Critical coupling

A resonator coupled to an input waveguide or beam loses its stored energy through two channels: intrinsic loss inside it (absorption, scattering, radiation) and the coupling port itself. Critical coupling is the case in which the two rates are equal. At resonance the light leaking back out through the coupler is then equal in amplitude and opposite in phase to the directly transmitted (or reflected) input, the two cancel, and all of the incident power is dissipated inside the resonator. The signature is a resonance dip that reaches zero. In terms of quality factors, the coupling Q equals the intrinsic Q, and the loaded Q is half of either: a silicon ring with $Q_i = 2 \times 10^5$ at 1550 nm, critically coupled, has $Q_L = 10^5$, a linewidth of 15.5 pm (1.93 GHz) and a photon lifetime of 82 ps.

Energy-balance formula

In the coupled-mode description, the stored field decays at an intrinsic rate $\kappa_0$ and a coupling rate $\kappa_e$ (energy decay rates, $\kappa = \omega_0/Q$). The amplitude transmission past a single-port resonator at resonance is

$$t_0 = \frac{\kappa_0 - \kappa_e}{\kappa_0 + \kappa_e} = \frac{Q_c - Q_i}{Q_c + Q_i},$$

so the power transmission is $T_0 = t_0^2$, which vanishes when $Q_c = Q_i$. With $1/Q_L = 1/Q_i + 1/Q_c$, the loaded Q at that point is $Q_i/2$. For an all-pass ring resonator written with round-trip amplitude transmission $a$ and self-coupling $t$, the same condition reads $a = t$.

How close is close enough

The extinction depends on the ratio $x = Q_c/Q_i$ alone, and $x$ and $1/x$ give the same dip depth:

A 10% mismatch still gives a dip deeper than 26 dB, which is why measured rings commonly show 20–40 dB extinction ratios near critical coupling; the infinite value of the ideal formula requires an exact match. Because $x$ and $1/x$ give the same depth, an under-coupled resonance and the over-coupled one obtained by exchanging $Q_c$ and $Q_i$ have identical power spectra. The phase response separates them. Across an over-coupled resonance the transmitted phase sweeps through a full $2\pi$; across an under-coupled one it makes a limited excursion and returns.

Different resonators, same condition

• Rings and disks on a bus waveguide. The gap sets the coupling coefficient through the overlap of evanescent fields, and coupling falls roughly exponentially with gap. Because the intrinsic loss is known only after fabrication, designers sweep the gap on a test die to bracket critical coupling.
• Add-drop rings. The through port goes to zero when the input coupling equals the sum of the drop coupling and the intrinsic loss; the fraction delivered to the drop port is then $\kappa_{e2}/\kappa_{e1}$, close to 1 only when intrinsic loss is small.
• Fabry-Perot cavities. A two-mirror Fabry-Perot resonator seen in reflection is critically coupled, or impedance matched, when the input mirror transmission equals all other round-trip losses. For an external enhancement cavity with 1% round-trip loss, a 1% input coupler gives zero reflected power and a circulating power 100 times the input; input couplers of 0.5% or 2% leave 11% reflected and reduce the buildup to 89.
• Whispering-gallery resonators. Microspheres and microtoroids are coupled through a tapered fiber or a prism whose gap is adjusted while watching the dip deepen.

Where it matters

Critical coupling maximizes the power delivered into the resonator, which is the goal in resonant detectors, enhancement cavities for second-harmonic generation and Kerr microcombs, and the stored energy at critical coupling is $2P_\text{in}Q_L/\omega_0$, about 0.16 pJ for 1 mW into the ring above. It also gives the deepest notch filters. Measurement of intrinsic loss from a ring spectrum, set out in ring resonator Q extraction, relies on knowing which side of critical coupling the device is on.

Pitfalls

The coupling condition holds at one wavelength. Directional-coupler coupling usually rises with wavelength while intrinsic loss changes slowly, so a ring critically coupled at 1550 nm may be noticeably over-coupled at 1600 nm, and its dips change depth across the spectrum. Thermal drift, laser power high enough to shift the resonance and backscattering that splits the mode into a doublet all make the dip shallower than the coupling alone would predict, so a shallow dip does not by itself indicate mismatched coupling.

Common questions

What is the difference between critical, under and over coupling?

Under-coupled resonators lose more energy internally than through the coupler ($Q_c > Q_i$); over-coupled ones lose more through the coupler ($Q_c < Q_i$). Critical coupling is the boundary, where the two are equal and the transmitted power at resonance is zero.

Why is the loaded Q at critical coupling half the intrinsic Q?

The loaded linewidth is set by the total loss rate, which is the sum of the intrinsic and coupling rates. At critical coupling these are equal, so the total is twice the intrinsic rate and the linewidth doubles.

References: H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984); A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321 (2000); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).