Optics 061 · Diffraction, Fourier optics, and computational imaging

Talbot Carpet

Slide an absorbing sensor through finite amplitude or phase gratings. Explore shifted and fractional Talbot images, crossed two-dimensional patterns, source tilt and real edge diffraction against an explicitly ideal periodic reference.

Full unshifted recurrence / actual sensor distance

z_{\rm T},z

$0.50$

Mask transmission / finite-window mean intensity

T,\overline I_{\rm window}

$50\%$

Finite carrier side / sensor pixel pitch

L,\Delta x

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Talbot Carpet

Grating: 0 absorbing, 1 phase

g

0

Actual grating pitch

d

80\,\mathrm{\mu m}

Coherent wavelength

\lambda

532\,\mathrm{nm}

Opening / phase-stripe fraction

D

0.25

Programmed stripe phase at current wavelength

\phi

0.50\pi

Complete periods per side (finite carrier)

N

16

Pattern: 0 stripes, 1 crossed grid

b

0

Sensor travel / full unshifted Talbot distance

z/z_{\rm T}

1

Horizontal source tilt

\theta_x

0\,\mathrm{mrad}

Display gain exponent (raw data unchanged)

e_G

0

A grating can image itself without a lens. Move the sensor to half distance: the pattern shifts half a pitch; at full distance it returns without the shift. With narrow openings, try one third: three coherent copies fit into one period. Phase-grating revival can look uniform, while a quarter-distance turns phase into strong intensity contrast. Compare the actual finite mask with the separately labelled infinite-periodic reference; change the number of physical periods to expose edge diffraction. The clickable carpet is a computed multi-plane scan of that same finite field, not a glowing object or a reconstruction of a single exposure. Raw sensor pixels, contrast and logarithmic display brightness remain separate.

Finite-carrier scalar paraxial Fresnel integrals, ideal programmed phase, coherent monochromatic illumination and area-averaged pixels. Periodic Fourier reference and separate axial point scan are not acquired reconstructions. Unequal apparatus scales; no vector edge physics, noise or partial coherence.

Start with the essentials

Focus question: Why does a periodic grating reproduce shifted and multiplied copies of itself during propagation?
One-sentence intuition: At half this distance every even harmonic keeps its sign and every odd harmonic reverses it: $u^{\rm per}_{z_{\rm T}/2}(x)=t^{\rm per}(x-d/2)$ . That sign pattern translates the whole field half a pitch. Pure phase transmission can still look uniform in intensity.

Core mathematical model

One mask, two physical behaviours

\begin{aligned}A(x)&=\mathbf1_{|x|<Nd/2}\\g_D(x)&=\sum_n\mathbf1_{|x-nd|<Dd/2}\\t_{\rm amp}&=A g_D\\t_{\rm phase}&=A[1+(e^{i\phi}-1)g_D]\end{aligned}

Opening fraction sets real slit width. The amplitude mask blocks light outside the openings; phase stripes change amplitude phase, not intensity transmission. The aperture envelope is finite. The programmed phase is specified at the chosen wavelength, not inferred from an unspecified material.

Actual finite acquisition

\begin{aligned}u_z(x)&=\frac{1}{\sqrt{i\lambda z}}\int_{-Nd/2}^{Nd/2}t(\xi)e^{\frac{i\pi(x-\xi)^2}{\lambda z}}\,\mathrm d\xi\\U_z(x,y)&=u_z(x)v_z(y)\end{aligned}

Common longitudinal carrier omitted. Slit endpoints give Fresnel C and S differences. A striped mask has a clear finite vertical window; a crossed mask is the separable product of two gratings. No periodic boundary is used for this acquisition.

Separate infinite-periodic reference

\begin{aligned}u^{\rm per}_z(x)&=\sum_{m=-M}^M c_m e^{2\pi imx/d-i\pi\lambda zm^2/d^2}\\c_m^{\rm amp}&=D\,\operatorname{sinc}(mD)\\c_m^{\rm phase}&=\delta_{m0}+(e^{i\phi}-1)c_m^{\rm amp}\\M&=96\end{aligned}

Sinc means sin of pi times its argument divided by that product. The zero order is essential. Periodic transmission omits the finite carrier envelope. This finite harmonic reference has edge ringing and a small omitted power tail; it is neither a finite-mask acquisition nor a full-wave prediction.

Fractional copies add as amplitudes

\begin{aligned}u_{z_{\rm T}/4}^{\rm per}(x)&=\frac{1-i}{2}t^{\rm per}(x)+\frac{1+i}{2}t^{\rm per}(x-d/2)\\u_{z_{\rm T}/3}^{\rm per}(x)&=\sum_{j=0}^2a_jt^{\rm per}(x-jd/3)\\a_j&=\frac13\sum_{m=0}^2e^{-2\pi im^2/3+2\pi imj/3}\end{aligned}

These equalities describe the untruncated infinite-periodic limit, not the finite acquisition or truncated reference. Quarter-plane coefficients follow the even/odd harmonic phases. With amplitude openings narrower than one third pitch, third-plane copies are separate. Wider copies overlap coherently, so multiplying intensity stripes is not a valid general rule. Finite edges spoil exact repetition.

Tilt, pixels and transmission are distinct

\begin{aligned}I_{\theta}(x,z)&=I_0(x-z\theta_x,z)\\I_{ab}&=\frac1{\Delta x^2}\iint_{\mathrm{pixel}_{ab}}|U_z|^2\,\mathrm dx\,\mathrm dy\\T_{\rm amp}&=D^{1+b},\quad T_{\rm phase}=1\end{aligned}

Tilt shears the carpet and moves the pattern on the sensor. Mean intensity within the six-pitch window is not total transmitted power. The camera and enlarged data use identical area-averaged pixels; the carpet is an independently evaluated point-intensity scan, not a one-frame reconstruction. Display gain never changes raw values.

Why real self-images are not perfect

\begin{aligned}k_z&=\sqrt{k^2-k_x^2-k_y^2}\\k_z&\simeq k-\frac{k_x^2+k_y^2}{2k}\\\Delta x&=d/32\end{aligned}

The solver uses the quadratic paraxial phase, ideal coherence and abrupt thin boundaries. High-angle edge components, finite bandwidth, limited coherence and pixel integration limit fine detail. Only finite-carrier diffraction and pixel integration are included here; no infinite fractal or external scientific certification is claimed.

Common difficulties

Uniform intensity does not mean no self-image

Typical misconceptionEvery Talbot plane must show bright copies of a phase grating.

Better mental modelA lossless phase grating has uniform intensity at the mask. Its ideal field revival restores that phase pattern, not visible amplitude stripes. Contrast can develop at fractional planes. Finite carrier edges add diffraction.

Run the experiment

01
Scene 1: Unshifted finite revival
Start with amplitude stripes. Compare half and full recurrence; inspect the finite mask openings and actual sensor front.
What to observe: Half distance shifts the ideal pattern half a pitch; full distance restores unshifted field.
02
Scene 2: One third · three copies
Use the third-plane preset. Widen openings until the three coherent copies overlap; compare with the periodic reference.
What to observe: Narrow amplitude openings separate fractional copies; broad copies interfere instead of simply multiplying intensity.
03
Scene 3: Phase to intensity · crossed grid
Switch to crossed phase stripes. Compare quarter-distance contrast with full-distance uniformity, then reduce the finite period count.
What to observe: Phase revival can look uniform. A finite carrier and area-averaged detector prevent perfect sharp repetition.