Chapter 13 Phys 3 Lecture 34 Modern optics
Blackbodies and Lasers* Blackbodies Stimulated Emission Gain and Inversion The Laser Four-level System Threshold Some lasers Pump Fast decay Laser Fast decay * Light Amplification by Stimulated Emission of Radiation
Blackbody radiation Definition of a Blackbody Motivation Blackbody Radation Laws 1- The Planck Law - The Wien Displacement Law 3- The Stefan-Boltzmann Law 4- The Rayleigh-Jeans Law
Blackbody radiation A blackbody is an ideal body that absorbs all electromagnetic radiation falling on it No light is reflected No light is transmitted This propety is valid for radiation corresponding to all wavelengths and to all angles of incidence. The black body is an ideal absorber and emitter of radiation.
Motivation The blackbody is important in thermal radiation theory and practice. The blackbody has universal characteristics. The ideal blackbody notion is important in studying thermal radiation and electromagnetic radiation transfer in all wavelength bands. The blackbody is used as a standard with which the absorption of real bodies is compared. Historic importance for quantum mechnics.
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 molecules that, together, have transitions at every wavelength.
Black-Body Radiation Laws The Rayleigh-Jeans Law It agrees with experimental measurements for long wavelengths. It predicts an energy output that diverges towards infinity as wavelengths grow smaller. The failure has become known as the ultraviolet catastrophe. I(, T ) ckt 4
Black-Body Radiation Laws Planck s Law We have two forms. As a function of wavelength. I(, T ) And as a function of frequency 3 h I(, T ) c hc 5 1 h e kt 1 hc ekt 1 1 The Planck Law gives a distribution that peaks at a certain wavelength, the peak shifts to shorter wavelengths for higher temperatures, and the area under the curve grows rapidly with increasing temperature.
Comparison between Classical and Quantum viewpoint Cosmic black body background radiation, T = 3K. Remindeer: Rayleigh-Jeans, but Blackbody 0.
Wein Diplancement law max b T - It tells us as we heat an object up, its color changes from red to orange to white hot. - You can use this to calculate the temperature of stars. The surface temperature of the Sun is 5778 K, this temperature corresponds to a peak emission = 50 nm = about 5000 Å. - b is a constant of proportionality, called Wien's displacement constant and equals.897 768 5(51) 10 3 m K =.897768 5(51) 10 6 nm K.
The Stefan-Boltzmann law P AT j T 4 4 Where P is the total radiant power at all wavelengths Gives the total energy being emitted at all wavelengths by the blackbody (which is the area under the Planck Law curve). Explains the growth in the height of the curve as the temperature increases. Notice that this growth is very abrupt. Sigma = 5.67 * 10-8 J s -1 m - K -4, Known as the Stefan-Boltzmann constant.
Blackbodies and Lasers* Blackbodies Stimulated Emission Gain and Inversion The Laser Four-level System Threshold Some lasers Pump Fast decay Laser Fast decay * Light Amplification by Stimulated Emission of Radiation
Reminder: Atomic and molecular vibrations correspond to excited energy levels in quantum mechanics. Energy levels are everything in quantum mechanics. Excited level Energy E = h Ground level The atom is vibrating at frequency,. The atom is at least partially in an excited state.
Excited atoms emit photons spontaneously. 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, phosphorescence.
Atoms and molecules can also absorb photons, making a transition from a lower level to a more excited one. Energy Excited level Ground level This is, of course, absorption. Absorption lines in an otherwise continuous light spectrum due to a cold atomic gas in front of a hot source.
In what energy levels do molecules reside? Boltzmann population factors Energy E 3 E E 1 N 3 Population density N exp E / k T i i B N N 1 N i is the number density of molecules in state i (i.e., the number of molecules per cm 3 ). T is the temperature, and k B is Boltzmann s constant.
Boltzmann Population Factors In the absence of collisions, molecules tend to remain in the lowest energy state available. Energy 3 Low T N exp E / kbt N exp E / k T 1 1 Collisions can knock a molecule into a higher-energy state. The higher the temperature, the more this happens. B Energy 3 High T 1 1 Molecules In equilibrium, the ratio of the populations of two states is: N / N 1 = exp( E/k B T ), where E = E E 1 = h Molecules As a result, higher-energy states are always less populated than the ground state, and absorption is stronger than stimulated emission.
In 1916, Einstein showed that another process, stimulated emission, can occur. Before After Spontaneous emission Absorption Stimulated emission
Einstein A and B coefficients In 1916, Einstein considered the various transition rates between molecular states (say, 1 and ) involving light of irradiance, I: Spontaneous emission rate = A N Absorption rate = B 1 N 1 I Stimulated emission rate = B 1 N I In equilibrium, the rate of upward transitions equals the rate of downward transitions: Solving for N /N 1 : B 1 N 1 I = A N + B 1 N I Recalling the Maxwell- Boltzmann Distribution (B 1 I ) / (A + B 1 I ) = N / N 1 = exp[ E/k B T ]
Einstein A and B coefficients and Blackbody Radiation Now solve for the irradiance in: (B 1 I ) / (A + B 1 I ) = exp[-e/k B T ] Multiply by A + B 1 I : B 1 I exp[e/k B T] = A + B 1 I Solve for I: I = A / {B 1 exp[e/k B T] B 1 } or: I = [A/B 1 ] / { [B 1 /B 1 ] exp[e/k B T] 1 } Now, when T I should also. As T, exp[e/k B T ] 1. So: B 1 = B 1 B Coeff up = coeff down! And: I = [A/B] / {exp[e/k B T ] 1} Eliminating A/B: I 3 hv exp hv / k T 1 B using E = h
Stimulated emission can lead 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.
The Laser A laser is a medium that stores energy, surrounded by two mirrors. A partially reflecting output mirror lets some light out. I 0 I 1 I 3 Laser medium I R = 100% with gain, G R < 100% A laser will lase if the beam increases in intensity during a round trip: that is, if I3 I0 Usually, additional losses in intensity occur, such as absorption, scattering, and reflections. In general, the laser will lase if, in a round trip: Gain > Loss This is called achieving Threshold.
Laser gain I(0) Laser medium I(L) Neglecting spontaneous emission: di dt di c dz The solution is: BN I -BN I 1 B N - N 1 I( z) I(0)exp N N z I 1 [Stimulated emission minus absorption] Proportionality constant is the absorption/gain cross-section, There can be exponential gain or loss in irradiance. Normally, N < N 1, and there is loss (absorption). But if N > N 1, there s gain, and we define the gain, G: 0 L z G exp N N L 1 If N > N 1 : If N < N 1 : 1 g N N N N 1
Inversion In order to achieve G > 1, stimulated emission must exceed absorption: B N I > B N 1 I Inversion Or, equivalently, N > N 1 Energy Negative temperature This condition is called inversion. It does not occur naturally. It is inherently a non-equilibrium state. Molecules In order to achieve inversion, we must hit the laser medium very hard in some way and choose our medium correctly.
Achieving inversion: Pumping the laser medium Now let I be the intensity of (flash lamp) light used to pump energy into the laser medium: I I 0 I 1 I 3 Laser medium I R = 100% R < 100% Will this intensity be sufficient to achieve inversion, N > N 1? It ll depend on the laser medium s energy level system.
Rate equations for a two-level system Rate equations for the densities of the two states: Pump 1 N Laser N 1 Absorption dn dt dn dt 1 Stimulated emission Spontaneous BI( N N ) AN 1 BI( N N ) AN 1 emission Pump intensity If the total number of molecules is N: N N N 1 N N1N d N N BI N ( N1N) ( N1N) AN dt N N dn BI N AN A N dt
Why inversion is impossible in a two-level system N Laser In steady-state: dn BI N AN A N dt 0 BIN AN AN 1 N 1 ( A BI) N AN N AN /( A BI) N N /(1 BI / A) N where: Isat A/B N 1 I / Isat I sat is the saturation intensity. N is always positive, no matter how high I is! It s impossible to achieve an inversion in a two-level system!
Rate equations for a three-level system Assume we pump to a state 3 that rapidly decays to level. Spontaneous emission dn dt dn dt 1 BIN BIN AN 1 Absorption AN 1 Pump The total number of molecules is N: N N N N N N 1 1 d N BIN 1 AN N N N dt N1 N N dn BIN BI N AN A N dt 3 1 Fast decay Laser Level 3 decays fast and so is zero.
Why inversion is possible in a three-level system dn BIN BI N AN A N dt In steady-state: 0 BIN BIN AN AN 3 Pump 1 Fast decay Laser ( A BI) N ( A BI) N N N( ABI)/( ABI) N N 1 I / I 1 I / I sat sat where: I sat A/ B I sat is the saturation intensity. Now if I > I sat, N is negative!
Rate equations for a four-level system 3 Fast decay Now assume the lower laser level 1 also rapidly decays to a ground level 0. As before: dn dt Because dn dt N1 0, BIN AN 0 BI( N N ) AN N N dn BIN BI N A N dt Pump 1 0 N N N Laser Fast decay The total number of molecules is N : 0 N N N 0 At steady state: 0 BIN BIN AN
Why inversion is easy in a four-level system (cont d) 0 BIN BIN AN ( A BI) N BIN 3 Pump 1 0 Fast decay Laser Fast decay N BIN /( ABI) N ( BIN / A)/(1 BI / A) N N I / I sat 1 I / I sat where: I sat A/ B I sat is the saturation intensity. Now, N is negative always!
What about the saturation intensity? Isat A/ B A is the excited-state relaxation rate: 1/ 3 Pump 1 0 Fast decay Laser Fast decay B is the absorption cross-section,, divided by the energy per photon, ħ: / ħ Both and depend on the molecule, the frequency, and the various states involved. I sat 10 5 to 10 13 W/cm ħ~10-19 J for visible/near IR light ~10-1 to 10-8 s for molecules ~10-0 to 10-16 cm for molecules (on resonance) The saturation intensity plays a key role in laser theory.
Two-, three-, and four-level systems It took laser physicists a while to realize that four-level systems are best. Two-level system Three-level system Four-level system Fast decay Fast decay Pump Laser Pump Laser Pump Laser Fast decay At best, you get equal populations. No lasing. If you hit it hard, you get lasing. Lasing is easy!