Question

A continuous-wave (cw) argon-ion laser beam has an average power of \(10.0 \mathrm{~W}\) and a beam diameter of \(1.00 \mathrm{~mm}\). Assume that the intensity of the beam is the same throughout the cross section of the beam (which is not true, as the actual distribution of intensity is a Gaussian function). a) Calculate the intensity of the laser beam. Compare this with the average intensity of sunlight at Earth's surface \(\left(1400 . \mathrm{W} / \mathrm{m}^{2}\right)\) b) Find the root-mean-square electric field in the laser beam. c) Find the average value of the Poynting vector over time. d) If the wavelength of the laser beam is \(514.5 \mathrm{nm}\) in vacuum, write an expression for the instantaneous Poynting vector, where the instantaneous Poynting vector is zero at \(t=0\) and \(x=0\) e) Calculate the root-mean-square value of the magnetic field in the laser beam.

Short answer

Question: Calculate the intensity of a 10 W laser beam with a 1-mm diameter and wavelength of 514.5 nm. Compare the intensity to the average intensity of sunlight. Find the root-mean-square electric field, average value of the Poynting vector, instantaneous Poynting vector, and root-mean-square value of the magnetic field in the laser beam.

Step-by-step solution

a) Calculating the intensity of the laser beam

First, we need to determine the intensity of the laser beam (I). Intensity (I) is the power per unit area i.e., \(I = \cfrac{Power}{Area}\) We are given: Power (P) = 10 W Beam diameter (d) = 1 mm = \(1\times10^{-3}\;\text{m}\) (convert to meters) As intensity is the same throughout the cross-section of the beam, we can consider the area as a circle. Area (A) \(= \text{π}\times (\cfrac{d}{2})^2\) Now, we can calculate the intensity of the laser beam: I = \(\cfrac{P}{A}\) We will also compare this intensity with the average intensity of sunlight, which is 1400 W/m².

b) Finding the root-mean-square electric field

To find the root-mean-square electric field (E_rms), we'll use the relationship between intensity and electric field: \(I = \cfrac{1}{2} \varepsilon_0 c E_\text{rms}^2\) Where: \(\varepsilon_0\) is the vacuum permittivity (\(\approx 8.854 \times 10^{-12} \;\mathrm{F/m}\)) c is the speed of light (\(\approx 3.0 \times 10^8 \;\mathrm{m/s}\)) Solve for \(E_\text{rms}\).

c) Calculating the average value of the Poynting vector

The Poynting vector (S) represents the power per unit area, so its average value will be equal to the intensity calculated in part a).

d) Writing an expression for the instantaneous Poynting vector

The instantaneous Poynting vector (S_ins) can be expressed as: \(\hat{S}_\text{ins}(x,t) = S_0 \sin^2 (kx - \omega t) \hat{z}\) Where S0 is the peak power, and k and ω are the wave number and angular frequency, respectively. Wavelength (λ) = \(514.5\times10^{-9} \text{m}\) (convert to meters) k = \(\cfrac{2\pi}{\lambda}\) c = \(\omega \lambda\) ω = \(\cfrac{2\pi c}{\lambda}\) Substitute these values into the expression for the instantaneous Poynting vector.

e) Calculating root-mean-square value of the magnetic field

To find the root-mean-square value of the magnetic field (B_rms), we can use the relationship between the electric and magnetic fields in a plane wave: \(E_\mathrm{rms} = cB_\mathrm{rms}\) Substitute the given values and the E_rms we calculated in part b) to solve for B_rms.

Key concepts

Poynting Vector

The Poynting vector is a central concept in electromagnetism that represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. In simple terms, it tells us how much electromagnetic power is passing through a certain area, and in what direction the energy is moving. The average value of the Poynting vector is particularly important for continuous-wave (cw) beams like lasers because it provides a measure of the average power being transferred.
In the context of our exercise, the average Poynting vector is equivalent to the intensity of the laser beam because the intensity is defined as the power delivered per unit area. Since the laser in the problem has a known power and a uniform circular cross-section, we can use these to calculate the intensity, and by extension, the average Poynting vector.
This concept comes into play in parts c) and d) of the exercise, where we determine the average value over time and the expression for the instantaneous Poynting vector, respectively. The instantaneous Poynting vector varies with position and time, reflecting the oscillating nature of the electromagnetic field in the laser beam.

Root-mean-square Electric Field

The root-mean-square (rms) electric field is a statistical measure of the magnitude of the oscillating electric field within a wave, such as the electromagnetic wave of a laser beam. This value is crucial because it allows us to express the strength of the electric field in a way that can be easily related to the measurable quantities like power and intensity.
When we deal with sinusoidally varying fields, such as in a laser, the rms value gives us a meaningful average field strength over time. For example, while the peak electric field tells us the maximum value the field can reach, the rms electric field is useful for calculating the power transmitted by the wave, specifically in step b) of the exercise.
Calculating the rms electric field involves using the given beam power and geometry to find the intensity, and then relating that intensity to the electric field strength through the relationship involving the vacuum permittivity and the speed of light. This demonstrates how an understanding of the rms electric field is essential for describing the behavior of electromagnetic waves, as well as for practical applications like determining the energy transmission in a laser.

Gaussian Beam Profile

The Gaussian beam profile is a fundamental concept in optics describing the spatial variation of the intensity of a particular type of laser beam. The intensity of a true Gaussian beam follows a Gaussian function, meaning it is highest at the center of the beam and decreases exponentially towards the edges.
This profile results from the way the light emerges from the laser and spreads as it propagates. Importantly, it is different from a beam with uniform intensity across its diameter, as assumed in step a) of our exercise for simplicity. In reality, lasers typically do have a Gaussian intensity distribution, which affects how the beam interacts with materials and how it should be handled safely.
In the context of understanding laser beam intensity, considering a Gaussian beam profile provides a more accurate picture of how the energy is distributed across the cross-section of the beam. This affects calculations of intensity and, consequently, the electric and magnetic fields derived from it. Though the exercise simplifies matters by assuming a uniform distribution, the concept of a Gaussian beam profile is vital for anyone working with lasers in practical applications or theoretical research.