36-2 Intensity in Single-Slit Diffraction#
Prompts
For single-slit diffraction, what is the intensity formula \(I(\theta)\) in terms of \(I_m\), \(\alpha\), and the angle \(\theta\)? What is \(\alpha\)?
At what values of \(\alpha\) do the minima occur? Show that this reproduces \(a\sin\theta = m\lambda\) from Section 36-1.
Where are the secondary maxima approximately located? Why is the central maximum much brighter than the secondary maxima?
Why does the factor \((\sin\alpha/\alpha)^2\) approach 1 as \(\alpha \to 0\)? What does that tell you about the center of the pattern?
In the double-slit pattern (Section 36-4), the single-slit factor \((\sin\alpha/\alpha)^2\) acts as an envelope. What does that mean for the relative intensities of the interference fringes?
Lecture Notes#
Overview#
The intensity \(I(\theta)\) in the single-slit diffraction pattern varies with angle \(\theta\) according to \(I = I_m (\sin\alpha/\alpha)^2\), where \(\alpha = (\pi a/\lambda)\sin\theta\) and \(I_m\) is the intensity at the center.
Minima occur when \(\alpha = m\pi\) (\(m = \pm 1, \pm 2, \ldots\)), reproducing \(a\sin\theta = m\lambda\) from Section 36-1.
Secondary maxima lie roughly halfway between minima; they are much weaker than the central maximum (the first is at \(\sim 4.5\%\) of \(I_m\)).
This diffraction factor appears as an envelope in the double-slit pattern (Section 36-4), modulating the intensity of the interference fringes.
The Intensity Formula#
For monochromatic light of wavelength \(\lambda\) passing through a slit of width \(a\), the intensity at angle \(\theta\) from the central axis is
where
and \(I_m\) is the intensity at the center of the pattern (\(\theta = 0\)).
Physical origin
The slit is treated as a continuous distribution of Huygens sources. Adding their contributions (e.g., via a phasor diagram) yields a resultant amplitude proportional to \(\sin\alpha/\alpha\). Since intensity \(\propto\) (amplitude)\(^2\), we obtain \(I \propto (\sin\alpha/\alpha)^2\).
Minima#
Intensity is zero when \(\sin\alpha = 0\) and \(\alpha \neq 0\) — i.e., when \(\alpha = m\pi\) for \(m = \pm 1, \pm 2, \ldots\):
This matches the minima condition from Section 36-1 (zone method).
Central Maximum#
At \(\theta = 0\), we have \(\alpha = 0\). The limit \(\lim_{\alpha \to 0} (\sin\alpha/\alpha) = 1\), so \(I(0) = I_m\). The central maximum is the brightest part of the pattern.
Secondary Maxima#
The secondary maxima occur roughly halfway between adjacent minima. They are found where \(d/d\alpha\,(\sin\alpha/\alpha)^2 = 0\) (excluding \(\alpha = 0\)).
Maximum |
Approximate \(\alpha\) |
Relative intensity \(I/I_m\) |
|---|---|---|
Central |
0 |
1 |
First side |
\(\approx 1.43\pi\) |
\(\approx 0.047\) |
Second side |
\(\approx 2.46\pi\) |
\(\approx 0.017\) |
Third side |
\(\approx 3.47\pi\) |
\(\approx 0.008\) |
The secondary maxima are much weaker than the central maximum (the first is only about 4.5% of the central intensity).
Intensity vs. Angle#
The intensity pattern has:
A broad central maximum at \(\theta = 0\);
Minima at \(a\sin\theta = m\lambda\) (\(m = \pm 1, \pm 2, \ldots\));
Secondary maxima between the minima, with decreasing height as \(|\theta|\) increases.
Connection to double-slit
In the double-slit pattern (Section 36-4), the intensity is \(I = I_m (\cos\beta)^2 (\sin\alpha/\alpha)^2\), where \(\beta = (\pi d/\lambda)\sin\theta\). The factor \((\sin\alpha/\alpha)^2\) is the diffraction envelope — it limits the intensity of the interference fringes and can eliminate some of them when a diffraction minimum coincides with an interference maximum.
Summary#
The single-slit intensity is \(I = I_m (\sin\alpha/\alpha)^2\) with \(\alpha = (\pi a/\lambda)\sin\theta\).
Minima occur at \(\alpha = m\pi\) \(\Leftrightarrow\) \(a\sin\theta = m\lambda\).
The central maximum is brightest; secondary maxima are much weaker (~4.5%, 1.7%, 0.8%, …).
This diffraction factor acts as an envelope in the double-slit pattern.