Gaussian Beams in Circular Hollow Metal Waveguides

One of the most important applications of the above equation is the propagation of Gaussian beams in hollow metal waveguides. In this case, the complex ABCD matrix is

$$M_{system}= \left( \begin{array}{cc} A & B\\ C & D \end{ar... ... \right) = \left( \begin{array}{cc} 1 & z\\ 0 & 1 \end{array} \right)$$ (5)

and the input field has the form

$$E_{x,in}(r_0^{\prime}) = E_0 exp(-r_0^{\prime 2}/w_0^{\prime 2})$$ (6)

If we further define

$$w_0^{\prime} \equiv w_0/D_0$$ (7)

then the input field can be written

$$E_0^\prime (r^\prime) \equiv i2 E_0 \frac{\hat{z}_0}{z} exp(-i\beta_0 z) exp \left( -i \frac{\hat{z}_0}{z} r^{\prime 2} \right)$$ (8)

The resulting output field components are

$$E_1^\prime (r^\prime) \equiv \int_0^\infty exp \left[ -\left( \f... ...ft( 2 \frac{\hat{z}_0}{z} r_0^\prime r^\prime \right) r_0^\prime dr_0^\prime$$ (9)

and

$$E_2^\prime (r^\prime) \equiv exp \left( - \frac{1}{w_0^{\prime ... ...eft( 2 \frac{\hat{z}_0}{z} r_0^\prime r^\prime \right) r_0^\prime dr_0^\prime$$ (10)

and it only remains to solve these two integrals.