Question 2
Let's consider the design of the Czerny-Turner spectrograph in a bit more detail using our newfound FT understanding. For optimal performance, one typically performs
(f)/(#)
-matching to ensure that the beam passing into the spectrograph expands to properly fill the diffraction grating (e.g., the Acton SpectraPro 150 or 300 operate with an
(f)/(#)
of 4.0 ). The
(f)/(#)
equals the focal length (in this case, 150 mm ) divided by the beam diameter. Under normal operation, the first curved mirror performs a spatial Fourier transform of the entrance slit to the spatial frequency domain
/bar (v)=(\theta )/(\lambda )
at the grating, with
\theta
equaling the angle away from the optical axis and
\lambda
the wavelength of light. The second curved mirror performs the inverse spatial FT to project the diffracted light back into the spatial domain at the exit slit. To describe the diffraction process, each "tooth" in the diffraction grating can be viewed as an impulsive point source, with the diffraction angle set by the Bragg condition. Under designed operations, the expanded beam interacts with a very large number of teeth, such that we can effectively interpret the grating as a constant function in our FT description in this question.
Next, let's consider the case of a sloppy experimentalist, in which an improperly shaped beam is passed into the entrance slit. Let's assume an
(f)/(H)
of 100 is used to very gently focus the light into the spectrograph instead of the anticipated f-number of 4 . Assume a Gaussian beam with a characteristic width
x_(0)
set by the
(f)/(\pi )
to describe the gently focused source. Convert from spatial frequency, to angle, and finally to physical position on the grating. Calculate the diameter of the beam upon projection onto the diffraction grating for an
(f)/(#)1
of 100 and an entrance slit width of
100\mu m
. For a diffraction grating with 150 grooves
/mm
, with how many "teeth" of the grating is the beam interacting? In this limit, the finite number of elements contributing to diffraction within the grating becomes relevant and will affect the projection on the exit slit. Using FT pictures, quantitatively describe the beam in the spatial frequency domain immediately after interacting with the diffraction grating for monochromatic light using your basis set functions. What will be the form of the function (in summation notation) at the exit slit of the monochromator after taking the spatial FT?
Write a Matlab script to perform the summation from the previous question and plot the function at the slit. How does your answer change at the
(f)/(#)
of the incident light is changed?