1. Introduction

Coherent stokes and anti-stokes Raman spectroscopy (CSRS and CARS) involve nonlinear wave-mixing mediated by the third order susceptibility χ ⁽³⁾ which contains vibrational “finger-prints” of molecules involved in the wave-mixing process. Therefore, CSRS and CARS measurements of the magnitude of χ ⁽³⁾ can provide information specific to the combination of molecules and their surrounding environment. The chemical specificity of CSRS and CARS is being exploited in various applications including, as examples, concentration analysis, monitoring chemical reactions, flame and temperature diagnostics, microscopy, biological imaging, and forensics. In addition to molecular identification, coherent Raman techniques provide several additional advantages. The high sensitivity permits noninvasive interrogation of samples and the nonlinearity provides enhanced spatial resolution in comparison to linear techniques. Furthermore, coherent Raman signals are strong and directional which allows them to be isolated from fluorescent signals.

Coherent Raman signals contain contributions from the resonant (${\chi }_r^{\left(3\right)}$(Ω)) and non-resonant (${\chi }_{\mathit{\text{nr}}}^{\left(3\right)}$(Ω)) parts of χ ⁽³⁾(Ω). The non-resonant component forms a background and limits the sensitivity of CARS or CSRS measurements. In order to increase sensitivity, techniques such as chirped delayed pulses [1, 2], and pulse shaping methods [3, 4] have been employed. The main idea in these techniques is to selectively probe resonant Raman levels to enhance the sensitivity using a single laser source (as opposed to two narrow-band lasers with one of them tunable). Variations of these techniques utilize long or shaped probe pulses to suppress the non-resonant contribution. Recently, spectral heterodyne interferometry [5] was demonstrated in an imaging geometry to measure the imaginary part of ${\chi }_r^{\left(3\right)}$(Ω) thereby isolating the resonant and non-resonant contributions. This technique relies on the fact that the non-resonant contribution has a negligible imaginary component.

The Raman susceptibility ${\chi }_r^{\left(3\right)}$(Ω) is a complex tensor that contains information regarding the oscillator strength and damping behavior for each vibrational or rotational resonance. Usually, information contained in the complex ${\chi }_r^{\left(3\right)}$(Ω) is extracted using two separate measurements. For example, using conventional Raman spectroscopy (CARS or CSRS) the resonance frequency (and oscillator strength) are determined, while a time-resolved technique such as pump-probe is applied to obtain the damping behavior and magnitude of nonlinearity. In principle, it is also possible to calculate the real and imaginary parts of the complex ${\chi }_r^{\left(3\right)}$(Ω), knowing the magnitude square of ${\chi }_r^{\left(3\right)}$(Ω) with some assumptions [6]. Recently, a temporal interferometric method [7, 8] and spectral heterodyne interferometry [5] have been discussed to determine the complex ${\chi }_r^{\left(3\right)}$(Ω) in a single experiment.

In this article, we propose a method to measure the complex ${\chi }_r^{\left(3\right)}$(Ω) using spectral interferometry and shaped femtosecond optical pulses. The shaped pulses selectively excite a Raman resonance resulting in an enhanced signal. Our proposed technique combines three main advantages: a single femtosecond laser source for the excitation pump and probe pulses, signal enhancement due to selective probing of resonant Raman levels, and spectral interferometry for determining the complex ${\chi }_r^{\left(3\right)}$(Ω) spectrum. We present the results of detailed numerical simulations which reveal this is indeed a promising approach that is, in principle, quite simple and eliminates the need for mechanical scanning of a delay line.

In coherent Raman spectroscopy (shown in Fig. 1), two laser fields (E ₁(ω ₁), and E ₂(ω ₂)) with a frequency difference equal to that of a Raman level (ω ₁ - ω ₂ = Ω _R ) are employed for resonant excitation. A probe (ω ₃) measures the interaction in which coherent temporal Raman gratings produced by the laser fields ω ₁ and ω ₂ change the frequency of the probe field. The probe beam undergoes an up-shift (CARS signal, (ω_AS ) and a down-shift (CSRS signal, ω_S ) in frequency, when the interaction length is small in comparison to the coherence length [9, 10].

Coherent Raman signals can also be generated using a broadband laser instead of tunable narrow-band lasers. Spectral selectivity in exciting a particular Raman active level is achieved by Fourier domain pulse shaping or by the use of chirped delayed pulses. In either of these methods, a set of periodic pulses are created to selectively excite a Raman resonance. If the frequency of this pulse train is a harmonic of the Raman frequency, it leads to a coherent population build-up of that particular Raman resonance and to selective excitation. In our experimental simulations, we use chirped delayed pulses for simplicity. A pulse shaper could be used instead of chirped pulses.

 

Fig. 1. Energy level diagram in coherent Raman spectroscopy.

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Fig. 2. Schematic of spectral interferometric coherent Raman imaging (SIRI) setup. BS, beam splitter; SHG, second harmonic generation crystal.

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Fig. 3. Polarizations of the input electric fields (E _1,2,3) in the SIRI instrument. The excitation pulses (E _1,2) and probe (E ₃) are polarized at an angle (optimum angle depends on the χ ⁽³⁾ coefficients of the sample) to each other. The input polarizer is oriented along the unprimed x-axis and the output analyzer is along the primed y-axis.

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2. SIRI setup

The schematic of the spectral interferometric coherent Raman imaging (SIRI) setup is shown in Fig. 2. The technique utilizes femtosecond pulses that are chirped to a few picoseconds. A fraction of the pulse is up-converted to second harmonic and acts as a probe beam. The remaining fundamental beam is split into two portions which are delayed with respect to each other resulting in a pulse-train-like interference pattern. The pulse-train is our E ₁ = E ₂ laser beam. In our simulations we use E ₁(t) = E ₂(t) = E(t)+E(t - τ_d ), where $E\left(t\right)=\mid \exp \left(-\frac{t^2\left(1+\mathit{iC}\right)}{2{\tau }_g^2\left(1+C^2\right)}\right).$Here C is the chirp parameter, τ_g is the transform limited Gaussian pulse width, τ_d is the relative delay and t represents time in the group velocity frame of the pulse. A particular Raman resonance, Ω _R , is selected by tuning the input chirp C, or by tuning the delay between the two pulses according to ${\tau }_d=\frac{{\Omega }_R{\tau }_g^2\left(1+C^2\right)}{C}.$

For the probe, we use a chirped pulse as opposed to normally used scanning femtosecond pulse [1] or a narrow band pulse [4]. By using a chirped probe, the complete temporal phase and amplitude information of the nonlinear polarization can be captured simultaneously without the need to scan the probe delay. Additional advantage of using the chirped probe is, that it allows us to use spectral interferometry between the frequency shifted Raman signals (diffracted) with the undiffracted probe since the probe and signal are broadband pulses. This form of spectral interferometry is analogous to the Gabor’s in-line spatial holography [11]. From the interferometric signal the complex nonlinear ${\chi }_r^{\left(3\right)}$(Ω) can be deconvolved. Usually the diffracted beam is very weak in comparison to the undiffracted beam. We use crossed polarizers to pass a small fraction of the undiffracted probe and interfere it with the Raman signals that are both frequency shifted and polarization rotated in the interaction process. Thus the components of nonlinear polarization involved in the wave-mixing process are ${\chi }_{r\mathit{,}\mathit{\text{yxxy}}}^{\left(3\right)}$(ω ₃±Ω,ω ₃,-ω ₂,ω ₁) and ${\chi }_{r\mathit{,}\mathit{\text{yxyx}}}^{\left(3\right)}$(ω ₃±Ω,ω ₃,-ω ₂,ω ₁) (for isotropic symmetry class ${\chi }_{r\mathit{,}\mathit{\text{yxxy}}}^{\left(3\right)}$ = ${\chi }_{r\mathit{,}\mathit{\text{yxyx}}}^{\left(3\right)}$). The excitation pump fields E ₁ = E ₂ are polarized at an angle to the probe, which is polarized along x. The susceptibility component ${\chi }_{\mathit{\text{yxxy}}}^{\left(3\right)}$ generates a signal that is y-polarized and is detected using an analyzer along the y′-axis as shown in the Fig. 3 (the angle δ in the Fig. 3 is made larger for clarity). We choose our probe beam to be the second harmonic so that the background from the fundamental beams can be easily separated by a spectral filter.

3. Theory

The electric field produced by the nonlinear polarization due to the third order interaction in a thin Raman sample (group velocity mismatch and dispersion are negligible) is given by

Here E _1,2,3 are complex fields after their respective carrier frequency term is removed. In Equation 1, ${\chi }_{\mathit{\text{yxxy}}}^{\left(3\right)}$(Ω) is the sum of resonant and non-resonant contributions ${\chi }_{r\mathit{,}\mathit{\text{yxxy}}}^{\left(3\right)}$(Ω) + ${\chi }_{\mathit{\text{nr}}\mathit{,}\mathit{\text{yxxy}}}^{\left(3\right)}$(Ω). In our model, we use a real non-resonant susceptibility while the resonant susceptibility is of the form ${\chi }_{r,\mathit{yxxy}}^{\left(3\right)}\left(\Omega \right)=\frac{R_0}{{\Omega }_R^2-{\Omega }^2-\mathrm{i\Omega }\gamma }.$This form contains both coherent stokes and coherent anti-stokes components. Here, R ₀ is the oscillator strength and γ is a phenomenological damping constant for the resonance at Ω _R . The Raman signal passes through an analyzer with its polarization axis (along y′) oriented at a small angle with respect to y. This allows a small portion of the probe, E _3y′ (ω ₃) = rE _3x (ω ₃), to leak through the analyzer. The interference pattern produced by the leaking probe with the Raman signal is

Here r accounts for the leakage of the probe. A typical (simulated) spectral interferogram is shown in Fig. 4(a).

3.1. Recovery algorithm

In order to deconvolve the susceptibility from the interferogram, we use the following approach. We also measure |$E_y^{\left(3\right)}$(ω ₃)|² separately by blocking the reference probe (i.e. rE _3x(ω ₃)) completely using the analyzer. Fig. 4(b) shows such a spectrogram used in our simulations. Since this spectrogram contains contributions from (χ ⁽³⁾)², it is subtracted from the interferogram. Equation 2 is re-written as,

In these equations * represents complex conjugation. The left hand side of Equation 4, ΔI(ω ₃) = I(ω ₃)- |$E_y^{\left(3\right)}$(ω ₃)|², is an experimental quantity. Now, ΔI(ω ₃) is completely linear in χ ⁽³⁾; this allows us to use linear deconvolution techniques. In this equation, ΔI ^± (ω ₃), refer to the contributions from the positive and negative frequency side-bands when ΔI(ω ₃) is Fourier transformed. Both the cross terms in Equation 3 have contributions to the terms ΔI ^± (ω ₃) and they can be written as,

As before, ± sign in the superscripts of the cross terms refer to the ± side-band contributions upon Fourier transformation of the corresponding terms. Since ΔI (ω ₃), is a real quantity, ΔI ⁺ (ω ₃) = ΔI ^- (ω ₃)^* and only one of the above two equations is independent. The ± contributions to the $E_y^{\left(3\right)}$ (ω ₃) come from ${\chi }_{\mathit{\text{yxxy}}}^{\left(3\right)\pm }$ (Ω)(${\int }_0^{\infty }$ dω ₁ E _1x(ω ₁ + Ω)$E_{2y}^{*}$ (ω ₁)^± portion in Equation 1. Using this, Equation 5 can be written in a matrix form as,

The elements of matrices B ^± are given by

Here

The resonant χ ⁽³⁾ spectrum is symmetric for the real part and anti-symmetric for the imaginary component, upon mirror reflection about zero frequency (that is the Hermitian property, χ ⁽³⁾(Ω) = χ ^(3)*(-Ω) as shown in Fig. 4 (c), (d)). In matrix notation we have ${\chi }_{r\mathit{,}\mathit{\text{yxxy}}}^{\left(3\right)+}$ (Ω) = ${M\chi }_{r\mathit{,}\mathit{\text{yxxy}}}^{\left(3\right)-*}$ (Ω), where M(i,j) = 1 for j = N - i (N is the matrix dimension) and 0 otherwise. Typically, the non-resonant susceptibility has small dispersion over the pulse bandwidth and we consider it constant. In such a situation, Equation 6 is

We use Equation 7–10 to recover ${\chi }_{\mathit{\text{yxxy}}}^{\left(3\right)+}$ . The complete susceptibility is

Note that in the above equation, ${\chi }_{\mathit{\text{yxxy}}}^{\left(3\right)+}$ and ${\chi }_{\mathit{\text{yxxy}}}^{\left(3\right)-}$ also represent the anti-stokes and stokes components of Raman susceptibility. In Equation 10, ΔI ⁺ is obtained from the interferogram and the matrix elements of B ^± are constructed from the well characterized input fields E ₁ = E ₂, and E ₃. Since all of these quantities have experimental noise, the inversion of Equation 10 is not a well posed problem. Therefore, we apply Tikhonov regularization [12] to this equation and re-write it as,

where B_T = (B ^† B+δ_T I), Δ$I_T^{+}$ =B ^†ΔI ⁺ and B = B ⁺ +B ^- M. Here ^† represents conjugate transpose, I is the identity matrix and δ_T is a regularization parameter which accounts for realistic noise in experiments. For our simulations, we set ${\delta }_T\approx \frac{\max \left(\mid B_T\left(i,j\right)\mid \right)}{20}$ (for N = 1024). This choice of δ_T can stabilize the inversion even in the presence of ≈5%intensity noise in experimental spectra. We solve the regularized equation using LU decomposition [12].

4. Numerical results and discussion

Results of numerical simulation of our technique are presented in Fig. 4. For the simulations, we obtain the excitation pulses by chirping a τ_g =10 fs transform limited pulse with a chirp of C = 140 (1.4 ps). The probe pulse is chirped using C = 280 (2.8 ps). Part (c) and (d) of Fig. 4 show the resonant susceptibility used in the simulations. For this example, ${\chi }_{\mathit{\text{nr}}\mathit{,}\mathit{\text{yxxy}}}^{\left(3\right)}$ = 0 is used. The susceptibility has two resonances centered at Ω _R = 159 cm^-1 and 190 cm^-1. The delay between the excitation pulses is set to address the 159 cm^-1 resonance. The real and imaginary parts of the recovered spectra are shown in parts (e) and (f) of Fig. 4. Notice that only one of the two resonances is recovered because of selective excitation. The magnitude of the real and imaginary parts closely agree within the narrow spectral excitation region. The shape of χ ⁽³⁾ in other regions differs. Complete spectral shapes of χ ⁽³⁾ can be obtained by scanning the excitation pulse delay to obtain the spectrum point by point. Fig. 4(g) and (h) show the spectrum obtained by scanning. For the simulations in Fig. 4, we added a 1% uniformly distributed random noise to I(ω ₃) before applying the deconvolution algorithm.

 

Fig. 4. (a) Spectral interferogram of the reference with the Raman signal, (b) spectrogram of the Raman signal, (c) and (d) real and imaginary parts of Raman susceptibility used in the simulations, (e) and (f) selectively recovered susceptibility using (a) and (b). (g) and (h) Reconstructed susceptibility by scanning the delay.

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Fig. 5. Comparison of original (solid lines) and recovered (dashed lines) susceptibility in presence of a large background. The ratio of non-resonant background to Raman signal is 50 in (a) and (b) and 250 in (c) and (d). The recovered function matches closely within the excitation region.

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4.1. Recovery in presence of a large background

In experimental measurements, we must be able to recover χ ⁽³⁾ in the presence of a large non-resonant background. We consider two examples with a background that is nearly two orders of magnitude larger than magnitude of resonant signal (i.e. at resonance, ${\chi }_{\mathit{\text{nr}}}^{\left(3\right),\mathit{\text{real}}}$/${\chi }_r^{\left(3\right),\mathit{\text{imag}}}$ ≈ 100). A 0.5% noise level is added to the interferograms in the simulations to account for experimental noise. Because of large background, direct regularization of Equation 10 yields poor recovery. Therefore, we first solve for the magnitude of the non-resonant contribution. We re-write Equation 10 as,

where B = (B ⁺ + B ^- M). Since the background is large, we approximate the real part of ΔI ⁺ - ${B\chi }_{\mathit{\text{rr}}\mathit{,}\mathit{\text{yxxy}}}^{\left(3\right)+}$ to zero and solve for the constant ${\chi }_{\mathit{\text{nr}}\mathit{,}\mathit{\text{yxxy}}}^{\left(3\right)+}$. Knowing the value of non-resonant χ ⁽³⁾, we regularize Equation 13 and solve it as before. Results of the simulations are shown in Fig. 5. For the two rows in Fig. 5, background to signal ratios are 50 and 250 respectively. Although the shapes of recovered spectra are more distorted, the magnitude of the real and imaginary parts of χ ⁽³⁾ agree closely to the original value within a small spectral range centered around the excitation frequency. As before, scanning the delay can reconstruct the complete spectrum.

 

Fig. 6. Red colored area is assigned a Raman active mode at 625 cm^-1 while the yellow area is Raman active at 682 cm^-1. Green region is Raman active at both the frequencies. (a) Original image (b) 50 times stronger (than peak Raman signal) background, (c) recovered image via selective excitation.

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4.2. Imaging

Spectrally selective excitation is especially attractive for medical imaging. The spectral selectivity makes the best use of inherent image contrast available in biological samples, without the need for external agents. Fig. 6 shows the simulations that illustrate the performance of the algorithm for a spatially varying image (computer generated) with large non-resonant background. The two colors for the cells in the Fig. 6 (a) represent two different Raman active modes at 625 cm^-1 (example, a cell transfected with Z-DNA) and 682 cm^-1 (example, a cell trans-fected with B-DNA). The green colored cells are also spatially overlapping with the red cell. In addition to resonant Raman, the original image also has a spatially varying (but spectrally constant) non-resonant background shown in Fig. 6 (b). The ratio of background to Raman signal is nearly 50 times. In the simulations we prepare the pulses to address the 625 cm^-1 resonance. The right most image in Fig. 6 is the recovered image. We can see that the shape of the cell is recovered quiet well. The noise in the recovered image is due to the approximate solution found by Tikhonov regularization.

4.3. Resolution

A broad range of Raman resonances can be selectively excited and probed with good resolution using this technique. The overall resolution of the technique has contributions from the probe and the spectral selectivity process. For obtaining high resolution in spectral selectivity, the chirp of the excitation pulse should be large and linear. The spectral selectivity resolution (ΔΩ_s) is approximately given by inverse of the temporal overlap of the two chirped excitation pulses according to $\Delta {\Omega }_s\propto \frac{1}{{\tau }_g{\left(1+C^2\right)}^{0.5}-{\tau }_d}\approx \frac{1}{{\tau }_{g}C\left(C-{\tau }_g{\Omega }_R\right)}$ (for C ≫ 1)This suggests that for selectively exciting a large Raman shift Ω _R , pulses chirped to long pulse widths (large C) result in high spectral selectivity. In this expression, maximum value of τ_g Ω _R is the time bandwidth product of the transform limited pulse (since maximum Raman shift measurable by the technique is limited by the laser bandwidth) which is typically less than 3. By chirping the pulse nearly 100 times (C ≈ 100), spectral selectivity of nearly 0.1 cm^-1 is possible. The overall resolution of the measured Raman spectrum is also determined by the probe beam. For small Raman shifts, the interferogram contains wide spectral overlap region between the probe and the Raman shifted beam resulting in several fringes in the interferogram and a high resolution reconstruction. For large Raman shifts, the spectral overlap decreases (lesser number of fringes) leading to lower resolution. Good resolution is possible for the entire measurable energy range, if the bandwidth of the probe is sufficiently larger (twice) than the excitation pulse bandwidth (E ₁, E ₂). In theory, the upper limit of measurable energy range of the instrument is limited by the bandwidth of the excitation pulses. For microscopy applications, we are interested in probing only a particular resonance, in such cases, if we choose ΔΩ_S to be same as the spectral width of the resonance $\left(\frac{1}{{\tau }_{g}C}\approx {\Omega }_R\right)$, we can obtain good sensitivity.

 

Fig. 7. Illustration of distortion effects due to a nonlinear chirp. Oscillator pulses are dispersed (using second and third order dispersion) by 10 cm of fused silica. (a)-(b) Original, (c)-(d) recovered χ ⁽³⁾ when the excitation is tuned to 159 cm^-1, and (e)-(f) χ ⁽³⁾ reconstructed by scanning the delay.

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4.4. Practical limits

An attractive approach to implement the SIRI technique is to replace the mirrors in Fig. 2 with bulk prisms and directly use femtosecond pulses from an oscillator. The oscillator pulses broaden due to material dispersion. A total of 10 cm of fused silica results in the required linear chirp. However, in addition to linear chirp, this approach results in third order pulse distortion. Deviation of the pulse shape from linear chirp can affect the spectral selectivity and resolution. Simulation results in Fig. 7 show the effect of third order dispersion from a 10 cm fused silica. For illustration we use a χ ⁽³⁾ with three resonances at 159 cm^-1, 190 cm^-1, and 288 cm^-1. The delay between the pulses in the excitation arm is tuned to address the 159 cm^-1 resonance. However, the two nearby resonances are simultaneously excited by the excitation pulse due to third order distortion. Since two nearby resonances are excited, the sensitivity of the technique to the Raman resonance at 159 cm^-1 is lower (by 23%) than the case when there is no third order distortion. In the reconstruction (Fig. 7(c), (d)) we notice that the χ ⁽³⁾ at 159 cm^-1 closely matches the original. Fig. 7(e)-(f) show the reconstruction obtained by scanning the pulse delay. The overall reconstructed χ ⁽³⁾ obtained by scanning closely resembles the original.

5. Conclusions

In conclusion we have presented a spectrally selective interferometric Raman imaging technique to recover the real and imaginary parts of χ ⁽³⁾. The technique uses chirped pulses which result in smaller non-resonant contributions than short pulses. Due to the spectrally selective excitation, the technique is very sensitive. Our simulations show that even in presence of a strong non-resonant background, over two orders of magnitude smaller imaginary part of Raman susceptibility is recovered. The technique only requires a single laser to selectively excite Raman resonance in the finger-print region. We are currently implementing this technique in our laboratory.