Theory of Holographic Relaxation Spectroscopy with Nonsinusoidal Gratings

The theory of holographic relaxation experiments is presented with special attention given to nonsinusoidal laser-induced concentration patterns. Diffraction efficiencies are derived at all Bragg orders for photochromic volume gratings created both with crossed laser beams and by means of square-wave intensity patterns. Analytical equations are obtained for the square-wave case. The equations include real and imaginary parts of the complex index of refraction, the wavelength of the probe beam, and time dependence resulting from mass diffusion. Numerical evaluations are presented for first- and second-order diffraction from pure absorption gratings with various optical thicknesses as functions of exposure. It is concluded that square-wave grating profiles with optimized duty factors produce higher diffraction efficiencies under almost all conditions than do profiles created holographically. Square-wave exposure profiles are recommended for use in grating relaxation experiments because of both efficiency and experimental convenience. In general, the time dependence of transient signals is very complicated; but for low efficiencies, i.e., weak gratings, diffraction at the jth Bragg incidence is dominated by a single exponential term having a characteristic time of 1/[D(jK)²] where D is the tracer diffusion coefficient and K is 2π divided by the grating period.