Fraunhofer Diffraction: Single Slit, Double Slit, and Gratings

1. Single-Slit Diffraction

Let's start with the simplest case: a single rectangular slit of width $a$. When a plane wave illuminates this slit, every point across the slit's width acts as a source of secondary Huygens wavelets. These wavelets interfere to create a characteristic diffraction pattern on a distant screen.

To find the resulting intensity at an angle $\theta$, we must sum the contributions from all these infinitesimal sources across the slit. This requires integration.

1.1 The Intensity Formula

As shown in the derivation, the total electric field is found by integrating the contributions from all the infinitesimal sources across the slit width $a$. The resulting intensity $I(\theta )$ at an angle $\theta$ is given by:

$I(\theta )=I_0{\left(\frac{\sin \alpha }{\alpha }\right)}^2$

where $I_0$ is the maximum intensity at the center of the pattern ($\theta =0$), and the phase parameter $\alpha$ is defined as:

$\alpha =\frac{\varphi }{2}=\frac{1}{2}\left(\frac{2\pi }{\lambda }a\sin \theta \right)=\frac{\pi a\sin \theta }{\lambda }$

Here, $\varphi$ represents the total phase difference between the wavelets originating from the top and bottom edges of the slit. The function $(\sin \alpha /\alpha {)}^2$ is often called the sinc-squared function and describes the characteristic diffraction pattern.

1.2 Analysis of the Pattern

The shape of the intensity curve is a large central maximum flanked by a series of much weaker secondary maxima.

Condition for Minima (Dark Fringes):
The intensity will be zero when $\sin \alpha =0$, but $\alpha \ne 0$ (since $\alpha =0$ corresponds to the central maximum, where ${\lim }_{\alpha \to 0}\frac{\sin \alpha }{\alpha }=1$).
The condition $\sin \alpha =0$ is met when $\alpha =m\pi$ for any non-zero integer $m$.

$\frac{\pi a\sin \theta }{\lambda }=m\pi$
$a\sin \theta =m\lambda ,m=\pm 1,\pm 2,\pm 3,\dots$
This simple equation gives the angular positions of all the dark fringes in the pattern.

Condition for Maxima (Bright Fringes):

• Central Maximum: The principal maximum occurs at $\theta =0$, which means $\alpha =0$. The intensity here is $I_0$.
• Secondary Maxima: These smaller peaks occur approximately halfway between the minima. The exact condition is found by differentiating the intensity function and setting the derivative to zero, which leads to the transcendental equation $\tan \alpha =\alpha$. The solutions are approximately $\alpha \approx \pm 1.43\pi ,\pm 2.46\pi ,\dots$, which is close to, but not exactly, $(m+\frac{1}{2})\pi$.

2. Double-Slit Diffraction

Now, what happens if we have two identical slits, each of width $a$, with their centers separated by a distance $d$?

The resulting pattern is a beautiful combination of two effects:

1. Diffraction due to each individual slit, as described above.
2. Interference between the waves coming from the two different slits, as in Young's double-slit experiment.

The overall intensity pattern is the product of the single-slit diffraction pattern (which acts as an "envelope") and the two-slit interference pattern.

The following video provides a clear step-by-step derivation of the intensity formula.

2.1 The Intensity Formula

As derived in the video and in more detail in the MIT notes (LINK, section 14.7), the intensity for a double-slit is:

$I(\theta )=I_{\text{max}}{\cos }^2\beta {\left(\frac{\sin \alpha }{\alpha }\right)}^2$

Let's break this down:

• The diffraction term, ${\left(\frac{\sin \alpha }{\alpha }\right)}^2$, is the same as for a single slit, with $\alpha =\frac{\pi a\sin \theta }{\lambda }$. It forms the overall envelope of the pattern.
• The interference term, ${\cos }^2\beta$, describes the rapid oscillations of the interference fringes. The phase parameter $\beta$ is given by:
  $\beta =\frac{\pi d\sin \theta }{\lambda }$
  where $d$ is the center-to-center separation of the slits.

2.2 Missing Orders

An interesting phenomenon occurs when a position that should be an interference maximum coincides with a diffraction minimum. When this happens, that bright fringe is "missing" because the diffraction envelope at that position is zero.

• Condition for Interference Maxima: $d\sin \theta =m\lambda$ (where $\beta =m\pi$, so ${\cos }^2\beta =1$).
• Condition for Diffraction Minima: $a\sin \theta =p\lambda$ (where $\alpha =p\pi$, so $\sin \alpha =0$).

A missing order occurs if both conditions are met for the same angle $\theta$. Dividing the two equations gives the condition for missing orders:
$\frac{d\sin \theta }{a\sin \theta }=\frac{m\lambda }{p\lambda }⟹\frac{d}{a}=\frac{m}{p}$
For example, if the slit separation is twice the slit width ($d=2a$), then $m=2p$. The interference maxima for $m=\pm 2,\pm 4,\pm 6,\dots$ will align with the diffraction minima for $p=\pm 1,\pm 2,\pm 3,\dots$ and will be absent from the pattern.

3. The Diffraction Grating (N Slits)

A diffraction grating is a direct extension of the double slit to a very large number, $N$, of parallel, equally spaced slits. Gratings are powerful tools in science and engineering, particularly for spectroscopy—the precise analysis of light's constituent wavelengths.

The derivation follows the same logic: we find the resultant amplitude by summing the contributions from all $N$ slits.

3.1 The Grating Intensity Formula

The intensity pattern for a grating with $N$ slits is given by:

$I(\theta )=I_0{\left(\frac{\sin \alpha }{\alpha }\right)}^2{\left(\frac{\sin (N\beta )}{\sin \beta }\right)}^2$

Again, this is a product of two terms:

• A diffraction term, ${\left(\frac{\sin \alpha }{\alpha }\right)}^2$, identical to the single-slit case.
• An N-slit interference term, ${\left(\frac{\sin (N\beta )}{\sin \beta }\right)}^2$, which governs the locations of the sharp bright lines.

The parameters $\alpha =\frac{\pi a\sin \theta }{\lambda }$ and $\beta =\frac{\pi d\sin \theta }{\lambda }$ are the same as in the double-slit case.

3.2 Analysis of the Grating Pattern

Principal Maxima:
The most important features are the very bright and sharp principal maxima. They occur when the denominator of the interference term is zero, i.e., $\sin \beta =0$. This happens when $\beta =m\pi$, which gives the famous grating equation:

$d\sin \theta =m\lambda ,m=0,\pm 1,\pm 2,\dots$

At these points, the interference term ${\left(\frac{\sin (N\beta )}{\sin \beta }\right)}^2$ is not undefined. Using L'Hôpital's rule, its value approaches $N^2$. This is a crucial result: the intensity of the principal maxima is proportional to $N^2$. A grating with 1000 slits produces peaks that are a million times more intense than the light from a single slit, and $1000^2/2^2=250,000$ times more intense than a double slit!

Minima and Secondary Maxima:

• Minima: Zero intensity occurs when the numerator is zero but the denominator is not: $\sin (N\beta )=0$ but $\sin \beta \ne 0$. This leads to $N-1$ minima between each pair of principal maxima.
• Secondary Maxima: Between these minima, there are $N-2$ small secondary maxima. For large $N$, these are negligible in intensity compared to the principal maxima.

The large number of minima crowded between the principal maxima is what makes the principal maxima extremely sharp and narrow. The angular width of a principal maximum can be shown to be proportional to $1/N$, which explains why gratings with more slits have higher resolving power—the ability to distinguish between two very close wavelengths. This is why diffraction gratings, not prisms, are the instruments of choice for high-resolution spectroscopy.

Conclusion

In this lesson, we have developed a comprehensive theory for Fraunhofer diffraction, building from the simplest case to the most powerful.

Key Takeaways:

• Diffraction is the result of interference from a continuous distribution of Huygens' wavelets within an aperture.
• The single-slit pattern is described by $I(\theta )=I_0(\sin \alpha /\alpha {)}^2$, with minima at $a\sin \theta =m\lambda$.
• The double-slit pattern is a product of two-slit interference and single-slit diffraction: $I(\theta )\propto {\cos }^2\beta (\sin \alpha /\alpha {)}^2$. This gives rise to the phenomenon of "missing orders" when an interference maximum coincides with a diffraction minimum.
• A diffraction grating extends this to $N$ slits, producing extremely sharp and intense principal maxima according to the grating equation $d\sin \theta =m\lambda$. The intensity of these peaks is proportional to $N^2$, and their sharpness is proportional to $1/N$.

Preview of the Next Lesson:
We have now analyzed how the wave nature of light governs the distribution of its intensity in space through interference and diffraction. In the next lesson, we will turn to another fundamental property of electromagnetic waves: polarization. We will explore the transverse nature of light waves, describe different states of polarization (linear, circular, elliptical), and analyze how materials like polarizers and wave plates can manipulate this property.