$$\require{cancel}$$

# 4: Diffraction

In the preceding chapter, we implicitly regarded slits as objects with positions but no size. The widths of the slits were considered negligible. When the slits have finite widths, each point along the opening can be considered a point source of light—a foundation of Huygens’s principle. Because real-world optical instruments must have finite apertures (otherwise, no light can enter), diffraction plays a major role in the way we interpret the output of these optical instruments. For example, diffraction places limits on our ability to resolve images or objects. This is a problem that we will study later in this chapter.

• 4.1: Prelude to Diffraction
Due to Huygens’s principle, we can imagine a wave front as equivalent to inﬁnitely many point sources of waves. Thus, a wave from a slit can behave not as one wave but as an infinite number of point sources. These waves can interfere with each other, resulting in an interference pattern without the presence of a second slit. This phenomenon is called diffraction.
• 4.2: Single-Slit Diffraction
Diffraction can send a wave around the edges of an opening or other obstacle. A single slit produces an interference pattern characterized by a broad central maximum with narrower and dimmer maxima to the sides.
• 4.3: Intensity in Single-Slit Diffraction
The intensity pattern for diffraction due to a single slit can be calculated using phasors as $$I = I_0 \left(\frac{sin \space \beta}{\beta}\right)^2,$$  where $$\beta = \frac{\phi}{2} = \frac{\pi D \space sin \space \theta}{\lambda}$$, D is the slit width, λλ is the wavelength, and θθ is the angle from the central peak.
• 4.4: Double-Slit Diffraction
With real slits with finite widths, the effects of interference and diffraction operate simultaneously to form a complicated intensity pattern. Relative intensities of interference fringes within a diffraction pattern can be determined. Missing orders occur when an interference maximum and a diffraction minimum are located together.
• 4.5: Diffraction Gratings
A diffraction grating consists of a large number of evenly spaced parallel slits that produce an interference pattern similar to but sharper than that of a double slit.     Constructive interference occurs when $$d \space sin \space \theta = m \lambda$$ form = 0, ± 1, ±2,..., where d is the distance between the slits, θ is the angle relative to the incident direction, and m is the order of the interference.
• 4.6: Circular Apertures and Resolution
Light diffracts as it moves through space, bending around obstacles, interfering constructively and destructively. This can be used as a spectroscopic tool—a diffraction grating disperses light according to wavelength, for example, and is used to produce spectra—but diffraction also limits the detail we can obtain in images.Diffraction limits the resolution in many situations. The acuity of our vision is limited because light passes through the pupil, which is the circular aperture of the eye.
• 4.7: X-Ray Diffraction
Since X-ray photons are very energetic, they have relatively short wavelengths. Thus, typical X-ray photons act like rays when they encounter macroscopic objects, like teeth, and produce sharp shadows. However, since atoms are on the order of 0.1 nm in size, X-rays can be used to detect the location, shape, and size of atoms and molecules. The process is called X-ray diffraction, and it involves the interference of X-rays to produce patterns.
• 4.8: Holography
A hologram is a true three-dimensional image recorded on film by lasers. Holograms are used for amusement; decoration on novelty items and magazine covers; security on credit cards and driver’s licenses (a laser and other equipment are needed to reproduce them); and for serious three-dimensional information storage. You can see that a hologram is a true three-dimensional image because objects change relative position in the image when viewed from different angles.