Diffraction (Fraunhofer Class)

In Fraunhofer diffraction, the source and the observation screen are effectively at infinite distances from the diffracting aperture (realized experimentally by placing the aperture between two collimating and imaging convex lenses). The incident and diffracted waves are strictly planar.

1. Integration Across Aperture Width

Consider a long slit of width $a$ oriented along the $y$-axis, illuminated normally by a monochromatic plane wave of wavelength $\lambda$. Divide the slit into differential strips of width $dx$ at position $x \in [-a/2, a/2]$. For a diffracted angle $\theta$, the path difference of the wavelet from strip $dx$ relative to the center ($x=0$) is: $$\Delta(x) = x \sin \theta$$ The phase difference relative to the center is: $$\phi(x) = k \Delta(x) = \frac{2\pi}{\lambda} x \sin \theta$$ The total diffracted complex electric field amplitude is obtained by integrating across the slit: $$E(\theta) = C \int_{-a/2}^{a/2} e^{i k x \sin \theta} dx$$ Evaluating the integral: $$E(\theta) = C \left[ \frac{e^{i k x \sin \theta}}{i k \sin \theta} \right]_{-a/2}^{a/2} = C \frac{e^{i \frac{k a \sin \theta}{2}} - e^{-i \frac{k a \sin \theta}{2}}}{i k \sin \theta}$$ Defining the dimensionless phase parameter: $$\beta = \frac{k a \sin \theta}{2} = \frac{\pi a \sin \theta}{\lambda}$$ Using Euler's formula $\sin \beta = \frac{e^{i\beta} - e^{-i\beta}}{2i}$: $$E(\theta) = C \frac{2i \sin \beta}{i (2\beta / a)} = (C a) \frac{\sin \beta}{\beta} = E_0 \operatorname{sinc}(\beta)$$

2. Irradiance Distribution

The diffracted intensity is proportional to $|E(\theta)|^2$: $$I(\theta) = I_0 \left( \frac{\sin \beta}{\beta} \right)^2$$

3. Extrema Analysis

• Central Maximum (Principal Maximum): As $\theta \to 0$, $\beta \to 0$, and $\lim_{\beta \to 0} \frac{\sin \beta}{\beta} = 1$. The central intensity is $I(0) = I_0$.
• Diffraction Minima (Zero Intensity): Occur when $\sin \beta = 0$ with $\beta \neq 0$: $$\beta = m \pi \implies \frac{\pi a \sin \theta}{\lambda} = m \pi \implies a \sin \theta_m = m \lambda, \quad m \in \{\pm 1, \pm 2, \pm 3, \dots\}$$
• Secondary Maxima: Occur where $\frac{dI}{d\beta} = 0$: $$\frac{d}{d\beta} \left( \frac{\sin^2 \beta}{\beta^2} \right) = \frac{2\sin\beta(\beta \cos \beta - \sin \beta)}{\beta^3} = 0 \implies \tan \beta = \beta$$ Solving transcendental equation $\tan \beta = \beta$:
  1. 1st Secondary Maximum: $\beta_1 = 1.4303\pi \implies I_1 = \frac{I_0}{(1.4303\pi)^2} \approx 0.0472 I_0 \ (4.72\%)$
  2. 2nd Secondary Maximum: $\beta_2 = 2.4590\pi \implies I_2 \approx 0.0165 I_0 \ (1.65\%)$

The central maximum contains over **$85\%$ of the total diffracted energy**, and its angular width $\Delta \theta_{\text{central}} = 2 \arcsin(\lambda/a)$ is inversely proportional to the slit aperture.

When light passes through two parallel slits of width $a$ separated by center-to-center distance $d$, the resulting pattern is the product of single-slit diffraction and two-beam interference.

1. Mathematical Derivation

Let the two slits be centered at $x = -d/2$ and $x = +d/2$. The diffracted field is: $$E(\theta) = C \left[ \int_{-d/2 - a/2}^{-d/2 + a/2} e^{i k x \sin \theta} dx + \int_{d/2 - a/2}^{d/2 + a/2} e^{i k x \sin \theta} dx \right]$$ Factoring out the phase of each slit center: $$E(\theta) = E_{\text{single}}(\theta) \left( e^{-i \frac{k d \sin \theta}{2}} + e^{+i \frac{k d \sin \theta}{2}} \right) = 2 E_0 \left(\frac{\sin \beta}{\beta}\right) \cos \alpha$$ where: $$\beta = \frac{\pi a \sin \theta}{\lambda} \quad (\text{Diffraction parameter}), \quad \alpha = \frac{\pi d \sin \theta}{\lambda} \quad (\text{Interference parameter})$$ The intensity distribution is: $$I(\theta) = 4 I_0 \left( \frac{\sin \beta}{\beta} \right)^2 \cos^2 \alpha$$ This represents a system of narrow two-beam interference fringes modulated by the broader single-slit diffraction envelope.

2. Phenomenon of Missing Orders (Absent Spectra)

An interference maximum occurs when: $$d \sin \theta = m \lambda \quad (m = 0, 1, 2, \dots)$$ A diffraction minimum occurs when: $$a \sin \theta = p \lambda \quad (p = 1, 2, 3, \dots)$$ If a diffraction minimum coincides with an interference maximum, the zero of the diffraction envelope suppresses that interference fringe entirely. Dividing the two equations: $$\frac{d \sin \theta}{a \sin \theta} = \frac{m \lambda}{p \lambda} \implies \frac{d}{a} = \frac{m}{p} \implies m = p \left(\frac{d}{a}\right)$$ For example, if $d = 3a$ (slit separation is three times the slit width): For $p = 1$, $m = 3$; for $p = 2$, $m = 6$. Thus, the 3rd, 6th, 9th, $\dots$ interference fringes are **completely missing** from the pattern.

A diffraction grating consists of an array of a large number $N$ of equidistant, parallel narrow slits of width $a$ separated by opaque spaces of width $b$. The grating element is $d = a + b$.

1. The $N$-Slit Grating Formula

Integrating over $N$ identical slits separated by distance $d$: $$E(\theta) = E_{\text{single}}(\theta) \sum_{n=0}^{N-1} e^{i n (2\alpha)} = E_0 \left(\frac{\sin \beta}{\beta}\right) \left( \frac{1 - e^{i N 2\alpha}}{1 - e^{i 2\alpha}} \right)$$ The intensity distribution is: $$I(\theta) = I_0 \left( \frac{\sin \beta}{\beta} \right)^2 \left( \frac{\sin N\alpha}{\sin \alpha} \right)^2$$ where $\alpha = \frac{\pi d \sin \theta}{\lambda}$.

2. Principal Maxima

When $\alpha = m \pi$ ($m \in \mathbb{Z}$), $\sin \alpha = 0$ and $\lim_{\alpha \to m\pi} \frac{\sin N\alpha}{\sin \alpha} = N$. The intensity reaches: $$I_{\text{principal}} = N^2 I_0 \left( \frac{\sin \beta}{\beta} \right)^2$$ The angular positions are governed by the fundamental **Grating Equation**: $$d \sin \theta = m \lambda \implies (a + b) \sin \theta = m \lambda, \quad m \in \{0, \pm 1, \pm 2, \dots\}$$

3. Angular Dispersion of a Grating

The angular dispersion measures the angular separation per unit wavelength change. Differentiating the grating equation with respect to $\lambda$: $$d \cos \theta \frac{d\theta}{d\lambda} = m \implies D = \frac{d\theta}{d\lambda} = \frac{m}{d \cos \theta} = \frac{N_g m}{\cos \theta}$$ where $N_g = 1/d$ is the number of ruling lines per unit length (e.g., 600 lines/mm).

The wave nature of light sets a fundamental diffraction limit on the ability of any optical instrument to separate two closely spaced objects or spectral lines.

1. Rayleigh’s Criterion for Two Spectral Lines

Lord Rayleigh established that two spectral lines of equal intensity are defined to be **just resolved** when the central diffraction maximum of the first line falls exactly on the first diffraction minimum of the second line. At this separation, the combined intensity profile displays a dip (saddle) between the two peaks of approximately: $$\frac{I_{\text{saddle}}}{I_{\text{peak}}} = \frac{8}{\pi^2} \approx 0.8106 \ (81.1\%)$$ The human eye and CCD sensors readily perceive this $19\%$ intensity dip.

2. Resolving Power of a Diffraction Grating

The principal maximum of order $m$ for wavelength $\lambda + \Delta \lambda$ occurs at: $$d \sin \theta = m(\lambda + \Delta \lambda)$$ The first minimum of order $m$ for wavelength $\lambda$ occurs when the phase increases by $\pi/N$: $$N d \sin \theta = N m \lambda + \lambda$$ Equating the two conditions at Rayleigh's limit: $$N m(\lambda + \Delta \lambda) = N m \lambda + \lambda \implies N m \Delta \lambda = \lambda$$ The **Chromatic Resolving Power** of a grating is: $$\mathcal{R} = \frac{\lambda}{\Delta \lambda} = m N$$ where $m$ is the spectral order and $N$ is the total number of illuminated grating lines.

3. Resolving Power of a Prism

For a triangular prism of base length $B$ made of a dispersive material with refractive index dispersion $\frac{dn}{d\lambda}$: $$\mathcal{R}_{\text{prism}} = \frac{\lambda}{\Delta \lambda} = B \left| \frac{dn}{d\lambda} \right|$$ Unlike gratings whose resolving power is governed purely by order and line count, a prism's resolving power is determined entirely by its physical base width and material dispersion.