An efficient way to numerically calculate diffraction from large volume holograms is developed using the first-order Born approximation. For this, everything except the propagating part of the Green's function is neglected, and the fact that the gratings have a slowly varying envelope is used. The results of the new method are compared with analytical solutions of plane-wave diffraction with absorption, with phase-conjugated readout of a hologram recorded with a point source, and with numerical simulations of shift multiplexing with high-numerical-aperture microscope objectives. We show that the new method gives correct results in all cases and is several orders of magnitudes faster than FFT-based integration.
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