22. Lasers. Stimulated Emission: Gain. Population Inversion. Rate equation analysis. Two-level, three-level, and four-level systems
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1 . Lasers Stimulated Emission: Gain Population Inversion Rate equation analysis Two-level, three-level, and four-level systems
2 What is a laser? LASER: Light Amplification by Stimulated Emission of Radiation "light" could mean anything from microwaves to x-rays Essential elements:. A laser medium - a collection of atoms, molecules, etc.. A pumping process - puts energy into the laser medium 3. Optical feedback - provides a mechanism for the light to interact (usually many times) with the laser medium Stimulated emission causes the number of photons in the laser beam to grow. The factor by which an input beam is amplified by a medium (during one pass through) is called the gain and is represented by G.
3 Reminder: Einstein A and B coefficients In 96, Einstein considered the various transition rates between two energy levels (say, and ) involving light of irradiance, I. Initial state Final state Absorption rate = B I Spontaneous emission rate = A Stimulated emission rate = B I, = populations of states and I = irradiance of the light field A, B = Einstein coefficients
4 Laser gain I(0) Laser medium I(L) eglecting spontaneous emission: di c di n dz The solution is: B I-BI B - I( z) I(0)exp z I There can be exponential gain or loss in irradiance. ormally, <, and there is loss (absorption). [Stimulated emission minus absorption] 0 Proportionality constant is called the absorption (or gain) cross-section L 0exp I z I z z But if >, there s gain, and we define the gain, G: G exp L If > : If < : g
5 Reminder: The Maxwell-Boltzman Distribution In the absence of collisions, molecules tend to remain in the lowest energy state available. Low T Collisions can knock a molecule into a higher-energy state. The higher the temperature, the more this happens. High T In equilibrium at a temperature T, the ratio of the populations of any two states is given by: exp E / k T E k T B exp / B exp E/ k T B Since T > 0, we always find <.
6 Populations and the cross-section exp E/ kbt At low temperatures, or for large values of E, we find <<. Example: The energy level separation is equal to the photon energy. For visible light, = 600 nm: h = hc/ = joules. At room temperature: k B T = joules. If > : If < : E k T exp / e ~ 0 g B So = 0, effectively. units of are m, area. is something like the absorption per molecule. It is the effective size (area) of a molecule as seen by an incoming photon.
7 Inversion In order to achieve G >, stimulated emission must exceed absorption: Or, equivalently, B I > B I A population inversion > This condition is called inversion. It does not occur naturally in steady-state. It is inherently a non-equilibrium state. Energy Molecules In order to achieve inversion, we must pump a lot of energy into the gain medium. And it needs to be the right medium.
8 Achieving inversion: Pumping the laser medium Suppose we pump energy into the laser medium, using another light source with intensity I: I I I 0 output Mirror with R < 00% I Laser medium I 3 Mirror with R = 00% The key question for the remainder of today s lecture: Will this intensity I be sufficient to achieve inversion, >? The answer depends on the laser medium s energy level configuration.
9 Rate equations for a two-level system Earlier we neglected spontaneous emission. Let s look again, and be a bit more careful. Rate equations for the population densities of the two states: Absorption Stimulated emission Spontaneous emission d d BI( ) A BI( ) A Pump intensity d BI A d BI A A Pump If the total number of molecules is : Laser ( ) ( )
10 How does the population difference depend on pump intensity? In steady-state the time derivative is zero: d BI A A 0 BI A A Pump Laser Solve for : ( A BI) A A /( A BI) B I A I / Isat where: Isat A/ B I sat is called the saturation intensity, a unique parameter for any gain medium.
11 What is the saturation intensity? Isat A/ B A is the excited-state relaxation rate due to spontaneous emission: / B is the absorption cross-section,, divided by the energy per photon, ħ: / ħ Both and depend on the molecule, and also the frequency of the light. I sat 0 5 to 0 3 W/cm ħ~0-9 J for visible/near IR light ~0 - to 0-8 s for molecules ~0-0 to 0-6 cm for molecules (on resonance) The saturation intensity plays a key role in laser theory. It is the intensity which corresponds to one photon incident on each molecule, within its cross-section, per recovery time.
12 Why inversion is impossible in a two-level system Population difference population difference I sat I / Isat a plot of this function Pump Recall that It s impossible to achieve a steady-state inversion in a two-level system! Why? Because absorption and stimulated emission are equally likely. Laser For population inversion, we require 0 4I sat Pump intensity Even for an infinite pump intensity, the best we can do is = (i.e., = 0) 6I sat is always positive, no matter how hard we pump on the system!
13 Rate equations for a three-level system So, if we can t make a laser using two levels, what if we try it with three? Assume we pump to a state 3 that rapidly decays to level. 3 Pump Fast decay Laser d d BI BI A A The total number of molecules is : Level 3 decays fast and so 3 = 0. d BI A
14 Why inversion is possible in a three-level system d BI BI A A 3 Pump Fast decay Laser In steady-state: 0 BI BI A A Solve for : A BI B A I A BI B A I I / I I / I sat sat where, as before: I sat A/ B I sat is the saturation intensity. ow if I > I sat, is negative!
15 Rate equations for a four-level system 3 Fast decay ow assume the lower laser level also rapidly decays to a ground level 0. As before: d Because d BI A 0 BI( ) A 0, d BI BI A Pump 0 Laser Fast decay The total number of molecules is : 0 0 At steady state: 0 BI BI A
16 Why inversion is easy in a four-level system 3 Pump Fast decay Laser 0 BI BI A 0 Fast decay Solve for : BI ABI B A I B A I I / I sat I / I sat where: I sat A/ B I sat is the saturation intensity. ow, is negative for any non-zero value of I!
17 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. o lasing. If you hit it hard, you get lasing. Lasing is easy!
18 Population inverstion in two-, three-, and four-level systems levels population difference 0 3 levels population inversion - 4 levels 0 I sat 4I sat 6I sat 8I sat 0I sat intensity of the pump
19 The first laser ever built was a three-level system
20 The Hee laser is a 4-level system. e excited by collisions with He atoms. lasing from 3s to p energy levels 3. fast decay to ground state
21 Many lasers are almost ideal 4-level systems For example, the green laser we use in class is an d +3 :YAG laser. The energy levels of d +3 constitute a classic example of a four-level system. Infrared laser emission is converted to green light by a nonlinear optical process: frequency doubling! (see lecture 38)
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