Question about Light incident at an angle on a diffraction grating.

[Dgray101]

Homework Statement

Show that the equation mλ=dsinθ becomes mλ=d[ sin(θ-κ) + sin(κ) ] when the light is incident on the diffraction grating at an angle κ to the normal.

Homework Equations

The Attempt at a Solution

I am not quite sure of the answer (we are just learning this in class and I am doing practice problems to help me get a better understanding) I have tried working it out but I can't seem to understand :S :(

 

[mfb]

I have tried working it out but I can't seem to understand :S :(

Then show your work please, so we can see what went wrong.

 

[Dgray101]

Okay I'll explain what I have done. The problem with this is, I haven't done this kinda of physics in so long... so my understanding might be off entirely but...

When the light comes in incident at some angle η, there must be a diffracted wavelength of θ' .
However because it is incident at an angle, I think that the diffraction is less then if it hit the grating perpendicular. So there is some relation between the incident and diffracted wavelength, much like there is between incident light and refracted light in different mediums.

Okay so... I am thinking that if the angle is to be diffracted at some θ' then it must be related somehow to the diffracted angle when light is perpendicular to the grating...

So I think that the diffractions should be located in the same spot. But now the angle would sin(θ-η)
But if the locations would be the same, you would have to add the sin of the incident angle η?

 

[mfb]

What is a "diffracted wavelength"?

I think that the diffraction is less

What does that mean?

So there is some relation between the incident and diffracted wavelength

The wavelength does not change.

It is all about length differences. You have to consider both sides here.

Okay so... I am thinking that if the angle is to be diffracted at some θ' then it must be related somehow to the diffracted angle when light is perpendicular to the grating...

So I think that the diffractions should be located in the same spot.

There is some relation, but the angles are not the same.

 

[Nickie]

I would approach this question by first understanding the concept of diffraction grating and how it works. A diffraction grating is a device with a large number of parallel slits or grooves, which are evenly spaced and act as a waveguide for light. When light passes through a diffraction grating, it gets diffracted or spread out into different directions, creating a pattern of bright and dark fringes.

Now, to answer the question, we need to consider the geometry of the diffraction grating. When light is incident on the grating at an angle κ, the normal to the grating is now at an angle of (90-κ) to the direction of the incident light. We can use this information to rewrite the equation mλ=dsinθ as mλ=dsin(90-κ). Using trigonometric identities, we can simplify this to mλ=dcosκ.

Next, we need to understand the significance of κ in this equation. κ represents the angle between the normal and the direction of incident light. This angle is also the angle of diffraction, which is the angle at which the diffracted light emerges from the grating. We can use this information to rewrite the equation as mλ=d(cos(θ+κ)).

Finally, we can use the trigonometric identity cos(θ+κ)=cosθcosκ-sinθsinκ to further simplify the equation to mλ=d(cosθcosκ-sinθsinκ). And since cosθ represents the horizontal displacement of the diffracted light, we can rewrite the equation as mλ=d(cosθ)-d(sinθsinκ). This can also be written as mλ=d(sin(π/2-θ))-d(sinκsin(π/2-θ)).

Therefore, we can conclude that mλ=d(sin(θ-κ)+sinκ) when the light is incident on the diffraction grating at an angle κ to the normal. This equation takes into account both the angle of incidence and the angle of diffraction, and helps us understand the behavior of light as it passes through a diffraction grating.