ECE 240a - Notes on Spontaneous Emission within a Cavity

Transcription

1 ECE 0a - Notes on Spontaneous Emission within a Cavity Introduction Many treatments of lasers treat the rate of spontaneous emission as specified by the time constant τ sp as a constant that is independent of the effect of the cavity. In this set of notes, we discuss the effect of the cavity on the rate of spontaneous emission and show that the emission can either be enhanced or inhibited. The enhancement of the spontaneous emission is called the Purcell effect. Mode Density The gain sites used for laser emission must emit into a mode of the system. An estimate of the modification of the spontaneous emission rate can be determined by estimating the ratio of the mode density in free space to the mode density for a cavity.. Mode Density in ree Space The electromagnetic mode density in free space is required for several aspects of lasers.the mode density dn/dν per unit frequency in a volume V for both polarizations is given by a modified form of..0) of M&E dn. = ρ ν = 8πν 0 dν c 3 V, where ν 0 is the operating frequency. Using ν = ω/π and dν = dω/π, the mode density in angular frequency is dn. = ρ ω = ω 0 dω π c 3 V, where c = c 0 /n is defined in the medium. Using λν = c, we can also write this in wavelength units as dn dλ = ρ λ = 8π where λ is defined in the medium. This density includes both polarizations. The mode density into a single polarization is half this value. V λ 3 ) λ, This mode density is derived in Section 7. of the Second edition of Ver or in Section. of M&E and will not be derived here.

2 . Mode Density for a Cavity Suppose that we have a single-mode cavity. This cavity is characterized by a photon lifetime τ p. Equivalently it can be characterized by the finesse, which is the ratio of the free-spectral range ν fsr = v q+ ν q = c nd to the bandwidth ν of a single longitudinal mode. See Lecture 0.). The cavity can also be characterized by a quality factor Q defined in the same way as that of a filter so that Q = ν 0 ν = ω 0 ω. ) The bandwidth ν of a single-mode cavity is the approximate frequency width of one mode. The reciprocal / ν is the bandwidth is an estimate of the number of modes per unit frequency, which is the mode densityρ cav for the cavity where we work with angular frequency units, which are more common..3 Purcell Effect ρ cav = ω = Q ω 0, ) Most derivations of the enhancement use a single polarization so that the free-space mode density is reduced by a factor of two. The ratio of the free space mode density to the cavity mode density is then ρ cav Q/ω 0 = ρ ω V ω0 /π c 3 = ) V π Q λ 3. 3) We see that the this factor depends on the volume of the cavity V as compared to the wavelength λ and the photon lifetime in the cavity as expressed by Q. In the next sections of the notes, we will place these back-the-envelope results on a firm foundation by determining the effect of a concentric resonator on the radiation emitted by a dipole. This derivation also provides expressions for the spontaneous emission lifetime. 3 Spontaneous Emission from a Classical Point Dipole In order to derive the effect of the cavity on the spontaneous emission rate, we start with a classical model based on Maxwell s equation for the rate of emission for a dipole in free space. Consider a harmonic dipole moment pt) = e sinω 0 t) x where e is the electron charge and x is a unit vector in the direction of the oscillation of the charge. We state without proof the transverse part of the electric field, called the radiation field E rad r, t), at a position r from a point dipole located at a position R is E rad r, t) = ω ) 0 e jk r R nr πɛ c x) n r r R, ) where n r is a unit vector in the direction of r R, c = c 0 /n is the speed of light in the medium, ɛ = ɛ 0 ɛ r is the permitivity in the medium, and r R λ. Note that this assumption will limit the size of the cavity for which the analysis is accurate. If the dipole is linearly polarized, then the radiated power per solid angle is given by See the second to last line of.5.3) in M&E) dp dω = πɛ π See Section.5 of M and E first edition for a complete derivation c 3 d ) pt) sin θ, 5) dt

3 where θ is the angle between x and r. If pt) = Re[p 0 e jω 0t ], then and d pt) dt = ω 0Re[p 0 e jω 0t ] 6) d pt) dt ) = ω 0 p 0 cos ω 0 t + arg p 0 ). 7) Averaging over a time interval that is long as compared to the period T = π/ω 0, we have Therefore, for a lossless time-harmonic dipole we have cos ω 0 t + arg p 0 ) = /. 8) dp dω = p 0 ω0 πɛ 8π c 3 sin θ, 9) where p 0 = ex is the magnitude of the dipole moment. The time-average total power radiated by the dipole in one polarization is determined by integrating over a solid angle of π steradians and is see.5.3) in M&E) P = p 0 ω ˆ π ˆ π 0 πɛ 8π c 3 dφ sin 3 θdθ, 0 0 = ω0 πɛ 3c 3 p 0. 0) Substituting p 0 ω 0 /c3 = πɛ)3p from 0) into 9), we have dp dω = 3P 8π sin θ. ) This expression relates the total time-averaged radiated power P to the power per unit solid angle dp/dω. 3. Classical spontaneous emission lifetime Now consider a classical model of a free non-driven) harmonic oscillator given in Lecture, which can be written in terms of the dipole moment p = ex d p dt + σ dp dt + ω op = 0. ) In this form, we interpret τ sp = /σ. The most general solution for this equation is in the form of a damped oscillation. The characteristic equation is cf. Lecture ) and has roots that can be written as x + σx + ω 0 = 0, 3) x i = σ/ ± jω 0 σ /ω0 ). ) Choosing the positive root to agree with our convention for the time dependence, e jω 0t, the general solution is 3

4 [ ] pt) = Re p 0 e σt/ e jω 0 σ /ω 0 )t 5) = p 0 cos ω 0 Xt + arg p 0 ), where p 0 is the complex amplitude of the harmonic dipole moment and X = σ /ω 0 ). or this lossy dipole, the time-averaged radiated power P given in 0) is modified by including a power loss term e σt so that the time-average power becomes a function of the damping rate P t) = ω0 πɛ 3c 3 p 0 e σt. 6) If the damping rate σ is much less than the resonant frequency so that σ ω 0, then we can define a time-average energy over a single cycle of the oscillation of the dipole. This condition defines a weakcoupling regime. Using p = ex, we can write the kinetic energy as KE = mvt) = m e ) dpt). 7) dt Similarly, the potential energy is PE = kxt) = m e ω 0pt), 8) where k = mω 0 has been used. Therefore, the total average energy is E = m ) ) dpt) e ω0pt) +. 9) dt Substituting 7) into this equation and averaging over a time interval longer that the oscillation so that cos ω 0 t) = /, the kinetic energy and the potential energy each have an equal contribution to the overall energy, which can be written as Et) = mω 0 e p 0 e σt. 0) Now equate the time-averaged energy loss given in 0) to the integral of the time-averaged power loss given in 6) so that ˆ Et) = P t)dt mω 0 e p 0 e σt = σ ω0 πɛ 3c 3 p 0 e σt. Solving for /σ and recalling that the spontaneous emission lifetime τ sp is defined at the time for which the energy or number of upper states) has decayed to /e of its initial value we can write = σ = A = e ω0 τ sp πɛ 3mc 3, ) which yields a lifetime of about 0 ns for an oscillating electron at a wavelength of one half a micron. The constant e /3mc 3 is a characteristic time that is on the order of the time it takes light to travel an electron radius r 0 = e /mc.

5 3. Quantum treatment of the spontaneous emission lifetime The spontaneous emission rate A nm between to energy levels n and m is presented in E&M Section 7.6 and is given by A nm = πɛ 3 D nmω 3 0 c 3 ) where D nm is the magnitude-squared of the dipole moment so that D nm = e r nm and r nm is the expected value of the overlap between the two wavefunctions that define the two energy states. See Lecture 3.) The expression derived from quantum mechanics is equal to the classical expression if the classical potential energy kx = mω 0 x is set equal to one half the vacuum state energy or ω/). Making this substitution, we have for the classical result given in ) = σ = A = e x ω0 3 τ sp πɛ 3 c 3. 3) Given this equivalence, we will derive the effect of the cavity on the spontaneous emission rate based on a classical analysis noting that we expect this analysis to break down for small cavities much less than the wavelength λ for two reasons. The first is that not all terms of the dipole field are used. The second is that the true quantum nature of the interaction is not accounted for. Dipole Radiation in a Resonator The sum of the solid angle Ω side out the sides of the resonator and twice the solid angle Ω mir that a single mirror subtends is equal to π or Ω side + Ω mir = π. ) Suppose that Ω mir is small. Then using the geometry given in igure??, we have Ω mir πb /L, 5) where the parameters are defined in igure??. If the resonator is stable, then the power incident on a mirror that subtends a solid angle Ω mir and the power in the solid angle Ω side is equal to the total radiated power. Therefore, using ) at an angle θ = π/, the power emitted into the total solid angle Ω mir subtended by the two mirrors can be expressed in terms of the total time-averaged power emitted from the dipole ) dp = Ω mir dω θ=π/ = 3P π Ω mir. 6) Ignoring the effect of the cavity, the power that is emitted out the side of the cavity is then P side = P P mir = P 3 ) π Ω mir. 7) Note that is only the effect of the solid angle subtended by the cavity mirrors and does not include the effect of the cavity. The presence of the cavity affects. 5

6 . Effect of the Cavity The power is the power emitted into the solid angle Ω mir subtended by the two mirrors in the absence of the effect of the cavity. The power P cav including the effect of the cavity has two features. At resonance and for a highly reflective cavity such that R, the power inside the cavity is related to the power outside the cavity emitted through a single mirror by = T/)P cav where T = R is the power transmission for one mirror. We then have P cav T = R where and the finesse of the cavity is = R R) = π π R R. See Lecture 0.) The second feature is the frequency dependence of the cavity, which is given by the Airy function for that cavity see Lecture 0). Therefore, we can write the ratio P cav / as 3 P cav }{{} Power enhancement by cavity + sin kl) }{{} requency dependence Airy function). 8) The first term in 8) is the enhancement of the power from the effect of the cavity and is the energy storage effect of the resonator. The second term in 8) is the frequency dependence as expressed by the Airy function. The maximum enhancement of the power caused by the effect of the cavity occurs if sin kl) = 0 with Pcav ) max = Q, 9) R where the relationship between the finesse and Q is discussed in Lecture 0 and Verdeyen Section 6.3, and R. The minimum value occurs if sin kl) = with Pcav ) min = R Q. 30) We see that relative to the radiation from a dipole in free space that subtends a solid angle Ω mir, the radiation into a cavity mode is enhanced or inhibited by a factor that is proportional to the Q of the cavity. The enhancement/inhibition factors change for different kinds of cavities and also change if the dipole is oriented exactly along the axis. or details of this effect, see Reference 3. 3 See Ref. 3 for a more detailed analysis.) 6

7 . Total Power Into the Cavity Using 6), we can write P cav + sin kl) 3P ) π + sin Ω mir. 3) kl) We see that the total radiation rate into the cavity depends on both the solid angle Ω mir and the finesse or the Q) of the cavity. The maximum total power in the cavity occurs if R. We then have cf.??)) ) 3 P max = P R π Ω mir. 3) The minimum total power into the cavity occurs P max = 3P R) Ω mir. 33) 8π Note that the minimum value is less than the rate if there was no cavity, so that P cav = so that P min = = 3P π Ω mir. 3) These expressions, derived using a classical analysis, are good starting points for an analysis. However, they are expected to break down for cavities on the order of the wavelength λ because they do not include all of the relevant physics. References Laser Electronics Verdeyen Chapter 6. Lasers Milonni and Eberly Chapter. Radiative decay and level shift on an atom in an optical resonator, Daniel Heinzen Massachusetts Institute of Technology, Ph.D. thesis 988) Chapter. 7