For bore radius (a), the transverse wavenumber is
[ k_c=\frac{x_{mn}}{a}. ]
For TE modes, (x_{mn}) is the (n)-th nonzero root of (J_m’(x)=0). For TM modes it is the (n)-th root of (J_m(x)=0). The cutoff frequency and propagation constant are
[ f_c=\frac{c x_{mn}}{2\pi a},\qquad \beta=\sqrt{k_0^2-k_c^2}. ]
The code evaluates both cosine and sine angular orientations, (\cos(m\phi)) and (\sin(m\phi)), when applicable.
The phasor convention is
[ \exp(+j\omega t-j\beta z). ]
For a TE mode, (H_z=\psi) and (E_z=0):
[ \mathbf E_t= \frac{j\omega\mu}{k_c^2}\hat{\mathbf z}\times\nabla_t\psi,\qquad \mathbf H_t=-\frac{j\beta}{k_c^2}\nabla_t\psi. ]
For a TM mode, (E_z=\psi) and (H_z=0):
[ \mathbf E_t=-\frac{j\beta}{k_c^2}\nabla_t\psi,\qquad \mathbf H_t=-\frac{j\omega\epsilon}{k_c^2} \hat{\mathbf z}\times\nabla_t\psi. ]
The forward modal power is calculated numerically:
[ P_m=\frac{1}{2}\operatorname{Re} \iint_A(\mathbf E_m\times\mathbf H_m^*)\cdot\hat{\mathbf z}\,dA. ]
If (w_L) is the 1/e field radius on a thin lens of focal length (f_L), the diffraction-limited paraxial waist is
[ w_0(f)=M^2\frac{\lambda f_L}{\pi w_L}. ]
The default uses (M^2=1) and (w_L=D/2=25.4) mm. The aperture-plane field is
[ \mathbf E_{\mathrm{inc}}(r) =\hat{\mathbf x}\exp\left(-\frac{r^2}{w_0^2}\right). ]
The Gaussian power fraction intercepted by the radius-(a) bore is
[ T_{\mathrm{ap}}=1-\exp\left(-\frac{2a^2}{w_0^2}\right). ]
The forward modal amplitude is estimated by power orthogonality:
[ a_m= \frac{\iint_A (\mathbf E_{\mathrm{inc}}\times\mathbf H_m^*)\cdot\hat{\mathbf z}\,dA} {2P_m}. ]
The launched modal power is (|a_m|^2P_m). A small energy guard removes finite-grid quadrature excess if the sum of projected modal power is slightly larger than the geometrically admitted Gaussian power.
This is an aperture mode-matching estimate. A rigorous discontinuity solution would include reflected free-space fields and evanescent waveguide modes.
For conductivity (\sigma), the good-conductor surface resistance is
[ R_s=\sqrt{\frac{\pi f\mu_0}{\sigma}}. ]
The wall power dissipated per unit propagation length is
[ P_{\mathrm{wall}}= \frac{R_s}{2}\oint_{\partial A}|\mathbf H_t|^2\,ds. ]
For field dependence (\exp(-\alpha z)),
[ \alpha=\frac{P_{\mathrm{wall}}}{2P_m}. ]
The power after guide length (L) is
[ P_m(L)=P_m(0)\exp(-2\alpha L), ]
and the corresponding power loss is
[ \mathcal L_{\mathrm{dB}}= \frac{20}{\ln 10}\alpha L. ]
For an ideal air-filled guide,
[ v_g=c\sqrt{1-\left(\frac{f_c}{f}\right)^2}. ]
The reported delay is relative to free-space propagation over the same length:
[ \Delta\tau=L\left(\frac{1}{v_g}-\frac{1}{c}\right). ]
At each selected frequency the output phasor is
[ \mathbf E_{\mathrm{out}}(x,y;f)= \sum_m a_m\mathbf E_m(x,y;f) \exp[-\alpha_mL-j\beta_mL]. ]
Because distinct modes retain their propagation phase, the transverse output pattern includes coherent multimode interference.
coupling_fraction_full_beam includes aperture clipping and mode overlap.coupling_fraction_aperture is referenced only to power entering the bore.copper_loss_db is wall propagation loss, not total experimental insertion
loss.output_power_fraction is referenced to the full Gaussian before the
waveguide aperture.