Coupled-Mode Theory and Directional Couplers

Bring two single-mode waveguides side by side and something the wave-equation article already prepared quietly takes over: each guided mode does not stop at its core wall but leaks an evanescent tail into the cladding, and where those tails overlap, the two guides stop being independent. Coupled-mode theory treats that overlap as a weak perturbation, writes the composite field as a sum of the isolated-guide modes with slowly varying amplitudes $a_1(z)$ and $a_2(z)$ , and — after one projection step — collapses all of Maxwell's equations for the pair into just two coupled first-order ordinary differential equations. It is one of the highest-leverage approximations in optics: a full electromagnetic boundary-value problem replaced by a two-by-two system a student can solve by hand. And the answer is one you already know from atomic physics. Those two equations are, symbol for symbol, the driven two-level Schrödinger equation, with the coupling coefficient $\kappa$ playing the Rabi frequency, the propagation-constant mismatch $\delta$ playing the detuning, and the distance $z$ standing in for time $t$ . So the light does not split once and stay split — it flops. Power injected into one guide crosses completely into the other after a coupling length $L_c=\pi/2\kappa$ , then crosses back, exactly as an on-resonant drive inverts a two-level atom and back again. Detune the guides and the transfer becomes incomplete and faster, capped at $\kappa^2/(\kappa^2+\delta^2)$ , the optical twin of off-resonant Rabi flopping. Because $\kappa$ falls exponentially with the gap and grows with wavelength, this one mechanism underwrites 3 dB splitters, electro-optic switches, wavelength multiplexers, and every ring-to-bus coupling on a photonic chip.

Key takeaways

Two adjacent dielectric waveguides exchange power through the exponential overlap of their evanescent mode tails; coupled-mode theory reduces the full problem to two first-order equations for the slowly varying amplitudes $a_1(z),a_2(z)$ .
For identical guides the exchange is complete and periodic — power sloshes fully back and forth with coupling length $L_c=\pi/2\kappa$ , where $\kappa$ is the field-overlap coupling coefficient.
A phase mismatch $\delta=(\beta_1-\beta_2)/2$ caps the transferred fraction at $\kappa^2/(\kappa^2+\delta^2)$ and shrinks the beat period to $\pi/\sqrt{\kappa^2+\delta^2}$ — off-resonant flopping.
The composite structure has two real normal modes, an even supermode $\beta_+$ and an odd supermode $\beta_-$ split by $2\sqrt{\kappa^2+\delta^2}$ ; their beat is the power transfer.
Because $\kappa$ depends exponentially on the gap and strongly on wavelength, a fixed-length coupler is a switch (via electro-optic $\delta$ ) and a wavelength filter at once.
Key idea: evanescent overlap couples two waveguide modes into even and odd supermodes whose beat produces periodic power transfer — the whole device is the driven two-level (Rabi) problem in disguise, governed by coupling $\kappa$ and mismatch $\delta$ .

The core insight

Evanescent field overlap couples two waveguide modes into even and odd supermodes whose beat produces periodic power transfer; the entire device physics is the driven two-level (Rabi) problem in disguise, governed by the coupling $\kappa$ and the phase mismatch $\delta$ , with distance $z$ in the role of time.

Where this lives today

The directional coupler is the single most reused element in integrated photonics. On a silicon or silicon-nitride chip, essentially every functional block is built from couplers: a Mach–Zehnder modulator is two 3 dB couplers with a driven phase shifter between them; a microring add–drop filter is a bus-to-ring coupler whose $\kappa$ sets the resonance linewidth and the through-versus-drop extinction; a switch fabric is a mesh of programmable 2×2 couplers. In lithium-niobate and indium-phosphide, electro-optic directional-coupler switches route wavelength channels with nanosecond speed and low crosstalk, driven by exactly the $\sqrt{3}\pi/L$ mismatch derived above. The reach extends past classical photonics. Because the coupled-mode equations are the two-level problem, a phase-matched coupler performs a perfect, lossless, unitary two-mode rotation — a beamsplitter. That makes coupler meshes the natural hardware for programmable linear-optical processors: a triangular or rectangular array of Mach–Zehnder couplers implements an arbitrary unitary matrix on the input modes, and these networks are the elementary gates of photonic quantum computing and optical neural-network accelerators. An idea born to describe travelling-wave tubes now sits inside quantum photonic chips.

The physical picture

Picture two swimming-pool lanes separated by a low wall. Set a wave running in one lane and a little of it spills over the wall each cycle and pushes the water in the other lane. If the two lanes are identical — same length, same natural rhythm — the sloshing builds coherently and, given time, all the energy crosses to the second lane, then all of it comes back. If the lanes have different natural rhythms, the push and the response fall out of step, and the second lane never fills completely before the transfer reverses. That is the whole of coupled-mode theory, mechanical model and all: resonance gives complete exchange, detuning gives partial exchange. The optics is the same story told with fields. A guided mode is not confined to its core; outside the high-index core the field decays exponentially, $E\propto e^{-\gamma x}$ , and that evanescent tail reaches across the gap into the neighbouring core. The coupling coefficient $\kappa$ is literally the integral of one guide's tail over the other guide's core, so it inherits the exponential: halve the gap and $\kappa$ leaps, lengthen the wavelength and the fatter tail raises $\kappa$ too. This is why couplers are the most fabrication-sensitive elements on a chip — a 20 nm error in a 200 nm gap visibly shifts the splitting ratio.

The composite structure has, in exact truth, two genuine normal modes — supermodes. The even (symmetric) one peaks over both cores with the same sign; the odd (antisymmetric) one has opposite sign over the two cores, with a node in the gap. These are the real eigenmodes and they propagate rigidly, each with its own constant $\beta_+$ or $\beta_-$ . Launching power into a single guide means launching an equal mix of the two; as they run down the coupler they accumulate a growing phase difference $(\beta_+-\beta_-)z$ , and when it reaches $\pi$ the interference pattern has slid entirely to the far core — so the supermode beat length is exactly the coupling length. Coupled-mode theory and the supermode picture are one physics in two languages: the amplitude equations track the individual guides, the supermodes track the eigenbasis. This is the optical twin of Rabi flopping. In a driven two-level atom the ground and excited amplitudes obey the same pair of equations, with the Rabi frequency where $\kappa$ sits and the detuning where $\delta$ sits, so everything an AMO student knows — complete periodic inversion on resonance, incomplete and faster oscillation off it — carries over verbatim. The next figure is that flopping, drawn for a real coupler.

Figure

Power exchange along a phase-matched coupler with κ = 0.5 per mm: P₁ in blue and P₂ in amber slosh completely, with the first full crossover at the coupling length L_c = π divided by 2κ ≈ 3.14 mm and the full period at 2L_c. A detuning δ = κ, the red dashed curve, caps the transfer at 50% and beats faster

The core equations

Start from the amplitudes. Each guide, alone, would carry its mode with a fixed propagation constant; the neighbour's presence drives a slow leakage of amplitude from one to the other. Keeping only the leading (slowly varying) terms, the two amplitudes obey a linear first-order pair — the coupled-mode equations, the master relation of the whole subject.

\frac{\mathrm{d}a_1}{\mathrm{d}z}=-i\kappa_{12}\,a_2\,e^{-i(\beta_2-\beta_1)z},\qquad \frac{\mathrm{d}a_2}{\mathrm{d}z}=-i\kappa_{21}\,a_1\,e^{+i(\beta_2-\beta_1)z} \qquad (1)

where: $a_1,a_2$ — slowly varying complex amplitudes of the guided modes of guides 1 and 2; $\kappa_{12},\kappa_{21}$ — coupling coefficients, equal for lossless reciprocal guides so that $\kappa_{12}=\kappa_{21}^{*}\equiv\kappa$ ; $\beta_1,\beta_2$ — propagation constants of the isolated modes; $z$ — distance along the coupler. The exponentials carry the phase mismatch: when the guides are identical they vanish and the coupling drives amplitude across on every cycle in step. The coupling coefficient is not a free parameter — it is a definite overlap integral of the two mode fields weighted by the index perturbation each guide represents to the other. This is where the exponential sensitivity to geometry enters, because the mode fields it multiplies are the evanescent tails.

\kappa=\frac{\omega\varepsilon_0}{4}\iint \Delta\varepsilon(x,y)\,\mathbf{E}_1^{*}\cdot\mathbf{E}_2\,\mathrm{d}x\,\mathrm{d}y \qquad (2)

where: $\Delta\varepsilon$ — the permittivity perturbation, i.e. the extra dielectric of guide 2 seen by mode 1 (and vice versa); $\mathbf{E}_1,\mathbf{E}_2$ — normalized transverse mode fields; $\omega,\varepsilon_0$ — angular frequency and vacuum permittivity. Because $\mathbf{E}_1$ evaluated at guide 2 is an evanescent tail $\propto e^{-\gamma d}$ , the integral gives $\kappa\propto e^{-\gamma d}$ in the gap $d$ — the origin of coupler sensitivity. Non-identical guides carry different propagation constants, and it is their difference, not their values, that governs the exchange. Two quantities package this: the half-mismatch $\delta$ and the generalized coupling constant $\beta$ that will set every beat length below.

\delta\equiv\frac{\beta_1-\beta_2}{2},\qquad \beta\equiv\sqrt{\kappa^2+\delta^2} \qquad (3)

where: $\delta$ — half the propagation-constant mismatch, the optical detuning; $\beta$ — generalized coupling constant. Note $2\beta=\beta_+-\beta_-$ is exactly the supermode splitting, so $\beta$ is simultaneously the beat rate and half the eigenvalue gap. When $\delta=0$ the guides are phase-matched and $\beta=\kappa$ . Solving the pair for power launched entirely into guide 1 gives the central experimental prediction — a sinusoidal transfer whose amplitude is throttled by the mismatch. This single formula contains the resonant and off-resonant cases at once.

P_2(z)=P_0\,\frac{\kappa^2}{\kappa^2+\delta^2}\,\sin^2\!\big(\sqrt{\kappa^2+\delta^2}\;z\big) \qquad (4)

where: $P_0$ — power launched into guide 1 at $z=0$ ; $P_2(z)$ — power in guide 2; conservation gives $P_1(z)=P_0-P_2(z)$ . The prefactor $\kappa^2/(\kappa^2+\delta^2)$ is the maximum reachable fraction — unity only when $\delta=0$ — and the argument shows detuning speeds the beat. The design length follows from asking where the first full (or, at $\delta=0$ , complete) transfer occurs. This coupling length is the number a fabricator sets the interaction region to.

L_c=\frac{\pi}{2\kappa}\quad(\delta=0),\qquad L_c=\frac{\pi}{2\sqrt{\kappa^2+\delta^2}}\quad(\text{general}) \qquad (5)

where: $L_c$ — the distance for maximum transfer to guide 2, complete when $\delta=0$ . Halving it, so that $\kappa L=\pi/4$ , gives the 50/50 (3 dB) split. The general form shows that detuning shortens the coupler as well as weakening it. For circuit design one wants the coupler as a single block relating output amplitudes to input amplitudes. Integrating (1) gives a unitary two-by-two transfer matrix — the object that snaps into any larger photonic network.

\begin{pmatrix}a_1(L)\\ a_2(L)\end{pmatrix}=\begin{pmatrix}\cos\beta L - i\frac{\delta}{\beta}\sin\beta L & -i\frac{\kappa}{\beta}\sin\beta L\\[2pt] -i\frac{\kappa}{\beta}\sin\beta L & \cos\beta L + i\frac{\delta}{\beta}\sin\beta L\end{pmatrix}\begin{pmatrix}a_1(0)\\ a_2(0)\end{pmatrix} \qquad (6)

where: $L$ — physical length of the interaction region; the off-diagonal magnitude squared, $(\kappa/\beta)^2\sin^2\beta L$ , is the cross-coupled power fraction, reproducing (4). The factor $-i$ on the cross terms is the $\pi/2$ phase every coupler stamps on transferred light — the fact that makes Mach–Zehnder interferometers work. Finally, a fully crossed coupler can be switched dark by an applied mismatch. Setting the off-diagonal term to zero at fixed $L=L_c$ fixes the required mismatch and hence the switch drive.

\Delta\beta_{\text{switch}}=\frac{\sqrt{3}\,\pi}{L}\;\Rightarrow\;P_2=0 \qquad (7)

where: $\Delta\beta=2\delta$ — the mismatch induced electro-optically to return all power to the input guide (the "bar" state); $L=L_c$ — the length that gives full cross at $\delta=0$ . The $\sqrt{3}$ is exact: it is where the shortened beat of the detuned coupler completes one half-period earlier and lands the light back home.

Derivation AFrom the wave equation to the coupled-mode equations

The goal is to show how the full electromagnetic problem of two neighbouring guides — in principle a boundary-value problem for the vector wave equation over the whole cross-section — collapses to the compact pair in equation (1). The move that makes it possible is to refuse to solve the composite structure exactly and instead expand its field in the modes we already know, those of each guide in isolation, letting only their amplitudes drift with $z$ . Projection and a slowly-varying assumption do the rest, and the coupling coefficient falls out as an overlap integral.

Step 1. Write the composite transverse field as a superposition of the two isolated-guide modes carrying their own rapid phases, $\mathbf{E}(x,y,z)=a_1(z)\mathbf{E}_1(x,y)e^{-i\beta_1 z}+a_2(z)\mathbf{E}_2(x,y)e^{-i\beta_2 z}$ , with the amplitudes $a_i(z)$ allowed to vary slowly. Why: the isolated-guide modes are an excellent basis when the guides barely perturb each other, so all the physics of coupling is pushed into the slow drift of $a_1,a_2$ rather than into the mode shapes.

Step 2. Insert this into the vector wave equation for the composite dielectric. That dielectric differs from guide $i$ alone by a perturbation $\Delta\varepsilon_i$ — the presence of the other core. Because each $\mathbf{E}_i$ already solves its own unperturbed equation, those terms cancel and only the perturbation terms and the $z$ -derivatives of the amplitudes survive. Why: subtracting the two problems each guide solves exactly leaves precisely the small residual that couples them, which is what we want to keep.

Step 3. Drop $\mathrm{d}^2a_i/\mathrm{d}z^2$ against $\beta_i\,\mathrm{d}a_i/\mathrm{d}z$ — the slowly-varying-envelope approximation. Why: weak, gradual coupling means the amplitude changes negligibly over one wavelength, so its second derivative is second-order small and discarding it linearizes the system to first order in $\mathrm{d}/\mathrm{d}z$ .

Step 4. Project: multiply through by $\mathbf{E}_j^{*}$ and integrate over the transverse plane. Orthonormality of the modes isolates a single $\mathrm{d}a_j/\mathrm{d}z$ on the left, and the surviving cross term on the right is the overlap integral that defines $\kappa$ in equation (2). Why: projecting onto each mode is the standard trick for turning a field equation into amplitude equations, and it hands us the coupling coefficient as a concrete integral rather than a fitted constant.

Step 5. The result is $\mathrm{d}a_1/\mathrm{d}z=-i\kappa\,a_2\,e^{i(\beta_1-\beta_2)z}$ and its partner; writing $\beta_1-\beta_2=2\delta$ gives the boxed pair $\mathrm{d}a_1/\mathrm{d}z=-i\kappa\,a_2\,e^{2i\delta z},\ \mathrm{d}a_2/\mathrm{d}z=-i\kappa\,a_1\,e^{-2i\delta z}$ . Why: the rapidly rotating self-coupling terms average to zero over a wavelength and only the resonant cross-coupling remains — the same rotating-wave logic that reduces a driven atom to its two-level equations.

Go deeper · Derivation B · 1 minSolving the coupled equations as Rabi flopping

Derivation BSolving the coupled equations as Rabi flopping

The aim here is to integrate the coupled pair with all the power launched into one guide and arrive at the sinusoidal transfer law (4), and in doing so to make the correspondence with the driven two-level atom explicit rather than merely asserted. The trick is to absorb the rotating exponentials into the amplitudes so the system becomes constant-coefficient, at which point it is a two-by-two eigenvalue problem identical in form to the two-level Schrödinger equation, and its solution is Rabi flopping with $z$ in the role of time.

Step 1. Substitute $b_1=a_1 e^{-i\delta z}$ and $b_2=a_2 e^{i\delta z}$ to remove the explicit $z$ -dependence from the coefficients. Why: the phase factors are a rotating frame; moving into it turns a time-dependent-looking problem into a static matrix problem that can be diagonalized once and for all.

Step 2. The equations become $\mathrm{d}b_1/\mathrm{d}z=-i\delta b_1-i\kappa b_2$ and $\mathrm{d}b_2/\mathrm{d}z=+i\delta b_2-i\kappa b_1$ — a linear system whose matrix has eigenvalues $\pm\sqrt{\kappa^2+\delta^2}=\pm\beta$ . Why: this is term-for-term the two-level Hamiltonian with $\delta$ on the diagonal (detuning) and $\kappa$ off it (Rabi coupling), so the generalized Rabi frequency $\beta$ appears as the eigenvalue split.

Step 3. Apply the boundary condition of all power in guide 1: $b_1(0)=1,\ b_2(0)=0$ . Why: this is the physical launch — light injected into one input port — and it selects the particular combination of the two eigen-solutions.

Step 4. Propagating gives $b_2(z)=-i(\kappa/\beta)\sin(\beta z)$ , so $P_2(z)=|b_2|^2P_0=P_0(\kappa^2/\beta^2)\sin^2(\beta z)$ , the boxed transfer law. Why: the modulus squared converts amplitude to power, and the prefactor $\kappa^2/\beta^2=\kappa^2/(\kappa^2+\delta^2)$ is exactly the off-resonant ceiling — the atom analogue of never fully inverting a detuned drive.

Step 5. Set $\delta=0$ : then $\beta=\kappa$ and $P_2=P_0\sin^2(\kappa z)$ , complete transfer with first maximum at $z=\pi/2\kappa=L_c$ . Why: on resonance the ceiling rises to one and the flop is total and periodic, the direct optical image of a $\pi$ -pulse fully inverting a two-level atom.

Go deeper · Derivation C · 1 minThe supermode basis and the switching condition

Derivation CThe supermode basis and the switching condition

This derivation earns the even/odd picture and the exact $\sqrt{3}$ of the switch. The goal is to show that the coupled system is diagonalized by two normal modes with propagation constants $\beta_\pm=\bar\beta\pm\beta$ , that ordinary power transfer is simply their beat, and that a coupler built one coupling-length long can be driven completely dark by a specific applied mismatch — the operating principle of the electro-optic switch. We diagonalize, read off the beat, then impose the dark-port condition.

Step 1. Diagonalize the constant-coefficient matrix of the previous derivation. Its eigenvectors are the symmetric and antisymmetric combinations of the guide amplitudes — the even and odd supermodes — with propagation constants $\beta_\pm=\bar\beta\pm\sqrt{\kappa^2+\delta^2}$ , where $\bar\beta=\tfrac12(\beta_1+\beta_2)$ . Why: the eigenvectors of the coupling matrix are by definition the modes that propagate without changing shape, so they are the true normal modes of the composite waveguide.

Step 2. A single-guide launch is an equal superposition of even and odd; after distance $z$ they differ in phase by $(\beta_+-\beta_-)z=2\beta z$ , and the cross power is $\sin^2(\beta z)$ — identical to (4). Why: this reproduces the transfer law from a completely different starting point, confirming that "power sloshing between guides" and "two supermodes beating" are one phenomenon in two languages.

Step 3. Build a cross coupler: choose $L=L_c=\pi/2\kappa$ at $\delta=0$ , so at zero drive $\beta L=\pi/2$ and all power crosses. Now apply a mismatch $\delta$ electro-optically and demand $P_2(L)=0$ , i.e. $\sin(\sqrt{\kappa^2+\delta^2}\,L)=0$ , so $\sqrt{\kappa^2+\delta^2}\,L=\pi$ . Why: driving the cross power to zero returns all light to the input guide — the "bar" state — and the first zero of the sine sets the smallest switching drive.

Step 4. Substitute $L=\pi/2\kappa$ : then $\sqrt{\kappa^2+\delta^2}=2\kappa$ , giving $\delta^2=3\kappa^2$ , so $\delta=\sqrt{3}\,\kappa$ and $\Delta\beta=2\delta=2\sqrt{3}\,\kappa=\sqrt{3}\,\pi/L$ . Why: this is the exact bar-state condition of equation (7); the $\sqrt{3}$ is not empirical but the geometry of shortening a half-period beat until it closes one period over the fixed length $L$ .

Limiting cases

Case	Result	Meaning
Phase-matched ( $\delta=0$ )	$P_2(z)=P_0\sin^2(\kappa z)$	Complete periodic transfer; full crossover at $L_c=\pi/2\kappa$ — on-resonance Rabi flopping.
Detuned ( $\delta\ne0$ )	$P_2(z)=P_0\dfrac{\kappa^2}{\kappa^2+\delta^2}\sin^2\!\big(\sqrt{\kappa^2+\delta^2}\,z\big)$	Incomplete transfer, reduced ceiling and faster beat — off-resonant flopping.
Strong mismatch ( $\lvert\delta\rvert\gg\kappa$ )	$P_2^{\max}\approx\kappa^2/\delta^2\to 0$	Guides decouple; the switch "bar" regime — a voltage holds light in one guide.
3 dB coupler	$\kappa L=\pi/4,\quad P_1=P_2=\tfrac12 P_0$	Half a coupling length gives the 50/50 split, with a $\pi/2$ phase — the Mach–Zehnder building block.
Full-cross ( $L=L_c,\ \delta=0$ )	$P_2=P_0,\ P_1=0$	All power crosses — the "cross" state of a directional-coupler switch.

Common trapCoupled-mode theory is perturbative — its own modes must stay a good basis

Coupled-mode theory assumes the overlap is weak, so that each guide's isolated mode remains a nearly-correct description and the coupling is a small correction. Push the guides too close, or let them touch, and that assumption fails: the true even and odd supermodes then differ substantially from the naive sum of isolated modes, the simple overlap integral for $\kappa$ becomes inaccurate, and you must diagonalize the full composite structure numerically instead. A second, related trap is to treat $\kappa$ and $\delta$ as fixed device constants. Both depend on wavelength and on the gap — $\kappa$ exponentially so — which is exactly why a coupler that splits 50/50 at one wavelength does something entirely different at another, and why $L_c$ is never wavelength-independent.

Worked examples

1 · A silicon-photonics directional coupler from its supermode split (confidence builder)

Two identical silicon strip waveguides run parallel over a coupling region on a chip. A mode solver reports that at $\lambda_0=1550\ \mathrm{nm}$ the composite structure has an even supermode of effective index $n_+=2.4510$ and an odd supermode $n_-=2.4490$ , a split of $\Delta n_{\text{eff}}=2\times10^{-3}$ . How long must the coupling region be for a full crossover, and how long for a 3 dB (50/50) splitter? What is $\kappa$ ?

Step 1. Convert the index split to a propagation-constant split. The supermode constants are $\beta_\pm=(2\pi/\lambda_0)n_\pm$ , so $\beta_+-\beta_-=(2\pi/\lambda_0)\Delta n_{\text{eff}}$ . Numerically $\beta_+-\beta_-=2\pi(2\times10^{-3})/(1550\times10^{-9}\ \mathrm{m})=8.11\times10^{3}\ \mathrm{rad/m}=8.11\ \mathrm{rad/mm}$ .

Step 2. At phase match $\beta_+-\beta_-=2\kappa$ , so $\kappa=\tfrac12(\beta_+-\beta_-)=4.05\ \mathrm{rad/mm}=4.05\ \mathrm{mm^{-1}}$ . The full crossover happens when the supermodes have slipped by half a beat, $\beta L_c=\pi/2$ , i.e. $L_c=\pi/2\kappa=\lambda_0/(2\,\Delta n_{\text{eff}})$ . This tidy identity is worth remembering: $L_c=\lambda_0/(2\Delta n_{\text{eff}})=1550\ \mathrm{nm}/(4\times10^{-3})=387.5\ \mathrm{\mu m}$ .

Step 3. A 3 dB splitter needs half of that, $\kappa L=\pi/4$ : $L_{3\,\mathrm{dB}}=L_c/2=193.8\ \mathrm{\mu m}$ . So a few-hundred-micron interaction region — routine on a silicon chip — turns one waveguide's light fully into its neighbour, and half that length splits it evenly. Note how directly the device length is fixed by a part-in-a-thousand index split: the whole coupler is an interferometer between two nearly degenerate supermodes.

2 · Detuning a coupler: the switch drive and the transfer ceiling

A pair of identical guides is fabricated with $\kappa=0.5\ \mathrm{mm^{-1}}$ and an interaction length set to one coupling length so that, unbiased, all power crosses. (a) What is $L$ ? (b) A control voltage introduces a mismatch $\delta=\kappa$ ; what maximum fraction can now cross? (c) What mismatch $\Delta\beta=2\delta$ switches the coupler fully dark (bar state)?

Step 1. The cross length is $L=L_c=\pi/2\kappa=\pi/(2\times0.5\ \mathrm{mm^{-1}})=\pi\ \mathrm{mm}\approx\mathbf{3.14\ mm}$ . At zero bias this coupler is in the cross state, $P_2=P_0$ .

Step 2. With $\delta=\kappa$ the transfer ceiling is $P_2^{\max}/P_0=\kappa^2/(\kappa^2+\delta^2)=\kappa^2/(2\kappa^2)=\tfrac12$ . Even at its best point the detuned coupler now sends at most 50% across — complete transfer is impossible once $\delta\ne0$ , precisely the off-resonant Rabi ceiling.

Step 3. For a full dark (bar) state at fixed $L=L_c$ we need $\sqrt{\kappa^2+\delta^2}\,L=\pi$ , i.e. $\delta=\sqrt{3}\,\kappa=\sqrt{3}(0.5)=0.866\ \mathrm{mm^{-1}}$ , so $\Delta\beta=2\delta=\mathbf{1.73\ mm^{-1}}$ , equivalently $\Delta\beta=\sqrt{3}\,\pi/L$ . A modest electro-optic index change delivering $\Delta\beta\approx1.73\ \mathrm{mm^{-1}}$ flips this coupler from cross to bar — the elementary 2×2 optical switch.

Physics of the portrait

Coupled-mode theory did not begin in optics. It was forged for microwave tubes and millimetre waveguides in the 1950s, and only carried into integrated optics two decades later, by the man who made it the working language of the field.

Origins · 1 minAmnon Yariv and how this was discovered

Amnon Yariv

b. 1930 · coupled-mode theory of guided-wave optics

Amnon Yariv, born in Tel Aviv in 1930 and a professor at Caltech since 1964, is one of the architects of modern photonics, and it was his 1973 paper "Coupled-Mode Theory for Guided-Wave Optics" that recast the microwave-era formalism of Pierce and Miller into the compact amplitude equations optical engineers still use. His insight was that the sprawling boundary-value problem of two interacting dielectric guides reduces, under a slowly-varying-envelope approximation, to exactly the two-by-two system of a driven two-level atom — the mathematics he knew intimately from quantum electronics. Yariv's textbooks, "Quantum Electronics" and (with Pochi Yeh) "Photonics: Optical Electronics in Modern Communications", trained a generation to see directional couplers, distributed-feedback lasers, phase-conjugate mirrors, and electro-optic modulators as instances of coupled modes. He extended the same overlap-integral logic to gratings, to semiconductor laser arrays, and to bus-to-resonator coupling — the ancestor of today's microring filters. That a photonic switch, a fibre grating, and a Rabi-flopping atom all obey one equation is, more than anyone's, Yariv's legacy: mode coupling as a single unifying idea running from vacuum tubes to silicon chips.

History

1954

— J. R. Pierce formulates coupled-mode theory for microwave and electron-beam systems

1969

— Stewart Miller proposes integrated optics, framing waveguide couplers on a single substrate

1973

— Amnon Yariv casts coupled-mode theory for guided-wave optics, giving the directional-coupler equations

1976

— Kogelnik and Schmidt demonstrate the reversed-Δβ electro-optic directional-coupler switch

1984

— Hermann Haus systematizes coupled-mode theory in Waves and Fields in Optoelectronics

Going further

Texts. Saleh and Teich, Fundamentals of Photonics, 3rd ed., Ch. 8 (Waveguides) and Ch. 10 — coupled-mode theory of directional couplers · Yariv and Yeh, Photonics: Optical Electronics in Modern Communications — the coupled-mode formalism and its device applications · Okamoto, Fundamentals of Optical Waveguides, Ch. 4 — supermodes and the transfer matrix · Haus, Waves and Fields in Optoelectronics — the systematic coupled-mode treatment · Hecht, Optics — evanescent waves and frustrated total internal reflection · Foot, Atomic Physics, Ch. 7 — the driven two-level (Rabi) problem, the exact mathematical analogue. Key relations and numbers to keep in mind: the supermode splitting is $\beta_+-\beta_-=2\sqrt{\kappa^2+\delta^2}$ , whose half-beat gives $L_c$ ; the bar-state switching condition is $\Delta\beta\,L=\sqrt{3}\pi$ at $L=L_c$ ; and the anchor numerics follow directly from $L_c=\pi/2\kappa$ and $L_c=\lambda_0/2\Delta n_{\text{eff}}$ : $\kappa=0.5\ \mathrm{mm^{-1}}\Rightarrow L_c=3.14\ \mathrm{mm}$ ; $\Delta n_{\text{eff}}=2\times10^{-3}$ at 1550 nm $\Rightarrow L_c=387.5\ \mathrm{\mu m}$ .

Prerequisites. Waveguide Modes and the Wave Equation · Total Internal Reflection and Optical Guiding · Single-Mode and Multimode Operation. Leads to. Integrated Photonic Waveguides and Circuits · Microring Resonators and Optical Cavities on Chip · Dispersion Management and Compensation.

Ask the tutor: Ask me to walk through why a detuned directional coupler is off-resonant Rabi flopping, and I will match every optical symbol to its two-level-atom twin step by step.

Is coupled-mode theory exact, or an approximation?

It is a perturbation theory, and a very good one when the guides are weakly coupled. Its central assumption is that each guide's isolated mode remains an accurate description of the field in that guide, so that the neighbour is a small correction handled by an overlap integral. For the well-separated guides of a typical coupler this is excellent — errors are a percent or two. It breaks down when the guides are so close their fields strongly reshape each other; then the isolated-mode basis is poor, the simple $\kappa$ integral is inaccurate, and the honest route is to compute the true even and odd supermodes of the whole structure numerically. Usefully, the supermode picture is exact and always available as a check.

What is the difference between the coupling κ and the mismatch δ?

They are the two independent knobs of the device and they do opposite jobs. The coupling $\kappa$ is how strongly the guides talk — an overlap integral fixed by the gap and the wavelength — and it sets how fast power can cross. The mismatch $\delta=(\beta_1-\beta_2)/2$ is how different the two guides are, and it throttles how much power can cross: any nonzero $\delta$ caps the transfer below 100%. In the atomic analogy $\kappa$ is the Rabi frequency (drive strength) and $\delta$ is the detuning (how far off resonance). Complete transfer needs strong coupling and zero mismatch; a switch works by holding $\kappa$ fixed and pushing $\delta$ from zero up to $\sqrt{3}\kappa$ .

Why does the transferred light pick up a 90° phase shift?

It comes straight from the $-i$ on the off-diagonal terms of the transfer matrix, equation (6). Physically, the coupling equation reads $\mathrm{d}a_2/\mathrm{d}z=-i\kappa a_1$ : the rate of growth of the second amplitude is $90°$ out of phase with the first, so light that crosses over lags the light that stays by a quarter cycle. This is not a nuisance — it is essential. In a Mach–Zehnder interferometer built from two 3 dB couplers, the $\pi/2$ phase from each coupler is exactly what makes the two output ports complementary, so that a controlled phase in the arms steers all the light to one port or the other. Lose track of the $-i$ and the interferometer's port assignments come out backwards.

How can a "directional" coupler be directional if power sloshes back and forth?

"Directional" refers to co-propagation, not to the sloshing. In an ordinary directional coupler both guides carry light in the same direction ( $+z$ ), and the coupler transfers power between these co-directional modes; light injected into port 1 emerges at the forward ports, never backward — reflection is negligible. The back-and-forth in $z$ is the transfer oscillating with distance, and you cut the device at the length that gives the split you want. Contradirectional couplers are different: a grating couples a forward mode to a backward one, giving a reflected, band-limited response rather than a periodic transfer. The plain evanescent coupler here is the co-directional kind.

Why is a fixed-length coupler also a wavelength filter?

Because $\kappa$ depends on wavelength — strongly. Longer wavelengths have fatter evanescent tails, so they overlap more and couple more strongly, raising $\kappa(\lambda)$ and shortening $L_c(\lambda)$ . A coupler of fixed physical length $L$ therefore satisfies $\kappa(\lambda)L=\pi/2$ (full cross) at one wavelength and something quite different at another. Choose $L$ so that one channel sees a full crossover while another sees a full period ( $\kappa L=\pi$ , back to the input), and the device separates the two wavelengths into different output ports — a directional-coupler multiplexer. Grating-assisted and contradirectional couplers sharpen this crude wavelength dependence into a narrow, well-defined passband for add–drop filtering.

How does the electro-optic switch actually turn the light off?

Start with a coupler built exactly one coupling length long so that, unbiased, all the light crosses to the second guide — the "cross" state. A voltage across one guide changes its refractive index (the electro-optic effect), shifting its propagation constant and creating a mismatch $\delta$ . As Worked Example 2 shows, when the induced $\Delta\beta=2\delta$ reaches $\sqrt{3}\pi/L$ the shortened, detuned beat completes a full period over $L$ and returns all the power to the input guide — the "bar" state, dark at the cross port. Sweeping the voltage between these settings routes the light between two outputs, a compact 2×2 switch. Reversed- $\Delta\beta$ electrode designs relax the tolerance on $L$ and are the standard practical form.