Phase

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The phase of a wave states where in its cycle the wave is, at a given place and time. Two waves from the same laser that travel along different paths arrive with different phases, and the difference decides how they combine when they meet. Every circuit in this series works by controlling such phase differences. This article follows the phase along a path and through materials, and shows why only differences in phase count.

The position in the cycle

The harmonic wave of the article Describing a wave has the field $E = A\cos(kx - \omega t + \varphi_0)$ . The whole argument of the cosine is the wave’s phase [1]. The constant $\varphi_0$ is set by the source; the terms $kx$ and $\omega t$ state how the phase changes with position and time. One cycle of the wave corresponds to a change of phase by $2\pi$ . A phase of $\pi$ is half a cycle: the field is then the negative of what it would be at a phase of 0.

The phase is the angle of the wave’s phasor, the arrow that represents its complex amplitude $a = A e^{i\varphi}$ [2]. The phasor clocks in the figures of this series show this arrow. Because an arrow turned by a full circle is the same arrow, phases that differ by a multiple of $2\pi$ describe the same state of the wave, and changing the phase by $\Delta\varphi$ amounts to multiplying the complex amplitude by $e^{i\Delta\varphi}$ .

Phase grows along a path

A wave that travels a distance $L$ gains the phase $kL = 2\pi L/\lambda$ : one full cycle for every wavelength of path. Two waves that leave a laser together and travel paths of different length therefore arrive with different phases. If one path is longer by half a wavelength, the wave on it arrives half a cycle behind the other, and its field is the negative of the other’s at every instant. If it is longer by a whole wavelength, the two arrive in step again.

Light is slower in a material

In a transparent material light travels more slowly than in vacuum, with the speed $c/n$ , where $n$ is the material’s refractive index (Maxwell's equations). For air $n$ is very close to 1, for silica glass about 1.444 [3] and for silicon about 3.47 at a wavelength of 1550 nm [3]. The frequency of the light does not change when it enters a material; its speed does, and so does its wavelength, which becomes $\lambda/n$ , where $\lambda$ is the wavelength in vacuum [2]. A given length of material therefore holds $n$ times as many cycles as the same length of vacuum, and the phase gained over a length $L$ is

\varphi = \frac{2\pi}{\lambda}\, n L. \tag{1}

The product $nL$ is the optical path length. Throughout the series, $\lambda$ without further qualification is the wavelength in vacuum.

For light of 1550 nm, 1 µm of air holds 0.65 cycles and 1 µm of silicon holds 2.24 cycles. Replacing 1 µm of air by 1 µm of silicon therefore delays the wave by 1.6 cycles.

Figure 1 shows two waves from one laser. Both travel 4 µm, one through air only, the other partly through a block of glass or silicon. The phasor clocks at the ends show the phase of each wave there. Both clocks turn at the same rate, so the angle between their arrows stays fixed: it is the phase difference $\Delta\varphi$ .

end of path 1

end of path 2

Block in path 1

Block length L0.000 µm

phase difference Δφ: 0.00 cycle = 0

Wavelength in air 1550 nm. Both clocks turn once per period; the angle between their arrows is the phase difference. The dotted line marks the field at that end of the path.

Path 1 contains 0.000 µm of silicon. At the end, the wave in path 1 is 0.00 cycle behind the wave in path 2.

Figure Two waves from one laser travel paths of equal length, one of them partly through a block of glass or silicon. The clocks show each wave's phasor at the end of its path; the dotted line marks its field at that instant.APX-002

The block needed for half a cycle follows from Eq. (1): the phase difference is $2\pi(n - 1)L/\lambda$ , which equals $\pi$ for $L = \lambda/(2(n - 1))$ . For silicon this is 0.31 µm, for glass 1.75 µm.

Only differences can be observed

Can the phase of a single wave be measured at all? A detector reports the power of a wave, $|a|^2 = A^2$ , which does not depend on the phase $\varphi$ of its complex amplitude. Delaying a single wave by any amount therefore changes nothing a detector can see. Adding the same phase to all waves in an experiment changes nothing either; it corresponds to starting the clock at a different moment.

What can be observed is the phase difference between two waves that come from the same laser. Their phasors keep a fixed angle, and when the two waves are brought together, this angle decides how they add, which the article Interference describes. A phase difference of 1.6 cycles, as in the example above, has the same effect as one of 0.6 cycles.

Key idea

A wave gains a phase of

2\pi

for every wavelength of optical path it travels, and only differences in phase between waves from the same source can be observed.

How do we model itOptional · why a common phase cannot be observed

In the complex notation of the article Describing a wave, propagation over a length $L$ of a material with index $n$ multiplies the complex amplitude by $e^{ikL}$ with $k = 2\pi n / \lambda$ , and a delay by a phase $\Delta\varphi$ multiplies it by $e^{i\Delta\varphi}$ . A longer path gives a larger positive phase.

A phase common to all waves is not observable because every measured quantity is built from products $a_j a_k^*$ of complex amplitudes, in which a common factor $e^{i\theta}$ cancels: $(e^{i\theta} a_j)(e^{i\theta} a_k)^* = a_j a_k^*$ . The phase difference between two paths of optical path lengths $n_1 L_1$ and $n_2 L_2$ is

\Delta\varphi = \frac{2\pi}{\lambda}\,(n_1 L_1 - n_2 L_2).

In short

The phase states where a wave is in its cycle; a full cycle is $2\pi$ , and changing the phase by $\Delta\varphi$ turns the phasor, which multiplies the complex amplitude by $e^{i\Delta\varphi}$ .
A wave gains the phase $2\pi nL/\lambda$ over a length $L$ of material with refractive index $n$ , because inside the material its wavelength is $\lambda/n$ .
A detector cannot see the phase of a single wave; the phase difference between two waves from one laser becomes visible when they are combined.

Next · FoundationsInterferenceWhen two waves of light meet, their fields add, and the power of the sum depends on their phase difference.Read next Build it in the labMZI switchZoom into the two arms of the switch and compare the crests of the wave at their ends while turning the heater.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
3Chrostowski, L., Hochberg, M. (2015). Silicon Photonics Design. Cambridge University Press. doi:10.1017/CBO9781316084168

Approximations used on this page: APX-002 no reflections.