1. Blackbodies and Lasers*
Prof. Rick Trebino, Georgia Tech
Boltzmann’s Law
Stimulated Emission
Gain
Inversion
The Laser
Four-level System
Threshold
Some lasers
Image from:
* Light Amplification by Stimulated Emission of Radiation

2. Calculations in Physics: Semi-Classical physics
The least precise calculations are performed completely classically and using Newton’s Laws.
The most precise computations are performed fully quantum- mechanically, using the precise potential, solving Schrodinger’s Equation for it, and including superpositions of states, when appropriate. But they are usually very difficult—or impossible.
Intermediate cases involve semi-classical computations, in which an atom’s energy levels are computed quantum-mechanically, but superpositions are neglected. Light waves may be treated classically or as photons.
We’ll now do two semi-classical calculations, one of the blackbody radiation spectrum and another of laser threshold, in which we’ll treat molecules as energy levels and light as photons.

3. Another Big Simplification: The Two-Level System
All atoms and molecules have infinitely many states and their corresponding energy levels.
That’s also too complicated.
So, if a photon whose energy matches a particular transition energy is present, physicists usually just consider two states and their corresponding energy levels.
Excited level
Energy
This doesn’t always work.
Ground level

4. Spontaneous emission
When an atom in an excited state falls to a lower energy level, it emits a photon of light.
Excited level
Energy
Ground level
Molecules typically remain excited for no longer than a few nanoseconds. This is often also called fluorescence or, when it takes longer (because the transition is “forbidden”), phosphorescence.

5. Absorption
When an atom encounters a photon of light, it can absorb the photon’s energy and jump to an excited state.
Excited level
This is, of course, absorption.
Energy
Ground level
Image from
Absorption lines in an otherwise continuous light spectrum due to a cold atomic gas in front of a hot broadband source.

6. Einstein showed that another process, stimulated emission, can also occur.
When a photon encounters an atom in an excited state, the photon can induce the atom to emit its energy as another photon of light, resulting in two identical photons.
Excited level
Energy
Ground level
Einstein first proposed stimulated emission in 1916.

7. Population density (Number of molecules per unit volume)
In what energy levels do molecules reside? Boltzmann Population Factors
Ni is the number density (also known as the population density) of molecules in state i (i.e., the number of molecules per cm3).
T is the temperature, and kB is Boltzmann’s constant = ×  J/K E3 N3 E2 N2
Energy N1 E1
Population density (Number of molecules per unit volume)

8. The Maxwell-Boltzman Distribution
In the absence of collisions,
molecules tend to remain
in the lowest energy state
available.
Collisions can knock a mole-
cule into a higher-energy state.
The higher the temperature,
the more this happens.
Low T
High T 3 3
Energy
Energy 2 2 1 1
Molecules
Molecules
In equilibrium, the ratio of the populations of two states is:
N2 / N1 = exp(-DE/kBT), where DE = E2 – E1 = hn
As a result, higher-energy states are always less populated than the ground state, and absorption is stronger than stimulated emission.

9. Blackbody Radiation
Blackbody radiation is emitted from a hot body. It's anything but black!
The name comes from the assumption that the body absorbs at every frequency and hence would look black at low temperature.
It results from a combination of spontaneous emission, stimulated emission, and absorption occurring in a medium at a given temperature.
It assumes that the box is filled with many different molecules that, together, have transitions (absorptions) at every wavelength.

10. Einstein A and B Coefficients
In 1916, Einstein considered the various transition rates between molecular states (say, 1 and 2) involving light of intensity, I:
Spontaneous emission rate = A N2
Absorption rate = B12 N1 I
Stimulated emission rate = B21 N2 I
In equilibrium, the rate of upward transitions equals the rate of downward transitions:
at frequency n
A N2 + B21 N2 I = Down = Up = B12 N1 I
Recalling the Maxwell-
Boltzmann Distribution
Dividing by N1 (A + B21I ) yields N2/N1:
(B12 I ) / (A + B21 I ) = N2 / N1 = exp(–DE/kBT)

11. Blackbody Radiation
Now solve for the intensity in: (B12 I) / (A + B21 I) = exp(-DE/kBT)
Multiply by (A + B21 I) exp(DE/kBT): B12 I exp(DE/kBT) = A + B21 I
Solve for I: I = A / [B12 exp(DE/kBT) – B21]
Dividing numerator and denominator by B21:
I = (A/B21) / [(B12/B21) exp(DE/kBT) – 1]
Now, when T ® ¥, I should also. As T ® ¥, exp(DE/kBT) ® 1.
So: (B12/B21) – 1 = 0
And: B12 = B21 º B ¬ Coeff up = coeff down!
And: I = (A/B) / [exp(DE/kBT) – 1]

12. Blackbody Radiation I = (A/B) / [exp(DE/kBT ) – 1]
We can eliminate A/B based on other information and, writing in terms of the intensity per unit frequency, In :
also using DE = hn
This is the total intensity per unit frequency (that is, in a range from n to n + dn ) emitted by an arbitrary blackbody.
We considered only two levels, but our approach was general and so applies to any two levels and hence to the entire spectrum.
Notice that it’s independent of the size of the object. To obtain the emitted power, multiply by the surface area of the blackbody.

13. Blackbody Emission Spectrum
The higher the temperature, the more the emission and the shorter the average wavelength.
Blue hot is hotter
than red hot.
The sun’s surface is 6000 degrees K, so its blackbody spectrum peaks at ~ 500 nm—in the green. However, blackbody spectra are broad, so it contains red, yellow, and blue, too, and so looks white.

14. We can tell how hot a star is by its blackbody emission spectrum.

15. The earth is a blackbody, too.

16. Cosmic Microwave Background
The 3° cosmic microwave background is blackbody radiation left over from the early universe!
Peak frequency is ~ 150 GHz
Interestingly, blackbody radiation retains a blackbody spectrum despite the expansion the universe.
It does get colder, however.
Wave-number (cm-1)

17. Emissivity
Our analysis assumed a perfectly black object. However, the object may not have transitions at all wavelengths.
The emissivity takes this into account. It is the ratio of the actual emitted spectrum and the theoretical blackbody spectrum. It depends on the medium and its density and is a measure of just how black the body is.

18. Stimulated emission leads to a chain reaction and laser emission.
If a medium has many excited molecules, one photon can become many.
Excited medium
This is the essence of the laser. The factor by which an input beam is amplified by a medium is called the gain and is represented by G.

19. Laser medium with gain, G
The Laser
A laser is a medium that stores energy, surrounded by two mirrors.
A partially reflecting output mirror lets some light out.
R = 100%
R < 100% I0 I1 I2 I3
Output mirror
Back mirror
Laser medium with gain, G
I1 = G I0
I3 = G I2
A laser will lase if the beam increases in intensity during a round trip:
that is, if
Usually, additional losses in intensity occur, such as absorption, scat-tering, and reflections. In general, the laser will lase if, in a round trip:
Gain > Loss This called achieving Threshold.

20. Inversion Inversion N2 > N1
In order to achieve G > 1, stimulated emission must exceed absorption:
B N2 I > B N1 I
Or, equivalently,
This condition is called inversion.
It does not occur naturally (it’s forbidden by the Boltzmann distribution). It’s inherently a non-equilibrium state.
In order to achieve inversion, we must hit the laser medium very hard in some way and choose our medium correctly.
Energy
Inversion
Molecules
“Negative temperature”
N2 > N1

21. Achieving Inversion: Pumping the Laser Medium
Now let I be the intensity of (flash lamp) light used to pump energy into the laser medium:
Back mirror I
Output mirror
Laser medium
Will this intensity be sufficient to achieve inversion, N2 > N1?
It’ll depend on the laser medium’s energy level system.

22. Rate Equations for a Two-Level System2 1 N2 N1
Laser
Pump
Rate equations for the densities of the two states:
Stimulated emission
Spontaneous emission
Absorption
If the total number of molecules is N:
Pump intensity

23. Why Inversion is Impossible in a Two-Level System2 1 N2 N1
Laser
In steady-state:
where:
Isat is the saturation intensity.
DN is always positive, no matter how high I is!
It’s impossible to achieve an inversion in a two-level system!

24. Rate Equations for a Three-Level System
Fast decay
Laser Transition
Pump Transition 1 2 3
Assume we pump to a state 3 that rapidly decays to level 2. No pump stimulated emission!
Spontaneous emission
The total number of molecules is N:
Level 3 decays fast and so is zero.
Absorption

25. Why Inversion is Possible in a Three-Level System
Fast decay
Laser Transition
Pump Transition 1 2 3
In steady-state:
Now if I > Isat, DN is negative!

26. Rate Equations for a Four-Level System
Laser Transition
Pump Transition
Fast decay 1 2 3
Now assume the lower laser level 1 also rapidly decays to a ground level 0. So ! And
As before:
The total number of molecules is N :
Because
At steady state:

27. Why Inversion is Easy in a Four-Level System (cont’d)
Laser Transition
Pump Transition
Fast decay 1 2 3
Why Inversion is Easy in a Four-Level System (cont’d)
Most laser materials are four-level systems.
Now, DN is negative—always!

28. What about the saturation intensity?
Laser Transition
Pump Transition
Fast decay 1 2 3
What about the saturation intensity?
A is the excited-state relaxation rate: 1/t
B is the absorption cross-section, s, divided by the energy per photon, ħw: s / ħw
ħw ~10-19 J for visible/near IR light
Both s and t depend on the molecule, the frequency, and the various states involved.
t ~ to 10-8 s for most molecules to 10-3 s for laser molecules
s ~ to cm2 for molecules (on resonance)
1 to 1013 W/cm2
The saturation intensity plays a key role in lasers.
For example, when the intensity approaches (or exceeds) it, the medium becomes less absorbing (N1 decreases).

29. The Ruby Laser
Invented in 1960 by Ted Maiman at Hughes Research Labs, it was the first laser.
Ruby is a three-level system, so you have to hit it hard.

30. A Common Laser Medium’s Energy Levels
The lower laser level can be almost any level in the S0 manifold.
S1: 1st excited
electronic state manifold
Metastable state
Pump Transition
Laser Transitions
S0: Ground
electronic state manifold
Dyes, for example, are such ideal four-level systems that it’s often difficult to stop them from lasing in all directions!

31. Dye Lasers
Duarte and Piper, Appl. Opt. 23,
Dye lasers are an ideal four-level system, and a given dye will lase over a range of ~100 nm.

32. Dyes cover the visible, near-IR, and near-UV ranges.

33. The Helium-Neon Laser
Energetic electrons in a glow discharge collide with and excite He atoms, which then collide with and transfer the excitation to Ne atoms, an ideal 4-level system.

34. The Carbon Dioxide Laser
The CO2 laser operates analogously. N2 is pumped, transferring the energy to CO2.
Vibrational energy level diagram depicting the 10.6 micron infrared transition in the carbon dioxide molecule. (The nitrogen vibrational levels shown on the right are used to enhance lasing in laboratory lasers)
Image from

35. CO2 Laser in the Martian Atmosphere
Detuning from line center (MHz)
The atmosphere is thin and the sun is dim, but the gain per molecule is high, and the pathlength is long.
The small red circle centered on Chryse Planitia represents the region over which the laser emissions were detected. Solar radiation is responsible for pumping a population inversion in the carbon dioxide of the tenuous upper levels of the atmosphere of Mars (Mumma et al., 1981) and Venus (Deming et al., 1983). Population inversions have also been found in comets (Mumma, 1993). On mars, the solar pump intensity is strongest near the solar point, and falls off gradually towards the terminator. There is some locus where the inversion vanishes, but it is difficult to say exactly where that is. The R(8) transition at microns in the infrared is produced from one vibrational quanta of asymmetric stretching to one quantum of symmetric stretching with a change of one quantum of rotational energy from J=8 to J=7. (mars image courtesy : Philip James, University of Toledo; Steven Lee, University of Colorado; and NASA Hubble Space Telescope)
Due to the low densities of the lasing species in the mesosphere and thermosphere of Mars the gain is very low, about 10 percent, comparable to single-pass gains in some earth based CO2 lasers. The low gain is partly compensated by the extremely large volumes of active lasing medium. Over the very long distances scales, the exponential properties of amplified spontaneous emission produce a significant spectral signature at the lasing frequency. The laser amplification has been confirmed by several groups (Gordiets et al., Stepanova et al. and Dickinson et al.)
Spectra of martian CO2 emission line as a function of frequency difference from line center (in MHz). Blue profile is the total emergent intensity in the absence of laser emission. Red profile is gaussian fit to laser emission line. Radiation is from a 1.7 arc second beam (half-power width) centered on Chryse Planitia (long +41 lat +23). (Mumma et al., 1981)
Image from

36. The Argon Ion Laser Argon ion laser lines:
Fast decay
The Argon Ion Laser
Argon ion laser lines:
Wavelength Relative Power Absolute Power
454.6 nm W
457.9 nm W
465.8 nm W
472.7 nm W
476.5 nm W
488.0 nm W
496.5 nm W
501.7 nm W
514.5 nm W
528.7 nm W
The Argon ion laser also has some laser lines in the UV.
But it’s very inefficient.
Population inversion is achieved in a two step process. First of all, the electrons in the tube collide
with argon atoms and ionize them according to the scheme:
Ar (ground state) + lots of energetic electrons
Ar+ (ground state) + (lots + 1) less energetic electrons .
The Ar+ ground state has a long lifetime and some of the Ar+ ions are able to collide with more
electrons before recombining with slow electrons. This puts them into the excited states according to:
Ar+ (ground state) + high energy electrons Þ Ar+ (excited state) + lower energy electrons .
Since there are six 4p levels as compared to only two 4s levels, the statistics of the collisional process
leaves three times as many electrons in the 4p level than in the 4s level. Hence we have population
inversion. Moreover, cascade transitions from higher excited states also facilitates the population
inversion mechanism. The lifetime of the 4p level is 10 ns, which compares to the 1 ns lifetime of the
4s level. Hence we satisfy tupper > tlower and lasing is possible.
Table from
Energy level diagram from
More detailed diagram:

37. The Krypton Ion Laser Krypton ion laser lines: Wavelength Power
406.7 nm .9 W
413.1 nm W
415.4 nm .28 W
468.0 nm .5 W
476.2 nm .4 W
482.5 nm .4 W
520.8 nm .7 W
530.9 nm 1.5 W
568.2 nm 1.1 W
647.1 nm 3.5 W
676.4 nm 1.2 W

38. Laser (Coherent) vs. Incoherent Light
1. It’s strong.
2. It’s uni-directional.
3. Total intensity N2 or 0.
4. Total intensity is the mag-square of the sum of individual fields.
Incoherent light:
1. It’s relatively weak.
2. It’s omni-directional.
3. Total intensity N.
4. Total intensity is the sum of individual intensities.
N = number of beamlets
Etotal = E1 + E2 + … + En
Itotal = I1 + I2 + … + In