Numerical Methods in Photonics

7.6.5  CALCULATING THE SOMMERFELD INTEGRAL

The integral expressions (7.94), (7.95), (7.98), and (7.99) are known as Sommerfeld integrals [3,4]. For the first part of the integrals (7.94) and (7.98), we might as well just use the analytic expression for g^(d) given in for example Equation (7.6). For the rest of the integral, we can observe that as long as y + y′ > 0, the factor e^{–ik_y1(y+y′)} will decrease exponentially with increasing k_x for large values of k_x, and thus, the integral will converge, and the maximum values of k_x that one needs to consider will decrease with increasing y + y′. The integral is, however, slowly convergent in the limit y + y′ = 0. In that case, convergence is still ensured due to the oscillating integrand cos(k_x [x – ...