Problem 55 - Fundamentals of Physics

Question

A pulsed laser emits light at a wavelength of \(694.4 \mathrm{~nm}\). The pulse duration is \(12 \mathrm{ps}\), and the energy per pulse is \(0.150 \mathrm{~J} .\) (a) What is the length of the pulse? (b) How many photons are emitted in each pulse?

Step-by-Step Solution

Answer

(a) The pulse length is \(3.6 \times 10^{-3}\) meters. (b) The number of photons per pulse is approximately \(5.24 \times 10^{17}\).

Step 1: Convert Wavelength to Meters

The given wavelength of the laser is \( 694.4 \text{ nm} \). To perform calculations, we need to convert this wavelength to meters. We know that \( 1 \text{ nm} = 1 \times 10^{-9} \text{ meters} \). Thus, \( 694.4 \text{ nm} = 694.4 \times 10^{-9} \text{ meters} \).

Step 2: Calculate the Speed of Light in the Medium

We assume the laser pulse travels in air, so we use the speed of light \( c = 3 \times 10^8 \text{ m/s} \).

Step 3: Calculate Pulse Length

The length of the pulse can be found using the formula \( \text{Pulse Length} = \text{Pulse Duration} \times \text{Speed of Light} \). The pulse duration is given as \( 12 \mathrm{ps} = 12 \times 10^{-12} \text{s} \). Substituting the values, we get \( \text{Pulse Length} = 12 \times 10^{-12} \text{ s} \times 3 \times 10^8 \text{ m/s} = 3.6 \times 10^{-3} \text{ m} \).

Step 4: Calculate Energy of Single Photon

The energy of a single photon is given by \( E = \frac{hc}{\lambda} \), where \( h = 6.63 \times 10^{-34} \text{ J⋅s} \) is the Planck constant. Substituting the values, we get \( E = \frac{6.63 \times 10^{-34} \text{ J⋅s} \times 3 \times 10^8 \text{ m/s}}{694.4 \times 10^{-9} \text{ m}} \approx 2.86 \times 10^{-19} \text{ J} \).

Step 5: Calculate Number of Photons Per Pulse

To find the number of photons emitted per pulse, we use the formula \( N = \frac{E_{\text{total}}}{E_{\text{photon}}} \). The energy per pulse is given as \( E_{\text{total}} = 0.150 \text{ J} \). Using the energy of a single photon calculated in Step 4, we have \( N = \frac{0.150 \text{ J}}{2.86 \times 10^{-19} \text{ J}} \approx 5.24 \times 10^{17} \).

Key Concepts

Wavelength to Meters Conversion

Understanding how to convert wavelength from nanometers to meters is essential in optics.The wavelength given in nanometers (nm) must be converted into meters (m) for scientific calculations.

• A nanometer is one-billionth of a meter, denoted as \(1 ext{ nm} = 1 \times 10^{-9} ext{ meters}\).
• To convert a wavelength from nanometers to meters, multiply the numerical value by \(10^{-9}\).

For example, the wavelength \(694.4 ext{ nm}\) converts to \(694.4 \times 10^{-9} ext{ m}\). This forms the cornerstone for further calculations, such as determining the energy of photons or speed calculations.

Speed of Light in Medium

In physics, the speed of light is a constant denoted by \(c\), typically approximated as \(3 \times 10^8 ext{ m/s}\) in vacuum.However, this exercise assumes propagation through air, which closely approximates a vacuum.

• While the speed of light remains constant in a vacuum, it can vary in different mediums due to refractive indices.
• In air, the variation is negligible, hence we use the standard value of \(c = 3 \times 10^8 ext{ m/s}\).

This knowledge is crucial when calculating properties like pulse length or energy distribution of light in different environments.

Pulse Length Calculation

Pulse length refers to the distance a pulse travels in a given duration.The formula to calculate pulse length is straightforward: Pulse Length = Pulse Duration \(\times\) Speed of Light.

• Pulsed lasers have a defined pulse duration, measured in time (e.g., picoseconds).
• To find the pulse length, first convert the pulse duration to seconds, such as \(12 ext{ ps} = 12 \times 10^{-12} ext{ s}\).

Next, use the speed of light value, resulting in a pulse length calculation of \(3.6 \times 10^{-3} ext{ m}\). The pulse length reveals how far the light travels during one pulse.

Number of Photons per Pulse

Determining the number of photons in a laser pulse relates directly to the pulse's total energy and the energy of individual photons.Photons, the elementary particles of light, carry energy that depends on the light's wavelength.

• The energy of a single photon can be calculated using \(E = \frac{hc}{\lambda}\), involving Planck's constant \(h\), speed of light \(c\), and wavelength \(\lambda\).
• For the wavelength of \(694.4 \text{ nm}\), find the photon energy ≈ \(2.86 \times 10^{-19} ext{ J}\).
• The total energy per pulse, \(0.150 ext{ J}\), helps calculate the number of photons by dividing the total pulse energy by per-photon energy.

Thus, the number of photons per pulse is approximately \(5.24 \times 10^{17}\), a measure of the light intensity of each emitted pulse.