1. Introduction

Vibrational spectroscopy provides the means to non-destructively assess the molecular composition of subcellular structures [1–3], trace metabolic and functional processes [4,5], and identify disease [6–8]. A variety of different methods have been demonstrated for extracting the molecular information, each with different characteristics (see Ref [9] for a recent review). In this work we focus on stimulated Raman scattering (SRS), which provides several advantages over other methods when it is applied to microscopy; for example, under appropriate conditions it has orders of magnitude better sensitivity over spontaneous Raman scattering [3,10,11], it is insensitive to the non-resonant background present in coherent anti-Stokes Raman scattering (CARS) [12,13], and has approximately an order of magnitude better spatial resolution than IR microscopy [14,15]. The advantages of SRS have enabled imaging of tissues over large areas based on intrinsic lipid content [16], video rate molecular imaging [17], and characterization/tracing of drug delivery [4,5]. However, due to the different sources of noise and technical constraints (discussed below), most SRS methods are limited to point scanning at a few Raman frequencies (most often, only one). This work presents a proof of concept for a novel approach based on dispersion to overcome such limitation that may enable spatial and spectral multiplexed SRS imaging.

In SRS the Raman signal is amplified by matching the frequency difference between two light fields to a particular molecular vibrational frequency, Ω₀. Under these conditions the higher-energy field experiences a loss (stimulated Raman loss), while the lower-energy field experiences a gain (stimulated Raman gain) [Fig. 1(a)]. Unlike spontaneous Raman scattering or CARS, where the signals of interest are emitted at a different wavelength than the input light, SRS relies on detecting small amplitude changes on the input fields (relative magnitude of 10⁻⁶ to 10⁻³). In order to extract the small nonlinear signal from the large background, transfer modulation schemes are employed (i.e., high-frequency lock-in detection) [18]. This also reduces the low-frequency laser noise that can otherwise obscure the nonlinear signals. However, this very same scheme that enables SRS imaging makes spatial and spectral multiplexing technically challenging, though some methods for rapid scanning across wavelength or spectral multiplexed detection have been reported [19–23]. Here we use the refractive index (RI) changes (i.e., dispersion) that occur simultaneously with the loss/gain processes in SRS to reduce the influence of noise, and enable efficient multiplexing even in cases where high-frequency lock-in detection is not possible or is cumbersome due to, for example, slow detector response. A detailed theoretical analysis is presented along with a noise characterization; and finally, spectral and spatial multiplexing are demonstrated using phase microscopy and optical coherence tomography.

 

Fig. 1 (a) SRS energy diagram with two fields tuned to a material’s vibrational resonance at a frequency Ω₀. (b) Stimulated Raman interaction between a broadband (probe) pulse and a pump result in amplifications (Stokes side) and loss (anti-Stokes side) of the probe. The material’s complex third-order nonlinear susceptibility is also plotted.

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2. Theory

When two input fields (e.g., Stokes and pump) coherently excite a molecular oscillation, the material undergoes a polarization change that is characterized by the third-order nonlinear optical susceptibility (χ⁽³⁾). After the Stokes (or anti-Stokes) field propagates through the perturbation, the field’s properties change according to [24],

While the imaginary part of χ⁽³⁾ can be assessed by monitoring intensity changes, detection of the real part requires a phase sensitive approach, such as Fourier-domain (FD) interferometry [25–28]. Thus, consider an additional reference field E_ref = E₀ × exp(-iωT), which is identical to the probe in absence of the pump but with a time-delay T (assume both fields follow the same path, see Fig. 2). The FD interferometric signal (expressed in complex form) between the probe and reference fields after a nonlinear interaction can be described as,

 

Fig. 2 Experimental system. A 4-f femtosecond pulse-shaper is used to generate two identical pulses (probe and reference) separated by time T. SP: spectrometer, AOM: acousto-optic modulator, RF: radio frequency.

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3. Experimental system and methods

The experimental system (Fig. 2) consists of a modified pump-probe setup with a narrow-band pump, a wide-band probe and time-delayed replica (reference), and Fourier-domain interferometric detection. We use a regenerative amplifier laser source (RegA, Coherent) with a repetition rate of 20 kHz (λ₀ = 808 nm, Δλ = 25 nm and duration ~75 fs). Eighty percent of the RegA output is used to pump an optical parametric amplifier (OPA, Coherent), which then generates the pump beam (λ₀ = 647 nm, filtered to Δλ = 1.8 nm corresponding to a duration of ~0.5 ps). The other twenty percent of the RegA output is sent to a 4-f pulse shaper to produce two identical pulses separated by 2 ps, which serve as the probe and reference. The probe and reference beams are combined with the pump using a dichroic mirror, the pump is temporally overlapped with the probe, and all three beams are focused onto a sample. Transmitted or scattered light is detected with a high-resolution spectrometer (spectral resolution δλ = 0.074 nm; BaySpec OCT spectrograph), triggered to collect every shot of the laser (integration time set to 20 µs). For more details see Refs [27,28] which describe a similar setup.

The interferometric data is processed using methods similar to holography [26,27]: The spectral data [black line in Fig. 3(a)] is Fourier-transformed (FT), then the peak in the time domain corresponding to the 2 ps delay between the probe and reference is digitally filtered [Fig. 3(b)]. This gives the complex interferometric signal described in Eq. (3). Next, the peak is shifted to DC, and transformed back to the frequency-domain, which yields the probe’s amplitude and phase relative to the reference field [red and blue lines on Fig. 3(a)]. The process is repeated with and without the pump beam. Figure 3(c) shows the amplitude and phase changes resulting from a nonlinear interaction with a liquid olive oil sample, using a transmission geometry with a 10X, 0.25NA objective, a pump power of 50 µW, and a probe-plus-reference power of 50 µW.

 

Fig. 3 Signal processing and representative data. The interferometric signal [black line in (a)] is Fourier transformed (b). The time-domain peak is filtered, shifted back to DC, and transformed back to the Fourier domain. This process gives both the amplitude and phase [red and blue lines, respectively, in (a)]. (c) The process is repeated with and without the pump, and the changes reveal the material’s nonlinear properties. (d) The conventional SRS (Raman) spectrum is obtained by normalizing by the source’s power spectral density. (e) The dispersive properties are obtained by dividing by ω/c. The plots (d-e) show three different probe powers, with the pump set to 50 µW. The error (gray area) is determined from the standard deviation of 10 measurements. Inset in (e) shows the modeled dispersion obtained by first modeling the attenuation using two Gaussian curves centered at −2922 cm⁻¹ and −2885 cm⁻¹, as shown in the inset in (d), and then applying a subtractive Kramers-Kronig relation.

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4. Results

4.1 Signal scaling and noise analysis

To confirm the theoretical analysis, we first follow conventional SRS signal detection schemes, which involve lock-in detection (pump modulated at 5 kHz). One hundred consecutive laser shots are used to generate one lock-in detection signal, which is equivalent to a 5 ms lock-in time constant (bandwidth of 200 Hz). Then the SRS spectrum (i.e., attenuation) is obtained by dividing the amplitude modulation by I₀(ω), assessed by averaging 1000 acquisitions without the pump. The phase information, ΔPhase(ω) = (ω/c)2z_rΔn(ω) [26,27], is divided by ω/c to obtain the changes in the refractive index (constant terms are ignored). The process is repeated 10 times to assess noise levels. The resulting attenuation (α) and dispersion (Δn) spectra for three probe powers are shown in Figs. 3(d) and 3(e). The attenuation measurements are in good agreement with the known Raman signature of olive oil [29]. To verify the dispersive behavior, we first model the attenuation using two Gaussian curves centered at −2922 cm⁻¹ and −2885 cm⁻¹, as illustrated in the inset of Fig. 3(d), and then apply a subtractive Kramers-Kronig relation to simulate the causal signal’s real component [30]. The model [inset of Fig. 3(e)] is in good agreement with the experimental results.

Next, we analyze the signal and noise scaling with probe power (Fig. 4). As expected, the amplitude changes scale linearly with probe power (fitted slope of log-log plot using all points = 0.96 +/− 0.01), while the dispersion is constant (slope = 0.00 +/− 0.03). Moreover, the noise of the amplitude measurement increases proportionally to √I₀ (slope = 0.59 +/− 0.04), while the dispersive noise decreases with √I₀ (slope = −0.50 +/− 0.07); both resulting in an effective SNR scaling proportional to √I₀. These results are in agreement with theory and indicate that the lock-in detection measurements are shot noise limited (see Appendix for more details).

 

Fig. 4 (a) Signal, (b) noise (assessed from 10 independent measurements), and (c) SNR scaling with varying probe power for both the phase and amplitude measurements using olive oil as the sample.

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We confirm the expected signal and noise dependence on the pump beam with and without lock-in detection (labeled modulated and unmodulated, respectively) using benzene as the sample [Fig. 5(a)-5(c)]. Representative attenuation and dispersion spectra are shown in Fig. 5(d), which are in good agreement with the expected Raman response [31] and modeled dispersion [Fig. 5(e)]. The model uses an attenuation spectrum consisting of two Gaussian responses centered at 2960 cm⁻¹ and 3049 cm⁻¹, and then applying the subtractive Kramers-Kronig relation to obtain the dispersion. To acquire the attenuation and dispersion spectra without lock-in detection (unmodulated pump), we take the average amplitude and phase with the pump ‘on’ using 100 consecutive acquisitions (same total number of acquisitions as the lock-in measurement), repeat 10 times, and compare to a single averaged measurement with the pump ‘off’ using 1000 acquisitions. As we have done in previous work investigating molecular reorientation [27,28] and linear dispersion [25,26], the random phase variations introduced by instabilities in the interferometer are removed by subtracting the average spectral phase from each acquisition, which does not alter the spectral features of interest. This was not done for the previous experiment since lock-in detection obviates the step (these fluctuations typically have a low-frequency). The power study [Fig. 5(a)-5(c)] shows the expected linear dependence of the signals with increasing pump power, and a constant noise value. It is important to highlight that the noise of the unmodulated amplitude signal is approximately an order of magnitude larger than all of the other types of measurements, including the unmodulated phase signal. This results from the fact that the unmodulated amplitude is highly affected by laser noise that persists without lock-in detection. The SNR plot [Fig. 5(c)] shows that the unmodulated phase signal has an equivalent SNR to the modulated phase signal, again indicating that the phase is unperturbed by other sources of noise that plague the unmodulated amplitude signal. The phase measurements also show a slight SNR improvement over the modulated amplitude signal, as expected (see Appendix), and the unmodulated amplitude exhibits the lowest SNR.

 

Fig. 5 (a) Signal, (b) noise (assessed from 10 independent measurements), and (c) SNR scaling with varying average pump power for both the phase and amplitude measurements using benzene as the sample. (d) Experimental and (e) modeled attenuation and dispersion spectra. The error (gray area) is determined from the standard deviation of 10 measurements.

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We further investigate the noise properties of the signals by analyzing the noise power spectrum (NPS) at two particular wavelengths. The NPS is obtained by acquiring 1000 consecutive amplitude or phase measurements (without the pump), and then taking an FT (amplitude measurements are normalized by their mean). The NPS of the amplitude [Fig. 6(a)] shows a number of features resulting from 1/f noise, as well as other periodic components (e.g., at 3 kHz) that arise from, for example, air currents or vibrations. This highlights the need for lock-in detection, which shifts the measurement to a clean region of the noise spectrum. The features on the phase NPS are much less pronounce [Fig. 6(b)], since, as previously discussed, it is a relative measure. Most importantly, all noise components (features in the spectrum) can be completely eliminated by simply subtracting the phase of one spectral point from the other [yellow line in Fig. 6(b)]—this is analogous to subtracting the average spectral phase. Self-referencing in the same manner for the amplitude signal does not produce the same outcome: taking the ratio of the two points [yellow line in Fig. 6(a)] or subtracting one from the other (data not shown) amplifies the noise. This suggests that the amplitudes of these two spectral points are anti-correlated (e.g., the spectrum is oscillating about a central point). Similar results are produced even when taking advantage of temporal multiplexing; that is, using the amplitude from the DC part of the interferometric spectrum and comparing (ratio or difference) to the component shifted by 2 ps at same wavelength. While these two points are correlated, the peaks are not completely removed (likely due to uncorrelated noise between the reference and probe). These results clarify why the unmodulated phase signal exhibits an SNR advantage over the unmodulated amplitude: the phase signal is more robust to laser and ambient noise. Further, the results pave the way for efficient multiplexing even in cases where high-frequency lock-in detection is not possible or is cumbersome due to, for example, slow camera response.

 

Fig. 6 (a) Noise power spectrum (NPS) of the amplitude at two wavelengths (blue and red lines). The yellow line is the NPS of the ratio of the normalized signals. A similar NPS is obtained from the difference of the two signals. Inset shows the measured signals (blue and red) and their ratio (yellow). (b) NPS of phase at two wavelengths (blue and red lines). The yellow line is the NPS of the differences between the two signals. Note that this removes all noise components. Inset shows the measured signals (blue and red) and the difference (yellow).

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4.2 Dispersion-based SRS imaging

4.2.1 Spectral multiplexing

As a basic demonstration of the molecular imaging capabilities of dispersion-based SRS, we image a thin sample composed of oil and water, pressed between two coverslips (~200 µm thick). Here we use a 20X, 0.5NA objective, with a pump power of 100 µW and a probe-plus-reference power of 10 µW. Spectrometer integration time is kept at 20µs. The sample is raster scanned along the x-y plane using a NanoCube piezo stage with a maximum scanning area of 120 X 120 µm². The transmission image [Fig. 7(a)] shows the two liquids and their interface, which appears dark because light refracts away from the entrance slit of the spectrometer. In the surrounding areas of the interface, light also couples slightly differently into the spectrometer, changing the initial spectral amplitude and dispersion conditions. These artifacts are removed by modulating the pump at 5 kHz as the sample is scanned which superposes a sinusoid pattern on the data. Then digital holography methods are used to extract the signals of interest [32]: Each row on the image is Fourier transformed, which produces a DC component and a component shifted by the spatial frequency produced by the pump modulation. The shifted component is digitally filtered and demodulated. The digital filter produces a signal that is comparable to lock-in detection with a 400 µs lock-in time-constant, and a 2.5 kHz bandwidth which spans a significant portion of the available bandwidth (10 kHz) and is too broad to remove many of the noise features in the amplitude NPS [see Fig. 6(a)], thus resulting in the observed periodic artifact in the images [Figs. 7(b), 7(d) and 7(e)]. A longer lock-in time-constant is required to suppress these artifacts, but this would significantly slow down imaging speeds. For thinner samples the dispersion measurement does not require this step (i.e., the pump modulation and image demodulation)—in such a case a single reference dispersion spectrum is sufficient, as has been demonstrated for dispersion-based molecular imaging in the linear regime [26].

 

Fig. 7 Broadband dispersion-based SRS imaging. (a) Transmission image of a thin sample composed of oil and water, pressed between two coverslips. (b) SRS molecular image based on nonlinear amplitude changes computed by projecting the spectrum in each spatial pixel of the image onto the average olive oil SRS spectrum. (c) Average SRS spectrum from the boxed two regions in (a). (d) SRS molecular image based on nonlinear dispersion. (e) Nonlinear dispersions image scaled by the signal intensity, which rejects noise caused by low intensity signals. (f) Average SRS dispersion spectrum from the boxed two regions in (a).

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The average spectral changes in amplitude and refractive index from two regions of interest, one in water and another in oil [see boxes in Fig. 7(a)], clearly demonstrate the superior SNR of the dispersion measurement [Fig. 7(f)] compared to the amplitude [Fig. 7(c)]. The average spectra are then used to render false color images of the molecular information. Specifically, the spectrum from each spatial pixel in the amplitude/dispersion image is projected onto the average oil amplitude/dispersion spectrum. The results show that, compared to the amplitude image [Fig. 7(b)], the dispersion image [Fig. 7(d)] is less noisy (due to better suppression of noise components) and has better contrast between the two substances. As discussed above [see Fig. 4(b)], the phase noise increases with decreasing collected light, thus dark regions produce noisy results that yield large errors in the projection image [truncated using neutral gray in Fig. 7(d)]. Figure 7(e) circumvents this problem by simply scaling (i.e., multiplying) the dispersion image by the intensity (note that the noise scales with √I₀, while the intensity scales with I₀).

4.2.2 Spatial multiplexing

With the understanding that dispersion-based SRS detects changes in materials’ refractive index produced by the presence of a pump beam, we can borrow spatial multiplexing methods that have also been recently applied to photothermal imaging. There are two possibilities: (1) using wide-field holography methods [33,34] or (2) FD-optical coherence tomography (OCT), which achieves axial multiplexing by coherence gating and can image relatively deep into highly scattering samples [35,36]. Here we use the later since it uses the same instrumentation as the experiments above with minimal adjustments.

The modified system consists of a Michelson interferometer with f = 50 mm lenses before the sample and reference arms. With our laser light source and spectrometer, we obtain an axial resolution of ~10µm and depth range of ~2.5mm. Pump and probe powers are set to 100µW each, measured at the back aperture of the sample arm lens. For this proof-of-concept experiment, the pump bandwidth is increased (~10 nm) to delivered higher laser powers to the sample, consequently broadening the spectral features of the SRS spectrum. More advanced methods can be used to extract the spectral information simultaneously with the spatial information (e.g., spectroscopic OCT [37,38]) if a narrow-band pump were used, however such methods are outside of the scope of this manuscript. Signal acquisition (1024 acquisitions per A-scan) and processing is identical to photothermal OCT [36], with the exception that the pump modulation rate is set to a much higher frequency (5 kHz) than is possible for photothermal imaging. Pump modulation frequencies for photothermal imaging are typically around a couple of hundred hertz to avoid heat pileup, which destroys the molecular signal and can also damage the samples [34,36].

Figure 8 shows molecular OCT images of 3 µm polystyrene beads suspended in agar, and freshly excised human adipose tissue from the abdomen. The anonymous and discarded tissue samples were procured from the Department of Plastic Surgery at Duke University and are not subject to an Institutional Review Board protocol. Here the molecular information is overlaid with the structure, all rendered from the same data. To enable a fair comparison of the signal strength of the nonlinear amplitude and phase changes, the molecular information is normalized by the noise. Note that the SNR scale ranges from 0 to 83dB for Fig. 8(a) and 0 to 78dB for Fig. 8(b), using the standard OCT definition of SNR = 20log(Signal/Noise). For the beads, the pump was shifted to 747 nm to target the 1000 cm⁻¹ vibrational mode of polystyrene. Again, for both samples the results show that the nonlinear phase provides superior SNR compared to the amplitude. Since no spectral information is recovered here, we confirm that the signals result from SRS and not a thermal response by reimaging the samples with a short pump-probe time-delay of τ = 2ps, which is insufficient for heat dissipation (i.e., thermal signals should persist). The results show that the molecular signals disappear, confirming that the changes in phase and amplitude result form an instantaneous nonlinear stimulated Raman response.

 

Fig. 8 SRS molecular OCT imaging. (a) Images of 3µm polystyrene beads and (b) freshly excised human adipose tissue from the abdomen taken with the pump and probe beams temporally overlapped (τ = 0) and with a 2ps time delay. The SNR scale ranges from 0 to 83dB for (a) and from 0 to 78dB for (b) using the standard OCT definition [SNR = 20log(Signal/Noise)].

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5. Discussion

In this work we have demonstrated (theoretically and experimentally) that dispersion effects that accompany the absorptive/gain processes in SRS can be used for spectroscopy and molecular imaging. Because the dispersive signals do not depend on the intensity of the detected light (I₀), they are more robust to laser and ambient noise compared to the amplitude changes. Thus, this method can potentially enable efficient spatial and spectral multiplexing even in cases where high-frequency lock-in detection is not possible or is cumbersome. We showed that it is possible to assess the full dispersion spectrum (currently limited by the laser source to ~400 cm⁻¹) with a single spectrometer acquisition or laser shot (20–50 µs). For imaging with broadband spectral information, we demonstrated pixel rates of 2.5 kHz (400 µs). For thin samples (few tens of microns) pixel rates of 20 kHz (50 µs) are possible for the dispersion-based measurement. We also demonstrated spatial multiplexed SRS imaging using OCT, which could have a significant impact in many areas of biology and medicine (e.g., breast or brain tumor margin detection). Finally, we outlined other opportunities for spatial multiplexing using wide-field holography and also spectroscopic OCT, which would massively parallelize the spatial and spectral information. Our future work will also explore the use of higher repetition-rate light sources (MHz), which are more commonly used in biophotonics.

It is important to highlight the difference between dispersion-based SRS and other related methods. First consider photothermal imaging: While the two methods measure changes in the refractive index, they are based on fundamentally different physical processes, with SRS offering a much wider set of endogenous molecular targets, some of which have already shown tremendous value for disease detection, among many other applications [5,6,16]. Dispersion-based SRS can also achieve faster imaging speeds since it is not limited by thermal diffusion. As briefly discussed in the results section, pump modulation rates for photothermal imaging have to be low to allow for the heat to dissipate from one cycle to the next. In practice the pump modulation frequency is typically set to ~200 Hz. SRS, on the other hand, is an instantaneous process, so our dispersion-based method is fundamentally limited by the repetition rate of the laser or detector speed (here ~20 kHz).

Broadband dispersion-based SRS imaging shares some commonalities with nonlinear interferometric vibrational imaging (NIVI), a CARS-based method [6,29]. NIVI uses a similar experimental setup (RegA with an OPA) and FD-interferometry to remove the non-resonant background present in CARS. However, NIVI detects the anti-Stokes radiation, thus the setup must supply three carefully tuned beams: pump, Stokes, and also a reference anti-Stokes beam to mix with the emitted radiation from the sample. Further, the system must be carefully calibrated for pump-induced chirp and the phase of the reference anti-Stokes field. These steps are not necessary for dispersion-based SRS. Further, NIVI has only been demonstrated for thin (5 µm) biopsy samples.

Dispersion-based SRS OCT is also similar to pump-probe OCT [39,40]. While the general system layout is the same, there are two major differences: First, pump-probe OCT has been applied to detect long-lived excited states of strong absorbers, and to our knowledge has never been used for molecular imaging based on coherent Raman scattering. Secondly, nonlinear changes in the phase of the OCT signals have not been investigated with pump-probe OCT.

A drawback of our approach is that the dispersion spectra are more difficult to interpret than the conventional Raman signals. However, because the two spectra (real and imaginary parts of χ⁽³⁾) are related via the Kramers-Kronig relation, it is possible to convert the dispersion signal back to the more conventional form. Moreover, almost all broadband vibrational imaging methods still require some type of mathematical interpretation, such as principal component analysis, to differentiate between chemical species [3,6,41,42]—such methods are also likely to be of more practical use for our approach.

Finally, it is worth pointing out that the same signal and noise analysis applied here also holds for linear absorption and dispersion. Recently substantial efforts have been made to quantify endogenous chromophores, such as hemoglobin, using phase imaging [26,43–45]. The analysis performed here can help better determine, based on the noise properties of the system, whether the added complexity of interferometric (phase) detection is warranted, or if simpler methods based on absorption (such as hyperspectral imaging) would provide sufficient sensitivity for their applications.

6. Conclusion

SRS has shown tremendous promise in the field of biology and medicine. This proof of concept shows that dispersion effects that accompany the conventional SRS loss/gain processes may offer new opportunities for spectral and spatial multiplexing. Though the quality of the images presented here are modest, better-suited light sources—particularly a stronger, narrow band pump—will likely provide drastic improvements. Such improvements will also be critical for exploring more complex biological samples. The combination of dispersion-based SRS and phase sensitive imaging methods has the potential to enable faster wide-area and volumetric molecular imaging. Such methods would be valuable in a clinical setting for a number of applications.

Appendix

In this section we derive an analytical expression for the signal and noise of the amplitude and phase signals. Note that our treatment applies to both linear and nonlinear absorptive and dispersive effects. Starting from the detected complex signal given by Eq. (3), the amplitude changes can be described as,

(4)$$\Delta \left|{\tilde{I}}_{\mathrm{int}}(\omega )\right|=I_0(\omega )-I_0(\omega )e^{-\frac{C_0}{2}\omega \mathrm{Im}\left\{-{\chi }^{(3)}\right\}{I}_{pu}}$$

which can be rewritten in the more familiar form of

(5)$$\Delta \left|{\tilde{I}}_{\mathrm{int}}(\omega )\right|=I_0(\omega )\cdot (1-e^{-\frac{1}{2}\alpha z_r})$$

with the attenuation coefficient α = 3ω/(2n₀²c²ε₀) × Im{-χ⁽³⁾} × I_pu. The factor of ½ in the exponential is a result of interferometric detection. For small α, the amplitude changes reduce to

(6)$$\Delta \left|{\tilde{I}}_{\mathrm{int}}(\omega )\right|=\frac{1}{2}{I}_0(\omega )\cdot \alpha z_r=\frac{3\omega z_r}{4n_0^2{c}^2{\epsilon }_0}\mathrm{Im}\left\{-{\chi }^{(3)}(\Omega )\right\}{I}_{pu}\cdot I_0(\omega ).$$

Next, we consider the wavelength-dependent phase changes,

(7)$$\Delta \angle {\tilde{I}}_{\mathrm{int}}(\omega )=C_0\omega \mathrm{Re}\left\{{\chi }^{(3)}(\Omega )\right\}{I}_{pu}=\frac{3\omega z_r}{2n_0^2{c}^2{\epsilon }_0}\mathrm{Re}\left\{{\chi }^{(3)}(\Omega )\right\}{I}_{pu}$$

where ∠ denotes the phase angle. To compare the two measurements we now turn our attention to the frequency-dependent susceptibility for a single resonant frequency at Ω₀, which can be expressed as a complex Lorentzian function,

(8)$${\chi }^{(3)}(\Omega )={\chi }_0^{(3)}\cdot \frac{{\Omega }_0^2}{{\Omega }_0^2-{\Omega }_{}^2+i\Omega \Delta \Omega }.$$

The real and imagery responses are illustrated in Fig. 1(b). At resonance (i.e., peak absorption/gain), the maximum value of the imaginary part is given by Im{χ⁽³⁾}_max = χ₀⁽³⁾ × Ω₀/ΔΩ. The maximum and minimum values for the real part are located slightly off resonance [Ω = Ω₀(1 ± ΔΩ/Ω₀)^½] with an amplitude of Re{χ⁽³⁾} = ± ½χ₀⁽³⁾ × Ω₀/ΔΩ, assuming Ω₀>>ΔΩ. Note that the non-resonant response is a constant term of the third-order nonlinear optical susceptibility (i.e., independent of Ω) and does not alter the dispersive measurements. As we have previously explored [27,28], other nonlinear effects, such as two-photon absorption and molecular reorientation, do not produce appreciable wavelength-dependent refractive index changes, particularly when a narrow-band pump beam is utilized.

Now we discuss how shot-noise affects the two signals. For the amplitude it is clear that noise scales with the square root of number of detected photons σ_shot~√I₀. However the analytical form of the phase noise (also commonly referred to as the phase sensitivity, δϕ) requires a few more steps. Following Choma et al. [46], the shot-noise has a random phase, ϕ_rand, with a uniform probability distribution from 0 to 2π, as illustrated in Fig. 9. Thus, the only parts of the noise that contribute to the phase sensitivity are those orthogonal to the complex signal ${\tilde{I}}_{int}$, mathematically σ_shot sin(ϕ_rand). The final expression for the sensitivity of the phase is given by the average angle spanned between the signal and noise [46],

(9)$$\delta \varphi =\frac{2}{\pi }{\displaystyle \underset{0}{\overset{\pi /2}{\int }}{\tan }^{-1}\left(\frac{{\sigma }_{noise}\sin {\varphi }_{rand}}{\left|{\tilde{I}}_{\mathrm{int}}\right|}\right)}\cdot d{\varphi }_{rand}=\frac{2}{\pi }\cdot \frac{1}{\sqrt{I_0}}$$

where $\left|{\tilde{I}}_{int}\right|=I_0$ and I₀>>σ_shot. Finally, the SNR of the amplitude and phase is given by,

(10)$$SN{R}_{amp}=\frac{\Delta \left|{\tilde{I}}_{\mathrm{int}}\right|}{{\sigma }_{shot}}~\frac{C_0\omega {\chi }_0^{(3)}{\Omega }_0{I}_{pu}\sqrt{I_0}}{2\Delta \Omega }$$

(11)$$SN{R}_{phase}=\frac{\Delta \angle {\tilde{I}}_{\mathrm{int}}}{\delta \varphi }~\frac{\pi C_0\omega {\chi }_0^{(3)}{\Omega }_0{I}_{pu}\sqrt{I_0}}{4\Delta \Omega }$$

Therefore, in the shot-noise limit both signals follow the same scaling, with the phase offering only a slight advantage, in agreement with the results shown in Figs. 4(c). However, our analysis of the SNR with and without high-frequency lock-in detection (Fig. 5) suggest that other noise sources, such as laser or ambient noise, do not possess a random phase and only contribute to the noise along the direction of the complex signal, that is σ_shot cos(ϕ_rand), which does not affect δϕ. This feature gives the dispersion measurement a significant SNR advantage over the amplitude and provides a number of alternatives for spatial and spectral multiplexing.

 

Fig. 9 Representation of the phase sensitivity, δϕ, given by the average angle spanned by the complex signal,${\tilde{I}}_{int}$,and orthogonal noise components, σ_shot sin(ϕ_rand). Adapted from [46].

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Acknowledgments

We gratefully acknowledge Dr. Simone Degan and Dr. Detlev Erdman for procuring tissue samples. This research was supported by the NIH R01-CA166555 (WSW), the Burroughs Wellcome Fund 1012639 (FER), NIH F32CA183204 (FER), NSF CHE-1309017 (MCF), and Duke University.

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