Lecture 8 Blackbody Radiation Einstein Coefficients and Homogeneous
Lecture #8: Blackbody Radiation, Einstein Coefficients, and Homogeneous Broadening Substitute Lecturer: Jason Readle Thurs, Sept 17 th, 2009 ECE 455: Optical Electronics
Topic #1: Blackbody Radiation ECE 455: Optical Electronics
What is a Blackbody? • Ideal blackbody: Perfect absorber – Appears black when cold! • Emits a temperature-dependent light spectrum ECE 455: Optical Electronics
Blackbody Energy Density • The photon energy density for a blackbody radiator in the ν → ν + dν spectral interval is ECE 455: Optical Electronics
Blackbody Intensity • The intensity emitted by a blackbody surface is (Units are or J/s-cm 2 or W/cm 2) ECE 455: Optical Electronics
Blackbody Peak Wavelength • The peak wavelength for emission by a blackbody is where 1 Å = 10– 8 cm ECE 455: Optical Electronics
Example – The Sun • Peak emission from the sun is near 570 nm and so it appears yellow – What is the temperature of this blackbody? – Calculate the emission intensity in a 10 nm region centered at 570 nm. Tk = 5260 K ECE 455: Optical Electronics
Example – The Sun • Also 570 nm → 17, 544 cm– 1 k. T (300 K) e. V ECE 455: Optical Electronics
Example – The Sun or Dn = 9. 23 · 1012 s– 1 = 9. 23 THz ECE 455: Optical Electronics
Example – The Sun ECE 455: Optical Electronics
Example – The Sun Since hν = 2. 18 e. V = 3. 49 · 10– 19 J → ρ(ν) d ν / hν = 1. 58 · 1010 ECE 455: Optical Electronics
Example – The Sun Remember, Intensity = Photon Density · c or = 4. 7 · 1020 photons-cm– 2 -s– 1 = 164 W-cm– 2 ECE 455: Optical Electronics
Example – The Sun ECE 455: Optical Electronics
Topic #2: Einstein Coefficients ECE 455: Optical Electronics
Absorption • Spontaneous event in which an atom or molecule absorbs a photon from an incident optical field • The asborption of the photon causes the atom or molecule to transition to an excited state ECE 455: Optical Electronics
Spontaneous Emission • Statistical process (random phase) – emission by an isolated atom or molecule • Emission into 4π steradians ECE 455: Optical Electronics
Stimulated Emission • Same phase as “stimulating” optical field • Same polarization • Same direction of propagation ECE 455: Optical Electronics
Putting it all together… • Assume that we have a two state system in equilibrium with a blackbody radiation field. ECE 455: Optical Electronics
Einstein Coefficients • For two energy levels 1 (lower) and 2 (upper) we have – A 21 (s-1), spontaneous emission coefficient – B 21 (sr·m 2·J-1·s-1), stimulated emission coefficient – B 12 (sr·m 2·J-1·s-1), absorption coefficient • Bij is the coefficient for stimulated emission or absorption between states i and j ECE 455: Optical Electronics
Two Level System In The Steady State… • The time rate of change of N 2 is given by: Remember, ρ(ν) has units of J-cm– 3 -Hz– 1 ECE 455: Optical Electronics
Solving for Relative State Populations • Solving for N 2/N 1: ECE 455: Optical Electronics
Solving for Relative State Populations But… we already know that, for a blackbody, ECE 455: Optical Electronics
Einstein Coefficients • In order for these two expressions for ρ(ν) to be equal, Einstein said: and ECE 455: Optical Electronics
Example – Blackbody Source • Suppose that we have an ensemble of atoms in State 2 (upper state). The lifetime of State 2 is • This ensemble is placed 10 cm from a spherical blackbody having a “color temperature” of 5000 K and having a diameter of 6 cm • What is the rate of stimulated emission? ECE 455: Optical Electronics
Example – Blackbody Source ECE 455: Optical Electronics
Example – Blackbody Source hν = 3. 2 e. V l = 387. 5 nm n = 7. 7 · 1014 s– 1 ECE 455: Optical Electronics
Example – Blackbody Source • Blackbody emission at the surface of the emitter is ECE 455: Optical Electronics
Example – Blackbody Source • Assuming dν = Δν = 100 MHz, 0(ν)dν = 3. 7 · 10– 5 J-cm– 2 -s– 1 7. 2 · 1013 photons-cm– 2 -s– 1 at 387. 5 nm • At the ensemble, the photon flux from the 5000 K blackbody is: = 6. 48 · 1012 photons-cm– 2 -s– 1 ECE 455: Optical Electronics
Example – Blackbody Source And or ρ(ν)dν = 3. 46 · 10– 17 J-cm– 3 ECE 455: Optical Electronics
Example – Blackbody Source • The stimulated emission coefficient B 21 is = 3. 5 · 1024 cm 3 -J– 1 -s– 2 ECE 455: Optical Electronics
Example – Blackbody Source • Finally, the stimulated emission rate is given by = – 3. 5 · 1024 cm 3 -J– 1 -s– 2 ECE 455: Optical Electronics
To reiterate… This is negligible compared to the spontaneous emission rate of A 21 = 106 s– 1 ! ECE 455: Optical Electronics
Example – Laser Source • Let us suppose that we have the same conditions as before, EXCEPT a laser photo-excites the two level system: Let Δνlaser = 108 s– 1 (100 MHz, as before). ECE 455: Optical Electronics
Example – Laser Source • If the power emitted by the laser is 1 W, then – Power flux, P = 127. 3 W-cm– 2 Since hν = 3. 2 e. V = 5. 1 · 10– 19 J → P = 2. 5 · 1020 photons-cm– 2 -s– 1 ECE 455: Optical Electronics
Example – Laser Source = 4. 24 · 10– 17 J-cm– 3 -Hz– 1 = 83. 3 photons-cm– 3 -Hz– 1 ECE 455: Optical Electronics
Example – Laser Source 3. 5 · 1024 cm 3 -J– 1 -s– 2 · 4. 24 · 10– 17 J-cm– 3 -s = 1. 48 · 108 s– 1 ECE 455: Optical Electronics
Example – Laser Source • Remember, in the case of the blackbody optical source: • What made the difference? ECE 455: Optical Electronics
Source Comparison Total power radiated by 5000 K blackbody with R = 0. 5 cm is 11. 1 k. W ECE 455: Optical Electronics
Key Points • Moral: Despite its lower power, the laser delivers considerably more power into the 1 → 2 atomic transition. • Point #2: To put the maximum intensity of the blackbody at 387. 5 nm requires T 7500 K! • Point #3: Effective use of a blackbody requires a process having a broad absorption width ECE 455: Optical Electronics
Ex. Photodissociation C 3 F 7 I + hν → I* ECE 455: Optical Electronics
Bandwidth • In the examples, bandwidth Δν is very important – Δν is the spectral interval over which the atom (or molecule) and the optical field interact. ECE 455: Optical Electronics
Topic #3: Homogeneous Line Broadening ECE 455: Optical Electronics
Semi-Classical Conclusion This diagram: suggests that the atom absorbs only (exactly) at ECE 455: Optical Electronics
The Shocking Truth! ECE 455: Optical Electronics
Line Broadening • The fact that atoms absorb over a spectral range is due to Line Broadening • We introduce the “lineshape” or “lineshape function” g(ν) ECE 455: Optical Electronics
Lineshape Function • g(ν) dν is the probability that the atom will emit (or absorb) a photon in the ν → ν + dν frequency interval. • g(ν) is a probability distribution and Δν / ν 0 << 1 ECE 455: Optical Electronics
Types of Line Broadening • There are two general classification of line broadening: – Homogenous — all atoms behave the same way (i. e. , each effectively has the same g(ν). – Inhomogeneous — each atom or molecule has a different g(ν) due to its environment. ECE 455: Optical Electronics
Homogeneous Broadening • In the homogenous case, we observe a Lorentzian Lineshape where ν 0 ≡ line center ECE 455: Optical Electronics
Homogeneous Broadening Δν = FWHM Bottom line: Homogeneous → Lorentzian ECE 455: Optical Electronics
Sources of Homogeneous Broadening • Natural Broadening — any state with a finite lifetime τ sp (τsp ≠ ∞) must have a spread in energy: • Collisional Broadening — phase randomizing collisions ECE 455: Optical Electronics
Natural Broadening • ΔE Δt ≥ Heisenberg’s Uncertainty Principle ECE 455: Optical Electronics
Natural Broadening • In the case of an atomic system: ECE 455: Optical Electronics
Natural Broadening • In general ECE 455: Optical Electronics
Example: Sodium (Na) (Both arrows indicate “resonance” transitions) ECE 455: Optical Electronics
Example: Sodium (Na) • Radiative lifetime of the 3 p 2 P 3/2 state is 16 ns = 9. 9 · 106 s– 1 ≈ 10 MHz ν 0 = 5. 1 · 1014 Hz ECE 455: Optical Electronics ~ 2 · 10– 8!
Example: Mercury (Hg) ECE 455: Optical Electronics
Example: Mercury (Hg) • Remember: In general, ECE 455: Optical Electronics
Collisional Broadening • An atom that radiates a photon can be described as a classical oscillator with a particular phase ECE 455: Optical Electronics
Collisional Broadening • Suppose now that we have collisions between atom A (the radiator) and a second atom, B… ECE 455: Optical Electronics
Collisional Broadening • Such collisions alter the phase of the oscillator. (Arrows indicate points at which oscillator suffers collision) ECE 455: Optical Electronics
Collisional Broadening • Result? Broadening of Transition! • The rate of phase randomizing collisions is: where: k. C (cm 3 – s– 1) is known as the rate constant of collisional quenching (deactivation of the excited atom) NC (cm-3) is the number density of colliding atoms ECE 455: Optical Electronics
Collisional Broadening ECE 455: Optical Electronics
Total Homogenous Broadening • Is calculated by summing the rates of the various homogeneous broadening processes: ECE 455: Optical Electronics
Example – Kr. F Laser • Kr. F laser (λ = 248. 4 nm) • τsp = 5 ns • k. C = 2 · 10– 10 cm 3 -s– 1 • 1 atmosphere ≡ 2. 45 · 1019 cm– 3 ECE 455: Optical Electronics
Example – Kr. F Laser Δνtotal = 31. 8 MHz + ECE 455: Optical Electronics · P(atm)
Example – Kr. F Laser Δνtotal = 31. 9 MHz + 1. 6 GHz · P spontaneous collisions Note that these terms are equal for P = 0. 02 atm! ECE 455: Optical Electronics
Next Time • Inhomogeneous broadening • Threshold gain ECE 455: Optical Electronics
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