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A helium-neon laser \((\lambda=633 \mathrm{nm}),\) shown in Figure \(\mathrm{P} 17.67,\) is built with a glass tube of inside diameter \(1.0 \mathrm{mm} .\) One mirror is partially transmitting to allow the laser beam out. An electrical discharge in the tube causes it to glow like a neon light. From an optical perspective, the laser beam is a light wave that diffracts out through a 1.0-mm-diameter circular opening. a. Explain why a laser beam can't be perfectly parallel, with no spreading. b. The angle \(\theta_{1}\) to the first minimum is called the divergence angle of a laser beam. What is the divergence angle of this laser beam? c. What is the diameter (in \(\mathrm{mm}\) ) of the laser beam after it travels \(3.0 \mathrm{m} ?\) d. What is the diameter of the laser beam after it travels \(1.0 \mathrm{km} ?\)

Step by step solution

01

Explanation for non-parallelism of laser beam

A laser beam cannot be perfectly parallel due to the wave property of light known as diffraction. When light waves pass through a small opening or around an obstacle, they spread out and deviate from their original path. This phenomenon causes the laser beam to spread and diverge rather than remaining in a perfectly parallel path.

02

Calculation of Divergence Angle

The divergence angle of the laser beam can be calculated using the sine-theta formula derived from the concept of diffraction. The formula is \(\sin \Theta = (\lambda / D)\), where \(\lambda\) is the wavelength and D is the diameter of the circular opening. Plugging in the values given in the exercise, we have \(\sin \Theta = (633 x 10^-9 m) / (1.0 x 10^-3 m)\). Solving this, we find that \(\Theta = \arcsin((633 x 10^-9 m) / (1.0 x 10^-3 m))\). The resulting angle is the divergence angle \(\Theta\).

03

Calculation of Laser Beam Diameter after Traveling 3.0 m

Using basic trigonometry in a right triangle, with the divergence angle being \(\Theta\) and the distance the laser beam travels being one side, we can find the diameter (twice the height of the triangle) of the laser beam after it travels 3.0 m. The diameter is \(2 \times 3.0 m \times \tan \Theta\). Calculate the numerical value using the previously found value of \(\Theta\).

04

Calculation of Laser Beam Diameter after Traveling 1.0 km

Similarly to the previous step, the diameter of the laser after it travels 1.0 km can be found using the same trigonometric approach. The diameter is \(2 \times 1000 m \times \tan \Theta\). Calculate the numerical value using the previously found value of \(\Theta\).

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Key Concepts

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

Laser Beam Divergence

When we talk about laser beams, one aspect that comes into play is divergence. It may seem like a laser beam should stay perfectly straight, like a ruler. However, due to diffraction, which is a fundamental property of light, it doesn't.
Diffraction occurs when light waves pass through a small opening, like the glass tube in a helium-neon laser. The waves spread out as they exit, causing the beam to diverge.
The angle at which the beam spreads is known as the divergence angle. This angle isn't constant but depends on the wavelength of light and the size of the opening. So, even though lasers are super precise, they can't avoid this natural wave behavior.

Wavelength

Wavelength is a critical factor in understanding light behavior, especially with lasers. It's the distance between identical points in consecutive cycles of a wave, like peaks.
For a helium-neon laser, the wavelength is given as 633 nm. This tiny measurement is key to calculating how light interacts with openings or obstacles.
The wavelength, combined with the opening's size, directly influences the divergence of the beam. Specifically, a longer wavelength relative to the opening size leads to more significant diffraction and wider beam divergence.
Understanding this helps explain why laser beams can't be perfectly parallel. They will always display some spreading due to their wave nature.

Trigonometry in Optics

Trigonometry is essential in analyzing how light behaves, especially when calculating the effects of diffraction.
To find out how a laser beam expands over a distance, we use trigonometric principles. First, determine the divergence angle \( \Theta \) using the formula: \( \sin \Theta = \frac{\lambda}{D} \), where \( \lambda \) is the wavelength, and \( D \) is the opening diameter.
Then, predict how much the laser beam spreads as it travels further. If you want to know the beam's diameter after traveling a distance, apply the formula:
\[ \text{Diameter} = 2 \times \text{Distance} \times \tan \Theta \]
This method lets us see how a beam evolves from its small origin to where you might encounter it.