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Lasers* Stimulated Emission Fast decay Gain Inversion Pump Laser The Laser Transition Transition Four-level System Threshold Some lasers Fast decay * Light Amplification by Stimulated Emission of Radiation 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. The Laser A laser is a medium that stores energy, surrounded by two mirrors. A partially reflecting output mirror lets some light out. I0 I1 I3 Laser medium I2 R = 100% with gain, G R < 100% A laser will lase if the beam increases in intensity during a round trip: that is, if I 3 I 0 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. 2 Calculating the gain: Einstein A and B coefficients 1 In 1916, Einstein considered the various transition rates between molecular states (say, 1 and 2) involving light of irradiance, I: Absorption rate = B N1 I Spontaneous emission rate = A N2 Stimulated emission rate = B N2 I where Ni is the number density of molecules in the ith state, and I is the irradiance. Laser medium Laser gain I(0) I(L) Neglecting spontaneous emission: z 0 L dI dI c BN 2 I - BN1I [Stimulated emission minus absorption] dt dz B N 2 - N1 I Proportionality constant is the The solution is: absorption/gain cross-section, I ( z ) I (0) exp N2 N1 z There can be exponential gain or loss in irradiance. Normally, N2 < N1, and there is loss (absorption). But if N2 > N1, there’s gain, and we define the gain, G: If N2 > N1: g N2 N1 G exp N2 N1 L If N2 < N1 : N1 N2 Inversion In order to achieve G > 1, stimulated emission must exceed absorption: B N2 I > B N1 I Inversion Or, equivalently, “Negative Energy N2 > N1 temperature” This condition is called inversion. It does not occur naturally. It is Molecules 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. 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 I0 I1 I3 Laser medium I2 R = 100% R < 100% Will this intensity be sufficient to achieve inversion, N2 > N1? It’ll depend on the laser medium’s energy level system. Rate equations for a 2 N2 two-level system Pump Laser 1 N1 Rate equations for the densities of the two states: Stimulated emission Spontaneous Absorption emission dN 2 BI ( N1 N 2 ) AN 2 If the total number dt of molecules is N: Pump intensity dN1 N N1 N 2 BI ( N 2 N1 ) AN 2 dt N N1 N 2 d N 2 N 2 ( N1 N 2 ) ( N1 N 2 ) 2 BI N 2 AN 2 dt N N d N 2 BI N AN AN dt Why inversion is impossible 2 N2 in a two-level system Laser d N 1 N1 2 BI N AN AN dt In steady-state: 0 2BI N AN AN ( A 2BI )N AN N AN /( A 2BI ) N N /(1 2BI / A) N where: I sat A / B N 1 2 I / I sat Isat 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 3 Fast decay three-level system 2 Pump Laser Assume we pump to a state 3 that Transition Transition rapidly decays to level 2. No pump stimulated emission! 1 Spontaneous emission dN 2 BIN1 AN 2 dt The total number Level 3 Absorption of molecules is N: decays fast and dN1 N N1 N 2 BIN1 AN 2 so is zero. dt N N1 N 2 d N 2 BIN1 2 AN 2 2N 2 N N dt 2N1 N N d N BIN BI N AN AN dt 3 Why inversion is possible 2 Fast decay in a three-level system Pump Laser Transition Transition d N BIN BI N AN AN 1 dt In steady-state: 0 BIN BI N AN AN ( A BI )N ( A BI ) N N N ( A BI ) /( A BI ) 1 I / I sat N N 1 I / I sat Now if I > Isat, N is negative! Rate equations for a 3 Fast decay four-level system 2 Now assume the lower laser level 1 Pump Laser also rapidly decays to a ground level 0. Transition Transition So N1 0 ! And N N 2 1 Fast decay dN 2 0 As before: BIN 0 AN 2 dt dN 2 The total number BI ( N N 2 ) AN 2 of molecules is N : dt N N0 N2 Because N N 2 N0 N N2 d N BIN BI N AN dt At steady state: 0 BIN BI N AN 3 Why inversion is easy Fast decay 2 in a four-level system (cont’d) Pump Transition Laser Transition 0 BIN BI N AN 1 Fast decay 0 ( A BI )N BIN N BIN /( A BI ) N ( BIN / A) /(1 BI / A) I / I sat N N 1 I / I sat Now, N is negative—always! 3 What about the Fast decay 2 saturation intensity? Pump Laser Transition Transition I sat A / B 1 Fast decay A is the excited-state relaxation rate: 1/t 0 B is the absorption cross-section, , divided by the energy per photon, ħw: / ħw ħw ~10-19 J for visible/near IR light Both and t depend on the w t ~10-12 to 10-8 s for most molecules molecule, the I sat 10-9 to 10-3 s for laser molecules frequency, and t ~10-20 to 10-16 cm2 for molecules (on the various resonance) states involved. 1 to 1013 W/cm2 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 Three-level Four-level system system system Fast decay Fast decay Pump Pump Laser Transition Laser Transition Transition Transition Laser Pump Transition Transition Fast decay At best, you get If you hit it hard, equal populations. Lasing is easy! you get lasing. No lasing. Achieving Laser Threshold An inversion isn’t enough. The laser output and additional losses in intensity due to absorption, scattering, and reflections, occur. I0 I1 Laser medium I3 Gain, G = exp(gL), and I2 R = 100% Absorption, A = exp(-L) R < 100% The laser will lase if the beam increases Gain > Loss in intensity during a round trip, that is, if: This called achieving Threshold. It means: I3 > I0. Here, it means: I 3 I 0 exp( gL) exp( L) R exp( gL) exp( L) I 0 2( g ) L ln(1/ R) Types of Lasers Solid-state lasers have lasing material distributed in a solid matrix (such as ruby or neodymium:yttrium-aluminum garnet "YAG"). Flash lamps are the most common power source. The Nd:YAG laser emits infrared light at 1.064 nm. Semiconductor lasers, sometimes called diode lasers, are pn junctions. Current is the pump source. Applications: laser printers or CD players. Dye lasers use complex organic dyes, such as rhodamine 6G, in liquid solution or suspension as lasing media. They are tunable over a broad range of wavelengths. Gas lasers are pumped by current. Helium-Neon lases in the visible and IR. Argon lases in the visible and UV. CO2 lasers emit light in the far-infrared (10.6 mm), and are used for cutting hard materials. Excimer lasers (from the terms excited and dimers) use reactive gases, such as chlorine and fluorine, mixed with inert gases such as argon, krypton, or xenon. When electrically stimulated, a pseudo molecule (dimer) is produced. Excimers lase in the UV. 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. 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. Carbon Dioxide Laser The CO2 laser operates analogously. N2 is pumped, transferring the energy to CO2. CO2 laser in the Martian atmosphere The atmosphere is thin and the sun is dim, but the gain per molecule is high, and the pathlength is long. Detuning from line center (MHz) The Helium Cadmium Laser The population inversion scheme in HeCd is similar to that in HeNe’s except that the active medium is Cd+ ions. The laser transitions occur in the blue and the ultraviolet at 442 nm, 354 nm and 325 nm. The UV lines are useful for applications that require short wavelength lasers, such as high precision printing on photosensitive materials. Examples include lithography of electronic circuitry and making master copies of compact disks. The Argon Ion Laser Argon ion laser lines: Wavelength Relative Power Absolute Power 454.6 nm .03 .8 W 457.9 nm .06 1.5 W The Argon 465.8 nm .03 .8 W ion laser 472.7 nm .05 1.3 W also has 476.5 nm .12 3.0 W some laser 488.0 nm .32 8.0 W lines in the 496.5 nm .12 3.0 W UV. 501.7 nm .07 1.8 W But it’s very 514.5 nm .40 10.0 W inefficient. 528.7 nm .07 1.8 W The Krypton Ion Laser Krypton ion laser lines: Wavelength Power 406.7 nm .9 W 413.1 nm 1.8 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 Dye lasers Dye lasers are an ideal four-level system, and a given dye will lase over a range of ~100 nm. A dye’s energy levels The lower laser level can be almost any level in the S0 manifold. S1: 1st excited electronic state manifold Pump Transition Laser Transitions S0: Ground electronic state manifold Dyes are so ideal that it’s often difficult to stop them from lasing in all directions! Dyes cover the visible, near-IR, and near-UV ranges. Titanium: Sapphire (Ti:Sapphire) Absorption and emission spectra of Ti:Sapphire Upper level lifetime: 3.2 msec Al2O3 lattice oxygen Ti:Sapphire lases from aluminum ~700 nm to ~1000 nm. Diode Lasers Some everyday applications of diode lasers A CD burner Laser Printer