For the preceding part where the paraxial condition is valid, scalar diffraction is satisfactory for the light propagation evaluation. For example, in typical systems for optical microscopy, fabrication and manipulation, a monochromatic beam propagates over a long distance by passing optical elements such as focusing lenses, expanders, and collimators before entering an objective lens with a high NA. However, in most practical apparatuses, scalar and vector diffractions co-exist for different parts of the optical system. In addition, scalar and vector diffractions are separately analyzed in conventional studies because different integral formulas are needed for each case. Therefore, the versatile computation of optical diffraction in an efficient and flexible fashion is highly demanded for wide applications. However, these methods can generate only the light field distribution within a fixed region of interest (ROI) and sampling numbers (i.e., resolution) determined by the intrinsic characteristic of the Fourier transform (FT), lacking flexibility in computing the desired local distribution with variable sampling intervals. Fast Fourier transform (FFT)-based algorithms have been developed to perform fast calculations of light diffraction 15, 16, 17, 18, 19. However, the point-by-point calculation fashion renders the computation extremely tedious and inefficient. The direct integration method was first used to calculate both scalar and vector diffraction 12, 13, 14. Although mathematical expressions for optical diffractions have been presented authoritatively for ages, fundamental breakthroughs have rarely been achieved in diffraction computations. For high-NA optical systems, polarization effects play a paramount role near the focal spot, and thus, vector diffraction must be adopted for light field tracing 9, 10, 11. Scalar diffraction can yield sufficiently accurate results if the diffracting aperture and observing distance are both far larger than a wavelength, which is most valid for optical systems with a low numerical aperture (NA). Scalar diffraction considers only the scalar amplitude of one transverse component of either the electric or the magnetic field with certain simplifications and approximations 8. The diffraction of electromagnetic (EM) waves can be cataloged into scalar diffraction and vector diffraction according to the validation of different approximation conditions. The efficient calculation of light diffraction is of significant value toward the real-time prediction of light fields in microscopy 1, laser fabrication 2, 3, 4, 5, and optical manipulation 6, 7. Based on these results, full-path calculation of a complex optical system is readily demonstrated and verified by experimental results, laying a foundation for real-time light field analysis for realistic optical implementation such as imaging, laser processing, and optical manipulation.ĭiffraction is a classic optical phenomenon accounting for the propagation of light waves. Furthermore, the region of interest and the sampling numbers can be arbitrarily chosen, endowing the proposed method with superior flexibility. The high efficiency facilitates the ultrafast evaluation of light propagation in diverse optical systems. The computation time can be substantially reduced to the sub-second level, which is 10 5 faster than that achieved by the direct integration approach (~hours level) and 10 2 faster than that achieved by the fast Fourier transform method (~minutes level). Here, we present a fast and flexible calculation method for computing scalar and vector diffraction in the corresponding optical regimes using the Bluestein method. However, existing calculation methods suffer from low computational efficiency and poor flexibility. Efficient calculation of the light diffraction in free space is of great significance for tracing electromagnetic field propagation and predicting the performance of optical systems such as microscopy, photolithography, and manipulation.
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