directions and phase
13.3 The Pupil Function
Let \(A(\rho, \theta)\) be the pupil amplitude (often 1 inside the pupil and 0 outside), and let \(W(\rho, \theta)\) be the wavefront aberration function in units of length (e.g., microns). The pupil function is:
\[ P(\rho, \theta) = A(\rho, \theta) \, e^{i \, \frac{2\pi}{\lambda} W(\rho, \theta)} \]
where
- \(\lambda\) is the wavelength
- \(\rho, \theta\) are polar coordinates over the pupil
The phase term encodes the optical path difference introduced by aberrations.
13.3.1 Phase: Wave direction
How we represent the phase
13.3.2 Amplitude: Pupil shape and transmission
To model different physical configurations:
- Stopping down the aperture Modify \(A(\rho, \theta)\) to reduce the pupil radius
- Adding obscurations, vignetting, or scratches Mask parts of \(A(\rho, \theta)\) or add amplitude modulation
- Simulating phase plates or diffractive elements Alter \(W(\rho, \theta)\) with custom phase profiles
These changes immediately reflect in the PSF via the Fourier transform. How we represent the amplitude Pupil shape can get introduced here.
Enables us to specify properties like scratches, aperture size.
13.3.3 Computing the PSF
The complex field in the image plane (at best focus) is proportional to the Fourier transform of the pupil function:
\[ U(x, y) \propto \mathcal{F}\{ P(\rho, \theta) \} \]
The PSF is the intensity of this field:
\[ \text{PSF}(x, y) = |U(x, y)|^2 \]
Hence, the wavefront shape directly influences the spatial structure of the PSF.
13.4 Complex wavefront models
We need phase and amplitude
13.4.1 Zernike
Efficient representations. Zernike polynomials. Explain the Zernike polynomials as an approximation to the wavefront. Born and Wolf description of derivation might be good here.
This pdf from the book might be what I mean: appVII circle_polynomials_of_zernike_921 It has a lot of functional equations.
The wavefront function \(W(\rho, \theta)\) is often expanded in terms of Zernike polynomials \(Z_n^m(\rho, \theta)\), which form an orthonormal basis over the unit circle:
\[ W(\rho, \theta) = \sum_{n,m} a_n^m Z_n^m(\rho, \theta) \]
Each coefficient \(a_n^m\) corresponds to a specific aberration mode:
- defocus
- astigmatism
- coma
- spherical aberration
- trefoil, etc.
This representation enables:
- systematic control over image degradation
- parametric simulations of optical quality
- statistical modeling (e.g., for manufacturing tolerances or biological variation)
13.4.2 Seidel
Seidel aberrations
13.5 Imperfections: Wavefront Aberrations and the PSF
To model the image of a point source (i.e., the PSF), we consider the pupil function, which describes the amplitude and phase of the light at the exit pupil of the system. Aberrations are incorporated into the phase component.
Model of plausible PSFs in this Zernike polynomial space is better than the free PSF description. Look for an explanation of why the wavefront at the lens is related to the image irradiance by a Fourier Transform. Goodman? ChatGPT? Come on, it must be out there.
Lead up to wavefront in the next section.
Complex valued description of the PSF. Why this rather than just the PSF?
What is a perfect wavefront for a thin lens, or really any lens? Collimated input, what is the output?
Deviations from the perfect wavefront. Notice that people measure aberrations in different ways (see below).
13.6 ISETCam and wavefronts
13.7 Examples
13.7.1 Flare
Scratches and HDR rendering illustrated
Here’s a Quarto-formatted explanation of the mathematical connection between wavefront aberrations and the point spread function (PSF), emphasizing how Zernike polynomials and pupil modifications fit into the framework.
13.7.2 Wavefront Sensing Hardware:
Systems like Shack-Hartmann sensors measure the wavefront directly. Using Zernike decomposition on the output allows immediate interpretation and simulation without needing to reconstruct PSFs from images.
13.7.3 Interfacing with Human Vision Models:
In visual optics (e.g., for modeling the human eye), Zernike representations are the standard. This makes it easy to compare and combine artificial optics (e.g., camera lenses) with biological optics models.