2D Femtosecond Spectroscopy - Walter Scott, Jr. College of

1

2D Femtosecond

Spectroscopy

Jesse Wilson Ph.D. Qualifying Exam

Advisor: Prof. Randy A. Bartels

SDG

2

Motivation

Who needs another dimension?

3

Broadening Mechanisms

Palese, et al. J. Phys. Chem. 1994.

Homogeneous Inhomogeneous

1D Raman

2D Raman

4

(ii)

(i)

Mode Coupling

*Okumura, et al. J. Chem. Phys. (1999)

Complex molecules

Raman-active dipoles

5

Mode Coupling (1D Raman)

*Okumura, et al. J. Chem. Phys. (1999)

6

2D Raman Spectrum

7

2D Raman Spectrum (Fundamentals)

8

2D Raman Spectrum (Coupling)

9

Another example

*Zhang, et al. J. Chem. Phys. (1999)

Linear Uncoupled

Coupled

10

2D IR Spectra Contain Structural

Information

Acetyleproline-NH2 in chloroform.

*Hochstrasser, et al. Bull. Chem. Soc. Jpn. (2002)

11


Information

Magnitude


12


Information

Real part


13

2D IR Spectrum Features


14

2D IR Spectrum: Broadening


Homogeneous/inhomogeneous width

15

2D IR Spectrum: Anharmonicity


Vibrational anharmonicity

16

2D IR Spectrum: Mode Coupling


Mode coupling

17

2D Spectroscopy Advantages

Discern homogeneous, inhomogeneous lines

Mode coupling

Vibrational anharmonicity

Structural information

18

Papers Reviewed

Steffen, Fourkas, and Duppen.

“Time resolved four-and six-wave mixing

in liquids. I. Theory.”

Journal of Chemical Physics (1996).

Blank, Kaufman, and Fleming.

“Fifth-order two-dimensional Raman

spectra of CS2 are dominated by third-

order cascades.”

Journal of Chemical Physics (1999).

19

Steffen, et al. (Part I. Theory)

20

Steffen, et al (Part II. Experiment)

21

Blank, et al.

22

Roadmap

1) One-dimensional Raman methods

a) Motion in Liquids

b) Third order response theory

c) Liouville space paths and Feynman diagrams

2) Two-dimensional Raman methods

a) Fifth order response theory

b) Homogeneous damping effects

c) Inhomogeneous damping and rephasing

d) Third order Cascades

e) Solutions to the problem of cascades

23

1D Methods

…and their shortcomings

24

Roadmap











25

Time-resolved ISRS

t = 0

26

Time-resolved ISRS: Pumping

t = 0

pump

Stokes

27

Time-resolved ISRS: Ultrafast

t = 0

pump

wwpumpwstokes

E(w)

•Ultrafast pulse spectrum

•Short pulses: pulse << vib

•Impulsive excitation

Stokes

28

Time-resolved ISRS: Coherence

t

29

Time-resolved ISRS: Probing

t =

30

1D Experimental SetupLaser

Oscill

ato

r

Detector

pump

probe

signal

sample

31

32

Liquid Intermolecular

Modes

Homogeneous or Inhomogeneous?

33

Roadmap











34

CS2

Instantaneous response

Diffusive tail

Steffen and Duppen. J. Chem. Phys. (1997)

Librations

35

H2O

Palese, et al. J. Phys. Chem. 1994.

36

Broadening

Homogeneous: g(w) rapidly fluctuates

Inhomogeneous: g(w) changes slowly

w

ww )cos()()( tgtR)cos()( ttR w

37

Inhomogeneous Model

2

20

2

)(

)(

ww

ww

eg

= degree of inhomogeneity

)()3( tR

0

)3(

inh

)3(

inh ),()()( tRgdtR www

38

Model fit to data

Steffen and Duppen. J. Chem. Phys. (1997)

Experiment

Homogeneous limit

= 1.00 rad/ps

= 2.53 rad/ps

39

2D Provides Discrimination

*Palese et al. J. Phys. Chem. 1994.

Homogeneous

Inhomogeneous

Intermediate

40

2D Simulations

Homogeneous Inhomogeneous

41

42

Four Wave Mixing

Theory

Third-order, or one-dimensional theory…

43

Roadmap











44

3rd Order Response

)()()()](~),(~[

)()()(),,,(

32211221

32211321

ttttHtttti

ttttttttttR

)()()(),,,(ddd)( 321321

)3(

321

)3( tEtEtEttttRttttP

Raman

Hyper-polarizability (THG…)

45

Simplified 3rd Order Response

)()()()](~),(~[

)()()(),,,(

32211221

32211321

ttttHtttti

ttttttttttR

)]0(~),(~[2

)()( 111

iR

132

1 0

tt

t

46

Harmonic Oscillator

Unperturbed Hamiltonian:

Raising, lowering operators:

Displacement operator

)(21

0 aaH BO w

)(2

aam

qw

1~ a 1~ a

47

Polarizability Depends on q

2

21)( qqq

)(2

aam

qw

)2(2

)(2

22 aaaaaam

aam

qww

One-level transitions

Two-level transitions

Zero-level transition

48

3rd order HO Response

ww

ww

)2sin()()12(2

)sin(2

)()(

122

2

2

1

2

1

11

)3(

Pm

m

R

Hyperpolarizability

2

1

49

50

Double-Sided Feynman

Diagrams

Illustrating density matrix evolution

51

Roadmap











52

Optical Polarization

)(Tr)( tVtP

)( rrqV

53

1101

1000

Density Matrix

Population densities

Coherencesbra

ket

54

Liouville Equation

HHi

HHi

Hi

t

,

55

Perturbation Expansion

)()()()( )2()1()0( tttt

E2E 43 EE

56

Time evolution

*Mukamel, Principles of Nonlinear Optical Spectroscopy

]]]]),0([),([[),([),()( 0111

)(

VtVttVtrEi

t nn

n

n

57

Initial State

aaVV ),0(]),0([ 0

11

00

aa

58

Commutator

baV )0(

11

00

aa

,12,10,11

,02,01,00

)0( abaaV

59

Commutator

)0()0(),0(]),0([ 0 VaaaaVaaVV

baV )0(

baabV ]),0([ 0

60

Double-sided Feynman Diagrams

*Yee and Gustafson, Optics Communications (1977)

a

t

a a a

b a

0t

a b

baabVaaaaVaaVV )0()0(),0(]),0([ 0

61

Second order commutator

]]),0([),([ 01 VtV aaVtV ),0(),( 1

caabtV ),( 1

)()()()( 1111 tVcacatVtVababtV

bacbcbac

62

Liouville Space Paths

aa ba

ac bc

ca

ab

]]),0([),([ 01 VtV

)()()()( 1111 tVcacatVtVababtV

bacbcbac

63

Second Order Diagrams

a a

t

0t

c

1tt

b

a

a a

b

a

c

ca

a a

b

b

c

a

a a

a b

ca

64

Third order (1 of 8)

a at

0t

c

2tt

d

b

b

1tt

d

65

Polarization

a

at

0t

c

2tt

d

b

b

1tt

d

)(Tr)( tVtP

a

66

Polarization

a

at

0t

b

2tt

d

b

b

1tt

d

3tt c

a

b

a

)(Tr)( tVtP

67

Time-resolved ISRS

a

at

0t

b

2tt

d

b

b

1tt

d

3tt c

a

b

a

01 t 32 tt

68

Time-resolved ISRS

a

at

0t

b

t

a

a

a a

01 t 32 tt

69

Time-resolved ISRS

a

at

0t

b

t

a

a

a a

a

a b

a

b b

70

Coherence

a

at

0t

b

t

a

a

a a

a

a b

a

b b

Cohere

nce

w

ab

ba

i

EEi

e

e

/)(

71

Time-resolved ISRS

a

0t

0t

1

t

0

0

0 0

012

72

Time-resolved ISRS

a

0t

0t

1

t

0

0

0 1

012

01

w

10

01 /)(2

EE

73

Time-resolved ISRS

t

0t

t

10

w

10

01 /)(2

EE

0

0 1

0

1 1

74

Time-resolved ISRS

a

0t

0t

1

t

0

0

0 1

1001

www

10sin1010 ii

ee0

0 1

0

1 1

75

76

2D Methods

Tanimura and Mukamel (1993)

77

Roadmap











78

2D Femtosecond Spectroscopy

t = 0

1 2

79

2D Femtosecond Spectroscopy:

Impulsive pumping

t = 0

1 2

80


Coherence propagation

t 1

1 2

81


Rephasing pulse

t = 1

1 2

82


Second coherence propagating

t 2

1 2

83


Probing

t = 2

1 2

84

2D Experimental SetupLaser

Oscill

ato

r

Detector

1

signal

sample

2

2

1

85

Example: CS2

*Astinov, et al. Chemical Physics Letters (2000).

86

87

Six-wave Mixing Theory

Steffen, Fourkas, and Duppen

88

Roadmap











89

Fifth-order Response

0

21

2

1

2

21

)5(

2

0

1

)5( )()(),(dd)()( tEtERtEtP

)0(~,)(~),(~

4

)()0(~),(~

2

)()0(~),(~

4

)()(),(

1212

21

11

2121

)5(

i

i

i

R

6WM

Raman / 2nd hyper-Raman

90

Conditions

Nonlinear polarizability (NP)

Anharmonic coupling (AN)

ij

ji

qjii

i

qi

qqqq

qq

qq

00

2

02

1)()(

ijk

kjiijk

i

ii qqqqqV )3(2)2(

6

1

2

1)(

Coupling of ortho modesOrdinary Raman

Tokmakoff, et al. Chem Phys. (1998)

91

Harmonic Oscillator

92

Roadmap











93

Phenomenological Damping

Weak systembath coupling

ww i

*2/)(

State lifetime Decoherence

94

Full 3rd Order Damped Response

95

Full 5th Order Damped Response

96

State-independent Damping

State decay

One quantum coherences

Two quantum coherences

1

1

*

1,

2

2

*

2,

97

Damped 3rd Order Response

decoherence rates

98

Damped 5th Order Response

Undamped

Damped

99

Brownian Oscillator Bath

System / Bath linear coupling

Tanimura and Mukamel’s approach

Equivalent to phenomenological model when:

damping is state dependent!

21

221

1

100

Coupled Bath Dephasing

First excited state decayOne-quantum dephasing

101

Level-dependent dephasing leads to a

new term in R(5)

102

Roadmap











103

Inhomogeneous Damping

),,()(d),(

),()(d)(

21

)5(

0

21

(5)

inh

1

)3(

0

1

(3)

inh

www

www

RgR

RgR

104

One term is invariant to g(w

105

Rephasing Pathways

106

Initial State

1t 1

107

Pump Interaction

1t

0t

1 0

1

w10ie01

108

Rephasing Pulse

1t

0t

1 0

1

1 2

w10ie01

1t

21 w12ie

109

Probe Pulse

1t

0t

1

21 t

0

1

2 2

1 2

w10ie01

1t

21 w12ie

110

Phase Cancellation

1t

0t

1

21 t

0

1

2 2

1 2

w10ie01

1t

21 w12ie

)()( 21102110212110 wwwww

iiee

111

112

Third-order Cascades

Blank, Kaufman, and Fleming

113

Roadmap











114

CS2 Predicted 2D Response

115

CS2 Predicted 2D Response

116

What was measured:

117

Sequential Cascades

t = 0

chromophore a

chromophore b

2 4

118

Sequential Cascades

t = 0

chromophore a

chromophore b

2 4

119

Sequential Cascades

t 2

chromophore a

chromophore b

2 4

120

Sequential Cascades

t = 2

chromophore a

chromophore b

2 4

121

Sequential Cascades

t 2 + 4

chromophore a

chromophore b

2 4

122

Sequential Cascades

t = 2 + 4

chromophore a

chromophore b

2 4

123

Sequential response

124

Eliminating Sequential Cascades

Theory Measured

125

Parallel Cascades

t = 0

chromophore a

chromophore b

2 4

126

Parallel Cascades

t = 0

chromophore a

chromophore b

2 4

127

Parallel Cascades

t 2

chromophore a

chromophore b

2 4

128

Parallel Cascades

t = 2

chromophore a

chromophore b

2 4

129

Parallel Cascades

t 2+4

chromophore a

chromophore b

2 4

130

Parallel Cascades

t = 2+4

chromophore a

chromophore b

2 4

131

Parallel Response

132

Measured Parallel Response

Simulated Measured

133

134

Eliminating Cascades

135

Roadmap











136

Diffractive Optics

Astinov, et al. Opt. Lett. (2000)

137

Diffractive Optic Results

Cascade

Astinov, et al. Opt. Lett. (2000)

138

More Accurate Phase Matching

sinc(Dkl/2) assumes

collinear propagation

Constant spatial overlap

Account for z-

dependent spatial

overlap:

Blank, et al. J. Chem. Phys. (2000)

139

Reduced Interaction Length

LEekL

REn

LiE kLi

s

s

D D

signal

2/

42

)5(5

signal2

sinc),( w

2

cascade

2/

2/

42

)3(

cas

5

int

int

cas

cas2

cascade

2sinc

2sinc

),(

LEeLk

eLk

REnn

iLE

Lkib

Lkia

b

a

D

D

D

D

ww

140

Heterodyne Detection

141

Heterodyne Detection Results

Astinov, et al. Chem. Phys Lett. (2000)

142

143

Conclusions

Steffen et al’s theory did not account for

parallel cascades

Blank, et al. found parallel cascades

dominate the signal

Eliminating parallel cascades reveals desired

signal

144

Thanks

145

Roadmap

1. One-dimensional Raman methods

1. Motion in Liquids

2. Third order response theory

3. Illustrating 3rd order response with Feynman

diagrams

2. Two-dimensional Raman methods

1. Fifth order response theory

2. Homogeneous damping effects

3. Inhomogeneous damping and rephasing

4. Third order Cascades

5. Solutions to the problem of cascades

146

147

2D Resonant IR (Part I)

148

2D Resonant IR (Part II)

Documents

2D Femtosecond Spectroscopy - Walter Scott, Jr. College of