Interference

6 min readBuilds on Describing a wave and PhaseOpen in the lab

When two waves of light overlap, their fields add. The power of the resulting wave is in general not the sum of the two powers: it depends on the phase difference between the waves, and for two waves of equal power it can take any value from zero to four times the power of one. This effect is interference. It turns a difference of phase into a difference of power, which a detector can measure, and every device in the rest of this series relies on it. This article shows how two waves add, what power their sum carries, and when light interferes at all.

Waves add

Where two waves are present at the same place and time, the field there is the sum of their fields. This is the principle of superposition; it holds for light in all the materials this series considers, because Maxwell’s equations, which govern light in them, are linear (Maxwell's equations) [1]. The two waves do not disturb each other: once they have passed the region where they overlap, each continues as if the other had not been there [2].

Superposition applies to the fields, and it carries over to the complex amplitudes of two waves of the same frequency [1]. The complex amplitude of the sum is

a = a_1 + a_2. \tag{1}

With the phasors of the article Describing a wave, Eq. (1) means adding arrows tip to tail: the arrow of the second wave is drawn from the tip of the first, and the sum runs from the start of the first to the tip of the second.

In step and out of step

If the two waves are in step, with a phase difference of zero, their crests coincide and the sum has the amplitude $A_1 + A_2$ : the waves reinforce each other. This is constructive interference. If they are half a cycle apart, a crest of one meets a trough of the other and the amplitude of the sum is $|A_1 - A_2|$ . This is destructive interference; for two waves of equal amplitude, the sum vanishes. The name comes from this case: waves out of step diminish, or interfere with, each other [2]. Between these extremes the arrows form a triangle, and the length of the sum lies in between.

Figure 1 shows two waves from one laser, their sum, and their phasors added tip to tail. The meters compare the power of the sum with the sum of the powers.

phasors: wave 1, then wave 2, and the sum

Phase difference Δφ0

Amplitude of wave 21.00

Power of the sum4.00 P₁

Sum of the powers2.00 P₁

P₁ is the power of wave 1. The dashed line is the sum of the powers; the curve is the power of the sum for every phase difference, the dot the current one.

Wave 2 has amplitude 1.00 and is 0.00 cycle behind wave 1. Their sum has a power of 4.00 P₁, while the sum of their powers is 2.00 P₁.

Figure Two waves of the same frequency and their sum. The phasor diagram adds the complex amplitudes tip to tail. The curve below gives the power of the sum for every phase difference; the dashed line is the sum of the two powers.APX-005

Power does not add

The power of the sum is the squared magnitude of Eq. (1). With $P_1 = |a_1|^2$ , $P_2 = |a_2|^2$ and the phase difference $\Delta\varphi$ between the two waves,

P = |a_1 + a_2|^2 = P_1 + P_2 + 2\sqrt{P_1 P_2}\,\cos\Delta\varphi. \tag{2}

This is the interference equation [1] [2]. The first two terms are the sum of the powers. The third, the interference term, can be positive or negative, and it is what makes the result depend on the phase difference.

For two waves of equal power $P_0$ , Eq. (2) becomes

P = 2P_0\,(1 + \cos\Delta\varphi) = 4P_0 \cos^2\frac{\Delta\varphi}{2}. \tag{3}

In step, $\Delta\varphi = 0$ , the sum carries $4P_0$ : twice the field gives four times the power. Half a cycle apart, $\Delta\varphi = \pi$ , it carries nothing. A quarter of a cycle apart, $\Delta\varphi = \pi/2$ , the interference term vanishes and the power is $2P_0$ , the sum of the two powers [1]. Complete cancellation is possible only for waves of equal power; for unequal waves the sum never falls below $(\sqrt{P_1} - \sqrt{P_2})^2$ .

Where the light goes

When two waves cancel, where does their light go? Interference does not create or destroy energy. It moves power from one place to another: where the sum of two waves is darker than the sum of their powers, it is brighter elsewhere, and the total power is conserved [1]. When two beams cross in free space, this produces the familiar pattern of bright and dark bands. In the circuits of this series, the light that is missing at one output of a device leaves by another output, as the article Splitters and combiners describes.

Figure 2 shows two beams of equal power crossing at an angle $\theta$ , as their summed field at one instant. Across the beams the phase difference between them changes steadily, so their power alternates between four times the power of one beam and zero. The bright bands are $\lambda/\sin\theta$ apart: 2.7 µm for beams 35° apart at 1550 nm [1]. On average across the bands, the power is the sum of the two powers.

the field (left); right, the power averaged over time−+

Angle between the beams θ35 °

Phase difference Δφ0

distance between bright bands λ/sin θ: 2.70 µm
brightest band: 4 × one beam

Two beams of 1550 nm and equal power in air, drawn as plane waves, at one instant. The strip at the right shows the power across the beams at the right edge, averaged over time: from zero in the dark bands to four times the power of one beam in the bright ones.

Two beams cross at 35 degrees with a phase difference of 0. Their power forms bright and dark bands 2.70 micrometres apart.

Figure Two plane waves of 1550 nm and equal power crossing in air, wave 2 at the angle

\theta

to wave 1, after Saleh and Teich [1]. Left: their summed field at one instant, red where it points one way, blue where it points the other. Right: the power across the beams, averaged over time, in bright and dark bands.APX-005

When light interferes

Equation (2) assumes that the phase difference $\Delta\varphi$ stays fixed while the detector averages. For waves that come from the same laser and are split into two paths, it does: both carry the same oscillation, delayed by different amounts. Such waves are coherent.

Ordinary light sources emit light whose phase changes randomly and rapidly. Between two such sources the phase difference takes all values during a measurement, $\cos\Delta\varphi$ averages to zero, and the interference term disappears [1]. The powers then add. Two independent lasers are an intermediate case: each keeps its phase steady for a comparatively long time, and their interference has been observed over short times [2]. In general their frequencies also differ slightly, so their phase difference drifts [1], and a detector that averages over a long time sees the sum of their powers. The lab treats light from separate lasers in this way.

Figure 3 shows two lasers of equal power whose frequencies differ by $\Delta f$ . Their power together is not steady: it beats between zero and four times the power of one laser, at the difference frequency $\Delta f$ [1]. A detector reports the average over its averaging time. If that time is short compared with one beat period, the reading follows the beat. If it spans many beat periods, the beat averages out and the reading is the sum of the two powers; over a whole number of beat periods it vanishes exactly [2]. Two lasers 1 GHz apart beat once per nanosecond, and their wavelengths at 1550 nm differ by only 8 pm.

Frequency difference Δf1.00 GHz

Detector's averaging time T316 ps

beat period 1/Δf: 1.0 ns
wavelength difference at 1550 nm: 8.0 pm
share of the beat the detector sees: 84 %

Violet: the power of the two lasers together; black: the detector's reading, the average over the last T; dashed: the sum of the two powers. The optical oscillation, 193 THz, is already averaged out. At Δf = 0 the lasers share one frequency and stay in step.

Two lasers 1.00 GHz apart beat with a period of 1.0 ns. A detector averaging over 316 ps sees 84 % of the beat; the rest averages to the sum of the two powers.

Figure Two lasers of equal power at 1550 nm whose frequencies differ by

\Delta f

. Violet: their power together, which beats at

\Delta f

; black: what a detector reports, the average over its averaging time

T

; dashed: the sum of the two powers.APX-005

Measuring a phase difference

Equation (2) makes a phase difference measurable. The power of two combined waves changes by the full range between constructive and destructive interference when their phase difference changes by half a cycle, so a detector that measures the power measures the phase difference [1]. An instrument that splits light into two paths, delays one relative to the other, recombines them and detects the result is an interferometer. The Mach-Zehnder interferometer is one, built on a chip (The Mach-Zehnder interferometer).

Key idea

Two coherent waves add their fields, not their powers; the power of their sum depends on the phase difference between them.

How do we model itOptional · the interference equation, and why two lasers do not interfere on average

With $a_1 = A_1 e^{i\varphi_1}$ and $a_2 = A_2 e^{i\varphi_2}$ ,

\begin{aligned} |a_1 + a_2|^2 &= (a_1 + a_2)(a_1 + a_2)^* \\ &= |a_1|^2 + |a_2|^2 + a_1 a_2^* + a_1^* a_2 \\ &= A_1^2 + A_2^2 + 2A_1A_2\cos(\varphi_2 - \varphi_1), \end{aligned}

which is Eq. (2) with $\Delta\varphi = \varphi_2 - \varphi_1$ .

Two waves of different frequencies $f_1$ and $f_2$ have a phase difference that grows in time, $\Delta\varphi(t) = 2\pi (f_2 - f_1) t + \text{const}$ . Their summed power oscillates at the difference frequency $|f_2 - f_1|$ [1]. A detector that averages over a time $T \gg 1/|f_2 - f_1|$ sees the mean of $\cos\Delta\varphi(t)$ , which is zero, and reports $P_1 + P_2$ . Two lasers whose frequencies differ by 1 GHz beat at 1 GHz; a detector that averages over a microsecond averages over a thousand beat periods.

In short

Where waves overlap, their fields add; their complex amplitudes add like arrows, tip to tail.
The power of two coherent waves together is $P_1 + P_2 + 2\sqrt{P_1 P_2}\cos\Delta\varphi$ : between zero and four times the power of one, for equal waves.
Interference moves power from one place or output to another; the total power is conserved.

Next · FoundationsPhotonic integrated circuitsChips that guide light along paths of their own, what they are used for today, what they cannot do, and what the rest of this series builds.Read next Build it in the labTwo-colour demultiplexerThe example has two lasers. With one laser, the outputs depend on the heater; with both, each colour interferes only with itself.Open the lab

References

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

Approximations used on this page: APX-005 separate lasers are incoherent.