Integrated Photonic Waveguides and Circuits

A photonic integrated circuit takes the waveguide physics of the three articles before it — total internal reflection at a dielectric step, the modal wave equation whose eigenvalue is a propagation constant $\beta$ , and the single-mode cutoff — and turns it into wiring. What a fiber does over a spool of glass, a chip does over a few square millimetres: it guides, splits, delays, filters, and interferes light using ridges of high-index material carved on a wafer. Silicon ( $n\approx3.48$ ), silicon nitride ( $n\approx2.0$ ), and lithium niobate ( $n\approx2.2$ ) form the cores, a silica cladding ( $n\approx1.44$ ) surrounds them, and because the index contrast dwarfs a fiber's, a guided mode is squeezed into a cross-section a few hundred nanometres across and bent around a corner only microns in radius without spilling out. One number does most of the work. The effective index $n_{\mathrm{eff}}=\beta/k_0$ is the modal analogue of the bulk index: it fixes the guided phase velocity, the on-chip wavelength, a resonator's resonance spacing, and the rate at which neighbouring guides trade power through overlapping evanescent tails. From that single mode-carrying transmission line one assembles everything else — the directional coupler that acts as an on-chip beam splitter, the Mach–Zehnder interferometer that turns a phase shift into a brightness change, and the microring that traps light in a whispering loop. This article treats the waveguide as a phase-and-amplitude line, derives the coupler and ring transfer functions from coupled-mode theory, and quantifies the loss that caps every circuit. The unifying claim: a passive PIC is nothing but a network of controlled phases $\beta L$ and controlled couplings $\kappa$ .

Key takeaways

A photonic integrated circuit routes light through micron-scale dielectric waveguides on a chip; guidance is the same total-internal-reflection physics as a fiber, but the huge index contrast ( $\Delta n\sim2$ for silicon-on-silica) shrinks modes to sub-wavelength size and bends to a few microns.
Every guided mode is summarized by its effective index $n_{\mathrm{eff}}=\beta/k_0$ , bounded by $n_{\mathrm{clad}}<n_{\mathrm{eff}}<n_{\mathrm{core}}$ ; its dispersion is the group index $n_g$ , and the two are not interchangeable.
The building blocks are the single-mode waveguide, the directional coupler (evanescent power exchange, full transfer at $L_c=\pi/2\kappa$ ), the Mach–Zehnder interferometer (phase-to-amplitude conversion), and the microring resonator.
A microring's through-port transmission is $t=(a-r e^{i\phi})/(1-a r e^{i\phi})$ ; it dips at resonances spaced by $\mathrm{FSR}=\lambda_0^2/n_g L$ , reaching zero at critical coupling $r=a$ .
Loss comes from sidewall scattering, bulk and substrate absorption, and bend radiation; silicon nitride reaches below 1 dB/m and silicon wires run 1–3 dB/cm, with $1\,\mathrm{Np}=4.343\,\mathrm{dB}$ .
Key idea: a photonic integrated circuit is a network of dielectric waveguides characterized by an effective index $n_{\mathrm{eff}}$ and a loss coefficient $\alpha$ ; every passive function — splitting, filtering, delay, interference — reduces to controlling the propagation phase $\beta L=n_{\mathrm{eff}}k_0 L$ and the evanescent coupling $\kappa$ between guides.

Where this lives today

The clearest place to see this physics working is the silicon-photonic transceiver inside a datacenter. On a chip smaller than a grain of rice, microring or Mach–Zehnder modulators encode data onto light, arrayed-waveguide gratings and ring filters multiplex dozens of wavelengths onto one bus, germanium detectors read them out, and grating couplers hand the light to fiber — the whole link is the coupled-mode and ring-transfer physics above, at tens of gigabits per second per wavelength, made in the same CMOS foundries that build processors. Ring modulators are equation (4) in action: a carrier-injection index shift nudges a resonance across the laser line, turning a voltage into amplitude modulation in a micron-scale footprint. The frontier pushes the same parts harder. High- $Q$ silicon-nitride microrings with engineered anomalous dispersion turn a single continuous-wave pump into a soliton frequency comb — hundreds of evenly spaced lines from one ring — now packaged as chip-scale optical clocks and the ranging engine of solid-state LiDAR. Meshes of thermo-optically tuned Mach–Zehnder interferometers implement arbitrary unitary transformations on many modes, serving as the matrix-multiply core of optical neural networks and the reconfigurable fabric of linear-optical quantum processors, where single photons through couplers and phase shifters run boson-sampling protocols. In every case the design still reduces to choosing $n_{\mathrm{eff}}$ , $\kappa$ , and $\alpha$ .

The physical picture

Think of a garden hose made of glass whose walls are a slightly less dense glass. Light zig-zagging inside strikes the walls beyond the critical angle, reflects totally, and cannot leak away. On a chip the "hose" is a rectangular ridge of silicon a few hundred nanometres tall, and because the index jump to the silica cladding is so large, the mode is held tightly enough to bend around a corner only microns in radius. A fiber, with a contrast a hundred times weaker, would radiate its light on any bend tighter than a centimetre. High contrast is what fits a whole circuit onto a chip instead of a benchtop. The mode does not stop abruptly at the core edge: its field decays into the cladding as an evanescent tail reaching a fraction of a wavelength beyond the glass, and that leaking tail is the secret of every integrated device. Bring a second identical guide within tail-reach and each mode feels the other, so power slowly sloshes back and forth — a directional coupler, the on-chip beam splitter, which needs no lossy junction, only proximity. The figure shows both ideas: how the index ladder $n_{\mathrm{clad}}<n_{\mathrm{eff}}<n_{\mathrm{core}}$ holds the mode, and how overlapping tails exchange power along the coupling region.

Figure

Left, a silicon strip waveguide confines a mode in a 450 nm × 220 nm core of index n ≈ 3.48 buried in silica n ≈ 1.44, giving an effective index n_eff ≈ 2.40 pinned between them. Right, two such guides 200 nm apart let their evanescent tails overlap, so power oscillates as P₁ = cos²κz and P₂ = sin²κz, splitting 50 slash 50 at the half-beat length and transferring fully at L_c = π divided by 2κ

A microring resonator is simply a waveguide biting its own tail. Light coupled in from a straight bus circulates, and only wavelengths whose round-trip phase is a whole multiple of $2\pi$ add up in step, building a strong circulating field. On those resonances the ring drains the bus and the through-port transmission drops, producing sharp dips whose narrowness measures how many times a photon laps before it is lost. Everything is phase: a tiny index change — heating the waveguide, a voltage across an electro-optic material, or free carriers injected into silicon — shifts every resonance and rebalances every interferometer, turning a passive linear circuit into an active, reconfigurable processor. The second figure is what a PIC designer actually measures: the periodic transmission spectrum of a microring.

Figure

Through-port transmission of a 10 μm-radius silicon microring, L = 2πR = 62.8 μm, n_eff ≈ 2.40, n_g ≈ 4.2, near λ = 1550 nm. Resonances fall where the round-trip phase is a multiple of 2π, one every free spectral range FSR = λ² divided by n_g L = 9.1 nm, at longitudinal order m ≈ 97; at critical coupling r = a the dips reach zero transmission because all the light is dissipated inside the ring

The core equations

The one quantity that ties a mode to a circuit is its effective index: the propagation constant $\beta$ from the modal wave equation, normalized to the free-space wavenumber. It is the modal cousin of the bulk index $n=c/v$ , and because a guided mode lives partly in core and partly in cladding, it is squeezed between the two indices.

n_{\mathrm{eff}} = \frac{\beta}{k_0},\qquad n_{\mathrm{clad}} < n_{\mathrm{eff}} < n_{\mathrm{core}} \qquad (1)

where: $\beta$ — propagation constant of the mode (rad/m); $k_0=2\pi/\lambda_0$ — free-space wavenumber; $n_{\mathrm{core}},n_{\mathrm{clad}}$ — core and cladding refractive indices. A tightly confined mode sits near $n_{\mathrm{core}}$ ; a weakly bound one drops toward $n_{\mathrm{clad}}$ and pushes more field into the cladding, where it can couple or radiate. The phase index sets how fast crests move, but a pulse — and every resonance spacing — travels at the group velocity. Because $n_{\mathrm{eff}}$ itself depends on wavelength, the group index is not $n_{\mathrm{eff}}$ but its dispersion-corrected partner.

n_g = n_{\mathrm{eff}} - \lambda_0\,\frac{\mathrm{d}n_{\mathrm{eff}}}{\mathrm{d}\lambda_0} \qquad (2)

where: $n_g$ — group index, setting the pulse velocity $v_g=c/n_g$ ; $\mathrm{d}n_{\mathrm{eff}}/\mathrm{d}\lambda_0$ — combined waveguide and material dispersion. In a strongly guiding silicon wire $n_g\approx4.2$ can far exceed $n_{\mathrm{eff}}\approx2.4$ — a distinction that bites the moment you compute a resonator's spacing. Two identical guides brought side by side exchange power sinusoidally as their evanescent fields overlap: the directional coupler, whose split is set entirely by the product of coupling coefficient and length.

P_1(z)=P_0\cos^2(\kappa z),\qquad P_2(z)=P_0\sin^2(\kappa z) \qquad (3)

where: $P_1,P_2$ — powers remaining in and transferred to the second guide; $\kappa$ — coupling coefficient per unit length, larger for a smaller gap; $z$ — coupling length. Complete transfer occurs at $L_c=\pi/2\kappa$ and a 50/50 split at $L_{3\mathrm{dB}}=\pi/4\kappa$ , the input coupler of every on-chip interferometer. Close a waveguide into a loop coupled to a bus and the through-port field is a rational function of the round-trip phase — a single transfer function holding all-pass filtering, the resonance dips, and critical coupling.

t = \frac{a - r\,e^{i\phi}}{1 - a r\,e^{i\phi}},\qquad \phi = \beta L = \frac{2\pi n_{\mathrm{eff}} L}{\lambda_0} \qquad (4)

where: $t$ — complex through-port field transmission; $r$ — bus–ring self-coupling coefficient, with $r^2+\kappa^2=1$ when lossless; $a=e^{-\alpha L/2}$ — single-pass amplitude around the ring; $L$ — ring circumference; $\phi$ — round-trip phase. At critical coupling $r=a$ the numerator vanishes on resonance and transmission drops to zero. Resonances occur where the round-trip phase is a multiple of $2\pi$ ; differentiating that condition — with $n_{\mathrm{eff}}$ dispersing — gives the free spectral range, which depends on the group index, not the phase index.

\lambda_m = \frac{n_{\mathrm{eff}} L}{m},\qquad \mathrm{FSR} = \frac{\lambda_0^2}{n_g L} \qquad (5)

where: $\lambda_m$ — resonance wavelength for integer order $m$ ; $\mathrm{FSR}$ — spacing between neighbouring resonances; $n_g$ — group index. Using $n_{\mathrm{eff}}$ here is the single most common ring-design error, overestimating the FSR by nearly a factor of two in silicon. A dip's narrowness is the loaded quality factor, the ratio of resonance wavelength to full width at half maximum; high $Q$ needs both low loss and weak coupling.

Q = \frac{\lambda_0}{\Delta\lambda_{\mathrm{FWHM}}} = \frac{\pi n_g L \sqrt{ra}}{\lambda_0\,(1-ra)} \qquad (6)

where: $Q$ — loaded quality factor; $\Delta\lambda_{\mathrm{FWHM}}$ — resonance full width at half maximum; $r,a$ — coupling and round-trip amplitude factors, near unity for high $Q$ . Equivalently $Q=(\lambda_0/\mathrm{FSR})\,\mathcal{F}$ with finesse $\mathcal{F}=\pi\sqrt{ra}/(1-ra)$ , which counts photon round trips. Two 3-dB couplers around a pair of arms make a Mach–Zehnder interferometer, converting the arm-to-arm phase difference into output brightness.

P_{\mathrm{out}}=\frac{P_{\mathrm{in}}}{2}\big[1+\cos(\Delta\phi)\big],\qquad \Delta\phi=\frac{2\pi n_{\mathrm{eff}}\,\Delta L}{\lambda_0} \qquad (7)

where: $\Delta\phi$ — arm-to-arm phase difference; $\Delta L$ — physical path imbalance (or effective imbalance from a local index change); $P_{\mathrm{in}},P_{\mathrm{out}}$ — input and one output power. A phase swing of $\pi$ carries the output from fully bright to fully dark — the basis of every Mach–Zehnder modulator and switch. Finally, loss is quoted two ways, and the conversion between the natural attenuation constant and the decibel figure is worth memorizing.

\alpha_{\mathrm{dB/cm}} = 10\log_{10}(e)\,\alpha \approx 4.343\,\alpha \qquad (8)

where: $\alpha$ — power attenuation constant in $\mathrm{cm}^{-1}$ , so $P(z)=P_0 e^{-\alpha z}$ ; $\alpha_{\mathrm{dB/cm}}$ — the same loss in decibels per centimetre. The factor $10\log_{10}(e)=4.343$ is why $1\,\mathrm{Np}=4.343\,\mathrm{dB}$ ; a silicon-nitride guide below 1 dB/m is $\alpha<2.3\times10^{-3}\,\mathrm{cm}^{-1}$ .

Derivation ACoupling length of a symmetric directional coupler

The goal is to show, from first principles, that two identical waveguides brought close together do not just leak into each other once but exchange power back and forth sinusoidally, and to pin down the two design lengths every PIC needs: the length $L_c$ for complete transfer and the length $L_{3\mathrm{dB}}$ for an even 50/50 split. We start from the coupled-mode equations for the two slowly varying mode amplitudes, decouple them into a harmonic oscillator in the propagation coordinate, apply the boundary condition that all power begins in one guide, and read off the powers and the design lengths.

Step 1. Let $a_1(z),a_2(z)$ be the slowly varying mode amplitudes in the two guides. Weak evanescent overlap couples them, $\mathrm{d}a_1/\mathrm{d}z=-i\kappa a_2$ and $\mathrm{d}a_2/\mathrm{d}z=-i\kappa a_1$ , with a single real $\kappa$ because the guides are identical and phase-matched ( $\beta_1=\beta_2$ ). Why: each guide's evanescent tail is a distributed source feeding the other, and phase-matching makes the exchange resonant rather than washing out.

Step 2. Differentiate and substitute to decouple: $\mathrm{d}^2 a_1/\mathrm{d}z^2=-\kappa^2 a_1$ . Why: eliminating $a_2$ leaves a simple-harmonic equation in $z$ , so power beats between the guides like energy between two coupled pendula.

Step 3. Solve with all power in guide 1, $a_1(0)=1$ , $a_2(0)=0$ : $a_1=\cos(\kappa z)$ , $a_2=-i\sin(\kappa z)$ . Why: the $-i$ quarter-wave lag on the crossed amplitude is exactly what makes the coupler a unitary beam splitter.

Step 4. Square to get the powers, $P_1=\cos^2(\kappa z)$ and $P_2=\sin^2(\kappa z)$ , summing to $1$ for all $z$ . Why: energy is conserved — power is relocated between two lossless guides, not dissipated at a junction.

Step 5. Read off the lengths: complete transfer at $\kappa L_c=\pi/2$ so $L_c=\pi/2\kappa$ , an even split at $\kappa L_{3\mathrm{dB}}=\pi/4$ so $L_{3\mathrm{dB}}=\pi/4\kappa$ . Why: $L_c$ is a full crossover; $L_{3\mathrm{dB}}$ is the 50/50 splitter feeding both arms of a Mach–Zehnder. *Note: energy is conserved at every $z$ since $P_1+P_2=1$ throughout, and the half-beat length $L_{3\mathrm{dB}}=\pi/4\kappa$ is exactly the 3-dB splitter used as the input coupler of a Mach–Zehnder.*

Go deeper · Derivation B · 1 minTransfer function of an all-pass microring resonator

Derivation BTransfer function of an all-pass microring resonator

The goal is to derive the through-port response of a single microring coupled to one bus waveguide and to watch three results fall out at once: the periodic resonance dips, the Lorentzian lineshape near each dip, and the critical-coupling condition at which a resonance goes perfectly dark. We model the bus–ring coupler as a lossless point coupler, propagate the circulating field once around the ring picking up loss and phase, close the feedback loop self-consistently, and simplify the through-port field.

Step 1. Write the lossless coupler as a unitary relation, $E_t=rE_{\mathrm{in}}+i\kappa E_{\mathrm{circ}}$ and $E_r=i\kappa E_{\mathrm{in}}+rE_{\mathrm{circ}}$ , with real $r,\kappa$ and $r^2+\kappa^2=1$ . Why: the ring and bus behave like a directional coupler with a very short interaction length; the $i$ on the cross term enforces unitarity, so the coupler loses no power.

Step 2. Propagate once around the loop, $E_{\mathrm{circ}}=a\,e^{i\phi}E_r$ , with amplitude decay $a=e^{-\alpha L/2}$ and round-trip phase $\phi=\beta L=2\pi n_{\mathrm{eff}}L/\lambda_0$ . Why: each lap attenuates and delays the ring field before it re-enters the coupler — the feedback that builds resonance.

Step 3. Close the loop: substituting into $E_r=i\kappa E_{\mathrm{in}}+r a e^{i\phi}E_r$ gives $E_r=i\kappa E_{\mathrm{in}}/(1-r a e^{i\phi})$ . Why: the circulating field is its own source after one lap; the denominator $1-r a e^{i\phi}$ goes small — and the field large — on resonance.

Step 4. Assemble $E_t=rE_{\mathrm{in}}+i\kappa a e^{i\phi}E_r$ and simplify with $\kappa^2=1-r^2$ to $t=(a-r e^{i\phi})/(1-r a e^{i\phi})$ , so $T=|t|^2=(a^2-2ar\cos\phi+r^2)/(1-2ar\cos\phi+a^2r^2)$ . Why: interference between the directly transmitted field and the light that leaked out after circulating produces the dips.

Step 5. Impose critical coupling, $a=r$ : on resonance $\phi=2\pi m$ the numerator $a-r=0$ , so $T=0$ . Why: when the fraction coupled in per lap equals the fraction lost per lap, the two through-port fields cancel and every photon is dissipated inside the ring. *As a check: setting $a=1$ (lossless ring) gives $|t|=1$ for all $\phi$ — the ring becomes a pure phase filter, adding wavelength-dependent phase without touching amplitude.*

Limiting cases

The behaviour of a PIC element is governed by the confinement regime of the guide and by the coupling strength of a resonator; the useful limits are:

Case	Result	Meaning
Strong confinement (Si photonic wire)	$\Delta n\gtrsim2,\quad w,h\sim0.2\text{–}0.5\,\mu\mathrm{m}$	Sub-micron single-mode guides, bend radii of a few microns, but sidewall-roughness scattering dominates loss (1–3 dB/cm) and the modes are strongly polarization-dependent.
Weak confinement (SiN, doped silica)	$\Delta n\sim0.5\text{–}0.05,\quad w\sim1\text{–}10\,\mu\mathrm{m}$	Larger, near-Gaussian modes, very low loss (below 1 dB/m in SiN), easy fiber coupling, but bend radii of 50 μm to millimetres.
Lossless all-pass ring	$a\to1$ , so $\lvert t\rvert=1$ for all $\phi$	The ring adds only a wavelength-dependent phase, never touching amplitude — a pure delay or dispersion element used in optical filters and buffers.
Critical coupling	$r=a$ (coupling loss equals round-trip loss)	On-resonance transmission drops exactly to zero; all input power dissipates in the ring — the operating point of a maximum-extinction filter or modulator.
Under- vs over-coupling	$r>a$ (under), $r<a$ (over)	Both give partial dips; the through-port phase response differs, and only crossing critical coupling flips the sign of the transmission slope — key for delay-line and sensing designs.

Common trapThe effective index is a fixed material constant

It is not. Unlike a bulk index, $n_{\mathrm{eff}}$ is a property of the mode, not the material: it depends on wavelength, on polarization (quasi-TE versus quasi-TM), and on the exact cross-section geometry down to nanometre sidewall variations. Two consequences bite designers constantly. First, it is the group index $n_g=n_{\mathrm{eff}}-\lambda_0\,\mathrm{d}n_{\mathrm{eff}}/\mathrm{d}\lambda_0$ — not the phase index — that sets a ring's free spectral range and a coupler's bandwidth; in silicon $n_g$ can be nearly double $n_{\mathrm{eff}}$ , so using the phase index overestimates the FSR by almost a factor of two. Second, the rectangular cross-section is birefringent, so quasi-TE and quasi-TM modes have different $n_{\mathrm{eff}}$ and hence different resonances and coupling ratios; a device designed for one polarization misbehaves for the other, which is why most PICs operate a single polarization with polarization-diversity circuitry at the input.

Worked examples

1 · A silicon microring, from geometry to quality factor (confidence builder)

A silicon strip waveguide is bent into a ring of radius $R=10\ \mu\mathrm{m}$ , with $n_{\mathrm{eff}}=2.40$ and group index $n_g=4.2$ at $\lambda_0=1550\ \mathrm{nm}$ . It is coupled to a bus with self-coupling $r=0.99$ and has round-trip amplitude $a=0.998$ . Find the ring's circumference, the longitudinal order of the resonance near 1550 nm, the free spectral range, and the loaded quality factor and finesse.

Step 1. The circumference is $L=2\pi R=2\pi(10\ \mu\mathrm{m})=\mathbf{62.8\ \mu m}$ , the length light travels per lap.

Step 2. The order is $m=n_{\mathrm{eff}}L/\lambda_0=(2.40)(62.8\ \mu\mathrm{m})/(1.550\ \mu\mathrm{m})=97.3$ , so the nearest resonance sits at integer $\mathbf{m\approx97}$ .

Step 3. The FSR uses the group index, not $n_{\mathrm{eff}}$ : $\mathrm{FSR}=\lambda_0^2/(n_g L)=(1.550\times10^{-6})^2/[(4.2)(62.8\times10^{-6})]=2.4025\times10^{-12}/2.639\times10^{-4}=\mathbf{9.1\ nm}$ . Using $n_{\mathrm{eff}}=2.40$ instead would give 15.9 nm — wrong by 75%.

Step 4. With $ra=(0.99)(0.998)=0.98802$ and $\sqrt{ra}=0.99399$ , the finesse is $\mathcal{F}=\pi\sqrt{ra}/(1-ra)=\pi(0.99399)/(0.01198)=\mathbf{261}$ .

Step 5. The loaded quality factor is $Q=\pi n_g L\sqrt{ra}/[\lambda_0(1-ra)]=\pi(4.2)(62.8\times10^{-6})(0.99399)/[(1.550\times10^{-6})(0.01198)]=\mathbf{4.4\times10^{4}}$ , matching $Q=(\lambda_0/\mathrm{FSR})\mathcal{F}=(1550/9.1)(261)$ ; the resonance width is $\Delta\lambda=\lambda_0/Q\approx\mathbf{35\ pm}$ . L = 62.8 μm, m ≈ 97, FSR = 9.1 nm, 𝓕 ≈ 261, Q ≈ 4.4×10⁴ (35 pm wide).

2 · Designing a directional coupler

A directional coupler has coupling coefficient $\kappa=0.15\ \mu\mathrm{m}^{-1}$ . Find the length for complete power transfer, the length for a 50/50 split, and the length at which 90% of the power has crossed to the second guide.

Step 1. Complete transfer needs $\kappa L_c=\pi/2$ , so $L_c=\pi/(2\kappa)=\pi/(2\times0.15\ \mu\mathrm{m}^{-1})=\mathbf{10.5\ \mu m}$ , a full crossover.

Step 2. A balanced splitter needs $\kappa L_{3\mathrm{dB}}=\pi/4$ , so $L_{3\mathrm{dB}}=\pi/(4\kappa)=\mathbf{5.24\ \mu m}$ — exactly half of $L_c$ , the 50/50 input coupler of a Mach–Zehnder.

Step 3. For 90% transfer set $\sin^2(\kappa z)=0.90$ , so $\kappa z=\arcsin(0.9487)=1.249\ \mathrm{rad}$ and $z=1.249/0.15\ \mu\mathrm{m}^{-1}=\mathbf{8.33\ \mu m}$ — past $L_{3\mathrm{dB}}$ but short of $L_c$ , since transfer slows as it nears completion. L_c = 10.5 μm, L_3dB = 5.24 μm, 90% at z = 8.33 μm.

3 · Critical coupling and extinction of a ring modulator

A microring has round-trip amplitude $a=0.98$ . Compute the on-resonance through-port transmission for three couplers: $r=0.98$ (critical), $r=0.95$ (over-coupled), and $r=0.99$ (under-coupled), and give the extinction ratio in decibels.

Step 1. On resonance $\cos\phi=1$ , so the transmission collapses to $T=(a-r)^2/(1-ar)^2$ . This is the number a modulator's extinction ratio is built on.

Step 2. Critical coupling, $r=a=0.98$ : the numerator $a-r=0$ , so $T=\mathbf{0}$ and the extinction is in principle infinite. All the input light is dissipated inside the ring — the deepest possible dip.

Step 3. Over-coupled, $r=0.95<a$ : $T=(0.98-0.95)^2/(1-0.931)^2=(0.03)^2/(0.069)^2=9.0\times10^{-4}/4.76\times10^{-3}=\mathbf{0.189}$ , an extinction of $-10\log_{10}(0.189)=\mathbf{7.2\ dB}$ .

Step 4. Under-coupled, $r=0.99>a$ : $T=(0.98-0.99)^2/(1-0.970)^2=(0.01)^2/(0.0298)^2=1.0\times10^{-4}/8.88\times10^{-4}=\mathbf{0.113}$ , an extinction of $\mathbf{9.5\ dB}$ . Both mistuned cases leave a partial dip; only critical coupling reaches zero, which is why a modulator is biased as close to $r=a$ as fabrication allows. Critical → T = 0; over-coupled → 0.19 (7.2 dB); under-coupled → 0.11 (9.5 dB).

Physics of the portrait

Integrated photonics is barely fifty years old, and it began not with a device but with a name and a vision: that the discrete lenses, prisms, and mirrors of an optics bench could be replaced by a monolithic dielectric circuit fabricated the way transistors are.

Origins · 1 minStewart Miller and how this was discovered

Stewart Miller

1918–1990 · Bell Labs; named integrated optics, 1969

Stewart E. Miller was an engineer at Bell Telephone Laboratories who spent his career on guiding light and microwaves through structures rather than free space. In 1969, in a short and now-famous Bell System Technical Journal paper titled "Integrated Optics: An Introduction," he coined the term and laid out the program: miniature, monolithically fabricated circuits of waveguides, couplers, and modulators that would do for photons what the integrated circuit had done for electrons. In the same issue his colleague Enrique Marcatili published the companion analysis of the dielectric rectangular waveguide and directional coupler — the backbone this article rests on — solving the non-circular, high-contrast geometry with no counterpart in fiber theory. The decisive turn came when Richard Soref argued in the 1980s that silicon, the material of the whole microelectronics industry, could itself be a photonic waveguide, and that free carriers injected into it (the plasma-dispersion effect) could modulate light. That seeded silicon photonics — carried forward by Intel's gigahertz modulators, Michal Lipson's high-index-contrast microrings, and the high- $Q$ silicon-nitride soliton combs of Tobias Kippenberg. Miller's vision of an optics bench shrunk onto a chip is now a commercial foundry technology, and the effective index, coupling coefficient, and loss coefficient he and Marcatili first quantified remain the three numbers that specify any linear circuit.

History

1969

— Stewart Miller coins "integrated optics"; Marcatili solves the rectangular dielectric waveguide and directional coupler

1987

— Soref and Bennett quantify silicon's plasma-dispersion effect for on-chip modulation

2004

— Intel demonstrates the first gigahertz-scale silicon Mach–Zehnder modulator

2010

— CMOS foundries begin offering silicon-photonics process design kits

2014

— dissipative Kerr soliton frequency combs are observed in high-Q microresonators

Going further

Texts. Saleh and Teich, Fundamentals of Photonics, 3rd ed., Ch. 8 (Guided-Wave Optics) and Ch. 10 (Photonic-Crystal and Integrated Optics) · Yariv and Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed., coupled-mode-theory chapters · Hecht, Optics, 5th ed., §5.6 (fiber and integrated optics) · Okamoto, Fundamentals of Optical Waveguides, 2nd ed. · Loudon, The Quantum Theory of Light, 3rd ed. (mode-quantization background) · Demtröder, Laser Spectroscopy (waveguide-spectroscopy applications) · W. Bogaerts et al., "Silicon microring resonators," Laser and Photonics Reviews 6, 47 (2012). The key results are: the all-pass through-port field $t=(a-re^{i\phi})/(1-are^{i\phi})$ from the lossless coupler matrix plus round-trip feedback; $L_c=\pi/2\kappa$ from the coupled ODEs; $\mathrm{FSR}=\lambda_0^2/(n_g L)$ with waveguide dispersion included; $1\,\mathrm{Np}=4.343\,\mathrm{dB}$ ; and the worked anchors $L=62.8\ \mu\mathrm{m}$ , $m\approx97$ , $\mathrm{FSR}=9.1\ \mathrm{nm}$ , $Q\approx4.4\times10^{4}$ for the $R=10\ \mu\mathrm{m}$ ring. Material indices at 1550 nm (Si $\approx3.48$ , SiN $\approx2.0$ , LiNbO $_3\approx2.2$ , SiO $_2\approx1.44$ ) and loss figures (SiN below 1 dB/m; Si wire 1–3 dB/cm) are standard values.

Prerequisites. Total Internal Reflection and Optical Guiding · Waveguide Modes and the Wave Equation · Single-Mode and Multimode Operation · Numerical Aperture and Acceptance Angle. Leads to. Coupled-Mode Theory and Directional Couplers · Microring Resonators and Optical Cavities on Chip · Nonlinear Optical Fibers and Effects · Fiber Attenuation and Loss Mechanisms.

Ask the tutor: ask me to size a microring for a target FSR and $Q$ , or to walk a signal through a full Mach–Zehnder modulator step by step.

Is the effective index just the average of the core and cladding indices?

No — it is a weighted eigenvalue, not an arithmetic average. It comes from solving the modal wave equation with the full two-dimensional cross-section, and its value reflects how much of the mode's field sits in the high-index core versus the low-index cladding. A tightly confined mode has $n_{\mathrm{eff}}$ near $n_{\mathrm{core}}$ ; a weakly bound one, with a large evanescent tail, drops toward $n_{\mathrm{clad}}$ . It is bounded by the two indices, but where it lands depends on wavelength, polarization, and geometry — which is why it must usually be computed numerically.

Why does the free spectral range use the group index instead of the effective index?

Because the FSR is the spacing between adjacent resonances, and finding it means differentiating the resonance condition with respect to wavelength — and $n_{\mathrm{eff}}$ itself changes with wavelength. Carrying that derivative through, the dispersion term $-\lambda_0\,\mathrm{d}n_{\mathrm{eff}}/\mathrm{d}\lambda_0$ combines with $n_{\mathrm{eff}}$ into the group index, giving $\mathrm{FSR}=\lambda_0^2/(n_g L)$ . Physically, resonance spacing is about how the round-trip delay shifts with wavelength, and delay is set by group velocity. In silicon $n_g\approx4.2$ against $n_{\mathrm{eff}}\approx2.4$ , so the phase index gives an FSR almost twice too large.

What is critical coupling, and why want a resonance that goes completely dark?

Critical coupling is $r=a$ : the fraction of light coupled into the ring per round trip exactly equals the fraction lost per round trip. On resonance the directly transmitted field and the light leaking back out after circulating are equal and opposite, cancel at the through port, and the transmission drops to zero — the input power ending up dissipated inside the ring. You want that dark dip whenever extinction matters: a ring modulator toggles between a bright off-resonance state and a dark on-resonance state, and a sensor tracks a sharp zero because it is far easier to locate precisely than a shallow, rounded dip.

How is a Mach–Zehnder modulator different from a ring modulator?

Both convert a phase shift into intensity, but they trade footprint against bandwidth. A Mach–Zehnder splits light into two arms, phase-shifts one, and recombines; its output $(P_{\mathrm{in}}/2)[1+\cos\Delta\phi]$ sweeps bright to dark as $\Delta\phi$ passes $\pi$ . It is broadband but accumulates phase slowly, so the arms are long (hundreds of microns to millimetres). A ring modulator instead slides a sharp resonance across the laser line, so it is compact (about 10 μm) and power-efficient, but works only near one resonance and drifts with temperature. Datacenter links use both, per the footprint-versus-wavelength-tolerance trade.

Why do quasi-TE and quasi-TM modes have different effective indices, and does it matter?

It matters a great deal. A rectangular waveguide is not symmetric under a 90° rotation, so the boundary conditions differ for light polarized mostly horizontally (quasi-TE) versus vertically (quasi-TM); the two see different confinement and acquire different effective indices — the guide is birefringent. Since $n_{\mathrm{eff}}$ sets resonance wavelengths, coupling ratios, and delays, a device designed for quasi-TE resonates and splits differently for quasi-TM. So a PIC is designed for a single polarization, and light in the wrong one is filtered out or split off by polarization-diversity circuitry at the input.

What limits how low the loss can go, and why is silicon nitride so much better than silicon?

Three mechanisms set the floor: scattering off etched-sidewall roughness, bulk and substrate absorption, and bend radiation. Sidewall scattering usually dominates and scales steeply with index contrast — the same high contrast that lets silicon bend on a micron radius also slams its mode against every nanometre of roughness, so silicon photonic wires sit at 1–3 dB/cm. Silicon nitride has a much lower index (about 2.0 versus 3.48), so its modes are larger and gentler on the sidewalls and its telecom-band absorption is tiny; careful SiN reaches below 1 dB/m. The price is weaker confinement and larger bend radii — the strong-versus-weak-confinement trade.