The cosmic microwave background (CMB), relic radiation from the Big Bang, had very faint temperature and polarization fluctuations imprinted on it as a structure formed in the early universe. By mapping the temperature fluctuations (Δ*T*/*T*≈10^{−5}), cosmologists have learned a great deal. But in order to determine cosmological parameters uniquely and to rigorously test theories of the very early universe, the even fainter polarization fluctuations must be mapped. This demands a significant increase in sensitivity over earlier experiments and an in-depth understanding and control of telescope optics, in particular their effect on polarized signals.

In order to maximize the CMB signal against foregrounds such as the atmosphere and galactic emission, most observations are made at frequencies between 0.02 and 0.9THz, which present their own set of unique challenges. Optical design in the terahertz waveband can be challenging and difficult to do with any confidence using techniques developed for visible wavelengths, especially for high-precision applications such as CMB astronomy. The Space Optics Design and Analysis group at the National University of Ireland, Maynooth (NUI Maynooth), specializes in the field of quasi-optical design for astronomical instrumentation.^{1}

**Figure 1.** A quasi-optical telescope (the QUaD telescope) set up in MODAL. The model includes 31 corrugated conical horns at the focal plane (modeled using electromagnetic mode matching), two lenses, and a Cassegrain telescope. The fundamental Gaussian mode is used for beam visualization.

Optical design is concerned with the problem of calculating an electromagnetic field over a surface in an optical system when the field, or currents, over some other surface is known. The full solution to Maxwell's equations is usually extremely difficult to find, and in practice approximations have to be made. Although physical optics (PO) can be used to characterize electromagnetic systems to high accuracy, it is computationally intensive at terahertz frequencies and often not suitable for the initial design or preliminary analysis of large, multielement optical systems. Ray tracing, used so successfully at visible wavelengths, is generally not suitable for systems where diffraction is expected to be important. An alternative is to decompose the source field into modes, each a solution to the paraxial wave equation. Propagation of the modes is usually straightforward and simply involves recombining scaled modes with an appropriate mode-dependent phase-slippage term included. Commonly used mode sets include Gaussian beam modes, Gabor modes, and plane waves. We have found agreement between the far-field beam predictions of these methods down to −40 dB in many cases, so long as care is taken with propagation steps involving fast beams.

In general there is a lack of dedicated software tools for modeling the range of components and propagation conditions encountered in typical quasi-optical systems, and we employ a variety of commercial and in-house software packages for this task. Our ideal software package would combine several techniques, the more approximate methods allowing features such as optimization during the design stages, with a more rigorous vector PO technique for the final analysis. The ability to model lenses in a rigorous way is also becoming important. To this end we have been developing our own software analysis package, MODAL.^{2}

**Figure 2.** A Gaussian beam illuminating a Fourier grating (designed by R. K. May). The grating was designed to produce a ring of eight beams in the far field.

MODAL aims to integrate advanced modeling techniques and access to high-performance computing into a powerful and user-friendly tool for the quasi-optical designer. It provides a range of optical elements. Those implemented so far include idealized sources, corrugated conical horns, mirrors, apertures, dielectric lenses, and phase gratings. The available simulation methods are scalar diffraction integrals, vector PO, and modal propagation using Gaussian-Hermite beam modes.

The range of elements implemented to date has allowed us to accurately model two CMB telescopes with MODAL,^{2} the QUaD (Quest at Dasi) telescope and the beam combiner of the Millimeter-Wave Bolometric Interferometer (MBI). Of particular interest to us are the polarization properties of these telescopes and the understanding of any elements that are likely to introduce instrumental or cross-polarization. In the near future MODAL will be further developed to allow easy operation in batch mode to facilitate tolerance analyses. The software framework has been designed so that it is straightforward to add new types of sources and elements or new propagation methods without affecting the main program. This is important as we aim to integrate other aspects of our research at NUI Maynooth, such as modeling of standing waves in quasi-optical systems.^{3}

The precise analysis of quasi-optical telescopes is especially important for CMB measurements. In general, we have found that more than one analysis technique is required during the course of a project, and thus we have tailored our own software package, MODAL, to this very specific requirement of optical design at terahertz frequencies.

*We acknowledge the support of all the members of the QUaD and MBI collaborations.*

Creidhe O'sullivan

Experimental Physics,

National University of Ireland, Maynooth

Maynooth, Ireland