Lecture 11- ECE 240a neous Emission from a (See Notes on - - PowerPoint PPT Presentation

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Lecture 11- ECE 240a

(See Notes on Spontaneous Emission)

ECE 240a Lasers - Fall 2019 Lecture 11 1

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Free Space Mode Density

The electromagnetic mode density in free space is required for several aspects

• f lasers.

The mode density dN/dν per unit frequency in a volume V for both polarizations is given by a modified form of (2.1.40) of M&E dN dν . = ρν = 8πν2 c3 V , where ν0 is the operating frequency. Using ν = ω/2π and dν = dω/2π, the mode density in angular frequency is dN dω . = ρω = ω2 π2c3 V , where c = c0/n is defined in the medium. Using λν = c, we can also write this in wavelength units as dN dλ = ρλ = 8π

V

λ3

1

λ, where λ is defined in the medium.

ECE 240a Lasers - Fall 2019 Lecture 11 2

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Mode Density for a Cavity

This cavity is characterized by a photon lifetime τp. Also characterized by the finesse F. (See Lecture 10.) The cavity can also be characterized by a quality factor Q defined in the same way as that of a filter so that Q = ν0 ∆ν = ω0 ∆ω . The bandwidth ∆ν of a single-mode cavity is the approximate frequency width

• f one mode.

The reciprocal 1/∆ν of the bandwidth is an estimate of the number of modes per unit frequency, which is the mode densityρcav for the cavity ρcav = 1 ∆ω = Q ω0 , where we work with angular frequency units, which are more common.

ECE 240a Lasers - Fall 2019 Lecture 11 3

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Simple Estimate of Effect of the Cavity - Purcell Effect

Most derivations of the effect of cavity use a single polarization. Then 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 ρcav ρω = Q/ω0 2V ω2

0/π2c3 =

1 4π Q ×

V

λ3

• .

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.

ECE 240a Lasers - Fall 2019 Lecture 11 4

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

“Classical” Spontaneous Emission from a Dipole

Now place “back-the-envelope” results on a firm foundation by determining the effect of a concentric resonator on the radiation emitted by a dipole. 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 p(t) = e sin(ω0t) x where e is the electron charge and x is a unit vector in the direction of the oscillation of the charge. The transverse part of the electric field, called the radiation field Erad(r, t), at a position r from a point dipole located at a position R can be written as1 Erad(r, t) = − 1 4πǫ ω2 c2

• (

nr × x) × nr

e−jk|r−R|

|r − R| , where nr is a unit vector in the direction of r − R, c = c0/n is the speed of light in the medium, ǫ = ǫ0ǫr 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.

1See Section 2.5 of M and E first edition for a complete derivation ECE 240a Lasers - Fall 2019 Lecture 11 5

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Power into a Solid Angle - 1

If the dipole is linearly polarized, then the radiated power per solid angle is given by (See the second to last line of (2.5.13) in M&E) dP dΩ = 1 4πǫ 1 4π 1 c3

• d2p(t)

dt2

2

sin2 θ, where θ is the angle between x and r. If p(t) = Re[p0ejω0t], then d2p(t) dt2 = −ω2

0Re[p0ejω0t]

Solve for

• d2p(t)

dt2

2

• d2p(t)

dt2

2

= ω4

0|p0|2 cos2(ω0t + arg p0).

ECE 240a Lasers - Fall 2019 Lecture 11 6

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Power into a Solid Angle - 2

Averaging over a time interval that is long as compared to the period T = 2π/ω0, we have cos2(ω0t + arg p0) = 1/2. Therefore, for a lossless time-harmonic dipole we have dP dΩ = 1 4πǫ |p0|2 8π ω4 c3 sin2 θ, where |p0| = ex is the magnitude of the dipole moment. This is the power per solid angle in one polarization of a dipole.

ECE 240a Lasers - Fall 2019 Lecture 11 7

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Total Time-Averaged Radiated Power

The time average total power radiated by the dipole in one polarization is determined by integrating over a solid angle of 4π steradians and is (See (see (2.5.13) in M&E) P = 1 4πǫ |p0|2 8π ω4 c3

2π

dφ

π

sin3 θdθ. = 1 4πǫ ω4 3c3 |p0|2 Substituting |p0|2ω4

0/c3 = (4πǫ)3P into the previous equation, we have

dP dΩ = 3P 8π sin2 θ. This expression relates the total time-averaged radiated power P to the power per unit solid angle.

ECE 240a Lasers - Fall 2019 Lecture 11 8

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

“Classical” Spontaneous Emission Lifetime

Start with classical model of a free (non-driven) harmonic oscillator given in Lecture 1, which can be written in terms of the dipole moment p = ex d2p dt2 + σ dp dt + ω2

• p = 0

In this form, we interpret τsp = 1/σ. The most general solution for this is in the form of a damped oscillation. The characteristic equation is (cf. Lecture 1) x2 + σx + ω2

0 = 0

The roots can be written as xi = −σ/2 ± jω0

• 1 − (σ

2/4ω2

0).

ECE 240a Lasers - Fall 2019 Lecture 11 9

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

General Solution

Choosing the positive root to agree with our convention for the time dependence, ejω0t, the general solution is p(t) = Re

• p0e−σt/2ejω0
• 1−(σ2 /4ω2

0)t

• = |p0| cos (ω0Xt + arg p0) ,

where p0 is the complex amplitude of the harmonic dipole moment and X =

• 1 − (σ

2/4ω2

0).

The time-averaged radiated power is modified by including a power loss term e−σt P(t) = 1 4πǫ ω4 3c3 |p0|2e−σt Ifσ ≫ ω0, then we can define a time-average energy over a single cycle of the

• scillation.

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.

ECE 240a Lasers - Fall 2019 Lecture 11 10

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Energy in a Damped Harmonic Oscillator

Using p = ex, we can write the kinetic energy KE = 1

2mv(t)2 = m

2e2

• dp(t)

dt

2

Similarly, we have for the potential energy PE = 1

2kx(t)2 = m

2e2 ω2

0p(t)2

where k = mω2

0 has been used. Therefore, the total average energy is

E = m 2e2

• ω2

0p(t)2 +

• dp(t)

dt

2

. Substituting p(t) into this equation and averaging so that cos2(ω0t) = 1/2, the KE and PE each have an equal contribution, which can be written as E(t) = mω2 2e2 |p0|2e−σt.

ECE 240a Lasers - Fall 2019 Lecture 11 11

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Equate the Energy with the Power

Now equate the time-averaged energy loss to the integral of the time-averaged power loss so that E(t) = −

• P(t)dt

mω2 2e2 |p0|2e−σt = 1 σ 1 4πǫ ω4 3c3 |p0|2e−σt Solving for 1/σ = tsp , we can write the “classical” spontaneous emission lifetime as 1 τsp = σ = A = 1 4πǫ 2e2ω2 3mc3 . Putting in values yields a lifetime of about 10 ns for an oscillating electron at a wavelength of one half a micron. The constant 2e2/3mc3 is a characteristic time that is on the order of the time it takes light to travel an electron radius r0 = e2/mc2.

ECE 240a Lasers - Fall 2019 Lecture 11 12

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Quantum Treatment of Spontaneous Emission

The spontaneous emission rate Anm between to energy levels n and m is presented in E&M Section 7.6 and is given by Anm = 1 4πǫ 4 3 D2

nmω3

¯ hc3 where Dnm is the squared-magnitude of the dipole moment so that Dnm = e2 |rnm|2 rnm 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 1

2kx2 = 1 2mω2 0x2 is set equal to

• ne half the vacuum state energy or 1

2(¯

hω/2). Making this substitution, we have for the classical result 1 τsp = σ = A = 1 4πǫ 4e2x2ω3 3¯ hc3 .

ECE 240a Lasers - Fall 2019 Lecture 11 13

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Validity of Classical Treatment Quantum Treatment of Spontaneous Emission

Given the equivalence between the classical description and the quantum description of spontanteous emission, the effect of the cavity on the spontaneous emission rate based on a classical analysis. We also adopt this approach. However, 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.

ECE 240a Lasers - Fall 2019 Lecture 11 14

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Dipole Radiation in a Resonator

We now consider how the classical dipole emission rate is affected by placing a dipole in a concentric resonator with equal radii of curvature placed at ±a.

rd Dipole Image L=2a Ω side Ω m

2b

The origin is defined at the focal point of the mirror and the center of the dipole is displaced from the origin by rd = xd x + zd

• z. This displacement

results in an image that is offset from the origin by −rd as is shown in the Figure.

ECE 240a Lasers - Fall 2019 Lecture 11 15

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Solid Angles

The solid angle Ωside out the sides of the resonator and twice the solid angle Ωmir that a single mirror subtends is equal to 4π or Ωside + 2Ωmir = 4π. Suppose that Ωmir is small. Then using the geometry given in the Figure Ωmir ≈ 4πb2/L2, where the parameters are defined in Figure. If the resonator is stable, then the power incident on a mirror that subtends a solid angle Ωmir and the power in the solid angle Ωside is equal to the total radiated power. Therefore, at an angle θ = π/2 the power Pmir emitted into the total solid angle 2Ωmir subtended by the mirrors can be expressed in terms of the total time-averaged power emitted from the dipole Pmir =

dP

dΩ

• θ=π/2

2Ωmir = 3P 4π Ωmir.

ECE 240a Lasers - Fall 2019 Lecture 11 16

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Total Radiated Power Ignoring the Cavity

Ignoring the effect of the cavity, the power that is emitted out the side of the cavity is then Pside = P − Pmir = P

• 1 − 3

4π Ωmir

• Note that Pmir is only the effect of the solid angle subtended by the cavity

mirrors and does not include the effect of the cavity.

ECE 240a Lasers - Fall 2019 Lecture 11 17

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Effect of the Cavity-1

The power Pcav including the effect of the cavity has two features. At resonance and for a highly reflective cavity such that R ≈ 1, the power inside the cavity is related to the power outside the cavity Pmir emitted through a single mirror by Pmir = (T /2)Pcav where T = 1 − R is the power transmission for one mirror. We then have Pcav Pmir ≈ 2 T = 2 1 − R ≈ √ F where F = 4R (1 − R)2 and the finesse of the cavity is F = π 2 √ F ≈ π √ R 1 − R (See Lecture 10.)

ECE 240a Lasers - Fall 2019 Lecture 11 18

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Effect of the Cavity-2

The second feature is the frequency dependence of the cavity, which is given by the Airy function for that cavity (see Lecture 10). Therefore, we can write the ratio Pcav/Pmir as2 Pcav Pmir ≈ √ F

• Power

enhancement by cavity 1 1 + F sin2(kL)

• Frequency dependence

(Airy function) . (1) The first term in (1) 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 (1) is the frequency dependence as expressed by the Airy function.

2(See Ref. 3 for a more detailed analysis.) ECE 240a Lasers - Fall 2019 Lecture 11 19

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Maximum Enhancement

The maximum enhancement of the power caused by the effect of the cavity

• ccurs if sin2(kL) = 0 with

Pcav

Pmir

• max

≈ √ F = 2 1 − R ∝ Q, where the relationship between the finesse and Q is discussed in Lecture 10 and Verdeyen Section 6.3.

ECE 240a Lasers - Fall 2019 Lecture 11 20

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Cavity Q and Finesse (repeat from Lecture 10)

The Q of a cavity is defined as Q = ν0 ∆ν = ω0 ∆ω This value differs from the definition of the finesse by a factor of Q F = ν0/∆ν νfsr/∆ν = ν0 νfsr = ν0 (c/2d) = 2d λ ≈ 104 − 106 for reasonable sized resonators. Using the definition of the finesse F ≈ π/(1 − R) for R ≈ 1, we have that Q ∝ 1 1 − R with the scaling factor being different than the scaling factor for the finesse.

ECE 240a Lasers - Fall 2019 Lecture 11 21

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Minimum Enhancement (Enhibited)

The minimum value occurs if sin2(kL) = 1 with

Pcav

Pmir

• min

≈ 1 √ F = 1 − R 2 ∝ Q−1. We see that relative to the radiation from a dipole in free space that subtends a solid angle Ωmir, the radiation into a cavity mode is enhanced or inhibited by a factor that is proportional to the Q of the cavity.

ECE 240a Lasers - Fall 2019 Lecture 11 22

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Total Power Into the Cavity

Using Pmir = 3P 4π Ωmir and Pcav Pmir ≈ √ F 1 + F sin2(kL), we can we can write Pcav ≈ 3P 4π

• √

F 1 + F sin2(kL)

• Ωmir

We see that the total radiation rate into the cavity depends on both the solid angle Ωmir and the finesse (or the Q) of the cavity.

ECE 240a Lasers - Fall 2019 Lecture 11 23

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Max/Min Total Power

The maximum total power in the cavity occurs if R ≈ 1. so that Pmax = P

• 1

1 − R

3

2π Ωmir. The minimum total power into the cavity occurs Pmax = 3P(1 − R) 8π Ωmir. Note that the minimum value is less than the rate if there was no cavity, so that Pcav = Pmir so that Pmin = Pmir = 3P 4π Ωmir.

ECE 240a Lasers - Fall 2019 Lecture 11 24

Lecture 11- ECE 240a Mode Density

Mode Density for a Cavity Purcell Effect

“Classical” Sponta- neous Emission from a Dipole

“Classical” Spontaneous Emission Lifetime Quantum Treatment of Spontaneous Emission

Dipole Radiation in a Concentric Resonator

Validity of Analysis

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

• rder of the wavelength λ because they do not include all of the relevant

physics.

ECE 240a Lasers - Fall 2019 Lecture 11 25