 # Introduction

In the mm-wave, the coupling between the E-field and the detector is often performed through a waveguide, or feed horn, which strongly selects a defined field distribution. This can be very beneficial for eliminating stray light, but is also important to model and characterize so that high efficiency in coupling to the correct modes of the input field are obtained. This type of analysis should be performed in tandem with the first order design of the instrument, since the two are interdependent.

A seminal reference on these types of calculations is Quasioptical Systems Gaussian Beam Quasioptical Propagation and Applications by Goldsmith.

In this reference, the field at the output aperture of the feed horn is derived for multiple styles of horns. To use this information to determine how well an optical system will couple the incident light to the detector, we must perform an overlap integral of the feed field and the optics field. A subtlety to performing the overlap integral is that the optics field is typically calculated at the focus of the optical system, but the focus is not placed at the feed horn aperture, instead it is placed at the phase center of the feed horn internally. To accommodate this shift, either the feed field or the optics field must be propagated to the common plane of evaluation.

In this post, we will employ the technique of angular spectrum propagation to the feed horn field, thereby calculating the field at the same plane as the optics field. Once we have the two fields, an overlap integral is performed to calculate the coupling. The coupling integral is dependent on wavelength as well as the spatial behavior of the fields.

# Feed Horn Aperture Fields

Some common feed horn architectures are:

• EH_{11} Mode in Hollow Circular Dielectric Waveguide
\begin{align} E(r) &= J_0 \left( \frac{2.405 r}{a} \right) \qquad &r \le a \\ &0 &r > a, \end{align}

where r is the radial distance in the aperture of the guide, J_0 is the Bessel function of zeroth order, 2.405 is the first zero of J_0, and a is the radius of the waveguide aperture.

• Corrugated Conical Waveguide
E_{ap}(r) = J_0 \left( \frac{2.405 r}{a} \right) \exp \left( \frac{-i \pi r^2}{\lambda R_h} \right) \hat{y}

where \lambda is the wavelength of the radiation, R_h is the slant length of the feed horn, i.e. the effective radius of curvature of the cone that defines the waveguide. The slant length is related to the aperture radius by a = R_h \sin \alpha where \alpha is the horn opening half-angle.

• Smooth-Walled Conical Waveguide
\begin{align} E_y(r) &= \frac{J_1(ur/a)}{ur/a)} \sin^2\phi + J'_1(ur/a) \cos^2\phi \\ E_x(r) &= \left[ \frac{J_1(ur/a)}{ur/a} - J'_1(ur/a) \right] \sin \phi \cos \phi \end{align}

where u = 1.841, r and \phi define the aperture coordinate system, J_1 is the Bessel function of the first order, and J'_1 is its derivative. To account for a slant length of R_h, the fields would be multiplied by a spherical wave phase factor \exp \left[ -i \pi r^2 / \lambda R_h \right]. Note that since the waveguide does not have a single linear polarization, there will be a polarization coupling factor.

# Optics Fields

The field produced by an imaging optical system can be very complex, but in this analysis we will only consider the effects of diffraction, i.e. the aberrations are sufficiently well controlled that they are negligible. In the mm-wave, this can be fairly readily realized due to the long wavelengths. The most common aperture employed is a circular one, which leads to an Airy disk. In the case of an elliptical aperture, which is common in off-axis systems, the output is an elliptical Airy. This more general formulation is

E = 2\pi \epsilon a^2 C \frac{J_1(\kappa)}{\kappa}

where C = \frac{\exp \left[ikf \right]}{ikf} \exp \left[\frac{ik}{2f} \left(x^2 + y^2 \right) \right],
\kappa = \frac{k}{f} \epsilon a \sqrt{ \frac{x^2}{\epsilon^2} + y^2},
\epsilon = b / a,
k = \frac{2\pi}{\lambda},
\lambda is the wavelength of radiation, f is the focal length of the system, 2a and 2b are the major and minor diameters of the aperture, and x and y are the aperture coordinate system.

# Angular Spectrum Propagation

The technique of angular spectrum propagation is powerful because it does not make any significant approximations to the transfer function of free space. This means that the results are valid at any distance from the input plane. To use the angular spectrum propagator, we multiply the Fourier transform of the input field by the transfer function of free space

\exp \left[2\pi i \gamma z \right]

where \gamma = \sqrt{ \frac{1}{\lambda^2} - \left( \xi^2 + \eta^2 \right) }, and z is the distance to propagate along the z-axis. To obtain the propagated field, we then compute the inverse Fourier transform

E = \mathcal F^{-1} \left[ \mathcal F \left[ E_0 \right] \exp \left(2\pi i \gamma z \right) \right] , where E_0 is the input field and E is the propagated field.

# Coupling Integral

The coupling integral between the optics field and feed horn field is performed at the same plane in space. It is the normalized 2D integral over the plane of the two fields multiplied together to account for any mismatch.

\iint \frac{E_{opt} E^*_{feed}}{\sqrt{ \iint E_{opt} E^*_{opt} * \iint E_{feed} E^*_{feed} } }

The coupling should be evaluated at all wavelengths that are of interest since the optics and feed horn fields depend on the wavelength. The spectral behavior can be significant.