## Spatial-spectral coupling in coherent anti-Stokes Raman scattering microscopy |

Optics Express, Vol. 21, Issue 13, pp. 15298-15307 (2013)

http://dx.doi.org/10.1364/OE.21.015298

Acrobat PDF (952 KB)

### Abstract

Coherent anti-Stokes Raman scattering (CARS) microscopy is a third-order nonlinear optical technique which permits label-free, molecule-specific hyperspectral imaging. The interference between coherent resonant and non-resonant terms leads to well known distortions in the vibrational spectrum, requiring the use of retrieval algorithms. It also leads to spatial imaging distortions, largely due to the Gouy phase, when objects are smaller than the Rayleigh range. Here we consider that the focal position and spectral contributions to the nonlinear image formation are intrinsically coupled and cannot be corrected by conventional retrieval methods.

© 2013 OSA

## 1. Introduction

1. A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. **82**4142–4145 (1999) [CrossRef] .

4. J. Cheng, A. Volkmer, L. D. Book, and X. S. Xie, “An epi-detected coherent anti-Stokes Raman scattering (E-CARS) microscope with high spectral resolution and high sensitivity,” J. Phys. Chem. B **105**, 1277–1280 (2001) [CrossRef] .

5. L. Li, H. Wang, and J. Cheng, “Quantitative coherent anti-Stokes Raman scattering imaging of lipid distribution in coexisting domains”, Biophys. J. **89**, 3480–3490 (2005) [CrossRef] [PubMed] .

8. A. F. Pegoraro, A. D. Slepkov, A. Ridsdale, D. J. Moffatt, and A. Stolow, “Hyperspectral multimodal CARS microscopy in the fingerprint region,” J. Biophotonics (2012) [CrossRef] [PubMed]

9. C. Chung, J. Hsu, S. Mukamel, and E. O. Potma, “Controlling stimulated coherent spectroscopy and microscopy by a position-dependent phase,” Phys. Rev. A **87**, 033833 (2013) [CrossRef] .

11. E. O. Potma, C. L. Evans, and X. S. Xie, “Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging,” Opt. Lett. **31**, 241–243 (2006) [CrossRef] [PubMed] .

*φ*(

*ω*) of the resonant response relative to that of the NRB leads to well known spectral distortions [12

12. W. M. Tolles, J. W. Nibler, J. R. McDonald, and A. B. Harvey, “A Review of the Theory and Applications of Coherent Anti-Stokes Raman Spectroscopy (CARS),” Appl. Spectrosc. **31**253–271 (1971) [CrossRef] .

13. Y. Liu, Y. J. Lee, and M. T. Cicerone, “Broadband CARS spectral phase retrieval using a time-domain Kramers-Kronig transform,” Opt. Lett. **34**, 1363–1365 (2009) [CrossRef] [PubMed] .

14. E. M. Vartiainen, H. A. Rinia, M. Müller, and M. Bonn, “Direct extraction of Raman line-shapes from congested CARS spectra,” Opt. Express **14**, 3622–3630 (2006) [CrossRef] [PubMed] .

*χ*

^{(3)}(

*ω*). It is commonly assumed that the CARS signal intensity is proportional to the squared modulus of the third order non-linear susceptibility

*χ*

^{(3)}(

*ω*) as follows [15] where

*χ*

^{(3)}(

*ω*). Having measured |

*χ*

^{(3)}(

*ω*)| using Eq. (1), one can deduce the spectral phase

*φ*(

*ω*)—defined by

*χ*

^{(3)}(

*ω*)= |

*χ*

^{(3)}(

*ω*)|exp(

*iφ*(

*ω*))—using the general properties of the susceptibility, and use this to retrieve the purely imaginary (i.e. resonant) response as discussed, for example, in Ref. [13

13. Y. Liu, Y. J. Lee, and M. T. Cicerone, “Broadband CARS spectral phase retrieval using a time-domain Kramers-Kronig transform,” Opt. Lett. **34**, 1363–1365 (2009) [CrossRef] [PubMed] .

16. E. O. Potma, W. P. de Boeij, and D. A. Wiersma, “Nonlinear coherent four-wave mixing in optical microscopy,” J. Opt. Soc. Am. B **17**, 1678–1684, (2000) [CrossRef] .

17. J. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B **19**, 1363–1375 (2002) [CrossRef] .

18. N. Djaker, D. Gachet, N. Sandeau, PF. Lenne, and H. Rigneault, “Refractive effects in coherent anti-Stokes Raman scattering microscopy,” Appl. Opt. **45**, 7005–7011 (2006) [CrossRef] [PubMed] .

19. K. I. Popov, A. F. Pegoraro, A. Stolow, and L. Ramunno, “Image formation in CARS microscopy: effect of the Gouy phase shift,” Opt. Express **19**, 5902–5911 (2011) [CrossRef] [PubMed] .

20. D. Gachet, F. Billard, N. Sandeau, and H. Rigneault, “Coherent anti-Stokes Raman scattering (CARS) microscopy imaging at interfaces: evidence of interference effects,” Opt. Express **15**10408–10420 (2007) [CrossRef] [PubMed] .

21. D. Gachet, F. Billard, and H. Rigneault, “Focused field symmetries for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A **77**061802(R) 1–4 (2008) [CrossRef] .

## 2. Experimental observations

*μ*m to ∼10

*μ*m, as highlighted in Fig. 1. NB has several Raman resonances in the fingerprint region, including a weak C=C stretch mode at 1585 cm

^{−1}[23

23. C. V. Stephenson, W. C. Coburn Jr., and W. S. Wilcox, “The vibrational spectra and assignments of nitrobenzene, phenyl isocyanate, phenyl isothiocyanate, thionylaniline and anisole”, Spectrochim. Acta **17**, 933–946 (1961) [CrossRef] .

8. A. F. Pegoraro, A. D. Slepkov, A. Ridsdale, D. J. Moffatt, and A. Stolow, “Hyperspectral multimodal CARS microscopy in the fingerprint region,” J. Biophotonics (2012) [CrossRef] [PubMed]

^{−1}to ∼1800 cm

^{−1}—using laser powers of 180 mW and 20 mW for pump and Stokes, respectively, measured at the input to the microscope scan head. The Raman spectral resolution was ∼30 cm

^{−1}, determined by the selected linear chirp [24]. The spectrum was calibrated by direct measurement of the pump anti-Stokes wavelengths using a spectrometer and correlating its frequency to the position on the translation stage.

*μ*m, shown in Fig. 1(a), and recorded CARS spectra as a function of

*z*displacement, using 0.5

*μ*m steps in

*z*ranging from −2.5

*μ*m to +2.5

*μ*m from the centre of the highlighted droplet. Negative

*z*corresponds to the NB droplet’s position being closer to the laser source. In Fig. 1(a), we show the highlighted NB droplet imaged at the best focus position, defined to be

*z*= 0

*μ*m. For comparison, in Fig. 1(b), we show the identical NB droplet but with the focus displaced to

*z*= −1

*μ*m.

*z*= 0

*μ*m is shown as open circles in Fig. 2(a). The CARS spectrum displays the expected red-shift of its maximum due to the interference between the resonant and NRB contributions [3

3. C. L. Evans and X. S. Xie, “Coherent anti-Stokes Raman scattering microscopy: chemical imaging for biology and medicine,” Annu. Rev. Anal. Chem. **1**, 883–909 (2008) [CrossRef] .

*z*= −1

*μ*m and

*z*= 1

*μ*m focus are also shown in Fig. 2(a). Despite the CARS images in Figs. 1(a) and 1(b) appearing nearly identical, the raw CARS spectra are considerably different. We observed distortions of the spectral amplitude and position of the peak and dip, even though magnitudes of the resonant and non-resonant signals should be approximately the same in each case (since the NB droplet is entirely within the Rayleigh range for all three cases). In Fig. 2(b), we show the Raman spectra at each focus position, retrieved using an established Kramers-Kronig algorithm [13

13. Y. Liu, Y. J. Lee, and M. T. Cicerone, “Broadband CARS spectral phase retrieval using a time-domain Kramers-Kronig transform,” Opt. Lett. **34**, 1363–1365 (2009) [CrossRef] [PubMed] .

*z*= 0 is consistent with the high resolution spontaneous Raman spectrum (solid line), albeit at our lower spectral resolution of ∼30 cm

^{−1}. However, it can be seen that displacing the focus from

*z*= −1

*μ*m to

*z*= +1

*μ*m causes a spectral shift in the maximum of order ∼10 cm

^{−1}.

## 3. Numerical simulations

19. K. I. Popov, A. F. Pegoraro, A. Stolow, and L. Ramunno, “Image formation in CARS microscopy: effect of the Gouy phase shift,” Opt. Express **19**, 5902–5911 (2011) [CrossRef] [PubMed] .

*μ*m wavelength in free space and the Stokes was down-shifted from the pump by a frequency

*ω*. Both laser beam components were assumed to be 2 ps in duration. Inside the simulation domain, the electromagnetic wave was propagated numerically by the standard Maxwell solver [25

25. K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Ant. Prop. **14**, 302–307 (1966) [CrossRef] .

*n*

_{0}= 1.34 and possessed a nonzero Kerr nonlinear susceptibility which generated the NRB signal. The Raman medium response was described by the third-order Raman polarization [26

26. M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media,” IEEE J. Quant. Electron. **40**, 175–182 (2004) [CrossRef] .

*μ*m-diameter spherical droplet located somewhere on the laser axis. The droplet could also have a linear refractive index generally different from that of the NRB (see below). The magnitudes of the nonlinear susceptibilities were chosen such that the strengths of the resulting NRB and resonant signals were comparable, as was the case for the experimental results presented above. The Raman resonant frequency Ω

*and line width Γ were chosen to be Ω*

_{R}*= 1600 cm*

_{R}^{−1}(0.144

*ω*, where

_{p}*ω*is the central frequency of the pump beam) and Γ = ∼15 cm

_{p}^{−1}(1.34 × 10

^{−3}

*ω*).

_{p}*z*displacement in a manner very similar to the

*z*variations seen in the experimental results in Fig. 2(a). The CARS spectrum of the NB droplet located 1

*μ*m before the focus is different from the spectrum at 1

*μ*m after the laser focus, despite the laser intensity being the same at these points. The retrieval of these spectra are shown in Fig. 3(b), yielding a spectral shift of order 10 cm

^{−1}(9 × 10

^{−4}

*ω*). We see that even an idealized model having a Gaussian laser envelope shape, a flat NRB spectrum and a matched linear refractive index reproduces the experimentally observed spatial-spectral coupling effect. The idealized model does not, however, quantitatively reproduce the intensities of the experimental CARS spectra, particularly at

_{P}*z*= +1

*μ*m. We note that accurately fitting the intensity profile requires detailed knowledge of the point spread function, as well as explicit knowledge of the resonant-to-nonresonant background ratio. Fortunately, the phases of these fields are relatively insensitive to these parameters and, therefore, they have minimal effect on the observed spectral shifts.

*n*= 1.34) and nitrobenzene (

*n*= 1.554) is accounted for in our simulations, further distortions, as shown in Fig 3(c), of the CARS spectrum are found as compared to the index-matched case; its subsequent retrieval is shown in 3(d), where the solid line assumes matching of the linear refractive indices and the dashed line includes the known differing linear refractive indices of NB and 1% agarose. Beyond the dominant effect of the Gouy phase, the presence of a linear refractive index mismatch within the sample leads to a further spectral shift of order 5.5 cm

^{−1}(5 × 10

^{−4}

*ω*) at the point of best focus. A similar spectral shift is expected at positions

_{P}*z*= ±1

*μm*that will be cumulative with the effects due to the Gouy phase; this may be especially problematic in specific cases, as the shift due to the refractive index will depend heavily on the size and shape of the particular object being probed.

## 4. Discussion

*z*= ±1

*μ*m of the focal position in Fig. 2(a), or the correpsonding simulation in Fig. 3(a), is the Gouy phase shift of the laser beam. As was demonstrated in Ref. [19

19. K. I. Popov, A. F. Pegoraro, A. Stolow, and L. Ramunno, “Image formation in CARS microscopy: effect of the Gouy phase shift,” Opt. Express **19**, 5902–5911 (2011) [CrossRef] [PubMed] .

*δϕ*= 2

*ϕ*−

_{Gp}*ϕ*+

_{Gs}*δϕ*, where

_{l}*ϕ*and

_{Gp}*ϕ*are the Gouy phase shifts of the pump and Stokes beams at the object location, respectively, and

_{Gs}*δϕ*is the extra phase shift due to any linear refractive index mismatches in the sample. Due to these additional propagation phase terms, the resonant response in Eq. (1) will no longer be purely imaginary. Rather, the following equation should be used to reveal these phases [21

_{l}21. D. Gachet, F. Billard, and H. Rigneault, “Focused field symmetries for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A **77**061802(R) 1–4 (2008) [CrossRef] .

*δϕ*is written explicitly. More generally, this can be written as where

*V*is the volume occupied by the resonant species. The

_{R}*δϕ*is a geometrical propagation factor and only varies slowly with

*ω*over the range relevant to the CARS spectrum. After the spatial integration is performed, Eq. (3) becomes equivalent to Eq. (2), yielding a certain phase

*δϕ*and magnitude of

*δϕ*are known, the Raman spectrum can be generally retrieved from the CARS spectrum using Eq. (2). Development of a general retrieval routine which can provide this information is a challenging task and will be a subject for future studies. In the present case, we have specific

*a priori*information about our simulated sample (i.e. there is only one isolated Raman line and an almost frequency-independent NRB) and, therefore, we can use a simple fitting procedure to retrieve the resonant Raman spectrum. Our fitting method is based on a simulated annealing search [27

27. D. Vanderbilt and S. G. Louie, “A Monte Carlo simulated annealing approach to optimization over continuous variables,” J. Comput. Phys. **56**, 259–271 (1984) [CrossRef] .

*δϕ*in the fit parameter space of the retrieval routine, the Raman spectrum can be reliably extracted as shown in Fig. 4, as can the phase shift information. Note that Fig. 3(b) is equivalent to the retrieved Raman spectrum using the explicit assumption of

*δϕ*= 0, all other parameters (

*, Γ) being fit by simulated annealing. This procedure (with assumption*

_{R}*δϕ*= 0) corresponds to a retrieval procedure which does not include spatial-spectral coupling.

*δϕ*results in the resonant term being no longer purely imaginary. Therefore, retrieval methods not accounting for this will generate an erroneous spectral phase

*φ*(

*ω*), leading to distortions in the retrieved Raman spectrum resulting from factors such as the Gouy phase shift or the linear refractive index mismatch within the sample. Importantly, this shows that the CARS spectrum is generally dependent on the internal composition of the inhomogeneous sample, even for objects of uniform chemical composition. It should be noted, however, that Eq. (2) can still be used without loss of generality for the retrieval of Raman data from the distorted CARS spectrum. From a phenomenological point of view, one can approximate the effects of

*δϕ*on the CARS spectral shape by substituting Eq. (4) directly into Eq. (2). This is shown in Fig. 5(a), where there is a family of CARS spectral line shapes, each corresponding to the same Raman line and the same relative amplitudes of resonant and nonresonant contributions to the CARS signal, but having a different phase

*δϕ*. Upon retrieval of the Raman spectra using the standard Kramers-Kronig algorithm, shown in Fig. 5(b), we see that the expected simple resonant Raman lineshape is not retrieved, but rather we find a family of dispersive Raman peaks associated with the various phases. Due to the intrinsic nature of the coupling, post-processing images with Raman retrieval will be unable to reliably remove the spatial distortions (e.g. shadows) in the CARS image and may, in fact, introduce new artefacts leading to the possible misinterpretation of hyperspectral CARS images.

## 5. Conclusion

*χ*

^{(3)}, with contributions due to the linear index mismatch. Consequently, care must be taken in assigning physio-chemical significance to CARS spectral variations within samples having regions of interest smaller than the Rayleigh range of the laser focus. Such systems will invariably experience significant spatial-spectral coupling.

21. D. Gachet, F. Billard, and H. Rigneault, “Focused field symmetries for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A **77**061802(R) 1–4 (2008) [CrossRef] .

*χ*

^{(3)}(

*ω*) via Eq. (2) rather than Eq. (1). In addition, we note that such efforts may provide a novel route to significantly enhanced axial resolution, if the position of the emitter within the laser focus can be elucidated from the relative phase. For example, the nanometer-scale axial displacements of a cell membrane component or other small object within the laser focal volume could be monitored in real time by fitting the phase shift variations in the retrieved Raman spectrum, as implicitly suggested by Fig. 5. We are currently investigating such super-resolution strategies for CARS microscopy.

## Acknowledgments

## References and links

1. | A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. |

2. | E. O. Potma and X. S. Xie, “CARS microscopy for biology and medicine,” Opt. Photon. News |

3. | C. L. Evans and X. S. Xie, “Coherent anti-Stokes Raman scattering microscopy: chemical imaging for biology and medicine,” Annu. Rev. Anal. Chem. |

4. | J. Cheng, A. Volkmer, L. D. Book, and X. S. Xie, “An epi-detected coherent anti-Stokes Raman scattering (E-CARS) microscope with high spectral resolution and high sensitivity,” J. Phys. Chem. B |

5. | L. Li, H. Wang, and J. Cheng, “Quantitative coherent anti-Stokes Raman scattering imaging of lipid distribution in coexisting domains”, Biophys. J. |

6. | M. Müller and J. M. Schins, “Imaging the thermodynamic state of lipid membranes with multiplex CARS microscopy,” J. Phys. Chem. |

7. | S. H. Parekh, Y. J. Lee, K. A. Aamer, and M. T. Cicerone, “Label-free cellular imaging by broadband coherent anti-Stokes Raman scattering microscopy,” Biophys. J. |

8. | A. F. Pegoraro, A. D. Slepkov, A. Ridsdale, D. J. Moffatt, and A. Stolow, “Hyperspectral multimodal CARS microscopy in the fingerprint region,” J. Biophotonics (2012) [CrossRef] [PubMed] |

9. | C. Chung, J. Hsu, S. Mukamel, and E. O. Potma, “Controlling stimulated coherent spectroscopy and microscopy by a position-dependent phase,” Phys. Rev. A |

10. | S. Maeda, T. Kamisuki, and Y. Adachi, |

11. | E. O. Potma, C. L. Evans, and X. S. Xie, “Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging,” Opt. Lett. |

12. | W. M. Tolles, J. W. Nibler, J. R. McDonald, and A. B. Harvey, “A Review of the Theory and Applications of Coherent Anti-Stokes Raman Spectroscopy (CARS),” Appl. Spectrosc. |

13. | Y. Liu, Y. J. Lee, and M. T. Cicerone, “Broadband CARS spectral phase retrieval using a time-domain Kramers-Kronig transform,” Opt. Lett. |

14. | E. M. Vartiainen, H. A. Rinia, M. Müller, and M. Bonn, “Direct extraction of Raman line-shapes from congested CARS spectra,” Opt. Express |

15. | G. L. Eesley, |

16. | E. O. Potma, W. P. de Boeij, and D. A. Wiersma, “Nonlinear coherent four-wave mixing in optical microscopy,” J. Opt. Soc. Am. B |

17. | J. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B |

18. | N. Djaker, D. Gachet, N. Sandeau, PF. Lenne, and H. Rigneault, “Refractive effects in coherent anti-Stokes Raman scattering microscopy,” Appl. Opt. |

19. | K. I. Popov, A. F. Pegoraro, A. Stolow, and L. Ramunno, “Image formation in CARS microscopy: effect of the Gouy phase shift,” Opt. Express |

20. | D. Gachet, F. Billard, N. Sandeau, and H. Rigneault, “Coherent anti-Stokes Raman scattering (CARS) microscopy imaging at interfaces: evidence of interference effects,” Opt. Express |

21. | D. Gachet, F. Billard, and H. Rigneault, “Focused field symmetries for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A |

22. | D. Gachet, S. Brustlein, and H. Rigneault, “Revisiting the Youngs double slit experiment for background-free nonlinear Raman spectroscopy and microscopy,” Phys. Rev. Lett. |

23. | C. V. Stephenson, W. C. Coburn Jr., and W. S. Wilcox, “The vibrational spectra and assignments of nitrobenzene, phenyl isocyanate, phenyl isothiocyanate, thionylaniline and anisole”, Spectrochim. Acta |

24. | A. F. Pegoraro, A. Ridsdale, D. J. Moffatt, Y. Jia, J. P. Pezacki, and A. Stolow, “Optimally chirped multimodal CARS microscopy based on a single Ti:sapphire oscillator,” |

25. | K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Ant. Prop. |

26. | M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media,” IEEE J. Quant. Electron. |

27. | D. Vanderbilt and S. G. Louie, “A Monte Carlo simulated annealing approach to optimization over continuous variables,” J. Comput. Phys. |

**OCIS Codes**

(020.3690) Atomic and molecular physics : Line shapes and shifts

(030.1670) Coherence and statistical optics : Coherent optical effects

(300.6230) Spectroscopy : Spectroscopy, coherent anti-Stokes Raman scattering

(180.4315) Microscopy : Nonlinear microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: April 30, 2013

Manuscript Accepted: June 9, 2013

Published: June 19, 2013

**Virtual Issues**

Vol. 8, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Aaron M. Barlow, Konstantin Popov, Marco Andreana, Douglas J. Moffatt, Andrew Ridsdale, Aaron D. Slepkov, James L. Harden, Lora Ramunno, and Albert Stolow, "Spatial-spectral coupling in coherent anti-Stokes Raman scattering microscopy," Opt. Express **21**, 15298-15307 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-13-15298

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### References

- A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett.824142–4145 (1999). [CrossRef]
- E. O. Potma and X. S. Xie, “CARS microscopy for biology and medicine,” Opt. Photon. News15, 40–45 (2004). [CrossRef]
- C. L. Evans and X. S. Xie, “Coherent anti-Stokes Raman scattering microscopy: chemical imaging for biology and medicine,” Annu. Rev. Anal. Chem.1, 883–909 (2008). [CrossRef]
- J. Cheng, A. Volkmer, L. D. Book, and X. S. Xie, “An epi-detected coherent anti-Stokes Raman scattering (E-CARS) microscope with high spectral resolution and high sensitivity,” J. Phys. Chem. B105, 1277–1280 (2001). [CrossRef]
- L. Li, H. Wang, and J. Cheng, “Quantitative coherent anti-Stokes Raman scattering imaging of lipid distribution in coexisting domains”, Biophys. J.89, 3480–3490 (2005). [CrossRef] [PubMed]
- M. Müller and J. M. Schins, “Imaging the thermodynamic state of lipid membranes with multiplex CARS microscopy,” J. Phys. Chem.106, 3715–3723 (2002).
- S. H. Parekh, Y. J. Lee, K. A. Aamer, and M. T. Cicerone, “Label-free cellular imaging by broadband coherent anti-Stokes Raman scattering microscopy,” Biophys. J.99, 2695–2704 (2010). [CrossRef] [PubMed]
- A. F. Pegoraro, A. D. Slepkov, A. Ridsdale, D. J. Moffatt, and A. Stolow, “Hyperspectral multimodal CARS microscopy in the fingerprint region,” J. Biophotonics (2012) [CrossRef] [PubMed]
- C. Chung, J. Hsu, S. Mukamel, and E. O. Potma, “Controlling stimulated coherent spectroscopy and microscopy by a position-dependent phase,” Phys. Rev. A87, 033833 (2013). [CrossRef]
- S. Maeda, T. Kamisuki, and Y. Adachi, Advances in Non-linear Spectroscopy, R. J. H. Clark and R. E. Hester, eds. (John Wiley and Sons Ltd., 1988) p. 253.
- E. O. Potma, C. L. Evans, and X. S. Xie, “Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging,” Opt. Lett.31, 241–243 (2006). [CrossRef] [PubMed]
- W. M. Tolles, J. W. Nibler, J. R. McDonald, and A. B. Harvey, “A Review of the Theory and Applications of Coherent Anti-Stokes Raman Spectroscopy (CARS),” Appl. Spectrosc.31253–271 (1971). [CrossRef]
- Y. Liu, Y. J. Lee, and M. T. Cicerone, “Broadband CARS spectral phase retrieval using a time-domain Kramers-Kronig transform,” Opt. Lett.34, 1363–1365 (2009). [CrossRef] [PubMed]
- E. M. Vartiainen, H. A. Rinia, M. Müller, and M. Bonn, “Direct extraction of Raman line-shapes from congested CARS spectra,” Opt. Express14, 3622–3630 (2006). [CrossRef] [PubMed]
- G. L. Eesley, Coherent Raman Spectroscopy(Pergamon Press, 1981).
- E. O. Potma, W. P. de Boeij, and D. A. Wiersma, “Nonlinear coherent four-wave mixing in optical microscopy,” J. Opt. Soc. Am. B17, 1678–1684, (2000). [CrossRef]
- J. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B19, 1363–1375 (2002). [CrossRef]
- N. Djaker, D. Gachet, N. Sandeau, PF. Lenne, and H. Rigneault, “Refractive effects in coherent anti-Stokes Raman scattering microscopy,” Appl. Opt.45, 7005–7011 (2006). [CrossRef] [PubMed]
- K. I. Popov, A. F. Pegoraro, A. Stolow, and L. Ramunno, “Image formation in CARS microscopy: effect of the Gouy phase shift,” Opt. Express19, 5902–5911 (2011). [CrossRef] [PubMed]
- D. Gachet, F. Billard, N. Sandeau, and H. Rigneault, “Coherent anti-Stokes Raman scattering (CARS) microscopy imaging at interfaces: evidence of interference effects,” Opt. Express1510408–10420 (2007). [CrossRef] [PubMed]
- D. Gachet, F. Billard, and H. Rigneault, “Focused field symmetries for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A77061802(R) 1–4 (2008). [CrossRef]
- D. Gachet, S. Brustlein, and H. Rigneault, “Revisiting the Youngs double slit experiment for background-free nonlinear Raman spectroscopy and microscopy,” Phys. Rev. Lett.104213905 1–4 (2010).
- C. V. Stephenson, W. C. Coburn, and W. S. Wilcox, “The vibrational spectra and assignments of nitrobenzene, phenyl isocyanate, phenyl isothiocyanate, thionylaniline and anisole”, Spectrochim. Acta17, 933–946 (1961). [CrossRef]
- A. F. Pegoraro, A. Ridsdale, D. J. Moffatt, Y. Jia, J. P. Pezacki, and A. Stolow, “Optimally chirped multimodal CARS microscopy based on a single Ti:sapphire oscillator,” œ172984–2996 (2009).
- K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Ant. Prop.14, 302–307 (1966). [CrossRef]
- M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media,” IEEE J. Quant. Electron.40, 175–182 (2004). [CrossRef]
- D. Vanderbilt and S. G. Louie, “A Monte Carlo simulated annealing approach to optimization over continuous variables,” J. Comput. Phys.56, 259–271 (1984). [CrossRef]

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