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33.8 Fourier optics⧉

So far in this book light has been a stream of rays: straight lines that a lens bends to meet at an image point, that an aperture clips, that a sensor pixel integrates over a cone. Geometric optics is a wonderful approximation and it carries most of photography. But it cannot explain why a perfect lens, stopped down, still blurs; why a smaller aperture is sharper up to a point and then softer past it; or why the resolution of a microscope has a hard ceiling that no amount of polishing the glass can lift. For those you have to admit the thing geometric optics suppresses: light is a wave. And the moment you treat it as a wave, imaging turns into linear systems in the spatial-frequency domain, a central idea in this part.

33.8.1 Light as a wave: amplitude, phase, and what "coherent" means⧉

Model the light in a plane as a complex field $U(x,y) = A(x,y)\,e^{i\phi(x,y)}$: an amplitude $A$ (how strong) and a phase $\phi$ (where in its oscillation cycle the wave is, at that point). A sensor cannot see the field. It sees intensity, $I = |U|^2$, the time-averaged irradiance, with the physical proportionality constant absorbed into the field normalization. The phase is lost at detection. That asymmetry, fields propagate but intensities are measured, is the source of every subtlety to come.

The decisive fork is coherence. If the illumination is coherent (for example, a single laser, a stable relative phase across the field, as in holography or a microscope's condenser set just so), then waves from different points have a fixed phase relationship and the fields add: $U_{\text{total}} = U_1 + U_2$, and only afterward do you square to get intensity. Adding fields means interference: two beams can cancel. If the illumination is incoherent (ordinary daylight, many extended LED sources, the diffuse glow of most photographic scenes, where the phase jitters randomly and fast), then the cross terms average to zero and the intensities add: $I_{\text{total}} = I_1 + I_2$. This is not a detail. It decides what kind of linear system the camera is: a coherent system is linear in the complex field, an incoherent system is linear in intensity. These are two different operators and two different transfer functions. Ordinary photography lives in the incoherent world.

33.8.2 Diffraction is a Fourier transform⧉

Send a wave through an aperture and let it travel. Geometric optics says it casts a sharp-edged shadow; the wave spreads, diffracts, and the further it goes the more the pattern reorganizes itself. Track the field across the gap and you get Fresnel diffraction (near field) and, far enough away, Fraunhofer diffraction (far field), and the Fraunhofer result is the key one to remember because it is simple: the far-field diffraction pattern of an aperture is the Fourier transform of the field just leaving that aperture. A narrow slit, whose transform is a wide sinc, spreads its light into a broad fan; a wide aperture, whose transform is narrow, stays collimated. Diffraction is a Fourier transform produced by free-space propagation. The catch is that "far enough away" can mean kilometers for a camera-sized aperture, which is exactly the problem a lens solves.

33.8.3 A lens computes a Fourier transform⧉

Here is the fact the whole chapter hangs on. A converging lens does to a wave in one focal length what free space would take infinity to do: place a transparency in the front focal plane of a lens, illuminate it coherently, and the field in the back focal plane is a scaled two-dimensional Fourier transform of the transmitted field, under the thin-lens, scalar, paraxial model and within the finite aperture. The lens adds precisely the curved phase factor that drags the far field in to a finite distance. The spatial frequencies of the input (the fine detail) are sorted by the lens into positions in the back focal plane: low frequencies land near the axis, high frequencies far out toward the edges. The image's spectrum, normally an abstraction, is suddenly a real pattern of light in a physical plane that can be filtered optically.

Because the spectrum is physically accessible, you can filter it optically. For equal focal lengths $f$, put a second lens $2f$ behind the first, with the shared Fourier plane $f$ from each lens, and it Fourier-transforms a second time, turning the spectrum back into an image. This is the 4f system (four focal lengths end to end), and the plane in the middle, the Fourier plane, is where optical computing was born. Drop a mask there and you multiply the spectrum by that mask: this is optical spatial filtering, convolution performed at the speed of light with no arithmetic at all.

fig-lens-fourier-transform
Figure 33.8.1. The 4f system makes a spectrum physical. Coherent light passes through an input transparency in the front focal plane; the first lens forms its exact 2-D Fourier transform in the shared Fourier plane one focal length behind, where low frequencies sit near the axis and high frequencies toward the rim. A mask placed in that plane multiplies the spectrum: a small central stop is a low-pass filter (softens, like stopping down), an opaque central dot is a high-pass filter (keeps only edges), a phase dot recovers invisible phase objects (phase contrast). The second lens transforms again, delivering the filtered image at the output plane. The hardware computes a convolution at the speed of light.

The set of standard filters is small. A small opaque disc in the center blocks the low frequencies and passes the rest: a high-pass filter that suppresses slowly varying structure and emphasizes edges. A small transparent hole that blocks everything but the center is a low-pass filter. In fact, this is exactly what stopping a lens down does. A transparent phase dot that retards only the undiffracted central beam converts otherwise-invisible phase objects (transparent cells, flow in air) into visible contrast: Zernike's phase contrast microscopy, a Nobel Prize, and the cousin of Schlieren imaging that photographs shockwaves and heat shimmer by filtering the Fourier plane. This is the optical-computing lineage: analog correlators and pattern recognizers built entirely from lenses, and it runs straight into modern optical neural networks; we follow that thread in Part 39, Adjacent fields and applications.

33.8.4 The pupil function is the transfer function⧉

Now fold the trick back into an ordinary camera. A camera is not literally a two-lens 4f relay, but its local coherent imaging response has a related pupil description. Define the pupil function $P(u,v)$ as the complex transmission across the aperture: amplitude transmission times the phase delay relative to an ideal wavefront. For an unaberrated circular stop it is $1$ inside the disc and $0$ outside. Under the scalar, paraxial, locally shift-invariant model, the pupil determines the transfer function; coherent and incoherent imaging require different formulas.

Take it carefully, because coherent and incoherent give different answers. Under coherent illumination the system is linear in the field, and the coherent transfer function is the pupil itself:

$$ H_{\text{coh}}(u,v) = P(u,v), \qquad U_i = \mathcal F^{-1}\!\big\{\, P \cdot \mathcal F\{U_o\} \,\big\}. $$

Here the pupil coordinates are understood to be rescaled into image spatial-frequency coordinates, with magnification and wavelength accounted for. The output field $U_i$ is the input field $U_o$ with its spectrum multiplied by $P$ and transformed back, a textbook linear filter, with the aperture acting directly as the filter mask.

Under incoherent illumination the system is linear in intensity, and the relevant object is the point spread function: the image of a single point source. A point source fills the pupil with a uniform field, and the field reaching the sensor is the Fourier transform of the pupil; the sensor squares it, so

$$ h(x,y) = \big|\,\mathcal F\{P\}\,\big|^2. $$

The PSF is the squared Fourier transform of the pupil. Its transform (the incoherent optical transfer function) then follows from the autocorrelation theorem (the transform of a squared magnitude is an autocorrelation):

$$ H(u,v) = \frac{P \star P}{\displaystyle\int |P|^2}, \qquad \text{MTF} = |H|. $$

Here $\star$ denotes complex autocorrelation, including conjugation of one shifted pupil; Fourier scaling and PSF normalization are implicit. The incoherent OTF is the (normalized) autocorrelation of the pupil, and its magnitude is the MTF that every lens-test chart measures. These three lines are the load-bearing identities of the chapter: $H_{\text{coh}}=P$, $h=|\mathcal F\{P\}|^2$, and $\text{OTF}=P\star P$. Memorize the pupil and you have computed the lens.

fig-pupil-otf-psf
Figure 33.8.2. One pupil, three faces. Each row takes a pupil function $P(u,v)$ (left) and reads off the PSF $h=|\mathcal F\{P\}|^2$ (center) and the MTF $=|P\star P|$, the magnitude of the pupil's autocorrelation (right). Top: a clear circular pupil: a tight Airy PSF and a smooth MTF falling to a hard cutoff. Middle: the same pupil with aberration phase $P = A\,e^{i2\pi W/\lambda}$: the PSF smears and the MTF sags, even though no light was blocked. Bottom: a deliberately coded pupil (a phase plate): a broadened but well-behaved, zero-free MTF, the engineered blur of Chapter 33.1, Wavefront coding. The pupil is the design surface; the PSF and MTF are consequences.

33.8.5 The diffraction limit⧉

The identities immediately give the most important fact in imaging optics: a finite aperture sets a hard ceiling on resolution that no perfect glass can exceed. The OTF is the pupil's autocorrelation: the overlap of the pupil with a shifted copy of itself. Shift two discs of radius $R$ apart and their overlap shrinks to nothing once the shift exceeds $2R$. So the MTF reaches exactly zero at a finite spatial frequency and stays zero beyond it: detail finer than that is not merely attenuated, it is gone, multiplied by zero, unrecoverable by any deconvolution (which is why even the cleanest deblurring has a ceiling; cf. Chapter 6.7, Image formation as a linear system). For an incoherent system the cutoff frequency is

$$ f_c = \frac{1}{\lambda N}, \qquad N = f/\# = \frac{f}{D}, $$

where $\lambda$ is the wavelength and $N$ the $f$-number. Read it: a faster lens (smaller $N$) resolves more, a shorter wavelength resolves more, and the relation is fixed: at $f/8$ in green light the cutoff is about 227 line-pairs per millimeter at $\lambda=550$ nm (the image-plane cutoff, where contrast reaches zero). This is why stopping down past a point softens an image: you have pushed $f_c$ below the sensor's own resolution, and diffraction, not the lens, now limits you.

The same wall, in the microscope's language, is Abbe's (1873) Abbe 1873: the spatial period at the corresponding incoherent cutoff is

$$ d_{\min} = \frac{\lambda}{2\,\mathrm{NA}}, $$

with $\mathrm{NA}$ the numerical aperture ($n\sin\alpha$, with medium refractive index $n$ and collection half-angle $\alpha$, the microscope's analog of aperture). In real space the PSF of a circular pupil is the Airy disc (a bright core ringed by faint haloes) and two points are "resolved" when their Airy cores are far enough apart to tell two bumps from one (the Rayleigh criterion). For two incoherent point sources and a circular pupil, Rayleigh's separation is instead $0.61\lambda/\mathrm{NA}$. Resolution claims must specify the criterion, illumination coherence, wavelength, and sampling; the cutoff period and two-point separation are not interchangeable.

fig-diffraction-limit-airy
Figure 33.8.3. The diffraction limit, from pupil to ceiling. A circular pupil (left) Fourier-transforms to the Airy disc PSF (center), a bright core and faint rings, and autocorrelates to an MTF (right) that falls to exactly zero at the cutoff $f_c = 1/(\lambda N)$; frequencies beyond it are erased, not dimmed. Two point sources are resolved only until their Airy cores merge, the Abbe/Rayleigh limit $d_{\min}=\lambda/2\,\mathrm{NA}$. Smaller $f$-number or shorter wavelength pushes the wall out; nothing else can.

33.8.6 Aberrations are pupil phase⧉

Real lenses are not the ideal that gives a flat-phase pupil; they have aberrations (spherical, coma, astigmatism, field curvature) and Fourier optics gives them a single, unified home. An aberration is nothing but a phase error across the pupil: instead of arriving as a perfect spherical wavefront converging on the focus, the wavefront is deformed, by an amount $W(u,v)$ called the wavefront error. Write it straight into the pupil:

$$ P(u,v) = A(u,v)\,\exp\!\Big(i\,\frac{2\pi}{\lambda}\,W(u,v)\Big). $$

The amplitude $A$ is the aperture shape; the phase carries every aberration. Opticians expand $W$ in the Zernike polynomials, an orthogonal basis on the disc whose low terms are the named aberrations (defocus, astigmatism, coma, spherical), so "this lens has $0.1\lambda$ of coma" is a statement about one coefficient of $W$. Feed that $P$ through $h=|\mathcal F\{P\}|^2$ and you get exactly how the aberration blurs the image; correcting a lens is the craft of flattening $W$ across the pupil.

This is the hinge of the whole part. An aberration is an accidental, unwanted pupil phase that designers fight to remove. But nothing stops you from putting a pupil phase there on purpose. That is precisely Chapter 33.1, Wavefront coding: a cubic $W(u,v)$, a deliberate aberration engineered so the PSF varies less with focus over the design range. Fourier optics is what tells you what that cubic phase does to the image, and it is the same machinery, $P = A\,e^{i2\pi W/\lambda}$, whether the phase came from a flaw in the glass or a designed plate in the pupil.

33.8.7 Every code in this part is a choice of pupil⧉

Now stand back: the chapter has unified the entire part. Coded imaging is the discipline of designing the operator $A$ (Chapter 6.7, Image formation as a linear system); Fourier optics tells you where that design lives. It lives in the pupil function $P$, and there are exactly as many ways to code as there are things you can do to a pupil:

Four chapters, one design surface. The pupil is the place computation reaches into the optics, and Fourier optics is the dictionary that says, for any code you write there, what it does to the image spectrum. That is why this theory chapter, though it could open the part, reads best after the codes: you have already met amplitude, phase, time, and learned pupils as separate tricks, and here they collapse into one.

33.8.8 Fourier ptychography: synthesizing a bigger pupil⧉

End on a striking use of the frame: computing your way around the diffraction limit. A microscope objective faces Abbe's trade-off Abbe 1873: a low-NA lens sees a wide field but coarse detail, a high-NA objective often combines finer detail with higher magnification and a smaller field of view; field size is also constrained by the optical design and sensor. The cutoff $f_c \propto \mathrm{NA}$ is a hole in frequency space, and the lens can only ever capture the band that fits inside its pupil. Fourier ptychography (Zheng, Horstmeyer & Yang 2013) enlarges that hole computationally Zheng, Horstmeyer & Yang 2013. Illuminate the sample with a plane wave from many different angles in turn: an LED array does it cheaply. A tilted illumination shifts the sample's spectrum across the fixed pupil, so each angle lets a different patch of high-frequency content sneak through the low-NA opening. Capture one low-resolution image per angle; each is a different shifted sub-region of frequency space. Then stitch them in the Fourier domain by iteratively solving for the field whose magnitude matches every measured image while tiling the patches into one wide spectrum (recovering the phase the sensor discarded along the way, the same phase-retrieval bind from the start of the chapter). The result is a synthetic frequency support larger than that of the collection objective alone, delivering a high-NA, gigapixel reconstruction and the wide field of the low-NA objective at once.

fig-fourier-ptychography
Figure 33.8.4. Fourier ptychography synthesizes a large pupil. A sample is lit from many angles by an LED array (left); each illumination angle shifts the sample's spectrum across the microscope's small, fixed low-NA pupil, so a different patch of high spatial frequencies passes through, giving one low-resolution image per angle (center). Iterative phase-retrieval stitches the patches in the Fourier domain into a wide synthetic aperture (right), reconstructing a high-resolution, wide-field image whose effective NA (and resolution) exceeds any single shot. Computation tiles a pupil the objective could never have.

The additional illumination angles supply measurements outside the objective's usual band; reconstruction does not recover those frequencies from an unchanged single low-NA measurement. The thin-sample, stable-scene, calibrated-illumination model and overlap between spectral patches are essential to the phase-retrieval step. The diffraction limit is the wall this part works against; the pupil is the surface every code is written on; and Fourier ptychography shows that with enough coded measurements you can build a pupil you never had and extend the effective pupil by coded acquisition and reconstruction.