# Introduction

In this post we will consider the first order design and some practical limitations of a reflective grating spectrometer in the Czerny-Turner configuration used in an instrument on a radio telescope to couple to a feed horn bank on top of absorbers.

Some background information is needed to be able to make informed decisions when choosing the parameters of the spectrometer since most choices involve trade-offs or are limited by manufacturing.

### Spectrometer Configuration

The grating spectrometer we choose to analyze is a Czerny-Turner configuration, which consists of a slit, collimating mirror, reflective grating, camera mirror, and feed horn array. The spectrometer slit is placed at the focal plane of a telescope with focal length f_{tele} and entrance pupil diameter D_{tele}. The collimating mirror of focal length f_{col} creates a pupil of size D_{G}, which is the location of the grating. The grating is illuminated at an angle \alpha, which creates an elliptical pupil. The camera mirror of focal length f_{cam} focuses the diffracted wavelengths and reflected spatial field points to the feed horn array, which couples the E-field to the absorber.

### Chromatic Resolving Power

A Grating spectrometer has a chromatic resolving power R defined by

R_{G} = m\text{N} = mT\frac{D_G}{\cos\alpha} = \frac{\lambda_{0}}{\delta \lambda},

where N is the number of illuminated grating lines, m is the diffraction order, T is the grating frequency (lines/\mu m), \lambda_{0} is the center wavelength of the band, and \delta \lambda is the diffraction-limited resolvable wavelength shift.

### Dispersion

A reflective grating diffracts incident light according to the grating equation

\theta_{G} = \sin^{-1}\left[m\lambda T + \sin\alpha \right]

which produces a spectral half-field-of-view (HFOV) of

2 \theta_\lambda = \sin^{-1}\left[m\lambda_2 T + \sin\alpha \right] - \sin^{-1}\left[m\lambda_1 T + \sin\alpha \right],

where \lambda_2 and \lambda_1 are the maximum and minimum wavelengths in the spectral bandwidth, respectively. It also relays the spatial HFOV \theta_x from the collimator unchanged via

\text{L}_{\text{slit}} = 2 f_{col} \tan\theta_x,

where \text{L}_{\text{slit}} is the length of the entrance slit to the spectrometer.

### Plate Scale

The camera mirror sets the plate scale on the focal plane based on the spectral and spatial FOV, where the length in the respective dimensions is

\text{L}_\lambda = 2 f_{cam} \tan\theta_\lambda

\text{L}_x = 2 f_{cam} \tan\theta_x.

The focal ratio (F) is different in the spatial and spectral dimensions due to the projection of the pupil on the grating, and the angle of diffraction off of the grating \theta_{G}

F_\lambda = \frac{f_{cam}}{D_G} \frac{\cos \alpha}{\cos \theta_{G}}

F_x = \frac{f_{cam}}{D_G} \cos \beta

where \cos \beta is the angle of incidence in the spatial dimension.

We can then express the spatial extent of the focal plane in the two dimensions in units of F \lambda

\tilde{\text{L}}_\lambda = \frac{\text{L}_\lambda}{F_\lambda \lambda} = \frac{2 f_{cam} \tan\theta_\lambda}{F_\lambda \lambda}

\tilde{\text{L}}_x = \frac{\text{L}_x}{F_x \lambda} = \frac{2 f_{cam} \tan\theta_x}{F_x \lambda}.

### Slit Length

The slit length is set by the magnification m of the spectrometer, which is the ratio of the camera to collimator focal length, and the spatial extent of the focal plane array L_{x}

m = \frac{f_{cam}}{f_{col}}

\text{L}_{x} = m \text{L}_{\text{slit}}

### Detectors

A typical mm-wave instrument will employ feed horns, which are a metal structure that acts as a waveguide for the E-field, coupling an absober to specific modes. The feed horn structures are physically large and require significant spacing to accommodate their size, which leads to the spectrometer chromatic resolution being limited by the pitch of the detector, not the grating.

Given a feed horn array with a pitch of \Delta\text{p}_x and \Delta\text{p}_\lambda, respectively, the focal plane is sampled with \text{N}_{x} spatial pixels and \text{N}_\lambda spectral pixels

\text{N}_{x} = \frac{\text{L}_x}{\Delta\text{p}_x}

\text{N}_{\lambda} = \frac{\text{L}_\lambda}{\Delta\text{p}_\lambda}.

In units of F\lambda, the horns are spaced by

\Delta\tilde{\text{p}}_x = \frac{\Delta\text{p}_x}{F_x\lambda}

\Delta\tilde{\text{p}}_\lambda = \frac{\Delta\text{p}_\lambda}{F_\lambda\lambda}.

### Telescopes

An instrument uses the collection capability and resolution afforded by the larger aperture of a telescope to enable its science goals. The diffraction limit of a telescope is given by

\theta_{res} = \frac{\lambda_0}{D_{tele}},

which means that over the length of the slit, there are N diffraction limited beams given by

\text{N} = \frac{\text{L}_\text{slit}}{f_{tele} \tan \theta_{res}},

resulting in a diffraction limited HFOV of

\theta = \frac{\text{N}}{2} \theta_{res}.

# First Order Design

### Driving Requirements

The achieved spectral resolution of a grating spectrometer is typically the biggest driving requirement in the first order design. Since most are also pitch-limited, this sets a requirement on the number of pixels needed in the detector.

When feed horn arrays are employed, a typical requirement is that the detectors are spaced by F \lambda or 2 F \lambda to achieve optimal coupling to a point source [1]. This requirement in combination with the chromatic resolution defines the size of the focal plane array. As an example, at \lambda = 500 \mu m, and F = 4, the feed horns are spaced by 1 - 2 mm. With a reasonable number of spectral pixels, on the order of 50 - 100, the spectral length of the detector is defined.

The large focal plane along the spectral dimension drives the grating to need to produce a wide range of diffracted angles, or the use of a very long focal length camera mirror. This outcome presents a significant challenge to the designer due to the spatial constraints involved when using reflective optics. To have unobscured optics, care must be employed when trading all the parameters in the first order design. The grating pitch T, angle of incidence \alpha, and diameter are the three knobs to tune to achieve the desired layout and optical behavior.

The gratings found in mm-wave spectrometers are typically fabricated by direct machining, which means that designing a custom grating is standard. This is not the case in other wavelength regimes where manufacturing a grating is difficult. This is why we have the flexibility to tune the pitch.

A further consideration of the spatial extent of the instrument must be given because typical mm-wave instruments must be chilled to cryogenic temperatures of 4K, and then even colder at the actual detector level, down to around 250 mK. To achieve this cold environment, a large cryostat must be built, which imposes physical constraints on the size and position of the optics. Therefore, the packaging envelope of the system is crucial to understand when laying out the system.

### Design Process

Given the above background information and understanding of the driving requirements, the overall process for setting the first order design of the spectrometer is feasible. To fully specify the system, the telescope parameters must be known, which are typically defined by the project. The number of spatial and spectral detectors, and their pitch must also be set, which can be influenced by manufacturing, but can also be flexible within the optical design. The diffraction order of the grating is typically set at the \pm 1 order. Once these inputs are determined, the optical first order layout can commence.

Choosing the grating pitch T, angle of incidence \alpha, and the pupil diameter D_G completely determine the system along with the requirement on the detector spacing in F \lambda units. Choosing the exact values of these three parameters is typically done in an iterative manner because they impact the physical layout as well as the optical behavior. Therefore, a modelling tool is useful to have on hand.

# Citations

[1] M. J. Griffin, J. J. Bock, and W. K. Gear, *Relative performance of filled and feedhorn-coupled focal-plane architectures*, App. Opt. 41 (**31**) 2002.