# Spectrometer slit width

1. Homework Statement
A visible light spectrometer has a grating of width D=10cm and a collimating lens of focal length L=1m. Show that the optimal width for the input slit is such that it will produce a diffraction pattern of width D.

2. Homework Equations
The diffraction pattern of a single slit of width b is given by $$I*Sinc(\beta)$$

Where $$\beta = \frac{kb}{2}$$

k is the wave number.

3. The Attempt at a Solution
I really have no idea where to begin here. I understand that the purpose of a collimator lens is to send parallel rays of light to the diffraction grating, so wouldn't that mean it would require an image of 0 width at it's focal point? So shouldn't the slit be made as small as possible?

The only other thing I can think of is this has to do with the dependence of the diffraction pattern on the wavelength of light. Is the slit supposed to diffract the light before it reaches the diffraction grating or something?

I'm completely lost and have been going in circles for hours so any help would be appreciated.