Nonlinear optical characterization of LiNbO_3 I Theoretical analysis of Maker fringe patterns for x-cut wafers

Abstract

Maker fringe analysis was adapted to x-cut LiNbO3 wafers to examine variations in birefringence, thickness, and photoelastic strain. The pump beam was polarized parallel to the crystalline y axis and produced e- and o-polarized Maker fringes, owing to d31 and d22, respectively, by rotation of the sample about the y axis. Fitting our model to the o-polarized data enabled computation of the sample thickness to an uncertainty of approximately ±0.01 μm. The accuracy was limited by an implicit ±2×10-4 uncertainty in no that exists in the commonly used Sellmeier equation of G. J. Edwards M. Lawrence , Opt. Quantum Electron. 16, 373 (1984). For a pump wavelength λp=1064 nm, fitting the model to the e-polarized fringes revealed that ne at 532 nm deviated from the Sellmeier result by typically -1.58×10-4. The uniformity of ne over a wafer 10 cm in diameter was approximately ±4×10-5. This result is consistent with that expected from compositional variations. Our model included multiple passes of the pump and second-harmonic waves. The effects of photoelastic strain in producing perturbations and mixing of the e- and o-polarized fringes was investigated. This was restricted to two experimentally motivated cases that suggested that strains produce rotations of the optic axis by typically ±0.05° about the x axis and y axis with the former assigned to an indeterminant combination of S1,S2, and S4 and the latter to an indeterminant combination of S5 and S6. In both cases the magnitude of the collective strains is of the order of 10-4. The birefringence variations that are due to strain are of the same magnitude as those expected from compositional variations. The formalism developed here is used in the subsequent mapping study of x-cut wafers.