The properties of waves flow from the wave equation, and interference is their signature

Suppose that a function of space and time y (x,y,z,t) satisfies the partial differential equation

where v is a constant with the dimensions of a velocity. The requirement that v be a constant is a very strict one, implying that the medium of propagation is homogeneous and isotropic (same in all directions). Then direct substitution shows that an arbitrary function y
= f(**n·
r** ±
vt) satisfies the equation, where **n** is a unit vector pointing in any direction and **r** is a position vector from an arbitrary origin. These solutions are waves of arbitrary shape whose surfaces of equal phase are planes, moving in any direction at speed v. Moreover, any linear combination of solutions

is also a solution, since the equation is linear in y. The solutions are said to be *superimposed*. From the obvious properties of these solutions, they are called *wave functions*, and the partial differential equation is called the *simple wave equation*. Do not be misled into thinking that *all* waves behave in this manner, or even that *many* do. In fact, most physical waves, such as those of sound in air or waves on the surface of water, or an electromagnetic wave moving along a transmission line, or light propagating down a fibre, are not precisely described by an equation of this type. The great virtue of the wave equation is that it well describes the behaviour of a wide variety of waves whenever the *phase velocity* v depends very little on the shape of the wave. This is quite accurately true for electromagnetic waves, including light, propagating in vacuum, and very close to true for sound waves of low intensity propagating in a homogeneous isotropic medium, for example, which explains the great utility of the simple wave equation.

The expression on the left of the simple wave equation is called the Laplacian, and expresses in some way the curvature or strain or departure from equilibrium of a medium whose displacement in some sense is y. If y gives the shape of an elastic membrane, then in equilibrium its Laplacian is zero. If y is the electrostatic potential in a region, then its Laplacian is zero. The expression on the right is in some way an acceleration, or mass times acceleration, so that the wave equation resembles Newton's second equation of motion. Any true physical interpretation depends on the type of wave under consideration, but it will involve the balancing of inertial and strain forces whose constants will appear in the expression for the phase velocity v. For mechanical waves, these will be the density and some elastic constant.

One unacquainted with the methods of science may well ask why recognising that a wave is described by some mathematical equation is so important. Isn't it just one of many properties of a wave, and not a very informative one at that? By no means. In the language of geometry, it is a *postulate*, something not proved but assumed as the basis for argument. Euclid's postulates defined Euclidean space, in which all his later results lived. By a slight change to one postulate, we get geometry on the surface of a sphere, which differs in some respects (two parallel lines meet at two points instead of none, for example). The verification that space is actually Euclidean, or that a wave satisfies the wave equation, follows from experience, not proof. Starting from the postulate of the wave equation, a great number of mathematical results can be obtained by logical proof, and we know all are valid provided the postulate is true. By recognizing that a wave satisfies the wave equation, we immediately acquire a gigantic kit of tools and properties established by mathematical analysis over the past several centuries.

Many properties of waves satisfying the wave equation will be familiar from different branches of Physics. Huyghens' Principle is one, that a later wavefront is the envelope of wavelets from a present wavefront. Fermat's Principle, that the optical path length is a minimum, is another. We have written the wave equation in rectangular coordinates, which gives the plane-wave solutions naturally. These solutions have the property that the amplitude of the wave does not change as the wave moves. We could also use other coordinate systems. Spherical coordinates give waves expanding from a centre, or contracting on a centre, whose amplitude varies inversely as the distance from the centre. Cylindrical coordinates give waves expanding from a line or contracting on a line, whose amplitude varies inversely as the square root of the distance from the line. In both cases there can be different angular dependences, and the various solutions can be superimposed to satisfy a variety of conditions.

The wave equation can also be generalized to the case of a velocity that depends on position or propagation direction, v = v(x,y,z,n_{x},n_{y},n_{z}). This introduces very great mathematical complexity (making the equation insoluble in terms of elementary functions), but has the expected results. Huyghens' and Fermat's principles still apply in many cases, as Huyghens' in double refraction (non-isotropic medium) and Fermat's in inhomogeneous media.

A further, and extremely useful generalization is to consider wave functions that are periodic functions of time, y
= g(x,y,z)e^{iw
t}, where w
= 2p
f is the *angular frequency* in radians per second, and f is the frequency in Hz. Now we can do the time derivative and reduce the equation to Ñ
^{2}g = -w
^{2}g/v^{2} = -k^{2}g. Here, Ñ
^{2} stands for the space derivatives (the Laplacian) and k = w
/v = 2pf/v = 2p
/l
, where l
is the *wavelength*, defined by the equation v = l
f. The *amplitude* g then satisfies the equation (Ñ
^{2} + k^{2})g = 0, called the Helmholtz wave equation. The Helmholtz equation is much easier to solve than the wave equation with the time. For a plane wave, a solution to this equation is g = e^{-ikn·
r} and we recover the same dependence on space and time as in a solution to the simple wave equation, but now the function is no longer arbitrary, but must be complex exponential, or *sinusoidal*. Such functions are also called *harmonic*. This is really not a great restriction, because Fourier's Theorem allows us to expand an arbitrary function as a linear combination of harmonic functions. If v is independent of frequency, we recover our previous results completely.

However, a very great flexibility has been introduced. Let us suppose that the phase velocity v is a function of the frequency. Now, at any given time, let us Fourier analyse a certain wave in terms of its sinusoidal or harmonic components. If the phase velocity were a constant, this superposition would describe the same wave at all later times. If the phase velocity depends on the frequency, the different frequencies move at different speeds, and their relation to one another changes. This can greatly change the shape of the wave as it moves along. In optical media, the *index of refraction* is the ratio of the speed of light in vacuum to the speed of light in the medium, n = c/v. It varies with the frequency of light, usually becoming larger as the wavelength decreases. This causes refracted rays of different frequencies to follow different angles. White light is analysed into its frequency components, which are spread out into a *spectrum*. For this reason, the phenomenon is called *dispersion*. Waves on the surface of deep water are also strongly dispersive, the phase velocity being proportional to the square root of the wavelength. The ocean performs a Fourier analysis of the disturbance due to a storm, the long waves arriving first, the short waves later, at great distances. Another example is the propagation of electromagnetic waves on a transmission line, such as a coaxial cable or optical fibre.

For any dispersive medium, the dependence of the phase velocity on the frequency is most conveniently given by a *dispersion relation* w
= w
(k), which can be plotted with w
and k along rectangular axes. The phase velocity is v = w
/k. The slope of the curve, u = dw
/dk, gives the speed of movement of a wave pulse consisting of a small range of frequencies, such as a brief burst of monochromatic light, and is called the *group velocity*.

There are waves for which the speed of propagation depends on the amplitude of the wave. One example is the propagation of strong sound waves in air. Whatever the initial shape of the wave, small variations tend to move forward when the pressure is higher than normal, and backward when the pressure is less than normal. The front and back of the wave become steep and eventually become shock waves, with a shape like the letter N. The speed of waves on water depends on their height when the water is shallow, so they steepen when moving onshore and break, all their energy dissipated in throwing turbulent water forward (and destroying any rock or structures in their way). Such waves obviously do not satisfy the principle of superposition, and cannot be described by the simple wave equation. They are called *nonlinear* waves, because their wave equation is not linear in the amplitude, and they may have very peculiar properties. In some cases, they exhibit *solitons*, which are pulse wave solutions that do not spread out indefinitely, but retain their shape and interact with one another in complicated ways.

A *monochromatic* wave is one whose time dependence is described by e^{iw
t} = sin w
t + i cos w
t. The use of complex numbers here may seem strange, but it is much easier than using sines and cosines individually, because the exponentials multiply and divide using the laws of exponents. We used it to derive Helmholtz's equation simply for convenience. We could have used the sine or cosine equally well, or even the sine with a phase constant d
, sin (w
t + d
). But there is another reason to choose the exponential. Suppose we have to find sin w
t + 2 cos (w
t + 45°
). I am sure you could do this with trigonometric formulas eventually, but I do not like to contemplate doing this myself. Instead, I represent the sine by e^{iw
t}, and the cosine by 2e^{i135°
} e^{iw
t}, and their sum by (1 + 2e^{i135°
}) e^{iw
t}. This is (2.4142 - i1.4142) e^{iw
t} = 2.7979e^{-i30.36°
} e^{iw
t}. What I want is just the real part of this, or 2.7979 sin (w
t - 30.36°
). The rectangular-polar conversion facility of your pocket calculator is convenient for this calculation. If you go ahead and find this trigonometrically, you will deeply appreciate the wonderful effects of complicating things with complex numbers. All we have to do in this case is really add the two vectors (1,0) and (1.4142 - i1.4142), and forget about the factor e^{iw
t}, which goes through with no change or effect.

The vectors we have just used are called *phasors* or *amplitudes*, and expressing them as complex numbers is a way to include both magnitude and phase in the same quantity. We use bold-face letters to stand for them, as **A** = Ae^{id
} = A_{r} + iA_{i}, where the equivalent polar and rectangular forms are shown. Scientific pocket calculators have special functions to convert from one form to the other quickly. They are used everywhere in alternating current calculations, and in quantum mechanics. The factor e^{iw
t} is imagined as a rotation with angular velocity w
in the 2-dimensional complex phasor space, and the actual physical quantites are the projections on one of the axes or the other as this rotation takes place. Our calculations, however, never include it.

Phasors are added and subtracted, and multiplied or divided by complex numbers, but are never multiplied or divided themselves, since e^{iw
t} x e^{iw
t} = e^{i2w
t}, and this introduces things at double frequencies. There is one very common case, however, in which we want to multiply two phasors, and that is in the calculation of the *intensity* of the wave, because this is proportional to the square of the amplitude of the wave. To see what's involved, consider a wave A sin (w
t + d
). Its square is A^{2}sin^{2}(w
t + d
) = (A^{2}/2)[1 + cos (2w
t + 2d
)], which consists of a constant term A^{2}/2, and a double-frequency term. In the long run, the double-frequency term is as often positive as negative, so the time average intensity is just A^{2}/2. How can we get this from the phasor **A** = Ae^{id
}? Well, **AA***/2 will do the job nicely, where **A*** is the complex conjugate of **A **(change sign of imaginary part or negate the angle). So we simply multiply the phasor by its complex conjugate, and divide by 2. There is more, however. In electrical circuits, the power is the product of the voltage and the current, p = vi. Suppose v = V sin w
t and i = I sin (w
t + d
). Then p = VI sin w
t sin (w
t + d
) = VI [sin^{2w
}t cos d
+ sin w
t cos w
t sin d
). Double-frequency terms again appear, and all average to zero. What is left is P = <p> = (VI/2) cos d
. In AC circuits, small letters stand for instantaneous values, and capital letters for amplitudes. Now, how can we get this from the phasors? The phasors are V e^{iw
t} and I e^{i(w
t + d
)}, and if we do what we did before, we get (VI/2)e^{-id
}. The real part of this is (VI/2) cos d
, which is what we wanted. Our formula for intensity is now P = Re(**AB***)/2 = Re(**A*****B**)/2. You can see why the second formula gives the same result, so it does not matter which phasor is conjugated. This formula is often required. Incidentally, the reason that j is used instead of i for Ö
-1 in electrical engineering is the possibility of confusion with the symbol i for current. Also in engineering, the magnitude of a phasor usually stands for V/Ö
2 instead of for V itself, so the power is simply P = Re(**VI***). That is, phasors are *rms* (root-mean-square) values, which are commonly used in specifying voltages and currents. That is, 240V means an amplitude, or maximum value, of 339V.

Consider the wave Ae^{i(-kx +w
t)}. This sinusoidal wave has amplitude A, and moves to the right (in the direction of the x-axis) with phase velocity x/t = w
/k. This is the speed of any point of constant phase on the wave. The wave Ae^{i(kx+w
t)} moves to the left at the same speed. The corresponding phasors are **A**_{+} = Ae^{-ikx} and **A**_{-} = Ae^{ikx}. A travelling wave is associated with a phase that varies linearly with distance, and an amplitude that does not. If these two waves are superimposed, we get **A** = A(e^{ikx} + e^{-ikx}) = 2A cos kx. The phase of this wave does not change at all, but the amplitude is a sinusoidal function of distance! This is called a *standing wave*. The points at which the amplitude is greatest are called the *loops*, and the smallest *nodes*.

Now let us superimpose two waves travelling in the same direction, one with wave number k - D
k/2 and frequency w
- D
w
/2, the other with wave number k + D
k/2 and frequency w
+ D
w
/2. Let both waves have the same amplitude, unity. Their superposition gives e^{i(-kx + w
t)}[e^{i(D
kx/2 - D
w
t/2)} + e^{i(-D
kx/2 + D
w
t/2)}] = 2 cos(D
kx/2 - D
w
t/2)e^{i(-kx +w
t)}. This is a wave with the average k and w
of the two waves travelling to the right, but with an amplitude that varies cosinusoidally between +2 and -2, and is itself a wave travelling with the velocity D
k/D
w
, which we recognize as the group velocity (when the difference D
w
is small). The waves are said to be *beating* at the angular frequency D
w
/2 at a fixed position x. When the waves are at exactly the same frequency, the beats disappear, and the situation is called *zero beat*. This is a powerful method of tuning two frequencies to be the same when the ear is used as the detector. In radiotelegraphy, a variable beat frequency oscillator's output is mixed with the received signal and adjusted to give a beat frequency that is in the audio range and comfortable to the ear.

Our analysis of waves so far has said nothing about the sources of the waves, but simply describes waves that have somehow come into being. The wave equation without a source term is called the *homogeneous wave equation*. The *inhomogeneous wave equation* contains an additional term that represents the radiation of waves from a source. For mechanical waves it may be the movement of a physical boundary, or the creation or annihilation of fluid at some point, or varying applied forces. The source of electromagnetic waves is an accelerated electric charge. When a source radiates, it sends energy into the surrounding medium, and so must exert a reaction on the source, the *radiation reaction*, that accounts for the energy transfer. These questions are much more difficult than those concerned merely with the propagation of waves that are already there, but at the same time extremely valuable.

We have not had to say anything about the nature of the wave function, beyond the fact that knowing it allows all properties of the wave to be determined. A wave dependent on a single parameter is called a *scalar wave*. There are more complicated waves in which more than one quantity must be known. Each of these quantities may separately satisfy the wave equation, but there are also relations between the several quantities. A sound wave is determined when the difference of the pressure from the equilibrium pressure in the medium, the *overpressure*, is known. Displacements of the medium are also involved, but these are always in the direction of propagation, normal to the wave front, and determined by the overpressure. The sound wave is a *longitudinal *wave. An electromagnetic wave involves vector electric and magnetic fields. If the electric field is known, the magnetic field can be found, so in general three quantities are required, and we have a *vector* wave. There is a condition that the electric field must be in the wave front, or *transverse*, so there are really only two independent wave functions. An electromagnetic wave is a *transverse* wave. The two components are the two polarizations of the wave, which may be specified in a number of ways. For electromagnetic waves in which only one polarization state exists, or the two states are mixed equally, as in white light, and for phenomena that are nearly independent of polarization, such as diffraction, scalar waves give an adequate description.

Let us again look at the superposition of two waves. If the two waves are not of the same frequency, the resultant amplitude will vary rapidly at any point, the waves being now in phase, now out of phase. If they are at nearly the same frequency, the amplitude variation will have the nature of beats. With waves of the same frequency, the resultant amplitude will be constant at any point. Representing the waves by phasors, the resultant amplitude is easily written down using the Law of Cosines for the vector addition:

a^{2} = a_{1}^{2} + a_{2}^{2} -2a_{1}a_{2} cos d

The intensities are proportional to the squares of the amplitudes, so this expression can be used at once. It says that the resultant intensity of two superimposed waves is equal to the sum of the intensities of the two individual waves, plus an *interference term* that can be positive or negative, and is proportional to the cosine of the phase difference between the two waves. If the waves are of equal amplitude, the intensity varies from four times the intensity of one wave alone, to zero. The phase difference is equal to k(s_{1} - s_{2}), where s_{1} and s_{2} are the optical path lengths from the two sources to the point under consideration. As one moves about, then, there should be great variations in resultant intensity, called *interference fringes*. The existence of interference fringes is conclusive evidence that waves are being observed.

In ordinary life we know that such fringes are not observed with light, though obvious in the surface waves on water. Their absence made a particle model of light quite plausible. It was not until the delicate diffraction phenomena were explained by Thomas Young as the result of interference that it became clear that light was, in fact, a wave. When we say 'wave' here, we mean that amplitudes and phases are involved, which is the case in both classical wave theory and quantum mechanics (which, indeed, is also called *wave mechanics*). It is important to explain why interference phenomena are not commonly observed, and why special arrangements are required to see them at all.

The first reason is the extremely high frequency of light, in the area of 10^{14} Hz. This means that we can only observe average intensities, not the amplitudes, and that the patterns must be constant in time. Secondly, light is emitted by charges moving on an atomic scale, in individual random bursts that last no longer than about 10^{-8} s, so that the corresponding wave is no longer than a metre or so in the best case. The wave bursts from two separate sources can never have constant phase relationships. We have only a chaos of overlapping, constantly changing, interference fringes that average out to uniformity, making the resultant intensity equal to the sum of the individual intensities. Finally, most light has a broad spectrum of frequencies. Indeed, this is the requirement for white light. Even if the interference from each frequently separately were stable, the fringes would be of different scales and would overlap. The only exception would be near where the path difference for the two waves was zero, which would be the same for any colour. Considering the nature of ordinary light, it is no wonder that interference is not observed.

The laser is a different case. Here, the atomic radiators are made to radiate in step, and to stay in step for a reasonable length of time, to a very good approximation. It has been possible to observe interference between light coming from separate lasers, including fringes and beats. Lasers are much more like the light sources one imagines when studying optics. The light from other kinds of sources, even when made nearly monochromatic by a spectroscope or filter, is at best only narrow-band noise. Two light beams that can make interference fringes when superimposed are called *coherent*. Coherence is limited both in time and space. For example, consider a wave train from some source that oscillates smoothly and predictably, except for arbitrary sudden changes of phase an average time t apart. Two points on the wave will be coherent at any fixed time if they are less than a distance ct apart (on average), and coherent at one point for time intervals of t or less. These numbers are the *coherence length* and the *coherence time*.

The search for interference in common light is not hopeless. A first step is to reduce the bandwidth of the light by passing it through a filter or spectrometer, or by using a nearly monochromatic souce like a sodium lamp or a discharge tube. This makes possible a coherence length of at least a few wavelengths. The second step is to divide each individual wave in the chaotic mixture into two coherent parts, send these parts over different paths, and recombine them later. Now the fringes for each wavelet will register, and may be visible. It is often necessary to restrict the size of the source by using a pinhole or slit. Before lasers, this was the only way to observe interference, and even with lasers it is still a very good idea, since lasers make adjustment and observation very much easier. The parts can be divided by taking different areas of the wavefront (*wavefront-splitting*) or else by partial reflection (*amplitude-splitting*). Each setup is analysed the same way, by evaluating the phase difference and using the simple equation above.

The most usual way of producing interference is to make two virtual images of the same source, and letting the waves from the two images interfere. This can be done by refraction (using a prism with two faces slightly inclined to each other), or reflection (using two mirrors inclined slightly to each other). Both of these methods carry the name of Fresnel (biprism and mirrors, respectively). An interesting variation of the latter method is to use only one mirror, and let the virtual image interfere with the actual source (Lloyd's mirror). This cannot be done with the prism, since one wave will pass through the glass, the other through air, and the path difference will be excessive (more than the coherence length). These methods are all wavefront-splitting, and the source is usually a fine slit. The slit must be parallel to the fringes, and helps to get a much higher intensity than would be obtained from a pinhole. Suitable mirrors are optical flats at grazing incidence. The Fresnel biprism and mirrors have a bright fringe at the centre of the pattern (zero path difference), but the Lloyd's Mirror has a dark fringe, showing that there is a phase change of half a wavelength on reflection from rare to dense. Both Fresnel's Biprism and Lloyd's Mirror are easy to set up and straightforward to analyse. The Fresnel Mirrors require a little more work.

The diagram shows these arrangments, and gives formulas for the fringe spacing D x. For the biprism, if a = 100 mm, b = 900 mm, n = 1.5 and a = 1° , the fringe spacing is 0.57 mm at 500 nm wavelength. For the Lloyd's mirror, if a = 0.5 mm and b = 1000 mm, the fringe spacing is 0.5 mm for 500 nm wavelength. The fringe spacing is measured with a micrometer eyepiece. This is not a precise way to measure wavelengths, but is a very instructive and satisfying experiment.

Every physics text presents another arrangement called Young's Experiment. Young did not actually carry it out in this way, because the resulting intensities are so feeble, but it can be explained very clearly. A slit in front of a source makes a cylindrical wavefront, which falls on two closely-spaced slits in a second screen. The waves diffracted through these slits fall on a screen, where they are superimposed and interfere. The double passage of slits and the dependence on diffraction from narrow slits render the intensity very feeble. The experiment can in principle be done with pinholes, but conditions are even worse. Young used a wire obstacle instead of slits. The light diffracted at top and bottom of the wire appears to come from line sources, just as if it came from two slits, but much brighter. In this way he first measured the wavelength of light, showing that it went from about 400 nm to 700 nm in going from blue to red. The fringe spacing is given by essentially the same formula as for Lloyd's Mirror.

Note that all the fringes mentioned above can actually be projected on a screen. Another kind of interference arises from a thin transparent film, for example a layer of oxidized oil on the surface of water. Fresh oil collects in droplets, but old oil dropped by motorcars will spread over the surface, gradually spreading and becoming thinner. When we look at any part of the film, we see the sky reflected in both the top and bottom interfaces of the oil layer, giving two coherent waves of roughly equal intensity but with a path length difference depending on the thickness of the oil. The light that passes through is absorbed by the bottom of the pool, if we are lucky, so it does not drown out the delicate reflected light. The two coherent waves are here produced by amplitude splitting, and are focused by our eyes on the retina, where they are superimposed and interfere. Such fringes cannot be projected on a screen. They are said to be localized in the film (but actually are produced on the retina). The colours we see are the result of the removal of the complementary colours by interference. Their pattern reveals the inequality in thickness of the film. In an ideal case, the film is thickest where the original drop fell and thinnest at the edges of the spot. The colours change with time as the film thins. The thinnest films disappear, because only the top reflection involves a half-wavelength phase change (the refractive index of the oil is greater than that of both the air and the water), and all wavelengths are removed. The film becomes 'black.' These brightly-coloured spots are really the most obvious evidence of interference. Perhaps they did not become at all common until the motor age. Remarkably, they do not demand a localized monochromatic source.

At the thin edge of the spot, the blue end of the spectrum will first not be destroyed by interference, and a greenish colour will be seen. When green is removed, the remainder will be purplish, perhaps magenta. Removal of red will leave a bluish green or cyan. A still thicker film might remove both blue and red, leaving a green. These colours have been extensively studied, and Minnaert gives a list of them.

If you set the convex surface of a lens of long focal length on an optical flat, there is a thin film of air between the glass surfaces whose thickness increases as the square of the distance from the point of contact. Colours can be seen near the point of contact, which appears as a small black disc, when white light is reflected. If monochromatic light is used, many concentric bull's-eye fringes can be seen. These are Newton's Rings, described by him in the Opticks, and whose explanation caused him considerable difficulty. In fact, he did not get it right, since his model of light was inadequate. He had to assume that the light particles were alternately reflectible and refractable as they travelled, lending a kind of 'wavelength,' but this was grasping at straws, and he knew it. The blackness of the central disc again shows the relative phase shift of half a wavelength on reflection. Like the colours of thin films on water, this manifestation of interference is easily seen, but was never noticed as anything unusual.

The same principle is used in the Michelson interferometer, in which the two beams are completely separated and can be subjected to any desired treatments. The thin film here is the space between two virtual images of the source. This space can be varied at will, and made wedge-shaped or parallel, so a variety of fringes can be produced. When the images are parallel, the fringes are circular, expanding from or contracting on the centre when the spacing is changed. When the film is angular, the fringes are straight lines (approximately) that move right or left as the spacing is changed. The colours that are produced in white light are the same as those made by oil films, but now the thickness of the film can be changed at will. The Michelson interferometer is suitable for accurate measurement of distance or wavelength. It was originally intended for detecting ether drift (since one arm could be in the direction of motion through the ether, and the other normal to it), and proved that there was no ether drift, consistent with relativity. Adjustment of the interferometer is straightforward, but not easy. In the diagram, BS is the beam splitter, and CP is the compensating plate, exactly the same as the beam splitter except not silvered. It is used so that both paths pass through the same thickness of glass.

When only two waves interfere, the intensity in the fringes varies sinusoidally, and the dark and light intervals seem equally wide. If the sides of the thin film are made more reflective than what arises simply from the difference of refractive indexes, then many more beams are produced, with equal phase delays. Now if N beams are in phase, the resultant intensity is proportional to N^{2}, which can be very big. A very small phase shift between successive beams (imagine the phasors rotated by a small amount) can cause the vector sum to be zero, the phasors forming a polygon of N sides, so we have darkness very close to the bright fringe. It is easy to see that the fringes have now become quite sharp, the sharper the more reflective the surfaces are made, separated by areas of weak intensity. Sketch vector diagrams of the phasors for various phase delays to see this clearly. It is much easier to measure the location of these sharp fringes, so they are preferred for accurate measurement. (The only way to achieve accuracy with a Michelson interferometer is to count hundreds or thousands of fringes, which can be tedious unless it is done by machine.)

There are good examples of multiple-beam interference. The *diffraction grating* consists of a large number of parallel scratches, each one of which makes a wavelet that can interfere with the others. When all the phasors line up, we get a sharp line. Two parallel optical flats, partly-silvered on the inner sides, make a *Fabry-Perot* étalon (interferometer), whose sharp fringes can even separate the hyperfine components of a spectral line, after a preliminary sorting out of the wavelengths by a prism. A thin film whose thickness is to be investigated can be lightly silvered on both sides. Then the fringes make a contour map of its thickness, giving answers to a fraction of a wavelength. An *interference filter* consists of a semitransparent metal coating, a transparent layer around a wavelength thick, and another semitransparent metal coating on a glass support. This is like a Fabry-Perot, with sharp pass bands widely spaced. All but the one wanted are eliminated with wide-band absorption filters. The result is a filter that passes only a very small wavelength interval, producing nearly monochromatic light.

Suppose we coat glass with a thin transparent film about 125 nm thick that has an index less than that of the glass. The light reflected normally from front and back of the film will be a half-wavelength out of phase, and so will interfere destructively to reduce the total light scattered. Note that there is no extra half-wavelength in this case, since both reflections are rare to dense. The cancellation will be exact only for light of wavelength 500 nm (green), but will be pretty good over the visual spectrum. If we can find a material whose index is the square root of the index of the glass, then the two reflected waves will be equal in amplitude, and the cancellation will be total: the glass will not reflect visible light, but transmit it all. MgF_{2} has an index of 1.38, which is a bit too large (1.22 is what is needed), but the film is hard and durable, and gives a considerable reduction. A lens with this coating appears a faint magenta in reflected light. Double and triple layers, using materials of higher indexes than glass as well as the low-index ones, can give more exact results. Antireflection coatings do not merely reduce light loss; reflected light in a multielement system can wind up where it should not, producing glare.

Finely and regularly layered transparent media can produce interesting effects due to interference of the many beams produced. Reflection and transmission can be strongly wavelength-dependent, producing bright colours. High reflectance is also possible, and multilayer coatings are used to make excellent mirrors, much better than metal ones. Bird feathers, butterfly wings, and beetles may acquire their colours in this way. The play of colours in opals and feldspars, and the bronzy reflectivity of biotite mica have a similar source. White-light holograms also make use of layered media.

The waves of wave mechanics do not obey the simple wave equation, but a similar one, the Schrödinger equation, which has the first derivative with respect to time on the right-hand side instead of the second. Nevertheless, the wave function is an amplitude with phase, its average square has a physical meaning (probability density) and is subject to interference and diffraction. In fact, waves are only one aspect of quantum mechanics. The nature of the interaction of light with matter is deeply modified, and the well-known quanta, photons, are introduced. These are definitely not Newton's light particles, since they have many strange properties not shared by material bodies. On the other hand, things like electrons are found to behave like waves in that they can be diffracted, and interfere, even making fringes like light. The lore of waves is an essential part of the universe.

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Composed by J. B. Calvert

Created 29 April 2000

Last revised