Geometrical analysis of ring fields for various numerical apertures is described to design an Offner concentric hyper-spectral imager utilizing a convex grating. A new efficient design of spectrometer optics is suggested in order to increase its spectral resolution. The design has been derived under the mathematical condition causing no ray obstruction since the original form of the “ring of zero aberration” by Offner causes ray obstruction with high numerical aperture. It is accomplished by the analysis of ray paths and using the wave-front aberration term between the common center and the object point in Lobb’s aberration theory. A ring field could be directly calculated using the new equation by entering the radius of curvature of the optical surfaces, numerical aperture, grating spacing and diffraction order. The analytical design was demonstrated using optical design software Code V to manufacture a HSI working in a wavelength range of 900-1700 nm. The simulation results show that the new concept offers an analytical initial design taking the least time of calculation. The simulated HSI exhibited Modulation Transform Function over 80% at Nyquist frequency, root-mean-square spot diameters as small as on quarter of a pixel, and a spectral resolution 3.2 nm. The equation and analytical design procedure shown here can be applied to most of the Offner concentric system regardless of the wavelength range.
Three dimensional data of three plastic bottles containing the different amounts of water and silicon wafer on the figure are measured by prototype. Analysis of the spectral line proved that these data can be used to measurement of spectral resolution, smile and keystone, and slit tilting.