# Optics question using ray matrices

• nateastle
In summary, the conversation discusses a calculation problem involving a long glass rod with a convex hemispherical surface and an object in the form of an arrow. The problem requires the use of ray matrices, with three matrices needing to be formed. The final matrix, mFinal, is computed as [m3][m2][m1]. The conversation also includes some discussion about the order of multiplying matrices and the potential for incorrect signs in the matrices.

## Homework Statement

The problem is: The left end of a long glass rod 10.0 cm in diameter, with an index of refraction 1.5, is ground and polished to a convex hemispherical surface with a radius of 5.0cm. An object in the form of an arrow 2.00mm tall, at right angles to the axis of the rod, is located on the axis 25.0 cm to the left of the vertex of the convex surface. Find the position and height of the image of the arrow formed by paraxial rays incident on the convex surface.

From this I know that R= 5cm, do = 25cm, ho=.2cm, n1 = 1, n2 = 1.5

## Homework Equations

n1/n2
-(nlarge - nsmall)/n2R

## The Attempt at a Solution

(and what an attempt it is.)
The teacher wants us to use ray matrices to figure out the answer. I figure there are 3 matrices that will need to be formed.

Here is what I have:
$$m1 = \left(\begin{array}{cc} 1 & 25 \\ 0 & 1\\ \end{array} \right)$$

$$m2 = \left(\begin{array}{cc} 1 & 0 \\ -0.067 & .67\\ \end{array} \right)$$

$$m3 = \left(\begin{array}{cc} 1 & di \\ 0 & 1\\ \end{array} \right)$$

When I multiply the 3 together I get my final result as:

$$mfinal = \left(\begin{array}{cc} 2.675 & 2.675di + 16.75 \\ -.067 & -.067di +.67\\ \end{array} \right)$$

For some reason I am not getting the right answers in my final matrix to lign up with the image matrix. Any help would be appreciated.

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What order did you use for the matrices when you multiplied them?

I multiplied [m1] [m2] I then took the result and multiplied it by [m3].

nateastle said:
I multiplied [m1] [m2] I then took the result and multiplied it by [m3].

Try [m3][m2][m1] and think about why you might want it that way

I did it that way and that is how I got mFinal. I don't know if my math is not right or if something else is not right.

nateastle said:
I did it that way and that is how I got mFinal. I don't know if my math is not right or if something else is not right.

I think you need to check it. Do you know that [m2][m1] is not the same as [m1][m2]?? I see no way you can have di in the bottom row of the final matrix. I also see no way to get any negatives.

I switched the result of [m2][m1] with where [m3] was suppose to be. Thanks for you help, so here is what I have for my mFinal:
$$mfinal = \left(\begin{array}{cc} 2.675 + di & 16.75 -.67di\\ -.067 & .67\\ \end{array} \right)$$

the negetive I have from my m2 matrix, I posted it wrong orignaly.

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nateastle said:
I switched the result of [m2][m1] with where [m3] was suppose to be. Thanks for you help, so here is what I have for my mFinal:
$$mfinal = \left(\begin{array}{cc} 2.675 + di & 16.75 -.67di\\ -.067 & .67\\ \end{array} \right)$$

the negetive I have from my m2 matrix, I posted it wrong orignaly.

The only way I can see you getting a 16.75 anywhere is from multiplying [m1][m2] in that order instead of [m2][m1]. Then [m3][m1][m2] gives your new bottom row but not your top row.

You still need to compute [m3][m2][m1]. You can do it as

{[m3][m2]}[m1] or [m3]{[m2][m1]}

Matrix multiplication is associative, but it is not commutative.

## 1. What are ray matrices in optics?

Ray matrices, also known as transfer matrices, are mathematical tools used to describe the behavior of light rays as they pass through various optical elements such as lenses, mirrors, and prisms. They allow us to predict the path of light rays and determine the properties of the image formed by an optical system.

## 2. How are ray matrices calculated?

Ray matrices are calculated by multiplying individual matrices representing each optical element in a system. The matrices are composed of values that describe the refractive index, thickness, and curvature of the optical elements. By multiplying these matrices together, we can obtain the overall transfer matrix for the system.

## 3. What is the significance of ray matrices in optics?

Ray matrices are essential for understanding the behavior of light in optical systems. They allow us to predict how light will behave, and they are used extensively in the design and optimization of optical systems such as cameras, telescopes, and microscopes.

## 4. Can ray matrices be used to correct for aberrations in optical systems?

Yes, ray matrices can be used to correct for aberrations in optical systems. By manipulating the values in the transfer matrices, we can adjust the behavior of light rays to compensate for aberrations caused by imperfections in the optical elements.

## 5. Are ray matrices limited to simple optical systems?

No, ray matrices can be used to describe the behavior of light in both simple and complex optical systems. However, for complex systems, the calculations can become quite complicated, and other techniques such as ray tracing may be used in conjunction with ray matrices to obtain more accurate results.