Rectangular Aperture Diffraction

Cut a small rectangular hole in an opaque card, shine a laser through it, and catch the light far away: the screen shows a bright cross. A vivid central rectangle of light sits at the centre, with a row of fainter bars marching out along the horizontal axis and another row along the vertical, while the diagonal directions stay almost dark. Rotate the card and the cross rotates with it, always aligned to the rectangle's edges — and, tellingly, the cross is taller where the aperture is narrower. That last fact is the whole story of aperture diffraction compressed into one image: a small dimension spreads the light widely, a large dimension keeps it tight. The reason the pattern is a clean cross is separability. A uniform rectangular aperture is the product of two independent top-hats, one in $x$ and one in $y$ , and the Fourier transform of a product of single-variable functions is the product of their transforms. So the two-dimensional diffraction integral splits exactly into two one-dimensional single-slit integrals, and the amplitude is a product of two sincs. Each axis behaves precisely like its own single slit — same zeros at $a\sin\theta=m\lambda$ , same $4.72\%$ side lobes — and the two patterns simply multiply. Bright bars along the axes are one slit's side lobe times the other slit's central peak; the diagonal squares are two faint side lobes multiplied together, which is why they nearly vanish. This separability is a small miracle of Cartesian geometry, and it fails the moment the aperture stops being a rectangle. A circular hole has no separable coordinates, so its diffraction integral must be done in polar form and yields the Airy pattern of rings instead of a cross of bars. The rectangle is therefore the natural bridge between the single slit and the circular aperture — the cleanest two-dimensional diffraction pattern, and the model for detector pixels, slit spectrographs, and rectangular telescope stops.

Key takeaways

The Fraunhofer pattern of a rectangular aperture (widths $a$ along $x$ , $b$ along $y$ ) is the two-dimensional Fourier transform of a separable field, so the amplitude factorises into a product of two single-slit sincs.
The intensity is $I=I_0\,\operatorname{sinc}^2\beta_x\,\operatorname{sinc}^2\beta_y$ , with $\beta_x=(\pi a/\lambda)\sin\theta_x$ and $\beta_y=(\pi b/\lambda)\sin\theta_y$ — a central bright maximum with orthogonal sidelobe structure along the two transverse directions.
Each axis is an independent single slit: zeros at $a\sin\theta_x=m\lambda$ and $b\sin\theta_y=n\lambda$ , central full widths $2\lambda/a$ and $2\lambda/b$ . A narrow edge diffracts the light widely.
A square aperture concentrates about $81.5\%$ of the transmitted power in its central rectangular lobe ( $0.9028^2$ ), slightly less than the circular Airy disk's $83.8\%$ — the circular aperture concentrates a somewhat larger fraction of the power in its central lobe.
Key idea: a uniform rectangular field is separable in $x$ and $y$ , so its far-field amplitude is the product of two single-slit sincs, giving an intensity $I_0\,\mathrm{sinc}^2\beta_x\,\mathrm{sinc}^2\beta_y$ whose lobe widths run inversely to the two aperture dimensions.

The core insight

A uniform rectangular aperture is the product of two top-hats, one per Cartesian axis, and the Fourier transform of a separable function is the product of its transforms. So the far-field amplitude of a rectangle is the product of two single-slit sincs, and the intensity is $I_0\,\mathrm{sinc}^2\beta_x\,\mathrm{sinc}^2\beta_y$ . Each axis diffracts exactly like its own slit — zeros at $a\sin\theta_x=m\lambda$ , $b\sin\theta_y=n\lambda$ — and the pattern is a cross whose central lobe is broad in the direction of the narrow aperture edge.

Where this lives today

The rectangular aperture is everywhere a detector or a stop has straight edges. Every square or rectangular pixel in a CMOS or CCD sensor integrates the incident field over its area, imposing a $\mathrm{sinc}$ modulation-transfer function $\mathrm{sinc}(\pi a f_x)\,\mathrm{sinc}(\pi a f_y)$ on the image's spatial frequencies — the pixel-aperture roll-off that governs sharpness and aliasing in every digital camera. Slit spectrographs use a tall, narrow entrance aperture ( $a\gg b$ ) precisely so the diffraction is confined to a thin line across the dispersion direction, and the sinc envelope from the narrow dimension sets a floor on spectral-line width. Rectangular field stops, rectangular laser-mode apertures, and rectangular sub-apertures of some segmented and space telescopes all produce sinc-product point-spread functions rather than Airy disks, giving deliberately anisotropic resolution. The separability that produces the cross also makes the mathematics cheap: because a rectangular pattern is a product of one-dimensional transforms, diffraction from rectangular gratings, coded apertures, and rectangular spatial-light modulators can be computed axis-by-axis with fast one-dimensional Fourier transforms — a shortcut used throughout computational imaging and display holography.

The physical picture

Treat the rectangle as two slits acting at right angles. The horizontal edges of width $a$ diffract light into a vertical spread governed by $\lambda/a$ ; the vertical edges of height $b$ diffract into a horizontal spread governed by $\lambda/b$ . Because the two edge pairs act on independent coordinates that never talk to each other, the total pattern is simply the product of two single-slit patterns — a bright central lobe crossed by rows of side bars along each axis, and near-darkness on the diagonals where two faint side lobes multiply.

Figure

A rectangular aperture (wide in x, narrow in y) and its far-field diffraction pattern: a cross of bright bars, with the central lobe elongated vertically — broad in the direction of the narrow edge, tight in the direction of the wide edge

The Fourier picture makes the cross inevitable. The far field samples the spatial-frequency content of the aperture; a sharp edge of extent $a$ carries frequencies up to $\sim1/a$ , so its diffracted light reaches to angles $\sim\lambda/a$ . Two orthogonal edges contribute independent frequency content, so the two-dimensional transform is the outer product of the two one-dimensional transforms — $\mathrm{rect}\times\mathrm{rect}$ transforms to $\mathrm{sinc}\times\mathrm{sinc}$ . The next figure plots the two orthogonal slices with real numbers, and shows the pattern is exactly twice as tall as it is wide when the aperture is twice as wide as it is tall.

Figure

Two orthogonal sinc² sections for a rectangle a = 0.2 mm × b = 0.1 mm at λ = 633 nm: the wide edge (a) gives the narrow blue section with central full width Δθ_x = 6.33 mrad; the narrow edge (b) gives the broad amber section with Δθ_y = 12.66 mrad — the pattern is 2× taller than wide

Most of the light stays in the bright central rectangle. The bars along the axes are one slit’s sidelobe multiplied by the other slit’s central peak, so their first sidelobe intensity reaches about $4.72\%$ of the central peak; the diagonal squares are a product of two first sidelobes, giving roughly $0.047^2\approx0.2\%$ of the central peak — which is why a photographed rectangular pattern can look like a bright plus sign on a dark field.

The core equations

The far field of the rectangle is the two-dimensional Fourier transform of its aperture function, and because that function is a product of independent top-hats the integral separates into two single-slit integrals. Carrying them out gives the amplitude as a product of two sincs.

E(\theta_x,\theta_y)=E_c\,\mathrm{sinc}\,\beta_x\,\mathrm{sinc}\,\beta_y,\qquad \beta_x=\frac{\pi a}{\lambda}\sin\theta_x,\ \ \beta_y=\frac{\pi b}{\lambda}\sin\theta_y \qquad (1)

where: $E_c$ — on-axis amplitude, $E_c=E_0\,ab\,e^{ikz}/(i\lambda z)$ ; $\mathrm{sinc}\,\beta\equiv\sin\beta/\beta$ ; $a,b$ — full aperture widths along $x,y$ ; $\theta_x,\theta_y$ — diffraction angles in the two planes; $\beta_x,\beta_y$ — half the edge-to-edge phase slip along each axis. The on-axis amplitude scales with the aperture area $ab$ ; each factor is exactly the single-slit amplitude of Single-Slit Diffraction. Squaring gives the observed intensity — the product of two $\mathrm{sinc}^2$ factors that draws the cross.

I(\theta_x,\theta_y)=I_0\left(\frac{\sin\beta_x}{\beta_x}\right)^2\left(\frac{\sin\beta_y}{\beta_y}\right)^2 \qquad (2)

where: $I_0=|E_c|^2$ — on-axis peak intensity, scaling as $(ab)^2$ ; $I$ — intensity at $(\theta_x,\theta_y)$ . Reading it physically: the pattern is bright only where both sincs are large — the central lobe and the two axial rows — and dim wherever either factor sits near a zero. The dark lines of the pattern are the zeros of either factor, and they reproduce the single-slit condition independently on each axis. These are full dark lines across the pattern, not isolated points.

a\sin\theta_x=m\lambda \quad\text{or}\quad b\sin\theta_y=n\lambda,\qquad m,n=\pm1,\pm2,\dots \qquad (3)

where: $m,n$ — nonzero integers labelling the dark lines along each axis. The central lobe is bounded by the $m=\pm1$ and $n=\pm1$ zeros, giving first-zero-to-first-zero angular widths $\Delta\theta_x\approx2\lambda/a$ and $\Delta\theta_y\approx2\lambda/b$ in the small-angle limit — inversely proportional to the aperture dimensions, so the narrower edge produces the broader diffraction spread. Far-field validity is the usual Fresnel-number condition applied to each axis (recall Fraunhofer Diffraction Regime and Fresnel Number): $N_{F,x}=a^2/(4\lambda z)\ll1$ and likewise for $b$ , so the aperture's own wavefront curvature is negligible and the clean transform holds.

Derivation AThe sinc-squared product from the separated Fraunhofer integral

The aim of this derivation is to show rigorously that the two-dimensional pattern is nothing more than two single slits multiplied together — that the separability of the aperture passes straight through the Fourier transform to give a product of sincs. We start from the Fraunhofer integral over the rectangle, use the fact that the integrand and the limits both factorise, evaluate each one-dimensional piece as a single-slit transform, and square. Separability does all the work; no genuinely two-dimensional integral is ever needed.

Step 1. Write the Fraunhofer integral for a uniform rectangle, $E\propto\displaystyle\int_{-a/2}^{a/2}\!\!\int_{-b/2}^{b/2} e^{-ik(x\sin\theta_x+y\sin\theta_y)}\,\mathrm{d}x\,\mathrm{d}y$ . The exponential splits as $e^{-ikx\sin\theta_x}\,e^{-iky\sin\theta_y}$ , a product of a function of $x$ alone and a function of $y$ alone. Why: the aperture transmission is $\mathrm{rect}(x/a)\,\mathrm{rect}(y/b)$ , separable in Cartesian coordinates, and the far-field kernel adds the two axes' phases — so the integrand is a clean product.

Step 2. Because the integrand factorises and the limits are independent constants, the double integral is the product of two one-dimensional integrals: $E\propto\big(\int_{-a/2}^{a/2}e^{-ikx\sin\theta_x}\mathrm{d}x\big)\big(\int_{-b/2}^{b/2}e^{-iky\sin\theta_y}\mathrm{d}y\big)$ . Why: $\iint f(x)g(y)\,\mathrm{d}x\,\mathrm{d}y=(\int f\,\mathrm{d}x)(\int g\,\mathrm{d}y)$ exactly when the region is a rectangle — the geometric fact that makes the whole problem one-dimensional twice over.

Step 3. Each integral is the single-slit transform of Single-Slit Diffraction: $\int_{-a/2}^{a/2}e^{-ikx\sin\theta_x}\mathrm{d}x=a\,\sin\beta_x/\beta_x$ with $\beta_x=(\pi a/\lambda)\sin\theta_x$ , and likewise $b\,\sin\beta_y/\beta_y$ for $y$ . So $E=E_c\,\mathrm{sinc}\,\beta_x\,\mathrm{sinc}\,\beta_y$ with $E_c\propto ab$ . Why: each factor is precisely the top-hat transform already derived for one slit, so the rectangle inherits both sincs and its on-axis amplitude scales with the area $ab$ .

Step 4. Square for intensity: $I=|E|^2=I_0\,\mathrm{sinc}^2\beta_x\,\mathrm{sinc}^2\beta_y$ , and read off the zeros — $\sin\beta_x=0$ ( $a\sin\theta_x=m\lambda$ ) or $\sin\beta_y=0$ ( $b\sin\theta_y=n\lambda$ ). Why: the modulus-squared of a product is the product of the moduli-squared, so the two-dimensional intensity is literally the two single-slit patterns overlaid multiplicatively — a cross with a dark line wherever either slit is dark. *As a check: the double integral separates to $ab\,\mathrm{sinc}\,\beta_x\,\mathrm{sinc}\,\beta_y$ , and taking $b\to\infty$ recovers the single-slit result.*

Go deeper · Derivation B · 1 minCentral-lobe energy of a square aperture, and the Airy contrast

Derivation BCentral-lobe energy of a square aperture, and the Airy contrast

This derivation answers a concrete question — how tightly does a square aperture concentrate light, and how does it compare with a round one? — and shows that separability makes the answer factor into a quantity we already know. The aim is to compute the fraction of the transmitted power in the central lobe of a square aperture as the square of the one-dimensional single-slit fraction, and then to contrast the result with the circular Airy disk to see which shape packs energy more tightly.

Step 1. For a separable pattern the enclosed-energy fraction factorises: the power in the central rectangle $|\beta_x|<\pi$ and $|\beta_y|<\pi$ divided by the total is $\big(\tfrac1\pi\int_{-\pi}^{\pi}\mathrm{sinc}^2\beta\,\mathrm{d}\beta\big)\big(\tfrac1\pi\int_{-\pi}^{\pi}\mathrm{sinc}^2\beta\,\mathrm{d}\beta\big)$ , because both the numerator and the total integral separate. Why: intensity is a product of one-dimensional factors, so a double integral over a rectangular region in $(\beta_x,\beta_y)$ splits into the product of two identical one-dimensional integrals.

Step 2. Insert the single-slit central-lobe fraction from Single-Slit Diffraction, $\tfrac1\pi\int_{-\pi}^{\pi}\mathrm{sinc}^2\beta\,\mathrm{d}\beta=0.9028$ (from the Dirichlet integral $\int_{-\infty}^{\infty}\mathrm{sinc}^2=\pi$ ). Why: each axis of the square is an independent single slit, so its central-lobe energy fraction is exactly the one-dimensional value already established.

Step 3. Multiply: the square aperture's central-lobe fraction is $0.9028^2=\mathbf{0.8151}$ — about $81.5\%$ of the transmitted light lands in the central rectangle. Why: the two axes are independent, so the joint probability of a photon landing in both central spans is the product of the two per-axis fractions.

Step 4. Contrast with the circle: the Airy disk of a circular aperture holds $0.8378$ ( $=1-J_0^2(x_1)-J_1^2(x_1)$ at the first Bessel zero) — slightly more than the square. Why: the circular aperture has rotational symmetry and a more compact central diffraction pattern than the square aperture, so a slightly larger fraction of its transmitted power lies inside the first Airy minimum. *As a check: $0.9028^2=0.8151$ , and the Airy encircled energy $0.8378$ follows from the first zero of $J_1$ at $3.8317$ .*

Limiting cases

Case	Result	Meaning
Slit limit, $b\to\infty$	$I\to I_0\,\mathrm{sinc}^2\beta_x$	the $y$ -sinc collapses to a line; the single slit is recovered
Square, $a=b$	$I=I_0\,\mathrm{sinc}^2\beta_x\,\mathrm{sinc}^2\beta_y$	symmetric cross; central lobe square, $2\lambda/a$ each way
Long slot, $a\gg b$	$\Delta\theta_x=2\lambda/a\ll\Delta\theta_y=2\lambda/b$	diffraction confined to a thin line along the short edge
Diagonal direction	$I\approx(4.7\%)^2\approx0.2\%$	product of two side lobes — the near-dark quadrants
Central-lobe energy (square)	$0.8151=0.9028^2$	slightly less concentrated than the Airy disk's $0.8378$
Circular aperture	not separable $\to$ Airy rings	polar geometry gives $[2J_1/x]^2$ , not a sinc product

Common trapTreating the rectangle like a circle, or misplacing the first zero

Two mistakes recur. First, the sinc-product result is specific to Cartesian-separable apertures; a circular hole is not separable in $x$ and $y$ , so its transform must be done in polar coordinates and gives the Airy function $2J_1(x)/x$ — a set of rings, not a cross of bars. Never apply $\mathrm{sinc}^2$ to a round aperture, and never apply the Airy $1.22\lambda/D$ to a rectangle. Second, watch where the first zero sits: $\mathrm{sinc}\,\beta=\sin\beta/\beta$ vanishes first at $\beta=\pi$ , i.e. $a\sin\theta_x=\lambda$ — not at $\lambda/2$ . The half-wavelength condition belongs to the half-slit pairing of two wavelets, not to the whole-aperture minimum; confusing them halves every predicted angle and doubles every predicted lobe count.

Worked examples

1 · The cross from a rectangular aperture (confidence builder)

A rectangular aperture $a=0.20$ mm (along $x$ ) by $b=0.10$ mm (along $y$ ) is illuminated by $\lambda=633$ nm light. Find the angular full widths of the central lobe along each axis and describe the pattern's shape.

Step 1. Along $x$ (the wide edge): $\Delta\theta_x=2\lambda/a=2(633\times10^{-9})/(0.20\times10^{-3})=\mathbf{6.33\ \text{mrad}}$ . The first dark line sits at $\sin\theta_x=\lambda/a=3.165$ mrad.

Step 2. Along $y$ (the narrow edge): $\Delta\theta_y=2\lambda/b=2(633\times10^{-9})/(0.10\times10^{-3})=\mathbf{12.66\ \text{mrad}}$ . The narrow edge diffracts twice as widely as the wide edge.

Step 3. So the central lobe is a rectangle of light $6.33$ mrad wide by $12.66$ mrad tall — twice as tall as it is wide, elongated in $y$ , perpendicular to the aperture's long ( $x$ ) edge. The aperture's aspect ratio inverts in the pattern.

Step 4. On a screen $L=1.5$ m away the central lobe measures $L\,\Delta\theta_x\times L\,\Delta\theta_y=9.5\ \text{mm}\times19.0\ \text{mm}$ , a bright upright rectangle crossed by faint side bars along each axis. Central lobe $6.33$ mrad $\times$ $12.66$ mrad; pattern $2\times$ taller than wide.

2 · Energy budget of a square aperture (exam level)

For a square aperture ( $a=b$ ), find the fraction of the transmitted light in the central lobe, the brightness of the first side lobe along an axis, and compare the concentration with a circular aperture.

Step 1. Each axis is an independent single slit holding $0.9028$ of its power in the central span $|\beta|<\pi$ . Because the square is separable, the two-dimensional central-lobe fraction is the product $0.9028\times0.9028=\mathbf{0.8151}$ — about $81.5\%$ of the light in the central square.

Step 2. The first sidelobe along an axis reaches $4.72\%$ of the central peak, because one slit contributes its first-sidelobe intensity ( $4.72\%$ ) while the other contributes its central intensity ( $100\%$ ). The first diagonal sidelobe reaches $(0.0472)^2\approx0.22\%$ of the central peak, so it is much fainter.

Step 3. A circular aperture of comparable size holds $\mathbf{0.8378}$ ( $83.8\%$ ) in its central Airy disk — slightly more than the square's $81.5\%$ . The round hole lacks the square's energetic axial bars, so it concentrates a little more light centrally.

Step 4. The lesson for instrument design: where a compact point-spread function matters most — a telescope, a microscope objective — a circular pupil packs energy marginally tighter, while a rectangular stop is chosen deliberately when anisotropic resolution is wanted (sharper along one axis). Square central lobe $81.5\%$ ; axial side lobe $4.72\%$ ; circular Airy disk $83.8\%$ .

Physics of the portrait

The object of study here is not a person but a thing you can photograph: the far-field intensity map itself, the diffraction pattern, which carries the aperture's shape encoded in light.

Origins · 1 minDiffraction Pattern and how this was discovered

Diffraction Pattern

the squared Fourier transform of an aperture · Fourier optics

A diffraction pattern is the angular distribution of intensity a wave forms after passing an aperture — and in the Fraunhofer regime it is exactly the squared modulus of the Fourier transform of the aperture. That identity turns the pattern into a readout of the aperture's shape: a slit's top-hat gives a $\mathrm{sinc}^2$ , a rectangle's separable top-hats give a cross of two sincs, a circle's radial disk gives the ringed Airy pattern, and a periodic array of holes gives sharp grating orders under a single-aperture envelope. Reciprocity is the pattern's defining habit — every small feature of the aperture spreads into a large feature of the pattern and vice versa, so a narrow slit paints a broad band and a broad aperture a tight spot. Because the pattern is the impulse response (point-spread function) of the optical system that forms it, it sets the resolution of every telescope, microscope, and camera, and its Fourier reading underpins crystallography, where the diffraction pattern of a lattice reveals the atomic arrangement, and coherent imaging, where recovering an object from its diffraction pattern is the phase-retrieval problem.

History

1665

— Francesco Maria Grimaldi records structured diffraction patterns and names the effect "diffractio"

1818

— Augustin-Jean Fresnel's memoir gives the wave computation of aperture diffraction patterns

1835

— Friedrich Magnus Schwerd analyses slit and rectangular-aperture patterns in quantitative detail

1835

— George Biddell Airy derives the circular-aperture pattern, completing the aperture-shape family

1873

— Ernst Abbe connects the far-field diffraction pattern to the resolving power of the microscope

Going further

Texts. Hecht, Optics, 5th ed., Ch. 10 (Fraunhofer diffraction, rectangular aperture) · Goodman, Introduction to Fourier Optics, 4th ed., Ch. 4 (Fourier optics) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 4 · Born & Wolf, Principles of Optics, 7th ed., §8.5 · Demtröder, Laser Spectroscopy, Vol. 1 (spectrograph apertures). Key numerical landmarks: the separated Fraunhofer integral gives $I_0\,\mathrm{sinc}^2\beta_x\,\mathrm{sinc}^2\beta_y$ with $\beta_{x,y}=\pi a_{x,y}\sin\theta_{x,y}/\lambda$ ; the transcendental side-lobe condition $\tan\beta=\beta$ gives the first secondary maximum at $\approx4.7\%$ (root $\beta\approx4.4934$ ); the central-lobe energy fraction for a square aperture is $\big(\pi^{-1}\int_{-\pi}^{\pi}\mathrm{sinc}^2\beta\,\mathrm{d}\beta\big)^2\approx0.903^2\approx0.815$ , contrasting with $\approx0.838$ for the Airy disk; and the central-lobe half-width gives $\Delta\theta=2\lambda/a=6.33$ mrad for the example above.

Prerequisites. Fraunhofer Diffraction Regime · Single-Slit Diffraction · Fresnel–Kirchhoff Diffraction Integral · Fresnel Number. Leads to. Circular Aperture Diffraction and the Airy Pattern · Rayleigh Resolution Criterion · Resolution Criteria: Sparrow and Abbe · Fresnel Diffraction Regime.

Why is the pattern a cross rather than a smooth blob?

Because the intensity is a product of two sincs, and a product is large only where both factors are large. Along the $x$ -axis the $y$ -sinc sits at its central peak, so the $x$ -sinc's own lobes show through as a row of bars; along the $y$ -axis the reverse happens. Off the axes, in the diagonal directions, both sincs are past their central lobes and near their side lobes, so their product is tiny — $0.047^2$ at the first diagonal maximum. The bright bars along the axes and the near-dark diagonals together read as a plus-sign, the visual signature of a separable rectangular aperture.

Does a narrow edge really diffract the light more widely?

Yes, and it is the crux of the whole pattern. The central-lobe angular full width along an axis is $2\lambda/(\text{aperture size along that axis})$ , so the smaller dimension gives the larger spread. A tall thin slot ( $a\gg b$ ) throws a broad vertical fan and a thin horizontal sliver — the diffraction is wide in the direction of the short edge. This inversion is Fourier reciprocity: a narrow function in space has a broad transform in angle. It is why a slit spectrograph makes its entrance aperture tall and narrow, spreading light only in the plane where dispersion needs room.

How does the rectangle relate to the single slit and the circular aperture?

It is the bridge between them. Make one dimension infinite ( $b\to\infty$ ) and the $y$ -sinc collapses to a line, leaving the pure single-slit $\mathrm{sinc}^2\beta_x$ . Keep both dimensions finite and you get the sinc-product cross. But you cannot deform the rectangle into a circle within this framework: the circle is not separable in Cartesian coordinates, so its transform must be done in polar form and yields the Airy pattern of concentric rings, governed by the Bessel zero $1.22\lambda/D$ rather than the sinc zero $\lambda/a$ . Rectangle and circle are genuinely different diffraction families joined only by the common Fraunhofer principle.

Why does a circular aperture concentrate slightly more energy than a square one?

Because the square wastes a little light in its axial bars. Both apertures put most of their power in the central lobe — $81.5\%$ for the square, $83.8\%$ for the circle — but the square's separable structure produces energetic side-lobe rows along the two axes, bleeding power out into the cross arms. The circular aperture has no such preferred directions; its side-lobe energy is spread thinly around rings, leaving marginally more in the central Airy disk. The difference is small, but it is one reason imaging optics favour round pupils when a compact point-spread function is the priority.

What sets the far-field distance for a rectangular aperture?

The same Fresnel-number condition as any aperture, applied to each dimension: $N_{F,x}=a^2/(4\lambda z)\ll1$ and $N_{F,y}=b^2/(4\lambda z)\ll1$ , recalling Fresnel Number. The larger dimension is the binding one, since it gives the larger Fresnel number and reaches the far field last. In practice a lens supplies the missing wavefront curvature (recall Fraunhofer Diffraction Regime) and places the sinc-cross in its focal plane, so the pattern is observed at finite distance no matter how small $z$ is — exactly as for the single slit or circular aperture.

Why do square pixels blur a digital image, and is that the same physics?

It is the same Fourier transform, applied to the pixel instead of a lens aperture. A square photosite of side $a$ averages the incident light over its area, and averaging over a top-hat is convolution with a rectangle — which multiplies the image's spatial-frequency spectrum by $\mathrm{sinc}(\pi a f_x)\,\mathrm{sinc}(\pi a f_y)$ , the transform of the pixel. High spatial frequencies are attenuated by the sinc roll-off, softening fine detail, and frequencies beyond the first sinc zero alias. So the same $\mathrm{rect}\to\mathrm{sinc}$ relation that draws the diffraction cross also sets the modulation-transfer-function limit of every pixelated sensor.