Question

A Fabry-Pérot interferometer consists of two parallel heavily silvered glass plates; light enters normally to the plates, and undergoes repeated reflections between them, with a small transmitted fraction emerging at each reflection. Find the intensity \(|B|^{2}\) of the emerging wave, where $$ B=A(1-r) \sum_{n=0}^{\infty} r^{n} e^{i n \phi} $$ with \(r\) and \(\phi\) real.

Short answer

\[ |B|^2 = \frac{|A(1 - r)|^2}{1 - 2r \cos{\phi} + r^2} \]

Step-by-step solution

- Understand the Given Equation

The problem provides the equation for the emerging wave: \[ B = A(1-r) \sum_{n=0}^{\infty} r^n e^{i n \phi} \] where \( A \) is the amplitude of the incident wave, \( r \) is the reflection coefficient, and \( \phi \) is the phase change upon reflection. Recognize that this is a geometric series.

- Identify the Geometric Series

The summation \( \sum_{n=0}^{\infty} r^n e^{i n \phi} \) represents an infinite geometric series with the first term as 1 (when \( n=0 \)) and the common ratio being \( r e^{i \phi} \).

- Sum of the Geometric Series

Use the formula for the sum of an infinite geometric series \( \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \) where \( |r| < 1 \). Here, \( a = 1 \) and the common ratio is \( r e^{i \phi} \). Therefore, \[ \sum_{n=0}^{\infty} r^n e^{i n \phi} = \frac{1}{1 - r e^{i \phi}} \]

- Substitute the Geometric Series

Substitute the sum back into the expression for \( B \): \[ B = A(1-r) \cdot \frac{1}{1 - r e^{i \phi}} \]

- Simplify the Expression for B

Combine the terms to simplify the expression: \[ B = \frac{A(1-r)}{1 - r e^{i \phi}} \]

- Find the Intensity |B|^2

The intensity is given by \( |B|^2 \), which is the magnitude squared of \( B \). So we need to calculate: \[ |B|^2 = \left| \frac{A(1-r)}{1 - r e^{i \phi}} \right|^2 \]

- Calculate the Magnitude

To find the magnitude, use properties of magnitudes: \[ \left| \frac{a}{b} \right| = \frac{|a|}{|b|} \] and recall that \( |e^{i \phi}| = 1 \). Then, \[ |B|^2 = \frac{|A(1-r)|^2}{|1 - r e^{i \phi}|^2} \]

- Simplify Magnitude Terms

Expanding the magnitudes: \[ |B|^2 = \frac{|A(1 - r)|^2}{|1 - r e^{i \phi}|^2} \] Calculate \( |1 - r e^{i \phi}|^2 \): \[ |1 - r e^{i \phi}|^2 = (1 - r e^{i \phi})(1 - r e^{-i \phi}) = 1 - r e^{i \phi} - r e^{-i \phi} + r^2 \] Using Euler's formulas, \[ |1 - r e^{i \phi}|^2 = 1 - r (e^{i \phi} + e^{-i \phi}) + r^2 = 1 - 2r \cos{\phi} + r^2 \]

- Final Intensity Expression

Substitute back into the intensity expression: \[ |B|^2 = \frac{|A(1 - r)|^2}{1 - 2r \cos{\phi} + r^2} \] Expand \( |A(1 - r)|^2 = |A|^2 |1 - r|^2 \) and solve.

Key concepts

Optical Physics

In optical physics, a Fabry-Pérot interferometer is a crucial device for measuring spectral properties of light. It consists of two parallel, highly reflective glass plates. Light enters these plates perpendicularly and undergoes multiple reflections between them. Some of this light gets transmitted through the plates and interferes constructively or destructively depending on the phase changes. By analyzing this interference, we can obtain information about the light's wavelength and intensity. The device is used in spectrometry, laser tuning, and telecommunications.

Geometric Series

A geometric series is a sequence of terms where each term is the previous term multiplied by a constant called the common ratio. In the problem, we encounter a geometric series of the form displayed as displayed as a summation displayed with the initial term (when n = 0) being 1 and the common ratio being displayed as This series has specific properties: it converges when displayed which means that we can find its sum using the formula displayed as displayed As seen, the summation of terms within the interferometer can be replaced by a gradual decrease in intensity, making it easier to analyze.

Complex Numbers

In mathematics, complex numbers extend the concept of one-dimensional real numbers to two-dimensional numbers represented as displayed displayed and denoted as. The given problem involves complex exponentials like displayed which are fundamental in wave physics. The term displayed represents a point on the unit circle in the complex plane, rotating by an angle displayed displayed each time. When dealing with the Fabry-Pérot interferometer, these complex exponentials account for the phase change (displayeddisplayed ) upon each reflection. Mastering Euler's formula, which connects complex exponentials to sines and cosines (displayed as), simplifies the understanding and manipulation of these calculations.

Intensity Calculation

The ultimate goal in the problem is to find the intensity of the light emerging from the interferometer, represented by displayed The intensity is derived from the magnitude squared of the wave amplitude displayed The simplified expression of B, after summing the geometric series, is displayed. To find the intensity, we must calculate. Using the properties of magnitudes and knowing the magnitude of an exponential term is always 1, we rewrite it as: displayed Expanding displayed displayed and solving, we ultimately get displayed displayed This fraction indicates how the intensity depends both on the initial amplitude and on the geometrical configuration (r, nwhich in turn defines the efficiency of the interferometer's design.)