Question

Lasers can be used to generate pulses of light whose durations are as short as 10fs. (a) How many wavelengths of light$\mathbf{\lambda }\mathbf{=}\mathbf{500}\mathbf{}\mathbf{nm}$ are contained in such a pulse? (b) In$\frac{\mathbf{10}\mathbf{}\mathbf{fs}}{\mathbf{1}\mathbf{}\mathbf{s}}\mathbf{=}\frac{\mathbf{1}\mathbf{}\mathbf{s}\mathbf{}}{\mathbf{X}}$ what is the missing quantity X (in years)?

Short answer

a) There are 6.0 wavelengths of light that are contained in such a pulse.

b) The missing quantity X (in years) is $3.2\times {10}^6\mathrm{years}$.

Step-by-step solution

The given data

a) Durations of pulses of light, $∆\mathrm{t}=10\mathrm{fs}$

b) Wavelength of the light, $\lambda =500\mathrm{nm}$

Understanding the concept of wavelength and time of pulse:

A pulse, like other waveforms, has a velocity and an amplitude. Because there is just one crest, there is no frequency or real wavelength, however the pulse width is related to wavelength.

From the concept of the length of a pulse train define the number of wavelengths present in the pulse by dividing the length value to the wavelength given. Next, for the X value in years, and divide the found time in seconds by the required number in years for the year conversion.

Formulae:

The length of a pulse train,

$\mathrm{L}=\mathrm{c}∆\mathrm{t}$ ….. (1)

Here the speed of lght is,

$\mathrm{c}=3\times {10}^8\mathrm{m}/\mathrm{s}$

The number of wavelengths in a given pulse,

$\mathrm{N}=\frac{\mathrm{L}}{\mathrm{\lambda }}$ ….. (2)

The value of seconds in years is,

$1\mathrm{s}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3.15\times {10}^7$}\right.\mathrm{years}$ ….. (3)

(a) Calculation of the number of wavelengths:

Substituting the value of length from equation (1) of pulse in equation (2), and using the given data, to get the required value of the wavelengths of the light that are contained in the pulse as follows.

$$\begin{gathered} \mathrm{N}=\frac{\mathrm{c}∆\mathrm{t}}{\mathrm{\lambda }} \\ =\frac{\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)\left(10\times {10}^{-15}\mathrm{s}\right)}{500\times {10}^{=9}\mathrm{m}} \\ =6.0 \end{gathered}$$

Hence, the value of the wavelengths is 6.0.

(b) Calculation of the value of X:

As given the equation,

$\frac{10fs}{1s}=\frac{1\mathrm{s}}{\mathrm{X}}$

Substitute known values in the above equation.

$$\begin{gathered} \mathrm{X}=\frac{{\left(1\mathrm{s}\right)}^2}{10\times {10}^{-15}\mathrm{s}} \\ =\frac{1{\mathrm{s}}^2}{{10}^{-14}\mathrm{s}\times 3.15\times {10}^7\mathrm{s}/\mathrm{years}}\left(\mathrm{from}\mathrm{equation}\left(1\right)\right) \\ =3.2\times {10}^6\mathrm{years} \end{gathered}$$

Hence, the value of is $3.2\times {10}^6\mathrm{years}$.