Lecture 13: Laser-Induced Fluorescence: Two-Level Model

1. Introduction and background Laser-Induced Fluorescence 2. Typical experimental setup 3. Signal level (steady & pulsed) Possible collisional change 4. Two-level model 2 LIF 5. Detection limits (pulsed laser) hν (spont. emission) 6. Characteristic times Laser absorption 7. Modifications to two-level model 1 1. Introduction

 What is Laser-Induced Fluorescence (LIF)?

 Laser-induced spontaneous emission

 Multi-step process: absorption followed by emission (fluorescence) Possible collisional change This is the laser- 2 induced fluorescence

hν (spont. Emission) = LIF signal Laser absorption (pulsed or CW laser) 1

 Notes . May occur from multiple states . Not instantaneous 2 1. Introduction

 Why is LIF of interest?

o  The LIF signal can be monitored at 90 to the exciting laser beam, thereby gives spatial resolution to the absorption measurement

 The signal rides on a dark background, rather than being based on signal differences as in absorption

 Species specific, and much stronger than Raman scattering

Volume element = A·L (~1mm3) L Waist

LIF Trans. Int. Ω l Collection optics d

λexc • Signal for scanned- LIF 3 1. Introduction

 LIF can be used to monitor multiple gas properties including

 ni (n,v,J) number density of species i in a state described by n, v, and J  T temperature (from the Boltzmann fraction)

 χi species concentration  P pressure (from line broadening)

 v velocity (from the Doppler shift of the absorption frequency)

 And species including

 Radicals: OH, C2, CN, NH, …

 Stable diatoms: O2, NO, CO, I2, …

 Polyatomics: NO2, NCO, CO2, Acetone (CH3COCH3), Biacetyl ((CH3CO)2), Toluene (C7H8), and other carbonyls and aromatic compounds

 As well as many atoms….

4 1. Introduction

Absorption LIF

Possible collisional 2 change 2 LIF Laser absorption hν (spont. emission) 1 Laser absorption 1

NOT instantaneous Instantaneous Multi-step process: absorption followed Single-step process after some delay by spontaneous emission

Line-of-sight Typically 90o to the excitation laser beam

Based on incident/transmitted Rides on a dark background signal difference

 Absorption and LIF can be performed with either CW or pulsed lasers 5 1. Introduction

 History & Some Key Accomplishments

 Flash-lamp pumped dye laser (Schafer, 1966)

 Tunable CW dye laser (1970)

 CW-doubled ring dye laser (1980)

 Nd-YAG and excimer-pumped dye lasers (1980)

 CW -scanned LIF (1980s)

 First PLIF imaging in laminar combustion (1982), turbulent (1984)

 Single-laser-pulse PLIF imaging of NO, T, and velocity (1989) OH PLIF in  Identification of ketone tracers for PLIF imaging (1991) laminar flame  M>1 Combustion: NO, T imaging (1993), OH imaging (1996)

 Extension of PLIF to high pressure/temperature (1993-2005)

 PLIF with vibrational transitions (IR-PLIF) (2000)

 Pulse-burst PLIF (2000)

 UV- PLIF of CO (2004) 2 OH PLIF in  High-speed PLIF (2005) spray flame  CW PLIF (2009) 6 2. Typical Experimental Setup

 Typical experimental setup for LIF systems Volume element Experiment (e.g. flame, flow, plasma) Excitation λex (~1mm3) Laser Trap or powermeter

Ω l Collection optics d PMT with spectral filter to record emission at

λex or at λdet (≠ λex), or grating monochromator  Recording modes to disperse the spectrum 1. Emission intensity at a fixed detection wavelength

2. Fluorescence spectrum (spectrometer) I 3. Excitation spectrum (scanned laser) Pulsed laser

4. Can be extended to 2-d with CCD LIF (decay rate array detector (PLIF) contains info on “quenching rate” 5. Can also measure temporal behavior 7 2. Typical Experimental Setup

 Measurement volume

Volume element 3 = A·L (~1mm ) L Waist Area = A

Ω l Collection optics d

. Pathlength: L ≈ 0.5 – 5mm lens area . Solid angle of collection:  Note:   2 l At f # = 2.5, Ω/4π = 0.01, d 2 / 4 1 or 1%. Collection process is   l 2 f # 2 relatively inefficient, even for l fast lenses. . f # of collection: f #  d 8 3. Signal Level

 Spontaneous emission corresponds to a decay in energy 2 Spontaneous Emission

A21 1  Steady conditions   S 21# photons/sec  N  A   n V  A  F 2 21 4 2 21 4 N2 = # molecules in the measurement volume in state 2

n2 = number density of molecules (#/cc) in state 2 V = volume (cc)

A21 = probability/sec of emission from state 2 to state 1 Ω/4π = collection fraction

21 SF power collected  SF # photons/sech S21 F  ∴ The LIF signal depends on constants and n2. Hence, the challenge of LIF is to find a way of specifying n2 quantitatively 9 3. Signal Level

 Spontaneous emission corresponds to a decay in energy 2 Spontaneous Emission

A21 1

 Pulsed conditions For unsteady conditions (e.g., short times or pulsed excitation):

 21  SF # photons  n2 t dtV  A21  0 4

n2(t) depends on the laser and the collision process (further explained later)

10 4. Two-level Model

 Entry-level model for LIF (two levels)

Induced Induced Collisional Collisional Fluorescence Absorption Emission Excitation De-Excitation 2

Energy n1W12 n2W21 n1Q12 n2Q21 n2A21

negligible “quenching”

 B I  12      d    Einstein theory prob/sec12  B12  d      d 1    I / c 

 Let B12  B12 / c 11 4. Two-level Model

 Entry-level model for LIF (two levels) -1 Iν Laser (~0.5cm FWHM) for typical pulses

Most pulsed lasers are spectrally broad compared ν0 ν with absorption lines  I ≈ constant over abs. line Absorption line  ν (~0.1cm-1 at STP)   Rate12  n1 B12 I  d  n1B12 I  n1W12 Probability per second of line absorption per unit spectral Rate (s-1) that individual molecules in intensity: B  B  / c state 1 undergo the transition to state 2 12 12 Rate analysis n  n I B  n I B  Q  A  S.S. W 2 1 12 2  21 21 21 n  n 12 2 SS 1  n1W12  n2 W21  Q21  A21 W21  Q21  A21 n2  0 So now we have a solution for n ! 2 12 4. Two-level Model

 Entry level model for LIF (two levels)

W12 n2 SS  n1 W21  Q21  A21

Two limits emerge from the steady-state analysis for n2:  Weak excitation (“linear LIF”)

 Strong excitation (“saturated LIF”)

Weak excitation limit: Induced emission from level 2 much weaker than the sum of collisional and spontaneous decay processes g1B12  g2 B21 W  B I  A  Q 21 21  21 21 g1W12  g2W21 0 0 0 0 Then n 2  n 1 , andn1  n1 (usually n 1  n where n is the conserved total 0 number density,n  n1  n2 ) 0 W12 n2  n1 A21  Q21 13 4. Two-level Model

 Weak excitation limit   Fluorescence signal SF  n2 V  A21  photons/s in weak excitation limit 4  A     0  21   n1V W12  B12 I        A21  Q21   4  photons absorbed/sec    "fluor. yield" fraction collected

Notes: 0 1. To find n 1, need to know Q21 (rate of the electronic quenching, a collisional process), A21, Iν,  and  0 2. SF  n1  ni fv,J T  LIF is directly proportional to the population density in the lower quantum level  LIF is typically viewed as a species diagnostic 3. Alternative view: # of absorptions/sec  # molec in 1prob/s of abs 0  n1V B12 I 14 4. Two-level Model

 Consider the saturation limit Induced emission rate >> collisional and spontaneous emission rates W21  A21  Q21 . Recall:

g1B12  g2 B21 W12 B12 g2 n2  n1  n1  n1 W B g so g1W12  g2W21 Notes: 21 21 1

1. If g2 = g1, then n2 = n1, when the transition is “saturated” 0 0 0 n1 n1 2. n1  total  n1  n2  n2   when g2  g1 1 g1 / g2 2  g   Fluorescence signal S  n0  2 V  A  photons/s in saturation limit F 1   21  g1  g2  4

Virtue: does not depend on Q21, laser intensity Complication: may be difficult to reach the saturation limit in all parts of the laser beam If intensity not sufficiently high, may reach intermediate situation between weak excitation and saturation 15 4. Two-level Model

 Intermediate result W 12 1 n2 SS  n1 0 B  Q  A  W21  Q21  A21 12 21 21 n2  n1 1  0 B12  B21  B12  B21 I  with n1  n2  n1

g1B12  g2 B21

B12 /B12  B21  g2 / g1  g2

S n 1 F  2  S n Q  A F ,sat 2,sat 1 21 21 B12  B21 I

16 4. Two-level Model

 Typical values for A and Q in electronic transitions 5 8 -1 6 -1 8 -1 1. A21 ≈ 10 ~ 10 s (10 s for NO, OH; 10 s for Na) Q  Zcollision frequency OH 2. 21 Note: A21 / Q21 P1atm,comb. T 2  n c  8kT / ~ 106 s1 /1010 s1   2 P / T ~ 104  9 10 1  A small number! Q21 10 10 s  at STP

3. Fluorescence yield = A21/(A21+Q21) << 1 except at low P  LIF is an inefficient process!

4. Spontaneous emission for IR is much weaker (smaller A21) than UV  LIF is weaker in the IR, unless Q21 is small (as may be the case!)

A21,IR / A21,UV 1 Q / Q  ? 21,IR 21,UV 17 4. Two-level Model

 Typical values for A and Q in electronic transitions

5. What is Iν for B21Iν >>A21+Q21? Call this (Iν)sat   . For low Q21, Iν,sat >>A21/B21 I  P / T 1/ B  . For high Q21, Iν,sat >>Q21/B21 so that  ,sat 21   3 3 I B . B  A /8h  B  B  21  A 21 21 21 21 c 21 8hc 2 @ λ = 500nm, B21  25A21cm /erg s

4 10 -1 6 -1 . Assume Q21 ≈ 10 A21 (e.g., Q21 = 10 s , A21 = 10 s ) 4 I ,sat 10 A21 / 25A21  erg/s   400 ergs/cm2   cm2 Hz  0.4 J/m2 18 4. Two-level Model

 Typical values for A and Q in electronic transitions - continued

5. What is Iν for B21Iν >>A21+Q21? Call this (Iν)sat

Power -1 10 -1 . But I  ; let d = 1mm, ∆νL = 0.5cm = 1.5 x 10 s Area  laser PW107 ergs/J  ergs  then I   PW 0.08  .8102 cm2 1.51010 s1  cm2W  400ergs/cm3  P W   5103 W  5kW! sat 0.08 . Far too high for a CW laser, but… . Note: Nd:YAG-pumped dye lasers give 1-10mJ per 10nsec pulse at 225 or 300nm, giving 105-106W! . Therefore, it’s easy to saturate within the limits of a 2-level model! In fact, with atoms it’s easy to reach full saturation; with molecules it is not so easy get full saturation, owing particularly to population exchanges with adjacent rotational states, but saturation is feasible at P ≤ 1atm 19 4. Two-level Model

 Typical values for A and Q: Vibrational transitions (IR)

1 2 -1 -1 1. A21 ≈ 10 ~ 10 s (30 s for CO, ∆v = 1) for strong IR bands

2. Q21 is either the vibration-translation (V-T) de-excitation rate or vibrational-vibrational (V-V) transfer rate (to another species). The dominant process is usually V-V.

5 -1 e.g., QV-V ≈ 10 s for CO with N2 near STP

3. Therefore, the fluorescent yield is typically in the range 10-3 –10-5. This is sufficient for IR LIF to be a useful diagnostic, providing a means of imaging many species not accessible via electronic transitions.

20 5. Detection Limits (Pulsed Laser)

 Fluorescence signal level in the weak excitation limit n0 nXf 1  i v,J S  S t dt photons F  F n = total number density A   n0V  I dt  B 21 X = mole fraction 1   12 i A21  Q21 4 fv,J = fraction of molecules initially in v and J E laser pulse energy I dt  p  but    L  Ac laser linewidth cross -sectional area of the exciting beam 

0 Ep A  SF  n1 L B12 assuming A21 << Q21  L Q 4

Length of the measurement zone L = V/Ac

21 5. Detection Limits (Pulsed Laser)

 Fluorescence signal level in the weak excitation limit: Example -5 2 . Laser pulse energy EP = 10 J (10μJ) = 10 ergs -8 . Pulse length τL = 10 s -1 . Pulse linewidth ∆νL = 1 cm . Number density n = 5 x 1018 molecules/cc (at 1 atm, 1620K)

. Boltzmann fraction fv,J = 0.01 (1% of the species is in the absorbing state) -2 2 -1 . Measurement volume V = Ac x L = (10 cm )(10 cm) = 0.001 cc . Collection angle Ω/4π = 10-3 (for optics with f # = 8) -4 2 6 -1 . Einstein coefficients A/Q ≈ 10 , B12 ≈ 20 · A21 [cm /erg·s], A21 = 10 s Using these values:  E ergs S  51018  X 102 101  P 20106 104 103 (photons) F  i    s1   0  L  B n1 12 E ergs X = mole fraction of the absorber i 1016  X  P i i 1cm1 31010 cm/s 7  310  X i 22 5. Detection Limits (Pulsed Laser)

 Fluorescence signal level in the weak excitation limit: Example # of e produced Quantum efficiency   # of photons input Total photoelectrons produced by photons S  incident on a quantum detector F

For ideal, shot-noise-limited detection SNR  SF 7 SF  310  X i   0.1 6 SNR = 1.7 for Xi = 1 ppm SNR  310  X i Xi = 0.3 ppm for SNR = 1

Impressive for a non-intrusive measurement made in a 1 mm3 volume in 10 nsec and with only 10 μJ of laser energy!

23 5. Detection Limits (Pulsed Laser)

 Saturation limit

  0 g2   laser  SF  n1V A21  dt  L  photons 0 g1  g2 4   For the same typical conditions

18 2 3 6 1 3 8 SF  510  X i 10 10 cc0.5  g2 /g1  g2 10 s 10 10 s 8  2.510  X i   0.1 7 SNR  2.510  X i SNR = 5 for Xi = 1 ppm

Xi,min = 0.04 ppm for SNR = 1

Thus, if we can saturate, the minimum detection limit ≈ 0.04ppm! But in molecules, the dominant collisional rate (in the rate equation analysis) tends to be rotational transfer, and because this rate is typically larger than

Qelec, saturation is difficult to achieve except at reduced pressures.

24 6. Characteristic Times

 Pulsed excitation Induced Induced Collisional Absorption Emission De-Excitation Fluorescence Laser pulse 2 Iν τlaser~5-20ns Energy W12 W21 Q21 A21

t 1  Recall:S t # photons/s  n t V  A  at any time t F 2 21 4 hence we need expressions for n2

W12 . n2 SS  n1 for any Iν W21  A21  Q21

0 W12  n1 W12 W21  A21  Q21 Applicable to pulsed experiments IF there is time to reach steady- weak 0 W12 . n2 SS  n1 state, i.e., τSS< τLaser A21  Q21 25 6. Characteristic Times

 Pulsed excitation: what is τSS

W12 0 W12 n2 SS  n1  n1 W21  A21  Q21 W12 W21  A21  Q21

weak 0 W12 n2 SS  n1 A21  Q21 Consider idealized step changes in I . ν n    2 SS Define τSS (characteristic time to reach steady-state) SS  n2 initial

1  SS  W12 W21  A21  Q21 1  decay  A21  Q21

26 6. Characteristic Times

 Two cases:

Weak Excitation Strong Excitation

W12 << Q21, A21 << Q21  τSS ≈ 1/Q21 W12 >> Q21 >> A21  τ ≈ 1/(W +W ) << 1/Q E.g., Q ≈ 109 –1010 s-1, τ ≈ 0.1 – 1ns SS 12 21 21 21 SS -9 For typical Q21, τSS << 10 s, SS SS relation applies at virtually all approximation is good on most time values of time in the laser pulse for scales of interest for strong excitation pulses ~ 10ns and longer (except, perhaps, for ultrafast lasers) 27 7. Modifications of the Two-level Model

 Recall: Entry level two-level model

W12 n2 SS  n1 W21  Q21  A21

Weak excitation limit Saturation limit

0  A21      g   S  n V W  B I     0  2  F 1 12 12    SF  n1  V  A21     A21  Q21   4  g  g 4 photons absorbed/sec   1 2    fraction collected "fluor. yield" photons/s photons/s 28 7. Modifications of the Two-level Model: Two Issues

 Hole-burning effects “Hole” burning (saturation due to Inhomogeneous depletion of a certain velocity class) (velocity) broadening can occur with intense, spectrally narrow lasers  Multi-level effects Upper and lower levels are really manifold states

Let’s consider multi-level effect 29 7. Modifications of the Two-level Model

 Multi-level effects Upper and lower levels are really manifold states

LIF signal and fluorescence quantum yield depend on collisional transfer rates among upper levels and the number of these monitored by the detection system. . Two limits: narrowband detection and broadband detection

 Narrowband detection: emission is collected only from v', J'

 Sum of A2i for allowed and collected transitions n2  n1B12 I  n2 B21I  A21  Qelec  Qrot  Qvib  Weak excitation B12 I 0 A21  n2 SS  n1 SF  n1 V  I B12   A21  Q A21  Q 4 In weak excitation limit The effective loss or “quenching” rate includes

Qrot, Qvib, Qelec (i.e., all processes removing molecules from the upper state observed by fluorescent emission) 30 7. Modifications of the Two-level Model

 Multi-level effects Upper and lower levels are really manifold states

 Broadband detection: from v' and all J'; i.e., the “2” state includes all rotational levels in v' (could also include collection from all v’ & J’) In weak excitation limit B I A  n  n 12  S  n V 0  I B  21  2 SS 1 A  Q F 1  12 21 A21  Q 4

Includes only Qelec! i.e., those collisions that preclude emission in collection bandwidth 31 7. Modifications of the Two-level Model

 Multi-level effects Upper and lower levels are really manifold states

 Conclusion: collection bandwidth defines “2”!

0 A21  SF  n1 V  I B12   A21  Qeff 4

Effective average of Effective transfer rate from states “2” emitting states “2” defined by the collection wavelength region

32 7. Modifications of the Two-level Model: Example Multi-Level Effects (OH)

 Fluorescence spectrum of OH LIF in a 30-torr flame at two different time delays relative to the beginning of the laser (Lucht et al., 1986)

2 2 + The P1(5) line of the OH X Π-A Σ (0,0) band was directly excited by the laser. The fluorescence lines labeled P12(5), P1(5), Q1(4), Q12(4), R1(3) all have the same upper level (Dicke and Crosswhite, 1962)

Delay = 4ns Delay = 10ns

. Note the evolution of fluorescence spectrum at different delays 33 7. Modifications of the Two-level Model Simple Models

 Fluorescence yield: multi-level models

-1 AP, AQ, and AR = A-coefficients (in s ) for P, Q, R branch emission from v', J' of an atom or molecule -1 Qel = electronic quench rate [s ] -1 Qrot = rotational transfer rate [s ] Assume these rates are the same for all J'

 “Narrowband collection” (single line) Fluorescence yield (FY)

= fraction of absorbed photons emitted into the collection bandwidth, e.g., to capture AP probability or rate of desired process FY  sum rate of all removal processes

A A FY  P  P usually AP  AQ  AR  Qrot  Qel Qrot

Qrot is usually  Qel  AP , AQ , AR But signal is weak! 34 7. Modifications of the Two-level Model

 Fluorescence yield: multi-level models

 “Broadband collection”; i.e., from P, Q & R of all J' in v' Consider a sequence of time steps, each with 1 of 3 outcomes: A, Qel, or Qrot, and let A = AP+AQ+AR A   A  Qrot  Qel Q …   rot A  Qrot  Qel   2 Absorption FY1   FY2   FY3   … Success rate for    n A remaining in v' FY   FYi     n0 1  A  Qel Note: Stimulated emission not included  “weak excitation” limit or linear LIF regime Conclusion: (1) Strong signals; (2) FY is the same as 2 level model! 35 Next Lecture – LIF/PLIF of Small Molecules

 LIF is spatially-resolved with signal from a point or a line

 Excite LIF with the laser expanded into a narrow sheet and collect the fluorescence with a camera : Planar Laser-Induced Fluorescence or PLIF

 Applications of PLIF to small (diatomic) radicals (e.g., OH & NO)