On the evaluation of modified Bessel functions of the second kind and fractional order for synchrotron radiation calculations

The modified Bessel functions of the second kind and fractional order K1/3(x) and K2/3(x) are of importance in the calculation of the frequency spectrum of synchrotron radiation. The parameter range of interest is typically 10−6<x<10. Recently, there has been particular interest in the generation of ‘terahertz’ radiation, which can be coherently enhanced by many orders of magnitude when the electron bunch length is shorter than the terahertz wavelength. This requires evaluation of the Bessel functions for small values of the argument. It is shown that the series commonly used to evaluate these functions has poor convergence properties under these conditions. An alternative series is derived which has much better convergence for x<1.