Fluorescence Recovery After Patterned Photobleaching is a well-known techniques to measure diffusion coefficient and drift velocity. We have developped a experimental set-up in the spirit of Davoust et al. [Davoust, J. and Devaux, P.F. and Léger, L., The EMBO journal (1982), **1** (1233)] pionneer work, which probe in reciprocal space diffusion equation.

Fluorescence bleaching of the labelled species in the illuminated fringes was achieved with a short (<1 s) full intensity light pulse, by means of a Pockels cell between nearly crossed polarizers (A and P). After photobleaching, the decay of the amplitude of the fluorescent molecule fringe pattern was detected by modulation of the position of the illuminating fringes at 1 kHz, using a piezoelectrically modulated mirror. The emitted fluorescence light was collected with an optical fiber and a Photomultiplier (PMT). Signals were analyzed by a lock-in amplifier, and averaged by integration over times varying from 0.1 to 0.5 s (100 to 500 periods). The recorded experimental signal gave the mean contrast *C*(*t*) between bleached and unbleached fringes, and vanished due to the lateral diffusion of the fluorescent probes in the bilayer.

The use of the same fringe pattern for bleaching and reading ensures that they are both characterized by the same spatial wavelength. Davoust et al. have shown that the component at the fundamental frequency of the fluorescence intensity detected by the lock-in amplifier varied with time as a single mode of the diffusion equation :

d*c*(*r*,t)/dt = *D* ▽^{2} *c*(*r*,t) [equation 1]

where *c*(*r*,t) is the local concentration in fluorescent molecules and *D* is the diffusion coefficient. Using a Fourier transform equation (1) became :

d*c*(*q*,t)/dt= -*q*^{2}*D* *c*(*q*,t) [equation 2]

A solution of equation (2) with proper boundary conditions is:

*c*(*q*,t) = *c*(*q*,0) exp(-t/τ_{q})

where *c*(*q*,0) is the fluorescence probe concentration just after photobleaching and τ_{q} is related to the diffusion coefficient *D* by :

τ_{q} = 1/*D**q*^{2}

As described in details by Davoust et al., the mean contrast *C*(*t*) is then directly related to the characteristic time constant τ_{q} by:

*C*(*t*)=*C*_{∞}+(*C*_{0}-*C*_{∞}) exp(-t/τ_{q})

where *C*_{0} (*C*_{∞}) corresponds to the contrast just (a long time) after the bleaching.

We shall stress here two important points regarding the specificity of this set-up. The use of a periodic sinusoidal pattern for bleaching and reading allows to probe a single mode of the diffusion equation. It is not the case in most usual FRAP setups, where a more complex pattern - generally a Gaussian spot - is photobleached. The analysis is here more simple and more precise. It is thus easier to detect small variations of the diffusion coefficient, or to discriminate between different populations. Finally, our experimental setup allowed to vary easily the interfringe value and, doing so, to directly check the validity of the diffusion law in the reciprocal space, probing equation (1).

Such an experimental setup (reading and bleaching with same pattern) had been used to investigate polymer diffusion at interfaces, DNA diffusion and electrophoresis, and lipid diffusion in supported bilayers.