Lecture 13: Laser-Induced Fluorescence: Two-Level Model · Lecture 13: Laser-Induced Fluorescence: Two-Level Model 1. Introduction and background 2. Typical experimental setup 3

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

1. Introduction and background

2. Typical experimental setup

3. Signal level (steady & pulsed)

4. Two-level model

5. Detection limits (pulsed laser)

6. Characteristic times

7. Modifications to two-level model

Laser-Induced Fluorescence

1

2

Laser absorption

Possible collisionalchange

hν (spont. emission)

LIF

What is Laser-Induced Fluorescence (LIF)? Laser-induced spontaneous emission Multi-step process: absorption followed by emission

(fluorescence)

Notes May occur from multiple states Not instantaneous

2

1. Introduction

2

1

Laser absorption (pulsed or CW laser)

hν (spont. Emission) = LIF signal

This is the laser-induced fluorescence

Possible collisional change

Why is LIF of interest?

The LIF signal can be monitored at 90o 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

3

1. Introduction

Trans. Int.

LIF

λexc

Collection optics

l

d

Ω

Volume element = A·L (~1mm3) L Waist

• Signal for scanned- LIF

4

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….

1. Introduction

1. Introduction

5

Absorption LIF

InstantaneousSingle-step process

NOT instantaneousMulti-step process: absorption followed after some delay by spontaneous emission

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

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

1

2

Laser absorption

Possible collisionalchange

hν (spont. emission)

LIF

1

2

Laser absorption

Absorption and LIF can be performed with either CW or pulsed lasers

6

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) Identification of ketone tracers for PLIF imaging (1991) 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 CO2 (2004) High-speed PLIF (2005) CW PLIF (2009)

1. Introduction

OH PLIF in spray flame

OH PLIF in laminar flame

Typical experimental setup for LIF systems

Recording modes1. Emission intensity at a fixed detection wavelength2. Fluorescence spectrum (spectrometer)3. Excitation spectrum (scanned laser)4. Can be extended to 2-d with CCD

array detector (PLIF)5. Can also measure temporal behavior

7

2. Typical Experimental Setup

Experiment (e.g. flame, flow, plasma)

Laser Trap or powermeter

PMT with spectral filter to record emission at λex or at λdet (≠ λex), or grating monochromatorto disperse the spectrum

Collection optics

l

d

Ω

Volume element (~1mm3)Excitation λex

IPulsed laser

LIF (decay rate contains info on “quenching rate”

Measurement volume

8

2. Typical Experimental Setup

Pathlength: L ≈ 0.5 – 5mm

Solid angle of collection:

f # of collection:

Area = A

Collection optics

l

d

Ω

Volume element = A·L (~1mm3) L Waist

22

2

2

# 14/

area lens

fldl

dlf #

Note:At f # = 2.5, Ω/4π = 0.01,

or 1%. Collection process is relatively inefficient, even for fast lenses.

Spontaneous emission corresponds to a decay in energy

Steady conditions

∴ 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

2

1

A21

Spontaneous Emission

44

cphotons/se # 21221221

AVnANSFN2 = # molecules in the measurement volume in state 2n2 = number density of molecules (#/cc) in state 2V = volume (cc)A21 = probability/sec of emission from state 2 to state 1Ω/4π = collection fraction

hSS FF cphotons/se #collectedpower 21

21FS

n2

Spontaneous emission corresponds to a decay in energy

Pulsed conditions

10

3. Signal Level

2

1

A21

Spontaneous Emission

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

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

4photons # 210 2

21 AVdttnSF

Entry-level model for LIF (two levels)

11

4. Two-level Model

2

1

n1W12

Induced Absorption

dBd1221prob/sec

n2W21 n1Q12 n2Q21 n2A21

Induced Emission

CollisionalExcitation

CollisionalDe-Excitation Fluorescence

Energy

negligible “quenching”

cI / 1 d

Einstein theory

IB12

Let cBB /1212

Entry-level model for LIF (two levels)

12

4. Two-level Model

Laser (~0.5cm-1 FWHM) for typical pulses

ν

Iν

Absorption line (~0.1cm-1 at STP)

ν0Most pulsed lasers are spectrally broad compared with absorption lines Iν ≈ constant over abs. line

121121line 12121Rate WnIBndIBn

Rate (s-1) that individual molecules in state 1 undergo the transition to state 2

Probability per second of absorption per unit spectral intensity: cBB /1212

2121212121

21212121212

AQWnWnAQBInBInn

Rate analysis

212121

121SS2 AQW

Wnn

S.S.

02 nSo now we have a solution for n2!

Entry level model for LIF (two levels)

13

4. Two-level Model

212121

121SS2 AQW

Wnn

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

Then , and (usually where is the conserved total number density, )

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

21212121 QAIBW

011 nn 12 nn 00

1 nn 0n21

0 nnn

2121

12012 QA

Wnn

212121

212121

WgWgBgBg

Weak excitation limit

14

4. Two-level Model

collectedfraction yield" "fluor.

2121

21

ecabsorbed/s photons

121201

212

4

photons/s 4

QAAIBWVn

AVnSFFluorescence signalin weak excitation limit

Notes:1. To find , need to know Q21 (rate of the electronic quenching,

a collisional process), A21, Iν, and2.

LIF is directly proportional to the population density in the lower quantum level LIF is typically viewed as a species diagnostic

3. Alternative view:

TfnnS JviF ,01

IBVn 12

01

abs of prob/s1in molec #s/secabsorption of #

01n

Consider the saturation limit

15

4. Two-level Model

Notes:1. If g2 = g1, then n2 = n1, when the transition is “saturated”

2. 12

01

21

01

22101 when

2/1total ggn

ggnnnnn

Induced emission rate >> collisional and spontaneous emission rates

212121 QAW

1

21

21

121

21

1212 g

gnBBn

WWnn

Recall:

so 212121

212121

WgWgBgBg

photons/s 421

21

201

AVgg

gnSFFluorescence signalin saturation limit

Virtue: does not depend on Q21, laser intensityComplication: 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

Intermediate result

16

4. Two-level Model

0121 nnn with

1

2112

2121

2112

12012 1

IBB

AQBB

Bnn

212121

121SS2 AQW

Wnn

212211212

212121

// gggBBBBgBg

IBBAQn

nSS

satsatF

F

2112

2121,2

2

, 1

1

Typical values for A and Q in electronic transitions1. A21 ≈ 105 ~ 108 s-1 (106 s-1 for NO, OH; 108 s-1 for Na)

2.

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!)

17

4. Two-level Model

TP

kTcn

ZQ

/

/8

frequencycollision

2

2

21

STPat s1010 110921

Q

Note:

A small number!

4

11016

OHT comb.atm,12121

10~s10/s10~

/

PQA

?/1/

UV,21IR,21

UV,21IR,21

QQAA

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

For high Q21, Iν,sat >>Q21/B21 so that

@ λ = 500nm,

Assume Q21 ≈ 104A21 (e.g., Q21 = 1010s-1, A21 = 106s-1)

18

4. Two-level Model

21, /1/ BTPI sat

hcA

cBBBhAB I

88/

3

2121

21213

2121

s/ergcm25 22121 AB

2

22

21214

,

J/m4.0Hzcm

erg/sergs/cm400

25/10

AAI sat

Typical values for A and Q in electronic transitions - continued

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

But ; let d = 1mm, ∆νL = 0.5cm-1 = 1.5 x 1010s-1

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

laserAreaPower

I

5kW!W1050.08ergs/cm400W 3

3

satP

then

Wcmergs08.0W

s105.1cm108.ergs/J10W

211022

7

PPI

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

1. A21 ≈ 101 ~ 102 s-1 (30 s-1 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.

e.g., QV-V ≈ 105 s-1 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

4. Two-level Model

Fluorescence signal level in the weak excitation limit

21

5. Detection Limits (Pulsed Laser)

42121

2112

01

QAABdtIVn

photonsdttSS FFJv,i

01 fnXn

n = total number density

Xi = mole fraction

fv,J = fraction of molecules initially in v and J

beam exciting theof area sectional-crosslinewidthlaser energy pulselaser

cL

p

AE

dtI

41201

QAB

ELnS

L

pF

Length of the measurement zone L = V/Ac

assuming A21 << Q21

but

Fluorescence signal level in the weak excitation limit: Example

22


EP = 10-5J (10μJ) = 102 ergs τL = 10-8 s ∆νL = 1 cm-1

n = 5 x 1018 molecules/cc (at 1 atm, 1620K) fv,J = 0.01 (1% of the species is in the absorbing state) V = Ac x L = (10-2 cm2)(10-1cm) = 0.001 cc Ω/4π = 10-3 (for optics with f # = 8) A/Q ≈ 10-4, B12 ≈ 20 · A21 [cm2/erg·s], A21 = 106 s-1

Laser pulse energyPulse length

Pulse linewidthNumber density

Boltzmann fractionMeasurement volume

Collection angleEinstein coefficients

i

Pi

BL

P

n

iF

X

EX

EXS

7

10116

3461

1218

103cm/s103cm1

ergs10

10101020s

ergs1010105120

1

Xi = mole fraction of the absorber i

Using these values:(photons)

Fluorescence signal level in the weak excitation limit: Example

23


Quantum efficiency

Total photoelectrons produced by photons incident on a quantum detector

input photons of #produced of #

e

FS

For ideal, shot-noise-limited detection FSSNR

1.0103 7

iF XS

iX 6103SNR SNR = 1.7 for Xi = 1 ppmXi = 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!

Saturation limit

24


photons 4

laser

02121

201

LF dtAgg

gVnS

For the same typical conditions

i

iF

X

gggXS

8

8316212

3218

105.2

s1010s10/5.0cc1010105

iX 7105.2SNR1.0

SNR = 5 for Xi = 1 ppmXi,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.

Pulsed excitation

25

6. Characteristic Times

2

1

W12

Induced Absorption

W21 Q21 A21

Induced Emission

CollisionalDe-Excitation Fluorescence

Energy

Laser pulseτlaser~5-20ns

t

Iν

Recall: at any time t

hence we need expressions for n2

for any Iν

2121

1201

weakSS2

21212112

1201

212121

121SS2

QAWnn

QAWWWn

QAWWnn

Applicable to pulsed experiments IF there is time to reach steady-state, i.e., τSS< τLaser

4

photons/s # 212

AVtntSF

Pulsed excitation: what is τSS

26


2121

1201

weakSS2

21212112

1201

212121

121SS2

QAWnn

QAWWWn

QAWWnn

Consider idealized step changes in Iν.Define τSS (characteristic time to reach steady-state)

initial2

SS2SS n

n

21212112SS

1QAWW

2121decay

1QA

Two cases:

27


Weak ExcitationW12 << Q21, A21 << Q21 τSS ≈ 1/Q21

E.g., Q21 ≈ 109 – 1010 s-1, τSS ≈ 0.1 – 1nsSS relation applies at virtually all values of time in the laser pulse for pulses ~ 10ns and longer

Strong ExcitationW12 >> Q21 >> A21

τSS ≈ 1/(W12+W21) << 1/Q21

For typical Q21, τSS << 10-9s, SSapproximation is good on most time scales of interest for strong excitation (except, perhaps, for ultrafast lasers)

Recall: Entry level two-level model

28

7. Modifications of the Two-level Model

Weak excitation limit

212121

121SS2 AQW

Wnn

photons/s

4collectedfraction yield" "fluor.

2121

21

ecabsorbed/s photons

121201

QA

AIBWVnSF

photons/s 421

21

201

AVgg

gnSF

Saturation limit

Hole-burning effects

Multi-level effects

29

7. Modifications of the Two-level Model: Two Issues

Inhomogeneous (velocity) broadening

“Hole” burning (saturation due to depletion of a certain velocity class)

can occur with intense, spectrally narrow lasers

Upper and lower levels are really manifold states

Let’s consider multi-level effect

Multi-level effects

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

30



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

vibrotelec QQQAIBnIBnn 212121212 Sum of A2i for allowed and collected transitions

Weak excitation

421

2112

01

QAABIVnSF

In weak excitation limit

QA

IBnn

21

121SS2

The effective loss or “quenching” rate includesQrot, Qvib, Qelec (i.e., all processes removing molecules from the upper state observed by fluorescent emission)

Multi-level effects

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’)

31




421

2112

01

QAABIVnSF

QAIBnn

21

121SS2

Includes only Qelec! i.e., those collisions that preclude emission in collection bandwidth

In weak excitation limit

Multi-level effects

Conclusion: collection bandwidth defines “2”!

32




421

2112

01

effF QA

ABIVnS

Effective average of emitting states “2”

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

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)

33

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

The P1(5) line of the OH X2Π-A2Σ+(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)

Note the evolution of fluorescence spectrum at different delays

Delay = 4ns Delay = 10ns

Fluorescence yield: multi-level models

“Narrowband collection” (single line)

34

7. Modifications of the Two-level ModelSimple Models

AP, AQ, and AR = A-coefficients (in s-1) for P, Q, R branch emission from v', J' of an atom or molecule

Qel = electronic quench rate [s-1]Qrot = rotational transfer rate [s-1]Assume these rates are the same for all J'

Fluorescence yield (FY)= fraction of absorbed photons emitted into the collection bandwidth, e.g., to capture AP

processes removal all of rate sumprocess desired of rateor y probabilitFY

usuallyFYrot

P

elrotRQP

P

QA

QQAAAA

RQPelrot AAAQQ ,,usually is

But signal is weak!

Fluorescence yield: multi-level models

“Broadband collection”; i.e., from P, Q & R of all J' in v'

35


AP, AQ, and AR = A-coefficients (in s-1) for P, Q, R branch emission from v', J' of an atom or molecule

Qel = electronic quench rate [s-1]Qrot = rotational transfer rate [s-1]Assume these rates are the same for all J'

Note: Stimulated emission not included “weak excitation” limit or linear LIF regime

1FY

elrot

rot

elrot

QQAQ

QQAA

…Absorption 2FY 2

3FY …eln

n

QAA

1FYFY

0i

Consider a sequence of time steps, each with 1 of 3 outcomes: A, Qel, or Qrot, and let A = AP+AQ+AR

Success rate for remaining in v'

Conclusion: (1) Strong signals; (2) FY is the same as 2 level model!

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)

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Documents

Lecture 13: Laser-Induced Fluorescence: Two-Level Model · Lecture 13: Laser-Induced Fluorescence: Two-Level Model 1. Introduction and background 2. Typical experimental setup 3