Optical Electronics - ECE 455 Lecture 3 Blackbody Radiation...

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Unformatted text preview: ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Blackbody Radiation, Gain and Broadening ECE 455 Optical Electronics Summary Tom Galvin Gary Eden ECE Illinois Introduction ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary In this section, we will learn how to do the following: Understand the properties of blackbody (thermal) light Describe absorption, spontaneous emission and stimulated emission with rate equations Predict the line width of atomic and molecular transitions Blackbody Radiation ECE 455 Lecture 3 Blackbody Radiation Blackbody radiation is emitted by objects because they are hot. The optical energy density for a blackbody radiator at temperature T between the frequencies of ν and ν + d ν is Lineshape Gain Summary ρ(ν )d ν = 8π h ν 3 c 3 exp 1 hν kB T dν (1) −1 where ν is the frequency, h is Planck’s constant, c is the speed of light, and kB is Boltzmann’s constant. The peak wavelength of this distribution can be found with the following equation λmax = 2.898 × 106 K·nm T (2) Blackbody Radiation Picture ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary Example: Human Blackbody ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary Problem: The temperature of a healthy person is about 37◦ C. If a human could be considered an ideal blackbody, at what wavelength would the blackbody spectrum peak? What are the energy and photon densities of visible light emitted by people immediately above the surface of the skin? What are the energy and photon densities of photons in ±1% bandwidth around the peak wavelength? Solution: The peak wavelength can be found with the aid of Equation 2 λp = 2.898 × 106 K·nm = 9344 nm (37 + 273.15) K (3) Example: Human Blackbody ECE 455 Lecture 3 To find the energy density of visible light, numerically integrate Equation 1 over the appropriate limits: Blackbody Radiation Lineshape ρvis = c 400 nm c 700 nm Gain 8π h ν 3 c 3 exp 1 hν kB T dν (4) −1 = 5.44 × 10−30 J-cm−3 Summary To convert energy density to photon density, divide by the energy per photon, hν . Numerical integration is once again required ρvis ¯ = c 400 nm c 700 nm 8πν 2 c 3 exp 1 hν kB T dν −1 = 1.87 × 10−11 photons-cm−3 (5) Example: Human Blackbody ECE 455 Lecture 3 For a ±1% bandwidth around the peak, the energy density is ρIR Blackbody Radiation = 1.01·c 9344 nm 0.99·c 9344 nm 8π h ν 3 c 3 exp Lineshape 1 hν kB T dν (6) −1 = 9.21 × 10−8 J-cm−3 Gain Summary Because the bandwidth of this integration region is much narrower, there is no need to perform another numerical integration; we can simply divide the above answer by the hc energy per photon λp = 2.13 × 10−20 J. 9.21 × 10−8 J-cm−3 = 4.33 × 1012 photons-cm−3 2.13 × 10−20 J/photon (7) The lesson: If you are looking for somebody in the dark, don’t use your eyes. Use an IR camera. ρIR = ¯ Absorption ECE 455 Lecture 3 Blackbody Radiation 2 Lineshape Gain Summary A two-level system absorbs a photon and is left in an excited state 1 Spontaneous Emission ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary Two-level system spontaneously decays from a higher energy level to a lower energy level and emits a photon. 2 The decay rate is characterized by the spontaneous emission lifetime τ Emission is uniform in all 4π steridians Spontaneous emission can only be explained by QED 1 Stimulated Emission ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary Photon interacts with two-level system in an excited state The system goes to a lower energy level and a photon with the corresponding energy is emitted Emitted photon has the same direction, polarization, phase and energy as the incident photon 2 hγ 2hγ 1 Three Fundamental Process ECE 455 Lecture 3 Blackbody Radiation Absorption dN2 dt = +B12 ρ(ν21 )N1 (8) Absorption Lineshape Gain Summary Spontaneous Emission dN2 dt = −A21 N2 (9) SpontaneousEmission Stimulated Emission dN2 dt = −B21 ρ(ν21 )N2 (10) StimulatedEmission Total population transfer dN2 = −A21 N2 − B21 ρ(ν21 )N2 + B12 ρ(ν21 )N1 dt (11) Relationship Between Einstein Coefficients I ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary Assume atoms are in equilibrium with a blackbody radiation d field. In steady state, dt () = 0. Solving Equation 11 for ρ, we find: A21 N2 ρ(ν ) = (12) B12 N1 − B21 N2 In thermal equillibrium, the populations of energy levels can be assumed to follow a Boltzmann distiribution. Thus: N2 g2 hν21 = exp − N1 g1 kB T (13) Relationship Between Einstein Coefficients II ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary Substitution of Equation 13 into 12 gives: ρ(ν21 ) = A21 B21 1 B12 g1 hν21 /kt B21 g2 e −1 (14) When the above equation is compared with the blackbody spectrum (Equation 1), ρ(ν ) = 8π h ν 3 c 3 exp 1 hν kB T the following relationships are discovered: g1 B21 = B12 g2 A21 8π hν 3 = B21 c3 (1) −1 (15) (16) Example: Relative Strength of Stimulated and Spontaneous Emission ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary Problem: Some sodium atoms contaminated the filament of a incandescent lightbulb filament. When the bulb is operating at T = 5XXX ◦ C, assume the sodium atoms and the optical field are in thermal equilibrium with the . Find the ratio of spontaneous emission to stimulated emission on the D1 line. Solution: A significant optical energy density is necessary to make stimulated emission as strong as spontaneous emission. Introduction to Lineshape ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary Atomic and molecular transition lines have a finite bandwidth; They are not perfectly sharp. Various physical processes broaden the spectrum lines. Homogenous line broadening occurs when all emitters are equally effected by broadening mechanisms. Inhomogenous line broadening occurs when all emitters are unequally effected by broadening mechanisms. Line broadening reduces the effective gain because not all atoms are capable of interacting with the radiation field. The Lineshape Function ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary For example, if there are N2 atoms in the upper energy level, then the number capable of emitting a photon with d ν of the frequency ν is: N (ν )d ν = g (ν )N2 d ν (17) If this equation is integrated over all frequencies, we must get N2 . Thus g (ν ) must be a probability distribution. In other words: ∞ g (ν )d ν = 1 (18) 0 Lifetime Broadening ECE 455 Lecture 3 Blackbody Radiation g (ν ) = Lineshape Gain Summary 1 ∆ν 2π (ν − ν0 )2 + where ∆ν = 1 2π 1 1 + τ1 τ2 ∆ν 2 2 (19) (20) The peak of the lineshape function is: g (ν0 ) = 2 π ∆ν (21) Collisional Broadening ECE 455 Lecture 3 Blackbody Radiation Lineshape Siegman 128 It can be shown from the principles of that the collision rate is: Gain Summary νcol = Nm σ 8kT π 1 1 − Mm M2 1/2 (22) An more phenominalogical description ∆ν = A + BP (23) Total Homogenous Broadening ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain The total linewidth from all homogenous broadening mechanisms is: Summary ∆νtotal = 1 [(A1 + k1 ) + (A2 + k2 ) + 2vcol ] 2π where the Ai ’s and ki ’s are the total radiative and non-radiative relaxation rates for level i . (24) Doppler Broadening I ECE 455 Lecture 3 Blackbody Radiation Lineshape Doppler broadening occurs in room temperature gases. In their rest frame all atoms will emit light with the same homogenous profile, gH (ν ). However, light is observed in the lab frame, where the profile will be doppler shifted. Gain Summary The doppler shift of a photon emitted from a non-relativistic particle moving toward the observer with velocity vz is ν =ν 1+ vz c (25) The observed lineshape profile in the lab frame is then gH (ν ) = gH ν 1 + vz c (26) Doppler Broadening II ECE 455 Lecture 3 It can be shown from thermodynamics that the distribution of particle velocities in the z direction is Blackbody Radiation M 2π kT Lineshape Gain 1/2 exp − 2 Mvz 2kT (27) Summary The total lineshape profile in the lab frame from all particles moving with all velocities is g (ν ) = M 2π kT 1/2 ∞ gH (ν ) · exp − −∞ 2 Mvz 2kT dvz (28) For light molecules at room temperature, doppler broadening dominates lifetime broadening, we may therefore make the approximation gH (ν ) ≈ δ (ν − ν0 ). Doppler Broadening Picture ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary Doppler Broadening III ECE 455 Lecture 3 The lineshape function of a Doppler broadened transition is: Blackbody Radiation Lineshape g (ν ) = Gain Summary 2 ∆νD ln2 π 1/2 where ∆νD = ν − ν0 ∆ν exp −4ln(2) 8kTln2 Mc 2 2 (29) 1/2 ν0 (30) The peak of the lineshape function is: g (ν0 ) = 2 ∆νD ln2 π 1/2 (31) Doppler Broadening and Laser Cooling ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary Tune a laser beam to slightly below an atomic resonance Atoms moving toward the laser see the light doppler shifted into resoannce The atom is free to re-emit the photon in any direction Six counter-propogating beams must be used to ensure zero net momentum transfer Laser COOLING PICTURE Example: Doppler Broadening in the Copper Vapor Laser ECE 455 Lecture 3 Blackbody Radiation Problem: The copper vapor laser (CVL) operates at T = 1750 K. Its wavelength is λ = 510.6 nm and the spontaneous emission lifetime is τsp = 5×10−7 s. Determine the form of the lineshape function and its FWHM. Lineshape Gain Summary Solution: The homogenous linewidth is ∆νH = 1 = 3.18 × 105 Hz 2πτsp (32) The doppler linewidth is ∆νD = ν0 8kTln2 = 2.20 × 109 Hz Mc 2 (33) Doppler broadening dominates this transition. Therefore the lineshape will be gaussian with a FHWM ∆νD . Energy Density and Intensity ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary If all radiation is travelling in a given direction, the following relationship exisists between the energy density and the intensity: I (ν ) = c ρ(ν ) n (34) Figure Demonstrating Relationship Stimulated Emission in Broadband Light ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary Equation 11 If the transition linewidth is much narrower than optical bandwidth, not all photons will be able to interact with the transition: dN2 dt ∞ (B12 N1 g (ν )ρ(ν ) − B21 N2 g (ν )ρ(ν )) d ν (35) = 0 ∞ = (B12 N1 − B21 N2 ) g (ν )ρ(ν )d ν 0 (36) ∞ ≈ (B12 N1 − B21 N2 ) ρ(ν21 ) g (ν )d ν (37) 0 = (B12 N1 − B21 N2 ) ρ(ν21 ) Here, ν21 is the center wavelength of the transition. (38) Stimulated Emission in Narrowband Light ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain If the bandwith of the optical field is much narrower than the transition linewidth, then not all atoms will be able to interact with the field. dN2 dt ∞ (B12 N1 g (ν )ρ(ν ) − B21 N2 g (ν )ρ(ν )) d ν (39) = 0 ∞ Summary = (B12 N1 − B21 N2 ) g (ν )ρ(ν )d ν 0 (40) ∞ ≈ (B12 N1 − B21 N2 ) g (νp ) ρ(ν )d ν (41) 0 = (B12 N1 − B21 N2 ) g (νp )ρp (42) Here νp is the wavelength of the stimulating optical field. Generally, lasers will oscillator at or very near the peak of the gain medium. It is this case we’ll be interested in for the rest of the semester. The Possibility of Gain ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary Consider the interaction of a gain medium with narrowband light N2 dN2 = −B21 N2 ρν g (ν ) + B12 N1 ρν g (ν ) − (43) dt τ2 g2 N2 = −B21 N2 ρν g (ν ) + B21 N1 ρν g (ν ) − (44) g1 τ2 N2 g2 = −B21 ρν g (ν ) N2 − N1 − (45) g1 τ2 λ2 N2 ρν c 1 g2 = −A21 (46) g (ν ) N 2 − N1 − 2 8π n n hν g1 τ2 σse (ν )Iν g2 N2 =− N 2 − N1 − (47) hν g1 τ2 where: σse (ν ) ≡ A21 λ2 g (ν ) 8π n 2 (48) The Possibility of Gain II ECE 455 Lecture 3 Notice that if Blackbody Radiation Lineshape Gain Summary N2 > g2 N1 g1 (49) The population of the upper state will decrease. Where does the energy go? Into the optical field! The term N2 − g2 N1 will appear repeatedly in our equations. g1 To simplify equations, we define the population inverison ∆N ≡ N 2 − g2 N1 g1 (50) The Stimulated Emission Cross Section ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary σse is not just a random collection of constants The stimulated emission cross section can be viewed as ’Optical thickness’ 1/(σse N ) Cross Section Picture Typical Values of the Stimulated Emisson Cross Section ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary Laser Argon Ion Nitrogen HeNe HeCd Copper Vapor GreeNe Dye (Rh6G) KrF CO2 Ti:Al2 O3 Nd:YAG Nd:Glass Ruby λ (nm) 488.0 337.1 632.8 441.6 510.5 543.5 577.0 248.0 10600 800.0 1064 1062.3 694.3 σse (m2 ) 2.6 × 10−16 4.0 × 10−17 3.0 × 10−17 9.0 × 10−18 8.6 × 10−18 2.0 × 10−18 2.9 × 10−20 2.6 × 10−20 3.0 × 10−22 3.4 × 10−23 2.8 × 10−23 2.9 × 10−24 2.5 × 10−24 Optical Amplification ECE 455 Lecture 3 Blackbody Radiation Consider a plane wave propagating in the +z direction through an atomic medium. Then the change in the density of photons Np after propagating through this medium is Lineshape Gain Summary dNp = N2 n σse (ν )Iν ∆N − η dz hν τ2 c (51) where η is an efficiency factor. For now, assume the beam is intense enough so that the second term on the right hand side is neglible. c Iν = Np hν (52) n dI = I σse ∆N (53) dz These equations are non-linear. Degeneracy and Gain I ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary Why does degeneracy affect the gain? An optical field will cause both stimulated emission and absorption; It is the net difference between these two processes which determines whether a beam is amplified or attenuated. Fermi’s Golden Rule (FGR) predicts a constant transition probability, W10 , per unit time between each individual pair of upper and lower states FGR also predicts that the absorption and stimulated rates are equal, so W01 = W10 . If there are multiple states at a given energy, the population at that energy will be evenly distributed among the states. Degeneracy and Gain II ECE 455 Lecture 3 Blackbody Radiation N1 Lineshape Gain Summary W10 W01 N0 The net stimulated emission is: dN1 dt = N1 W10 − N0 W01 = W10 (N1 − N0 ) (54) Degeneracy and Gain III ECE 455 Lecture 3 ½N1 Blackbody Radiation ½N1 Lineshape Gain W10 Summary W01 W10 W01 N0 The net stimulated emission is: dN1 dt N1 N1 W10 − N0 W01 + W10 − N0 W01 2 2 = W10 (N1 − 2N0 ) = (55) Degeneracy and Gain IV ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary The argument on the previous slide can be extended to any number of upper and lower states. We will always find g1 dN1 = W10 N1 − N0 dt g0 (56) where g1 and g2 are the upper and lower state degeneracies. We happen to know W10 = I σse hν10 Warning!! In some situations with very intense fields, the population will not have time to equilibrate among the degenerate energy levels. This analysis will then fail. (57) The Gain Coefficient ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary The γ (ν ) ≡ σ (ν )∆N (58) Summary ECE 455 Lecture 3 Blackbody Radiation Lineshape Gain Summary Blackbodies There are three primary mechanisms of broadening in gas: lifetime, doppler, and collisional. Broadening mechanisms in solids are more dificult to predict. A system will absorb light if ∆N < 0 and amplify it if ∆N > 0 ...
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