A method based on the Fresnel diffraction of light from the phase step is introduced for measuring effective focal length (EFL) and back focal length (BFL) of optical imaging systems. It is shown that, as a transparent plane-parallel plate is illuminated at a boundary region by a monochromatic beam of light, Fresnel diffraction occurs because of the abrupt change in phase imposed by the finite change in refractive index at the plate boundary. Variation of the incident angle in a convergent (or divergent) beam of light causes the periodic intensity along the central fringe of the diffraction pattern. The measurement of the extrema position of the intensity distribution accurately provides the EFL and BFL. The technique is easy to apply and can measure a wide range of both positive and negative focal lengths. The measuring setup can be very compact with low mechanical and optical noises. As examples of this technique, the EFLs of five different lenses are experimentally obtained. The results are quite consistent with the values indicated by the lens manufacturer.
In this Letter, we investigate the Fresnel diffraction of vortex beams from a phase plate and propose a novel (to the best of our knowledge) method to determine the fractional part of the topological charge of vortex beams. When a vortex beam with a fractional topological charge illuminates the edge region of a transparent plate, the visibility of the diffraction pattern on two sides of the beam is different. Rotation of the phase plate changes the visibility on the left and right sides of the beam, periodically. By measuring three consecutive angles of the minimum visibilities, the fractional part of the topological charge is obtained. The proposed method is verified experimentally and is shown to be independent of the phase plate and vortex beam parameters. The precision of the method is obtained better than 0.01.
In the present work, the Talbot effect of a square grating is analyzed when light is reflected from a rough surface. It is shown theoretically that the scattered light intensity in the Fresnel diffraction limit depends on statistical properties of the rough surface, the angle of incidence of the light, the grating period, and a geometric coefficient, related to the ratio of distance of the rough surface and the observation plane from the grating. At Talbot distances of the grating, the surface height difference function, in terms of multiplication of the Talbot number, the grating period, and the geometric coefficient is the modulation transfer function (MTF) of the scattering in reflection from the rough surface. If the argument of the height difference function is larger than twice the surface correlation length, the height difference function is constant for different spatial frequencies. Therefore, the square wave is reproduced with smaller contrast. The surface roughness can be obtained by measuring the contrast at different incident angles. It is also shown that the contrast measurements in both reflection and transmission, provide the refractive index of transparent samples with a rough surface. In experimental studies, the roughness of three metal standard rough surfaces are determined at different angles of incidence. Also, the refractive index of a sheet glass with a rough surface is obtained. The results are quite consistent.
In this work, a linear grating is used to project a periodic light intensity distribution on a rough interface, and the near field transmitted light scattering is studied. It is shown theoretically that the intensity in the Fresnel regime depends on statistical properties of the rough interface and the light intensity period. The self-image contrast exponentially depends on the interface height-height correlation function. The correlation is obtained in terms of multiplication of the self-image number and the period of the light intensity distribution. Therefore, the roughness and the correlation length of the interface can be obtained by determining the contrast of the self-images when the light intensity period is smaller than the interface correlation length. For periods longer than twice the correlation length, the contrast measurements only provide the interface roughness. In experimental studies, the roughness of interfaces is determined by square gratings with periods much longer than the correlation lengths. The rough interfaces are prepared by roughening sheet glass by powders of different grit numbers. The results for different gratings and light wavelengths are quite consistent.
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