Describing a wave

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The article Maxwell's equations showed that light is a wave of the electric field, and that one number, the field $E$ at each point and instant, is enough to describe it. Every calculation in the later articles starts from the description of a wave set out here: how the field depends on position and time, how it is written down, and what a detector measures of it.

What a wave is

A wave is a disturbance that travels, while the medium it travels through stays where it is. Wind blowing across a field of grain sets up waves that move across the field, although each stalk only sways back and forth in its place [1]. What moves along is the pattern of the disturbance, and with it energy. In a light wave the disturbance is the electric field, and no medium is needed: at every point the field grows, falls to zero, reverses and returns, while the pattern moves on at the speed of light.

Wavelength, frequency and speed

A wave that repeats regularly is described by three quantities. The wavelength $\lambda$ is the distance between two neighbouring crests at one instant. The frequency $f$ is the number of crests that pass a fixed point per second, and its inverse, the period $T = 1/f$ , is the time between two crests passing that point. In one period the wave moves on by one wavelength, so its speed is

c = \lambda f. \tag{1}

For the light used on silicon chips, $\lambda \approx 1550\,\mathrm{nm}$ , Eq. (1) gives a frequency of 193 THz and a period of 5.2 fs. The field at any point therefore reverses about $4 \times 10^{14}$ times per second.

One wave, two views

The simplest wave, and the one this series uses throughout, is a harmonic wave, whose field is a cosine:

E(x, t) = A \cos(kx - \omega t + \varphi), \qquad k = \frac{2\pi}{\lambda}, \quad \omega = 2\pi f. \tag{2}

Here $A$ is the amplitude, the largest value the field reaches, $k$ is the wave number and $\omega$ the angular frequency. The whole argument of the cosine is the wave’s phase, and the constant $\varphi$ is set by the source [1]. The phase is the subject of the next article; here it is enough that it says where in its cycle the wave is.

Equation (2) can be read in two ways. At one instant, $t$ fixed, it is a cosine in space with period $\lambda$ : a snapshot of the wave. At one place, $x$ fixed, it is a cosine in time with period $T$ : what an observer at that place sees. A crest, where the phase is zero, moves forward with the speed $\omega/k = \lambda f = c$ [1].

Figure 1 shows both views of one wave, slowed down by a factor of about $3 \times 10^{14}$ . The upper plot is the snapshot, $t$ fixed; the lower one records the field at the fixed place $x = 1\,\mu\mathrm{m}$ over the last 20 fs, with the newest value at the right. The dot at the probe place in the snapshot and the dot at “now” are the same number. A shorter wavelength shortens the period in space and, since $f = c/\lambda$ , the period in time as well.

Amplitude A1.00

Wavelength λ1550 nm

Phase φ0

period T = 1/f: 5.17 fs
frequency f = c/λ: 193 THz
speed of a crest, λ/T: 3.00 × 10⁸ m/s
range: infrared

Time runs about 3 × 10¹⁴ times slower than in reality. The field is given in units of the amplitude 1.

A harmonic wave of amplitude 1.00, wavelength 1550 nm and phase 0. At one instant the field repeats along the path every 1550 nm; at the fixed place x = 1 µm it repeats in time every 5.17 fs.

Figure One harmonic wave read in two ways. Above,

t

fixed: the field along the path at one instant, with the probe place

x = 1\,\mu\mathrm{m}

marked. Below,

x

fixed: the field at the probe place over the last 20 fs, the newest value at the right. The two dots are the same value.

Two ways to write a wave

The cosine of Eq. (2) is the quantity a probe would measure: a real number at every point and instant. For calculations, however, the cosine is awkward. Adding two cosines with different phases takes several trigonometric identities, and from the article Interference on, almost every step adds waves [1].

Euler’s formula, $e^{i\theta} = \cos\theta + i\sin\theta$ , offers a way out. The cosine is the real part of a complex exponential, so the same wave can be written as

E(x, t) = \operatorname{Re}\!\left[a\, e^{i(kx - \omega t)}\right], \qquad a = A\, e^{i\varphi}. \tag{3}

We calculate with the complex exponential throughout and take the real part only at the end. This is allowed because the operations applied to waves in this series, adding them, delaying them and sending them through components, act on the field linearly [1]. The imaginary part carries no physical meaning of its own; it is bookkeeping that makes the phases easy to handle.

The complex number $a$ in Eq. (3) is the wave’s complex amplitude. It holds everything that distinguishes this wave from others of the same frequency: its magnitude $|a|$ is the amplitude $A$ , its angle is the phase $\varphi$ [2]. It can be pictured as an arrow in the complex plane, of length $A$ and at the angle $\varphi$ from the real axis. This arrow is called a phasor [1], and the series draws it as a phasor clock: a dial with the arrow on it, the angle counted anticlockwise from the right as usual in the complex plane. As time passes, the factor $e^{-i\omega t}$ turns the arrow clockwise, once per period, and its projection onto the real axis runs through the values of the cosine. The factor $e^{ikx}$ turns it by the place: one wavelength further along the path, the arrow has turned once more.

Figure 2 shows the phasor clock next to the wave it stands for. The dashed arrow is the complex amplitude $a$ . The solid arrow is $a\,e^{i(kx_0 - \omega t)}$ at a probe place $x_0$ , and its projection onto the horizontal axis is the field at that place, the dot in the plot.

the arrow at x₀ = 0.50 µm

Amplitude A1.00

Phase φ0.25π

Probe place x₀0.50 µm

complex amplitude a = A eⁱᵠ: 0.71 + 0.71i
the arrow at x₀ now: −0.95 + 0.32i
field E(x₀, t) = its real part: -0.95

Dashed: the complex amplitude a. Light of 1550 nm, slowed down about 6 × 10¹⁴ times.

The complex amplitude a has length 1.00 and angle 0.25π. At the probe place x₀ = 0.50 µm the arrow is turned further by kx₀ and turns clockwise once per period; its projection on the horizontal axis is the field at that place, marked in the snapshot of the wave.

Figure The phasor clock of a wave of 1550 nm and the wave itself. The dashed arrow is the complex amplitude

a

; the solid arrow, the phasor at the probe place

x_0

, turns clockwise once per period. Its projection on the horizontal axis is the field at

x_0

, marked by the dot in the plot.

With this picture, the operations of the following articles become geometry. Delaying a wave turns its arrow; adding two waves adds their arrows tip to tail.

One restriction comes with the complex form: it may not be used for products of fields. Squaring the complex expression in Eq. (3) does not give the square of the real field [1]. The power of a wave, which is such a product, is therefore computed from the magnitude of the complex amplitude, as the next section shows.

What a detector measures

What, then, does a detector measure? No ordinary detector follows a field that reverses $4 \times 10^{14}$ times per second. A detector collects the energy the light delivers over some finite time and reports the average rate, the power of the light [1]. Even a microsecond spans about $10^9$ periods of light, so the average covers an enormous number of oscillations [1].

The energy of a light wave is stored in its fields. In vacuum the energy per unit volume is $u = \varepsilon_0 E^2$ , shared equally between the electric and the magnetic field [1]. The energy that arrives at a detector at each instant is therefore proportional to the square of the field [1]. The square is never negative, and it oscillates twice as fast as the field itself [1].

The power of a harmonic wave is therefore proportional to the time average of the squared field [2]. The square of a cosine averages to one half, so

P \propto \langle E^2 \rangle = \frac{A^2}{2} = \frac{|a|^2}{2}. \tag{4}

The power grows with the square of the amplitude. Doubling the amplitude of a wave quadruples its power, and a wave with half the amplitude carries a quarter of the power. The field can be positive or negative; the power is never negative, and it does not depend on the phase $\varphi$ . Figure 3 shows the difference. The field at the detector averages to zero; its square oscillates between 0 and $A^2$ , and its average, $A^2/2$ , lies where the humps above it fill the troughs below it.The oscillating field of light has been recorded directly, for a pulse of only a few cycles, with an electron probe in a specialised experiment [1]. Detectors in circuits measure power.NoteThe oscillating field of light has been recorded directly, for a pulse of only a few cycles, with an electron probe in a specialised experiment [1]. Detectors in circuits measure power.

Amplitude A1.00

Phase φ0

Detector (average power)1.00 × P₁

average of E², A²/2: 0.50
average of E: 0

Dashed: the average of E². Light of 1550 nm, one period 5.17 fs. P₁ is the power of a wave with amplitude 1.

The field at the detector oscillates between -1.00 and 1.00 and averages to zero. Its square oscillates twice as fast between 0 and 1.00 and averages to 0.50. The detector reads 1.00 times the power of a wave with amplitude 1.

Figure The field of 1550 nm light at a detector over four periods, and its square, to which the energy arriving at each instant is proportional. The dashed line is the average of the square,

A^2/2

, drawn after Hecht [1]. The phase shifts both curves but not the average.

The difference between field and power matters in every later article: waves add their fields, but detectors report powers. The quantities that decide what happens in a circuit, the amplitudes and phases of the fields, are not what a detector shows; the circuits in later articles are built to turn them into differences in power.

Throughout the series, as in the lab, powers are given relative to a reference: a percentage of the laser’s power, or a multiple of the power of a wave with amplitude 1. In these units the power of a wave is simply $|a|^2$ .

Key idea

A detector measures the average power of light, which is proportional to the squared magnitude of the complex amplitude; the field itself oscillates too fast for any ordinary detector to follow.

How do we model itOptional · the time average behind Eq. (4), and the sign convention

The power carried by a light wave per unit area is its intensity (in optics also called irradiance). In vacuum it is $I = \varepsilon_0 c \langle E^2 \rangle$ , where $\varepsilon_0$ is the vacuum permittivity and $\langle \cdot \rangle$ the average over a time much longer than one period [1]. For the harmonic wave of Eq. (2), at a fixed position $x$ and with $\theta = kx + \varphi$ ,

\begin{aligned} \langle E^2 \rangle &= \frac{A^2}{T_\text{avg}} \int_0^{T_\text{avg}} \cos^2(\theta - \omega t)\, \mathrm{d}t \\ &= \frac{A^2}{2 T_\text{avg}} \int_0^{T_\text{avg}} \bigl[1 + \cos(2\theta - 2\omega t)\bigr]\, \mathrm{d}t \\ &= \frac{A^2}{2} + A^2\,\frac{\sin 2\theta - \sin(2\theta - 2\omega T_\text{avg})}{4\omega T_\text{avg}}, \end{aligned}

where the second line uses $\cos^2 u = \tfrac12 (1 + \cos 2u)$ . The oscillating term in the last line is at most $A^2/(2\omega T_\text{avg})$ , so it vanishes once the average spans many periods. An average over one microsecond spans about $10^9$ periods [1], and the oscillating term is then at most $1/(2\pi \cdot 10^9) \approx 2 \times 10^{-10}$ of $A^2/2$ .

Only ratios of powers matter in this series, so the constants are dropped: a wave of complex amplitude $a$ is said to have power $|a|^2$ , and powers are compared with that of a wave of amplitude 1. The lab does the same and gives every power relative to its laser.

The sign in the exponent is a convention. Physics texts, and this series, write the time dependence as $e^{-i\omega t}$ , so that propagation over a length $L$ multiplies the complex amplitude by $e^{+ikL}$ . Many engineering texts write $e^{+j\omega t}$ instead, which flips the sign of every phase [2]; Hecht uses both forms [1]. Powers are the same in both conventions. When a formula for a phase is taken from a book, its convention has to be checked first.

In short

A harmonic wave is described by its wavelength $\lambda$ , its frequency $f$ and its amplitude $A$ , with $c = \lambda f$ ; light of 1550 nm oscillates at 193 THz.
The same wave can be written as a cosine, $A\cos(kx - \omega t + \varphi)$ , or as the real part of $a\,e^{i(kx - \omega t)}$ ; the complex amplitude $a = A e^{i\varphi}$ is pictured as an arrow, the phasor.
Detectors measure the average power, proportional to $|a|^2 = A^2$ : doubling the amplitude quadruples the power, and the phase of a single wave does not show.

Next · FoundationsPhaseThe position of a wave in its cycle, how it grows along a path and in a material, and why only differences in phase can be observed.Read next Build it in the labMZI switchZoom into a waveguide until the lab draws the wave itself; its crests move along the waveguide.Open the lab

References

1Hecht, E. (2017). Optics (5th edition, global edition). Pearson Education Limited. link
2Saleh, B. E. A., Teich, M. C. (2019). Fundamentals of Photonics (3rd edition). Wiley. link