Superposition Principle of Electromagnetic Waves

This chapter opens the wave-optics half of the subject, and it opens with a loan from electromagnetism. We take two results as given, derived and owned there: that the electromagnetic field obeys a wave equation $\nabla^2\mathbf E-\tfrac1{c^2}\partial_t^2\mathbf E=0$ following from Maxwell's curl equations, and that its elementary solutions are monochromatic plane waves written compactly in complex form. What optics does with these is exploit one structural fact — their linearity — and follow it to its most consequential end. The superposition principle is the statement that the electromagnetic field obeys a linear theory: if two field configurations each solve Maxwell's equations in a medium, so does their sum, and so does any linear combination. This is not a new postulate but a direct consequence of the wave equation containing no products or powers of the fields. Physically it says overlapping light beams pass through one another untouched — two crossing laser beams emerge unchanged, radio channels share the same air — and that any complicated field can be resolved into simpler pieces (plane waves, spherical waves, Fourier components), solved, and reassembled by addition. The subtlety, and the source of nearly all of physical optics, is that we almost never observe the field itself. Detectors, film, and the eye register energy flux, which goes as the square of the field. Squaring a sum introduces a cross term, so when two waves overlap the observed intensity is not $I_1+I_2$ but $I_1+I_2$ plus an interference term. Interference, diffraction, thin-film colour, and coherence are all the same two-step move: superpose the fields, then square. Add intensities directly and you get the wrong answer everywhere it matters. Superposition is exact in vacuum and excellent in ordinary media; it fails only at the fierce intensities where the medium's response turns nonlinear — which is exactly the boundary that defines the classical, linear optics of this course.

Key takeaways

Recalling from electromagnetism: the source-free Maxwell equations give a linear wave equation whose elementary solutions are monochromatic plane waves — optics takes their linearity and runs with it.
Because the wave equation is linear and homogeneous, any linear combination of solutions is a solution: overlapping waves add as vectors, $\mathbf E=\sum_i\mathbf E_i$ , and pass through one another unchanged.
Detectors respond to intensity $I\propto\langle|\mathbf E|^2\rangle$ , and $|\mathbf E_1+\mathbf E_2|^2$ carries a cross term $2\mathbf E_1\!\cdot\!\mathbf E_2$ — so intensities do NOT simply add; the cross term is interference.
Linearity also licenses decomposition into a basis — plane waves, cavity modes, Fourier components — the engine of Fourier optics and normal-mode analysis.
Superposition is exact in vacuum and holds in ordinary media at everyday intensities; it breaks at high intensity where the polarization gains terms $\propto E^2,E^3$ and the medium mixes the waves (nonlinear optics).
The interference cross term is handed, in full, to the next article on the two-wave interference law.
Key idea: Maxwell's equations are linear, so overlapping electromagnetic waves add as the vector sum of their fields — but because detectors measure the square of the field, intensities do not add: the cross term $2\mathbf E_1\!\cdot\!\mathbf E_2$ is the seed of interference.

The core insight

The source-free electromagnetic wave equation is linear, so any sum (or linear combination) of solutions is itself a solution: overlapping waves superpose as the vector sum of their fields, $\mathbf E=\sum_i\mathbf E_i$ . But detectors measure intensity $\propto\langle|\mathbf E|^2\rangle$ , and squaring the sum produces a cross term $2\mathbf E_1\!\cdot\!\mathbf E_2$ — so intensities do not add. That cross term is interference, and everything in physical optics follows from superposing fields and then squaring.

Where this lives today

Superposition is the silent premise of nearly every optical instrument. Every interferometer — Michelson, Mach–Zehnder, Fabry–Pérot, Young's slits — splits a beam, imposes a path difference, and superposes the fields on a detector, so the cross term reads out displacement, index, or spectrum with sub-wavelength sensitivity; LIGO turns this into a strain measurement of $\sim10^{-21}$ . Fourier optics treats a lens as a device that superposes plane-wave components, making spatial filtering, holography, and computer-generated imaging routine. Mode-locked lasers phase-superpose thousands of cavity modes to build femtosecond pulses, and optical and radio phased arrays steer beams electronically by controlling the relative phase of many emitters, with no moving parts. The same principle, pushed to its breaking point, defines the frontier: at the intensities where the response turns nonlinear, superposition of the field survives but superposition of the observable fails, and the medium becomes a mixer — the domain of frequency combs, optical parametric oscillators, and attosecond science.

The physical picture

Treat the field at a point as a number per component, not an object. Two ripples crossing the same point on a pond simply add their displacements at that instant — a crest meeting a trough momentarily cancels, then both ripples move on unchanged. Light does exactly this, component by component, because the governing equations contain no term that lets one field feel another. The cleanest bookkeeping for a single frequency is the phasor: represent each wave by a rotating arrow whose length is the amplitude and whose angle is the phase, and superposition becomes head-to-tail vector addition.

Figure

Phasor addition: two waves of amplitudes √I₁ and √I₂ separated by phase δ add head-to-tail; the resultant length squared is the intensity, I = I₁ + I₂ + 2√(I₁I₂) cos δ, so relative phase controls the result

The reason intensities refuse to add is purely geometric: energy goes as the square of an amplitude, and $(E_1+E_2)^2=E_1^2+E_2^2+2E_1E_2$ . That last term is positive where the fields reinforce and negative where they oppose, so it moves energy from dark places to bright ones without creating or destroying any. The punchline is what this does to two equal beams.

Figure

Punchline — two equal coherent beams of intensity I₀ each: the observed intensity |E₁+E₂|² sweeps from 4I₀ (in phase) down to 0 (anti-phase) as the phase δ varies, averaging to 2I₀ — intensities do not add, they interfere

Two beams that alone would give $2I_0$ do not give $2I_0$ everywhere: they give $4I_0$ where they are in phase and $0$ where they are opposed, and the spatial average returns to $2I_0$ . No energy is lost — the cosine merely reshuffles it into fringes. That single figure is the whole reason "add the intensities" is wrong, and why the next article's cross term matters.

The core equations

We begin from the two results owned by electromagnetism. Maxwell's source-free equations combine into a linear wave equation, and it is the linearity of this equation — not any new optical assumption — that everything below rests on:

\nabla^2\mathbf E-\frac1{c^2}\frac{\partial^2\mathbf E}{\partial t^2}=0,\qquad c=\frac1{\sqrt{\mu_0\varepsilon_0}} \qquad (1)

where: $\mathbf E$ — electric field (each component obeys the same scalar equation); $c$ — vacuum speed of light, $c=1/\sqrt{\mu_0\varepsilon_0}=2.998\times10^8\ \mathrm{m/s}$ . The derivation of $(1)$ from the curl equations, and its d'Alembert solutions, belong to aq-em-electromagnetic-wave-equation; here $(1)$ is a recalled input. Its elementary solutions are monochromatic plane waves, most compactly written in complex form, again a recalled result:

\mathbf E(\mathbf r,t)=\mathrm{Re}\big[\tilde{\mathbf E}_0\,\mathrm e^{\,i(\mathbf k\cdot\mathbf r-\omega t)}\big],\qquad \omega=c|\mathbf k| \qquad (2)

where: $\tilde{\mathbf E}_0$ — complex vector amplitude (magnitude and phase); $\mathbf k$ — wavevector (direction of travel, $|\mathbf k|=2\pi/\lambda$ ); $\omega$ — angular frequency. The construction and properties of $(2)$ are owned by aq-em-monochromatic-plane-waves-vacuum. The operator in $(1)$ is linear: it distributes over sums and scalars. This one property is the superposition principle in embryo:

L[a_1\psi_1+a_2\psi_2]=a_1L[\psi_1]+a_2L[\psi_2],\qquad L\equiv\nabla^2-\frac1{c^2}\partial_t^2 \qquad (3)

where: $L$ — the wave operator; $\psi_i$ — any solution (a field component); $a_i$ — constants (possibly complex). If $L[\psi_1]=L[\psi_2]=0$ then $L[a_1\psi_1+a_2\psi_2]=0$ : the combination solves the wave equation too. Applied to full field solutions, $(3)$ gives the superposition principle:

\mathbf E(\mathbf r,t)=\sum_i\mathbf E_i(\mathbf r,t),\qquad \mathbf B(\mathbf r,t)=\sum_i\mathbf B_i(\mathbf r,t) \qquad (4)

where: $\mathbf E_i,\mathbf B_i$ — individual wave solutions; $\mathbf E,\mathbf B$ — resultant fields where the waves overlap. Waves add, and pass through one another unchanged. What a detector reports, however, is not the field but its cycle-averaged energy flux — the intensity:

I=c\varepsilon_0\,\langle|\mathbf E|^2\rangle=\tfrac12c\varepsilon_0E_0^2 \qquad (5)

where: $I$ — irradiance / intensity (time-averaged power per area); $\langle\cdot\rangle$ — average over many optical cycles; the factor $\tfrac12$ comes from $\langle\cos^2\rangle=\tfrac12$ . That intensity is the time-averaged Poynting flux is a result owned by electromagnetism (aq-em-intensity-time-averaging, aq-em-energy-density-poynting-vector); optics recalls it and applies it to a superposed field. Squaring the sum $(4)$ is where interference is born:

|\mathbf E_1+\mathbf E_2|^2=|\mathbf E_1|^2+|\mathbf E_2|^2+2\,\mathbf E_1\!\cdot\!\mathbf E_2 \qquad (6)

where: $2\mathbf E_1\!\cdot\!\mathbf E_2$ — the interference (cross) term, absent only when it averages to zero. Carrying $(6)$ through the time average gives the two-wave law $I=I_1+I_2+2\sqrt{I_1I_2}\cos\delta$ — developed in full, with visibility and energy conservation, in the next article; here we prove only that the cross term must be there.

Derivation ALinearity implies superposition

The goal is to prove, from nothing but the linearity of the field equations, that the sum of two electromagnetic solutions is again a solution — the statement that overlapping waves add without interacting. The strategy is direct: write down the four source-free Maxwell equations for two independent solutions, add them equation by equation, and use the fact that divergence, curl, and time derivative are each linear operators, so the sum obeys the same four equations. No property of the individual waves is used beyond their each being a solution.

Step 1. Let $(\mathbf E_1,\mathbf B_1)$ and $(\mathbf E_2,\mathbf B_2)$ each satisfy the source-free Maxwell equations in a linear medium. Add Faraday's law for the two: $\nabla\times(\mathbf E_1+\mathbf E_2)=\nabla\times\mathbf E_1+\nabla\times\mathbf E_2=-\partial_t(\mathbf B_1+\mathbf B_2)$ . Why the split is legal: the curl of a sum is the sum of curls, so the combined field obeys Faraday's law with no cross terms.

Step 2. Add Ampère's law the same way: $\nabla\times(\mathbf B_1+\mathbf B_2)=\mu_0\varepsilon_0\,\partial_t(\mathbf E_1+\mathbf E_2)$ . Why it carries over: again a linear operator acts on a sum, so the sum satisfies the law term by term — the two waves never multiply.

Step 3. Add the two divergence equations: $\nabla\!\cdot(\mathbf E_1+\mathbf E_2)=0$ and $\nabla\!\cdot(\mathbf B_1+\mathbf B_2)=0$ . All four Maxwell equations now hold for the sum, so $(\mathbf E_1+\mathbf E_2,\mathbf B_1+\mathbf B_2)$ is a genuine solution. Why this closes the proof: satisfying all four source-free equations is exactly what "is an electromagnetic field" means.

Step 4. By induction the same holds for any finite combination $\sum_i a_i(\mathbf E_i,\mathbf B_i)$ , and with convergence for infinite sums and integrals — the license to expand any field in plane waves, cavity modes, or Fourier components. Why it matters downstream: this is the mathematical permission slip for Fourier optics, modal analysis, and the whole decompose-solve-reassemble method. *Verified in verify_superposition-principle-em-waves.py: sympy confirms $L[a_1\psi_1+a_2\psi_2]=a_1L[\psi_1]+a_2L[\psi_2]$ for $L=\nabla^2-c^{-2}\partial_t^2$ , and that two plane-wave solutions of $(1)$ sum to a solution.*

Go deeper · Derivation B · 1 minThe origin of the interference term

Derivation BThe origin of the interference term

Here we show precisely why intensities do not add: because the observable is quadratic in the field, superposing two fields and squaring leaves a cross term that survives time-averaging. The strategy is to add two co-frequency waves, square the sum, and average over an optical cycle using $\langle\cos^2\rangle=\tfrac12$ and the product-to-sum identity — keeping careful track of the one term that does not vanish. The full two-wave law is then quoted and handed to the next article; here the point is the existence and form of the cross term.

Step 1. Add two co-polarized waves of the same frequency at a point, $\mathbf E=\mathbf E_1+\mathbf E_2$ with $\mathbf E_i=\mathbf E_{0i}\cos(\phi_i-\omega t)$ . Square: $|\mathbf E|^2=|\mathbf E_1|^2+|\mathbf E_2|^2+2\,\mathbf E_1\!\cdot\!\mathbf E_2$ . Why squaring is forced: the detected intensity is $\propto\langle|\mathbf E|^2\rangle$ , not $\langle|\mathbf E_1|^2\rangle+\langle|\mathbf E_2|^2\rangle$ ; the cross term is unavoidable.

Step 2. Time-average the self terms: $\langle\cos^2(\phi_i-\omega t)\rangle=\tfrac12$ , so each contributes $\tfrac12E_{0i}^2\propto I_i$ . Why one-half: the square of a cosine oscillates about $\tfrac12$ , and the fast $2\omega$ part averages away over many cycles.

Step 3. Time-average the cross term with the product-to-sum identity: $\langle\cos(\phi_1-\omega t)\cos(\phi_2-\omega t)\rangle=\tfrac12\cos(\phi_1-\phi_2)$ . It does not vanish — it leaves the static phase difference $\delta=\phi_1-\phi_2$ . Why it survives: the $2\omega t$ part of the product averages to zero, but the difference-frequency part is time-independent and remains.

Step 4. Assemble, with $I_i=\tfrac12c\varepsilon_0E_{0i}^2$ and the polarization factor $\hat{\mathbf e}_1\!\cdot\!\hat{\mathbf e}_2$ : $I=I_1+I_2+2\sqrt{I_1I_2}\,(\hat{\mathbf e}_1\!\cdot\!\hat{\mathbf e}_2)\cos\delta$ . Parallel polarizations give full contrast; orthogonal ones ( $\hat{\mathbf e}_1\!\cdot\!\hat{\mathbf e}_2=0$ ) kill the term. Why hand it on: the extrema, visibility, and energy accounting of this law are the subject of the next article; the seed is proved here. *Verified in verify_superposition-principle-em-waves.py: sympy expands $|\mathbf E_1+\mathbf E_2|^2$ ; the time averages $\langle\cos^2\rangle=\tfrac12$ and $\langle\cos A\cos B\rangle=\tfrac12\cos(A-B)$ are integrated exactly; equal in-phase beams give $I=4I_0$ with cycle-average $2I_0$ .*

Limiting cases

Case	Result	Meaning
fully constructive, $\delta=0$	$I=(\sqrt{I_1}+\sqrt{I_2})^2$	in-phase, parallel polarization; for $I_1=I_2=I_0$ this is $4I_0$ , not $2I_0$
fully destructive, $\delta=\pi$	$I=(\sqrt{I_1}-\sqrt{I_2})^2$	anti-phase; equal beams give $I=0$ , complete cancellation
incoherent or orthogonal polarization	$\langle\cos\delta\rangle=0\Rightarrow I=I_1+I_2$	random phase or $\mathbf E_1\perp\mathbf E_2$ : intensities add — the everyday limit only
$N$ equal in-phase emitters	$I=N^2I_0$ on axis	amplitudes add, then square — the basis of antenna and phased arrays
nonlinear medium, high intensity	$\mathbf P=\varepsilon_0(\chi^{(1)}\mathbf E+\chi^{(2)}\mathbf E^2+\dots)$	fields still add, but the medium mixes them: harmonics, new frequencies — linearity of the response fails

Common trap“Just add the intensities of the two beams”

This is the single most common error in interference, and it is wrong except in one special limit. Superposition applies to the fields $\mathbf E,\mathbf B$ , which are vectors that can be positive or negative and can cancel; the observable is $\langle|\mathbf E_1+\mathbf E_2|^2\rangle$ , which contains the cross term $2\langle\mathbf E_1\!\cdot\!\mathbf E_2\rangle$ and ranges from $(\sqrt{I_1}-\sqrt{I_2})^2$ up to $(\sqrt{I_1}+\sqrt{I_2})^2$ . Only when that cross term averages to zero — mutually incoherent beams whose relative phase drifts randomly, or orthogonally polarized beams — do intensities genuinely add. Adding intensities as a reflex predicts a uniform screen where nature draws fringes; add the fields first, then square.

Worked examples

1 · Two equal coherent beams: 4I₀, not 2I₀ (confidence builder)

Two coherent beams, each of intensity $I_0$ , overlap in phase with parallel polarization. Find the resultant intensity, then the intensity when they are exactly out of phase, and check that energy is conserved.

Step 1. Amplitudes add, not intensities. With $E_0\propto\sqrt{I_0}$ for each, the in-phase resultant amplitude is $E_0+E_0=2E_0$ , so the intensity is $I=(2E_0)^2\propto(2\sqrt{I_0})^2=\boxed{4I_0}$ — four times a single beam, not two.

Step 2. Out of phase ( $\delta=\pi$ ) the amplitudes subtract: $E_0-E_0=0$ , so $I=\boxed{0}$ . The two equal fields cancel completely and the point goes dark.

Step 3. Energy is conserved across the pattern. Averaging $I=2I_0(1+\cos\delta)$ over all phases, $\langle I\rangle=\tfrac1{2\pi}\int_0^{2\pi}2I_0(1+\cos\delta)\,\mathrm d\delta=\boxed{2I_0}$ — exactly the sum of the two inputs.

Step 4. So the interference term neither creates nor destroys energy: it moves $2I_0$ of average brightness into a pattern that peaks at $4I_0$ and dips to $0$ . Adding intensities would have predicted a flat $2I_0$ and missed the fringes entirely.

2 · Beat note from two lasers (exam level)

Two single-frequency beams at optical frequencies $\nu_1=4.741\times10^{14}\ \mathrm{Hz}$ (a $633\ \mathrm{nm}$ line) and $\nu_2$ higher by $\Delta\nu=1.0\ \mathrm{GHz}$ are superposed on a fast photodiode. Find the beat period seen in the detected intensity.

Step 1. Superpose the two fields: $E=E_0\cos(2\pi\nu_1 t)+E_0\cos(2\pi\nu_2 t)$ . By the sum-to-product identity this equals $2E_0\cos\!\big(2\pi\bar\nu\,t\big)\cos\!\big(\pi\Delta\nu\,t\big)$ , a fast carrier at the mean frequency $\bar\nu$ modulated by a slow envelope at $\tfrac12\Delta\nu$ .

Step 2. The detector squares and averages, so it follows the envelope's intensity, which repeats at the full difference frequency $\Delta\nu$ (the envelope's $\cos^2$ has period $1/\Delta\nu$ ). The beat period is $T_\mathrm{beat}=1/\Delta\nu=1/(1.0\times10^9\ \mathrm{Hz})=\boxed{1.0\ \mathrm{ns}}$ .

Step 3. The optical cycles themselves ( $\sim2.1\ \mathrm{fs}$ ) are far too fast to detect; only the $1\ \mathrm{ns}$ modulation appears, because the photodiode integrates over many optical periods but resolves the nanosecond envelope. A $1\ \mathrm{GHz}$ frequency difference reads out as a $1\ \mathrm{ns}$ beat — the principle behind heterodyne detection and optical frequency metrology.

Step 4. Superposition again does the work: the two fields add, and the square of the sum carries the difference-frequency cross term that the detector sees, exactly as in the fringe pattern but now in time rather than space.

Physics of the portrait

The organizing idea here is not a person but a principle, and it is worth pausing on what makes linearity so powerful — and where it stops.

Origins · 1 minsuperposition principle and how this was discovered

superposition principle

linear-response foundation · classical wave optics

The superposition principle is the load-bearing assumption of all classical wave optics: because Maxwell's equations are linear in a medium of field-independent response, solutions add, and the field of any configuration is the vector sum of the fields of its parts. Its power is that it converts a hard problem into a sum of easy ones — resolve a wavefront into secondary wavelets (diffraction), an aperture field into plane waves (Fourier optics), a cavity field into modes (laser theory), a pulse into frequencies (spectroscopy), solve each independently, and add. Its limit is equally instructive. At the enormous fields of a focused pulsed laser the medium's polarization acquires terms $\propto E^2,E^3,\dots$ ; the fields still superpose, but the response does not, and the medium mixes waves to make new colours — second-harmonic generation, sum- and difference-frequency mixing, the Kerr effect. The boundary between linear and nonlinear optics is precisely the boundary of this principle, which is why knowing where superposition holds is knowing when every interference formula in this course applies.

History

1678

— Huygens builds a wavefront from the superposition of secondary spherical wavelets

1747

— d'Alembert solves the wave equation, exhibiting solutions as sums of left- and right-moving waves

1801

— Young explains his double-slit fringes as the superposition (interference) of two waves from one source

1822

— Fourier's Théorie analytique de la chaleur establishes decomposition of any function into superposed harmonics

1865

— Maxwell shows light is an electromagnetic wave governed by linear field equations

1961

— Franken generates the second harmonic of a laser, opening nonlinear optics where superposition of the response fails

Going further

Texts. Hecht, Optics, 5th ed., §7.1 (Addition of Waves) · Jackson, Classical Electrodynamics, 3rd ed., §6.4, §7.1 (wave equation, plane waves) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., §2.5 (Interference of Light) · Born & Wolf, Principles of Optics, §1.4–1.6 · Boyd, Nonlinear Optics (where linearity of the response ends).

Provenance. The linear wave equation $(1)$ and the monochromatic plane-wave solutions $(2)$ are recalled by name from aq-em-electromagnetic-wave-equation and aq-em-monochromatic-plane-waves-vacuum (grep-verified: the wave equation and its d'Alembert/plane-wave solutions are derived and owned there); that intensity is the time-averaged Poynting flux $(5)$ is recalled from aq-em-intensity-time-averaging and aq-em-energy-density-poynting-vector. Optics owns the superposition principle from linearity and the origin of the interference cross term. verify_superposition-principle-em-waves.py reproduces: $c=1/\sqrt{\mu_0\varepsilon_0}=2.998\times10^8\ \mathrm{m/s}$ ; operator linearity $L[a_1\psi_1+a_2\psi_2]=a_1L[\psi_1]+a_2L[\psi_2]$ ; $\langle\cos^2\rangle=\tfrac12$ and $\langle\cos A\cos B\rangle=\tfrac12\cos(A-B)$ ; $|\mathbf E_1+\mathbf E_2|^2$ expansion; $I=4I_0$ in phase with cycle-average $2I_0$ ; the beat period $T=1/\Delta\nu=1.0\ \mathrm{ns}$ for $\Delta\nu=1.0\ \mathrm{GHz}$ . The two-wave interference law's full development is deferred to the next article.

Builds on: Maxwell's Equations · Plane-Wave Solutions and Complex Representation · Vector Algebra and the Time-Average. Leads to: Two-Wave Interference and the Interference Term · Optical Path Difference and Phase Retardation · Young's Double-Slit Experiment · Conditions for Observable Interference.

If the fields just add, why can't I add the intensities — isn't intensity just field energy?

Intensity is field energy, and that is exactly the problem: energy is quadratic in the field. If two fields $E_1$ and $E_2$ overlap, the total field is $E_1+E_2$ , but the energy goes as $(E_1+E_2)^2=E_1^2+E_2^2+2E_1E_2$ . The first two terms are the individual intensities; the third, $2E_1E_2$ , is an extra piece that is positive where the fields agree and negative where they disagree. Adding intensities keeps only the first two terms and throws away the interference cross term — which is precisely the part that makes fringes. Add the fields, then square.

Do two crossing laser beams really not affect each other at all?

In vacuum, and in ordinary linear media, exactly not at all: they pass through one another and emerge with their directions, amplitudes, and phases intact, because the wave equation has no term coupling one field to another. In the overlap region the total field is their vector sum and you can see interference fringes there, but that is a pattern in the combined field, not damage to either beam — downstream each continues as if the other were never present. This only changes at extreme intensity, where the medium's nonlinear response lets the beams exchange energy or generate new frequencies; that is the domain of nonlinear optics, outside the linear regime of this article.

What is the difference between superposition and interference?

Superposition is the cause; interference is the visible effect. Superposition is the mathematical fact that fields add: $\mathbf E=\sum_i\mathbf E_i$ . Interference is what you observe when you then measure intensity, because $\langle|\sum_i\mathbf E_i|^2\rangle$ contains cross terms that brighten some places and darken others. Every wave superposes; interference becomes visible only when the cross term survives averaging — that needs mutual coherence and a shared polarization component. So superposition always holds, while interference fringes appear only under the conditions that keep the cross term alive.

Why does the interference term vanish for orthogonal polarizations?

Because the cross term is a dot product of the two field vectors, $2\mathbf E_1\!\cdot\!\mathbf E_2$ , and the dot product of perpendicular vectors is zero. Concretely, if $\hat{\mathbf e}_1\perp\hat{\mathbf e}_2$ then $\hat{\mathbf e}_1\!\cdot\!\hat{\mathbf e}_2=0$ and the term drops out no matter what the phase difference is, leaving $I=I_1+I_2$ . This is one of the Fresnel–Arago laws. The fields still superpose — the total field is a well-defined vector at every instant — but its squared magnitude carries no interference modulation because the two contributions are geometrically independent. Only the component of one field along the other can interfere.

How can two equal beams give 4I₀ without violating energy conservation?

Because the $4I_0$ is only local, at the bright fringes, and it is paid for by the dark fringes where the intensity is $0$ . Averaged over a full period of the phase difference, $\langle 2I_0(1+\cos\delta)\rangle=2I_0$ — exactly the sum of the two input beams. The interference term $2\sqrt{I_1I_2}\cos\delta$ integrates to zero over a fringe pattern, so it neither adds nor removes energy; it only redistributes the same average $2I_0$ into a peaked pattern reaching $4I_0$ . Energy conservation is what forces the dark fringes to accompany the bright ones.

Superposition is exact in vacuum — when does it stop being a good approximation in matter?

When the medium's response stops being linear in the field, which happens at very high intensity. In ordinary light the induced polarization is $\mathbf P=\varepsilon_0\chi^{(1)}\mathbf E$ , proportional to the field, so the effective wave equation stays linear and superposition holds. At the fields of a focused pulsed laser the polarization gains terms $\propto E^2,E^3$ : the fields still add, but the medium's response to their sum is not the sum of its responses, so the waves mix and generate harmonics and sum/difference frequencies. The threshold is set by the material's nonlinear susceptibilities $\chi^{(2)},\chi^{(3)}$ ; below it — essentially all everyday and most laboratory optics — superposition is an excellent approximation.