Conventional diffraction calculations typically employ complex Fourier transforms in which the source and target fields are represented by complex values. However, this approach is inefficient for certain applications. To address this problem, this study introduces diffraction calculations for three fields: real-to-complex, complex-to-real, and real-to-real. These calculations utilize a real-valued fast Fourier transform and Hermite symmetry, enabling accelerated computation by eliminating half of the spectra. This study also demonstrates the practical applications of these diffraction calculations. These applications include image reproduction in digital holography, speckle reduction in holographic projection, and accelerated hologram computation in holographic displays.