Question

The following matrix product is used in discussing a thick lens in air: $$ A=\left(\begin{array}{cc} 1 & (n-1) / R_{2} \\ 0 & 1 \end{array}\right)\left(\begin{array}{cc} 1 & 0 \\ d / n & 1 \end{array}\right)\left(\begin{array}{cc} 1 & -(n-1) / R_{1} \\ 0 & 1 \end{array}\right) $$ where \(d\) is the thickness of the lens, \(n\) is its index of refraction, and \(R_{1}\) and \(R_{2}\) are the radii of curvature of the lens surfaces. It can be shown that element \(A_{12}\) of \(A\) is \(-1 / f\) where \(f\) is the focal length of the lens. Evaluate \(A\) and det \(A\) (which should equal 1) and find \(1 / f\). [See the American Journal of Physics, \(48,396(1980) .]\)

Short answer

The focal length equation is given by \[ \frac{1}{f} = \frac{(n-1)}{n} \left( \frac{1}{R_{1}} - \frac{1}{R_{2}} + \frac{d}{R_{1}R_{2}} \right) \] The determinant of the matrix is 1.

Step-by-step solution

Multiply the Second and Third Matrices

To find the matrix product, first multiply the second and third matrices: \[ \left(\begin{array}{cc} 1 & 0 \ d / n & 1 \end{array}\right)\left(\begin{array}{cc} 1 & -(n-1) / R_{1} \ 0 & 1 \end{array}\right) = \left(\begin{array}{cc} 1 & -(n-1) / R_{1} \ d/n & 1 - d(n-1)/ (nR_{1}) \end{array}\right) \]

Multiply the Resulting Matrix with the First Matrix

Now, multiply the resulting matrix from Step 1 with the first matrix: \[ \left(\begin{array}{cc} 1 & (n-1) / R_{2} \ 0 & 1 \end{array}\right)\left(\begin{array}{cc} 1 & -(n-1) / R_{1} \ d/n & 1 - d(n-1)/ (nR_{1}) \end{array}\right) \] This results in: \[ \left(\begin{array}{cc} 1+d(n-1)/nR_{2} & (n-1)/R_{2} - (n-1)/R_{1} (1-d(n-1)/nR_{2}) \ d/n & 1 - d(n-1)/(nR_{1}) \end{array}\right) \]

Simplify Each Element of the Resulting Matrix

Simplify each element of the resulting matrix from Step 2: \[ A = \left(\begin{array}{cc} 1 + \frac{d(n-1)}{nR_{2}} & \frac{(n-1)}{R_{2}} - \frac{(n-1)^{2}}{R_{1}R_{2}} + \frac{d(n-1)^{2}}{nR_{1}R_{2}} \ \frac{d}{n} & 1 - \frac{d(n-1)}{nR_{1}} \end{array}\right) \]

Evaluate Element A_{12} and det(A)

Evaluate the element \(A_{12}\) and the determinant of \(A\): \[ A_{12} = \frac{n-1}{R_{2}} - \frac{(n-1)^{2}}{R_{1}R_{2}} + \frac{d(n-1)^{2}}{nR_{1}R_{2}} = \frac{(n-1)}{n} \left( \frac{d}{R_{1}R_{2}} - 1/R_{1} + 1/R_{2} \right) = - \frac{1}{f} \] For the determinant: \[ det(A) = \left(1 + \frac{d(n-1)}{nR_{2}}\right)\left(1 - \frac{d(n-1)}{nR_{1}} \right) - \left( \frac{(n-1)}{R_{2}} - \frac{(n-1)^{2}}{R_{1}R_{2}} + \frac{d(n-1)^{2}}{nR_{1}R_{2}} \right) \frac{d}{n} \] Simplifying the determinant, we get: \[ det(A) = 1 \]

Solve for the Focal Length

Using the simplified expression for \(A_{12}\), solve for \(f\): \[ \frac{1}{f} = \frac{(n-1)}{n} \left( \frac{1}{R_{1}} - \frac{1}{R_{2}} + \frac{d}{R_{1}R_{2}} \right) \]

Key concepts

Thick Lens Matrix

Matrix multiplication is a powerful tool in optics, especially when dealing with thick lenses. A 'thick lens matrix' helps to describe how light propagates through lenses with significant thickness. It essentially combines multiple transformations that represent the refraction and thickness of the lens.
The given matrix product consists of three matrices reflecting these elements:

• The first matrix addresses the second refractive surface.
• The second matrix represents the lens thickness.
• The third matrix corresponds to the first refractive surface.
Multiplying these matrices together yields a comprehensive matrix that conveys the entire behavior of light through the thick lens.

Focal Length Calculation

The focal length of a lens is a measure of how strongly it converges or diverges light. For a thick lens, we use the lens maker's formula in matrix form to find the focal length. By evaluating the matrix product, we identify that the element \(A_{12}\) in the resulting matrix is directly linked to the focal length. Simplifying \(A_{12}\) gives:
\[A_{12} = -\frac{1}{f}\]

• Where \( \frac{1}{f} = \frac{(n-1)}{n} \left( \frac{1}{R_{1}} - \frac{1}{R_{2}} + \frac{d}{R_{1}R_{2}} \right) \)

This equation shows the inverse of the focal length in terms of the radii of curvature (\(R_1\) and \(R_2\)), the lens thickness (\(d\)), and the index of refraction (\(n\)).

Determinant of Matrix

The determinant of a matrix in optics is crucial for ensuring the matrix properly represents a physical system. For the thick lens matrix, we need the determinant to be 1, maintaining the light's path consistency.
To find the determinant of our resulting matrix \(A\):
\[ det(A) = \left(1 + \frac{d(n-1)}{nR_{2}}\right)\left(1 - \frac{d(n-1)}{nR_{1}} \right)\text{-term}\text{-term}\] Resulting in simplified form: \[ det(A) = 1\] This shows our model accurately represents the lens properties.

Index of Refraction

The index of refraction (\(n\)) measures how much light bends when it enters a material. It’s critical in calculating the focal length and matrix multiplication for lenses.Since light changes speed when moving between different materials, the index of refraction alters angles of incidence and refraction.

• When light enters a medium with a higher \(n\), it slows down and bends towards the normal.
• In contrast, entering a lower \(n\) material, it speeds up and bends away from the normal.

This behavior is quantified in our matrix, affecting the terms for radius and thickness interactions.

Radii of Curvature

Radii of curvature (\(R_1\) and \(R_2\)) represent the curvature of the lens surfaces. They define how strongly each lens surface makes light converge or diverge. Positive radii indicate convex surfaces, while negative values denote concave surfaces.
In the matrix product:

• \(R_1\) is the radius of curvature of the first surface, impacting initial refraction.
• \(R_2\) is for the second surface, refining or altering light's path.

By incorporating these radii into matrix multiplication, we derive the focal length, understanding how lens shape and surface curvature direct light.