Purcell effect

The Purcell effect is the change in the rate of spontaneous emission of an atom, molecule or quantum dot when it is placed in an optical cavity. The rate of spontaneous emission depends on the surroundings as well as the emitter: it is proportional to the density of optical modes at the emitter's frequency and position. A resonant cavity with a high quality factor and a small mode volume concentrates that density of modes into a narrow band, and an emitter tuned to it radiates faster. The enhancement is the Purcell factor, $F_P$. For an InAs quantum dot at 930 nm, which has a radiative lifetime of about 1 ns in bulk GaAs, Purcell factors of a few to about ten shorten the lifetime to a few hundred or about a hundred picoseconds. Off resonance, or inside a photonic bandgap, the rate is suppressed instead.

The Purcell factor

E. M. Purcell gave the result in 1946 for nuclear magnetic transitions at radio frequencies. For a dipole exactly on resonance with a single cavity mode, placed at the field maximum and aligned with the field, the enhancement over emission in a uniform medium of index $n$ is

$$F_P = \frac{3}{4\pi^2}\left(\frac{\lambda}{n}\right)^3\frac{Q}{V},$$

where $\lambda$ is the free-space wavelength, $Q$ the cavity quality factor, and $V$ the mode volume. The prefactor $3/4\pi^2$ is 0.076. The formula assumes that the emitter's own linewidth is narrower than the cavity's, $\lambda/Q$; when the emitter is broader, the emitter linewidth replaces the cavity linewidth in $Q$ and the enhancement drops.

Mode volume

The mode volume measures how tightly the cavity field is confined:

$$V = \frac{\int \varepsilon(\mathbf r)\,|\mathbf E(\mathbf r)|^2\,d^3r}{\max\left[\varepsilon(\mathbf r)\,|\mathbf E(\mathbf r)|^2\right]}.$$

It is quoted in units of the cubic wavelength in the material, $(\lambda/n)^3$. At 930 nm in GaAs, with $n \approx 3.5$, $\lambda/n = 0.266$ µm and $(\lambda/n)^3 = 0.0188$ µm³. Micropillar cavities have mode volumes of a few to tens of $(\lambda/n)^3$; photonic-crystal nanocavities such as the L3 design reach about $0.7\,(\lambda/n)^3$. Because the field maximum appears in the denominator, the definition gives the enhancement for an emitter at that maximum; an emitter elsewhere sees less.

Worked values

For a GaAs micropillar with $Q = 5000$ and $V = 5\,(\lambda/n)^3$,

$$F_P = 0.076 \times \frac{5000}{5} \approx 76.$$

The cavity linewidth at 930 nm is $930/5000 = 0.19$ nm, so the quantum dot must be held within a fraction of that, by temperature or electric field tuning. A photonic-crystal L3 cavity with $Q = 10^4$ and $V = 0.7\,(\lambda/n)^3$ gives an ideal $F_P$ of about 1100.

Measured enhancements fall well short of these ideal numbers, typically by a factor of several or more, for four reasons: the dot is rarely at the field maximum, its dipole is not perfectly aligned with the field, its linewidth broadens through dephasing, and the cavity may be slightly detuned. The effective factor is reduced by the square of the normalized field amplitude at the dot, by the squared cosine of the dipole-field angle, and by a Lorentzian detuning term.

Measuring the enhancement

The Purcell factor is measured by time-resolved photoluminescence, usually time-correlated single-photon counting after a picosecond excitation pulse. The decay time of an emitter tuned onto the cavity resonance is compared with the same emitter detuned from it, or with emitters in unpatterned material. If emission into all other modes proceeds at a fraction $\gamma$ of the bulk rate, the on-resonance lifetime is $\tau_0/(F_P + \gamma)$; with $F_P = 10$ and $\gamma = 1$, a 1 ns lifetime becomes 91 ps.

Where it matters

The same ratio sets how much of the emission goes into the cavity mode, the spontaneous emission factor,

$$\beta = \frac{F_P}{F_P + \gamma},$$

which is 0.91 for the values above. That is the reason quantum-dot single-photon sources are built in micropillars and photonic-crystal cavities: the Purcell effect both speeds up emission, raising the repetition rate the source can support and reducing the effect of dephasing on photon indistinguishability, and directs the photons into a mode that can be collected. In nanolasers the same physics raises $\beta$ toward one. In VCSELs and microcavity LEDs the mode volume is large and the effect on the total rate is weak.

Pitfalls

• The formula holds in the weak-coupling regime. When the emitter-cavity coupling rate exceeds both the cavity and emitter decay rates, the system enters strong coupling and shows vacuum Rabi splitting instead of a faster exponential decay.
• Lifetime shortening can also come from non-radiative channels, such as surface recombination near etched sidewalls; comparison with detuned emitters in the same structure separates the two.

Common questions

What is the difference between the Purcell factor and β?

The Purcell factor is the rate enhancement for emission into the cavity mode; $\beta$ is the fraction of all emission that ends up in that mode. A large $F_P$ produces a large $\beta$.

Can the Purcell effect suppress emission?

Yes. An emitter detuned from all cavity modes, or placed inside a photonic bandgap, sees a lower density of modes than in free space, and its emission rate falls below the bulk value.

References: E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Physical Review 69, 681 (1946); J. M. Gérard et al., "Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity," Physical Review Letters 81, 1110 (1998); K. J. Vahala, "Optical microcavities," Nature 424, 839 (2003); D. Englund et al., "Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal," Physical Review Letters 95, 013904 (2005).