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Chapter 7: Problem 39

Derive an expression for the coherence length (in vacuum) of a wavetrain that has a frequency bandwidth \(\Delta \nu\) : express your answer in terms of the linewidth \(\Delta \lambda_{0}\) and the mean wavelength \(\bar{\lambda}_{0}\) of the train.

Short Answer

Expert verified
The coherence length is \( L_c = \frac{\bar{\lambda}_0^2}{\Delta \lambda_0} \).

Step by step solution

01

Understanding Coherence Length

Coherence length, denoted typically as \( L_c \), is the distance over which two waves remain correlated, and it's inversely related to the bandwidth \( \Delta u \) of the wave source in frequency. The general formula is \( L_c = \frac{c}{\Delta u} \), where \( c \) is the speed of light.
02

Relationship between Frequency Bandwidth and Wavelength

The change in wavelength \( \Delta \lambda \) can be related to the frequency bandwidth \( \Delta u \) using the relationship \( u = \frac{c}{\lambda} \), where \( \lambda \) is the wavelength. Taking the derivative with respect to \( \lambda \) gives \( du = -\frac{c}{\lambda^2} d\lambda \). Thus, \( \Delta u = -\frac{c}{\lambda_0^2} \Delta \lambda \).
03

Express Coherence Length in Terms of Wavelength

Substitute \( \Delta u = -\frac{c}{\bar{\lambda}_0^2} \Delta \lambda_0 \) from Step 2 into the coherence length formula: \( L_c = \frac{c}{\Delta u} = \frac{c}{ -\frac{c}{\bar{\lambda}_0^2} \Delta \lambda_0} = \frac{\bar{\lambda}_0^2}{\Delta \lambda_0} \).
04

Final Expression Adjustment

Since wavelength bandwidths (\( \Delta \lambda_0 \)) are positive and coherence length is also inherently a positive quantity, we adjust to take the absolute value, ensuring no negative values. Therefore, the coherence length in terms of wavelength is \( L_c = \frac{\bar{\lambda}_0^2}{\Delta \lambda_0} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Frequency Bandwidth
Frequency bandwidth is a critical concept in various fields, including optics, telecommunications, and signal processing. It refers to the range of frequencies within which a wave can oscillate around its central or carrier frequency.
  • A broader frequency bandwidth allows more frequencies to be transmitted, which could improve data transmission rates but may also increase signal distortion.
  • This bandwidth is represented as \( \Delta u \), which denotes the difference between the maximum and minimum frequencies of the wave.
For waves, like light in optics, frequency bandwidth relates to how different frequencies interfere and mix to form the overall wave train. The coherence length of a wave, especially in optics, is inversely proportional to its frequency bandwidth. This means that the narrower the frequency bandwidth, the longer the coherence length, resulting in limited dispersion and a more pronounced wave pattern over distance.
Mean Wavelength
Mean wavelength is a term used in optics to describe the average wavelength in a wavetrain. It is often denoted as \( \bar{\lambda}_0 \) and is crucial for determining the characteristics of the wave.
  • The mean wavelength is calculated by averaging all the wavelengths present in the wave train.
  • This helps in understanding phenomena like diffraction and interference patterns.
For a wavetrain, knowing the mean wavelength helps in simplifying calculations related to coherence length and other optical properties. It acts as the center point for wavelength distribution and is useful for assessing the overall behavior of the wave in vacuum conditions.
Wavelength Bandwidth
Wavelength bandwidth describes the spread of different wavelengths in a wave train. It is denoted by \( \Delta \lambda_0 \), representing the difference between the longest and shortest wavelengths present.
  • A smaller wavelength bandwidth usually implies a more coherent and monochromatic light source.
  • The relationship between frequency bandwidth and wavelength bandwidth is given by the formula \( \Delta u = -\frac{c}{\bar{\lambda}_0^2} \Delta \lambda_0 \), where \( c \) is the speed of light.
Understanding wavelength bandwidth is crucial for determining the coherence of light. A narrow wavelength bandwidth means that light waves will tend to remain more aligned over longer distances, leading to more pronounced interference and diffraction effects.
Speed of Light
The speed of light, usually denoted by \( c \), is a fundamental constant in physics. It plays a crucial role in the field of optics and many other scientific and engineering calculations.
  • The speed of light in vacuum is approximately \( 299,792,458 ext{ m/s} \), which is often rounded to \( 3 \times 10^8 ext{ m/s} \) for simplicity in calculations.
  • This constant is used to relate frequency and wavelength of any electromagnetic wave, with the formula \( c = \lambda \cdot u \).
Significantly, the speed of light is used to calculate the coherence length of light waves. It's a cornerstone for converting frequency bandwidth to wavelength bandwidth and vice versa, helping to understand how light waves behave in various environments.
Optics
Optics, the branch of physics dealing with light, encompasses the behavior and properties of light propagation. This includes wavelengths, frequencies, and the way light interacts with different media.
  • Key areas of optics include geometric optics, which studies light as rays, and physical optics, which addresses wave characteristics.
  • Practical applications of optics span from simple lenses in cameras to complex laser systems in various technologies.
In optics, understanding concepts like coherence length helps improve the design of optical instruments and systems. This understanding is instrumental in everything from enhancing the performance of communication devices to developing advanced medical imaging techniques.

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Most popular questions from this chapter

An ionized gas or plasma is a dispersive medium for EM waves. Given that the dispersion equation is $$ \omega^{2}=\omega_{p}^{2}+c^{2} k^{2} $$ where \(\omega_{p}\) is the constant plasma frequency, determine expressions for both the phase and group velocities and show that \(v v_{x}=c^{2}.\)

Microwaves of frequency \(10^{10} \mathrm{Hz}\) are beamed directly at a metal reflector. Neglecting the refractive index of air, determine the spacing between successive nodes in the resulting standing-wave pattern.

Take the function \(f(\theta)=\theta^{2}\) in the interval \(0<\theta<2 \pi\) and assume it repeats itself with a period of \(2 \pi .\) Now show that the Fourier expansion of that function is $$ f(x)=\frac{4 \pi^{2}}{3}+\sum_{m=1}^{x}\left(\frac{4}{m^{2}} \cos m \theta-\frac{4 \pi}{m} \sin m \theta\right) $$

Show that the Fourier series representation of the function \(f(\theta)=|\sin \theta|\) is $$ f(\theta)=\frac{2}{\pi}-\frac{4}{\pi} \sum_{m=1}^{\infty} \frac{\cos 2 m \theta}{4 m^{2}-1} $$

A magnetic-field technique for stabilizing a He-Ne laser to 2 parts in \(10^{10}\) has been patented. At \(632.8 \mathrm{nm},\) what would be the coherence length of a laser with such a frequency stability?
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