Problem 111 - Fundamentals of Physics

Question

Figure 34-56 shows a beam expander made with two coaxial converging lenses of focal lengths \(f_{1}\) and \(f_{2}\) and separation \(d=f_{1}+f_{2}\) The device can expand a laser beam while keeping the light rays in the beam parallel to the central axis through the lenses Suppose a uniform laser beam of width \(W_{i}=2.5 \mathrm{~mm}\) and intensity \(I_{i}=9.0 \mathrm{~kW} / \mathrm{m}^{2}\) enters a beam expander for which \(f_{1}=12.5 \mathrm{~cm}\) and \(f_{2}=30.0 \mathrm{~cm} .\) What are (a) \(W_{f}\) and (b) \(I_{f}\) of the beam leaving the expander? (c) What value of \(d\) is needed for the beam expander if lens 1 is replaced with a diverging lens of focal length \(f_{1}=-26.0 \mathrm{~cm} ?\)

Step-by-Step Solution

Answer

(a) The expanded beam width \(W_f = 6.0\) mm. (b) The intensity \(I_f = 1.56\) kW/m². (c) The required separation \(d = 56.0\) cm.

Step 1: Understand the Working of Beam Expander

A beam expander is used to increase the diameter of a laser beam. It achieves this by using two lenses with different focal lengths. In an ideal beam expander, the incoming parallel beam stays parallel after passing through the system, although the diameter can be altered.

Step 2: Calculate the Expanded Beam Width

The beam width is expanded by the ratio of the focal lengths of the lenses. The formula for the final width of the beam \(W_{f}\) is:\[ W_{f} = \left( \frac{f_{2}}{f_{1}} \right) W_{i} \]Substitute \(W_{i} = 2.5 \text{ mm}\), \(f_{1} = 12.5 \text{ cm}\), and \(f_{2} = 30.0 \text{ cm}\):\[ W_{f} = \left( \frac{30.0}{12.5} \right) \times 2.5 = 6.0 \text{ mm} \]

Step 3: Calculate the Intensity of the Expanded Beam

The intensity of the beam is related to its width. According to the inverse relationship between the area and intensity, the formula for the intensity \(I_{f}\) is:\[ I_{f} = I_{i} \left( \frac{W_{i}}{W_{f}} \right)^{2} \]Substitute \(I_{i} = 9.0 \text{ kW/m}^2\), \(W_{i} = 2.5 \text{ mm}\), and \(W_{f} = 6.0 \text{ mm}\):\[ I_{f} = 9.0 \times \left( \frac{2.5}{6.0} \right)^{2} = 1.56 \text{ kW/m}^2 \]

Step 4: Calculate Required Separation for Diverging Lens Setup

When replacing lens 1 with a diverging lens, the formula for the required separation \(d\) of lenses becomes:\[ d = f_{2} - |f_{1}| \]Substitute \(f_{1} = -26.0 \text{ cm}\) and \(f_{2} = 30.0 \text{ cm}\):\[ d = 30.0 + 26.0 = 56.0 \text{ cm} \]

Key Concepts

Laser Beam

A laser beam is a coherent stream of electromagnetic radiation, often in the visible spectrum. The term ``coherent" signifies that the light waves travel in phase, which is what gives lasers their remarkable properties, including high directionality and concentration of energy. This coherence is generated by the stimulated emission process in a laser cavity, which ensures the light waves maintain a constant phase relationship.
This precision allows lasers to be used in applications requiring high accuracy and energy concentration, such as cutting, measuring, and communications.

• Laser beams are usually very narrow, which allows them to travel long distances without significant spread.
• The width or diameter of the beam can be adjusted using systems like beam expanders.
• Beam expanders utilize lenses to increase the diameter, affecting the intensity and applicability of the laser beam.

Using beam expanders helps in improving the quality and efficiency of laser applications, making it easier to focus or redirect the beam as needed.

Focal Length

Focal length is a fundamental property of lenses, integral to their function in systems like beam expanders. It is defined as the distance from the lens to the focal point, where collimated light rays converge. In terms of a lens' capability:

• Positive focal lengths indicate converging lenses, which bring light rays together.
• Negative focal lengths denote diverging lenses, which spread light rays apart.

In a beam expander, the focal lengths of the lenses determine how much the laser beam will expand. The ratio of the focal lengths of the two lenses used affects the new width of the beam:
\[W_{f} = \left( \frac{f_{2}}{f_{1}} \right) W_{i} \]
This means that to control the expansion of the laser beam, engineers select lenses with appropriate focal lengths, achieving desired adjustments in width and intensity.

Intensity

Intensity is the amount of energy a laser beam carries over a specific area. It is a critical parameter in laser applications because it dictates how much energy can be delivered to a target. The formula for intensity in a beam is given by:
\[I = \frac{P}{A} \]
where \( P \) is the power and \( A \) is the area over which the power is distributed. When applied to beam expanders:

• As the width of a laser beam increases, its intensity decreases for a constant power level, due to the increase in the area over which the power is spread.
• This relationship can be calculated with:
  \[I_{f} = I_{i} \left( \frac{W_{i}}{W_{f}} \right)^{2} \]
  This inverse relationship is key to many applications, where both intensity and beam width must be optimized for the task at hand.

Understanding intensity adjustments through beam expansion helps in ensuring lasers are neither too dispersed nor too concentrated for specific applications.

Lens System

A lens system, like that used in a beam expander, is composed of multiple lenses arranged to manipulate light in specific ways. This configuration is crucial in optical devices to direct and focus laser beams accurately:

• In beam expanders, lenses are arranged to increase the beam's diameter while keeping it parallel to its original path.
• The lenses used can be converging or diverging, depending on the desired outcome in terms of beam size and quality.

By carefully selecting and positioning lenses, a lens system can refocus, redirect, and reshape a laser beam in complex ways. Engineers require precise calculations, including:
\[ d = f_{2} - |f_{1}| \]
for determining the distance between lenses, ensuring functionality and performance. A strong understanding of lens systems allows for innovative optical designs and effective use of lasers in various scientific, medical, and industrial fields.