Intensity in Single-Slit Diffraction

To calculate the intensity of the diffraction pattern, we follow the phasor method used for calculations with ac circuits in Alternating-Current Circuits. If we consider that there are N Huygens sources across the slit shown in CNX_UPhysics_37_01_Angle, with each source separated by a distance a/N from its adjacent neighbors, the path difference between waves from adjacent sources reaching the arbitrary point P on the screen is $\left(a\text{/}N\right)\text{sin}\theta .$ This distance is equivalent to a phase difference of $\left(2\pi a\text{/}\lambda N\right)\text{sin}\theta .$ The phasor diagram for the waves arriving at the point whose angular position is $\theta$ is shown in CNX_UPhysics_37_02_PhasorArb. The amplitude of the phasor for each Huygens wavelet is $\text{Δ}{E}_0,$ the amplitude of the resultant phasor is E, and the phase difference between the wavelets from the first and the last sources is

With $N\to \infty$, the phasor diagram approaches a circular arc of length $N\text{Δ}{E}_0$ and radius r. Since the length of the arc is $N\text{Δ}{E}_0$ for any $\varphi$, the radius r of the arc must decrease as $\varphi$ increases (or equivalently, as the phasors form tighter spirals).

The phasor diagram for $\varphi =0$ (the center of the diffraction pattern) is shown in CNX_UPhysics_37_02_PhasorN30(a) using $N=30$. In this case, the phasors are laid end to end in a straight line of length $N\text{Δ}{E}_0,$ the radius r goes to infinity, and the resultant has its maximum value $E=N\text{Δ}{E}_0.$ The intensity of the light can be obtained using the relation $I=\frac{1}{2}c{\epsilon }_0{E}^2$ from Electromagnetic Waves. The intensity of the maximum is then

where ${\epsilon }_0=1\text{/}{\mu }_0{c}^2$. The phasor diagrams for the first two zeros of the diffraction pattern are shown in parts (b) and (d) of the figure. In both cases, the phasors add to zero, after rotating through $\varphi =2\pi$ rad for $m=1$ and $4\pi$ rad for $m=2$.

The next two maxima beyond the central maxima are represented by the phasor diagrams of parts (c) and (e). In part (c), the phasors have rotated through $\varphi =3\pi$ rad and have formed a resultant phasor of magnitude $E_1$. The length of the arc formed by the phasors is $N\text{Δ}{E}_0.$ Since this corresponds to 1.5 rotations around a circle of diameter $E_1$, we have

so

and

where

In part (e), the phasors have rotated through $\varphi =5\pi$ rad, corresponding to 2.5 rotations around a circle of diameter $E_2$ and arc length $N\text{Δ}{E}_0.$ This results in $I_2\approx 0.016I_0$. The proof is left as an exercise for the student (fs-id1172100729513).

These two maxima actually correspond to values of $\varphi$ slightly less than $3\pi$ rad and $5\pi$ rad. Since the total length of the arc of the phasor diagram is always $N\text{Δ}{E}_0,$ the radius of the arc decreases as $\varphi$ increases. As a result, $E_1$ and $E_2$ turn out to be slightly larger for arcs that have not quite curled through $3\pi$ rad and $5\pi$ rad, respectively. The exact values of $\varphi$ for the maxima are investigated in fs-id1172100896762. In solving that problem, you will find that they are less than, but very close to, $\varphi =3\pi ,5\pi ,7\pi ,\text{…}\text{rad}.$

To calculate the intensity at an arbitrary point P on the screen, we return to the phasor diagram of CNX_UPhysics_37_02_PhasorArb. Since the arc subtends an angle $\varphi$ at the center of the circle,

and

where E is the amplitude of the resultant field. Solving the second equation for E and then substituting r from the first equation, we find

Now defining

we obtain

This equation relates the amplitude of the resultant field at any point in the diffraction pattern to the amplitude $N\text{Δ}{E}_0$ at the central maximum. The intensity is proportional to the square of the amplitude, so

where $I_0={\left(N\text{Δ}{E}_0\right)}^2\text{/}2{\mu }_{0}c$ is the intensity at the center of the pattern.

For the central maximum, $\varphi =0$, $\beta$ is also zero and we see from l’Hôpital’s rule that ${\text{lim}}_{\beta \to 0}\left(\text{sin}\beta \text{/}\beta \right)=1,$ so that ${\text{lim}}_{\varphi \to 0}I=I_0.$ For the next maximum, $\varphi =3\pi$ rad, we have $\beta =3\pi \text{/}2$ rad and when substituted into fs-id1172101777014, it yields

in agreement with what we found earlier in this section using the diameters and circumferences of phasor diagrams. Substituting $\varphi =5\pi$ rad into fs-id1172101777014 yields a similar result for $I_2$.

A plot of fs-id1172101777014 is shown in CNX_UPhysics_37_02_plot and directly below it is a photograph of an actual diffraction pattern. Notice that the central peak is much brighter than the others, and that the zeros of the pattern are located at those points where $\text{sin}\beta =0,$ which occurs when $\beta =m\pi$ rad. This corresponds to

or

which is fs-id1172098449014.

Intensity in Single-Slit Diffraction

Strategy

Solution

1. Solving fs-id1172098449014 for $\theta$ gives us ${\theta }_m={\text{sin}}^{\mathrm{-1}}\left(m\lambda \text{/}a\right),$ so that
   ${\theta }_1={\text{sin}}^{\mathrm{-1}}\left(\frac{\left(+1\right)\left(550\times {10}^{\mathrm{-9}}\text{m}\right)}{2.00\times {10}^{\mathrm{-6}}\text{m}}\right)=+16.0\text{°}$
   and
   ${\theta }_2={\text{sin}}^{\mathrm{-1}}\left(\frac{\left(+2\right)\left(550\times {10}^{\mathrm{-9}}\text{m}\right)}{2.00\times {10}^{\mathrm{-6}}\text{m}}\right)=+33.4\text{°}.$
2. The halfway point between ${\theta }_1$ and ${\theta }_2$ is $\theta =\left({\theta }_1+{\theta }_2\right)\text{/}2=\left(16.0\text{°}+33.4\text{°}\right)\text{/}2=24.7\text{°}.$

fs-id1172102022017 gives

From fs-id1172101777014, we can calculate

Significance

If the slit width a is varied, the intensity distribution changes, as illustrated in CNX_UPhysics_37_02_VaryWidth. The central peak is distributed over the region from $\text{sin}\theta =\text{−}\lambda \text{/}a$ to $\text{sin}\theta =+\lambda \text{/}a$. For small $\theta$, this corresponds to an angular width $\text{Δ}\theta \approx 2\lambda \text{/}a.$ Hence, an increase in the slit width results in a decrease in the width of the central peak. For a slit with $a\gg \lambda ,$ the central peak is very sharp, whereas if $a\approx \lambda$, it becomes quite broad.