Nonlinear Interferometric Vibrational Imaging - ACS Symposium

Chapter 16

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Nonlinear Interferometric Vibrational Imaging A Method for Distinguishing Coherent Anti-Stokes Raman Scattering from Nonresonant Four-Wave-Mixing Processes and Retrieving Raman Spectra Using Broadband Pulses Daniel L. Marks and Stephen A . Boppart Biophotonics Imaging Laboratory, Beckman Institute for Advanced Science and Technology, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, 405 North Mathews, Urbana, IL 61801

When utilizing broadband, short pulses to stimulate Coherent Anti-Stokes Raman Scattering (CARS), frequently the peak power is sufficient to excite other nonlinear, nonresonant processes in the material. These processes produce a four— wave-mixing component in the same frequency band as the C A R S signal, so that the two signal types cannot be distinguished on the basis of frequency band alone. Typically in biological materials, the nonresonant component produced by the bulk medium can overwhelm the C A R S signals produced by the usually much lower concentration target molecular species. Resonant processes can be distinguished from nonresonant processes in that molecular vibrations and rotations typically last longer than a picosecond, while nonresonant processes are not persistent and last shorter than 10 fs. Interferometry allows the arrival time of the signal to be determined to extremely high accuracy, limited by the bandwidth of the reference pulse. Because resonances are persistent, they can produce anti-Stokes radiation that persists after the nonresonant excitation is produced. Interferometry can distinguish the later arrival of the anti-Stokes radiation and therefore distinguish the resonant and nonresonant signals.

236

© 2007 American Chemical Society In New Approaches in Biomedical Spectroscopy; Kneipp, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2007.

237 More complicated pulse-shaping and interferometry schemes can allow additional flexibility to allow simultaneous sampling of the Raman spectrum while rejecting the nonresonant component. This technique enables an entire region of the Raman spectrum of a sample to be measured with a single brief broadband pulse.


Theory of CARS Interferometry Nonlinear Interferometric Vibrational Imaging (NIVI) is a previously described method used to measure the three-dimensional distribution of molecular species in various samples (biological or otherwise) (7-5/ Its basic operation is to stimulate the excitation of molecular bonds with particular resonance frequencies, and then use these excitations to produce radiation distinct from the excitation that can be measured. The physical process of excitation and stimulation of radiation is called Coherent Anti-Stokes Raman Scattering (CARS). Unlike previous methods that use C A R S in microscopy to probe for the presence of molecular species (4-6), NIVI utilizes a heterodyne approach where a reference signal is separately generated and interferomerically compared to the signal received from the sample. In this way, additional information can be inferred from the emitted radiation such as the distance to the sample and phase relationships containing additional details about the sample molecular composition (7,8). It also has other advantages in sensitivity and the ability to screen out background radiation that is not produced by the sample. This technique also can allow more flexibility in the choice of laser illumination source, because the coherent detection process does not rely on photon frequency alone to discriminate emitted radiation. In this chapter we describe how NIVI can be used to estimate the Raman spectrum of a sample, which can be specifically applied to distinguishing resonant C A R S processes and nonresonant four-wave-mixing processes (5). We are concerned with how to measure the concentration of several vibrational states simultaneously, and how to distinguish these signals from the nonresonant background. A t each point in a three-dimensional sample, a further dimension of Raman frequency specificity can be extracted. This can achieve further contrast between the constituents of biological tissues enabling a more thorough determination of molecular contents. We consider a focused optical pulse, with an electric field at the focus given in the frequency domain by i? (ω). At the focus there is a medium with a ( 3 )

complex Raman spectrum given by / ( Ω ) where Ω is the Raman frequency. The anti-Stokes/Stokes electric field produced by the C A R S / C S R S (Coherent Stokes Raman Scattering) process at the focus is given by:

In New Approaches in Biomedical Spectroscopy; Kneipp, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2007.

238

ο 0)

£ » = ]ρ (Ω)ΕΧω-Ω}ΚΙ Ο ω Ë (a>)= Downloaded by NORTH CAROLINA STATE UNIV on August 6, 2012 | http://pubs.acs.org Publication Date: August 2, 2007 | doi: 10.1021/bk-2007-0963.ch016

s

jP^iQjÉXû) ο

(CARS)

(2)

+Ω>/Ω (CSRS)

(3)

These equations assume that the optical field is strong and can be treated classically, a perturbative interaction with the sample that begins and ends with the vibrational ground state, and that there are no levels directly resonant with any of the individual frequencies in the pulse (resonant one-photon interactions do not occur). Near-infrared light (800-1400 nm), which is only weakly absorbed by biological tissues, contains frequencies typically well above the vibrational frequencies of molecular bonds, but below the electronic excitation frequencies, and so is suitable for NIVI. These equations give the time evolution of a C A R S / C S R S process involving many possible simultaneous Raman-active vibrations. In C A R S , the molecule is excited by two frequencies CD and C0 that X

are separated by the Raman vibrational frequency ω - ω = Ω . Χ

2

2

The upper

frequency ω is called the pump, and the lower frequency CC> the Stokes. In general, the signal consists of not just two discrete frequencies but a continuous spectrum. In this case, Eq. 1 represents the excitation of the molecule through stimulated Raman scattering (SRS). A vibration at frequency Ω is excited by every pair of optical frequencies with a frequency difference of Ω . The excitation at any given Raman frequency is given by the polarization Ρ ( Ω ) . λ

2

( 3 )

This excitation is converted to anti-Stokes radiation Ε (ω) by a second SRS Α

process whereby the pump signal at frequency G) is partially converted to the x

anti-Stokes signal at frequency ω = ω + Ω = 2ω - ω . Α

Χ

Χ

2

For broadband

signals, the anti-Stokes signal must be summed over all pump frequencies, and so Eq. 2 results. The CSRS process is likewise modeled by Eq. 1 and Eq. 3. In general, both C A R S and CSRS are generated simultaneously for any given pulse. The information about the sample is contained in its Raman spectrum ^

( 3 )

(Ω).

By sending in a pulse with a known electric field £ , ( # ) , and

measuring the returned anti-Stokes signal Ε (ω), Α

^

( 3 )

(Ω).

we would like to infer

Unfortunately, at optical frequencies, the electric field Ε (ω) is not Α

directly measurable because it oscillates on a time scale too fast to be electronically

demodulated,

requiring interferometric

demodulation.

schematic of the setup to do this is given in Figure 1.


A

239 Beam \^

Oscillator

\

Temporal Pulse Shaper \

Microscope

Filter for iUumination frequencies

Reference Generator Downloaded by NORTH CAROLINA STATE UNIV on August 6, 2012 | http://pubs.acs.org Publication Date: August 2, 2007 | doi: 10.1021/bk-2007-0963.ch016

CARS Sample

5 » Anti-Stokes Frequencies

Inteiferctmetric Demodulator

Reference Frequencies (same as anti-Stokes bardwidth)

Figure 1. Basic schematic of Nonlinear Interferometric Vibrational Imaging.

To demodulate the anti-Stokes field, we generate a reference field R(co) from the oscillator field that contains frequencies same as the anti-Stokes radiation. The interferometric demodulator measures the intensity of the crosscorrelation of the anti-Stokes and reference fields, which is given by the frequency spectrum 7(ω)= R(Û))*Ê (Û))> A good method of interferometric demodulation for these techniques is spectral interferometry (7,9-10). Spectral interferometry may be especially attractive because it allows the signal to be sampled in one shot to minimize transient effects, and can achieve higher signalto-noise ratio than conventional temporal interferometry. Putting the steps of generating the anti-Stokes radiation and interferometry together, the crosscorrelation spectrum is given by: a

7(^)= £ ( ^ ) ' ^

0

ω

Τ(ω)= R(œï ^ΧΩ^ΕΧω

0

(

0

C A R S

)

() 4

oo

+ ^ΐΕΧω'+Ω^ΕΧω^ω'άΩ

(CSRS)

0

(5)

A n important consequence of Eqs. 4 and 5 is that the cross-correlation spectrum Τ (ω) is a linear function of the Raman spectrum ^ ( Ω ) . Because of (3)

this, linear estimation can be used to infer ^

( 3 )

( Ω ) fromI(ω).

To better see


240 this, we reform Eqs. 4 and 5 as linear operators A and S with kernels

Α(ω,Ω)

and S(co Q) respectively: 9

ω

7(ω)=λχ=

{2)

w

Ιζ (Ω)Α(ω,Ω)άΩ

h

e

r

e

ο σο ,

,

Α(ω,Ω) = R(w)' £,(ω·-Ω)|£ (ώ> +Ω)£,.(ΰ) )'^· (CARS)

0


ί

(6)

ω

7(ω)=&

(3)

= ]> (Ω)*5(ω,Ω>/Ω where ο οο

S(m, Ω ) = R(coY Ε, (ω'+Ω) jÈ {ω'+Qf Ε, (ω'}ΐω'

0

(

(CSRS)

(7)

With the C A R S interferometry process now expressed as a linear operator A mapping ^

( 3 )

( Ω ) to l(co),

an inverse operator can be found to estimate

( 3 )

/ ( Ω ) from 7(ω). Many inverse operators are possible and should be chosen on the basis of stability, ability to model the physical system and noise, and ability to incorporate confidence measures of the data. We present an inverse operator based on the weighted Tikhonov regularized least-squares solution. The least-squares solution by itself attempts to find a solution {οτχ^{θ)

that minimizes the Euclidean distance

system / = Αχ.

r

f ° the linear

Unfortunately, this in practice tends to produce poor results due

to poor conditioning and the inability to incorporate confidence information about the data. A solution is to augment the linear system with a weighting operator (8)

ΨΙ = Ψ(ω)Ι(ω)

where W(co) is a weighting function that is of unit value inside the measured anti-Stokes bandwidth and zero outside. The weighting operator ensures that only data in the measured bandwidth contributes to the estimate of the Raman spectrum. The weighted linear system becomes WI = ΨΑχ . Formally, the Tikhonov regularized solution to this system is X = {A W WA + sÎY A W WI H

H

H

H

H

H

where the operator / is the identity operator λ

(for CSRS it is χ = (s W WS + εί)

H

H

S W WI).

The constant ε > 0 is set to

account for the magnitude of noise in the measurement, e.g. from thermal or


241

photon noise. In practice, this operator can be inverted numerically using iterative linear solution methods such as the preconditioned conjugate gradient method including the Fast Fourier Transform to implement the numerical crosscorrelations.


Resonant CARS versus Nonresonant Four-Wave-Mixing While the above formalism specifies how the measured interferometric cross-correlation is related to the Raman spectrum, and an inverse operator for this relation, it does not tell us what input signal £ , ( # ) or reference signal R(co) will enable reconstruction of the desired features of the Raman spectrum *

( 3 )

(Ω)·

Two important features of the Raman spectrum in practice that need to be mutually distinguished are the resonant and nonresonant components. The resonant components are specific to features of a molecule, which include vibrational frequencies, rotational frequencies, and electronic resonances. Nonresonant features are not specific to a particular molecule, and are weakly dependent on frequency. A typical Raman spectrum can be decomposed into a sum of resonance and nonresonant components: α )

( 9 )

^ ( Ω ) =^(Ω) ΣΛ (Ω) +

The

(ω)

nonresonant component

is a slowly changing function of Ω ,

and is usually approximated by a real-valued constant χ^ ( ω ) = χ . κ

resonant components χ^{θ)

ΝΚ

The

are sharply peaked at vibrational frequencies, and

are typically described by a homogeneously broadened Lorentzian spectrum: ν(3)( χ ' Ω

Λ η

2

V

2^ Ω -2/Γ Ω-(Ω^-Γ ) 2

(10)

2

η

η

2

where Ω = ^Q - 2Γ is the center frequency of the resonance, and Γ is the line n

Λ

η

width. A resonance has a Raman magnitude that is sharply peaked around the center frequency. Interferometric detection can distinguish the real-valued nearly constant nonresonant spectra and the sharply peaked resonant C A R S spectra. In addition, the imaginary part Ο ^ ( Ω ) is determined by only the ( 3 )

resonant Raman component, while the real part is determined by both resonant and nonresonant components. Another feature distinguishing resonant components is the phase reversal of 7Γ that occurs through the center frequency.


242 Noninterferometric instruments cannot distinguish the phase, while the interferometric method measures the complex ^ ( Ω ) . A S well as being a 3 )

signature of C A R S , this phase reversal may be used to separate several resonances close together in frequency. Distinguishing C A R S resonance from nonresonant four-wave-mixing has become problematic with the advent of ultrafast laser sources. These sources produce pulses on the order of 5-200 fs, much shorter than the lifetime of the resonance, which is 2π/Γ - If transform-limited ultrafast pulses are used, only a Downloaded by NORTH CAROLINA STATE UNIV on August 6, 2012 | http://pubs.acs.org Publication Date: August 2, 2007 | doi: 10.1021/bk-2007-0963.ch016

η

0)

small polarization p (n)

can be produced in the molecule, while the

nonresonant components are enhanced. Since the lifetime of the resonance is typically 1-100 ps, transform-limited ultrafast pulses excite resonant transitions inefficiently and nonresonant transitions efficiently. Many current noninterferometric C A R S instruments utilize narrowband pump and Stokes pulses of picosecond length for this reason (11,12). There are compelling reasons to use broadband sources to excite C A R S (1, 13-17). A n ultrafast pulse can be shaped into a longer picosecond pulse that can excite C A R S more efficiently. In addition, it can be used to excite many resonances simultaneously. Unfortunately, unlike narrowband pulses, the antiStokes radiation produced will not be narrowband. If many resonances are excited simultaneously, then the anti-Stokes radiation they produce have overlapping spectra. Because noninterferometric detection can only measure the spectrum of the anti-Stokes radiation, the contributions of each resonance to the anti-Stokes radiation are difficult to separate. Interferometric detection allows the demodulation of the anti-Stokes field so the Raman spectrum can be inferred when exciting multiple simultaneous resonances. In addition, broadband sources should allow for more Raman-frequency agile imaging instruments. When utilizing narrowband pump and Stokes pulses, the frequency difference between them must be tuned to the Raman frequency of interest. Retuning lasers or amplifiers is often difficult to make reliable and automatic. Pulse shaping, however, is achieved by movable gratings, prisms, or mirrors to adjust dispersion and delay, or by acousto-optic or liquid-crystal Fourier-plane pulse shapers without moving parts (14-20). Because pulse shapers do not typically involve feedback, oscillation, or overly sensitive alignment, pulse shapes can be changed much more easily. In addition, a computer can control these mechanisms automatically, so that changing the pulse shape should be much easier than retuning a laser source. One of the aims of this work is to manipulate the pulse shape of a broadband laser to stimulate specific Raman frequency bands, and use the resulting anti-Stokes radiation can be decoded to estimate the Raman frequency spectrum.


243


Distinguishing resonant CARS and nonresonant four-wavemixing using interferometry To see how interferometry can detect the difference between resonant and nonresonant processes, consider the experimental set-up shown in Figure 2 (a) and the shape of the anti-Stokes pulses produced by the long pump and short Stokes pulses previously mentioned. This combination of pulses is illustrated in Figure 2 (b). When the pump and Stokes pulses overlap, the molecule will be excited by SRS. This excitation will remain after the Stokes pulse passes. At the moment of the overlap, nonresonant four-wave-mixing processes can also be excited. However, because there is no persistent state associated with nonresonant processes, the nonresonant emission will end quickly after the Stokes pulse passes. With a Raman-active resonance, however, the pump can produce anti-Stokes radiation via SRS even after the Stokes has passed, because the resonance persists. Therefore, the nonresonant component can be discarded by rejecting any antiStokes radiation that occurs coincident with the Stokes pulse. The benefit of interferometry is that the time of arrival of the anti-Stokes radiation can be found very precisely by cross-correlating a broad bandwidth reference pulse with the anti-Stokes radiation. Incoherent detection can detect the interference between the resonant and nonresonant components of the anti-Stokes radiation, but does not directly detect the time of arrival of the anti-Stokes light. Figure 2 shows the actual measured cross-correlations between a short reference pulse and the anti-Stokes radiation for a cuvette of acetone with a resonance at 2925 cm' as the frequency difference between the pump and Stokes is tuned (5). The apparatus to acquire these interferogram is shown in Figure 2 (a) and consists of several components. A regenerative amplifier produces a pulse at 808 nm and 30 nm bandwidth that is used as a seed pulse for a secondharmonic-generation optical parametric amplifier, and also as the pump for the C A R S sample. The pump is lengthened to approximately 200 fs by passing it through a dispersive Dove prism made of B K 7 glass of 105 mm length. The idler of the optical parametric amplifier produces a Stokes pulse at 1056 nm and 70 fs in length, which is combined with the pump at a dichroic mirror. The pump is delayed so the Stokes arrives on the leading edge of the pump pulse. The signal of the optical parametric amplifier is at 653 nm and serves as the reference pulse because it matches the frequency of the anti-Stokes produced in the C A R S sample when the pump and Stokes are focused into it. The antiStokes and reference are cross-correlated by delaying them relative to each other and detecting their interference power on a silicon photodetector. Figure 2(c) shows interferograms acquired with various frequency differences between the pump and Stokes pulses. When this difference matches a Raman resonance, a resonant "tail" is generated, otherwise only the non-resonant impulse remains. 1



244

(b)

Pump Amplitude Stokes Amplitude CARS/ FWM FWM only -400

-200

0 200 T i m e (fs)

400

600

(c)

(Ék

1

'

j|h

M66cm j

'

:

llMliililiiiiii

•200

0 200 Time (&i

2916 cm"

^JkuÉfc». 400 -200

0

200 Time ffsrt

400

Figure 2. (a) Schematic of interferometer used to cross-correlate the reference and anti-Stokes signals, (b) Illustration of the anti-Stokes andfour-wave-mixing signals generated by resonant and nonresonant media illuminated by a long pump and short Stokes pulse, (c) Cross-correlation interferograms between anti-Stokes generatedfrom acetone and a reference pulse with various pump/Stokes frequency separations.


245


Note that to detect the difference between resonant and nonresonant processes did not require that the pump be lengthened to the lifetime of the resonance of the acetone. This is because the nonresonant component stops abruptly at the end of the Stokes pulse. Unlike the incoherent detection case, one can actually reject the nonresonant component based on its time of arrival, and not just depend on minimizing the amount of nonresonant four-wave-mixing that is generated.

Raman spectra retrieval with a chirped pump pulse We now present a method using a pulse sequence to recover the Raman spectrum ^ ( Ω ) in a particular frequency range. The method uses a pump that is broadband but dispersed to be lengthened in time, while using a short, nearly transform-limited Stokes pulse. These two pulses will be overlapped so that the Stokes pulse is on the leading edge of the pump pulse. (3)

The idea is to stimulate a broad bandwidth of Raman transitions using a short Stokes pulse overlapped with the leading edge of a pump pulse. After the Stokes pulse is over, the pump pulse continues to produce stimulated Raman scattering, which is demodulated with the reference pulse. The portion of the cross-correlation that overlaps the Stokes pulse arrival can be discarded to remove the nonresonant four-wave-mixing component of the anti-Stokes signal. Because the field used to produce the resonant C A R S is only due to the pump pulse, the cross-correlation amplitude and the pump pulse can be used to estimate the Raman spectrumχ (Ω), perhaps using an inverse of Eq. 6. A n alternate method of measuring the Raman spectrum with chirped pulses without interferometry is has also been described (21). To understand in better detail how this method can recover the Raman spectrum, consider a dispersed pump pulse and a Stokes pulse with an electric field {3)

Ε (ω) = Α (ω)&φ(ΐφ(ω)) ρ

for ω < ω < ω + Αω

ρ

0

0

(11)

Ρ

Ε (ω) = 0 otherwise ρ

Ë (co) = A for ω - Ω - Α ω s

s

0

0

3

12 < ω < ω - Ω + Αω Ι2 0

0

8

(12)

Ε (ω) = 0 otherwise 5

The frequency Ω corresponds to the center of the resonant Raman frequency 0

bandwidth of interest, while the bandwidths Αω and Aco give the bandwidths ρ

s

of the pump and Stokes signals, respectively. The amplitudes Α (ω) and A ρ

s

correspond to the real, positive amplitudes of the frequency components of the pump and Stokes signals, respectively. To find these pulses in the time domain,


246 we use the stationary phase approximation (which should apply well to dispersed pulses).

x

We define άφΙάω -ΐ\ώ),

function of ί\ω)),

œ\t)-f~ (œ)

(the inverse

and d Note that ?(ω) must be a strictly

increasing or decreasing function of ω.

The time-domain versions of these

signals are:

E (t) =

A {œ\t))^


p

exp(/O(0) for t'(co ) < t < t'(a> + Αω )

p

0

0

Ρ

(13)

E (t) = 0 otherwise p

E (t) = A exp(/(< (/) can be recovered by conjugate

of the

phase of the

probe


field

247 il/2 .

2

άΦ in the time domain. The ρ ( Ω ) and therefore ^ (n)canbe exp(-/O(0)|^ \dt ' recovered from P (t) by means of a Fourier transform. (3)

(3)

2


(3)

To demonstrate the measurement of high-resolution Raman spectra with broadband pulses, we acquired the complex Raman spectra of isopropanol using the set-up in Figure 3(a) (10). A regenerative amplifier produced pulses at 250 kHz repetition-rate, 805 nm center wavelength, and 25 nm fiill-width-halfmaximum ( F W H M ) bandwidth. The pulses were divided into two, with 90% of the light used as the pump a second-harmonic-generation optical parameter amplifier (SHG-OPA). The other 10% was dispersed to 6-10 ps pulse length by propagation through an 85 cm bar of B K 7 optical glass. The dispersed pulse acted as the C A R S pump and probe and was 45 mW average power at the sample. The SHG-OPA generated nearly transformed-limited signal pulses at 655 nm center wavelength and 25 nm F W H M , which were used to interferometrically demodulate the anti-Stokes signal. The idler was 2 mW average power, centered at 1050 nm, with a bandwidth of 30 nm F W H M , which served as the Stokes pulses. The pump pulse and Stokes pulse were combined using a dichroic beamsplitter so that the Stokes pulse overlapped the leading edge of the pump pulse. The combined pulses were focused into the sample liquid by a 30-mm focal length near-infrared achromatic objective lens. A gold mirror at the bottom of the liquid sample dish reflected the generated anti-Stokes light backwards to be recollected by the objective. The anti-Stokes light was separated from the remaining pump and Stokes radiation using a dichroic beam splitter, and then was superimposed onto the reference SHG-OPA signal pulse with a broadband 50/50 beam splitter. A spectral interferogram was sampled on the photodetector linear array, which corresponded to the real part of the Fourier transform of the temporal cross-correlation between the anti-Stokes and S H G O P A signal pulses. The interferogram was numerically resampled so that the spacing of the cross-correlation samples was uniform in frequency. From the resampled interferogram, the complex analytic signal was computed by assuming the cross-correlation signal was zero for negative time delays, so that the Hubert transform could be used. Effectively this was accomplished by taking the inverse Fourier transform of the interferogram and setting the signal to zero for negative delays. (10,22). To calibrate the dispersion of the interferometer and the spacing of frequencies on the linear array, a nonresonant sample (sapphire) was used. It was found that the ΟΡΑ signal and the anti-Stokes are almost transform limited and nearly identical, and the interferometer had negligible dispersion. Therefore, only the angular dispersion of the wavelengths on the array needed to be characterized. To calibrate the chirp introduced by the 85 cm B K 7 bar, the interferogram was sampled from a sample of acetone. Acetone has a narrowly


248


1

peaked resonance at 2925 cm" . The chirp was calibrated by searching for the chirp value that reconstructed the computed Raman spectrum as a sharp peak. Finally, we experimentally demonstrated our method by measuring the Raman spectrum of isopropanol, which was chosen because the fine detail in the spectrum would test the spectral resolution of the instrument. Figure 3 (b) shows the spectral interferogram obtained for the anti-Stokes signal, from which the Raman spectrum was inferred. The primary noise source was photon noise because the number of generated photoelectrons was much greater than the thermally generated charge. The spectral interferogram signal was integrated over a 25 ms interval by averaging together 100 line scans captured at 4000 Hz. The time-resolved anti-Stokes decay signal magnitude of Figure 3 (c) was computed from the inverse Fourier transform of the spectral interferogram. One can see the envelope of the signal shaped by the beats between various Raman resonances. The chirp of the probe pulse was numerically conjugated out of the time-resolved complex-valued decay signal, from which the Raman spectrum was computed by using a Fourier transform. Figure 3 (d) is a plot of the real part of the nonlinear susceptibility, which contains the nonresonant component. Figure 3 (e) contains the imaginary component of the nonlinear susceptibility, which is compared to the spontaneous Raman spectrum obtained by a commercial spectrometer given in Figure 3 (f). There is a close resemblance between the imaginary component of the nonlinear susceptibility and the spontaneous Raman spectrum. In fact, the magnitude of the imaginary part of the Raman susceptibility is proportional to the spontaneous Raman spectrum (23), so these should be similar. The differences in heights of spectral features are due to the nonuniform Stokes pulse spectrum. These results show that broadband pump pulses can be used to measure spectra with high resolution. This resolution is manifest in the two peaks at 2928 cm' and 2942 cm' that are clearly discerned by this method. Even though the pump bandwidth is almost 400 cm" , a high-resolution Raman spectrum can be obtained because the chirp of the pump can be interferometrically measured and numerically removed. 1

1

1

Raman Spectra Retrieval using Broadband Pulses One of the primary benefits of these methods is that they enable the use of ultrabroadband lasers with thousands of wavenumbers of bandwidth to be used to measure either narrow (less than 10 cm" ) or a very wide range of Raman frequencies. Therefore, the same laser can be used to produce light that both acts as the pump and Stokes frequencies (rather than requiring a separate Stokes to be derived), and the entire power spectral bandwidth of the laser can be used to contribute to the SRS excitation of the sample. There has been increasing interest in using these ultrabroadband sources in C A R S imaging and microscopy (24-30). When using narrowband pulses, the power spectral density of the pump 1


249 and Stokes must be very high to ensure sufficient excitation. With ultrabroadband excitation one can use its relatively low power spectral density to excite many vibrations simultaneously, but a one-to-one correspondence between anti-Stokes and Raman frequencies is not preserved. The essence behind these methods is a reinterpretation of Eq. 1. In the time domain, Eq. 1 becomes: (18)


0 Eq. χ (ή 0)

18 expresses the causal convolution of the impulse Raman response = Γ z (Q)exp(-iQt)dt 0)

with the instantaneous intensity of the incoming

pulse. If there is a modulation of the intensity at the same frequency as a Raman resonance, the Raman resonance will become excited. By using a pulse that contains a modulation of the intensity of the excitation wave that spans a range of frequencies, an entire range of the same Raman frequencies can be excited and studied. For example, pulses have been shaped with a periodic phase modulation in the Fourier domain (14-18,31), and by interfering chirped pulses together (32), have created the intensity modulation needed for SRS. Previously, narrowband lasers were the primary source available for stimulating C A R S , so that the intensity modulation was created by the beats between two narrowband pulses. Broadband sources have the benefit of providing a much more easily tunable source of beats of a particular frequency, because the laser need not be retuned. Only the pulse need be reshaped. A n ultrabroadband source of pulsed radiation can be used to both stimulate C A R S and provide the reference pulse. The pulses from the source will be divided by a frequency-selective element such as a dichroic beam splitter into lower and higher frequency pulses. The higher frequency pulse bandwidth will correspond to the anti-Stokes frequencies emitted by the sample, and will act as the reference pulse to demodulate the anti-Stokes signal. The lower frequency pulse will be shaped to stimulate the Raman frequencies of the sample in a particular bandwidth. The strategy used here is to split the lower frequency pulse into two copies. Each copy will propagate through separate dispersive elements, and then recombined afterwards with a time delay between them. When the two pulses are overlapped, a beat frequency is produced at the difference between the instantaneous frequencies of the two pulses at a given instant. By causing the frequency difference of the two pulses to vary between a lower and higher difference frequency during the time interval they are overlapped, the beats can stimulate the Raman frequencies in the same range. B y varying the instantaneous frequency of the two pulses, and their relative delay, the range of Raman frequencies that the overlapped pulses stimulate can be varied. A similar scheme was proposed to stimulate a single Raman-active vibration (32).



250



e

2900 1

3000

Wave number (cm")

2800

3100 ^'

(f\

2700

2800

2900 1

3000

Wave number (cm")

3100

Figure 3. (a) Schematic of interferometer used to sample the spectral interferograms of anti-Stokes signals, (b) A spectral interferogram sampledfrom isopropanol. (c) Time-resolved Raman decay signal from isopropanol. (d) Real part of Raman susceptibility of isopropanol. The shaded band indicates negative values of the susceptibility, (e) Imaginary part of the Raman susceptibility of isopropanol (f) Spontaneous Raman spectrum of isopropanol sampled with a commercial spectrometer.

^ ^

( )2700


(SI

252

To see how two dispersed pulses can produce a beat frequency spectrum that ranges from Q < Ω < Ω „ , consider two bandlimited pulses with different dispersion phases imposed on them : L

Ε (ω) = Ε εχρ(ΐφ (ω)) λ

χ

for ω -Δω/2

χ

0

with the added constraint of Eq. 25. In addition, one can filter the output signal from the sample for frequencies less than the minimum frequency of the illumination ω -Αω/2. In this case, one captures the CSRS output of the 0

sample. A reference pulse of this same frequency band will then demodulate this signal, and a regularized least-squares inversion operator for Eq. 7 can be employed to find the Raman spectrum. As an example of a pulse combination that can stimulate a band of Raman frequencies, we design a pulse that sweeps the beat frequency linearly from Q to Ω „ . To do this, we will split a transform-limited pulse of bandwidth L

ω - Αω/2 < ω < ω + Αω/2 0

0

into two components.

Each component will

have a linear chirp (quadratic phase) applied to it in the frequency domain. This chirp is imparted by a dispersive pulse shaper such as optical glass or a prism pair. The spectrum of the pulse combination that produces beats from Q to L

q over an interval Τ is given by: h

(31)

Ε(ω)-Ε c o / ^ - ^ ï f ^ ^ c x n i - ^ - ^ o V fa-*>of

„ Αω for ω ——

Nonlinear Interferometric Vibrational Imaging - ACS Symposium

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