Spontaneous Emission Enhancement and the Purcell Factor

2.3 Electron-Photon Coupling

2.3.4 Spontaneous Emission Enhancement and the Purcell Factor

The analysis of the previous two sections is rather tedious due to the complex nature of the semiconductor medium. In 1946 Edward Purcell [1], in working with a much simpler (and ideal) atomic medium, was able to derive a simple expression for the enhancement of the spontaneous emission rate in a generic microcavity. The method used to derive this result is very insightful and is presented below.

The local spontaneous emission rate of an atom, regardless of the cavity geometry, is proportional to the product of the spatial electric field intensity and spectral energy density; the local density of modes. We know from the analysis in sub-section 2.1.1 that the average spatial electric field intensity⁶⁷ of a given mode is inversely propor- tional to volume, and that the spectral density of modes is directly proportional to volume. Therefore, the average (in both space and frequency) local mode density remains approximately constant as cavity walls are brought inwards and the volume shrinks. This leads to the conclusion that the average emitter within the cavity will see an average local density of modes which is the same as that in free-space.

67Here we assume that within the cavity volume the dielectric function is constant, i.e., the cavity consists of a uniform dielectric material surrounded by mirrors.

Cavity enhancement of spontaneous emission is a result of the fact that the (non- averaged) local mode density is neither homogeneous in space or frequency. If the emitters are placed inside the cavity at anti-nodes of a cavity resonance, or if a cavity resonance aligns spectrally with an atomic transition, then an enhanced spontaneous emission rate over that in free-space is possible. Following the analysis of Purcell [1] and Kleppner [4] we can find an approximate formula for the cavity enhanced spontaneous emission rate of an active material source embedded in a microcavity.

In order to make connection to the work of Purcell, consider the active material to be independent, identical, randomly oriented atoms⁶⁸ which have an optically active transition centered at Wa with a line-shape much narrower than the optical cavity resonance linewidth:

llWa

«

^llWm (2.146)

where Llwa is the atomic linewidth and Llwm is the linewidth of the mth cavity resonance.

In the absence of a cavity, the electric field intensity is uniform and each atom has a spontaneous emission rate given by:

Ro _sp,a ^<X

J

^ue,opo _{ph a} ^P dw'""' ue,o(w )po (w ) '""' a ph a (2.147) where Pa is the narrow atomic line-shape, ue,o is the spatial electric field intensity per mode, p~h is the mode density per unit frequency in a homogeneous dielectric material (free-space) of refractive index n. The local mode density is ()~h

=

ue,o p~h.

Now consider a cavity resonant mode with an average spatial electric field intensity

tJ:;,,,

center frequency aligned with wa, and linewidth Llwm. The average spectral density of the cavity is approximately given by the product of the cavity volume and

68This allows us to assume that the atoms couple to different field polarizations equally on average.

the free-space mode density per unit frequency per unit volume,

(2.148) The average local mode density for the cavity in the vicinity of the atomic line fre- quency is then

(2.149) Equating tJ~h and

e;h

we can solve for the average electric field intensity in mode m:

ue,opo

tJe ,...._,

^ph

m ,...._,

^V

^(w~n3)

c 7r2 c3

(2.150)

The spontaneous emission rate of the average atom in the microcavity, assuming the atoms are uniformly distributed throughout the entire cavity, is

(2.151) where Pm ( w) is the Lorentzian line-shape of the cavity mode m, normalized such that

J

₀^{00 Pm(w) dw = 1.} Substituting the value of Uem given in eq. (2.150) into eq. (2.151), and taking the ratio of the cavity spontaneous emission rate to that in free-space one finds:

(2.152)

where P is the Purcell enhancement factor. If the atomic line is exactly centered with the mth cavity resonance then Pm(wa) ~ 1/(7r.6..wm) = Qm/(7rwa), and

(2.153)

where >.0 is the optical wavelength of the emitted light in vacuum. The Purcell enhancement factor is the product of a normalized spectral density Qm, and a nor- malized spatial density

1/Vc,

where

Ve=

Vc/(>.0/2n)³ is the cavity volume in units of cubic half-wavelengths in the material. If the atoms were not distributed uniformly within the cavity mode, but were selectively positioned at an anti-node of the cavity mode standing wave, then the average atom would see an electric field intensity larger than

u:n.

In general the average electric field intensity seen by the atoms within the active region volume, Va, is:

(2.154) (2.155)

The Purcell factor for the average atom is then given by

p~ ]:__~

2 T-7 ' 7f Veff,m

(2.156)

where the normalized effective volume for mode m is

(2.157)

When considering the effects of the active material distribution within the cavity,

it is helpful to define a spontaneous emission confinement factor,

r:.;:,,

between the active material and the mth cavity mode⁶⁹:

(2.158)

In terms of the confinement factor,

- 1 ( Va )

VetJ,m =

r:.;:,

(>..o/2n)3 ' ^(2.159) and the spontaneous emission rate for the average atom in the active region is,

(2.160) Comparing eq. (2.156) and (2.141) we see that the major modification to Purcell's original formula for the cavity enhanced spontaneous emission rate is to replace the cavity Q with the effective Q of the entire system of electrons and photons:

69The confinement factor defined above is different than the confinement factor typically used to calculate waveguide modal gain. The difference stems from the fact that the modal gain for a waveguide is typically written as a gain per unit length and the dispersive properties of the waveguide must be taken into account [64]. What we are effectively considering in this section is the average gain per unit time, (gf ( x, y, z)), for which the confinement factor defined above is correct. Unfortunately, what can be more directly measured in semiconductor amplifiers, and thus is typically referenced, is the local material gain per unit length, gf(x, y, z). For example, take gf(x, y, z) to be the (local) material gain per unit length (no waveguide) in a non-dispersive material with index n. The group velocity in this case is c/n, and the gain per unit length can be written in terms of the local gain per unit time: g'/ = (n/c)gf. Now if the field is propagating in a waveguide with an effective index of propagation for the guided mode of

n,

then the group velocity (due to waveguide dispersion only) is

vJ:B

^{= c/(n2} /n). This gives for the waveguide local gain per unit length: gI°⁹ = (n/n)(n/c)gf. The local gain per unit length of the waveguide in terms of the local material gain per unit length is then, gI°⁹(x,y,z)

=

(n(x,y,z)/n)gf(x,y,z). As described in this section, the average gain per unit time, (gf), can be calculated by averaging gf(x, y, z) using the electric field intensity as a weighting function. To calculate the average gain per unit length of a waveguide mode then gI°⁹(x,y,z) = (n(x,y,z)/n)g'/(x,y,z) must be averaged using the electric field intensity as a weighting function, not gf(x,y,z). This is the source of the difference in the confinement factor definitions.

1 1 1 1

-=-~-+--+

'

Q Qm Qhom. Qinhom.

(2.161) where Qm represents the cavity mode losses, Qhom. the homogeneous broadening, and

Qinhom. the inhomogeneous broadening of the excited state levels.