Light from first principles: Maxwell's equations, light in matter and waveguides

This is the first article in the series and needs no prior optics, only a little calculus. Later parts build on it: Vibrational spectroscopy, plasmonics and SERS continues from the Drude model and the surface plasmon, and The many faces of Raman uses the nonlinear optics introduced here. Numbers in square brackets refer to the references.

01What is light? Waves, particles and photons

For two centuries the question had two answers. Newton described light as a stream of corpuscles, which explained straight-line propagation and sharp shadows; Huygens described it as a wave, each point of a wavefront sending out secondary wavelets. In 1801–1803 Thomas Young passed light through two narrow slits and saw bright and dark bands on a screen — interference, which only waves can do [1]. Fresnel's theory of diffraction and the measurement that light travels more slowly in water than in air settled the matter for the nineteenth century: light is a wave. In 1865 Maxwell showed what is waving — electric and magnetic fields — and that their speed is the speed of light [2].

The answer did not last. Light ejects electrons from a metal only above a threshold frequency, however intense it is, and the electrons' energy grows with the frequency, not the intensity. Einstein explained this photoelectric effect in 1905 by proposing that light is absorbed in packets — photons — each with energy and momentum [3]

$$E = h\nu = \hbar\omega, \qquad p = \frac{h}{\lambda} = \hbar k ,$$

where \(h = 6.626\times10^{-34}\) J s is Planck's constant, \(\hbar = h/2\pi\), \(\nu\) the frequency, \(\omega = 2\pi\nu\) the angular frequency, \(\lambda\) the wavelength and \(k = 2\pi/\lambda\) the wavenumber (the magnitude of the wave vector). A green photon (530 nm) carries \(3.7\times10^{-19}\) J, or 2.3 eV.

So which is it? Both, in a precise sense. Light is emitted and absorbed in whole photons, but how it gets from source to detector is described by a wave whose intensity gives the probability of finding a photon. Send photons through two slits one at a time, and each arrives as a single dot; after many, the dots build up Young's fringes [4]. A single photon cannot be split between the slits, yet the pattern needs both open: it is the probability amplitudes from the two paths that add. The same is true of electrons and atoms, which have a de Broglie wavelength \(\lambda = h/p\) [5].

Figure 1 · Interactive

For almost everything in this article — refraction, reflection, guiding — the wave description is exact and the photon picture adds nothing. Photons matter when light is created or absorbed, and when it is counted: in detectors, lasers and Raman scattering. We therefore build the wave theory, keeping in mind that its intensity is a photon flux.

02Charges and fields: the language of the theory

A charge \(q\) at rest pushes or pulls other charges. Instead of an action at a distance, we say the charge creates an electric field \(\mathbf E\) everywhere around it, and a test charge feels the force \(\mathbf F = q\mathbf E\). Moving charges — currents — also create a magnetic field \(\mathbf B\), which pushes on other moving charges. Together these give the Lorentz force on a charge moving with velocity \(\mathbf v\):

$$\mathbf F = q\,(\mathbf E + \mathbf v\times\mathbf B) .$$

Fields are described locally with two derivatives. The divergence \(\nabla\cdot\mathbf E\) measures how much field flows out of a small volume — a source or sink. The curl \(\nabla\times\mathbf E\) measures how much the field circulates around a point. With \(\nabla = (\partial_x, \partial_y, \partial_z)\),

$$\nabla\cdot\mathbf E = \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z}, \qquad \nabla\times\mathbf E = \Big(\frac{\partial E_z}{\partial y} - \frac{\partial E_y}{\partial z},\ \frac{\partial E_x}{\partial z} - \frac{\partial E_z}{\partial x},\ \frac{\partial E_y}{\partial x} - \frac{\partial E_x}{\partial y}\Big) .$$

Two theorems connect these local statements to measurements on loops and surfaces: Gauss's theorem (\(\oint_S \mathbf E\cdot d\mathbf A = \int_V \nabla\cdot\mathbf E\,dV\), the flux through a closed surface equals the total divergence inside) and Stokes' theorem (\(\oint_C \mathbf E\cdot d\boldsymbol\ell = \int_S (\nabla\times\mathbf E)\cdot d\mathbf A\), the circulation around a loop equals the curl through any surface it bounds).

03Maxwell's equations, one by one

Four experimental laws, put together and completed by Maxwell [2, 6, 7]:

Here \(\rho\) is the charge density (C m⁻³), \(\mathbf J\) the current density (A m⁻²), \(Q_\text{in}\) the charge enclosed, \(I\) the current through the loop, \(\varepsilon_0 = 8.854\times10^{-12}\) F m⁻¹ the permittivity of vacuum and \(\mu_0 = 4\pi\times10^{-7}\) H m⁻¹ (to excellent approximation) the permeability of vacuum.

The term Maxwell added

Ampère's original law read \(\nabla\times\mathbf B = \mu_0\mathbf J\). Take its divergence: the divergence of a curl is always zero, so it demands \(\nabla\cdot\mathbf J = 0\) — charge could never pile up anywhere. But charge is conserved, \(\nabla\cdot\mathbf J = -\partial\rho/\partial t\) (the continuity equation), and while a capacitor charges, \(\rho\) on its plates changes. Using Gauss's law, \(\partial\rho/\partial t = \varepsilon_0\,\nabla\cdot\partial\mathbf E/\partial t\), so the combination \(\mathbf J + \varepsilon_0\partial\mathbf E/\partial t\) always has zero divergence. Maxwell added \(\varepsilon_0\,\partial\mathbf E/\partial t\) — the displacement current — and in doing so made light possible: a changing \(\mathbf E\) now makes \(\mathbf B\), just as a changing \(\mathbf B\) makes \(\mathbf E\). Each can sustain the other in empty space, without any charges at all.

04The wave equation and the speed of light

In empty space \(\rho = 0\) and \(\mathbf J = 0\). Take the curl of Faraday's law and substitute Ampère–Maxwell:

$$\nabla\times(\nabla\times\mathbf E) = -\frac{\partial}{\partial t}\nabla\times\mathbf B = -\mu_0\varepsilon_0\frac{\partial^2\mathbf E}{\partial t^2} .$$

The vector identity \(\nabla\times(\nabla\times\mathbf E) = \nabla(\nabla\cdot\mathbf E) - \nabla^2\mathbf E\), with \(\nabla\cdot\mathbf E = 0\) in empty space, gives the wave equation

$$\nabla^2\mathbf E = \mu_0\varepsilon_0\,\frac{\partial^2\mathbf E}{\partial t^2}, \qquad c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 2.998\times10^{8}\ \mathrm{m\,s^{-1}} ,$$

where \(\nabla^2 = \partial_x^2 + \partial_y^2 + \partial_z^2\) is the Laplacian. The same equation holds for \(\mathbf B\). Any disturbance satisfying it travels at speed \(c\) — and \(c\) comes out as a combination of two constants measured with capacitors and coils. When Maxwell found that this number equalled the measured speed of light, he concluded that light is an electromagnetic wave [2].

Plane waves

The simplest solution is a plane wave travelling along \(z\),

$$\mathbf E(z, t) = \mathbf E_0\cos(kz - \omega t + \varphi) = \mathrm{Re}\big[\tilde{\mathbf E}_0\,e^{i(kz - \omega t)}\big], \qquad \omega = c\,k ,$$

with amplitude \(\mathbf E_0\), phase \(\varphi\) and \(\tilde{\mathbf E}_0 = \mathbf E_0 e^{i\varphi}\) the complex amplitude (the complex form is used throughout because derivatives become multiplications: \(\partial_t \to -i\omega\), \(\nabla \to i\mathbf k\)). Putting it into Maxwell's equations shows three things:

• \(\nabla\cdot\mathbf E = 0\) gives \(\mathbf k\cdot\mathbf E_0 = 0\): the field is transverse, perpendicular to the direction of travel.
• Faraday's law gives \(\mathbf B = \frac{1}{\omega}\mathbf k\times\mathbf E\): \(\mathbf B\) is perpendicular to both, in phase with \(\mathbf E\), and \(|\mathbf B| = |\mathbf E|/c\).
• \(\mathbf E\), \(\mathbf B\) and \(\mathbf k\) form a right-handed set.

Figure 2 · Interactive

Frequencies span an enormous range — the electromagnetic spectrum: radio (metres), microwaves (centimetres), infrared (micrometres), visible light (400–700 nm), ultraviolet, X-rays and gamma rays. They are all the same wave equation; only the wavelength, and hence how the wave interacts with matter, changes.

05Polarisation and Jones calculus

Because the field of a plane wave is transverse, it has two independent directions in which to oscillate. How the two components move together — in step, out of step, with equal or unequal amplitudes — is the polarisation. Almost every optical instrument manipulates it, deliberately (polarisers, wave plates, isolators, ellipsometers, liquid-crystal displays) or accidentally (mirrors, fibres, stressed glass). A compact algebra introduced by R. C. Jones in 1941 [25] describes all of this with 2-component vectors and 2 × 2 matrices.

The polarisation ellipse

For a wave travelling along \(z\), write the complex amplitude as a two-component Jones vector,

$$\mathbf J = \begin{pmatrix}E_x\\ E_y\end{pmatrix} = E_0\begin{pmatrix}\cos\theta\\ \sin\theta\,e^{i\delta}\end{pmatrix}, \qquad E_x(t) = E_0\cos\theta\cos(\omega t),\ \ E_y(t) = E_0\sin\theta\cos(\omega t - \delta)$$

(at a fixed point \(z = 0\), with fields written as \(\mathrm{Re}[\mathbf J\,e^{i(kz - \omega t)}]\)). Here \(\theta\) sets the ratio of the amplitudes and \(\delta\) the phase difference between \(E_y\) and \(E_x\). The tip of \(\mathbf E\) traces a line if \(\delta = 0\) or \(\pi\) (linear polarisation), a circle if \(\theta = 45°\) and \(\delta = \pm 90°\) (circular), and an ellipse otherwise. The orientation \(\psi\) of its major axis and its ellipticity angle \(\chi\) (\(\tan\chi\) = minor/major axis ratio, signed by the sense of rotation) follow from

$$\tan 2\psi = \tan 2\theta\,\cos\delta, \qquad \sin 2\chi = \sin 2\theta\,\sin\delta .$$

Figure 3 · Interactive

Jones vectors: the standard states

Only relative amplitudes and the relative phase matter; an overall phase factor \(e^{i\varphi}\) changes nothing measurable, and vectors are usually normalised so that the intensity \(I \propto |E_x|^2 + |E_y|^2 = \mathbf J^\dagger\mathbf J\) equals 1.

Two states are orthogonal if \(\mathbf J_1^\dagger\mathbf J_2 = 0\) (\(\dagger\) = complex-conjugate transpose): H and V, D and A, R and L. Any orthogonal pair is a basis, so every state can be written in any of them — for example \(\mathrm H = (\mathrm R + \mathrm L)/\sqrt2\): linear light is an equal mixture of the two circular states, which is why a medium that delays one circular state relative to the other (an optically active sugar solution, or a Faraday rotator) rotates linear polarisation. The intensity that a polariser for state \(\mathbf J_a\) transmits from state \(\mathbf J\) is \(|\mathbf J_a^\dagger\mathbf J|^2\).

Jones matrices of optical elements

A linear, non-depolarising element changes \(\mathbf J\) into \(\mathbf J' = \mathbf M\mathbf J\), where \(\mathbf M\) is a 2 × 2 complex matrix. An element whose axis is at angle \(\theta\) to \(x\) is obtained from its matrix \(\mathbf M_0\) in its own axes by rotating into those axes, acting, and rotating back:

$$\mathbf M(\theta) = \mathbf R(-\theta)\,\mathbf M_0\,\mathbf R(\theta), \qquad \mathbf R(\theta) = \begin{pmatrix}\cos\theta & \sin\theta\\ -\sin\theta & \cos\theta\end{pmatrix} .$$

A train of elements multiplies from right to left in the order the light meets them: \(\mathbf J_\text{out} = \mathbf M_N\cdots\mathbf M_2\mathbf M_1\mathbf J_\text{in}\). Matrices do not commute — a polariser followed by a wave plate is not a wave plate followed by a polariser. The eigenvectors of \(\mathbf M\) are its eigenpolarisations, the states it leaves unchanged apart from a factor: the fast and slow axes of a wave plate, the transmission axis of a polariser, the two circular states of a rotator.

Worked examples

1. Malus's law. Linear light along \(x\) through a polariser at \(\theta\): \(\mathbf J' = \cos\theta\,(\cos\theta, \sin\theta)^\mathsf T\), so \(I' = I\cos^2\theta\). Unpolarised light through any polariser keeps half its intensity.

2. Three polarisers. Crossed polarisers (0° and 90°) transmit nothing. Insert a third at 45° between them: \(I = I_1\cos^2 45°\cos^2 45° = I_1/4\), where \(I_1\) is the intensity after the first polariser. Adding a polariser lets light through — because the middle one projects the light onto a direction that has a component along 90° again.

3. A half-wave plate rotates linear polarisation. With the matrix above, linear light at angle \(\alpha\) leaves at \(2\theta - \alpha\). Turning the plate by \(\theta\) turns the polarisation by \(2\theta\): a half-wave plate at 22.5° rotates horizontal light to 45°, at 45° to vertical. Combined with a polarising beam splitter, this is the standard variable attenuator and power divider of a laser lab.

4. A quarter-wave plate turns linear into circular — and back. For horizontal light and the plate's fast axis at 45°, the two components along the plate's axes are equal and the plate delays one by 90°, so the light leaves circular. At other angles it leaves elliptical. Conversely, circular light through a quarter-wave plate leaves linear at ±45° to its axes, which is how circular polarisation is analysed.

5. A retarder between crossed polarisers. With a retarder of retardance \(\Gamma\) at angle \(\theta\) between a polariser at 0° and an analyser at 90°,

$$T = \sin^2 2\theta\,\sin^2\frac{\Gamma}{2} .$$

Nothing passes when the retarder's axis is along either polariser; at \(\theta = 45°\) the transmission is \(\sin^2(\Gamma/2)\). A liquid-crystal pixel changes \(\Gamma\) with voltage and switches from dark to bright; stressed glass or plastic between crossed polarisers shows coloured fringes of equal retardance (photoelasticity); a polarising microscope identifies birefringent crystals the same way.

6. Optical isolators. A laser must be protected from its own reflected light. In a Faraday isolator, light passes a polariser at 0°, a Faraday rotator turns it by +45°, and an output polariser at 45° transmits it. Reflected light returns through the rotator and is turned by another +45° — the Faraday effect, set by a magnetic field, does not reverse with the direction of travel — arriving at 90° to the input polariser and blocked. A cheaper, polarisation-specific version uses a polariser and a quarter-wave plate at 45°: passing the plate twice is equivalent to a half-wave plate, which turns the returning light into the orthogonal state, rejected by the polariser.

7. Mirrors and ellipsometry. Reflection multiplies the s and p components by different complex Fresnel coefficients. A metal mirror at 45° therefore turns linear light at 45° into elliptical light, while pure s or pure p stay linear — keep the polarisation along s or p at every mirror of a polarisation-sensitive setup. Ellipsometry turns this into a measurement: it records \(\tan\Psi\,e^{i\Delta} = r_p/r_s\) and from it the thickness and optical constants of thin films.

8. Turning circular polarisation round. At normal incidence a mirror reverses the direction of travel but not the rotation of the field in space, so — seen from the new direction of travel — right circular light comes back left circular. This is used in circular-polarisation optical isolators, and is the reason why, in the Jones description of a reflection, the coordinate system must be defined carefully.

The calculator below assembles these examples: choose an input state and up to three elements, and follow the state after each one — as the path of the field vector and as a point on the Poincaré sphere introduced next.

Figure 4 · Interactive

Partially polarised light: Stokes parameters and the Poincaré sphere

Jones vectors describe light that is fully polarised. Sunlight, LEDs and light scattered from tissue are partly or completely unpolarised: the relative phase wanders randomly faster than any detector can follow. Such light is described by the four Stokes parameters, which are measurable intensities [26]:

$$S_0 = I_x + I_y, \quad S_1 = I_x - I_y, \quad S_2 = I_{+45} - I_{-45}, \quad S_3 = I_L - I_R ,$$

where \(I_x\), \(I_y\), \(I_{\pm45}\) are the intensities transmitted by polarisers at 0°, 90°, ±45°, and \(I_L\), \(I_R\) those of left and right circular analysers. For fully polarised light \(S_0^2 = S_1^2 + S_2^2 + S_3^2\), and from the Jones vector \(S_1 = |E_x|^2 - |E_y|^2\), \(S_2 = 2\,\mathrm{Re}(E_x^*E_y)\), \(S_3 = 2\,\mathrm{Im}(E_x^*E_y)\). In general the degree of polarisation is \(P = \sqrt{S_1^2 + S_2^2 + S_3^2}/S_0\), between 0 (unpolarised) and 1.

The normalised vector \((S_1, S_2, S_3)/S_0\) is a point on (or, for partial polarisation, inside) the Poincaré sphere. Linear states lie on the equator (H and V opposite, D and A between them), circular states at the poles, elliptical states in between, and orthogonal states at opposite ends of a diameter. Its coordinates are the ellipse angles of the start of this section: longitude \(2\psi\), latitude \(2\chi\). Every wave plate is a rotation of the sphere by its retardance \(\Gamma\) about the axis through its fast-axis state — which makes the effect of any sequence of plates easy to picture. Elements that depolarise or partially polarise need 4 × 4 Mueller matrices acting on the Stokes vector; for a fully polarised beam through non-depolarising elements they give the same result as Jones matrices.

Measuring polarisation

• Linear analysis: rotate a polariser in front of a detector; the intensity varies as \(\tfrac12[S_0 + S_1\cos2\phi + S_2\sin2\phi]\) for analyser angle \(\phi\), giving \(S_0, S_1, S_2\) but not the handedness.
• Rotating quarter-wave plate polarimeter: a quarter-wave plate rotating in front of a fixed polariser modulates the intensity at \(2\omega\) and \(4\omega\); the Fourier coefficients give all four Stokes parameters.
• Polarisation cameras carry a micro-polariser grid at 0°, 45°, 90° and 135° over the pixels and image \(S_0\)–\(S_2\) in one shot.

06Energy, momentum and photons in a beam

Electric and magnetic fields store energy with density \(u = \tfrac12\varepsilon_0E^2 + \tfrac{1}{2\mu_0}B^2\). From Maxwell's equations follows an energy balance (Poynting's theorem): energy flows with the Poynting vector

$$\mathbf S = \frac{1}{\mu_0}\,\mathbf E\times\mathbf B, \qquad I = \langle |\mathbf S|\rangle = \tfrac12\,c\,\varepsilon_0\,n\,|E_0|^2 ,$$

where \(I\) is the intensity (time-averaged power per area, W m⁻²) and \(n\) the refractive index of the medium (1 in vacuum; introduced below). The wave also carries momentum, \(I/c\) per unit area and time, which appears as radiation pressure — the basis of optical tweezers and solar sails. Dividing by the photon energy gives the photon flux \(I/\hbar\omega\): a 1 mW laser pointer at 532 nm emits \(2.7\times10^{15}\) photons per second.

A useful sense of scale: focusing that 1 mW into a 1 µm spot gives about 10⁹ W m⁻², and a field amplitude of about 10⁶ V m⁻¹ — still about a million times smaller than the fields inside atoms (~5 × 10¹¹ V m⁻¹), which is why ordinary optics is linear and why nonlinear optics (section 12) needs pulsed lasers.

07Light in a linear medium: refractive index and dispersion

Polarisation of matter

Matter is made of charges. A light field pushes electrons one way and nuclei the other, creating tiny dipoles. Their dipole moment per unit volume is the polarisation \(\mathbf P\) (not to be confused with the polarisation of the light). In a linear medium it is proportional to the field,

$$\mathbf P = \varepsilon_0\,\chi\,\mathbf E, \qquad \mathbf D = \varepsilon_0\mathbf E + \mathbf P = \varepsilon_0\,\varepsilon\,\mathbf E, \qquad \varepsilon = 1 + \chi ,$$

with \(\chi\) the electric susceptibility, \(\varepsilon\) the relative permittivity (dielectric function) and \(\mathbf D\) the displacement field. The bound charges enter Maxwell's equations through \(\mathbf P\); repeating the derivation of section 4 with \(\varepsilon_0 \to \varepsilon_0\varepsilon\) (and \(\mu \approx 1\) at optical frequencies) gives waves travelling at

$$v = \frac{c}{n}, \qquad n = \sqrt{\varepsilon\mu} \approx \sqrt{\varepsilon}, \qquad k = n\,\frac{\omega}{c} = n\,k_0 ,$$

where \(n\) is the refractive index and \(k_0 = \omega/c = 2\pi/\lambda_0\) the vacuum wavenumber. The frequency stays the same in the medium; the wavelength shortens to \(\lambda_0/n\).

Absorption: a complex index

If the medium absorbs, \(\varepsilon = \varepsilon' + i\varepsilon''\) is complex, and so is the index, \(\tilde n = n + i\kappa\), with \(\kappa\) the extinction coefficient. A plane wave then becomes

$$E \propto e^{i(\tilde n k_0 z - \omega t)} = e^{-\kappa k_0 z}\,e^{i(n k_0 z - \omega t)}, \qquad I(z) = I_0\,e^{-\alpha z}, \quad \alpha = \frac{4\pi\kappa}{\lambda_0} ,$$

the Beer–Lambert law, with \(\alpha\) the absorption coefficient (its chemistry form appears in the spectroscopy article).

Where n comes from: the Lorentz oscillator

Model a bound electron (charge \(-e\), mass \(m\)) as a damped spring with resonance \(\omega_0\) and damping \(\gamma\), driven by the light field:

$$m\ddot x + m\gamma\dot x + m\omega_0^2 x = -eE_0e^{-i\omega t} \quad\Rightarrow\quad x = \frac{-e E/m}{\omega_0^2 - \omega^2 - i\gamma\omega} .$$

With \(N\) such electrons per volume, \(P = -Nex\) and

$$\varepsilon(\omega) = 1 + \frac{Ne^2}{\varepsilon_0 m}\,\frac{1}{\omega_0^2 - \omega^2 - i\gamma\omega} = 1 + \frac{f\,\omega_0^2}{\omega_0^2 - \omega^2 - i\gamma\omega} ,$$

where \(f\) collects the oscillator strength. Below the resonance (\(\omega < \omega_0\)) the electrons follow the field, \(\varepsilon > 1\) and \(n\) rises with frequency — normal dispersion: blue light bends more than red in a prism. Near \(\omega_0\) the medium absorbs strongly and \(n\) falls with frequency (anomalous dispersion). Above it the electrons cannot keep up. Glass has its resonances in the ultraviolet and the infrared, so it is transparent and normally dispersive in the visible. Real materials have several resonances; summing them gives the Sellmeier formula \(n^2 = 1 + \sum_j B_j\lambda^2/(\lambda^2 - C_j)\) used for optical glasses [8]. Because \(\varepsilon(\omega)\) is causal, its real and imaginary parts are linked by the Kramers–Kronig relations: absorption somewhere implies dispersion everywhere.

Figure 5 · Interactive

Phase velocity, group velocity and dispersion

A pulse is a group of frequencies. Its envelope travels at the group velocity, not at \(c/n\):

$$v_g = \frac{d\omega}{dk} = \frac{c}{n_g}, \qquad n_g = n - \lambda_0\frac{dn}{d\lambda_0}, \qquad \beta_2 = \frac{d^2k}{d\omega^2} ,$$

with \(n_g\) the group index and \(\beta_2\) the group-velocity dispersion: different colours of a short pulse travel at different speeds, and the pulse spreads. For fused silica at 800 nm, \(n = 1.453\) but \(n_g \approx 1.467\). Dispersion limits ultrafast optics and fibre communications, and is put to use in pulse compressors and spectrometers.

08Anisotropic media: birefringence

In a crystal the electrons may be easier to displace along some directions than others; then \(\chi\) is a tensor and \(\mathbf P\) need not be parallel to \(\mathbf E\). In a uniaxial crystal (calcite, quartz, lithium niobate) light polarised perpendicular to the optic axis sees the ordinary index \(n_o\), and light with a component along it sees an extraordinary index between \(n_o\) and \(n_e\). Two polarisations travel at two speeds — birefringence — which splits beams, makes wave plates (a plate of thickness \(d\) delays one polarisation by \(\Delta\varphi = 2\pi(n_e - n_o)d/\lambda\)) and, as section 12 shows, makes efficient frequency conversion possible.

09At an interface: Snell, Fresnel, Brewster and total internal reflection

At a boundary between media 1 and 2, Maxwell's equations in integral form demand that the tangential components of \(\mathbf E\) and \(\mathbf H = \mathbf B/\mu\) are continuous. Matching a plane wave on both sides at every point of the boundary requires the tangential wave vector to be the same — which is Snell's law — and fixes the amplitudes of the reflected and transmitted waves — the Fresnel equations [6, 8]:

$$n_1\sin\theta_1 = n_2\sin\theta_2, \qquad r_s = \frac{n_1\cos\theta_1 - n_2\cos\theta_2}{n_1\cos\theta_1 + n_2\cos\theta_2}, \qquad r_p = \frac{n_2\cos\theta_1 - n_1\cos\theta_2}{n_2\cos\theta_1 + n_1\cos\theta_2} ,$$

where \(\theta_1, \theta_2\) are the angles of incidence and refraction from the normal, \(r_s\) and \(r_p\) the amplitude reflection coefficients for s polarisation (\(\mathbf E\) perpendicular to the plane of incidence, also called TE) and p polarisation (\(\mathbf E\) in the plane, TM), and the reflectance is \(R = |r|^2\). At normal incidence \(R = \big((n_1 - n_2)/(n_1 + n_2)\big)^2\): 4 % for air–glass.

• Brewster's angle, \(\tan\theta_B = n_2/n_1\): \(r_p = 0\), so reflected light is purely s-polarised — used in laser windows and polarisers.
• Total internal reflection: going from a denser to a rarer medium (\(n_1 > n_2\)), above the critical angle \(\sin\theta_c = n_2/n_1\) there is no real refraction angle, and \(R = 1\) for both polarisations.
• Even then the field does not stop abruptly: an evanescent wave decays into medium 2 as \(e^{-z/d_p}\) with \(d_p = \lambda_0 / \big(2\pi\sqrt{n_1^2\sin^2\theta_1 - n_2^2}\big)\), typically a few hundred nanometres. It carries no energy on average, but anything placed within it can absorb or scatter light — the basis of ATR spectroscopy, TIRF microscopy, SPR sensors (how an SPR sensor sees binding) and of light guiding.

Figure 6 · Interactive

10TE and TM: which is which, in any structure

“TE” and “TM” appear everywhere in optics — reflection, gratings, waveguides, fibres, photonic crystals, plasmonics — and they cause more confusion than almost any other pair of words. The reason is in the names: transverse electric means the electric field has no component along some reference direction (it is transverse to it); transverse magnetic means the magnetic field has none. Every light wave is transverse to its own direction of travel, so the names only mean something once the reference is fixed — and different structures, and different communities, choose different references. The first question is therefore always: transverse to what?

The recipe

Figure 7

Case 1 — a plane surface (reflection, refraction, thin films)

The plane of incidence contains \(\mathbf k\) and the surface normal. TE = s: \(\mathbf E\) is perpendicular to this plane and therefore parallel to the surface. TM = p: \(\mathbf E\) lies in the plane, \(\mathbf H\) is perpendicular to it. These are the two polarisations of the Fresnel equations in section 9: only TM has a Brewster angle, and TM is the one that, in a metal, can drive charge across the surface. In a laboratory with the optical table horizontal and the beam reflected sideways off a vertical mirror, the plane of incidence is horizontal: s light is vertically polarised, p light horizontally. Turn the mirror so that the beam goes up, and the names swap — the polarisation of the laser has not changed, the plane of incidence has. Multilayer coatings, ellipsometry (\(\Psi\) and \(\Delta\) compare \(r_p\) and \(r_s\)) and SPR prisms all use this convention.

Case 2 — a slab waveguide

Take propagation along \(z\), layers normal to \(x\), and the structure uniform along \(y\) (\(\partial/\partial y = 0\)). Writing out Maxwell's curl equations, the six field components split into two independent groups:

$$\text{TE:}\ \ (E_y,\ H_x,\ H_z), \qquad \text{TM:}\ \ (H_y,\ E_x,\ E_z) .$$

TE has no electric field along the propagation direction and its \(\mathbf E\) is parallel to the layers; TM has no magnetic field along it, and its \(\mathbf E\) points mainly across the layers. The two have different effective indices (TE is usually slightly higher) and different cut-offs — the modes of section 14.

Case 3 — rectangular (strip, rib) waveguides: quasi-TE and quasi-TM

A wire confines light in two directions, so no direction is uniform and the modes are hybrid: all six components are non-zero. They are named after their dominant electric field: quasi-TE when \(\mathbf E\) is mostly parallel to the substrate (across the width), quasi-TM when it is mostly perpendicular to it. In a standard 450 × 220 nm silicon wire the quasi-TE mode has the higher effective index and is the one most devices are designed for; grating couplers, ring resonators and directional couplers behave very differently for the two, which is why on-chip polarisation splitters and rotators exist.

Case 4 — hollow metal waveguides

Here the reference is the propagation direction alone. TE (or H) modes have \(E_z = 0\) and \(H_z \ne 0\); TM (or E) modes have \(H_z = 0\) and \(E_z \ne 0\). The indices \(m, n\) count half-wave variations across the width and height: TE₁₀ has its electric field across the short side, largest in the middle and zero at the side walls, and is the fundamental mode of a rectangular guide; the lowest TM mode is TM₁₁. A guide with two separate conductors (coaxial cable, a pair of wires) also supports a TEM mode, with neither field along the propagation direction.

Case 5 — optical fibres

The cylindrical symmetry makes most fibre modes hybrid, labelled HE and EH according to which longitudinal field dominates; the fundamental mode is HE₁₁. Only the rotationally symmetric modes are pure: TE₀₁, with \(\mathbf E\) circling the axis (azimuthal), and TM₀₁, with \(\mathbf E\) pointing radially. In weakly guiding fibres the modes are grouped into linearly polarised LP modes (section 15) — TE₀₁, TM₀₁ and HE₂₁ together form LP₁₁ — and “TE/TM” is rarely used for them.

Case 6 — gratings and periodic surfaces

In classical mounting the plane of incidence is perpendicular to the grooves. Then TE: \(\mathbf E\) parallel to the grooves; TM: \(\mathbf E\) perpendicular to them, \(\mathbf H\) parallel to the grooves. The two behave very differently:

• On a metal grating, only TM has a field component that crosses the ridges and piles up charge, so only TM couples to surface plasmons (grating-coupled SPR) and shows the sharp Wood–Rayleigh anomalies.
• A wire-grid polariser reflects TE (E along the wires drives currents in them) and transmits TM.
• Diffraction efficiencies of ruled and holographic gratings are quoted separately for TE and TM (sometimes called “s” and “p”, or “parallel” and “perpendicular to the grooves” — check the definition in the data sheet).
• For sub-wavelength gratings the two polarisations see different effective indices: form birefringence, used for wave plates and polarisation-sensitive metasurfaces.

In conical mounting the plane of incidence is rotated away from the grating vector. TE and TM are still defined relative to the plane of incidence (\(\mathbf E\) perpendicular or parallel to it), but the grating now mixes them: each diffracted order contains both, and rigorous solvers such as RCWA must treat them together.

Case 7 — photonic crystals: the convention flips

A two-dimensional photonic crystal (a lattice of rods or holes, uniform along \(z\)) separates, for light travelling in the periodic \(x\)–\(y\) plane, into two families. The photonic-crystal literature names them relative to the plane of periodicity [22]: TE has \(\mathbf E\) in the plane and \(\mathbf H\) along the rods; TM has \(\mathbf E\) along the rods and \(\mathbf H\) in the plane. A lattice of dielectric rods in air tends to open a band gap for TM, a lattice of air holes in a dielectric for TE. In photonic-crystal slabs (finite thickness) the modes become TE-like and TM-like by their symmetry about the slab's mid-plane. Compared with a grating, this is the opposite name for the same physical field — the most common source of TE/TM mistakes.

Case 8 — plasmonics, nanowires and scattering

• Surface plasmon polaritons on a flat metal surface are TM only: a TE wave has no field normal to the surface and cannot create surface charge (section 11 and the spectroscopy article).
• Metal nanorods have longitudinal and transverse plasmon resonances — named by whether \(\mathbf E\) is along or across the rod, not TE/TM.
• Scattering by long cylinders and wires: many texts call the case with \(\mathbf E\) parallel to the cylinder axis “TM” (others call it “case I” or “E-polarisation”), relative to the cross-sectional plane — like photonic crystals. Saying “E parallel to the axis” avoids the ambiguity.

The figure below draws each case in three dimensions, with the reference plane or direction marked and the oscillating field vectors of the chosen polarisation.

Figure 8 · Interactive

Test yourself

Figure 9 · Interactive

In the laboratory

• To set s or p at a sample, decide the plane of incidence first (the plane containing the beam and the sample normal), then rotate a half-wave plate or polariser so that \(\mathbf E\) is perpendicular (s) or parallel (p) to that plane. “Vertical” and “horizontal” are only meaningful once that plane is known.
• A polarising beam-splitter cube transmits p and reflects s relative to its own internal reflecting surface.
• For a grating, write down whether \(\mathbf E\) is parallel or perpendicular to the grooves; for a waveguide, which way the dominant \(\mathbf E\) points relative to the substrate. These statements are unambiguous, whatever the names.

11Light in metals

A metal has free electrons with no restoring force: set \(\omega_0 = 0\) in the Lorentz model and the result is the Drude model,

$$\varepsilon(\omega) = \varepsilon_\infty - \frac{\omega_p^2}{\omega^2 + i\gamma\omega}, \qquad \omega_p^2 = \frac{Ne^2}{\varepsilon_0 m} ,$$

where \(\omega_p\) is the plasma frequency (about 9 eV, ultraviolet, for gold and silver), \(\gamma\) the electron collision rate and \(\varepsilon_\infty\) the contribution of the bound electrons (derived in more detail in the spectroscopy article). Below \(\omega_p\) the real part of \(\varepsilon\) is negative. Then \(\tilde n\) is almost purely imaginary: the wave cannot propagate, it decays inside the metal within the skin depth \(\delta = c/(\kappa\omega) = \lambda_0/(2\pi\kappa)\) (for the field) — about 25 nm for gold in the red — and is almost entirely reflected. This is why metals are shiny and why a few tens of nanometres of gold make a mirror. Above \(\omega_p\) the electrons cannot follow and metals become transparent (alkali metals transmit ultraviolet). Real gold also absorbs in the blue through interband transitions, giving its colour [9].

Negative permittivity also allows a new kind of wave bound to the surface of a metal: the surface plasmon polariton, and in small particles the localised surface plasmon — the subject of the spectroscopy article and the basis of SPR sensors and SERS. Switch Figure 5 to the metal model to see \(n\), \(\kappa\), the reflectance and the penetration depth of a Drude metal.

12Nonlinear media

The linear relation \(\mathbf P = \varepsilon_0\chi\mathbf E\) is the first term of an expansion valid while the light field is small compared with the fields inside atoms (~10¹¹ V m⁻¹). Intense laser pulses reach a fraction of that, and the next terms appear [10, 11]:

$$P = \varepsilon_0\left(\chi^{(1)}E + \chi^{(2)}E^2 + \chi^{(3)}E^3 + \dots\right) ,$$

with \(\chi^{(2)}\) and \(\chi^{(3)}\) the second- and third-order susceptibilities (each term roughly \(10^{-11}\)–\(10^{-12}\) times smaller per V m⁻¹ than the previous one). Products of fields oscillate at sums and differences of their frequencies, so a nonlinear medium creates new colours.

Second-order effects

For \(E = E_0\cos\omega t\), the term \(\chi^{(2)}E^2 = \tfrac12\chi^{(2)}E_0^2(1 + \cos 2\omega t)\) radiates at \(2\omega\): second-harmonic generation (SHG), first seen in 1961 by focusing a ruby laser into quartz [12]. With two input frequencies the same term gives sum- and difference-frequency generation and, with one strong pump, optical parametric amplification and oscillation (tunable sources). \(\chi^{(2)}\) vanishes in any material with a centre of inversion symmetry — glasses, liquids, silicon — so it needs non-centrosymmetric crystals such as β-barium borate, potassium titanyl phosphate or lithium niobate.

Phase matching

The second harmonic generated at each point of the crystal adds up only if it stays in step with the harmonic already generated. Because of dispersion, \(n(2\omega) \ne n(\omega)\), and the phase mismatch is

$$\Delta k = k(2\omega) - 2k(\omega) = \frac{2\omega}{c}\big[n(2\omega) - n(\omega)\big], \qquad I_{2\omega}(L) \propto I_\omega^2\,L^2\,\mathrm{sinc}^2\!\Big(\frac{\Delta k L}{2}\Big) ,$$

where \(L\) is the crystal length and \(\mathrm{sinc}\,x = \sin x/x\). Unless \(\Delta k = 0\), the harmonic grows only over the coherence length \(L_c = \pi/\Delta k\) (typically micrometres) and then flows back into the fundamental. Two solutions exist: birefringent phase matching, choosing the angle and polarisations so that \(n_e(2\omega) = n_o(\omega)\), and quasi-phase matching, reversing the sign of \(\chi^{(2)}\) every \(L_c\) (periodically poled crystals) so that the harmonic is reset in step before it can flow back [13].

Figure 10 · Interactive

Third-order effects

\(\chi^{(3)}\) is present in every material. Its most important consequence is the optical Kerr effect: the refractive index depends on the intensity, \(n = n_0 + n_2 I\) (for silica \(n_2 \approx 2.6\times10^{-20}\) m² W⁻¹). A pulse then changes its own phase (self-phase modulation, which broadens its spectrum), a beam with a bell-shaped profile focuses itself (self-focusing), and in fibres the Kerr effect balanced against dispersion gives solitons — pulses that keep their shape [14]. \(\chi^{(3)}\) also gives third-harmonic generation, four-wave mixing and the coherent Raman methods CARS and SRS, in which the vibrational resonance appears in \(\chi^{(3)}\) (coherent Raman).

13Why guide light, and how: total internal reflection

A beam cannot stay narrow in free space. A Gaussian beam of waist radius \(w_0\) doubles its area after the Rayleigh range \(z_R = \pi w_0^2 n/\lambda_0\): a beam of 5 µm waist radius at 1.55 µm doubles its area after about 50 µm. To carry light over kilometres, around corners on a chip, or into a millimetre-long nonlinear crystal, it has to be guided. The idea is total internal reflection: a core of higher index \(n_1\) surrounded by a cladding of lower index \(n_2\) traps rays that meet the boundary beyond the critical angle.

Figure 11

Two quantities follow directly. The largest angle of acceptance from air is the numerical aperture, \(\mathrm{NA} = \sin\theta_\text{max} = \sqrt{n_1^2 - n_2^2}\). And a guided ray at angle \(\theta\) to the axis travels along the guide with propagation constant \(\beta = n_1k_0\cos\theta\); one defines the effective index \(n_\text{eff} = \beta/k_0\), which always lies between \(n_2\) and \(n_1\). Rays are not enough, though: a ray bouncing between the walls overlaps with itself, and only angles for which the round trip adds up in phase survive,

$$2\,n_1k_0\,d\,\sin\theta - 2\phi_r = 2\pi m, \qquad m = 0, 1, 2, \dots ,$$

where \(d\) is the core thickness and \(\phi_r\) the phase shift on total internal reflection (from the Fresnel coefficients). Each integer \(m\) is a mode: a field pattern that keeps its shape as it travels.

14Modes of a slab waveguide

The wave picture is exact. For a slab (core of thickness \(d\) in \(x\), infinite in \(y\)), look for fields \(E_y(x)\,e^{i(\beta z - \omega t)}\) (TE modes). The wave equation becomes

$$\frac{d^2E_y}{dx^2} + \big(n^2(x)k_0^2 - \beta^2\big)E_y = 0 .$$

In the core \(n_1^2k_0^2 > \beta^2\) and the solutions oscillate (cosines and sines); in the cladding \(\beta^2 > n_2^2k_0^2\) and they decay exponentially — the evanescent tails of section 9. Matching \(E_y\) and \(dE_y/dx\) at \(x = \pm d/2\) gives, with \(u = \tfrac{d}{2}\sqrt{n_1^2k_0^2 - \beta^2}\) and \(w = \tfrac{d}{2}\sqrt{\beta^2 - n_2^2k_0^2}\),

$$w = u\tan u\ \ \text{(even modes)}, \qquad w = -u\cot u\ \ \text{(odd modes)}, \qquad u^2 + w^2 = V^2 = \Big(\frac{k_0 d}{2}\Big)^2(n_1^2 - n_2^2) .$$

The single number \(V\) — the normalised frequency — decides everything: the number of guided modes is the integer part of \(2V/\pi\) plus one, and the guide is single-mode for \(V < \pi/2\). The fundamental mode TE₀ is always guided in a symmetric slab; it has no cut-off. TM modes follow similarly with a factor \(n_1^2/n_2^2\) in the matching condition.

Figure 12 · Interactive

15Optical fibres

An optical fibre is a cylindrical waveguide: a glass core of radius \(a\) inside a glass cladding of slightly lower index (index difference typically 0.3–1 %). In 1966 Kao and Hockham argued that if the impurities in glass could be removed, losses below 20 dB/km were possible, enough for communication [15]; modern fibres reach 0.15–0.2 dB/km at 1550 nm — light loses half its power only after 15–20 km.

Modes of a step-index fibre

Solving the wave equation in cylindrical coordinates gives Bessel functions in the core and modified Bessel functions in the cladding. For the small index differences of real fibres the modes group into nearly degenerate linearly polarised (LP) modes [16], and the controlling parameter is again a V number,

$$V = \frac{2\pi a}{\lambda_0}\,\mathrm{NA} = \frac{2\pi a}{\lambda_0}\sqrt{n_1^2 - n_2^2} .$$

For \(V < 2.405\) (the first zero of the Bessel function \(J_0\)) only the fundamental LP₀₁ mode is guided: the fibre is single-mode, which is what telecommunications use (core diameter about 8–9 µm at 1550 nm). Its intensity profile is close to a Gaussian with mode-field diameter \(2w\), \(w/a \approx 0.65 + 1.619V^{-3/2} + 2.879V^{-6}\) [17]. Multimode fibres (50–62.5 µm cores, \(V\) of tens) carry about \(V^2/2\) modes, are easy to couple into, and are used for short links, illumination and spectroscopy.

Figure 13 · Interactive

Other fibres

• Graded-index fibres: the core index falls smoothly from the centre, so rays travelling further also travel faster; modal dispersion almost cancels.
• Polarisation-maintaining fibres: stress rods make the fibre birefringent, so a polarisation launched on an axis stays there.
• Photonic-crystal and hollow-core fibres: a pattern of air holes forms the cladding (section 18), allowing endlessly single-mode guiding, extreme nonlinearity or guiding in air [18].
• Fibre probes for Raman and fluorescence, fibre Bragg gratings (a periodic index change in the core that reflects one wavelength — sensors of strain and temperature), and fibre lasers doped with erbium or ytterbium.

16Integrated waveguides and couplers

On a chip, guides are patterned by lithography into thin films. Their cross-sections give them their names:

Figure 14

High index contrast is the key design choice. Silicon (\(n \approx 3.48\)) on silica (\(n \approx 1.44\)) confines a mode to a wire about 450 × 220 nm and allows bends of a few micrometres radius, packing thousands of components on a chip — silicon photonics. Silicon nitride (\(n \approx 2.0\)) gives lower loss and works in the visible, which matters for on-chip Raman and biosensing; lithium niobate adds fast electro-optic modulation and \(\chi^{(2)}\) nonlinearity.

Coupled waveguides

Two guides placed close together share light through their overlapping evanescent tails. Coupled-mode theory [19] describes the amplitudes \(a_1, a_2\) of the two modes along \(z\):

$$\frac{da_1}{dz} = i\delta\,a_1 + i\kappa\,a_2, \qquad \frac{da_2}{dz} = i\kappa\,a_1 - i\delta\,a_2 \quad\Rightarrow\quad P_2(z) = \frac{\kappa^2}{\kappa^2 + \delta^2}\,\sin^2\!\big(\sqrt{\kappa^2 + \delta^2}\,z\big) ,$$

with \(\kappa\) the coupling coefficient (set by the gap) and \(2\delta = \beta_1 - \beta_2\) the mismatch of the propagation constants. Identical guides exchange all their power over the coupling length \(L_c = \pi/2\kappa\); stopping halfway makes a 50 : 50 splitter. Directional couplers, Mach–Zehnder interferometers and ring resonators (a loop coupled to a straight guide that transmits all wavelengths except those fitting a whole number of times around it [20]) are the building blocks of integrated optics, and also of evanescent-field biosensors.

Figure 15 · Interactive

17Metallic and plasmonic waveguides

Before fibres, electromagnetic waves were guided in hollow metal tubes. At microwave frequencies metals are almost perfect conductors, the tangential \(\mathbf E\) vanishes at the walls, and a rectangular guide of width \(a\) and height \(b\) supports modes TEₘₙ and TMₘₙ with cut-off frequencies

$$f_{c,mn} = \frac{c}{2}\sqrt{\Big(\frac{m}{a}\Big)^2 + \Big(\frac{n}{b}\Big)^2} ;$$

below the lowest, TE₁₀ (\(f_c = c/2a\)), nothing propagates. Such guides carry radar and the waves in MRI scanners and particle accelerators; they fail at optical frequencies because metals become lossy.

At optical frequencies the metal itself can guide: the surface plasmon polariton travels along a single metal–dielectric interface, and a thin dielectric gap between two metals (metal–insulator–metal, MIM) or a metal stripe or nanowire confines light to dimensions far below the wavelength [21]. The price is loss: propagation lengths are micrometres to millimetres, and the tighter the confinement, the shorter. Plasmonic guides are therefore used where confinement matters more than distance — in sensors (SPR), nanofocusing tips (TERS), and compact modulators and detectors.

18Photonic-crystal waveguides and fibres

Total internal reflection is not the only way to trap light. A material whose index varies periodically on the scale of the wavelength — a photonic crystal — reflects light by Bragg interference, and for a range of frequencies (a photonic band gap) no light can propagate through it in any direction, as electrons cannot propagate at energies in the band gap of a semiconductor [22]. Removing one row of holes from a two-dimensional photonic crystal leaves a line defect in which light is guided because it has nowhere else to go — even around sharp corners, and even in a core of lower index than its surroundings.

The same idea in fibres gives photonic-crystal fibres [18]: silica fibres with a lattice of air holes. With a solid core they guide by a modified total internal reflection and can be single-mode at every wavelength; with a hollow core they guide light in air by band-gap or anti-resonant reflection, with low nonlinearity, low latency, and the possibility of filling the core with gases or liquids for spectroscopy. Near the band edge, photonic-crystal waveguides also slow light down by factors of tens, enhancing light–matter interaction.

19Loss and dispersion in waveguides

Waveguide loss is quoted in decibels, \(\text{loss (dB)} = -10\log_{10}(P_\text{out}/P_\text{in})\): 3 dB halves the power. The contributions are

• Absorption by the material (ultraviolet electronic and infrared vibrational absorption; OH impurities near 1.38 µm in silica).
• Scattering: Rayleigh scattering from frozen density fluctuations in glass, scaling as \(\lambda^{-4}\) (why fibre loss is lowest near 1550 nm), and scattering from sidewall roughness in chip waveguides, worse at high index contrast.
• Bending loss: on a bend, the outer part of the evanescent tail would have to travel faster than light in the cladding and radiates away; it grows exponentially as the radius shrinks.
• Coupling loss at inputs, outputs and junctions, from mismatch of mode sizes.

Dispersion has three sources in a guide: material dispersion (\(n(\lambda)\), section 7), waveguide dispersion (the mode's \(n_\text{eff}\) changes with wavelength because the field distribution does) and, in multimode guides, modal dispersion (different modes travel at different speeds). In standard single-mode fibre, material and waveguide dispersion cancel near 1310 nm; dispersion-shifted designs move the zero to 1550 nm, and the same engineering in silicon and silicon nitride waveguides is what makes on-chip frequency combs possible.

20The waveguide family at a glance

From here the series turns from how light travels to what it tells us about molecules: Vibrational spectroscopy, plasmonics and SERS.

21References

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The figures evaluate the closed-form results given in the text (exact slab-mode equations, Fresnel equations, Lorentz and Drude models, coupled-mode theory, the Marcuse mode-field formula). Corrections and questions are welcome at rakesh_n@cpqct.iitr.ac.in.