Formulating the design of two freeform lens surfaces for point-like light sources.

Two freeform surfaces provide more degrees of freedom in designing illumination optics and can yield a better solution. The existing methods for point-like sources are mostly valid in designing one freeform surface. Designing two freeform surfaces for point-like sources still remains a challenging issue. In this Letter, we develop a general formulation of designing two freeform lens surfaces for point-like sources. The proposed method is very robust in designing freeform lenses with two elaborately designed surfaces. The examples clearly show that using two freeform surfaces yields better solutions to challenging illumination problems with ultra-high energy efficiency.

[The analysis of the sampling modulation transfer function and the influence on the Gaussian spectra].

As the parameters of the photoelectric detector have important effects on the performance of the dispersive spectrometers, it is necessary to detail the discrete sampling process of the photoelectric detector array. In the present paper, the sampling model was setup, and the effects on the sampling results caused by the spatial frequency of the cosine signal, the width of the sampling pixel, and the initial phase of the sampling pixel position to the crest of the input cosine signal were discussed thoroughly in the frequency domain. By introducing the integral function, a general expression of the sampling modulation transfer function was given, and the concept and expression of the average sampling modulation transfer function was proposed. Since that expression eliminates the effect of initial phase, it is much more convenient to the practical applications. For the typical Gaussian spectrum produced by the dispersive spectrometer, the Fourier transform result of that spectrum was multiplied by the average sampling modulation transfer function to produce a functional expression of the modulation transfer function of the whole system. The average aliasing error of the sampling process was expressed as a function of spatial frequencies; the relationship between the peak value of the average aliasing error and the width of the Gaussian spectrum was discussed; and the critical value of the spectrum width to restore this spectrum precisely was proposed. That critical value is significant for providing guidance to the design and fabrication of dispersive spectrometers.