Laser refraction through curved surface and water

[bregazzi]

Hello everyone, hope you can help me! I am doing research involving illumination of an object in a cylindrical acrylin tube filled with water using a Nd:Yag laser.

Homework Statement

I am trying to figure out the refraction of the laser as it passes from air through the curved acrylic then on into the water.
See attached image ;)

Homework Equations

So far I have tried to apply Snells law in several different variations but cannot seem to make sense of the results. Ideally I want to be a ble to tabulate the affect of the laser entering at any angle from 0-90Deg to the acrylic surface.

The Attempt at a Solution

n for air = 1.000277
n for acrylic = 1.492
n for water = 1.3330
inner dia of tube = 100mm
outer dia of tube = 110mm
thickness of target black body 30mm

 

[bregazzi]

https://www.physicsforums.com/showthread.php?t=402031

Is this the direction I need to be going in with this? That matrix looks a bit daunting though any sugestions as to how I shoud tackle it?

 

[Redbelly98]

Welcome to Physics Forums.

[STRIKE]If the acrylic is reasonably thin, you could ignore it and just use the refractive index of water. What is the acrylic thickness, and what is the diameter of the cylinder?[/STRIKE] (EDIT: just read Post #1 more carefully. Acrylic thickness is 10% of the radius, so ignoring it wouldn't be a terrible approximation. I'll have to think about this some more.)

Does the laser always hit the cylinder at the point directly opposite the object, and only the angle will change? Or will the location where the laser hits the cylinder change as well?

p.s. I have moved this thread out of the Homework area of the forums, since it is not homework.

 

[bregazzi]

Thanks for the reply and sory about the thread being in the wrong place...wasn't too sure where to put it!

The laser will always enter as in the diagram, perpendicular to the cylinder if that makes sense!

I want to find the effect if any the acrylic has on the beam in this position and then for other angles should the laser not be positioned correctly. I have done out a table for refraction from air to acrylic then onto water for planar surfaces but will the fact that the acrylic is not planar have a large impact on the refraction?

AIR TO ACRYLIC
Angle of Incidence Angle of Refraction
0.00 0.00
5.00 3.35
10.00 6.69
15.00 9.99
20.00 13.26
25.00 16.46
30.00 19.59
35.00 22.62
40.00 25.53
45.00 28.30
50.00 30.90
55.00 33.31
60.00 35.49
65.00 37.42
70.00 39.05
75.00 40.36
80.00 41.32
85.00 41.90
90.00 42.10

ACRYLIC TO WATER
Angle of Incidence Angle of Refraction

0.00 0.00
3.35 3.75
6.69 7.49
9.99 11.20
13.26 14.87
16.46 18.49
19.59 22.04
22.62 25.49
25.53 28.84
28.30 32.05
30.90 35.09
33.31 37.93
35.49 40.53
37.42 42.85
39.05 44.84
40.36 46.45
41.32 47.65
41.90 48.38
42.10 48.62

 

[Redbelly98]

Sorry I haven't had much time this week to look into this more. If the laser beam is going to be close to the central "optic axis" of your system, then using ray matrices would be a good way to solve the problem.

Besides being deflected, the acrylic and water could also change the amount of spread in the beam, since they act as a cylindrical lens -- especially the water.

I should have some time later to explain more about the ray matrices, if you like. You can also try googling:

"abcd matrix" optics​

 

[Redbelly98]

I'm assuming you're unfamiliar with the ray transfer matrix approach, so I'll give a a little introduction first.

At any point, a ray is characterized by two parameters: (1) it's displacement from the optical axis, and (2) the angle it makes w.r.t. the optical axis. So we write the ray in terms of the column vector,

v = \left[ \ \stackrel{y}{\theta} \ \right]​

A few things to know about the ray vector:

• The angle θ is small, so θ ≈ sinθ ≈ tanθ
• At a medium interface, the angle of the ray changes due to refraction.
• When the ray travels some distance through a medium without encountering an interface, the displacement changes but the angle does not

Any of these changes -- encountering an interface, or traveling some distance -- are described by a matrix that multiplies the ray vector. For example, over a distance D, the ray's displacement changes from y₁ to y₁+D·θ. We can write

y₂ = y₁ + Dθ₁
θ₂ = θ₁​

Or, in matrix form,

<br /> \left[ \stackrel{y_2}{\theta _2} \right] = \left[ \stackrel{1}{0} \ \stackrel{D}{1} \right] \ \left[ \stackrel{y_1}{\theta _1} \right]<br />​

Now on to your problem. A ray is incident on the acrylic with initial displacement and angle y_i and θ_i. Taking things one step at a time:

• First, the ray is refracted at the air-acrylic interface
• Then the ray travels in a straight line through the thickness of the acrylic
• Next, the ray is refracted at the acrylic-water interface
• Finally, the ray travels in a straight line through the water to the sample

Each of those steps is described by a matrix, so we can write for the final ray

v_f = M_water · M_{acrylic-water} · M_acrylic · M_air-acrylic · v_i
​

Where each M is the matrix for either a refraction or traveling in a straight line. Note that M_air-acrylic is the first matrix to operate on v_i, then M_acrylic, and so on.

That is a basic outline of how to solve this problem. For completeness I'll write the matrix for traversing a curved interface, from a medium of refractive index n₁ into a medium with index n₂, where R is the radius of curvature of the interface:

M =<br /> \left[ \stackrel{1}{\frac{1}{R}\left(\frac{n_1}{n_2}-1\right)} \ \ \ \ \stackrel{0}{\frac{n_1}{n_2}}\right]<br />​

I'm not sure how to write the matrix more neatly, but the upper-left element is 1 and the upper-right element is 0. Also, take R to be a positive value for your example.

A more complete list of matrices can be found here:

http://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis#Table_of_ray_transfer_matrices

Hope that helps; post back with questions if you have any.