Experimental cancellation of aberrations in intensity correlation in classical optics

We study the classical correlation function of spatially incoherent beams with a phase aberration in the beam path. On the basis of our experimental measurements and in the optical coherence theory, we show that the effects of phase disturbances, independently of their kind and without need of coordinate inversion, can be canceled out if the same phase is aligned in the signal and reference beam path. These results can be useful for imaging and microscopy through random media.

Figure 1

Experimental setup: ${\mathrm{M}}_1-{\mathrm{M}}_6$ are mirrors, ${\mathrm{L}}_1-{\mathrm{L}}_6$ are lenses, $\mathrm{B}{\mathrm{S}}_1-\mathrm{B}{\mathrm{S}}_4$ are beam splitters, RGGD is a rotating ground glass disk, $\mathrm{A}$ is a double-slit aperture, SLM is a spatial light modulator, and CCD is a charge-coupled (CCD) camera. Polynomial aberrations of the form ${\psi }_n\left(x\right)={\alpha }_n{x}^n\left(n=1,2,3,4\right)$ and a random phase are displayed on the SLM. $\mathrm{I}$ and $\mathrm{NI}$ stand for inverted and noninverted beams, respectively.

Figure 2

Detection planes for $S$ and $R$ intensities in Eq. (1).

Figure 3

(a–e) Measured high-order normalized (by its sum) intensity correlation profiles for $\delta =0.06\mathrm{mm}$ for different phase aberrations. Black (squares): nonaberrated curve for the case without aberrations in the SLM. Red circles: aberrated curves with coordinate inversion of the phase of the signal beam. Blue triangles: corrected curves (without coordinate inversion of the phase of the signal beam). (f) Comparison between the correlation with $|{\tilde{A}}_{\varphi }{|}^2$ and without background subtraction $\mathrm{\Gamma }$.

Figure 4

Simulation results of the first-order correlation curves for the noninverted configuration where a complete aberration cancellation is expected, (a) $n=2$ and (b) $n=4$; simulation results of the difference between the aberrated and nonaberrated first-order correlation curves varying the coherence length $\delta$ for $\alpha =0.3\mathrm{mm}$ (c) and $\alpha =0.8\mathrm{mm}$ (d).