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Population Inversion • in Thermal Equilibrium Lower Level Always Greater Population

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Population Inversion • in Thermal Equilibrium Lower Level Always Greater Population What are Lasers? What are Lasers? • Light Amplification by Stimulated Emission of Radiation LASER • Light emitted at very narrow wavelength bands (monochromatic) • Light emitted in a directed beam • Light is coherenent (in phase) • Light often Polarized • Diode lasers much smaller but operate on similar principals Why Study Lasers: Market & Applications • Market $6.0 billion (2006) (just lasers) Major areas: • Market Divided in laser Diodes (56%) & Non diode lasers (44%) Traditional Non Diode Laser • Materials Processing (30%) • Medicine (8%) Diode Lasers • Entertainment/CD/DVD/Printers (~21%) • Telecommunications (21%) Why Study Lasers: Laser Types Traditional Lasers • Solid State laser (Infra Red to Visible) • CO2 Gas laser (Far Infra Red) • Eximer Lasers (UV light) • These mostly used in material processing Diode Lasers • Near Infra Red diodes dominate • Mostly used in telecommunications and CD’s • Visible diode use is increasing • DVD’s driving this History of the Laser • 1917: Einstein's paper showing "Stimulated Emission" • 1957: MASER discovered: Townes & Schawlow • 1960: First laser using Ruby rods: Maiman first solid state laser • 1961: gas laser • 1962: GaAs semiconductor laser • 1964: CO2 laser • 1972: Fiber optics really take off • 1983: Laser CD introduced • 1997: DVD laser video disks World’s First Laser: Ruby Laser Dr. Maiman: Inventor of the World’s First Laser (on left) Electromagnetic Spectrum Light and Atoms • Light: created by the transition between quantized energy states c =νλ hc E = hν = λ c = speed of light ν = frequency hc = 1.24 x 10-6 eV m • Energy is measured in electron volts 1 eV = 1.602 x 10 -19 J • Atomic Energy levels have a variety of letter names (complicated) • Energy levels also in molecules: Bending, stretching, rotation Black Body Emitters • Most normal light emitted by hot "Black bodies" • Classical radiation follows Plank's Law 2π hc2 1 E( λ,T ) = W m3 λ 5 ⎡ ⎛ hc ⎞ ⎤ ⎢exp⎜ ⎟ − 1⎥ ⎣ ⎝ λ KT ⎠ ⎦ h = Plank's constant = 6.63 x 10-34 J s c = speed of light (m/s) λ = wavelength (m) T = Temperature (oK) K = Boltzman constant 1.38 x 10-23 J/K = 8.62 x 10-5 eV/K Black Body Emitters: Peak Emission • Peak of emission Wien's Law 2897 λ = μ m max T T = degrees K • Total Radiation Stefan-Boltzman Law E(T ) = σ T 4 W m2 σ = Stefan-Boltzman constant = 5.67 x 10-8 W m-2 K-4 Example of the Sun • Sun has a surface temperature of 6100 oK • What is its peak wavelength? • How much power is radiated from its surface 2897 2897 λ = = = 0.475μ m max T 6100 • or Blue green colour E(T ) = σ T 4 = 5.67x10-8 x 61004 = 7.85x107 W m2 • ie 78 MW/m2 from the sun's surface Black Body, Gray Body and Emissivity • Real materials are not perfectly Black – they reflect some light • Called a Gray body • Impact of this is to reduce the energy emitted • Reason is reflection at the surface reduces the energy emitted • Measure this as the Emissivity ε of a material ε = fraction energy emitted relative to prefect black body E ε = material Eblack body • Thus for real materials energy radiated becomes E(T ) = εσ T 4 W m2 • Emissivity is highly sensitive to material characteristics & T • Ideal material has ε = 1 (perfect Black Body) • Highly reflective materials are very poor emitters Electro-Magnetic Nature of Light • Classic light in vacuum has Electric field and magnetic field at 90o • Obtained from Maxwell’s Equations • Electric wave ⎡ ⎛ x ⎞⎤ ⎛ ⎡ 2π ⎤⎞ Ey ()x,t = E0 cos⎢ω⎜t − ⎟⎥ = E0 exp()i[]ω t − kx = E0 exp⎜i ω t − x ⎟ ⎣ ⎝ c ⎠⎦ ⎝ ⎣⎢ λ ⎦⎥⎠ 2π Where k = Wave vector k = λ c = velocity of light t = time (sec) λ = wavelength 2π ω= angular frequency (radians/sec) ω = 2πf = τ E0 ⎡ ⎛ x ⎞⎤ • Magnetic wave Bz ()x,t = cos⎢ω⎜t − ⎟⎥ c ⎣ ⎝ c ⎠⎦ hc • Photons are quantized wave packets with energy = λ • Coherent light: all the photons have waves aligned • Photons waves are behaving as sections of the continuous wave • Phase and E field direction are aligned but are discrete packets Irradiance or Light Intensity • What we see is the time averaged energy S, not E or B field t +T / 2 S()t = ∫ S(t)dt t −T / 2 • Where T is the period of the wave • Called the irradiance I in Watts/unit area/unit time 2 c 2 I == ε0c E = B μ0 • For sin waves this results in cε I = S = ε c E 2 = 0 E 2 0 2 • Not true in absorbing materials because • E & B have different relationship & phase there • If just a black body light expands in all directions • Thus intensity falls with inverse square of distance • Consider a sphere radius r0 with intensity I0 at surface • Then at distance r from the center of the sphere get I I(r) = 0 r 2 • Laser sources, or sources with optics behave differently Equilibrium Energy Populations • Laser are quantum devices • Assume gas in thermal equilibrium at temperature T • Some atoms in a Gas are in an excited state • Quantization means discrete energy levels -3 • Atoms Ni (atoms/m ) at a given energy level Ei • E0 is the ground state (unexcited) • Fraction at a given energy follows a Boltzmann distribution N ⎛ [E − E ]⎞ i = exp⎜− i 0 ⎟ N0 ⎝ KT ⎠ T = degrees K K = Boltzman constant 1.38 x 10-23 J/K = 8.62 x 10-5 eV/K Spontaneous and Stimulated Emission • Consider 2 energy levels E0 (ground state) and E1 (excited state) • Photon can cause Stimulated Absorption E0 to E1 • Excited state has some finite lifetime, τ10 (average time to change from state 1 to state 0) • Spontaneous Emission of photon when transition occurs • Randomly emitted photons when change back to level 0 • Passing photon of same λ can cause "Stimulated Emission" • Stimulated photon is emitted in phase with causal photon • Stimulated emission the foundation of laser operation Einstein's Rate Equations • Between energy levels 2 and 1 the rate of change from 2 to 1 is dN 21 = −A N dt 21 2 -1 • where A21 is the Einstein Coefficient (s ) • After long time energy follows a Boltzmann distribution N ⎛ [E − E ]⎞ 2 = exp⎜− 2 1 ⎟ N1 ⎝ KT ⎠ • If (E2 - E1) >> KT then over a long time N2( t ) = N2( 0 ) exp(A21t) • Thus in terms of the lifetime of the level τ21 sec, 1 A21 = τ 21 • illuminated by light of energy density ρ = nhν (J/m3) 3 (n= number of photons/m ) of frequency ν12 the absorption is • At frequency ν12 the absorption is dN 1 = N B ρ()ν emissions dt 1 12 12 m3s • B12 is the Einstein absorption coefficient (from 1 to 2) • Similarly stimulated emission rate (with B21=B12) is dN 2 = N B ρ()ν emissions dt 2 21 21 m3s Two level system: Population Inversion • In thermal equilibrium lower level always greater population • N1 >> N2 • Can suddenly inject energy into system - pumping • Now not a equilibrium condition • If pumped hard enough get "Population Inversion" • Upper level greater than lower level: N2 >> N1 • Population Inversion is the foundation of laser operation Creates the condition for high stimulated emission • In practice difficult to get 2 level population inversion • Best pumping with light gives is equal levels • Reason is Einstein’s rate equations dN dN 2 = N B ρ()ν = N B ρ ()ν = 1 emissions dt 2 21 21 1 12 21 dt m3s • Since B21=B12 then N1=N2 with light pumping • Need more levels to get population inversion Three level systems • Pump to E0 level E2, but require E2 to have short lifetime • Rapid decay to E1 • E1 must have very long lifetime: called Metastable • Now population inversion readily obtained with enough pumping • Always small amount of spontaneous emission (E1 to E0) • Spontaneous create additional stimulated emission to E0 • If population inversion: stimulated emission dominates: Lasing • Common example Nd:Yag laser • Problem: E0 often very full Four Level Systems • Pump to level E3, but require E3 to have short lifetime • Rapid decay to E2 • E2 must have very long lifetime: metastable • Also require E1 short lifetime for decay to E0 • Now always have E1 empty relative to E2 • Always small amount of spontaneous emission (E2 to E1) • Spontaneous photons create additional stimulated emission to E1 • If population inversion: stimulated emission dominates: Lasing • In principal easier to get population inversion • Problem: energy losses at E3 to E2 and E1 to E0 Absorption in Homogeneous Mediums • Monochromatic beam passing through absorbing medium homogeneous medium • Change in light intensity I is ΔI = I(x + Δx)− I(x) ΔI = −αΔxI(x) where α = the absorption coefficient (cm-1) • In differential form dI(x) = −αI(x) dx • This differential equation solves as I(x) = I0 exp(−αx) Gain in Homogeneous Mediums • If we have a population inversion increase I • Stimulated emission adds to light: gain I(x) = I0 exp(gx) g = small signal gain coefficient (cm-1) • In practice get both absorption and gain I(x) = I0 exp([g − a]x) • Gain is related directly to the population inversion g = g0 (N1 − N0 ) g0 = a constant for a given system • This seen in the Einstein B Coefficients • Thus laser needs gain medium to amplify signal .
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