Optics 056 · Diffraction, Fourier optics, and computational imaging

Huygens–Fresnel Propagation Lab

How does a diffraction pattern change from close to the aperture to far away? Gradually, with no sudden switch. Every point of the aperture sends out a wavelet, and the field at the screen is their complex sum. Close up, the pattern looks like the aperture with rippled edges. As the screen moves away, the Fresnel number falls smoothly and the pattern slowly reshapes into the far-field Fraunhofer pattern.

Move a detector continuously through the complex field behind one square aperture. See how weighted Huygens contributions build the near-field structure, how much power falls in a finite window and how the pattern approaches the Fraunhofer limit.

Fresnel number / physical distance

N_F,z

$0.50$

Detector probe / raw intensity

x_p,I_p/I_0

$50\%$

Window power / far-field profile difference

\eta_W,\epsilon_{\mathrm{FF}}

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Huygens–Fresnel Propagation Lab

Detector distance (moves on logarithmic rail)

z

0.15\,\mathrm m

Full side of the square aperture (changes the opening)

w

0.40\,\mathrm{mm}

Vacuum wavelength

\lambda

532\,\mathrm{nm}

Probe position on the centre line (fraction of window)

x_p/L

0.15

Display exposure decimal exponent (data unchanged)

\log_{10}G

0

Slide the detector from the rippled near field out to the broad far-field pattern. The square opening resizes with the aperture width. Compare the cyan propagated intensity with the gold Fraunhofer reference, which is labeled separately; the two agree only in the Fraunhofer limit. Move the gold probe to see the weighted amplitudes add before they are squared. Shift-drag the receiving face in the bench view, or click the propagation carpet, to move the same detector. Detail views show the mask and detector on their own. Play traces mathematical contributions, not photons, and exposure changes the display, not the data.

The calculation uses the scalar monochromatic Fresnel approximation over the exact square opening, with separable 96-node quadrature. It does not solve the full Maxwell equations, and it leaves out loss of coherence and camera noise. The scene is logarithmic along the axis, the transverse window adapts, and supports are enlarged; detail views show the apparatus parts alone. Exposure is set explicitly, and raw power is never rescaled to follow the peak.

Physics tutorial

Audit the physics of Huygens–Fresnel Propagation Lab

BackgroundEach illuminated element of the aperture contributes a complex amplitude, not a positive amount of brightness. The detector measures $I_z=|U_z|^2$ . A spot can therefore be dark even though every element sending light to it is illuminated: the contributions cancel. The Lab keeps one finite square aperture, so the question stays on propagation, not on a catalogue of aperture shapes.

Why it mattersHow does a diffraction pattern change from close to the aperture to far away?

Start with the essentials

Focus question: How does a diffraction pattern change from close to the aperture to far away?
One-sentence intuition: With the aperture half-width as the length scale, the propagation is characterized by $N_F=\frac{(w/2)^2}{\lambda z}$ . This number varies continuously. A small value means the far-field approximation works well, but it is not a switch: the Fresnel calculation still produces the pattern.

Core mathematical model

A complex contribution from every aperture element

\begin{aligned}U_z&=\frac{1}{i\lambda z}\iint_\Omega\\&\quad e^{i\pi[(x-\xi)^2+(y-\eta)^2]/(\lambda z)}\,\mathrm d\xi\,\mathrm d\eta\\I_z/I_0&=|U_z|^2\end{aligned}

The aperture sits at axial position zero. Distance is positive along the incoming plane wave, and transverse coordinates are measured from the centre. The incident amplitude is one and the wavelength is the vacuum value. Only the common carrier phase along the axis is removed. All coordinates in the integrals are in metres. The coloured curves in the scene stand for contributions to the integral; they are not bent light rays.

Square aperture: two one-dimensional factors

\begin{aligned}F_z(x)&=\int_{-w/2}^{w/2}e^{i\pi(x-\xi)^2/(\lambda z)}\,\mathrm d\xi\\U_z(x,y)&=\frac{F_z(x)F_z(y)}{i\lambda z}\end{aligned}

Within this Fresnel model the field splits exactly into a horizontal factor times a vertical factor. Each factor is computed with 96 Gauss–Legendre nodes across the exact opening, with positive, unequal weights. The phasor polygon shows only the horizontal factor divided by the aperture width; the detector intensity also needs the vertical factor and the propagation prefactor. Drawing only 24 of the terms does not change the detector data.

The far field drops the quadratic phase across the aperture

\begin{aligned}I_{\mathrm{FF}}/I_0&=\left(\frac{w^2}{\lambda z}\right)^2\\&\quad\operatorname{sinc}^2\!\left(\frac{\pi wx}{\lambda z}\right)\\&\quad\operatorname{sinc}^2\!\left(\frac{\pi wy}{\lambda z}\right)\\\operatorname{sinc}(t)&=\sin(t)/t\end{aligned}

The gold reference is this analytic limit, computed separately in the same units and observation window. Its first zero along the axis lies at the wavelength times the distance, divided by the full aperture width. A near-field pattern need not have that zero or look like this one. The reported discrepancy is the RMS difference of the two centre profiles divided by the larger peak. It is neither an experimental error nor a measure of accuracy over the whole field.

Paraxial validity is not the same as far field

\begin{aligned}N_F&=\frac{(w/2)^2}{\lambda z}\\|\delta\phi_4|&\le\frac{\pi r_{\max}^4}{4\lambda z^3}\\r_{\max}&=\sqrt2(L+w/2)\end{aligned}

The phase-bound readout puts an upper limit on the fourth-order path term that the Fresnel model leaves out, using the largest separation between the aperture and a corner of the window. In the near field, Fresnel can stay accurate while Fraunhofer fails badly. The aperture is always hundreds of wavelengths wide, and the control ranges are limited on purpose. The model does not solve for evanescent waves, vector fields at the edges, the thickness of the mask or light radiated backward.

Power that falls outside the detector window

\begin{aligned}\eta_W&=\frac{\int_{-L}^{L}\int_{-L}^{L}|U_z|^2\,\mathrm dx\,\mathrm dy}{w^2}\\P_{\mathrm{all}}/I_0&=w^2\\B&=\min\!\left(1,\frac{\ln(1+GI/I_0)}{\ln5}\right)\end{aligned}

With unit incident amplitude, the power through the square equals the aperture area, and it spreads over the whole observation plane. The finite square detector catches only part of it; the tails fall outside. The power inside the window is estimated from point samples by trapezoidal integration. The window half-width adapts: it is the larger of 0.7 aperture widths and three predicted far-field zero spacings. Exposure and logarithmic brightness change only the images; the readouts keep the raw values. The profile plot uses a labelled logarithmic axis with ticks in raw values, not intensities normalized to the peak.

How the scene maps physical distances

\begin{aligned}X_{\mathrm{scene}}(z)&=-240+600\frac{\ln(z/0.04)}{\ln100}\\Z_{\mathrm{scene}}(x)&=140x/L\\Y_{\mathrm{scene}}(y)&=155+140y/L\end{aligned}

The scene uses different scales along the axis and across it, and both are labelled. The mask stays at scene coordinate minus 320, leaving a gap before the first simulated distance. Aperture and detector share one transverse scale, so their sizes compare correctly within a view; the detail cameras isolate and magnify them without changing the model. Each row of the propagation carpet has its own transverse window, so its horizontal axis is not a fixed length in millimetres. Playback only reveals the contributions and moves slowed-down markers.

Common difficulties

A dark spot still receives contributions

Typical misconceptionIf more of the aperture is lit, the detector has to get brighter.

Better mental modelComplex amplitudes can cancel. Add the weighted fields first, then take the squared magnitude. A coarse sum with equal weights can also alias the rapidly oscillating phase, so drawing more dots does not by itself make the result correct.

Run the experiment

01
Scene 1: Point-wavelet construction
In the wavelet-construction scene, move the probe and look at the weighted phasor polygon. Reveal the partial sums step by step; the detector data do not change.
What to observe: When the phasors cancel, the intensity drops. Showing part of the sum is only a display; it is not a new measurement.
02
Scene 2: Fresnel near field
Choose the near field. Change the aperture width and slide the detector, then compare the Fresnel pattern with the gold reference.
What to observe: The near-field ripples change smoothly, and here the far-field approximation can fail.
03
Scene 3: Fraunhofer far field
Choose the far field. Put the probe near the predicted first zero and adjust the wavelength. Then change the exposure and check that the raw intensity stays the same.
What to observe: The Fraunhofer pattern appears as a limit. The power caught in the finite window does not depend on the image brightness.

Huygens–Fresnel Propagation Lab

Audit the physics of Huygens–Fresnel Propagation Lab

Core mathematical model

A complex contribution from every aperture element

Square aperture: two one-dimensional factors

The far field drops the quadratic phase across the aperture

Paraxial validity is not the same as far field

Power that falls outside the detector window

How the scene maps physical distances

Common difficulties

A dark spot still receives contributions

Run the experiment

Scene 1: Point-wavelet construction

Scene 2: Fresnel near field

Scene 3: Fraunhofer far field

Related experiments and articles

Aperture, propagation, measurement