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Chapter 15: Problem 24

A CO laser operating at 9.6 \(\mu \mathrm{m}\) uses an electrical power of \(5.00 \mathrm{kW}\). If this laser produces 100 -ns pulses at a repetition rate of \(10 \mathrm{Hz}\) and has an efficiency of \(27 \%,\) how many photons are in each laser pulse?

Short Answer

Expert verified
Each laser pulse contains approximately \(6.51 \times 10^{21}\) photons.

Step by step solution

01

Find the Energy per Pulse

The laser operates with a 27% efficiency, meaning only 27% of the electrical power is converted into laser energy. The laser power is 5 kW, or 5000 J/s, at a 10 Hz repetition rate. This means that a pulse occurs every 0.1 seconds. The energy used for one pulse is the power times the time:\[\text{Energy per pulse} = (5000 \text{ J/s}) \times (0.1 \text{ s/pulse}) \times 0.27\]\[\text{Energy per pulse} = 135 \text{ J}\]
02

Calculate Energy per Photon

The energy of a photon is given by the formula:\[E = \frac{hc}{\lambda}\]where \(h\) is Planck's constant \(6.626 \times 10^{-34}\) Js, \(c\) is the speed of light \(3 \times 10^8\) m/s, and \(\lambda\) is the wavelength. For this laser, \(\lambda = 9.6 \times 10^{-6}\) m:\[E_{photon} = \frac{(6.626 \times 10^{-34} \text{ J s}) \times (3 \times 10^8 \text{ m/s})}{9.6 \times 10^{-6} \text{ m}}\]\[E_{photon} = 2.073 \times 10^{-20} \text{ J}\]
03

Calculate Number of Photons per Pulse

To find the number of photons per pulse, divide the energy per pulse by the energy per photon:\[\text{Number of photons per pulse} = \frac{135 \text{ J}}{2.073 \times 10^{-20} \text{ J/photon}}\]\[\text{Number of photons per pulse} \approx 6.51 \times 10^{21}\]

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Key Concepts

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

Laser Efficiency
When we talk about laser efficiency, we are discussing how well a laser converts electrical power into light energy. A laser's efficiency is essentially a ratio that compares the useful output of the system (the laser light) to the input (the electricity used). For instance, a laser with 27% efficiency means that 27% of the electrical power is transformed into laser energy, while the rest is lost, often as heat or other forms of energy.
Here are some key points about laser efficiency:
  • Efficiency is typically expressed as a percentage.
  • High-efficiency lasers waste less energy and are often more cost-effective to operate.
  • The efficiency can depend on the type of laser and its operating conditions.
Understanding laser efficiency helps in designing systems that maximize the energy converted to useful laser light, minimizing power consumption and cost.
Photon Energy
Photons are the basic units of light, and their energy can be calculated using the formula: \( E = \frac{hc}{\lambda} \). Here, \( h \) represents Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the light.
Let's break down each part of this equation:
  • Planck's constant (\( h \)): This is a fundamental constant in physics, with a value of \( 6.626 \times 10^{-34} \) Js.
  • Speed of light (\( c \)): In a vacuum, this is approximately \( 3 \times 10^8 \) m/s.
  • Wavelength (\( \lambda \)): This is the distance between consecutive peaks of a wave. In this scenario, the laser wavelength is given as \( 9.6 \mu \mathrm{m} \) (which is \( 9.6 \times 10^{-6} \) m).
Each photon's energy depends on the wavelength of the light; shorter wavelengths correspond to higher energy photons. This relationship is key in understanding how laser systems function, especially in fields like fiber optics and telecommunications.
Electromagnetic Waves
Electromagnetic waves are waves of combined electric and magnetic fields that travel through space. Visible light, microwaves, and x-rays are all forms of electromagnetic waves, each with different wavelengths and frequencies.
Some important facts about electromagnetic waves include:
  • They travel at the speed of light in a vacuum (about \( 3 \times 10^8 \) m/s).
  • They do not require a medium to travel; they can move through the vacuum of space.
  • The electromagnetic spectrum includes a range of waves from radio waves (longest wavelength) to gamma rays (shortest wavelength).
Understanding these waves' properties enables us to harness them in technologies like lasers. Lasers produce light at specific wavelengths, which is why understanding electromagnetic waves is crucial for applications across various sectors like medicine, communication, and manufacturing.

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Most popular questions from this chapter

Which laser pulse contains more photons, a 10 -ns, 1.60 -mJ pulse at 760 nm or a 500 -ms, \(1.60-\mathrm{mJ}\) pulse at \(532 \mathrm{nm} ?\)

The decadic absorbance A of a sample is defined by \(A=\log \left(I_{0} / I\right),\) where \(I_{0}\) is the light intensity incident on the sample and \(I\) is the intensity of the light after it has passed through the sample. The decadic absorbance is proportional to \(c,\) the molar concentration of the sample, and \(l\), the path length of the sample in meters, or in an equation \\[A=\varepsilon c l\\] where the proportionality factor \(\varepsilon\) is called the molar absorption coefficient. This expression is called the Beer-Lambert law. What are the units of \(A\) and \(\varepsilon ?\) If the intensity of the transmitted light is \(25.0 \%\) of that of the incident light, then what is the decadic absorbance of the sample? At \(200 \mathrm{nm},\) a \(1.42 \times 10^{-3} \mathrm{M}\) solution of benzene has decadic absorbance of \(1.08 .\) If the pathlength of the sample cell is \(1.21 \times 10^{-3} \mathrm{m},\) what is the value of \(\varepsilon ?\) What percentage of the incident light is transmitted through this benzene sample? (It is common to express \(\varepsilon\) in the non SI units \(\mathrm{L} \cdot \mathrm{mol}^{-1} \cdot \mathrm{cm}^{-1}\) because \(l\) and \(c\) are commonly expressed in \(\mathrm{cm}\) and \(\mathrm{mol} \cdot \mathrm{L}^{-1},\) respectively. This difference in units leads to annoying factors of 10 that you need to be aware of.)

Hydrogen iodide decomposes to hydrogen and iodine when it is irradiated with radiation of frequency \(1.45 \times 10^{15} \mathrm{Hz}\). When \(2.31 \mathrm{J}\) of energy is absorbed by \(\mathrm{HI}(\mathrm{g}), 0.153 \mathrm{mg}\) of \(\mathrm{HI}(\mathrm{g})\) is decomposed. Calculate the quantum yield for this reaction.

The ground-state term symbol for \(\mathrm{O}_{2}^{+}\)is \({ }^{2} \Pi_{g}\). The first electronic excited state has an energy of \(38795 \mathrm{~cm}^{-1}\) above that of the ground state and has a term symbol of \({ }^{2} \Pi_{u}\). Is the radiative \({ }^{2} \Pi_{u} \rightarrow{ }^{2} \Pi_{g}\) decay of the \(\mathrm{O}_{2}^{+}\)molecule an example of fluorescence of phosphorescence?

A titanium sapphire laser operating at 780 nm produces pulses at a repetition rate of \(100 \mathrm{MHz}\). If each pulse is \(25 \mathrm{fs}\) in duration and the average radiant power of the laser is 1.4 \(\mathrm{W}\), calculate the radiant power of each laser pulse. How many photons are produced by this laser in one second?
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