### 27.1 The Wave Aspect of Light: Interference

- Wave optics is the branch of optics that must be used when light interacts with small objects or whenever the wave characteristics of light are considered.
- Wave characteristics are those associated with interference and diffraction.
- Visible light is the type of electromagnetic wave to which our eyes respond and has a wavelength in the range of 380 to 760 nm.
- Like all EM waves, the following relationship is valid in vacuum: $c=f\lambda $, where $c=3\times {\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}$ is the speed of light, $f$ is the frequency of the electromagnetic wave, and $\lambda $ is its wavelength in vacuum.
- The wavelength ${\lambda}_{\text{n}}$ of light in a medium with index of refraction $n$ is ${\lambda}_{\text{n}}=\lambda /n$. Its frequency is the same as in vacuum.

### 27.2 Huygens's Principle: Diffraction

- An accurate technique for determining how and where waves propagate is given by Huygens’s principle: Every point on a wavefront is a source of wavelets that spread out in the forward direction at the same speed as the wave itself. The new wavefront is a line tangent to all of the wavelets.
- Diffraction is the bending of a wave around the edges of an opening or other obstacle.

### 27.3 Young’s Double Slit Experiment

- Young’s double slit experiment gave definitive proof of the wave character of light.
- An interference pattern is obtained by the superposition of light from two slits.
- There is constructive interference when $d\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =\mathrm{m\lambda}\phantom{\rule{0.25em}{0ex}}\text{(for}\phantom{\rule{0.25em}{0ex}}m=\mathrm{0,}\phantom{\rule{0.25em}{0ex}}\mathrm{1,}\phantom{\rule{0.25em}{0ex}}-\mathrm{1,}\phantom{\rule{0.25em}{0ex}}\mathrm{2,}\phantom{\rule{0.25em}{0ex}}-\mathrm{2,}\phantom{\rule{0.25em}{0ex}}\dots )$, where $d$ is the distance between the slits, $\theta $ is the angle relative to the incident direction, and $m$ is the order of the interference.
- There is destructive interference when $d\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =\left(m+\frac{1}{2}\right)\lambda \phantom{\rule{0.25em}{0ex}}\text{(for}\phantom{\rule{0.5em}{0ex}}m=\mathrm{0,}\phantom{\rule{0.25em}{0ex}}\mathrm{1,}\phantom{\rule{0.25em}{0ex}}-\mathrm{1,}\phantom{\rule{0.25em}{0ex}}\mathrm{2,}\phantom{\rule{0.25em}{0ex}}-\mathrm{2,}\phantom{\rule{0.25em}{0ex}}\dots )$.

### 27.4 Multiple Slit Diffraction

- A diffraction grating is a large collection of evenly spaced parallel slits that produces an interference pattern similar to but sharper than that of a double slit.
- There is constructive interference for a diffraction grating when $d\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =\mathrm{m\lambda}\phantom{\rule{0.25em}{0ex}}\text{(for}\phantom{\rule{0.25em}{0ex}}m=\text{0,}\phantom{\rule{0.25em}{0ex}}\text{1,}\phantom{\rule{0.25em}{0ex}}\text{\u20131,}\phantom{\rule{0.25em}{0ex}}\text{2,}\phantom{\rule{0.25em}{0ex}}\text{\u20132,}\phantom{\rule{0.25em}{0ex}}\dots )$, where $d$ is the distance between slits in the grating, $\lambda $ is the wavelength of light, and $m$ is the order of the maximum.

### 27.5 Single Slit Diffraction

- A single slit produces an interference pattern characterized by a broad central maximum with narrower and dimmer maxima to the sides.
- There is destructive interference for a single slit when $D\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =\text{m\lambda},\phantom{\rule{0.25em}{0ex}}\text{(for}\phantom{\rule{0.25em}{0ex}}m=\text{1,}\phantom{\rule{0.25em}{0ex}}\text{\u20131,}\phantom{\rule{0.25em}{0ex}}\text{2,}\phantom{\rule{0.25em}{0ex}}\text{\u20132,}\phantom{\rule{0.25em}{0ex}}\text{3,}\phantom{\rule{0.25em}{0ex}}\dots )$, where $D$ is the slit width, $\lambda $ is the light’s wavelength, $\theta $ is the angle relative to the original direction of the light, and $m$ is the order of the minimum. Note that there is no $m=0$ minimum.

### 27.6 Limits of Resolution: The Rayleigh Criterion

- Diffraction limits resolution.
- For a circular aperture, lens, or mirror, the Rayleigh criterion states that two images are just resolvable when the center of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other.
- This occurs for two point objects separated by the angle $\theta =1\text{.}\text{22}\frac{\lambda}{D}$, where $\lambda $ is the wavelength of light (or other electromagnetic radiation) and $D$ is the diameter of the aperture, lens, mirror, etc. This equation also gives the angular spreading of a source of light having a diameter $D$.

### 27.7 Thin Film Interference

- Thin film interference occurs between the light reflected from the top and bottom surfaces of a film. In addition to the path length difference, there can be a phase change.
- When light reflects from a medium having an index of refraction greater than that of the medium in which it is traveling, a $\text{180\xba}$ phase change (or a $\lambda /2$ shift) occurs.

### 27.8 Polarization

- Polarization is the attribute that wave oscillations have a definite direction relative to the direction of propagation of the wave.
- EM waves are transverse waves that may be polarized.
- The direction of polarization is defined to be the direction parallel to the electric field of the EM wave.
- Unpolarized light is composed of many rays having random polarization directions.
- Light can be polarized by passing it through a polarizing filter or other polarizing material. The intensity $I$ of polarized light after passing through a polarizing filter is $I={I}_{0}\phantom{\rule{0.25em}{0ex}}{\text{cos}}^{2}\phantom{\rule{0.25em}{0ex}}\mathrm{\theta ,}$ where ${I}_{0}$ is the original intensity and $\theta $ is the angle between the direction of polarization and the axis of the filter.
- Polarization is also produced by reflection.
- Brewster’s law states that reflected light will be completely polarized at the angle of reflection ${\theta}_{\text{b}}$, known as Brewster’s angle, given by a statement known as Brewster’s law: $\text{tan}\phantom{\rule{0.25em}{0ex}}{\theta}_{\text{b}}=\frac{{n}_{2}}{{n}_{1}}$, where ${n}_{1}$ is the medium in which the incident and reflected light travel and ${n}_{2}$ is the index of refraction of the medium that forms the interface that reflects the light.
- Polarization can also be produced by scattering.
- There are a number of types of optically active substances that rotate the direction of polarization of light passing through them.

### 27.9 *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light

- To improve microscope images, various techniques utilizing the wave characteristics of light have been developed. Many of these enhance contrast with interference effects.