Electromagnetic Waves

Consider a plane electromagnetic wave propagating through a vacuum in the $z$-direction. Electromagnetic waves are, incidentally, the only commonly occurring waves which do not require a medium through which to propagate. Suppose that the wave is polarized in the $x$-direction: i.e., its electric component oscillates in the $x$-direction. It follows that the magnetic component of the wave oscillates in the $y$-direction. According to standard electromagnetic theory, the wave is described by the following pair of coupled partial differential equations:

$$\frac{\partial E_x}{\partial t} = - \frac{1}{\epsilon_0}\,\frac{\partial H_y}{\partial z},$$ (458)

$$\frac{\partial H_y}{\partial t} = -\frac{1}{\mu_0}\,\frac{\partial E_x}{\partial z},$$ (459)

where $E_x(z,t)$ is the electric field-strength, and $H_y(z,t)$ is the magnetic intensity (i.e., the magnetic field-strength divided by $\mu_0$). Observe that Equations (458) and (459), which govern the propagation of electromagnetic waves through a vacuum, are analogous to Equations (402) and (403), which govern the propagation of electromagnetic signals down a transmission line. In particular, $E_x$ has units of voltage over length, $H_y$ has units of current over length, $\epsilon_0$ has units of capacitance per unit length, and $\mu_0$ has units of inductance per unit length.

Equations (458) and (459) can be combined to give

$$\frac{\partial^2 E_x}{\partial t^2} = \frac{1}{\epsilon_0\,\mu_0}\,\frac{\partial^2 E_x}{\partial z^2},$$ (460)

$$\frac{\partial^2 H_y}{\partial t^2} = \frac{1}{\epsilon_0\,\mu_0}\,\frac{\partial^2 H_y}{\partial z^2}.$$ (461)

It follows that the electric and the magnetic components of an electromagnetic wave propagating through a vacuum both separately satisfy a wave equation of the form (351). Furthermore, the phase velocity of the wave is clearly

$$c = \frac{1}{\sqrt{\epsilon_0\,\mu_0}}=2.998\times 10^8\,{\rm m\,s}^{-1}.$$ (462)

Let us search for a traveling wave solution of (458) and (459), propagating in the positive $z$-direction, whose electric component has the form

$$E_x(z,t)=E_0\,\cos(k\,z-\omega\,t-\phi).$$ (463)

As is easily demonstrated, this is a valid solution provided that $\omega=k\,c$. According to (458), the magnetic component of the wave is written

$$H_y(z,t)=Z^{-1}\,E_0\,\cos(k\,z-\omega\,t-\phi),$$ (464)

where

$$Z= Z_0 \equiv \sqrt{\frac{\mu_0}{\epsilon_0}},$$ (465)

and $Z_0$ is the impedance of free space [see Equation (420)]. Thus, the electric and magnetic components of an electromagnetic wave propagating through a vacuum are mutually perpendicular, and also perpendicular to the direction of propagation. Moreover, the two components oscillate in phase (i.e, they have simultaneous maxima and zeros), and the amplitude of the magnetic component is that of the electric component divided by the impedance of free space.

Multiplying (458) by $\epsilon_0\,E_x$, (459) by $\mu_0\,H_y$, and adding the two resulting expressions, we obtain the energy conservation equation

$$\frac{\partial{\cal E}}{\partial t} + \frac{\partial{\cal I}}{\partial z} =0,$$ (466)

where

$${\cal E} = \frac{1}{2}\left(\epsilon_0\,E_x^{\,2} + \mu_0\,H_y^{\,2}\right)$$ (467)

is the electromagnetic energy per unit volume of the wave, whereas

$${\cal I} = E_x\,H_y$$ (468)

is the wave electromagnetic energy flux (i.e., power per unit area) in the positive $z$-direction. The mean energy flux associated with the $z$-directed electromagnetic wave specified in Equations (463) and (464) is thus

$$\langle {\cal I} \rangle = \frac{1}{2}\,\frac{E_0^{\,2}}{Z}.$$ (469)

For a similar wave propagating in the negative $z$-direction, it is easily demonstrated that

$$E_x(z,t) = E_0\,\cos(k\,z+\omega\,t-\phi),$$ (470)

$$H_y(z,t) = -Z^{-1}\,E_0\,\cos(k\,z+\omega\,t-\phi),$$ (471)

and

$$\langle {\cal I} \rangle =- \frac{1}{2}\,\frac{E_0^{\,2}}{Z}.$$ (472)

Consider a plane electromagnetic wave, polarized in the $x$-direction, which propagates in the $z$-direction through a transparent dielectric medium, such as glass or water. As is well known, the electric component of the wave causes the neutral molecules making up the medium to polarize: i.e., it causes a small separation to develop between the mean positions of the positively and negatively charged constituents of the molecules (i.e., the atomic nuclii and the electrons). (Incidentally, it is easily shown that the magnetic component of the wave has a negligible influence on the molecules, provided that the wave amplitude is sufficiently small that the wave electric field does not cause the electrons and nuclii to move with relativistic velocities.) Now, if the mean position of the positively charged constituents of a given molecule, of net charge $+q$, develops a vector displacement ${\bf d}$ with respect to the mean position of the negatively charged constituents, of net charge $-q$, in response to a wave electric field ${\bf E}$ then the associated electric dipole moment is ${\bf p} = q\,{\bf d}$, where ${\bf d}$ is generally parallel to ${\bf E}$. Furthermore, if there are $N$ such molecules per unit volume then the dipole moment per unit volume is written ${\bf P} = N\,q\,{\bf d}$. Now, in a conventional dielectric medium,

$${\bf P} = \epsilon_0\,(\epsilon-1)\,{\bf E},$$ (473)

where $\epsilon>1$ is a dimensionless quantity, known as the relative dielectric constant, which is a property of the medium in question. In the presence of a dielectric medium, Equations (458) and (459) generalize to give

$$\frac{\partial E_x}{\partial t} = - \frac{1}{\epsilon_0}\left(\frac{\partial P_x}{\partial t}+\frac{\partial H_y}{\partial z}\right),$$ (474)

$$\frac{\partial H_y}{\partial t} = -\frac{1}{\mu_0}\,\frac{\partial E_x}{\partial z}.$$ (475)

When combined with Equation (473), these expressions yield

$$\frac{\partial E_x}{\partial t} = - \frac{1}{\epsilon\,\epsilon_0}\,\frac{\partial H_y}{\partial z},$$ (476)

$$\frac{\partial H_y}{\partial t} = -\frac{1}{\mu_0}\,\frac{\partial E_x}{\partial z}.$$ (477)

It can be seen that the above equations are just like the corresponding vacuum equations, (458) and (459), except that $\epsilon_0$ has been replaced by $\epsilon\,\epsilon_0$. It immediately follows that the phase velocity of an electromagnetic wave propagating through a dielectric medium is

$$v = \frac{1}{\sqrt{\epsilon\,\epsilon_0\,\mu_0}} = \frac{c}{n},$$ (478)

where $c=1/\sqrt{\epsilon_0\,\mu_0}$ is the velocity of light in vacuum, and the quantity

$$n = \sqrt{\epsilon}$$ (479)

is known as the refractive index of the medium. Thus, an electromagnetic wave propagating through a transparent dielectric medium does so at a phase velocity which is less than the velocity of light in vacuum by a factor $n$ (where $n>1$). Furthermore, the impedance of a transparent dielectric medium becomes

$$Z = \sqrt{\frac{\mu_0}{\epsilon\,\epsilon_0}} = \frac{Z_0}{n},$$ (480)

where $Z_0$ is the impedance of free space.

Suppose that the plane $z=0$ forms the boundary between two transparent dielectric media of refractive indices $n_1$ and $n_2$. Let the first medium occupy the region $z<0$, and the second the region $z>0$. Suppose that a plane electromagnetic wave, polarized in the $x$-direction, and propagating in the positive $z$-direction, is launched toward the boundary from a wave source of angular frequency $\omega$ situated at $z=-\infty$. Of course, we expect the wave incident on the boundary to be partly reflected, and partly transmitted. The wave electric and magnetic fields in the region $z<0$ are written

$$E_x(z,t) = E_i\,\cos(k_1\,z-\omega\,t) + E_r\,\cos(k_1\,z+\omega\,t),$$ (481)

$$H_y(z,t) = Z_1^{-1}\,E_i\,\cos(k_1\,z-\omega\,t)- Z_1^{-1}\,E_r\,\cos(k_1\,z+\omega\,t),$$ (482)

where $E_i$ is the amplitude of (the electric component of) the incident wave, $E_r$ the amplitude of the reflected wave, $k_1=n_1\,\omega/c$, and $Z_1=Z_0/n_1$. The wave electric and magnetic fields in the region $z>0$ take the form

$$E_x(z,t) = E_t\,\cos(k_2\,z-\omega\,t),$$ (483)

$$H_y(z,t) = Z_2^{-1}\,E_t\,\cos(k_2\,z-\omega\,t),$$ (484)

where $E_t$ is the amplitude of the transmitted wave, $k_2=n_2\,\omega/c$, and $Z_2=Z_0/n_2$. According to standard electromagnetic theory, the appropriate matching conditions at the boundary ($z=0$) are simply that $E_x$ and $H_y$ both be continuous. Thus, continuity of $E_x$ yields

$$E_i + E_r = E_t,$$ (485)

whereas continuity of $H_y$ gives

$$n_1\,(E_i-E_r) = n_2\,E_t,$$ (486)

since $Z^{-1}\propto n$. It follows that

$$E_r = \left(\frac{n_1-n_2}{n_1+n_2}\right)E_i,$$ (487)

$$E_t = \left(\frac{2\,n_1}{n_1+n_2}\right)E_i.$$ (488)

The coefficient of reflection, $R$, is defined as the ratio of the reflected to the incident energy flux, so that

$$R = \left(\frac{E_r}{E_i}\right)^2 = \left(\frac{n_1-n_2}{n_1+n_2}\right)^2.$$ (489)

Likewise, the coefficient of transmission, $T$, is the ratio of the transmitted to the incident energy flux, so that

$$T = \frac{Z_2^{-1}}{Z_1^{-1}}\left(\frac{E_t}{E_i}\right)^2=... ...ac{E_t}{E_i}\right)^2 = \frac{4\,n_1\,n_2}{(n_1+n_2)^2} = 1-R.$$ (490)

It can be seen, first of all, that if $n_1=n_2$ then $E_r=0$ and $E_t=E_i$. In other words, if the two media have the same indices of refraction then there is no reflection at the boundary between them, and the transmitted wave is consequently equal in amplitude to the incident wave. On the other hand, if $n_1\neq n_2$ then there is always some reflection at the boundary. Indeed, the amplitude of the reflected wave is roughly proportional to the difference between $n_1$ and $n_2$. This has important practical consequences. We can only see a clean pane of glass in a window because some of the light incident on an air/glass boundary is reflected, due to the different refractive indicies of air and glass. As is well known, it is a lot more difficult to see glass when it is submerged in water. This is because the refractive indices of glass and water are quite similar, and so there is very little reflection of light incident on a water/glass boundary.

According to Equation (487), $E_r/E_i<0$ when $n_2> n_1$. The negative sign indicates a $\pi$ radian phase shift of the (electric component of the) reflected wave, with respect to the incident wave. We conclude that there is a $\pi$ radian phase shift of the reflected wave, relative to the incident wave, on reflection from a boundary with a medium of greater refractive index. Conversely, there is no phase shift on reflection from a boundary with a medium of lesser refractive index.

Note that Equations (487)-(490) are analogous to Equations (429)-(432), with refractive index playing the role of impedance. This suggests, by analogy with earlier analysis, that we can prevent reflection of an electromagnetic wave normally incident at a boundary between two transparent dielectric media of different refractive indices by separating the media by a thin transparent layer whose thickness is one quarter of a wavelength, and whose refractive index is the geometric mean of the refractive indices of the two media. This is the physical principle behind the non-reflective lens coatings used in high-quality optical instruments.