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We present three methods for calculating the spatio-temporal electric field of ultra-short pulses near the focal point of an optical system. The first method is the Riemann’s sum-based direct-integration method, which is used as a comparison for the other two methods. The other methods are a triple fast Fourier transform (FFT) algorithm and a combined FFT/quadrature algorithm. The proposed triple FFT method reduces dramatically the numerical integration time compared to the other two methods. In the FFT, the pixel size in the focal plane depends on the wavelength, so the pixel size changes for each frequency of the pulse. In this paper, we propose a way to overcome this problem giving a reliable, fast focus-field calculation. As an example, the electric field of the pulse at the focal region of a low numerical aperture mirror is evaluated by using the scalar diffraction theory, and results of the focusing pulse with the three methods are presented for a one-optical-cycle unchirped pulse beam, 2.7 fs at 810 nm, incident on the mirror. However, the FFT algorithm also works for the focusing of more complex optical systems such as high numerical aperture optical systems with aberrations and dispersion.
© 2017 Optical Society of America
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