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The optical setup was built according to Figure 1. The light from an optical fiber output is condensed on a slit and the resulting light is collimated by a plano-convex lens of focal lens f_{1}. The collimated beam is apertured and eventually filtered before hitting a blaze diffraction grating which separates the different wavelengths following the two major diffraction orders +1 and -1. The +1 diffraction order is focused by a second plano-convex lens of focal lens f_{2} to image the spectrum on a camera sensor.

In this article, we will analyze how the various optical elements affect the performance of our spectrophotometer. In future articles, we will try to make some modifications and see how the new setups performs compared to the one describe here. The filter is optional for this article and will be discussed in later posts.

The overall resolution of the spectrophotometer is limited by both the diffraction grating fineness and the slit size. A diffraction grating of N lines per mil will allow a resolution of ∆λ=λ/N where λ is the centre wavelength that can be discriminated from both λ+∆λ and λ-∆λ. The slit size (ε) also limits the resolution as large slit size will image overlapping wavelengths and then produce a low-pass-like effect.

The wavelengths range is bounded by the magnification ratio of the condensing/focusing lens pair and the scattering of the diffraction grating. Obviously, the detection will be limited by the lenses and filter and by the camera sensor. UV light will be filtered by conventional glass and camera sensor will become insensitive at wavelength higher than 800-100 nm (NIR limit).

A diffraction grating hit by a beam of light with an incident angle α will deviate the light at an angle β depending on the wavelength λ following the equation:

with m the diffraction order and α, β relative to the diffraction grating normal. It is possible to rearrange the equation as:

The wavelength range λ_{min}...λ_{max} will span with an opening angle ζ of:

which will image on the sensor as a distance d:

The span is then limited by the focal length of the focusing lens and the camera sensor size since d must never exceed it.

The sensitivity of the function β(λ), that is, the overall change of output angle with an infinitesimal change of wavelength, is given by the derivative:

This allows computing the divergence of the output beam, ∆β, for a known resolution ∆λ which enables to know what the smallest image dimension ∆d on the camera sensor will be. This can be obtained by computing ∆β at λ_{min}:

Ideally, we would like to focus as much light as possible on a minimum number of pixels to increase the S/N ratio. This means that ∆d should be equal to the size of 1 pixel of our camera sensor. However, it is often considered safer to oversample a little bit and to acquire each spectrum fringe ∆λ as 2 pixels to respect Shannon theorem. With pixel size p_{size}, this means a strict relation on f_{2}:

Finally, we should adapt the slit size ε such that its image will be equal to ∆d to ensure optimal usage of optical power. We know that infinity-correct setup built from the collimating/focusing lenses pair will image ε by a f_{2}/f_{1} ratio. That is:

For my tests, I have chosen an incident angle of zero (α=0) and the +1 diffraction order (blaze diffraction grating are optimized for first diffraction orders) which simplifies the previous equations. The diffraction grating had a fineness of 1200 lines/mm and the desired wavelengths range was the visible band from 400 to 800 nm. I am using an unfiltered 1280x1024 greyscale CMOS camera with 5.2 µm pixels with an 8 bits ADC. I do not recommend colour camera as their Bayer matrix will reduce resolution and make things more complicated (colour camera usually also have smaller pixels and aggressive NIR filter).

From this, we infer that the minimal resolution of the system is about 0.8 nm (we will round ∆λ to 1 nm) and that the angular span is about 45°. This limits f_{2} to a maximum focal length of 8 mm knowing the camera sensor dimension. With a resolution of 1 nm, the minimum deviation angle ∆β_{min} is about 0.078°. With a 5.2 µm pixel size, this leads us to an optimal focal length for f_{2} of 7.6 mm. Since this is smaller than the upper boundary of 8 mm computed before, we can validate the spectral range of 400-800 nm. It would actually be difficult to make any better than this as it almost maximize the sensor area with the smallest ray images recommended.

Unfortunately, we were not able to find lenses with such small focal lengths and the camera objective tested did not give satisfactory results. It seems that, in my experience, camera or DSLR objectives do not behave as perfect single lenses and cannot be used for infinity corrected optical setups. As a consequence, I had to rely on a sub-optimal 30 mm stock 1" plano-convex lens which restricted the wavelength ranges to about a 150 nm span. The wavelength boundaries can be modified by adjusting the incident angle on the diffraction grating by a few degrees.

Using a stock lens of 50 mm for the collimating lens lead us to the choice of a 50 µm slit (75 µm would have been ideal but was not available at the vendor). Other combinations of slit size and focal lengths are possible but larger focal length (for larger slit size) would require larger optics to cope with the NA issue and they quickly become expensive.

The condenser lens used was an aspheric lens of short focal length (f=20 mm) used to concentrate as much light as possible from the fiber output on the slit. The distances were adjusted according to the lens-maker equation and the available space remaining on the breadboard.

Concerning the aperture, a small iris was used to increase depth of field and limit lenses aberrations. This limits the amount of light reaching the camera but without it the lights rays are like spread on their side and mix with the rest of the spectrum.

Calibration of the spectrum can be made using a sample which emits known rays, such as a neon discharge lamp for which the location of the peaks can be obtained with precisions from a source like the[∞] astrosurf neon spectrum.

A typical output of the camera sensor is given on Figure 2. Outer regions of the bitmap tend to be a bit blurry due to aspheric aberrations of the focusing lens. Careful tuning of the lenses positions can reduce it to some degree. The wavelength-intensity graph can be obtained by scanning the grey levels according to the X axis (we are then limited by the camera ADC resolution to a 1:256, 1:1024 or 1:4096 ratio depending on the bit depth). I also recommend to scan each wavelength along the Y axis on a few pixels to increase the S/N ratio through averaging (noise will decrease as the square root of the number of pixels scanned).

The resulting graph is then plotted in excel and the extraction of peaks is made through, for example, a simple threshold value and local maxima selection macro. Each ray is then annotated with the matching wavelength value obtained by a careful comparison with the reference data (this may requires some experience). Plotting the pixel-wavelength allows identification of a mathematical relationship to change the X axis from pixels to wavelengths. I recommend using polynomial or local-adaptative methods for larger wavelengths span but my 150 nm range was quite well fitted by a simple linear regression with a r^{2} factor of 0.9998.

The final spectrum for a Neon discharge lamp is given on Figure 3. Despite the feint light of the lamp, the peaks are clearly visible and it shows a relatively good separation of rays, including the 638.299 nm and 640.225 nm ones which are distant of less than 2 nm.

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