17.3.2 Standing Electromagnetic Waves

Because electromagnetic waves are transverse waves, a standing electromagnetic wave is very much like a standing wave on a string. Standing electromagnetic waves can be established between two parallel mirrors that reflect light back and forth. The mirrors are boundaries, analogous to the boundaries at the ends of a string. In fact, this is exactly how a laser operates. The two facing mirrors in FIGURE 17.12 form what is called a laser cavity.

The figure illustrates a laser containing a standing light wave between two parallel mirrors.

FIGURE 17.12 Full Alternative Text

Because the mirrors act like the points to which a string is tied, the light wave must have a node at the surface of each mirror. One of the mirrors is only partially reflective, to allow some light to escape and form the laser beam, but this doesn’t affect the boundary condition.

Because the boundary conditions are the same, Equations 17.13 and 17.14 for $λ_{m}$ and $f_{m}$ apply to a laser just as they do to a vibrating string. The primary difference is the size of the wavelength. A typical laser cavity has a length $L \approx 30 cm,$ and visible light has a wavelength $λ \approx 600 nm$ . The standing light wave in a laser cavity has a mode number m that is approximately

m = \frac{2 L}{λ} \approx \frac{2 \times 0.30 m}{6.00 \times 10^{- 7} m} = 1, 000, 000

In other words, the standing light wave inside a laser cavity has approximately one million antinodes! This is a consequence of the very short wavelength of light.

EXAMPLE 17.3 The standing light wave inside a laser

Helium-neon lasers emit the red laser light commonly used in classroom demonstrations and supermarket checkout scanners. A helium-neon laser operates at a wavelength of precisely 632.9924 nm when the spacing between the mirrors is 310.372 mm.

In which mode does this laser operate?
What is the next longest wavelength that could form a standing wave in this laser cavity?

MODEL The light wave forms a standing wave between the two mirrors.

VISUALIZE The standing wave looks like Figure 17.12.

SOLVE

We can use $λ_{m} = 2 L / m$ to find that m (the mode) is

$m = \frac{2 L}{λ_{m}} = \frac{2 (0.310372 m)}{6.329924 \times 10^{- 7} m} = 980, 650$

There are 980,650 antinodes in the standing light wave.
The next longest wavelength that can fit in this laser cavity will have one fewer node. It will be the $m = 980, 649$ mode and its wavelength will be

$λ = \frac{2 L}{m} = \frac{2 (0.310372 m)}{980, 649} = 632.9930 nm$

REVIEW The wavelength increases by a mere 0.0006 nm when the mode number is decreased by 1.

Microwaves, with a wavelength of a few centimeters, can also set up standing waves. This is not always good. If the microwaves in a microwave oven form a standing wave, there are nodes where the electromagnetic field intensity is always zero. These nodes cause cold spots where the food does not heat. Although designers of microwave ovens try to prevent standing waves, ovens usually do have cold spots spaced $λ / 2$ apart at nodes in the microwave field. A turntable in a microwave oven keeps the food moving so that no part of your dinner remains at a node.