Deriving Focal Length for a Thick Lens

[Oijl]

Homework Statement

Derive
1/f = (n-1)*( (1/r1)-(1/r2)+(((n-1)t))/(nr1r2)))

using the ray matrix method.

Homework Equations

Matrix for curved dielectric interface:
M1 = [1, 0]
[(1/r)(n1/n2 - 1), n1/n2 ]

Matrix for translation through homogeneous medium:
M2 = [1, d]
[0, 1]

The Attempt at a Solution

A ray moving through a thick lens first passes through a curved dielectric interface, then travels through the lens (translation in a homogeneous medium), and then passes through another curved dielectric interface as the ray exits the lens.

So using the ray matrix method, which is

[M11, M12] [p1] = [p2]
[M21, M22] [a1] [a2],

where p1 is the height (from the lens' axis) at which the ray enters the lens, and p2 is the height at which it exits, and
where a1 is the angle at which the ray enters the lens with respect to the lens' axis, and a2 is the angle at which the ray exits.

I say

M2M1M2 = [M11, M12]
[M21, M22]

So now I'm set with dealing with this thick lens with the ray matrix method, but what I really want is to use it to derive the focal length.

Should I be looking at the principle planes, maybe? Obviously, with this ray matrix method I have two equations, and they have the variables I want, I think, except I also have the p's and a's, and I don't know how to handle them if I want

1/f = (n-1)*( (1/r1)-(1/r2)+(((n-1)t))/(nr1r2))).
I really feel as if I'm on the right track, finding this matrix, but...

What should I be considering in order to move from my ray matrix equation to the equation for the focal length

Thanks.

 

[mark726]

To derive the equation for the focal length using the ray matrix method, you can start by considering a thin lens with a single curved surface. In this case, the matrix for the curved dielectric interface will be:

M = [1, 0]
[(n-1)/r, n]

where r is the radius of curvature of the lens surface.

Next, you can consider the translation through the lens as a sequence of two translations through homogeneous media, with distances d1 and d2. This can be represented by the following matrices:

M1 = [1, d1]
[0, 1]

M2 = [1, d2]
[0, 1]

Combining these matrices with the curved dielectric interface matrix, you can derive the overall matrix for the thin lens:

M = M2*M1*M = [1, d2][1, 0][1, 0]
[0, 1][0, 1][(n-1)/r, n]

Simplifying this matrix multiplication, you will get:

M = [1, d1+d2]
[(n-1)/r, n]

Now, using the ray matrix method, you can find the height and angle at which a ray enters and exits the lens, given the focal length f. This can be represented by the following equations:

p2 = p1 + f*tan(a1)

a2 = a1 + (p1/f)

where p1 and a1 are the height and angle at the entrance surface, and p2 and a2 are the height and angle at the exit surface.

Substituting these equations into the overall matrix for the thin lens, you will get:

M = [1, d1+d2]
[(n-1)/r, n]

which can be rearranged to get the equation for the focal length:

1/f = (n-1)*[(1/r)-(1/(d1+d2))]

This is the desired equation for the focal length of a thin lens, derived using the ray matrix method. You can then extend this derivation to a thick lens by considering multiple curved surfaces and translations through homogeneous media.