Abstract

Second-harmonic (SID cross-correlation is a substantially new method for wave-field transformation that shares many properties with holographic methods [1,2]. In this case the reconstructed wave-field is formed by the frequency-doubled radiation, while both fields impinging on the SH crystal, which acts as the holographic plate, arc at the fundamental frequency. In fact, for a thin SH crystal (ß-barium borate in phase matching I, BBO I in Fig. 1) these fields are E_o= A_o(x, y) exp{i[k_lL_o(x, y) + ωt]} and E_R= A_Rexp{i[k_lL_R(x, y) + ωt]} where E_ois diffused by the illuminated object and E_Ris a fraction of the incident-pulse field serving as the reference field. The iconais L_o(x,y)and L_R(x,y)represent the distribution functions of the optical paths from Oand Rto tlic BBO I plane, that is they describe the positions of the wavefronts W_oand W_R. The SH cross-correlation produces a field term

$E_{2\omega }^{"}=A_{\text{o}}\left(x,y\right)A_R\exp \left\{i\left[2k_l\left(L_o\left(x,y\right)+L_R\left(x,y\right)\right)/2+2\omega t\right]\right\}$

which, if the interaction of the phase-matched fields E_oand E_Ris slightly non-collinear (angle αin Fig. 1), propagate along the αbisector, B_s. This term reconstructs what we call second-harmonic generated (SHG) holographic image of the object O.The image is a virtual one in all cases. By using a formalism typical of holography and techniques of ray tracing of simple lens systems we could determine the position and the scale of the reconstructed images (O'in Fig. 1) for both plane and spherical reference wave-fronts W_RFor instance, in the case of plane W_Rthe SHG hologram reconstructs a virtual image O’whose transversal size is equal to that of the object and longitudinal size (zdirection) is two times larger than that of O.The use of a spherical W_Rbrings about a focusing effect on the $E_{2\omega }^{"}$wave-fronts. In all experiments an imaging lens was used to transform O'into a real O"image detectable with a CCD camera.

© 2000 IEEE