[PPT] - Lecture 5- ECE 240a Density A and B coefficients Ver Chap. PowerPoint Presentation

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Lecture 5- ECE 240a

Ver Chap. 7&8

ECE 240a Lasers - Fall 2019 Lecture 5 1

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

E-M Field Matter Interactions

hν2 hν1 hν1 hν1

An incident photon is absorbed. An incident photon stimulates the creation of an identical, second

photon. This is stimulated emission.

Fundamental quantum-level amplification process. Must have N2 is greater than N1. Need external energy source that is not in thermal equilibrium with the system - pump.

A photon is emitted spontaneously. This is spontaneous emission.

No classical counterpart. Emitted photons are regarded as noise. Noise can be amplified by subsequent stimulated emission - amplified spontaneous emission (ASE).

ECE 240a Lasers - Fall 2019 Lecture 5 2

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Thermal Equilibrium

Probability distribution, p(Em), that an atom is in state Em is the Boltzmann distribution p(Em) ∝ exp

− Em

kBT

where kB is the Boltzmann constant. (k = 1.38 × 10−23 J/K) and T is

the temp. in Kelvin. The ratio N2/N1 then N2 N1 = g2 g1 exp

−E2 − E1

kBT

,

(1) where g is the degeneracy factor which is number of states at E2 and E1. Probability that the energy state En is occupied is p(n) = (1 − exp[−hf/kT0]) exp

−n

hf

kT0

ECE 240a Lasers - Fall 2019 Lecture 5

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Energy per E-M Mode

Energy in a mode is the energy in a quantized harmonic oscillator discussed in Lecture 2 and Problem Set 2 E(n) = hνn where zero-point energy is not included. Mean energy E is then E =

∞

n=0

E(n)p(n) = hν (1 − exp[−hf/kT0])

∞

n=0

n exp

−n

hf

kT0

=

hf exp[hf/(kT0)] − 1

ECE 240a Lasers - Fall 2019 Lecture 5 4

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Mode Density and Energy Density

The number of modes per frequency in 3-D is derived in Section 7.2 of Verdeyen p(ν)dν = 8πn3 c3 ν2dν where c is speed of light in vacuum and n is index of refraction. Spectral energy density ρ(ν) is mode density × number photons/mode × energy per photon ρ(ν) = 8πν2 c3 # modes/frequency × 1 ehν/kBT − 1

# photons/mode

× hν

energy per photon

= energy/frequency

ECE 240a Lasers - Fall 2019 Lecture 5 5

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Einstein A and B coefficients

Now we use Verdeyen notation Write the rate for both stimulated emission and absorption as R21 = σ(ν)Iν hν = B21ρ(ν) where ρ(ν) is the energy density and B21 is Einstein’s B coefficient. In thermal equilibrium for a closed system no external influence or pump so that Γ1 = Γ2 = 0 dN2 dt = −dN1 dt Then dN2 dt = R2(t) − N2 τ2 − σ(ν)Iν hν (N2 − N1) becomes dN2 dt = − A21N2

spon. emission

+ B21N1ρ(ν)

absorption

− B21N2ρ(ν)

stim. emission

= −dN1 dt

ECE 240a Lasers - Fall 2019 Lecture 5 6

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Relationship Between A21 and B21

Find ratio N2/N1in steady-state (equilibrium) N2 N1 = B21ρ(ν) A21 + B21ρ(ν) Equate with Boltzmann distribution N2 N1 = g2 g1 exp

− hν

kBT

=

B21ρ(ν) A21 + B21ρ(ν) Solve for ρ(ν) ρ(ν) = A21e−hν/kBT B21

B12g1

B21g2

(1 − e−hν/kBT )

ECE 240a Lasers - Fall 2019 Lecture 5 7

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Relationship -Cont.

Let g1/g2 = 1 and B12 = B21so that B12g1 B21g2 = 1 Now divide through by e−hν/kBT ρ(ν) = A21 B21 1 (ehν/kBT − 1) Substitute the expression for the energy density ρ(ν) = 8πν2 c3 × 1 ehν/kBT − 1 × hν Equate expressions and solve 8πν2 c3 × 1 ehν/kBT − 1 × hν = A21 B21 1 (ehν/kBT − 1)

r

A21 B21 = 8πhν3n3 c3 = 8πhn3 λ3

ECE 240a Lasers - Fall 2019 Lecture 5 8

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Cross-Section and Lineshape function

Assume for rest of class that g1 = g2 and ng = n (group index is same as index) Look at frequency dependence of stimulated emission (absorption) rate R21 = σ(ν)Iν hν = B21ρ(ν) where ρ(ν) = Iν vg = Iνng c ≈ Iνn c energy volume = intensity velocity = power/area length/time = energy/time/area length/time = energy volume Now use B21 = A21 λ3 8πh where A21 = 1/τ21 is decay time from upper to lower state. Equate and solve for cross-section σ(ν) = hν Iν

A

λ3 8πhn3

Iνn

c = A21 λ2 8πn2

Cross-section

× g(ν)

lineshape function

ECE 240a Lasers - Fall 2019 Lecture 5 9

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Atomic Lineshape Function

Now write rate in terms of density matrix formulation R21 = p2E2 6¯ h2 β β2 + (ω21 − ω)2 = πp2E2 6¯ h2

β/π

β2 + (ν21 − ν)2

g(ν)

= A21 λ2 8πn2

cross-section

Iν hν g(ν) Thus the Einstein A coefficient A21, which governs the rate of spontaneous emission can be derived (at least in principle) from quantum dipole moments p, which depend on the wavefunctions of the material the field interacts with Expression in M&E

ECE 240a Lasers - Fall 2019 Lecture 5 10

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Broadening

Gain sites can interact in same manner with different frequencies or each site may have slightly different gain characteristics. Gain sites that have the same characteristics are called homogeneous When the gain characteristics of each site are different are called inhomogeneous All materials exhibit both types of gain characteristics on different time scales. Examples of materials that are mostly homogeneously broadened

Semiconductors such as GaAs; isolated atoms or ions in crystalline hosts.

Materials that are mostly inhomogeneously broadened

Gases (Doppler) Atoms or ions in glass or other noncrystalline materials

ECE 240a Lasers - Fall 2019 Lecture 5 11

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Atoms in Crystal vs. atom in glass

Example: Atom (ion) glass:

Each Er ion experiences a different local field because of the inhomogeneous nature of the glass. Within the subset of Er ions that experience the same local field, the gain is homogeneously broadened. However there is an overall distribution of local fields and the total gain profile is inhomogeneously broadened producing a large gain bandwidth and wavelength dependence.

Atom in Crystal

Each atom experiences regular long range order.

ECE 240a Lasers - Fall 2019 Lecture 5 12

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Picture

Crystal (regular long-range order) Glass (short-range order) (no long range order)

ECE 240a Lasers - Fall 2019 Lecture 5 13

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Homogeneous Lifetime Broadening

Now determine peak cross-section for a lineshape function given as g(ν) = β/π β2 + (ν0 − ν)2 = ∆ν 2π

(ν0 − ν)2 − ∆ν

2

2 Now β = 1/τ where τ is total lifetime for both states (not de-phasing time of a single state) β = 1 τ1 + 1 τ2 Then ∆ν = β 2π = 1 2π

1 τ1 + 1 τ2

=

1 2π

1 τ1 + 1 τ2

=

1 2π (A1 + A2) Very fast and not usually an issue - Sets fundamental limit on how narrow a transition can be used. (NIST tables.) Pressure (collisional broadening) has the same functional form of where β = 1/T2 is the de-phasing time using in the density matrix equations. (This is Eq. 7.6.12)

ECE 240a Lasers - Fall 2019 Lecture 5 14

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Doppler Broadening (Inhomogeneous gas)

Gas (atoms or molecules) has a distribution of velocities. Distribution function is Gaussian for each component p(vz) = 1 √ 2πσ exp

− v2

z

2σ2

where the variance σ2 = kT /M where M is mass, k is Boltzmann

constant and T is temperature. Called pure Doppler broadening - due only to temperature distribution Total lineshape function is convolution of pressure-broaden lineshape (Lorentzian) and Doppler line shape (Gaussian) where center frequency of transition is modified to ν0 → ν0(1 + vz/c) Lineshape is then Voigt function. At high pressure (near standard temperature and pressure) lines are mostly pressure broadened At low pressure lines are mostly Doppler broadened

ECE 240a Lasers - Fall 2019 Lecture 5 15

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Variance and Full Width Half Maximum (FWHM)

If Full Width at Half Maximum ∆νD is used instead of variance σ2 then we can relate the two via σ2 = ∆ν2

D

8 ln 2 so that g(ν) =

4 ln 2

π

1/2

1 ∆νD exp

−4 ln 2(ν − ν0)2

∆ν2

D

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Lecture 5- ECE 240a Back to Basics Einstein Coefficients

Thermal Equilibrium Mode and Energy Density A and B coefficients

Cross Section and Lineshape Broadening

Homogeneous Broadening Inhomogeneous Broadening

Doppler Broadening in Atomic Sodium

3 2 1 1 2 3 2 4 6 8 10

150 K 200 K 250 K Frequency (GHz) Absorption Cross Section (x 1016 m2)

Values for graph from the NIST database.

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