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Magazine Chapter 13: Problem 28
A He-Ne laser operating at \(632.8 \mathrm{nm}\) has an internal beamwaist diameter of \(0.60 \mathrm{mm}\). Calculate the full-angular width, or divergence, of the beam.
Short Answer
Expert verified
The divergence of the beam is approximately \(1.34 \times 10^{-3}\) radians.
Step by step solution
01
Identify Given Values
The wavelength of the He-Ne laser beam is given as \( \lambda = 632.8 \text{ nm} \). The internal beam waist diameter is given as \( d_0 = 0.60 \text{ mm} \). Convert these to coherent units: \( \lambda = 632.8 \times 10^{-9} \text{ m} \) and \( d_0 = 0.60 \times 10^{-3} \text{ m} \).
02
Calculate Beam Waist Radius
The beam waist radius \( w_0 \) is half of the given beam waist diameter. Calculate it using:\[ w_0 = \frac{d_0}{2} = \frac{0.60 \times 10^{-3}}{2} = 0.30 \times 10^{-3} \text{ m} \].
03
Use the Divergence Formula
The full-angular width or divergence \( \theta \) of the laser beam is calculated using the formula:\[ \theta = \frac{2 \lambda}{\pi w_0} \].
04
Plug in the Values
Substitute the values of \( \lambda \) and \( w_0 \) to find \( \theta \):\[\theta = \frac{2 \times 632.8 \times 10^{-9}}{\pi \times 0.30 \times 10^{-3}}\].
05
Calculate the Result
Perform the calculation:\[ \theta \approx \frac{1265.6 \times 10^{-9}}{0.942 \times 10^{-3}} = \frac{1265.6}{942} \times 10^{-6} \approx 1.3447 \times 10^{-3} \text{ radians} \].So, the divergence is approximately \(1.34 \times 10^{-3} \text{ radians}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding He-Ne Lasers
He-Ne lasers, short for Helium-Neon lasers, are a type of gas laser that are commonly used in a variety of applications due to their ability to produce a highly coherent light beam. The main features of He-Ne lasers include:
- Emission Wavelength: Typically emit light in the visible spectrum, most commonly at a wavelength of 632.8 nm, which gives them a characteristic red light.
- Construction: Utilize a mixture of helium and neon gases contained within a sealed tube, excited by an electrical discharge.
- Applications: Used in applications such as bar code scanning, holography, and instructional demonstrations because of their stability and ease of application.
The Role of Wavelength in Laser Optics
The wavelength of a laser is a crucial parameter that defines the color of the light and its behavior during transmission. In the case of the He-Ne laser:
- Wavelength ( \( \lambda \) ): This is the distance over which the light's wave shape repeats. For a He-Ne laser, the wavelength is 632.8 nm, characterizing it as red light.
- Relevance to Divergence: The wavelength affects how a beam travels through space. Longer wavelengths lead to greater spreading of the beam, known as divergence.
Exploring the Beam Waist
The beam waist is a critical feature of any laser beam, representing the narrowest point of the beam as it propagates through space. Here's what you need to know:
- Definition: Beam waist is the location where the beam diameter is at its smallest value.
- Significance in Calculations: In calculations of beam divergence, the beam waist directly influences the amount of spreading or divergence of the laser as it travels. It鈥檚 vital for understanding focusing characteristics.
- Measurement: For He-Ne lasers, the beam waist is usually expressed in terms of diameter, but calculations often require converting this to a radius for precision, such as halving the diameter value.
Mastering Optics Calculations
Optics calculations might seem tricky at first, but with a systematic approach, they become much simpler to tackle. Here is a breakdown of key steps, emphasizing the exercise's context:
- Unit Conversion: Begin by converting all measurements into coherent units (e.g., nanometers to meters, millimeters to meters) to ensure consistency in calculations.
- Identify Formulas: Use relevant formulas, like the divergence formula for lasers, which relates the wavelength and beam waist radius to the beam spread.
- Substitution and Calculation: Carefully substitute the known values into your formulas and solve using basic algebraic principles.
- Verification: Always double-check your results for accuracy, ensuring that your units make sense and that calculations are consistent with expected physical results.
