In digital holography with a quasi-Fourier setup,6- 7 both a reference point source and an object are placed at the input plane and an array photodetector at the hologram plane. Thus, digital holograms are recorded optically and stored in computer memory as image files of primary interference fringes. Two symmetrical hologram reconstructions are obtained numerically by calculating the Fourier transform of the recorded data. If we denote the dynamic object wave front with and the position of the point source with , then the input can be described as: . If only out-of-plane harmonic vibrations of the object surface occur and the illumination is normal to the surface, then , where denotes the static object wave front, the vibration amplitude, the wavelength, and the vibration frequency. The diffracted field at a distance from the input plane can be calculated according to Fresnel approximation as:Display Formula
1where denotes the Fourier transform operation, the spatial frequencies, and where the constant terms are omitted. The exposure is obtained by integrating the instantaneous intensity , i.e., , where is the exposure time. For the integration time , one of the additive exposure terms is in the form:8Display Formula
2yielding for the reconstruction (modulus of the inverse FT):Display Formula
3In Eqs. 2- 3, is the zero-order Bessel function of the first kind. The Bessel function shows that the object reconstructions are covered with the modulation fringes. The number and distribution of fringes depend on the amplitude , which is generally a function of spatial coordinates and the frequency at which the investigated surface is vibrating, i.e., . Therefore, the hologram reconstruction reveals the main characteristics of the vibration mode: the node structure and vibration amplitudes. In the case of vibrating surfaces of most musical instruments, detection of very regular fringe patterns is expected due to the symmetry of their resonant bodies. For example, irregularities in fringe patterns of percussion instruments indicate detuning of their membranes.