Utilizing the asymptotic method of stationary phase, I derive expressions for the Fourier transform of a two-dimensional fringe pattern. The method assumes that both the amplitude and the phase of the fringe pattern are well-behaved differentiable functions. Applying the limits in two distinct ways, I show, first, that the spiral phase (or vortex) transform approaches the ideal quadrature transform asymptotically and, second, that the approximation errors increase with the relative curvature of the fringes. The results confirm the validity of the recently proposed spiral phase transform method for the direct demodulation of closed fringe patterns.
© 2001 Optical Society of America
(070.2590) Fourier optics and signal processing : ABCD transforms
(100.2650) Image processing : Fringe analysis
(100.5090) Image processing : Phase-only filters
(110.6980) Imaging systems : Transforms
(120.2650) Instrumentation, measurement, and metrology : Fringe analysis
Kieran G. Larkin, "Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform," J. Opt. Soc. Am. A 18, 1871-1881 (2001)