Fourier Optics and the Complex Pupil Function

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 wavefront aberration function $\Delta W(u,v)$ as described there.

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.

If the wavefront aberration function $\Delta W$ is zero throughout the $uv$ pupil plane the system is said to be diffraction limited at the relative Gaussian point. If it is not zero in any part of the pupil it is an indication that the lens has some aberrations that are fully described by phase distortions introduced by the imaginary plate for the given Gaussian point.

The Pupil Function

$\Delta W$ is assumed to be part of the $uv$ pupil plane, its extent limited by the shape and size of the aperture (the virtual image of the mechanical stop there), which so far we have assumed to be a circular disk. However the aperture can be any shape whatsoever and in practice it is often polygonal when stopped down.

unaberrated poligonal pupil function — Figure 2. Central portion of image of an unaberrated Pupil Function with octagonal aperture of 5mm diameter. It represents the transmission of the exit pupil, with zero values in black stopping all light and values of 1 inside the aperture in white letting 100% of it through.

Nor does it have to be all or nothing. Some lens designs sport a graduated stop in order to obtain desirable qualities like softer bokeh for example (think of a radial neutral density filter within the mechanical stop itself). These characteristics are akin to a variable radial attenuation at the stop and can be represented as the grayscale image of a transparency within the $uv$ plane, say with zero indicating full photon stoppage in black, one full throughput in white and values in between indicating varying transmission. Such a description of the aperture is referred to as the Pupil Function $P(u,v)$ .

Since the Pupil Function limits the bundle of light that makes it through the lens, it is responsible for determining the amount of diffraction introduced into the incoming wavefront.

The Generalized Pupil Function

In optics the instantaneous amplitude ( $a$ ) and phase ( $\varphi$ ) of an analytic signal like a light wave can be expressed compactly in complex notation as a phasor: $a e^{\frac{2\pi i}{\lambda}\varphi}$ , with $\lambda$ the wavelength of light.

Wavefront aberration Function $\Delta W(u,v)$ represents phase shifts so multiplying element-wise ( $\odot$ ) the real transmission Pupil Function $P(u,v)$ by it in complex notation produces the Generalized (or complex) Pupil Function, as described in Goodman^[1]

(1) $\begin{equation*} \mathcal{P}(u,v) = P(u,v) \odot e^{\frac{2\pi i}{\lambda}\Delta W(u,v)} \end{equation*}$

$\mathcal{P}(u,v)$ captures both the transmittance of the pupil and the phase errors introduced by aberrations in the lens relevant to the projected image point. It represents the complex amplitude transmittance within the Exit Pupil relative to the Gaussian point on the image plane predicted by geometrical optics. It can be thought of as the phase plate mentioned in the introduction. It is a self-contained description of the wave-optical performance of the lens.

generalized complex pupil function PSF — Figure 3. Pupil Function, letting through all rays/photons in the white disk-like area and blocking those outside of it in gray (complete stoppage of light here is shown in gray instead of black to allow for annotations). If the aberrated wavefront function ΔW(u,v) is not zero inside the aperture, it becomes the Complex Pupil Function. Cartesian and polar coordinates are shown for the Exit Pupil plane. In pink we can imagine the intensity Point Spread Function on the sensing plane resulting from the complex field present in the Pupil as a result of an impulse of light. Not drawn to scale.

If the lens does not have any aberrations, ΔW is equal to zero and the exponential term evaluates to 1, leaving just the real transmission component. In that case $\mathcal{P}(u,v) = P(u,v)$ .

Fourier Optics: Point Spread Function

The Generalized Pupil Function modifies the incoming light field by element-wise multiplication with it. For instance on-axis light from a distant source shows up as a plane wave at the exit pupil, it can be represented as a normalized white image of ones. In this case after multiplication with the Generalized Pupil Function the resulting field after the pupil would be equal to the Generalized Pupil Function.

More generally, with an on-axis unit impulse of light and several other assumptions typically valid in photography, Goodman demonstrates that the magnitude squared of the Fourier Transform $\mathcal{F}$ of such a complex Exit Pupil is the expected intensity Point Spread Function projected onto the Gaussian point in the ( $x,y$ ) sensing plane by the relative aberrated lens

(2) $\begin{equation*} PSF(x,y) \propto |\mathcal{F}[\mathcal{P}(u,v)]|^2, \end{equation*}$

normalizing factors dropped because normalization tends to be application dependent. This is the image that a distant star against an otherwise black sky would project onto the center of our camera’s ( $x,y$ ) sensing plane. Its deviation from the single point of light predicted by geometrical optics represents degradation of the signal by diffraction and aberrations introduced by the lens.

In fact, as a result of linearity and superposition, the image projected onto the camera’s sensing plane is effectively the ideal image predicted by geometrical optics convolved with the local PSF of the lens as defined. The physically larger the PSF, the fuzzier and less sharp the image.

Optical Transfer Function

What does this mean in terms of loss of detail and resolution in the image? Enter the so-called Optical Transfer Function, the normalized Fourier Transform of the PSF

(3) $\begin{equation*} OTF(f_x,f_y) = \frac{\mathcal{F}[PSF(x,y)]}{\iint \limits_{-\infty}^{+\infty} PSF(x,y)dxdy} \end{equation*}$

The OTF describes how detail at all spatial frequencies ( $f_x,f_y$ ) makes it through the lens on the way to the image plane. It is valid with incoherent lighting, mostly true in nature. The normalizing factor in the denominator is simply the sum total of the intensity of the PSF.

The OTF is a complex function and in typical photographic applications will always have its maximum at zero frequency with a value of one. Since the intensity PSF on the image plane is real and the Fourier transform of a real function is symmetric about the origin, the OTF is also symmetric about the origin. This property is known as Hermitian symmetry and it means that the value of the OTF at negative frequencies is the mirror image of its values at positive frequencies. If in addition the PSF is symmetric about the origin (an even function), the OTF will only be real also, this being the case for pure Defocus, Spherical Aberration and Axial Color for instance.

Modulation Transfer Function

The OTF is a complex function but for photographic spatial frequency considerations we are typically just interested in its effects on the intensity projected onto the image plane, for which phase is unnecessary. We can concentrate on the intensity of the effect by expressing the resulting OTF in complex notation as a magnitude multiplying a phase term, the Phase Transfer Function, which will be ignored. The magnitude (aka modulus, absolute value) of the OTF represents the Spatial Frequency Response of the lens otherwise known as its Modulation Transfer Function

(4) $\begin{equation*} MTF(f_x,f_y) = |OTF(f_x,f_y)| \end{equation*}$

It is calculated by taking the quadrature sum of the real and imaginary parts of the OTF at every spatial frequency. MTF provides a measure of the performance of the optics on the image plane at all possible spatial frequencies ( $f_x,f_y$ ). It inherits the properties of the OTF, with Hermitian symmetry and a maximum of one at zero frequency (also known as DC). The MTF near the center of an ‘in-focus’, in-spec photographic lens at working f-numbers $N_w$ will decrease monotonically from DC all the way to diffraction extinction at $\frac{1}{\lambda N_w}$ ; if it doesn’t it means that something else is at play, such as a less than ideal set up, capturing process or image processing.

All Together Now

The MTF is seen to just be the magnitude of the normalized Fourier transform of the PSF, which is the magnitude squared of the Fourier transform of the complex Exit Pupil function. It stems from the few photographic parameters covered in the last few articles: mean light wavelength $\lambda$ , working f-number $N_w$ and an estimate of the aberrations affecting the point of interest on the image plane in the form of the complex Pupil Function of the lens $\mathcal{P}(u,v)}$ .

In the example in the Figure below those would respectively be 0.5 $\mu m$ , $10$ and a clear circular aperture of 5 $mm$ diameter with $\Delta W = 0$ .

Figure 4. From Generalized Pupil Function to intensity Point Spread Function to Modulation Transfer Function of a perfect lens with f-number 10 under 0.5um light. Note diffraction extinction at 1/lambdaN.

See the articles on Fourier Optics for more detail on the theory and how numerical methods can be used to easily generate the intensity Point Spread Function and Spatial Frequency Response of complex lenses starting from the shape of the aperture and the wavefront aberration function.^[*]

The spatial resolution performance of our own photographic equipment can instead be estimated by measuring the relative OTFs at home. One of the better ways to perform that feat is to use the previously discussed slanted edge method.

MTF curves are usually the basis of many single figure metrics for perceived sharpness that take into account the response of the Human Visual System. They have names like MTFA, SQF, SQRI, MTF50, CPA Acutance, etc. See for example this article for a discussion of MTF50 as a proxy for perceived sharpness.

This concludes the short series of articles for photographers on how photons reflected by a scene and collected by a lens end up on the relative image plane.

Notes and References

_{1. Much inspiration in this series of articles comes from Introduction to Fourier Optics, 3rd Edition. Joseph W Goodman.}

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