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

The wavelength of the laser,$\lambda =694.4\mathrm{nm}$

The duration of the pulse,$t=12\mathrm{ps}$

The energy per pulse,$E=0.150\mathrm{J}$

Consider the known data below.

The Plank’s constant is,

$$\begin{gathered} \mathrm{h}=6.63\times {10}^{-34}\mathrm{J}.\mathrm{s} \\ =\left(6.242\times {10}^{18}\times 6.63\times {10}^{-34}\right)\mathrm{eV}.\mathrm{s} \\ =41.384\times {10}^{-16}\mathrm{eV}.\mathrm{s} \end{gathered}$$

The speed of light is,

$$\begin{gathered} \mathrm{c}=3\times {10}^8\mathrm{m}/\mathrm{s} \\ =\left(3\times {10}^8\times {10}^9\right)\mathrm{nm}/\mathrm{s} \\ =3\times {10}^{17}\mathrm{nm}/\mathrm{s} \end{gathered}$$

Photon energy is the energy carried by a single photon. The amount of energy is directly proportional to the magnetic frequency of the photon and thus, equally, equates to the wavelength of the wave. When the frequency of photons is high, its potential is high.

Using the distance formula get the length of the pulse. Now, for the number of emitted photons, calculate the energy of each photon using the given wavelength. Now, the ratio of the energy of the pulse and the energy of the photon gives the required answer.

Formula:

The length or the distance traveled of a light wave,

L =ct ….. (1)

The energy of the photon due to Planck’s relation,

$∆\mathrm{E}=\frac{\mathrm{hc}}{\mathrm{\lambda }}$ ….. (2)

Using the given data in equation (1), the length of the pulse can be calculated as follows:

$$\begin{gathered} \mathrm{L}=\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)\left(12\times {10}^{-12}\mathrm{s}\right) \\ =3.6\times {10}^{-3}\mathrm{m} \\ =3.6\mathrm{mm} \end{gathered}$$

Hence, the value of the length is 3.6 mm.

At first, the energy of the pulse is converted into as follows:

$$\begin{gathered} {\mathrm{E}}_{\mathrm{p}}=\frac{\mathrm{E}}{\mathrm{Q}} \\ =\frac{0.150\mathrm{J}}{1.6\times {10}^{-19}\mathrm{C}} \\ =\frac{0.150\mathrm{J}}{1.6\times {10}^{-19}\mathrm{J}/\mathrm{eV}} \\ =9.36\times {10}^{17}\mathrm{eV} \end{gathered}$$

Now, using the given data in equation (1), the energy of a single emitted photon can be calculated as follows:

$$\begin{gathered} \mathrm{E}=\frac{\left(41.384\times {10}^{-16}\mathrm{eV}.\mathrm{s}\right)\left(3\times {10}^{17}\mathrm{nm}/\mathrm{s}\right)}{694.4\mathrm{nm}} \\ =\frac{1240\mathrm{eV}.\mathrm{nm}}{694\mathrm{eV}} \\ =1.786\mathrm{eV} \end{gathered}$$

Thus, the number of photons emitted by the laser light can be given as:

$$\begin{gathered} \mathrm{N}=\frac{{\mathrm{E}}_{\mathrm{p}}}{\mathrm{E}} \\ =\frac{9.36\times {10}^{17}\mathrm{eV}}{1.786\mathrm{eV}} \\ =5.24\times {10}^{17}\mathrm{eV} \end{gathered}$$

Hence, the number of emitted photons is $5.24\times {10}^{17}\mathrm{eV}$.