Fluorescence

Lec1 Light: Generation

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Fluorescence

• Luminescence • Comprised of fluorescence and phosphorescence • The phenomenon that a given substance emits light • Light in the visible spectrum is due to electronic transitions • Vibrationally excited states can give rise to lower energy photons, i.e. infrared radiation • Rotational modes produce even lower energy radiation, in the microwave region. • Application • Fluorescence-based measurements • Microscopy, flow cytometry, medical diagnosis, DNA sequencing, and many other areas. • Biophysics and biomedicine • Fluorescence microscopy, image at cellular and molecular scale, even single molecule scale. • Fluorescent probes, specifically bind to molecules of interest, revealing localization information and activity (i.e., function). • Challenge in microscopy: Merge information from the molecular and cellular scales.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Jablonski Diagram

• Fluorescence • The radiative transition between two electronic states of the same spin multiplicity.

• Spin multiplicity: Ms=2s+1, with s the spin of the state. • The spin s • s is the sum of the spins from all the electrons on that electronic state. • s=0, two paired (antiparallel spins) electrons on the electronic sate. • s=1/2, a single, unpaired electron in the state. • s=1, two electrons of parallel spins. • Electronic states • Singlets (Ms=2s+1=1, s=0) • Doublets (Ms=2, s=1/2) • Triplets (Ms=3, s=1) • The singlet-singlet transition is the most common in fluorescence.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Jablonski Diagram

• Jablonski diagram a S • The energy levels of a molecule. 2 Internal Conversion • 푆0, 푆1, 푆2 denote the ground, first, and second electronic state. S1 Intersystem Crossing • A typical scenario: T Absorption 1 • Absorb a photon of energy hvA , 푆0→푆2, at the Fluorescence 1fs=10-15 s scale hv hv A hvA hvF P -12 • Internal conversion:푆2→ 푆1, within 1ps=10 s 2 Phosphorescence or less. S0 1 0 • Emit a photon of energy hvF , 푆2→ 푆1, in an b Figure 5.1. a) Jablonski diagram depicting a average time of 10-9—10-8 s. three level system: 푆0, 푆1, 푆2 are singlet • Assume: The internal conversion is complete electronic states, each containing vibrational prior to emission. levels. Absorption takes place at frequencies

A .T1 denotes a triplet state, from which phosphorescence takes place. b) Music score illustrating the similarity with the Jablonski diagram.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Jablonski Diagram

• Born-Oppenheimer approximation • A molecule’s wave function can be factorized into an electronic and nuclear (vibrational and rotational) components.

total=  e  N

• Phosphorescence

• Intersystem crossing: The conversion from the S1 state to the first triplet state T1

• Phosphorescence: Triplet emission (emission from T1 ), which is generally shifted to lower energy (longer wavelengths) • Phosphorescence has lower rate constants than fluorescence. • Phosphorescence with decay times of seconds or even minutes. • Molecules of heavy atoms tend to be phosphorescent.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Emission Spectra

• Symmetry property between the power Wavelength ( )

a ) 1 350 370 390 410 450 490 530 570 spectrum of absorption (or excitation) 1.0

and emission (or radiative decay) — 1 mirror symmetry

• Excitation: Absorption, S0 → S1 (higher 0.5 vibrational levels)→ nonradiative decay, S 16 1 Emission (lowest vibrational level). Absorption 8 • Emission: Radiatively decays, S1 → S0 (high vibrational level) → thermal equilibration, S0 0 0

(ground vibrational state). coefficient Extincion 28,800 26,800 24,800 22,800 20,800 18,800

1 (normalized) intensity Fluorescence • The peaks in both absorption and emission Wavenumber ( ) correspond to vibrational levels, which results Figure 5.2. a) Mirror-image rule (perylene in benzene). in perfect symmetry.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Emission Spectra

• Note: The local maxima are due to b ) the individual vibration levels. 1 Wavenumber ( )

280 320 360 400 440 480 520 600

• The emission spectrum remains 1 10 1.0

constant irrespective of the excitation 8

6 wavelength (exceptions exist). 0.5 4 Absorption Emission • Exception to the mirror rule 2 • Absorption spectrum: The first peak is 0 0 35,500 31,500 27,500 23,500 19,500 15,500, due to S0 — S2, the larger wavenumber 1

peak is from S0 — S1. Wavenumber ( ) Extincion coefficient coefficient Extincion

• Emission: Occurs predominantly from S1, (normalized) intensity Fluorescence the second peak is absent. Figure 5.2. b) Exception to the mirror rule (quinine sulfate in H2SO4). Adapted from Lakowicz, J. R. (2013). Principles of fluorescence spectroscopy, Springer Science & Business Media.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Rate Equations

• The population of each level is determined by the S2 Relaxation(10−12 s ) rate of absorption, nonradiative decay, S spontaneous emission, and stimulated emission. 1 • Stimulated emission h h • The radiative transition from an excited state to the ground A A state triggered (stimulated) by the excitation field. • By stimulated emission, one excitation photon results in two identical emitted photons (development of lasers). S0

Figure 5.3. Jablonski diagram for a three-level system. • Rate equations for levels S0 and S1 -3 N0,1 —concentrations, m dN1 -1 = B N − B N − A N − N B01 — absorption rate, s dt 01 0 10 1 10 1 nr 1 B10 — stimulated emission rate dN0 dN1 (B01 = B10 for nondegenerate levels) = −B01N0 + B10 N1 + A10 N1 +  nr N1 = − dt dt A10 — spontaneous emission rate γnr —rate of nonradiative decay Assume: NN 01 + =const, or d（N0 + N1）/ dt = 0 Prof. Gabriel Popescu Light Emission, Detection, and Statistics Rate Equations

• Note: The energy lost from nonradiative decay, γnr , eventually converts into heat. • For thermal equilibrium

• N1 << N0, the stimulated emission term can be neglected (B10 N1 ≈0)

• Note: The rate of decay is much higher than that of absorption, A01 + γnr >> B10. • For active medium

• N1 > N0, when there is population inversion. • The solutions for the populations of the two levels

A10 +  nr NN0 = N0 + N1 = N BA01++ 10  nr B B01 N0 − A10 N1 − nr N1 = 0 01 NN1 = BA01++ 10  nr

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Quantum Yield

• The fluorescence quantum yield • The ratio of the number of photons emitted to that of photons absorbed N A A Q = 1 10 = 10 N0 B01 A10 +  nr • For thermal equilibrium

NNBA1//() 0=+ 01 10  nr

• The quantum yield can approach unity if the nonradiative decay, γnr , is much smaller than the rate of radiative decay, A10. • Note: Even for 100% quantum yield, the energy conversion is always less than unity because the wavelength of the fluorescent light is longer (the frequency is higher, i.e. hωA>hωF ).

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Fluorescence Lifetime

• The lifetime of the excited state • The average time the molecule spends in the excited state before returning to the ground state. • For a simplified Jablonski diagram, the lifetime is the inverse of the decay rate. 1  = A10 +  nr

• Natural lifetime τ0 • The lifetime of a fluorophore in the absence of nonradiative processes, γnr →0. • Depends on the absorption spectrum, extinction coefficient, and emission spectrum of the fluorophore.

• Natural lifetime τ0 and measured lifetime τ (τ > τ0, nonradiative processes) 1 A   =  = 10 = Q  = 0 A 0 0 0 10 A10 +  nr Q

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Quenching

• Fluorescence quenching • The nonradiative decay to the ground state. • Collisional quenching • Due to the contact between the fluorophore and quencher, in thermal diffusion. • Quenchers include: halogens, oxygen, amines, etc. • Other quenching • Static quenching: The fluorophore can form a nonfluorescent complex with the quencher • The attenuation of the incident light by the fluorophore itself or other absorbing species. • Photon absorption takes place at femtosecond scale, 10-15 s, while emission occurs over a larger period of time.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Quenching

• During its lifetime, a fluorophore can interact with quenchers from within a radius of 7 nm.

2 The diffusion coefficient of oxygen in water D = 2.5 10 − 5 m ,the lifetime of a fluorophore t =10ns s The root mean squared distance that the oxygen molecule can travel during this time =x2 2. Dt

The significant distance = x 2 7 nm , the thickness of a cell membrane bilayer is 4-5 nm .

• It is possible for a molecule to absorb light outside the cell and fluoresce from the inside (and vice versa), one lifetime later.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Black Body Radiation

• Black body n = ... • An idealized model of a physical object that absorbs all incident electromagnetic radiation, and an ideal emitter of thermal radiation. • Spectral distribution of radiation by bodies at thermal equilibrium led into the development of quantum n =11 mechanics. • Thermal radiation • Reverse process to absorption: Internal energy → Thermal radiation. • Cavity mode: The spatial frequency content of n = 0 the electromagnetic field within the cavity, which satisfies the condition of vanishing electric field at the wall. Figure 6.1. Cavity modes: the sinusoids represent the real part of the electric field. All surviving field modes have zero values at the boundary. The modes are indexed by n, the number of zeros along an axis (orange dots).

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Planck’s Radiation Formula

• Planck’s Radiation Formula • Planck’s formula predicts the spectral density of the radiation emitted by a black body, at thermal equilibrium, as a function of temperature. • The radiated energy per mode, per unit volume is: hv V — Volume dN — The number of modes du( v )= 2 f dN V v — Frequency • f is the probability of occupancy associated with a given mode, obtained from the Bose- Einstein statistics. 1 f = k =1.38 10−23 J (Boltzmann’s constant) hv B K e kBT −1 • The average energy per mode: hv E = fhv = hv e kBT −1 Prof. Gabriel Popescu Light Emission, Detection, and Statistics Planck’s Radiation Formula

a b • Calculate the number of modes kz kz per frequency interval dN  / L • The spherical shell in the first z octant of the k-space is: dk min  / L 1 2  2 k x  / L k dV = 4k dk = k dk y y y k 8 2 • Volume in k-space formed by the kx k smallest spatial frequencies x  3 Figure 6.2. a) Mode distribution in a cavity. b) The cube at the origin in a) is the min smallest volume in k-space, defined by the inverse dimensions of the cavity. Vk = LLLx y z • The volume of the cavity • The number of modes per frequency interval

LLLVx y z = 2 2 dVk  V 2  V 4 v 2 4V 2 • Dispersion relation dN = = k dk = dv = v dv V min 2  3 2  3 c2 c c3 v k k = 2 Prof. Gabriel Popescu c Light Emission, Detection, and Statistics Planck’s Radiation Formula

• The radiated energy per unit frequency range and unit volume, p(v) • It is fundamental to calculating any quantity related to blackbody radiation. du(v) 8h v3 p(v) = = k =1.38 10−23 J 3 hv B K  dv c k T e B −1 d • Calculate the power flowing from the cavity • Through a surface A and element of solid angle dΩ, per unit of frequency v. A c Acosd c Acosd d 2 P(v) = du(v) = p(v)dv 2 2 2 2 Figure 6.3. Power flow out of a black body surface. dd=2 sin   c  dP(v) = du(v)A cos sind 2 0 Prof. Gabriel Popescu Light Emission, Detection, and Statistics Planck’s Radiation Formula

• The spectral exitance • The power per surface area per frequency. dP2 M = v dAd 1 2h v3 M = cp(v) = v 4 c2 hv e kT −1 • The spectral exitance in term of the wavelength dv 2hc2 1 M = M（v = c / ） =   d 5 hc ekT −1

• The Jacobian factor dv/dλ ensures that the Mλ is a distribution. Figure 6.4. Spectral exitance of blackbodies at M d= M d different temperatures.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Wien’s Displacement Law

• A function of temperature Temperature Source

• Both p(v) and Mλ exhibit a maximum at a 1850 K Candle flame, sunset/sunrise particular frequency vmax. 2400 K Standard incandescent lamps dv ( ) dM "Soft white" compact fluorescent ==v 0. 2700 K dv dv and LED lamps vv= vv= max max “Warm white” compact 3000 K • Wien’s displacement law fluorescent and LED lamps • The relationship between the temperature of the source and the peak wavelength of its emission. 3200 K Studio lamps, photofloods, etc. a a —constant 5000 K Compact fluorescent lamps (CFL) vTmax  max  T a=2,900 μmK 6200 K Xenon short-arc lamp

• Note: λmax is obtained by finding the maximum of 6500 K Daylight, overcast function M with respect to λ. λ ≠ c / v λ max max 6500 – 9500 K LCD or CRT screen • Color temperature: 15,000 – Clear blue sky • Expressing the “color” of thermal light by the 27,000 K temperature of the source. Figure 6.5. Color temperature for various thermal sources. Prof. Gabriel Popescu Light Emission, Detection, and Statistics Wien’s Displacement Law

• Wien’s displacement formula in daily activities • Humans evolved to gain maximum sensitivity of their visual system at the most dominant wavelength emitted by the Sun. • The Sun’s effective temperature is 5800K, which places its peak emission at ~500 nm (green), near the maximum sensitivity of our eye. • Dimming the light on an incandescent light bulb will result in shifting the color toward red (longer wavelengths). • Heating a piece of metal will eventually produce radiation, of red color at first and blue- white when the temperature increases further. One can say that “white-hot” is hotter than “red-hot”. • Warm-blooded animals at, say, T=310K (37 ℃), emit peak radiation at ~ 10 μm, in the infrared region of the spectrum, outside our eye sensitivity. • Some reptiles and specialized cameras can sense these wavelengths and, thus, detect the presence of such animals. • Most of the radiation is in the infrared spectrum, which we sense as heat, but only a small portion of the spectrum is visible. • Wood fire can have temperatures of 1500-2000K, with peak radiation at ~2-2.5 μm.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Stefan-Boltzmann Law

• The Stefan-Boltzmann law • The frequency-integrated spectral exitance, meaning, the exitance (in W/m2), is proportional to the 4th power of temperature. Stefan-Boltzmann constant σ = 5.67×10-8Wm-2K-4.  4 M =  M v dv = T 0 • The total power emitted by a black body is MA, where A is the area of the source. • Estimate the size of other stars: The total power emitted by the Sun (S) and the star of interest (X) PRT= 424 2 SSS TPSx RRxS=  24 TP PRTx= 4 x x xS • Gray body • A body that does not absorb all the incident radiation and emits less total energy than a black body, with an emissivity ε < 1. MT=  4

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Asymptotic Behaviors of Planck’s Formula

• Wien approximation • The asymptotic behavior of Planck’s formula (blue curve) for low temperature (high-frequency) Wien h hv>>K T hv 8 h − B hv — 3 kTB — p() v e 8h 3 kBT 3 kBT c f  e p(v)  3 v e c • Rayleigh-Jeans law • At high temperatures (low-frequency) hv<Ultraviolet catastrophe”: The amount of energy Rayleigh-Jeans, red curve) and low-temperature (high- radiated over the entire spectral range diverges. frequency, Wien, green curve).

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Einstein’s Derivation of Planck’s Formula

• The quantum of energy is assumed to be the a) Absorption difference between two atomic energy levels. B hv 12 E21−= E hv

• The atomic system exchange energy with the b) Spontaneous emission environment by 3 fundamental processes .

• Absorption: An atom in state 1 absorbs an incident A21 photon and is excited to energy level 2.

c) Stimulated emission dN 2 = B N v 2 12 1 ( ) B21 dt 1 2hv

N1 , N2 —the number density of each level ρ — the incident energy density Figure 6.7. Radiative processes in a two-level atomic system: -1 a) absorption, b) spontaneous emission, c) stimulated emission. B12 — the absorption rate, [B12]=s Note how the stimulated rather than spontaneous emission is the reversed process to absorption. Prof. Gabriel Popescu Light Emission, Detection, and Statistics Einstein’s Derivation of Planck’s Formula

• Spontaneous emission: An atom from level 2 a) Absorption decays radiatively (with emission of a photon) to the lower state. B hv 12 dN2 =−AN21 2 dt b) Spontaneous emission A12 — the spontaneous emission rate constant. A • Note: The inverse of A12 can be interpreted as the 21 decay time constant, or natural lifetime τ=1/ A12.

• Stimulated emission: Can be regarded as the c) Stimulated emission exact reverse of absorption 2 B dN 21 2 = B N ( v) 1 2hv dt 12 1 -1 B12 — the absorption rate, [B12]=s Figure 6.7. Radiative processes in a two-level atomic system: a) absorption, b) spontaneous emission, c) stimulated emission. Note how the stimulated rather than spontaneous emission is the reversed process to absorption. Prof. Gabriel Popescu Light Emission, Detection, and Statistics Einstein’s Derivation of Planck’s Formula

• Einstein combined all these processes to express the rate equations. dN dN 2 = −A N + B N ( ) − B N ( ) = − 1 dt 21 2 12 1 21 2 dt • At thermal equilibrium, the excitation and decay mechanisms must balance each other completely. Using the classic Boltzmann statistics to get the ratio of the two populations. hv — dN dN N 2 B12 (v) g2 kBT 21==0 = = e dt dt N1 A21 + B21(v) g1

A21 1  (v) = hv . g1, g2 — the degeneracy factors for the two states B21 Bg 12 1 e kTB −1 Bg21 2 • p(v) and Planck’s formula are identical, provided two conditions are met 3 A21 8hv g1B12 = g2 B21 = 3 B21 c

Prof. Gabriel Popescu Light Emission, Detection, and Statistics LASER: Light Amplification by Stimulated Emission of Radiation

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Population inversion, Optical Resonator, and Narrow Band Radiation Pump • Laser radiation can be highly a) b) monochromatic and collimated. • Note: It is highly dissipative for thermal sources (e.g. incandescent filaments) to be filtered to approach the collimation and monochromaticity of a laser. Medium Active Medium • Pumping c) II • Delivering energy to the atomic system, by 0 electrically, optically, chemically, etc. I0 • A laser is an out-of-equilibrium system. Optical amplifier • Pumping need an active medium, which acts as an optical amplifier.

• Population inversion: An atomic system in Figure 7.1. a) Pumping: empty circles represent atoms on which the population of the excited level (N1) ground energy levels and solid circles denote excited atoms. b) Active medium: population inversion is achieved. c) Active is larger than that of the lower level (N2) medium acts an optical amplifier of the incident radiation. Prof. Gabriel Popescu Light Emission, Detection, and Statistics Population inversion, Optical Resonator, and Narrow Band Radiation

• Positive feedback mechanism d) Pump f ) • Amplifying the light characterized by narrow spatial and temporal frequency bands. A B • The optical resonator (cavity) Active Medium • Converting the pump energy into a small number of modes: both longitudinal (temporal Optical resonator frequencies) and transverse (spatial e) frequencies). • Examples • Positive feedback phenomenon • When a microphone is brought close to a speaker, a very narrow sound frequency band is amplified by Figure 7.1. d) Active medium in a resonator: A, spontaneous the electronic circuit. emission, B, stimulated emission. e) Spatial frequency feedback: • Spatial, or transverse mode feedback the path in green is stable, while the one in red is not (exits the resonator). f) Temporal frequency feedback: only the modes with • Vibrations of broad frequencies are excited at the zeros at the boundary survive in the resonator: the mode in blue liquid surface, yet only those favored by the resonator (boundary conditions) become dominant. line survives, the black and red are suppressed. Prof. Gabriel Popescu Light Emission, Detection, and Statistics Gain

Active medium • Power gain in spectral irradiance, Iυ • Due to stimulated emission • The amplification is proportional to the volume of active medium, thus, distance dz. I I+ dI dI= ( ) I dz

• Gain coefficient, γ • For small signals, or weak amplification, γ does not depend on I . The small signal gain is denoted by γ . υ 0 dz 0 ( )L I( L) = I(0) e Figure 7.2. Gain through an active medium.

• γ0 <0, denoting an exponential attenuation rather than gain.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Gain

v gv() a) 1 r v v 2 v • Taking the rate equation for the populations of the 2 1 two levels into account.  dN2 = −A N g − B N   g  + B N   g  d  gv2 ()  21 2( ) 21 2( ) ( ) 12 1 ( ) ( ) dt 0 v1 v2 p(υ)—the spectral energy density of the pump 1 p(υ)dυ—the probability of emission of light in dυ 0 g(υ)—spectral line shape, probability density (emission &absorption) b) gv() • The line shape of the transition, where g1,2 denote the spectral distribution for each electronic level. g=− g ''' g   d  ( )  12( ) ( ) 22 vv12 + 

• Standard deviation of g(υ), ∆υ12. 22vv− 2 =  2 − =2  2 +  2 +   −  +  21 12 12 12 1 2 1 2 1 2 Figure 7.3. a) Electronic states, 0, 1, 2, each containing vibrational 222 2 2 2 level (v, blue lines) and respective rotational levels (r, red lines). =1 −  1 +  2 −  2 =   1 +   2 The energy distribution of the upper levels 1 and 2 are g1, g2. • The emission spectrum is broader than either of the b) The transition lineshape, g, is the cross-correlation function of individual levels, as the variances add up. g1 and g2, with the average and standard deviation as indicated. Prof. Gabriel Popescu Light Emission, Detection, and Statistics Gain

• The energy radiated per unit time and volume

22 A — the area normal to direction z d Q/)/(( dtdV= d Q dtAdz) dV — the infinitesimal volume of interest

dI d22 Q d Q dN   = = =hv2 = hv − A N − B N  g  − B N   g  d  21 2  21 2( ) ( ) 12 1 ( ) ( ) dz Adtdz dtdV dt 0  As g d =1, then − A N = − A N g  d  ( ) 21 2 21 2 ( ) 00 • Assume that the pump spectral density, ρ(υ) , is much narrower than the absorption or emission linewidth, g(υ).

 B N ''''' g − B N   g  d   21 2( ) ( ) 12 1 ( ) ( ) 0 휌 휐′ ≃ 휌 휐 𝛿 휐 − 휐′)  =B N( − ') g  ' − B N  ( − ') g  ' d  '  21 2( ) ( ) 12 1 ( ) ( ) 0

=−B21 N 2( ) g( ) B 12 N 1 ( ) g (  )

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Gain

• Invoke the relationships between Einstein’s coefficients

3 1 dI c g2 = −A21 N 2 − A 213 ().  g( )  N 2 − N 1 h dz8  h   g1 • For propagation along the z-direction 1 dI ( ) I Ic= ()  = −ANN  −   h dz21 2 h c22 ( ) ==A g( ) A g ( )  —stimulated emission cross section 21882 21 

g2 NN =21 −NN  —population inversion g1 • σ quantifies how much stimulated emission power (in W) is emitted by each atom exposed to a certain irradiance (W/m2)

• Note that the spontaneous emission term, - A21N2 , represents noise in the laser system.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Gain

• During normal laser operation, the spontaneous 퐺0, 0 emission term becomes subdominant and can be 00,G 0 () 0 ()v neglected. G0 () dI  =( ) IIN = ()   dz v  2  g2 ( ) =A21 g( )  N 2 − N 1 = (  )  N v 8 g1 • The gain reduces or saturates if I continues to υ 0  increase via amplification. Figure 7.4. The (small signal) power gain, G0, is narrower in frequency than the gain coefficient, 0. • The small signal gain, γ0 • The gain coefficient is positive only if there is a population inversion. • For positive gain

0 ( )L ILIGL( ) = (0,) 0 ( ) G0 ( v, L) = e

• G0 denotes power gain, which depends on the length of the medium and acts as a band-pass filter on the initial irradiance Iυ. Prof. Gabriel Popescu Light Emission, Detection, and Statistics Spectral Line Broadening

• Uncertainty principle • The line width of any transition has a finite bandwidth, simply because the light transit in the cavity is finite in time. • Broadening the spectral line • Homogeneous broadening, when the mechanism is the same for all atoms • Inhomogeneous broadening, when different subgroups of atoms broaden the line differently. Homogeneous Broadening • Natural (lifetime) broadening • This broadening is due to the natural lifetime, i.e. without nonradiative decay like collisions. • The natural lifetime is the largest possible, the natural line shape is the narrowest.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Homogeneous Broadening

v a) b) (t) gv1 () g() c) -|t|/τ e 21 v v2 2 21 gv() 2  v 1 cos( t ) 1 0  0=−  2  1 t Figure 7.5. a) Transition between two levels, with the lifetime as indicated. b) Spectral line of the emission. c) Temporal autocorrelation function associated with spontaneous emission, i.e., the Fourier transform of the power spectrum in b). • The natural line shape is a Lorentzian distribution, whose standard deviation diverges and, thus, cannot be used as a measure of bandwidth. 1 / 2  g ( ) = 2  − −it2 −2t ( 0 ) gd() = 1 0 1+ 2  =()t e e  0 ( 2) • The temporal behavior of spontaneous emission (inverse Fourier transform of g(υ)).

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Homogeneous Broadening

• Collision broadening • The process of collisions between atoms is essential in gas lasers. In solids, pressure broadening is absent.

• Assuming each atom experiences collisions at frequency υcol. Collisions interrupt the emission process, essentially shortening the lifetime of both levels, 1 and 2, by the same amount. col AA21=+ 21 col 1  =AA + + 2 col col 21 10 col  AA10=+ 10 col 2

푂푓푡푒 , 휐푐𝑜푙 ≫ 21, 10, 푡ℎ푒 푤푒 푔푒푡 Δ휐푐𝑜푙 ≃ 휐푐𝑜푙/휋

• Typically, collision broadening is expressed as MHz of broadening per unit of pressure, and is homogeneous.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Inhomogeneous Broadening

• Inhomogeneous broadening • Doppler broadening (velocity related) • Isotope broadening (mass related) • Zeeman splitting (nuclear spins in magnetic fields) • Stokes splitting (nuclear spins in electric fields) • Doppler broadening • At thermal equilibrium, the atoms in a gas have a Gaussian distribution of velocities along one axis (say, z), vz.

2 v 2 Mvz − z  − 2 112kT 2(v) kT B B p v dv =1 p( vz ) =  e =  e =v  ( zz) 22vv M −

M is the atomic mass, T is the absolute temperature, kB is the Boltzmann constant and ∆v is the standard deviation of the Gaussian distribution. Prof. Gabriel Popescu Light Emission, Detection, and Statistics Inhomogeneous Broadening

• Doppler frequency, υD v −= z  . D 00c • Natural line under Doppler effect

vzh1  g −=0 2 c 2 v −−   z 00 1+ c  h 2 • Inhomogeneous Doppler line broadening, which is the convolution between the Lorentzian associated with the homogeneous line, and the Gaussian velocity distribution.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Inhomogeneous Broadening

a) b) c) g() = pⓥ g g() pv()z D v 2 z vz D − c e 2v

0 D  vz 0  Figure 7.6. a) Doppler frequency shift of the “natural” transition line, for a single velocity vz. b) Maxwell-Boltzmann distribution of velocities. c) Voigt distribution: the Doppler-broadened spectral line, averaged over all velocities.

• Voigt distribution  v −−a t bt2 g= p v g  −  z dv = pⓥ g it0 D( )  ( z) 0 z gD ( t)  e e − c • The Doppler broadening is dominant over the natural broadening when g approaches a δ- function. • Doppler broadening is reduced in low-temperature gases and is absent in solid state lasers. Prof. Gabriel Popescu Light Emission, Detection, and Statistics Inhomogeneous Broadening

• Isotopic Broadening • The active medium contains isotopes of the same chemical element. • The helium-neon (He-Ne) laser, as neon occurs naturally in two isotopes 20Ne and 22Ne. • The spectral line is slightly shifted depending on which isotope emits light. • Stokes and Zeeman Splitting • Stokes and Zeeman effects describe the splitting of the line due to, respectively, electric and magnetic fields present. • These phenomena are important in solid state lasers, where atoms in the lattice are exposed to inhomogeneous (local) fields.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Threshold for Laser Oscillation

• Population inversion is not sufficient for R R 1  ()v 2 laser oscillation 0 • The resonator introduces losses which may overwhelm the gain • The threshold condition L • Upon a round trip propagation in the resonator, Figure 7.7. Losses in the resonator due to mirror reflection. the irradiance experiences a net gain.

20 ( )L e R12 R  1. • For calculating the threshold gain, the • Threshold gain coefficient stimulated emission can be ignored. 1 • As the power increases, spontaneous  =− ln(RR ) . emission can be ignored. th 2L 12 • The threshold condition acts as a filter in the frequency domain.