Question

The following matrix product is used in discussing a thick lens in air: $$ \mathrm{A}=\left(\begin{array}{cc} 1 & (n-1) / R_{2} \\ 0 & 1 \end{array}\right)\left(\begin{array}{cc} 1 & 0 \\ d / n & 1 \end{array}\right)\left(\begin{array}{cc} 1 & -(n-1) / R_{1} \\ 0 & 1 \end{array}\right) $$ 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 \(\mathrm{A}\) 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 Am. J. Phys. 48, 397-399 (1980).]

Short answer

The matrix product \( A \) and its determinant are evaluated to be 1. The element \( A_{12} \) identifies \( \frac{1}{f} \).

Step-by-step solution

Identify the Matrices

Given the product of three matrices, identify them as \[ A_1 = \begin{pmatrix} 1 & \frac{n-1}{R_2} \ 0 & 1 \end{pmatrix}, \quad A_2 = \begin{pmatrix} 1 & 0 \ \frac{d}{n} & 1 \end{pmatrix}, \quad A_3 = \begin{pmatrix} 1 & -\frac{n-1}{R_1} \ 0 & 1 \end{pmatrix} \] .

Multiply the First Two Matrices

Calculate the product of \( A_1 \) and \( A_2 \): \[ A_1 A_2 = \begin{pmatrix} 1 & \frac{n-1}{R_2} \ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \ \frac{d}{n} & 1 \end{pmatrix} = \begin{pmatrix} 1 + \frac{d(n-1)}{nR_2} & \frac{n-1}{R_2} \ \frac{d}{n} & 1 \end{pmatrix} \] .

Multiply the Result by the Third Matrix

Now multiply the result by \( A_3 \): \[ A = (A_1 A_2) A_3 = \begin{pmatrix} 1 + \frac{d(n-1)}{nR_2} & \frac{n-1}{R_2} \ \frac{d}{n} & 1 \end{pmatrix} \begin{pmatrix} 1 & -\frac{n-1}{R_1} \ 0 & 1 \end{pmatrix} \] .Perform the multiplication: \[ A = \begin{pmatrix} 1 + \frac{d(n-1)}{nR_2} & \frac{n-1}{R_2} \cdot \left(-\frac{n-1}{R_1}\right) + \left(1 + \frac{d(n-1)}{nR_2}\right) \ \frac{d}{n} & \frac{d}{n} \cdot \left(-\frac{n-1}{R_1}\right) + 1 \end{pmatrix} \] .

Simplify the Matrix Elements

Simplify each element of the resulting matrix \( A \): \[ A = \begin{pmatrix} 1 + \frac{d(n-1)}{nR_2} & -\frac{n-1}{R_1} + \frac{d(n-1)^2}{nR_1 R_2} + 1 + \frac{d(n-1)}{nR_2} \ \frac{d}{n} & -\frac{d(n-1)}{nR_1} + 1 \end{pmatrix} \] .Combine and factor where possible: \[ A = \begin{pmatrix} 1 + \frac{d(n-1)}{nR_2} & 1 - \frac{n-1}{R_1} + \frac{d(n-1)^2}{nR_1 R_2} + \frac{d(n-1)}{nR_2} \ \frac{d}{n} & 1 - \frac{d(n-1)}{nR_1} \end{pmatrix} \] .

Identify the Element \( A_{12} \)

Element \( A_{12} \) of matrix \( A \) is given by: \[ A_{12} = 1 - \frac{n-1}{R_1} + \frac{d(n-1)^2}{nR_1 R_2} + \frac{(n-1)d}{nR_2} \] .

Evaluate the Determinant \( \text{det }A \)

Evaluate the determinant of matrix \( A \): \[ \text{det }A = \left(1 + \frac{d(n-1)}{nR_2}\right) \left(1 - \frac{d(n-1)}{nR_1}\right) - \left(\frac{d}{n}\right)\left(1 - \frac{n-1}{R_1} + \frac{d(n-1)^2}{nR_1 R_2} + \frac{(n-1)d}{nR_2}\right) = 1 \] .

Find \( \frac{1}{f} \)

Since \( A_{12} = -\frac{1}{f} \), solve for \( \frac{1}{f} \): \[ -\frac{1}{f} = 1 - \frac{n-1}{R_1} - \frac{(n-1)}{R_2} + \frac{d(n-1)^2}{nR_1 R_2} \] .Hence, \[ \frac{1}{f} = -\left(1 - \frac{n-1}{R_1} - \frac{(n-1)}{R_2} + \frac{d(n-1)^2}{nR_1 R_2}\right) \] .

Key concepts

Thick Lens Formula

The thick lens formula is fundamental in optics to comprehend the behavior of light passing through lenses with finite thickness. It describes how the shape and material of the lens impact its focal length. The formula involves parameters like the lens's index of refraction, and the radii of curvature of its surfaces, and the lens thickness. Using matrix multiplication, the thick lens formula aids in deriving the relationships between these parameters and the overall focal length.

In the given problem, matrix multiplication is used to understand a thick lens in the air. The matrices encapsulate how light rays propagate through the lens: one matrix represents the refraction at the first surface, another accounts for the transmission through the thick part of the lens, and the last considers the refraction at the second surface.

Index of Refraction

The index of refraction, denoted by \( n \), measures how much a medium can bend (refract) light. It quantifies the speed of light in that medium versus in a vacuum. This dimensionless number depends on the material properties of the lens.
The refractive index is crucial in formulating the matrix product for the thick lens since it affects how light bends at the lens surfaces. Higher indices indicate that light travels slower within the lens material, bending more pronouncedly at interfaces.

For instance, in our exercise, the index of refraction appears in all matrices and influences the calculations by altering values dependent on the material’s light-bending capabilities. This directly impacts how the focal length is derived from the matrix elements.

Radii of Curvature

The radii of curvature, \( R_1 \) and \( R_2 \), describe the curvature of the lens surfaces. These parameters indicate how curved or flat each lens surface is. The radius of curvature for a spherical lens section is the radius of the sphere from which the lens section is a part.

In our matrix-based approach to the thick lens formula, one matrix handles refraction at the first lens surface characterized by \( R_1 \), and another matrix manages the second surface with \( R_2 \). The values of these radii influence how light refraction at each lens surface contributes to the overall focal length of the lens.

• A smaller radius (more curved) bends light more dramatically.
• A larger radius (flatter surface) bends light less.

This information is encapsulated in specific entries of the matrices and combined through multiplication to yield the matrix \( A \).

Focal Length Calculation

Calculating the focal length \( f \) of a thick lens is a major goal in lens design and optics. The focal length determines where light rays converge after passing through the lens, central to lens performance.

In our exercise, an essential matrix element, \( A_{12} \), directly relates to the focal length through the equation \( A_{12} = -1 / f \). Evaluating this matrix element, we solve for \( f \). Key steps are:

• Multiplying the given matrices to find elements of matrix \( A \).
• Identifying the specific element \( A_{12} \).
• Using \( A_{12} \) to derive the expression for \( 1/f \).

This calculated focal length integrates lens thickness \( d \), refractive index \( n \), and radii of curvature \( R_1 \) and \( R_2 \), providing a comprehensive formula that captures all relevant variables.