Fluorescence, Study notes of Law

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Light Emission, Detection, and StatisticsProf. Gabriel Popescu Lec1 Light: Generation Light Emission, Detection, and StatisticsProf. Gabriel Popescu • 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. Fluorescence Light Emission, Detection, and StatisticsProf. Gabriel Popescu • Born-Oppenheimer approximation • A molecule’s wave function can be factorized into an electronic and nuclear (vibrational and rotational) components. • 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. Jablonski Diagram total e N  =  Light Emission, Detection, and StatisticsProf. Gabriel Popescu • Symmetry property between the power spectrum of absorption (or excitation) and emission (or radiative decay) — mirror symmetry • Excitation: Absorption, S0 → S1 (higher vibrational levels)→ nonradiative decay, S1 (lowest vibrational level). • Emission: Radiatively decays, S1 → S0 (high vibrational level) → thermal equilibration, S0 (ground vibrational state). • The peaks in both absorption and emission correspond to vibrational levels, which results in perfect symmetry. Emission Spectra Ex ti n ci o n c o ef fi ci en t 1 1 ) Fl u o re sc en ce in te n si ty ( n o rm al iz ed ) Wavenumber ( 1) EmissionAbsorption 350 370 390 410 450 490 530 570 28,800 26,800 24,800 22,800 20,800 18,800 1.0 0.5 0 32 24 16 8 0 a Wavelength ( ) Figure 5.2. a) Mirror-image rule (perylene in benzene). Light Emission, Detection, and StatisticsProf. Gabriel Popescu • Note: The local maxima are due to the individual vibration levels. • The emission spectrum remains constant irrespective of the excitation wavelength (exceptions exist). • Exception to the mirror rule • Absorption spectrum: The first peak is due to S0 — S2, the larger wavenumber peak is from S0 — S1. • Emission: Occurs predominantly from S1, the second peak is absent. Emission Spectra 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. 1.0 0.5 0 , Wavenumber ( 1) Absorption Emission 280 320 360 400 440 480 520 600 10 8 6 4 2 0 35,500 31,500 27,500 23,500 19,500 15,500 Ex ti n ci o n c o ef fi ci en t 1 1 ) Fl u o re sc en ce in te n si ty ( n o rm al iz ed )b Wavenumber ( ) Light Emission, Detection, and StatisticsProf. Gabriel Popescu • The fluorescence quantum yield • The ratio of the number of photons emitted to that of photons absorbed • For thermal equilibrium • 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 ). Quantum Yield nr A A BN AN Q + == 10 10 010 101 1 0 01 10/ / ( )nrN N B A = + Light Emission, Detection, and StatisticsProf. Gabriel Popescu • 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. • 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) Fluorescence Lifetime 10 1 nrA   = + 0 10 1 A  = 0 10 10 0    Q A A nr = + = Q   = 0 Light Emission, Detection, and StatisticsProf. Gabriel Popescu • 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. Quenching Light Emission, Detection, and StatisticsProf. Gabriel Popescu • Black body • 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 mechanics. • Thermal radiation • Reverse process to absorption: Internal energy → Thermal radiation. • Cavity mode: The spatial frequency content of the electromagnetic field within the cavity, which satisfies the condition of vanishing electric field at the wall. Black Body Radiation 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). 11n = 0n = ...n = Light Emission, Detection, and StatisticsProf. Gabriel Popescu • 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: • f is the probability of occupancy associated with a given mode, obtained from the Bose- Einstein statistics. • The average energy per mode: Planck’s Radiation Formula ( ) 2 hv du v f dN V = 1 1 − = Tk hv Be f 231.38 10B Jk K −=  (Boltzmann’s constant) V — Volume dN — The number of modes v — Frequency 1− == Tk hv Be hv fhvE Light Emission, Detection, and StatisticsProf. Gabriel Popescu • Calculate the number of modes per frequency interval dN • The spherical shell in the first octant of the k-space is: • Volume in k-space formed by the smallest spatial frequencies • The volume of the cavity • Dispersion relation Planck’s Radiation Formula dk zk yk xk a xk yk zk / zL / yL/ xL b min Figure 6.2. a) Mode distribution in a cavity. b) The cube at the origin in a) is the smallest volume in k-space, defined by the inverse dimensions of the cavity. dkkdkkVd k 22 2 4 8 1   == 3 min k x y z V L L L  = dvv c V dv cc vV dkk V V Vd dN k k 2 32 22 3 2 3min 424 22      ==== x y zL L L V= 2 v k c = • The number of modes per frequency interval Light Emission, Detection, and StatisticsProf. Gabriel Popescu • A function of temperature • Both p(v) and Mλ exhibit a maximum at a particular frequency vmax. • Wien’s displacement law • The relationship between the temperature of the source and the peak wavelength of its emission. • Note: λmax is obtained by finding the maximum of function Mλ with respect to λ. λmax ≠ c / vmax • Color temperature: • Expressing the “color” of thermal light by the temperature of the source. Wien’s Displacement Law ( ) maxmax 0.v v vv v d v dM dv dv  == = = maxv T max a T   a —constant a=2,900 μmK Figure 6.5. Color temperature for various thermal sources. Temperature Source 1850 K Candle flame, sunset/sunrise 2400 K Standard incandescent lamps 2700 K "Soft white" compact fluorescent and LED lamps 3000 K “Warm white” compact fluorescent and LED lamps 3200 K Studio lamps, photofloods, etc. 5000 K Compact fluorescent lamps (CFL) 6200 K Xenon short-arc lamp 6500 K Daylight, overcast 6500 – 9500 K LCD or CRT screen 15,000 –  27,000 K Clear blue sky Light Emission, Detection, and StatisticsProf. Gabriel Popescu • 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. Wien’s Displacement Law Light Emission, Detection, and StatisticsProf. Gabriel Popescu • 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. • 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) • 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. Stefan-Boltzmann Law 4 0 TdvMM v ==   2 44S S SP R T = 2 44x x xP R T = 2 S x x S x S T P R R T P   =     4M T= Light Emission, Detection, and StatisticsProf. Gabriel Popescu • Spontaneous emission: An atom from level 2 decays radiatively (with emission of a photon) to the lower state. • Note: The inverse of A12 can be interpreted as the decay time constant, or natural lifetime τ=1/ A12. • Stimulated emission: Can be regarded as the exact reverse of absorption Einstein’s Derivation of Planck’s Formula 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. c) b) a) hv 2hv1 2 12B 21B 21A Absorption Spontaneous emission Stimulated emission 1 2 1 2 1 2 1 2 1 2 hv hv ( )2 12 1 dN B N v dt = A12 — the spontaneous emission rate constant. 2 21 2 dN A N dt = − B12 — the absorption rate, [B12]=s-1 Light Emission, Detection, and StatisticsProf. Gabriel Popescu • Einstein combined all these processes to express the rate equations. • 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. • p(v) and Planck’s formula are identical, provided two conditions are met Einstein’s Derivation of Planck’s Formula dt dN NBNBNA dt dN 1 221112221 2 )()( −=−+−=  2 1 0 dN dN dt dt = = Tk hv Be g g vBA vB N N — = + = 1 2 2121 12 1 2 )( )(   ( ) 21 21 12 1 21 2 1 . 1B hv k T A v B B g e B g  = − 212121 BgBg = 3 3 21 21 8 c hv B A  = g1, g2 — the degeneracy factors for the two states Light Emission, Detection, and StatisticsProf. Gabriel Popescu LASER: Light Amplification by Stimulated Emission of Radiation Light Emission, Detection, and StatisticsProf. Gabriel Popescu • Power gain in spectral irradiance, Iυ • Due to stimulated emission • The amplification is proportional to the volume of active medium, thus, distance dz. • Gain coefficient, γ • For small signals, or weak amplification, γ does not depend on Iυ. The small signal gain is denoted by γ0. • γ0 <0, denoting an exponential attenuation rather than gain. Gain Figure 7.2. Gain through an active medium. I Active medium I dI + dz ( )dI I dz  = ( ) ( ) ( )00 L I L I e    = Light Emission, Detection, and StatisticsProf. Gabriel Popescu • Taking the rate equation for the populations of the two levels into account. • The line shape of the transition, where g1,2 denote the spectral distribution for each electronic level. • Standard deviation of g(υ), ∆υ12. • The emission spectrum is broader than either of the individual levels, as the variances add up. Gain Figure 7.3. a) Electronic states, 0, 1, 2, each containing vibrational level (v, blue lines) and respective rotational levels (r, red lines). The energy distribution of the upper levels 1 and 2 are g1, g2. b) The transition lineshape, g, is the cross-correlation function of g1 and g2, with the average and standard deviation as indicated. ( ) ( ) ( ) ( ) ( )2 21 2 21 2 12 1 0 dN A N g B N g B N g d dt          = − − +   p(υ)—the spectral energy density of the pump p(υ)dυ—the probability of emission of light in dυ g(υ)—spectral line shape, probability density (emission &absorption) 2 22 2 2 2 12 12 12 1 2 1 2 1 2 2 22 2 2 2 1 1 2 2 1 2 = 2                = − + + − + = − + − =  +  ( ) ( ) ( )1 2' ' 'g g g d    = − )a 2 1 0 v 2v 1v 1v 2v 1( )g v 2( )g v v r )b v ( )g v 2 1v v− 2 2 1 2v v +  Light Emission, Detection, and StatisticsProf. Gabriel Popescu • The energy radiated per unit time and volume • Assume that the pump spectral density, ρ(υ) , is much narrower than the absorption or emission linewidth, g(υ). Gain ( )2 2/ ) / (d Q dtdV d Q dtAdz= ( ) ( ) ( ) ( ) 2 2 2 21 2 21 2 12 1 0 dI dNd Q d Q hv hv A N B N g B N g d dz Adtdz dtdV dt           = = = = − − −       ( ) ( )21 2 21 2 0 0 1,As g d then A N A N g d      = − = −  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 21 2 12 1 0 21 2 12 1 0 21 2 12 1 ' ' ' ' ' ( ') ' ( ') ' ' B N g B N g d B N g B N g d B N g B N g                             −   = − − −   = −   A — the area normal to direction z dV — the infinitesimal volume of interest 𝜌 𝜐′ ≃ 𝜌 𝜐 𝛿 𝜐 − 𝜐′) Light Emission, Detection, and StatisticsProf. Gabriel Popescu • 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. Spectral Line Broadening 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. Light Emission, Detection, and StatisticsProf. Gabriel Popescu Homogeneous Broadening • The natural line shape is a Lorentzian distribution, whose standard deviation diverges and, thus, cannot be used as a measure of bandwidth. • The temporal behavior of spontaneous emission (inverse Fourier transform of g(υ)). 2 1 v 2v 1v 1( )g v 2( )g v v 21 a) 0 2 1  = − b)   ( )g  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). ( ) ( ) ( ) 2 0 2 1 / 2 1 2 g        = − +  02 2( ) i t tt e e   − −  = e-|t|/τ 21 0cos( )t c) t (t) ( ) 0 1g d   = Light Emission, Detection, and StatisticsProf. Gabriel Popescu 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. • Typically, collision broadening is expressed as MHz of broadening per unit of pressure, and is homogeneous. 21 21 10 10 col col col col A A A A   = + = +  21 10 1 2 2 col colA A    = + + 𝑂𝑓𝑡𝑒 , 𝜐𝑐𝑜𝑙 ≫ 21, 10, 𝑡ℎ𝑒 𝑤𝑒 𝑔𝑒𝑡 Δ𝜐𝑐𝑜𝑙 ≃ 𝜐𝑐𝑜𝑙/𝜋 Light Emission, Detection, and StatisticsProf. Gabriel Popescu Inhomogeneous Broadening • Voigt distribution • 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. )a ( )g   D0 z D v c  )b ( )zp v zv 2 2 zv ve −  )c ( )Dg p g = ⓥ 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. ( ) ( ) 0 z D z z v g p v g dv p g c     −   = − =     ⓥ ( ) 2 0 a t bt i t Dg t e e − −  Light Emission, Detection, and StatisticsProf. Gabriel Popescu 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. Light Emission, Detection, and StatisticsProf. Gabriel Popescu Threshold for Laser Oscillation • Population inversion is not sufficient for laser oscillation • The resonator introduces losses which may overwhelm the gain • The threshold condition • Upon a round trip propagation in the resonator, the irradiance experiences a net gain. • Threshold gain coefficient • The threshold condition acts as a filter in the frequency domain. 1R 2R L 0 ( )v Figure 7.7. Losses in the resonator due to mirror reflection. ( )02 1 2 1. L e R R    ( )1 2 1 ln . 2 th R R L  = − • For calculating the threshold gain, the stimulated emission can be ignored. • As the power increases, spontaneous emission can be ignored. Light Emission, Detection, and StatisticsProf. Gabriel Popescu Laser Kinetics • Various particular cases can be solved, depending on which terms can be safely neglected. • P1 =0, no pumping of level 1. • 1/τ2 =0, no spontaneous decay of level 2 (large signal gain). • σ =0, no stimulated emission (weak signal). • 1/τ21 =0, no spontaneous emission. • dN1 /dt= dN2 /dt=0, steady state. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 22 2 2 1 2 1 21 1 2 1 1 21 N sP I sN s N s N s s hv N s N sP I sN s N s N s s hv       = − − −   = − + + −  Light Emission, Detection, and StatisticsProf. Gabriel Popescu Laser Kinetics • Example 1: P1 =0, σ =0. • Using partial fraction decomposition (or expansion) • The shift property of the Laplace transform ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 22 2 2 1 2 1 21 1 2 1 1 21 N sP I sN s N s N s s hv N s N sP I sN s N s N s s hv       = − − −   = − + + −  ( ) ( ) ( ) 2 2 2 2 1 1 21 1 1 P s N s s N s s N s      + =      + =    ( ) 2 2 2 2 2 2 1 1 11 P N s P s s s s       = = −    + +       ( ) ( )atf s a e f t−+ → ( ) 2 2 2 2 1 t N t P e  −  = −      Light Emission, Detection, and StatisticsProf. Gabriel Popescu Laser Kinetics • Expressing N1 (s) as a sum of simple fractions, of the form a/(s+b). 2 1 1 2 1 1 1 1 1A s s B s s C s s            + + +  + +  + =              ( ) 2 1 21 2 1 1 1 1 P N s s s s    =    + +      ( ) 2 1 21 2 1 1 1 P A B C N s s s s         = + +  + +    2 1 1 2 1 2 1 2 1 2 1 1 2 21 12 1 1 20 0 0 1 1 1 1 1 1 1 11 1 s s s A B C s s s s s s                  = + = + = = = = = = =        − −+ + + +              ， ， Light Emission, Detection, and StatisticsProf. Gabriel Popescu Gain Saturation • The effect of stimulated emission (σ ≠0) • Assume τ1 =0, level 1 decays infinitely fast, i.e., N1 =0 • The population dynamics of level 2 in the Laplace domain. • New lifetime τ2’, adjusted for the effects of stimulated emission. • Whenever I approaches Is , the gain saturates. (I=Is, τ2’= τ2 /2, faster depletion of level 2) ( ) 2 2 2 1 . I P N s s h s      + + =    ( ) ( ) ( ) ( )22 2 2 1 2 N sP I sN s N s N s s hv    = − − −  2 2 2 2 1 1 1 1 1 ' s I I h I          = + = +       2 s saturation irrad n ehI ia c  = ( ) 2 ' 2 2 2 ' 1 t N t P e  −  = −      2N 2N t sI I 𝑃2𝜏2/3 𝑃2𝜏2 Figure 7.10. Effects of the saturation intensity on the N2 kinetics. Light Emission, Detection, and StatisticsProf. Gabriel Popescu Gain Saturation • Saturation at steady state • The general equations at steady state. • The gain is obtained as σ(N2－N1). • Note: γ0 is small signal gain, i.e. I →0, γ → γ0. • For τ2 >> τ1, or τ21 = τ2 , Is=hv/στ2, the gain drops to ½ γ0. ( ) ( ) 2 2 2 1 2 2 1 1 2 1 21 1 0 0 N I P N N hv N NI P N N hv      − − − = + + − − = 1 2 2 1 1 21 2 1 1 2 1 2 21 1 1 P P N N I hv             − −    − =   + + −    ( ) ( )0 1 s v v I I   = + ( ) ( ) ( ) 1 0 2 2 1 1 21 2 1 2 2 21 1 1 1 1 s hv v v P P I v                = − − =        + −   