Lecture 11- ECE 240a Mode Density Mode Density for a Cavity Purcell Effect “Classical” Sponta- Lecture 11- ECE 240a neous Emission from a (See Notes on Spontaneous Emission) Dipole “Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission Dipole Radiation in a Concentric Resonator 1 ECE 240a Lasers - Fall 2019 Lecture 11
Free Space Mode Density Lecture 11- ECE 240a The electromagnetic mode density in free space is required for several aspects Mode of lasers. Density Mode Density for a Cavity The mode density dN / dν per unit frequency in a volume V for both Purcell Effect polarizations is given by a modified form of (2.1.40) of M&E “Classical” Sponta- neous = ρ ν = 8 πν 2 dN . Emission 0 V , from a c 3 dν Dipole “Classical” Spontaneous where ν 0 is the operating frequency. Emission Lifetime Using ν = ω / 2 π and dν = dω / 2 π , the mode density in angular frequency is Quantum Treatment of Spontaneous Emission ω 2 dN . 0 Dipole = ρ ω = π 2 c 3 V , Radiation in dω a Concentric where c = c 0 / n is defined in the medium. Resonator Using λν = c , we can also write this in wavelength units as � V � 1 dN dλ = ρ λ = 8 π λ , λ 3 where λ is defined in the medium. 2 ECE 240a Lasers - Fall 2019 Lecture 11
Mode Density for a Cavity Lecture 11- ECE 240a Mode Density This cavity is characterized by a photon lifetime τ p . Mode Density for a Cavity Also characterized by the finesse F . (See Lecture 10.) Purcell Effect “Classical” The cavity can also be characterized by a quality factor Q defined in the same Sponta- neous way as that of a filter so that Emission from a Dipole Q = ν 0 ∆ ν = ω 0 ∆ ω . “Classical” Spontaneous Emission Lifetime Quantum The bandwidth ∆ ν of a single-mode cavity is the approximate frequency width Treatment of Spontaneous of one mode. Emission Dipole Radiation in The reciprocal 1 / ∆ ν of the bandwidth is an estimate of the number of modes a per unit frequency, which is the mode density ρ cav for the cavity Concentric Resonator ∆ ω = Q 1 ρ cav = , ω 0 where we work with angular frequency units, which are more common. 3 ECE 240a Lasers - Fall 2019 Lecture 11
Simple Estimate of Effect of the Cavity - Purcell Effect Lecture 11- ECE 240a Mode Density Mode Density for a Cavity Purcell Effect Most derivations of the effect of cavity use a single polarization. “Classical” Sponta- neous Then the free-space mode density is reduced by a factor of two. Emission from a Dipole The ratio of the free space mode density to the cavity mode density is then “Classical” Spontaneous � V � Emission ρ cav Q / ω 0 1 Lifetime = 0 / π 2 c 3 = 4 π Q × . 2 V ω 2 λ 3 Quantum ρ ω Treatment of Spontaneous Emission Dipole We see that the this factor depends on the volume of the cavity V as Radiation in compared to the wavelength λ and the photon lifetime in the cavity as a Concentric expressed by Q . Resonator 4 ECE 240a Lasers - Fall 2019 Lecture 11
“Classical” Spontaneous Emission from a Dipole Lecture 11- ECE 240a Now place “back-the-envelope” results on a firm foundation by determining Mode the effect of a concentric resonator on the radiation emitted by a dipole. Density Mode Density for a Cavity Start with a classical model based on Maxwell’s equation for the rate of Purcell Effect emission for a dipole in free space. “Classical” Sponta- Consider a harmonic dipole moment p ( t ) = e sin ( ω 0 t ) � x where e is the neous Emission electron charge and � x is a unit vector in the direction of the oscillation of the from a Dipole charge. “Classical” Spontaneous Emission The transverse part of the electric field, called the radiation field E rad ( r , t ) , at Lifetime a position r from a point dipole located at a position R can be written as 1 Quantum Treatment of Spontaneous Emission � e − jk | r − R | Dipole � ω 2 E rad ( r , t ) = − 1 Radiation in 0 ( � n r × � x ) × � n r , a c 2 4 πǫ | r − R | Concentric Resonator 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, ǫ = ǫ 0 ǫ 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. 1 See Section 2.5 of M and E first edition for a complete derivation 5 ECE 240a Lasers - Fall 2019 Lecture 11
Power into a Solid Angle - 1 Lecture 11- ECE 240a Mode If the dipole is linearly polarized, then the radiated power per solid angle is Density given by (See the second to last line of (2.5.13) in M&E) Mode Density for a Cavity Purcell Effect � � 2 d 2 p ( t ) dP 1 1 1 “Classical” sin 2 θ , d Ω = Sponta- c 3 dt 2 4 πǫ 4 π neous Emission from a Dipole where θ is the angle between � x and � r . “Classical” Spontaneous If p ( t ) = Re [ p 0 e jω 0 t ] , then Emission Lifetime Quantum Treatment of d 2 p ( t ) Spontaneous = − ω 2 0 Re [ p 0 e jω 0 t ] Emission dt 2 Dipole Radiation in � � 2 a Concentric d 2 p ( t ) Solve for Resonator dt 2 � � 2 d 2 p ( t ) 0 | p 0 | 2 cos 2 ( ω 0 t + arg p 0 ) . = ω 4 dt 2 6 ECE 240a Lasers - Fall 2019 Lecture 11
Power into a Solid Angle - 2 Lecture 11- ECE 240a Mode Density Mode Density for a Cavity Purcell Effect Averaging over a time interval that is long as compared to the period “Classical” T = 2 π / ω 0 , we have Sponta- neous Emission cos 2 ( ω 0 t + arg p 0 ) = 1 / 2. from a Dipole “Classical” Spontaneous Emission Therefore, for a lossless time-harmonic dipole we have Lifetime Quantum Treatment of | p 0 | 2 ω 4 dP 1 Spontaneous c 3 sin 2 θ , 0 d Ω = Emission 4 πǫ 8 π Dipole Radiation in where | p 0 | = ex is the magnitude of the dipole moment. a Concentric Resonator This is the power per solid angle in one polarization of a dipole. 7 ECE 240a Lasers - Fall 2019 Lecture 11
Total Time-Averaged Radiated Power Lecture 11- ECE 240a Mode Density The time average total power radiated by the dipole in one polarization is Mode Density for a Cavity determined by integrating over a solid angle of 4 π steradians and is (See (see Purcell Effect (2.5.13) in M&E) “Classical” Sponta- � 2 π � π neous | p 0 | 2 ω 4 1 Emission sin 3 θdθ . 0 = P dφ from a c 3 4 πǫ 8 π Dipole 0 0 “Classical” ω 4 Spontaneous 1 3 c 3 | p 0 | 2 0 Emission = Lifetime 4 πǫ Quantum Treatment of Spontaneous Emission 0 / c 3 = ( 4 πǫ ) 3 P into the previous equation, we have Substituting | p 0 | 2 ω 4 Dipole Radiation in a d Ω = 3 P dP 8 π sin 2 θ . Concentric Resonator This expression relates the total time-averaged radiated power P to the power per unit solid angle. 8 ECE 240a Lasers - Fall 2019 Lecture 11
“Classical” Spontaneous Emission Lifetime Lecture 11- ECE 240a Mode Density Start with classical model of a free (non-driven) harmonic oscillator given in Mode Density for a Cavity Lecture 1, which can be written in terms of the dipole moment p = ex Purcell Effect “Classical” d 2 p dt 2 + σ dp Sponta- dt + ω 2 o p = 0 neous Emission from a Dipole In this form, we interpret τ sp = 1 / σ . “Classical” Spontaneous Emission The most general solution for this is in the form of a damped oscillation. Lifetime Quantum Treatment of The characteristic equation is (cf. Lecture 1) Spontaneous Emission x 2 + σx + ω 2 Dipole 0 = 0 Radiation in a Concentric Resonator The roots can be written as � 2 / 4 ω 2 x i = − σ / 2 ± jω 0 1 − ( σ 0 ) . 9 ECE 240a Lasers - Fall 2019 Lecture 11
General Solution Lecture 11- ECE 240a Choosing the positive root to agree with our convention for the time dependence, e jω 0 t , the general solution is Mode Density Mode Density for a Cavity � � � Purcell Effect 1 − ( σ 2 / 4 ω 2 0 ) t p 0 e − σt / 2 e jω 0 “Classical” p ( t ) = Re = | p 0 | cos ( ω 0 Xt + arg p 0 ) , Sponta- neous Emission from a where p 0 is the complex amplitude of the harmonic dipole moment and � Dipole 2 / 4 ω 2 “Classical” X = 1 − ( σ 0 ) . Spontaneous Emission Lifetime The time-averaged radiated power is modified by including a power loss term Quantum e − σt Treatment of Spontaneous ω 4 Emission 1 3 c 3 | p 0 | 2 e − σt 0 P ( t ) = Dipole 4 πǫ Radiation in a Concentric If σ ≫ ω 0 , then we can define a time-average energy over a single cycle of the Resonator oscillation. This condition defines a weak-coupling regime . In this regime, it is unlikely that the emitted radiation is re-absorbed by the dipole. That condition defines the strong-coupling regime . 10 ECE 240a Lasers - Fall 2019 Lecture 11
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