4.2 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 Figure 4.4, 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 $open parenthesis a / N close parenthesis sin θ.$ This distance is equivalent to a phase difference of $open parenthesis 2 π a / λ N close parenthesis sin θ.$ The phasor diagram for the waves arriving at the point whose angular position is $θ$ is shown in Figure 4.7. The amplitude of the phasor for each Huygens wavelet is $Δ E sub 0,$ the amplitude of the resultant phasor is E, and the phase difference between the wavelets from the first and the last sources is

$ϕ = (\frac{2 π}{λ}) a sin θ .$

With $N → ∞$ , the phasor diagram approaches a circular arc of length $N Δ E sub 0$ and radius r. Since the length of the arc is $N Δ E sub 0$ for any $ϕ$ , the radius r of the arc must decrease as $ϕ$ increases (or equivalently, as the phasors form tighter spirals).

Figure a shows an arc with phasors labeled delta E subscript 0. This subtends an angle at the center of the circle, through two lines labeled r. This angle is bisected and each half is labeled phi by 2. The endpoints of the arc are connected by an arrow labeled E. The tangent at one endpoint of the arc is horizontal. The tangent at the other endpoint of the arc makes an angle phi with the horizontal. Figure b shows the arc and the angle phi subtended by it. A dotted line extends from one endpoint of the arc to the opposite line r. It is perpendicular to r. It makes an angle phi with the arc and an angle 90 minus phi with the adjacent line r. — **Figure 4.7** (a) Phasor diagram corresponding to the angular position $θ$ in the single-slit diffraction pattern. The phase difference between the wavelets from the first and last sources is $ϕ equals open parenthesis 2 π / λ close parenthesis a sin θ$ . (b) The geometry of the phasor diagram.

The phasor diagram for $ϕ equals 0$ (the center of the diffraction pattern) is shown in Figure 4.8(a) using $N equals 30$ . In this case, the phasors are laid end to end in a straight line of length $N Δ E sub 0,$ the radius r goes to infinity, and the resultant has its maximum value $E equals N Δ E sub 0.$ The intensity of the light can be obtained using the relation $I = \frac{1}{2} c ε_{0} E^{2}$ from Electromagnetic Waves. The intensity of the maximum is then

$I_{0} = \frac{1}{2} c ε_{0} {(N Δ E_{0})}^{2} = \frac{1}{2 μ_{0} c} {(N Δ E_{0})}^{2},$

where $ε sub 0 equals 1 / μ sub 0 c squared$ . 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 $ϕ equals 2 π$ rad for $m equals 1$ and $4 π$ rad for $m equals 2$ .

Figure a shows 30 phasors in a line of length N delta E subscript 0. The length of a phasor is delta E subscript 0. Figure b shows a circle with phasors pointing in the anticlockwise direction. This is labeled m equal to 1, E equal to 0. Figure c shows phasors along a circle. They start from the bottom and go one and a half times around the circle in the anticlockwise direction. An arrow from the starting point to the ending point is labeled E1. It forms a diameter of the circle. Figure c is labeled 3 by 2 pi E1 equal to N delta E0. Figure d shows phasors along a circle. They start from the bottom and go twice around the circle in the anticlockwise direction. The figure is labeled m equal to 2, E equal to 0. Figure e shows phasors along a circle. They start from the bottom and go two and a half times around the circle in the anticlockwise direction. An arrow from the starting point to the ending point is labeled E2. It forms a diameter of the circle. Figure c is labeled 5 by 2 pi E2 equal to N delta E0. — **Figure 4.8** Phasor diagrams (with 30 phasors) for various points on the single-slit diffraction pattern. Multiple rotations around a given circle have been separated slightly so that the phasors can be seen. (a) Central maximum, (b) first minimum, (c) first maximum beyond central maximum, (d) second minimum, and (e) second maximum beyond central maximum.

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 $ϕ equals 3 π$ rad and have formed a resultant phasor of magnitude $E sub 1$ . The length of the arc formed by the phasors is $N Δ E sub 0.$ Since this corresponds to 1.5 rotations around a circle of diameter $E sub 1$ , we have

$\frac{3}{2} π E_{1} \approx N Δ E_{0},$

$E_{1} = \frac{2 N Δ E_{0}}{3 π}$

and

$I_{1} = \frac{1}{2 μ_{0} c} E_{1}^{2} = \frac{4 {(N Δ E_{0})}^{2}}{(9 π^{2}) (2 μ_{0} c)} \approx 0.045 I_{0},$

where

$I_{0} = \frac{{(N Δ E_{0})}^{2}}{2 μ_{0} c} .$

In part (e), the phasors have rotated through $ϕ equals 5 π$ rad, corresponding to 2.5 rotations around a circle of diameter $E sub 2$ and arc length $N Δ E sub 0.$ This results in $I sub 2 approximately equals 0.016 I sub 0$ . The proof is left as an exercise for the student (Exercise 4.119).

These two maxima actually correspond to values of $ϕ$ slightly less than $3 π$ rad and $5 π$ rad. Since the total length of the arc of the phasor diagram is always $N Δ E sub 0,$ the radius of the arc decreases as $ϕ$ increases. As a result, $E sub 1$ and $E sub 2$ turn out to be slightly larger for arcs that have not quite curled through $3 π$ rad and $5 π$ rad, respectively. The exact values of $ϕ$ for the maxima are investigated in Exercise 4.120. In solving that problem, you will find that they are less than, but very close to, $ϕ equals 3 π, 5 π, 7 π, … rad.$

To calculate the intensity at an arbitrary point P on the screen, we return to the phasor diagram of Figure 4.7. Since the arc subtends an angle $ϕ$ at the center of the circle,

$N Δ E sub 0 equals r ϕ$

and

$sin (\frac{ϕ}{2}) = \frac{E}{2 r} .$

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

$E = 2 r sin \frac{ϕ}{2} = 2 \frac{N Δ E_{o}}{ϕ} sin \frac{ϕ}{2} .$

Now defining

$β = \frac{ϕ}{2} = \frac{π a sin θ}{λ}$

(4.2)

we obtain

$E = N Δ E_{0} \frac{sin β}{β}$

This equation relates the amplitude of the resultant field at any point in the diffraction pattern to the amplitude $N Δ E sub 0$ at the central maximum. The intensity is proportional to the square of the amplitude, so

$I = I_{0} {(\frac{sin β}{β})}^{2}$

(4.4)

where $I sub 0 equals open parenthesis N Δ E sub 0 close parenthesis squared / 2 μ sub 0 c$ is the intensity at the center of the pattern.

For the central maximum, $ϕ equals 0$ , $β$ is also zero and we see from l’Hôpital’s rule that $lim sub β → 0 open parenthesis sin β / β close parenthesis equals 1,$ so that $lim sub ϕ → 0 I equals I sub 0.$ For the next maximum, $ϕ equals 3 π$ rad, we have $β equals 3 π / 2$ rad and when substituted into Equation 4.4, it yields

$I_{1} = I_{0} {(\frac{sin 3 π / 2}{3 π / 2})}^{2} \approx 0.045 I_{0},$

in agreement with what we found earlier in this section using the diameters and circumferences of phasor diagrams. Substituting $ϕ equals 5 π$ rad into Equation 4.4 yields a similar result for $I sub 2$ .

A plot of Equation 4.4 is shown in Figure 4.9 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 $sin β equals 0,$ which occurs when $β = m π$ rad. This corresponds to

$\frac{π a sin θ}{λ} = m π,$

$a sin θ = m λ,$

which is Equation 4.1.

Figure a shows a graph of I by I0 versus beta. There is a crest at the center of the graph at beta equal to 0. The y-value of this is 1. The graph has ripples on both sides of this which grow smaller as you go outwards. The graph has zeroes at minus 3 pi, minus 2 pi, minus pi, pi, 2 pi, 3 pi. Figure b shows a strip with alternating light and dark regions. The central portion is brightest. — **Figure 4.9** (a) The calculated intensity distribution of a single-slit diffraction pattern. (b) The actual diffraction pattern.

Example 1 Intensity in Single-Slit Diffraction

Light of wavelength 550 nm passes through a slit of width $2.00 μ m$ and produces a diffraction pattern similar to that shown in Figure 4.9. (a) Find the locations of the first two minima in terms of the angle from the central maximum and (b) determine the intensity relative to the central maximum at a point halfway between these two minima.

Strategy The minima are given by Equation 4.1, $a sin θ = m λ$ . The first two minima are for $m equals 1$ and $m equals 2.$ Equation 4.4 and Equation 4.2 can be used to determine the intensity once the angle has been worked out.

Solution

Solving Equation 4.1 for $θ$ gives us $θ_{m} = {sin}^{−1} (m λ / a),$ so that

$θ_{1} = {sin}^{−1} (\frac{(+ 1) (550 \times 10^{−9} m)}{2.00 \times 10^{−6} m}) = + 16.0 °$

and

$θ_{2} = {sin}^{−1} (\frac{(+ 2) (550 \times 10^{−9} m)}{2.00 \times 10^{−6} m}) = + 33.4 ° .$
The halfway point between $θ sub 1$ and $θ sub 2$ is
$θ = (θ_{1} + θ_{2}) / 2 = (16.0 ° + 33.4 °) / 2 = 24.7 ° .$

Equation 4.2 gives

$β = \frac{π a sin θ}{λ} = \frac{π (2.00 \times 10^{−6} m) sin (24.7 °)}{(550 \times 10^{−9} m)} = 1.52 π or 4.77 rad .$

From Equation 4.4, we can calculate

$\frac{I}{I_{o}} = {(\frac{sin β}{β})}^{2} = {(\frac{sin (4.77)}{4.77})}^{2} = {(\frac{−0.9985}{4.77})}^{2} = 0.044 .$

Significance This position, halfway between two minima, is very close to the location of the maximum, expected near $β equals 3 π / 2, or 1.5 π$ .

If the slit width a is varied, the intensity distribution changes, as illustrated in Figure 4.10. The central peak is distributed over the region from $sin θ equals − λ / a$ to $sin θ = + λ / a$ . For small $θ$ , this corresponds to an angular width $Δ θ approximately equals 2 λ / a.$ Hence, an increase in the slit width results in a decrease in the width of the central peak. For a slit with $a ≫ λ,$ the central peak is very sharp, whereas if $a \approx λ$ , it becomes quite broad.

Function graph showing y = 100*(sin(pi*awid*sin(theta*pi/180)/0.55+0.0000001)/(pi*awid*sin(theta*pi/180)/0.55+0.0000001))^2 on theta in [-90, 90], the parametric curve (asin(0.55/awid)*180/pi, t) for t in [0, 100] and the parametric curve (-asin(0.55/awid)*180/pi, t) for t in [0, 100]. Adjustable parameter: Slit width a (wavelength fixed at λ = 550 nm) (awid) = 2 μm. Viewing window: x from -92.17 to 92.17, y from -12 to 102. — The intensity this section derives, I = I₀ (sin(β/2)/(β/2))² with β/2 = πa sin θ/λ, drawn as a percentage of the central maximum against the diffraction angle θ. The wavelength is held at the example's λ = 550 nm and the slider is the slit width a, starting at the example's a = 2.00 μm; the first two minima the example computes, 16.0° and 33.4°, are the first two places this curve touches zero, and the inner pair is marked by the vertical lines that bound the central peak. The side peaks are almost nothing beside the centre — the section's Check Your Understanding puts the third maximum at 74.3° with 0.0083 I₀, which is 0.83% on this axis. Drag a up toward 4.4 μm and the central peak narrows while the zeros crowd in on it; drag it down to 0.6 μm and the central peak swells until its first zeros are out at ±66°, almost the whole screen, which is the section's remark that a slit with a ≈ λ diffracts very broadly.

Figures a through c show graphs of I by I0 versus theta in degrees. Each has a wave crest with y value 1 at x=0. Figure a, labeled a equal to lambda has a broad arc. Figure b, labeled a equal to 5 lambda has a narrower crest. It has zeroes roughly between 10 and 15 and between minus 10 and minus 15. Figure c, labeled a equal to 10 lambda has a narrow crest. It has zeroes at plus and minus 5, roughly between 10 and 15 and between minus 10 and minus 15. — **Figure 4.10** Single-slit diffraction patterns for various slit widths. As the slit width a increases from $a equals λ to 5 λ$ and then to $10λ$ , the width of the central peak decreases as the angles for the first minima decrease as predicted by Equation 4.1.

Summary

The intensity pattern for diffraction due to a single slit can be calculated using phasors as

$I = I_{0} {(\frac{sin β}{β})}^{2},$

where $β = \frac{ϕ}{2} = \frac{π a sin θ}{λ}$ , a is the slit width, $λ$ is the wavelength, and $θ$ is the angle from the central peak.

Conceptual Questions

In Equation 4.4, the parameter $β$ looks like an angle but is not an angle that you can measure with a protractor in the physical world. Explain what $β$ represents.

The parameter $β equals ϕ / 2$ is the arc angle shown in the phasor diagram in Figure 4.7. The phase difference between the first and last Huygens wavelet across the single slit is $2 β$ and is related to the curvature of the arc that forms the resultant phasor that determines the light intensity.

Problems

A single slit of width $3.0 μ m$ is illuminated by a sodium yellow light of wavelength 589 nm. Find the intensity at a $15 °$ angle to the axis in terms of the intensity of the central maximum.

A single slit of width 0.1 mm is illuminated by a mercury light of wavelength 576 nm. Find the intensity at a $10 °$ angle to the axis in terms of the intensity of the central maximum.

$I / I_{0} = 2.2 \times 10^{−5}$

The width of the central peak in a single-slit diffraction pattern is 5.0 mm. The wavelength of the light is 600 nm, and the screen is 2.0 m from the slit. (a) What is the width of the slit? (b) Determine the ratio of the intensity at 4.5 mm from the center of the pattern to the intensity at the center.

Consider the single-slit diffraction pattern for $λ equals 600 nm$ , $a equals 0.025 mm$ , and $x equals 2.0 m$ . Find the intensity in terms of $I_{o}$ at $θ equals 0.5 °$ , $1.0 °$ , $1.5 °$ , $3.0 °$ , and $10.0 °$ .

$0.63 I sub 0, 0.11 I sub 0, 0.0067 I sub 0, 0.0062 I sub 0, 0.00088 I sub 0$