Spectrometer design guide

This spectrometer design guide provides some simple and easy to use design guidelines and formulas for designing, evaluating and comparing various diode array, diffraction grating based spectrometers designs.

### Introduction

The input to the design process is the wavelength range, you want to cover, and the optical resolution, by which you need to resolve the various structures in your spectrum (often peaks).

### How a diode array spectrometer works

Basically, a spectrometer is an optical system consisting of two lenses or mirrors that produce an image of the input slit on the detector. In between the lenses or mirrors, a diffraction grating is placed that disperses different wavelengths from different angles. This causes different wavelengths of light entering the input slit to be imaged to different positions on the detector array.

Below, two common spectrometer geometries are illustrated; the transmission grating based and the Czerny-Turner. Also, the figures define the key design parameters of a spectrometer.

### Czerny-Turner

### Transmission grating based

### Flow chart

### The Spectrometer guide 8 steps

**Step 1: Choose the geometry**

The first step is to choose between the Czerny-Turner or LGL type geometry. For the Czerny Turner a typical value for Θ is around 30° wheras transmission gratings are generally used in the Littrow configuration and –1^{st} order where α = -β => Θ = 0°.

**Step 2: Choose grating**

The second step is to choose a diffraction grating. Most grating vendors have an on-line catalog where you can find one or more grating options to try in your design. You should choose a grating that has high diffraction efficiency in your wavelength. The important parameter that you shall use for the design in the next steps is the groove density *G*.

**Step 3: Calculate diffraction angle**

The angle of incidence α on the grating and the diffraction angle β for the center wavelength λ_{c} are key parameters in the spectrometer design. These angles can be calculated once the grating groove density *G* and the total deflection Θ are chosen.

**Step 4: Choose detector**

The purpose of the spectrometer design is to disperse the wavelength range across the width of the detector array *L*_{D}. There is a large range of diode array detectors specifically designed for spectrometers. In general, if you need a compact spectrometer you should aim for a short detector (typically 1/4” or 6.4 mm). However, if you require a broad spectral range and/or a high resolution you should aim for a wide detector (typically 1/1” or 25.8 mm).

**Step 5: Calculate focal length of focus lens**

Once the width of the detector is known you can calculate the focal length of the focusing mirror or lens.

**Step 6: Choose a magnification**

As mentioned earlier, the spectrometer is imaging the input slit to the detector and we generally want to have the slit as wide as possible to collect as much light through the input slit as possible. Therefore, the magnification in the system Μ should preferably be close to 1 which means that the width of the input slit ideally is imaged 1:1 onto the detector array.

**Step 7: Calculate focal length of collimation lens**

As in any imaging system, the magnification is determined by the ratio between the focal lengths of the two lenses in the system. For a spectrometer, this ratio has to be slightly modified due to the deflection along the beam path in the grating. However, once the magnification is chosen the focal length of the collimation mirror or lens can easily be calculated.

**Step 8: Calculate input slit width**

The input slit width ω_{slit} is determined by the required optical resolution Dl and the magnification. Once you know your input slit width you are ready to evaluate if your spectrometer design is viable or you have to go back and change some of your choices for grating, detector, or magnification for instance.

### Evaluation of design

Once you have done a design iteration using the 8 steps described in the previous pages you should check whether this design is practical at all. Three things that you can easily check are **the input slit width, diffraction limit of optics, and diffraction limit of grating** as described below.

### Is the input slit width practical?

Input slits come in widths down to 5 microns but such narrow slits will only allow a very limited amount of light to enter your spectrometer. So, if your design requires a slit of 5 – 10 microns or less you could consider the following:

- loosening your requirement for the resolution
- choosing a wider detector and choosing a grating with a higher groove density

### Diffraction limit of optics

The formulas on the previous pages do not take into account that the optics can never produce a spot smaller than the diffraction limit. You can use the following formula to calculate the FWHM in wavelength of the smallest possible spot your optics can produce:

If this value is larger than your required Δλ, your system is diffraction limited by the optics and you will not be able to obtain a better resolution than Δλ_{diffraction}

### Diffraction limit of grating

The grating itself also has a diffraction limited spot size (referred to as resolving power of the grating). The more grating lines are illuminated in a grating, the better is the resolution of the grating. The following formula gives the FWHM in wavelength of the smallest possible spot your grating can produce:

If this value is larger than your required Δλ, your system is diffraction limited by the grating and you will not be able to obtain a better resolution than Δλ_{diffraction}

## Want to know more?

For further information see below.