Light incident on a grating for spectroscopy

[KaiserBrandon]

Homework Statement

if light incident on a grating makes an angle phi with the normal of the grating, show that the equation m(lambda)=d(sin(theta)) becomes m(lambda)=d[sin(theta-phi)+sin(phi)]

Homework Equations

m(lambda)=d(sin(theta)) d=spacing of slits, theta=angle displaced from normal of the grating
m(lambda)=d[sin(theta-phi)+sin(phi)]

The Attempt at a Solution

I've been working on trying to get a solution to this equation for quite some time now, and the only thing that makes sense is the equation m(lambda)=d*sin(theta-phi). don't know where +sin(phi) comes from.

 

[Delphi51]

I don't get either of those answers!
Maybe we can put our heads together and sort out our differences!

 

[kerimek]

I would first verify the given equation m(lambda)=d(sin(theta)) by checking its derivation and ensuring that all the necessary assumptions and approximations have been made. Then, I would proceed to solve the given problem by considering the geometry of the situation.

When light is incident on a grating, it is diffracted into different orders (m) depending on the angle of incidence (theta) and the spacing of the grating (d). This can be described by the equation m(lambda)=d(sin(theta)). However, if the angle of incidence is not directly perpendicular to the grating, but at an angle phi, the light will be diffracted at an angle of theta-phi. Therefore, the equation becomes m(lambda)=d(sin(theta-phi)). This explains the first part of the given solution.

Now, we need to consider the effect of the angle phi on the diffracted light. At an angle of phi, the light will also be diffracted, but at an angle of theta. This is because the angle of incidence is now perpendicular to the grating, and therefore the light will not be diffracted further. This explains the second part of the given solution, d[sin(phi)].

In summary, the equation m(lambda)=d[sin(theta-phi)+sin(phi)] takes into account the overall diffraction of the light at an angle phi from the normal of the grating. It is important to note that this equation is an approximation and may not hold true for all cases. Further analysis and experimentation may be needed to fully understand the effects of incident angle on grating spectroscopy.