Diffraction

4.1 Huygens-Fresnel Principle

Every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront is the Envelope of these wavelets, accounting for both amplitude and phase.

Kirchhoff diffraction integral. The field at point $P$ due to an aperture in a screen is:

$E(P) = \frac{i}{\lambda}\iint_{\mathrm{aperture} E(Q)\,\frac{e^{-ikr}}{r}\cos\theta\,dS}$

Where $E(Q)$ is the field at the aperture point $Q$, $r$ is the distance from $Q$ to $P$And $\theta$ Is the angle between the normal to the aperture and the direction to $P$. The obliquity factor $\cos\theta$ ensures that wavelets do not propagate backwards. In the Fraunhofer limit ($r \to \infty$), This integral reduces to the Fourier transform of the aperture function (see Sections 4.8 and 7.1).

4.2 Single-Slit Diffraction

A slit of width $a$ is illuminated by plane waves of wavelength $\lambda$.

Intensity distribution (Fraunhofer diffraction):

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

Where $\alpha = \frac{\pi a \sin\theta}{\lambda}$.

Derivation. Divide the slit into infinitesimal elements of width $dy$ at position $y$. Each Element contributes a wavelet. The field at angle $\theta$ on a distant screen:

$E(\theta) = \int_{-a/2}^{a/2} E_0\, e^{iky\sin\theta}\,dy = E_0 \frac{\sin\left(\frac{ka\sin\theta}{2}\right)}{\frac{k\sin\theta}{2}} = E_0 a \frac{\sin\alpha}{\alpha}$

Where $\alpha = ka\sin\theta/2 = \pi a\sin\theta/\lambda$. Since $I \propto |E|^2$:

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

$\blacksquare$

Minima: $\alpha = m\pi$I.e., $a\sin\theta = m\lambda$ for $m = \pm 1, \pm 2, \ldots$

Central maximum: at $\theta = 0$With width (first zero to first zero) $\Delta\theta = 2\lambda/a$.

The secondary maxima occur approximately midway between consecutive minima. Their intensities are: $I_1/I_0 \approx 0.045$ (first secondary), $I_2/I_0 \approx 0.016$ (second), decreasing rapidly.

:::caution Common Pitfall The condition for the first minimum in single-slit diffraction is $a\sin\theta = \lambda$ (not $\lambda/2$). The factor of 2 difference from the double-slit maximum condition ($d\sin\theta = \lambda$) Reflects the fundamentally different geometry: in single-slit diffraction, the minimum occurs when Wavelets from the edges cancel, requiring a path difference of one full wavelength between them.

4.3 Double-Slit with Finite Width

Combining single-slit diffraction and double-slit interference:

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

Where $\alpha = \pi a\sin\theta/\lambda$ (diffraction envelope) and $\beta = \pi d\sin\theta/\lambda$ (interference fringes).

The interference fringes are modulated by the diffraction envelope. Missing orders occur when $\beta = m\pi$ coincides with $\alpha = n\pi$I.e., when $d/a$ is a ratio of integers.

4.4 Diffraction Gratings

A grating with $N$ slits, each of width $a$Separated by distance $d$:

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

Principal maxima: $d\sin\theta = m\lambda$ ($m = 0, \pm 1, \pm 2, \ldots$).

The angular width of a principal maximum is $\Delta\theta = \lambda/(Nd\cos\theta)$. The resolving power of a grating is:

$R = \frac{\lambda}{\Delta\lambda} = mN$

Where $N$ is the total number of illuminated slits.

4.5 The Rayleigh Criterion

Two point sources are just resolvable when the central maximum of one coincides with the first minimum Of the other:

$\theta_{\mathrm{min} = 1.22\frac{\lambda}{D}}$

Where $D$ is the aperture diameter (for a circular aperture).

4.6 Fresnel vs. Fraunhofer Diffraction

Fraunhofer (far-field): Source and screen are at infinity (or at the focal plane of a lens). Requires $a^2/\lambda \ll L$ (Fresnel number $N_F \ll 1$).

Fresnel (near-field): Source and/or screen are at finite distances. The Fresnel number $N_F = a^2/(\lambda L)$ characterises the regime. Fresnel diffraction uses Fresnel integrals and Produces patterns that depend on the distance.

4.7 Circular Aperture and the Airy Disk

For a circular aperture of diameter $D$The Fraunhofer diffraction pattern is an Airy pattern:

$I(\theta) = I_0 \left[\frac{2J_1(\beta)}{\beta}\right]^2$

Where $\beta = \pi D \sin\theta / \lambda$ and $J_1$ is the first-order Bessel function of the first Kind.

Derivation. The field in the Fraunhofer limit is the Fourier transform of the circular aperture Function $t(r) = 1$ for $r \leq D/2$ and $0$ otherwise. In polar coordinates:

$E(\theta) \propto \int_0^{D/2} J_0(kr\sin\theta)\, r\,dr = \frac{D}{2}\frac{J_1(\beta)}{\beta}$

Where we used the identity $\int_0^a J_0(\rho r)\,r\,dr = aJ_1(\rho a)/\rho$. Since $I \propto |E|^2$ The result follows. $\blacksquare$

The first zero of $J_1(\beta)$ is at $\beta = 1.22\pi$Giving:

$\sin\theta_1 = 1.22\frac{\lambda}{D}$

The bright central disk (the Airy disk) subtends an angle:

$\theta_{\mathrm{Airy} = 1.22\frac{\lambda}{D}}$

This is the basis of the Rayleigh criterion for resolving power of circular apertures (telescopes, Microscopes, the eye). Approximately 84% of the total transmitted power falls within the Airy disk.

4.8 Introduction to Fourier Optics

The Fraunhofer diffraction integral has a deep connection with Fourier analysis. For an aperture With transmission function $t(x, y)$The far-field diffraction pattern is:

$E(\theta_x, \theta_y) \propto \iint t(x,y)\, e^{-i(k_x x + k_y y)}\,dx\,dy$

Where $k_x = k\sin\theta_x$ and $k_y = k\sin\theta_y$. This is precisely the two-dimensional Fourier transform of $t(x,y)$Evaluated at spatial frequencies $k_x/(2\pi)$ and $k_y/(2\pi)$.

Key consequences:

1. A lens of focal length $f$Placed one focal length after the aperture, produces the Fourier transform at its back focal plane — it performs an optical Fourier transform.
2. Narrow features in the aperture (small $a$) produce broad diffraction patterns (large spread in $k$-space), and vice versa — the optical analogue of the uncertainty principle.
3. Spatial filtering: by placing masks in the Fourier plane, one can selectively remove or enhance spatial frequency components, modifying the image (the basis of optical image processing).

Example. A single slit of width $a$ has aperture function $t(x) = \mathrm{rect}(x/a)$. Its Fourier Transform is $\mathrm{sinc}(\pi a \sin\theta/\lambda)$Directly giving the single-slit diffraction Pattern. A periodic grating has sharp peaks in the Fourier transform (the diffraction orders), each Corresponding to a spatial harmonic of the grating.

Spatial filtering. A powerful application of Fourier optics is the manipulation of images by Modifying their spatial frequency content:

• Low-pass filter: A small aperture in the Fourier plane passes only the zeroth and low-order diffraction, removing fine detail (smoothing).
• High-pass filter: An opaque spot blocking the zeroth order removes the DC component, enhancing edges and fine structure (phase contrast microscopy).
• Band-pass filter: Selective removal of specific spatial frequencies (e.g., removing periodic noise from an image).

Phase contrast microscopy (Zernike, 1953) is a celebrated application. Biological specimens are Mostly transparent (phase objects) and produce no intensity contrast in ordinary microscopy. By Introducing a $\pi/2$ phase shift to the undiffracted (zeroth-order) light in the Fourier plane, Phase variations are converted to intensity variations, making transparent structures visible.

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