Anyone who has spent too much time in the sun receives a stinging reminder of the power of solar radiation. Enough energy reaches us across the vacuum of space to redden our noses. But how? Without a physical medium to transmit the energy, how does it get here? For a long time, many scientists thought there was such a medium, which they called the "ether." After all, if sound needs a medium in which to travel from one place to another, why not light?
One of the first to propose the existence of an ether was René Descartes (1596-1650), who wrote, "Objects of sight communicate themselves to us only through the fact that they move locally by the intermission of transparent bodies which are between them and us." Later, Robert Hooke (1635-1703) improved on this idea by imagining light energy to be pulsed disturbances traveling through an elastic ether as ever-expanding spheres. Christian Huygens (1629-1695) went even further, postulating that each point on a sphere of disturbance in the ether creates a new, secondary disturbance that propagates outwardly from that point. This far-reaching concept came to be known as Huygens' principle.
About the same time, however, Isaac Newton (1642-1727) proposed that light itself consisted of tiny particles moving through space and subject to the mechanical forces of matter. Newton's "corpuscular theory" validated the older and simpler belief that light is composed of physical rays that move in straight lines—an effect called rectilinear propagation.
The distance between crests defines the wavelength λ. The maximum displacement of the wave above or below the equilibrium point (the water level in calm seas) specifies the amplitude a. And the product of wavelength and frequency determines the speed of the wave, v.
Huygens' "wave/wavelet" theory could only explain rectilinear propagation through a more complicated geometrical construction of wavefronts, normals, and tangents. But his principle offered a better solution than Newton's theory to two very puzzling subtleties of light: diffraction and double refraction.
Two other peculiar effects, polarization and Newton's rings, gave trouble to Huygens and Newton, respectively. But Huygens' principle, though flawed, was eventually vindicated, and all of these strange effects (except double refraction) are better explained if light is thought of as a wave, particularly a transverse wave.
Transverse waves
If you have sailed the high seas, you'll remember the stomach-churning experience of transverse waves. If the waves are high enough, and the boat is dead in the water (stationary), the boat bobs up and down as the swells move past at a constant velocity. One instant you're on top of the world and the next you are in a valley of water.
The important feature to remember here is that the boat moves at right angles to the direction in which the wave travels. This defines a transverse wave. And since the water under the boat only moves up and down along a vertical line, the wave can be described as linearly polarized. But if the water just moves vertically, then what is it that moves horizontally? The answer is: all the points of constant phase.
The concept of phase enables us to keep track of where we are on the wave. If, for example, our boat could somehow surf on one of the wave crests, it would be at a point of constant phase. Consequently, it would stop oscillating vertically and move sideways with the wave at a steady speed. While surfing along with the wave, we might discover that we are traveling on a seemingly infinite train of crests extending as far as the eye can see. If we look to the side, we might see that the crest we are riding on also extends as far as the eye can see. The set of all points of constant phase in this direction defines the wavefront.
If the wavefront forms along a straight line, then the wave is called a plane wave because the line lies in a plane perpendicular to the direction of travel. But in general, any shape is possible. The velocity of the wavefront establishes the velocity of the wave: therefore, "wave velocity" and "phase velocity" mean the same thing.
There can be no doubt that transverse waves contain energy. That's why anything not nailed down on our sea-tossed craft soon ends up over the side. In fact, our boat makes a convenient energy detector for ocean waves.
Electromagnetic waves
Characterizing the energy-storage properties of the ether occupied much of the time of 19th-century physicists. But it was the British physicist James Clerk Maxwell (1831-1879) who made the extraordinary discovery that light energy is stored as electric and magnetic fields in space. No omnipresent metaphysical ether is needed.
Maxwell came to this unexpected conclusion through a carefully constructed synthesis of the known theories of electricity and magnetism. He combined these concepts into four famous simultaneous equations that describe electric and magnetic fields in space and time. Light waves, according to Maxwell, are electromagnetic disturbances that travel through space as a kind of waltz performed between the electric and magnetic waves.
The waltz begins, for example, when an electric charge vibrates up and down. The continuously accelerating charge creates changing electric and magnetic fields around it, which in turn induce changing electric and magnetic fields around themselves, and so on. In this way, the energy propagates through space at the speed of light. Potential energy is stored in a capacitive form via the permittivity of space, while "inertial" energy is stored in an inductive form via the permeability of space.
Electromagnetic energy quickly evolves into spherical waves (wavefronts that lie on the surface of spheres) that move outward from the charge in all directions at the speed of light. A linearly moving charge will generate a linearly polarized electromagnetic wave train with a frequency that matches the oscillation frequency of the charge.
Electromagnetic waves are transverse, with the electric field, magnetic field, and direction of propagation, all at right angles to one another. The electric field and direction of propagation define the plane of polarization. At remote distances from the source, the curvature of a spherical wave dwindles to the point where it becomes a plane wave.
Multiple wave
Until now, we have focused mostly on solitary wave trains with unique frequencies. But nature is infinitely more diverse than that. Light waves, for example, literally come in all sizes, shapes, and colors. And they come from all directions, too. The chaos created by the simultaneous contributions of so many different waves passing through each point in space would ordinarily leave us hopelessly confused. Thankfully, the principle of superposition simplifies the situation by allowing each wave's contribution to be considered separately from the others.
Any two sine waves of the same frequency and wavelength will combine as another sine wave whose phase and amplitude differ by a predictable amount. In fact, any number of sine waves with the same frequency and wavelength will add up to a single sine wave in the same predetermined way, although the math becomes more involved.
An especially useful superposition of waves for optical applications involves two waves of the same amplitude and frequency traveling in opposite directions. The sum of these two waves is called a standing wave because it doesn't actually go anywhere. The resultant wave pattern has a maximum amplitude twice that of the constituent waves but all of the energy is consumed by a sinusoidally changing amplitude between stationary notes of zero amplitude. The antinodes define positions of maximum amplitude.
A true standing wave cannot transmit energy unless there is some mechanism for energy loss, and all real-world standing waves suffer some kind of energy loss. In these circumstances they are called stationary waves. Linear laser cavities, for instance, create stationary waves because the output mirror reflects less than 100% of the energy back on itself.
When the amplitude, phase, and frequency of two or more waves differ, the combined effect produces a complex wave pattern. Such waves form a variety of periodic patterns depending on the magnitude of the frequency and phase differences. Unscrambling these waveforms into their sinusoidal constituents is the essence of Fourier analysis.
Interference
In 1801, Thomas Young (1773-1829) performed a convincing experiment that demonstrated the principle of superposition and the wave nature of light. Young's famous experiment actually demonstrated two wave effects of light: interference and diffraction. (This description is simplified for teaching purposes. I correct the simplifications later in the article.)
One version of Young's experimental setup consists of just two parallel opaque screens: one has two narrow slits in it and the second serves as a projection screen. The distance between the two screens, D, usually exceeds the distance between the slits, d, by a factor of 1000 or more.
Monochromatic light (light of a single frequency) enters from the left as a plane wave perpendicular to the screen with the slits. The plane wave encounters the screen, and light passes through the slits. If the wavelength of the light is significantly longer than the slits are wide, then light passing through the two slits will diffract into wave patterns that diverge from the slits with cylindrical wavefronts. The slits, in effect, obey Huygens' principle by acting as secondary sources from their positions on the original wavefront.
Rather than two sharp lines of light beaming straight out from the slits, the light instead fills the entire shadow region between the screens. Each slit, acting alone, illuminates the projection screen uniformly. Acting together, the two slits create a condition involving the superposition of the waves with the same amplitude and frequency.
Characterizing the illumination pattern at each point on the projection screen, then, reduces to a problem of finding the superposition of the waves whose phases differ by an amount dependent on the relative distance from the slits. A little trigonometry tells us that for a very large D and a small d this relative distance can be expressed as dy/D, where y is the distance of the point from the center of the projection screen. Therefore, the phase difference between the two waves is δ = 2πdy/λD.
The superposition of two sine waves tells us that the peak resultant amplitude is A = 2acos(δ/2). To relate this to the intensity that we would actually see projected on the screen, we square the amplitude to get the energy density:
I ≈ A2 ≈ 4a2cos2(πdy/λD)
This shows that the intensity pattern on the projection screen will appear as equally spaced bright and dark lines called fringes. The locations and brightness the fringes are governed by the value of the cos2 as it varies between 1 and 0.
Interferometers
Young's double-slit experiment has become a milestone of experimental physics, because the apparatus itself also identifies a basic type of interferometer. Interferometers split waveforms into two or more parts and recombine them in a variety of ways for an endless array of practical applications. Optical interferometers are especially useful in making exacting measurements in metrology, optics, spectrometry, and even astronomy.
Interferometers generally fall into one of two categories defined by either wavefront division or amplitude division. Young's experiment exemplifies wavefront-division interferometry, in which separate portions of the original wavefront are intercepted and then combined in some way. The Michelson interferometer, however, illustrates amplitude division, in which a single segment of the original wavefront is essentially duplicated by reflection from a partially reflecting surface. The amplitude of the original wavefront is therefore divided between the reflected and transmitted components, which are then recombined.
The American physicist Albert Michelson (1852-1931) introduced this instrument in 1881, and its contribution to physics is legendary. The basic design consists of a partial reflector (a beamsplitter) and two totally reflective mirrors at right angles to each other. A plane wave of monochromatic light enters the instrument from the left, striking the beamsplitter first. For a 50% beamsplitter, half of the light reflects towards one mirror, while the other half passes through the beamsplitter toward the other mirror.
Huygens, Fresnel, and diffraction
The diffraction of light, first noted by Francesco Grimaldi (1618-1663), proves that we really can see around corners. Therefore, the principle of rectilinear propagation cannot always be true. Young's double-slit experiment clearly shows this. But Young's apparatus, as described here, is a heuristic idealization, because slit widths that are significantly narrower than a wavelength of light cannot pass enough light to be of practical use. Apertures that do pass enough light span dimensions of several wavelengths. Instead of the uniform intensity distribution described earlier, the intensity pattern formed by these wider apertures is very different, and its explanation demands a more rigorous application of Huygens' principle.
The first successful description of the diffraction pattern of a single aperture was worked out by Augstin-Jean Fresnel (1788-1827) around 1818 using Huygens' principle. Like Young before him, Fresnel applied the principle of superposition, but with a twist. He assumed that the diffracted wavefront was the superposition of all the Huygens wavelets emanating from the entire width of the aperture.
Using this insight, Fresnel was able to accurately calculate the diffraction pattern of a single slit illuminated by a plane monochromatic wave. But he accomplished more than that, because the Huygens-Fresnel principle can successfully describe the diffraction of any number of slits or apertures.
The most important benefit of using a diffraction grating lies with its capacity to separate various wave frequencies in space. This ability can be understood from what we already know about the interference pattern in Young's experiment with two slits. The distance between the fringes is directly proportional to the wavelength; therefore longer wavelengths will generate fringes with wider spacings. Consequently, if the light passing through the two slits is composed of two or more wavelengths, then two or more interference patterns will appear on the projection screen. The ultimate expression of this property occurs when sunlight encounters a diffraction grating. The resulting diffraction/interference pattern appears to the eye as a rainbow of color.
All of the intensity patterns described here illustrate Fraunhofer diffraction (or far-field diffraction), because they are formed from plane waves. Such patterns must be images on the projection screen by a lens in order to be seen. In this discussion, the lens has been omitted for simplicity.
The most general (and common) form of diffraction in which the wavefronts are not planar is called Fresnel diffraction (or near-field diffraction). The intensity patterns formed by Fresnel diffraction do not require imaging, and the mathematical description of this kind of diffraction can lead to some of the most complex problems in optics. Nevertheless, the Huygens-Fresnel principle applies equally well to all these situations. But it is the principle of superposition that provides the key to a deeper understanding of all wave phenomena. The influence of this powerful concept can be felt in nearly every branch of physics.