Talbot Effect Self-Imaging Lab - 2D Intensity Carpet, Profiles and Fringe Contrast

Presets

Speed

Live readouts

Graphs

Numerical checks

running...

How to use

Light enters from the left through a grating (yellow slits) and the colour map shows the intensity behind it, x vertical and distance z horizontal. The blue line is the observation plane and the first graph shows the profile there. Try: (1) Medium grating - watch the pattern blur and then sharpen again at z_T; (2) Freeze at z_T/2 - the image returns shifted by half a period; (3) Freeze at z_T/4 - twice as many fringes; (4) Few harmonics - fewer terms smooth the edges (Gibbs ripples at many terms); (5) change the period or wavelength and read the new z_T.

The science

Paraxial propagation. A grating of period d is periodic, so its transmitted field is a Fourier series. Each diffraction order n travels at its own angle, and in the Fresnel (paraxial) approximation acquires an extra phase proportional to n squared:

U(x,z) = sum_n c_n exp(i 2 pi n x / d) exp(-i pi lambda z n^2 / d^2), I = |U|^2

with c_0 = a and c_n = sin(n pi a)/(n pi) for an open fraction a.

Revivals. The extra phase equals 2 pi n^2 (a whole number of turns for every order) when

z_T = 2 d^2 / lambda

so the intensity repeats exactly - a lens-free image. At z_T/2 the phase is pi n^2 = n pi (mod 2 pi), a sign alternation that is the same as a lateral shift by d/2. At z_T/4 and a 50% grating the pattern has half the period (frequency doubling); other rational fractions give finer fractional Talbot images.

Use. Talbot-Lau interferometers use the effect in X-ray phase-contrast imaging, and it underlies lithography proximity printing and atom optics.

Checks. (1) z_T for 6 um at 550 nm; (2) I(z_T) equals I(0); (3) I(z_T/2, x) equals I(0, x + d/2); (4) frequency doubling at z_T/4; (5) the mean intensity is the same at all z (energy conservation, Parseval).

Simplifications. Infinite periodic grating (a finite aperture adds edge diffraction and limits the carpet to a trapezoid), thin amplitude grating, scalar paraxial theory valid when d is much larger than lambda, perfectly coherent monochromatic light, and a truncated number of harmonics (a user setting).