Help needed with Polarisation (Rotated Waveplates)

[catedral3000]

Homework Statement

a) Plot, as a function of time, the horizontal and vertical components of
E(0,t) for 0 ≤ ωt ≤ 2π. Plot the trajectory of the point (Ex,Ey) for a constant z as a function of time.
A_H = 1, A_V = 1, ϕ_H = 0, ϕ_V = π/4
b) How would you convert the polarization pattern in part (a) to horizontal using
one half-and one quarter-wave plates? What should be the angles of both plates?

Relevant Equations

n/a

(a) The polarisation pattern is elliptical with maximum (1,1) and minimum (-1,-1), and anticlockwise in direction.

(b) I know the solution is a quarter-wave plate oriented π/4, and half-wave plate at π/16, but don't understand how to reach there. I've obtained the polarisation vector (cos π/8, isin π/8) so far.

I can't find much online guidance or textbook material working through this topic, so I'd appreciate any help I can get. Also, if anyone could let me know where I can get more practice on these kinds of problems that'd be great!

 

[kuruman]

Can you plot the trajectory of point (Ex,Ey) as suggested in part (a)? What shape do you get?
If so, can you do the same for for part (b)? What shape do you get in this case?

Also, please provide the statement of the problem as was given to you. What comes before parts (a) and (b)?

 

[catedral3000]

Can you plot the trajectory of point (Ex,Ey) as suggested in part (a)? What shape do you get?
If so, can you do the same for for part (b)? What shape do you get in this case?

Also, please provide the statement of the problem as was given to you. What comes before parts (a) and (b)?

I've now edited the original problem so it is complete - that all the information given though!

For part (a) I get an elliptical shape with maximum (1,1) and minimum (-1,-1), and anticlockwise in direction.

Part (b) is asking to convert this to horizontal polarisation using 1/2 and 1/4 wave plates.

 

[kuruman]

You know that the phase difference between the two components is $\frac{\pi}{4}.$

1. What phase difference do you need in order to get linear polarization?
2. What does a half-wave plate do to the phase difference? How would it change the trajectory?
3. What does a quarter-wave plate do to the phase difference? How would it change the trajectory?
4. What does combining the two plates do to the phase difference?

 

[TensorTronic-270]

“Geometric structural address for maximum stability state— the only unit-circle chart you need to solve every single wave-plate polarisation problem in 10 seconds flat.”

Then overlay the two red dots:

• One at π/8 (22.5°) → major axis of the ellipse
• One at π/16 (11.25°) → half-wave plate angle
• 

 

[kuruman]

“Geometric structural address for maximum stability state— the only unit-circle chart you need to solve every single wave-plate polarisation problem in 10 seconds flat.”

Then overlay the two red dots:

• One at π/8 (22.5°) → major axis of the ellipse
• One at π/16 (11.25°) → half-wave plate angle
• View attachment 367890

Where did you get this stuff?
What does it mean in relation to your question?
What red dots?

 

[catedral3000]

You know that the phase difference between the two components is $\frac{\pi}{4}.$

1. What phase difference do you need in order to get linear polarization?
2. What does a half-wave plate do to the phase difference? How would it change the trajectory?
3. What does a quarter-wave plate do to the phase difference? How would it change the trajectory?
4. What does combining the two plates do to the phase difference?

1. π
2. π, reflects in the fast axis
3. π/2, linear to cicular and vice versa
4. If their fast axes are aligned, would produce total phase difference 3π/2

 

[kuruman]

OK, let's backtrack and write some equations. For part (a) you can write $$\mathbf E=\left[\sin\left(t+\frac{\pi}{4}\right),\sin(t)\right].$$ A parametric plot gives the figure on the right which is what you have shown in post #3.

What equations will you get if you line up:
1. The quarter-wave plate with the x-axis?
2. The quarter-wave plate with the y-axis?
3. The half-wave plate with the x-axis?
4. The half-wave plate with the y-axis?

Does any one of these give you a phase difference of $\pi$ or zero which would mean linear polarization?

 

[Steve4Physics]

For part (a) I get an elliptical shape with maximum (1,1) and minimum (-1,-1), and anticlockwise in direction.

I
___________________________
@catedral3000, it’s worth addressing some problems with your answers to part a).

From the first graph it appears that you are using cosines. On this basis...

For convenience take $\omega =1$ and $z=0$, then the question effectively specifies that $E_H = \cos (t)$ and $E_V = \cos (t + \pi/4)$.

In this case:
- the first graph is correct;
- the second graph is wrong; it is a graph of $E_V = \cos (t + 3\pi/4)$;
- the direction on the third graph should be clockwise (e.g. find the points on the ellipse corresponding to t=0 and t=[a little bit bigger than 0]).

Also, (1,1) and (-1,-1) are not the ‘maximum’ and ‘minimum’ of this ellipse.

Minor edits,