Optics 056 · Diffraction, Fourier optics, and computational imaging

Huygens–Fresnel Propagation Lab

Move a detector continuously through one square-aperture complex field; connect weighted Huygens contributions, near-field structure, finite-window power and 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 $\text{valid model regime}$

Validated scientific apparatus

Huygens–Fresnel Propagation Lab

Detector distance (moves on logarithmic rail)

z

0.15\,\mathrm m

Square aperture full side (changes actual opening)

w

0.40\,\mathrm{mm}

Vacuum wavelength

\lambda

532\,\mathrm{nm}

Centre-line probe (fraction of window)

x_p/L

0.15

Display exposure decimal exponent (not data)

\log_{10}G

0

Slide the detector from a rippled near field into a broad far-field pattern. The real square opening resizes with aperture width. Compare the cyan propagated intensity against the separately labelled gold Fraunhofer reference; they agree only in its limit. Move the gold probe to see weighted amplitudes add before squaring. Shift-drag the receiving face in the bench view, or click the propagation carpet, to move the same detector. Detail views isolate actual mask and detector for inspection. Play traces mathematical contributions, not photons; exposure changes the display, not the data.

Scalar monochromatic Fresnel approximation, exact square domain and separable 96-node quadrature. No full Maxwell solver, coherence loss or noisy camera. Logarithmic axial scene, adaptive transverse window, enlarged supports; detail views isolate apparatus. Explicit exposure never rescales raw power to a moving peak.

CONTINUE EXPLORING

Aperture, propagation, measurement

Change one physical assumption at a time: slit count, coherence, aperture shape or propagation distance.

Physics tutorial

Audit the physics of Huygens–Fresnel Propagation Lab

BackgroundEach illuminated aperture element contributes a complex amplitude, not a positive brightness. The detector measures $I_z=|U_z|^2$ . Destructive interference can create darkness even though every contributing element is illuminated. One finite square aperture lets propagation, rather than aperture catalogues, stay the central question.

Why it mattersHow does a field evolve continuously from near-field wavelets to a far-field diffraction pattern?

Start with the essentials

Focus question: How does a field evolve continuously from near-field wavelets to a far-field diffraction pattern?
One-sentence intuition: Using the half-side convention, the propagation scale is $N_F=\frac{(w/2)^2}{\lambda z}$ . This changes continuously. A small number supports the far-field approximation; it is not a switch that replaces the Fresnel calculation.

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 is at physical axial coordinate zero. Positive distance follows the incident plane wave; transverse coordinates are measured from the centre. Unit incident amplitude and vacuum wavelength define the normalization. Only the common longitudinal carrier is removed. All coordinates in the integrals are metres. The coloured scene curves denote integration contributions, not bent geometrical 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}

Separation is exact within this Fresnel model, not a row copied across an image. Each factor uses 96 Gauss–Legendre nodes with positive, unequal weights over the exact opening. The phasor polygon shows only the horizontal factor divided by aperture width; the vertical factor and propagation prefactor are still needed for detector intensity. Revealing 24 terms does not change the acquisition.

Far field drops the aperture quadratic phase

\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 an independent analytic limit in the same units and observation window. Its first axial zero is one wavelength times distance divided by full aperture width. Near field need not obey that zero or resemble its pattern. The reported discrepancy is centre-profile RMS divided by the larger profile peak; it is not an experimental error or a full-field accuracy certificate.

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 estimates the omitted fourth-order path term over every aperture-to-window-corner separation. Fresnel can remain valid in the near field even while Fraunhofer fails badly. Aperture features stay hundreds of wavelengths wide. The controls are deliberately bounded; no evanescent waves, vector boundary fields, material thickness or backward radiation are solved.

Power leaves the window, not the theory

\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}

Unit-amplitude transmission through the square sends aperture-area power into the full plane. The finite square detector catches only a fraction; tails remain outside. Trapezoidal integration estimates its power from point samples. The window adapts to the larger of 0.7 aperture widths and three predicted far-field zero spacings. Exposure and logarithmic brightness affect images only; readouts retain raw values. The profile uses a labelled logarithmic axis with raw-value ticks, not peak-normalized intensities.

Read the coordinate map before the animation

\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}

Longitudinal and transverse scene scales are different and labelled. The mask stays at scene coordinate minus 320, with a gap before the first simulated distance. Aperture and detector share one transverse scale, so their relative sizes are honest within a view; detail cameras isolate and magnify them without changing the model. The carpet has a new transverse window on each row, not a fixed-millimetre x axis. Playback reveals contributions and slowed markers only.

Common difficulties

Darkness does not mean missing sources

Typical misconceptionAdding more illuminated aperture points must always brighten the detector.

Better mental modelComplex amplitudes can cancel. Sum weighted fields before taking squared magnitude. A coarse equal-weight sum can also alias the oscillatory phase; more dots are not a scientific validation.

Run the experiment

01
Scene 1: Point-wavelet construction
In wavelet construction, move the probe and inspect the weighted phasor polygon. Reveal partial sums without altering detector data.
What to observe: Vector cancellation changes intensity; partial display is not a new acquisition.
02
Scene 2: Fresnel near field
Choose near field. Change actual aperture width and slide the detector; compare Fresnel structure against the gold reference.
What to observe: Near-field ripples evolve continuously, while the far-field approximation can fail.
03
Scene 3: Fraunhofer far field
Choose far field. Place the probe near the predicted first zero, adjust wavelength, then change exposure without changing raw intensity.
What to observe: Fraunhofer emerges as a limit, and finite-window power remains separate from image brightness.