PHYS112 : Labs

12 Wave Optics

Introduction

Electromagnetic Waves

**Figure 26:** An electromagnetic wave.
$\scalebox{0.7}{ \includegraphics{wOptics-em.eps} }$

Electromagnetic waves such as light are transverse waves consisting of oscillating electric and magnetic fields oriented perpendicular to each other and to the direction of travel of the wave as shown in Figure 26. The fields are related to the intensity vector $\vec{S}$ via

$\begin{displaymath} \vec{S} = \frac{1}{\mu_o} \vec{E} \times \vec{B} \end{displaymath}$

(41)

The intensity vector $\vec{S}$ is also called the Poynting vector. It follows from Eq. 41 that the directions of the fields and the intensity vector are related by the right hand rule ( $\vec{E}: \mathrm{thumb}$ , $\vec{B}: \mathrm{fingers}$ , $\vec{S}: \mathrm{palm}$ ).

Polarization

The polarization of an electromagnetic wave is defined as the axis along which its electric field oscillates. Unpolarized electromagnetic waves consist of a superposition of equal amounts of waves of all possible polarizations. Light from an incandescent or fluorescent light bulb is unpolarized. Some sources produce polarized waved. For example, a linear broadcast antenna produces waves polarized parallel to the antenna.

**Figure 27:** Two polarizing filters oriented at a relative angle $\phi$ to each other.
$\scalebox{0.7}{ \includegraphics{wOptics-pols.eps} }$

Polarized waves can be produced by a polarizing filter. A polarizing filter absorbs light polarized along a specific axis. The light emerging from the filter is polarized perpendicular to the absorption axis of the filter. Unpolarized light incident on a polarizing filter loses half of its incident intensity,

$\begin{displaymath} I = \frac{1}{2} I_o \end{displaymath}$

(42)

Polarized light incident on a polarizing filter is partially absorbed according to

$\begin{displaymath} I = I_o \cos^2 \phi \end{displaymath}$

(43)

where $\phi$ is the angle between the filter axis and polarization of the incident light.

Light reflected from an interface between two media can be partially or fully polarized. The polarization of reflected light is complete when the reflected and refracted rays are perpendicular. The angle of incidence at which this is true is known as the polarizing angle or Brewster's angle and is given by

$\begin{displaymath} \theta_B = tan^{-1}\left(\frac{n_2}{n_1}\right). \end{displaymath}$

(44)

At Brewster's angle, the reflected light is polarized in the plane of the surface.

Diffraction

Light scattered from an object (or objects), such as a slit or a solid sphere is coherent - the individual electromagnetic waves composing the light are in phase with each other. Light from two or more coherent sources separated in space can interfere constructively or destructively. The scattered light recombines to produce bright and dark regions, often called fringes, corresponding to constructive and destructive interference between waves scattered from different parts of the object(s). This scattering of light is called diffraction, and the pattern of light and dark fringes is called a diffraction pattern. The spacing of the light and dark fringes in a diffraction pattern depends on the wavelength of the light. Hence diffraction patterns are most easily observed with monochromatic light - light of a single wavelength or a narrow band of wavelengths.

The dark fringes in the diffraction pattern produced by light of wavelength $\lambda$ on a single slit of width are described by

$\begin{displaymath} a \sin \theta = m \lambda \hspace{1cm} \mathrm{(minima)} \end{displaymath}$

(45)

where

is the slit width, and $m = \pm 1, \pm 2, \pm 3, \ldots$ .

The bright fringes corresponding to constructive interference between two slits separated by a distance are described by

$\begin{displaymath} d \sin \theta = m \lambda \hspace{1cm} \mathrm{(maxima)} \end{displaymath}$

(46)

where $m = 0, \pm 1, \pm 2, \pm 3, \ldots$ . In fact, Eq. 46 works for any number of slits with a constant spacing

, including a diffraction grating.

Experiments

Each of the experiments described below is set up on one of the six tables in the lab. Make your way through the stations, in any order, with your lab group. Write your responses to the prompts for each experiment on a printout of the assignment, and hand it in before you leave lab today.

Diffraction Grating

Look through a diffraction grating at the incandescent light bulb and the hydrogen lamp. Describe the diffraction patterns you see in each case and explain as many features of the patterns as you can. In addition to the differences between the patterns from the two sources, explain the ordering of the colors and the number of visible orders ( values).

Polarization

Shine the light bulb through three polarizing filters in lens holders. Remove the middle filter and adjust the orientations of the outer two so that no light is transmitted. Place the middle filter back in its holder, and observe the transmission through the stack of three filters for various orientations of the middle filter. Explain.
Look at reflections of ceiling lights on the floor through a polarizing filter. Try to find an approximate value of Brewster's angle for the floor. Describe your observations and give your result.
Use a polarizing filter to study the polarization of light from the sky. Describe your observations.

Single-slit and Double-slit Diffraction

The diagram above is a map of the slit patterns on the slit plate. The top number beside each slit combination is the number of slits. The middle number is the slit width. The bottom number is the slit separation.

$\scalebox{0.75}{ \includegraphics{wOptics-slideb.eps} }$

Look at the red and blue light from the source through the various slit combinations, and you should be able to see interference patterns. Describe and explain the patterns produced by the two columns on the right side of the plate. Explain the variation in the diffraction patterns with both color and slit separation and number.

Diffraction by an Obstruction

Use the laser ( $\lambda = 633$ nm) to measure the diameter of the wire provided. Describe your method, give your result, and compare your result with a Vernier caliper measurement.

CD vs. DVD

Use the laser ( $\lambda = 633$ nm) to determine the ratio of line densities of the DVD to the CD. Explain your method, and compare your result with the ratio of storage capacities 4.7 GB/0.74 GB = 6.4.

Michelson Interferometer

Electromagnetic waves in the wavelength range $3 \mathrm{cm} \leq \lambda \leq 30 \mathrm{cm}$ are called microwaves. The arrangement of a microwave source, a microwave detector, two mirrors, and a partial reflector shown in the diagram below is called a Michelson interferometer.

$\includegraphics{wOptics-michelson.eps}$

Plug in the microwave source, and set the detector sensitivity to $\times 1$ . Slowly slide one of the mirrors along its track and observe the response of the detector. Describe and explain your observations. As part of your explanation, add arrows to the diagram showing the paths taken by the light from the source to the detector.