A grating $30$ mm wide, with $900$ lines per millimeter, is illuminated with collimated light at normal incidence. The grating is illuminated with a wide range of wavelengths in the visible regime, from $400$ nm to $600$ nm $.$

a$)$ Calculate the diffraction angle for ordersM$=0,1$ and $2$ for the center wavelength of $500$ nm $.[10$ points$]$

b$)$ Sketch the diffraction pattern for the wavelength range, showing orders M$=0,1$ and $2.$ Within your schematic, show the two ends of your bandwidth $400$ nm and $600$ nm $.$ Which, if any, diffraction orders overlap? $[15$ points$]$

c$)$ Describe why one would want to use a blazed grating over a grating with a slit profile? $[5$ points$]$