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Magazine Chapter 3: Problem 8
What is the line width in hertz and in nanometers of the light from a helium- neon laser whose coherence length is \(5 \mathrm{~km}\) ? The wavelength is \(633 \mathrm{~nm}\).
Short Answer
Expert verified
The line width is 60,000 Hz and 0.00802 nm.
Step by step solution
01
Understand Coherence Length
The coherence length describes the length over which a coherent beam (like a laser) maintains a specified degree of coherence. It relates to the frequency stability of the source. The formula for coherence length \( L_c \) in terms of the wavelength \( \lambda \) and the line width \( \Delta u \) is given by \( L_c = \frac{c}{\Delta u} \), where \( c \) is the speed of light.
02
Rearrange the Formula to Find Line Width in Hertz
From \( L_c = \frac{c}{\Delta u} \), we can rearrange to find the line width \( \Delta u \) as \( \Delta u = \frac{c}{L_c} \). Substitute the known values: \( c = 3 \times 10^8 \mathrm{~m/s} \) and \( L_c = 5 \times 10^3 \mathrm{~m} \).
03
Calculate Line Width in Hertz
Substitute the values into the equation: \( \Delta u = \frac{3 \times 10^8}{5 \times 10^3} = 6 \times 10^4 \text{ Hz} \). The line width of the helium-neon laser in hertz is \( 60,000 \text{ Hz} \).
04
Convert Line Width to Nanometers
Use the relationship between frequency change and wavelength change given by \( \Delta u = \frac{c \Delta \lambda}{\lambda^2} \). Rearrange to find \( \Delta \lambda \) as \( \Delta \lambda = \frac{\lambda^2 \Delta u}{c} \). Substitute known values: \( \lambda = 633 \times 10^{-9} \mathrm{~m} \) and \( \Delta u = 6 \times 10^4 \text{ Hz} \).
05
Calculate Line Width in Nanometers
Substitute into the formula: \( \Delta \lambda = \frac{(633 \times 10^{-9})^2 \times 6 \times 10^4}{3 \times 10^8} \). Simplify to find \( \Delta \lambda \approx 8.02 \times 10^{-9} \text{ m} \), converting to nanometers gives \( \Delta \lambda \approx 0.00802 \text{ nm} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coherence Length
Coherence length is a fundamental concept in optics, especially in the study of lasers. It measures the distance over which a wave (such as a light wave from a laser) remains coherent and consistent in phase. Coherence is crucial for applications like holography and interferometry.
Longer coherence lengths mean the wave maintains its order over a greater distance, which is desirable for precision tasks.
Longer coherence lengths mean the wave maintains its order over a greater distance, which is desirable for precision tasks.
- Coherence length is related to the temporal stability of the light source's frequency.
- A stable frequency results in a longer coherence length.
- You can calculate it using the formula: \[ L_c = \frac{c}{\Delta u} \] where \( L_c \) is coherence length, \( c \) is the speed of light, and \( \Delta u \) is the line width.
Line Width
Line width refers to the range of frequencies emitted by a light source, like a laser. It's a measure of how pure or monochromatic the light is. For lasers, a narrow line width indicates a small range of emitted frequencies, which is ideal for precision applications.
In terms of coherence, a narrower line width correlates to a longer coherence length, meaning the emitted light retains its regularity over a larger distance.
In terms of coherence, a narrower line width correlates to a longer coherence length, meaning the emitted light retains its regularity over a larger distance.
- Line width is inversely proportional to coherence length: \( \Delta u = \frac{c}{L_c} \).
- For the helium-neon laser in the exercise, the calculated line width is 60,000 Hz.
- This small line width is what makes lasers precise in cutting, measuring, and data transmission.
Helium-Neon Laser
The helium-neon laser is a type of gas laser that typically emits a red beam with a wavelength of 633 nm. It's one of the most common types of lasers used in educational, industrial, and research settings due to its stability and ease of use.
Helium-neon lasers work by exciting a gas mixture of helium and neon to produce coherent light.
Helium-neon lasers work by exciting a gas mixture of helium and neon to produce coherent light.
- These lasers are known for their long coherence length, making them ideal for tasks like holography and high precision measurement.
- They produce light with narrow line widths, which contributes to their precision and effectiveness in different applications.
- The standard 633 nm wavelength is in the visible range, making them suitable for optical demonstrations and experiments.
Wavelength
Wavelength is the distance between successive peaks of a wave, such as light. In optics, the wavelength is fundamental in determining the color and energy of light. The helium-neon laser emits light at a wavelength of 633 nm, characteristic of its signature red color in visible light.
- The relationship between frequency and wavelength is given by the equation: \( c = \lambda u \), where \( c \) is the speed of light, \( \lambda \) is the wavelength, and \( u \) is the frequency.
- Given the wavelength, we can calculate changes in wavelength caused by line width using: \( \Delta \lambda = \frac{\lambda^2 \Delta u}{c} \).
- For the helium-neon laser, a change in wavelength due to line width was calculated to be approximately 0.00802 nm, demonstrating the narrow spectrum of its emission.
