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2) Calculate for the wavelength 𝜆 = 632.8 𝑛𝑚, a minimum beam waste of 𝜔0 = 1.19 ...

Jul 1, 2024

2) Calculate for the wavelength 𝜆 = 632.8 𝑛𝑚, a minimum beam waste of 𝜔0 = 1.19 𝑚𝑚, and a divergence angle Θ𝑅 = 339 𝜇𝑟𝑎𝑑 (i) the beam parameter product, (2 points) (ii) the 𝑀2 value, and (2 points) (iii) the minimum divergence angle Θ0 for an ideal beam in unit of degrees. (3 points) (iv) Given the 𝑀2 value from part (ii), what can be said about the beam shape. How would an ideal beam look like? (2 points) (v) Neglecting effects associated with the mirrors and lenses, what is the reason for a nonideal laser beam? How can it be counteracted. (4 points)

Answer

The answers to the questions are as follows:

Solution

The beam parameter product (BPP) is given by the product of the minimum beam waist

\omega_0

and the divergence angle

\Theta_R

\text{BPP} = \omega_0 \cdot \Theta_R

Given

\omega_0 = 1.19 \, \text{mm}

and

\Theta_R = 339 \, \mu\text{rad}

\text{BPP} = 1.19 \times 10^{-3} \, \text{m} \cdot 339 \times 10^{-6} \, \text{rad} = 4.0341 \times 10^{-7} \, \text{m} \cdot \text{rad}

The

M^2

value is calculated using the formula:

M^2 = \frac{\text{BPP}}{\text{BPP}_{\text{ideal}}}

For an ideal beam,

\text{BPP}_{\text{ideal}} = \frac{\lambda}{\pi}

. Given

\lambda = 632.8 \, \text{nm} = 632.8 \times 10^{-9} \, \text{m}

\text{BPP}_{\text{ideal}} = \frac{632.8 \times 10^{-9}}{\pi} = 2.014 \times 10^{-7} \, \text{m} \cdot \text{rad}

Thus,

M^2 = \frac{4.0341 \times 10^{-7}}{2.014 \times 10^{-7}} \approx 2

The minimum divergence angle

\Theta_0

for an ideal beam is given by:

\Theta_0 = \frac{\lambda}{\pi \omega_0}

Given

\lambda = 632.8 \, \text{nm}

and

\omega_0 = 1.19 \, \text{mm}

\Theta_0 = \frac{632.8 \times 10^{-9}}{\pi \times 1.19 \times 10^{-3}} \approx 1.69 \times 10^{-4} \, \text{rad}

Converting to degrees:

\Theta_0 \approx 1.69 \times 10^{-4} \times \frac{180}{\pi} \approx 0.0097^\circ

Given the

M^2

value from part (ii), the beam shape is not ideal. An ideal beam would have

M^2 = 1

, indicating a perfect Gaussian beam. Since

M^2 \approx 2

, the beam is more divergent and less focused than an ideal Gaussian beam

Nonideal laser beams can result from several factors, including imperfections in the laser cavity, thermal effects, and aberrations in optical components. These can be counteracted by improving the quality of the laser cavity, using better optical components, and implementing thermal management techniques

(i) Answer

The beam parameter product (BPP) is

4.0341 \times 10^{-7} \, \text{m} \cdot \text{rad}

(ii) Answer

The

M^2

value is approximately 2.

(iii) Answer

The minimum divergence angle

\Theta_0

for an ideal beam is approximately

0.0097^\circ

(iv) Answer

The beam shape is not ideal; it is more divergent and less focused than an ideal Gaussian beam.

(v) Answer

Nonideal laser beams can result from imperfections in the laser cavity, thermal effects, and aberrations in optical components. These can be counteracted by improving the quality of the laser cavity, using better optical components, and implementing thermal management techniques.

Key Concept

Beam parameter product (BPP) and

M^2

value

Explanation

The BPP is a measure of the beam quality, and the

M^2

value indicates how close the beam is to an ideal Gaussian beam.