Optics 061 · Diffraction, Fourier optics, and computational imaging

Talbot Carpet

Why does light behind a grating repeat the grating’s pattern at certain distances? The grating's diffraction orders pick up different phases as they travel, and at the Talbot distance those phases line up again, so the pattern reappears. Halfway there, the odd orders are flipped in sign and the copy is shifted by half a period. At simple fractions of the distance, several shifted copies overlap and the pattern looks finer. A finite grating makes every revival imperfect.

Slide an absorbing sensor behind finite amplitude or phase gratings. Find shifted and fractional Talbot images, crossed two-dimensional patterns, the effect of tilting the source and diffraction from the grating edges, and compare each with an 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 $\text{valid model regime}$

Model checked against analytic results

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 the Talbot distance and the pattern shifts by half a period; at the full distance it returns with no shift. With narrow openings, try one third of the distance: three coherent copies fit into each period. A phase grating can look uniform at its revival, while at a quarter of the distance its phase turns into strong intensity contrast. Compare the finite grating with the infinite-periodic reference, which is labeled separately, and change the number of periods to see diffraction from the edges. The clickable carpet is a computed scan of that same finite field over many planes; it is not a glowing object or a reconstruction from one exposure. Raw sensor pixels, contrast and logarithmic display brightness are kept separate.

The field is computed with scalar paraxial Fresnel integrals over a finite grating, with an ideal programmed phase, coherent monochromatic light and pixels that average over their area. The periodic Fourier reference and the separate axial point scan are calculations, not reconstructions of recorded data. Apparatus parts use unequal scales. Vector effects at edges, noise and partial coherence are not included.

Physics tutorial

Audit the physics of Talbot Carpet

BackgroundA field that repeats forever across the beam contains only a discrete set of transverse spatial frequencies. Fresnel propagation shifts their phases at different rates, and they all line up again after $z_{\rm T}=2d^2/\lambda$ . At this distance the whole field comes back with no shift. Some texts use half this distance as their definition instead.

Why it mattersWhy does light behind a grating repeat the grating’s pattern at certain distances?

Start with the essentials

Focus question: Why does light behind a grating repeat the grating’s pattern at certain distances?
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 pattern of signs shifts the whole field sideways by half a period. If the grating changes only the phase, the intensity can still look uniform.

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}

The opening fraction sets the actual slit width. The amplitude mask blocks light outside the openings. The phase stripes change the phase of the field but not the transmitted intensity. The aperture around the grating is finite. The phase step is set directly for the chosen wavelength; it is not derived from any unspecified material.

The field behind the finite mask

\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}

The common phase from propagation along the axis is left out. Each slit contributes differences of the Fresnel integrals C and S taken at its two edges. A striped mask is clear over a finite vertical window; a crossed mask is the product of two gratings, one in each direction. This simulated measurement uses no periodic boundary.

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}

Here sinc of an argument means the sine of pi times that argument, divided by pi times the argument. The zero order must be included. The periodic transmission has no envelope from the finite carrier. Because the reference keeps only a finite number of harmonics, it rings near edges and misses a small tail of power. It is neither the finite-mask simulation 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. They do not describe the finite simulation or the truncated reference. The quarter-plane coefficients come from the phases of the even and odd harmonics. When the amplitude openings are narrower than one third of a period, the copies at the third plane stay separate. Wider copies overlap coherently, so multiplying intensity stripes does not work as a general rule. Finite edges spoil exact repetition.

Tilt, pixels and transmission mean different things

\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 Talbot carpet and moves the pattern across the sensor. The mean intensity inside the six-period window is not the total transmitted power. The camera view and the enlarged data use the same area-averaged pixels. The carpet is a separate scan of point intensities, computed on its own and not rebuilt from one frame. Display gain never changes the 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, perfectly coherent light and thin masks with sharp edges. High-angle components from the edges, finite bandwidth, limited coherence and pixel integration all limit fine detail. Of these, only diffraction from the finite carrier and pixel integration are included here. The Lab claims neither an infinitely fine fractal nor any outside scientific certification.

Common difficulties

Uniform intensity does not mean no self-image

Typical misconceptionEvery Talbot plane should show bright copies of the phase grating.

Better mental modelRight behind a lossless phase grating, the intensity is already uniform. An ideal revival brings back that phase pattern, so it shows no visible amplitude stripes either. Intensity contrast can appear at fractional planes. The edges of the finite carrier add extra diffraction.

Run the experiment

01
Scene 1: Unshifted finite revival
Start with the amplitude stripes. Compare the half-distance and full-distance revivals, then look at the finite openings in the mask and the front face of the sensor.
What to observe: At half the distance the ideal pattern is shifted by half a period; at the full distance the field returns unshifted.
02
Scene 2: One third · three copies
Use the third-plane preset. Widen the openings until the three coherent copies overlap, and compare the result with the periodic reference.
What to observe: Narrow amplitude openings keep the fractional copies apart. Wide copies interfere with each other; they do not simply multiply the intensity stripes.
03
Scene 3: Phase to intensity · crossed grid
Switch to crossed phase stripes. Compare the contrast at a quarter of the distance with the uniform intensity at the full distance, then reduce the number of periods.
What to observe: A revived phase pattern can look uniform in intensity. The finite carrier and the area-averaging detector keep the repetition from being perfectly sharp.