## Raman Microscopy based on Doubly-Resonant Four-Wave Mixing (DR-FWM)

Optics Express, Vol. 17, Issue 19, pp. 17044-17051 (2009)

http://dx.doi.org/10.1364/OE.17.017044

Acrobat PDF (286 KB)

### Abstract

Doubly-resonant four-wave mixing (DR-FWM) is a nondegenerate four-wave mixing process in which four photons interact to coherently probe two distinct Raman resonances. We demonstrate DR-FWM microscopy as a label-free and nondestructive molecular imaging modality with high chemical specificity on the submicron scale by imaging alkyne-substituted oleic acid in both aqueous and lipid-rich environments. DR-FWM microscopy is contrasted to coherent anti-Stokes Raman scattering (CARS) microscopy and it is shown that the coherent addition of two simultaneously probed Raman resonances leads to a significant increase in signal without increasing the non-resonant background. Thus, this scheme enables the detection of weak Raman signals through amplification by a strong Raman resonance, potentially increasing the overall detection sensitivity beyond what has been demonstrated by either CARS or stimulated Raman scattering (SRS).

© 2009 OSA

## 1. Introduction

1. J. X. Cheng and X. S. Xie, “Coherent anti-Stokes Raman scattering microscopy: Instrumentation, theory, and applications,” J. Phys. Chem. B **108**(3), 827–840 (2004). [CrossRef]

5. S. Wachsmann-Hogiu, T. Weeks, and T. Huser, “Chemical analysis in vivo and in vitro by Raman spectroscopy--from single cells to humans,” Curr. Opin. Biotechnol. **20**(1), 63–73 (2009). [CrossRef] [PubMed]

*ω*, and a Stokes photon,

_{P}*ω*, interact simultaneously with a molecular vibrational mode to produce an anti-Stokes photon,

_{S}*ω*. The energy difference between the Stokes and pump photons,

_{CARS}*ℏω*, matches the vibrational energy of the molecular mode and the resulting anti-Stokes photon is produced at frequency

_{P}−ℏω_{S}*ω*(see Fig. 1(a) , Case 1). While CARS microscopy, based on diverse Raman bands has been reported, the broad use of CARS across the entire chemical fingerprint region has been limited to just a few fairly strong resonances [6

_{CARS}=(ω_{P}−ω_{S})+ω_{P}6. C. L. Evans and X. S. Xie, “Coherent Anti-Stokes Raman Scattering Microscopy: Chemical Imaging for Biology and Medicine,” Ann. Rev. Anal. Chem. **1**(1), 883–909 (2008). [CrossRef]

1. J. X. Cheng and X. S. Xie, “Coherent anti-Stokes Raman scattering microscopy: Instrumentation, theory, and applications,” J. Phys. Chem. B **108**(3), 827–840 (2004). [CrossRef]

7. F. Ganikhanov, C. L. Evans, B. G. Saar, and X. S. Xie, “High-sensitivity vibrational imaging with frequency modulation coherent anti-Stokes Raman scattering (FM CARS) microscopy,” Opt. Lett. **31**(12), 1872–1874 (2006). [CrossRef] [PubMed]

9. E. O. Potma, C. L. Evans, and X. S. Xie, “Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging,” Opt. Lett. **31**(2), 241–243 (2006). [CrossRef] [PubMed]

*ω*. If, however, a third source is introduced with another, distinct frequency,

_{P}*ω*, then several possibilities for four-wave mixing arise. In particular, the combination

_{P’}*ω*(Fig. 1(a), Case 3) can now probe another Raman resonance that is distinct from the original resonance

_{P’}−ω_{S}*ω*(Fig. 1(a), Case 2). If, in each of these processes the second pump photon is distinguishable from the first pump photon, then the signals generated in both these processes interfere constructively and lead to a stronger overall resonant signal,

_{P}−ω_{S}*ω*. As shown in detail below, this process can be utilized to amplify the signal from a weak Raman resonance in the presence of a much stronger Raman resonance.

_{DR-FWM}## 2. Theory

### 2.1 Doubly-resonant four-wave mixing

*ω*,

_{p}*ω*, and

_{p’}*ω*are non-degenerate. In this case, where

_{s}*ω*is distinguishable from

_{p}*ω*, photons with these frequencies can mix with

_{p’}*ω*in two ways to produce the same resulting photon,

_{s}*ω*(see Fig. 1(a), Case 2 & 3). Lotem et al. first observed that these two processes add coherently and showed that two resonant terms,

_{DRFWM}=(ω_{p}−ω_{s})+ω_{p’}=(ω_{p’}−ω_{s})+ω_{p}*χ*and

^{(3)}_{R}*χ*, had to be considered in order to distinguish the Raman resonance probed by

^{(3)}_{R’}*(ω*as opposed to

_{p}−ω_{s})*(ω*[10

_{p’}−ω_{s})10. H. Lotem, R. T. Lynch, and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A **14**(5), 1748–1755 (1976). [CrossRef]

*χ*, then becomes:

^{(3)}*χ*represents the 1- and 2-photon non-resonant electronic contributions,

^{(3)}_{NR}*χ*represents the resonant components of

^{(3)}_{j}*χ*,

^{(3)}*χ*denotes their complex conjugates (

^{(3)*}_{j}*j=R,R’*), and

*I*is the intensity of the DR-FWM signal. Note in Eq. (1) that as

_{DR-FWM}*ω*and

_{p}−ω_{s}*ω*both approach Raman resonances the non-resonant cross-term (in the second line) approaches zero as

_{p’}−ω_{s}*χ*and

^{(3)}_{R}*χ*both become purely imaginary.

^{(3)}_{R’}11. S. A. J. Druet, B. Attal, T. K. Gustafson, and J.-P. Taran, “Electronic resonance enhancement of coherent anti-Stokes Raman scattering,” Phys. Rev. A **18**(4), 1529–1557 (1978). [CrossRef]

12. Y. J. Lee, Y. Liu, and M. T. Cicerone, “Characterization of three-color CARS in a two-pulse broadband CARS spectrum,” Opt. Lett. **32**(22), 3370–3372 (2007). [CrossRef] [PubMed]

10. H. Lotem, R. T. Lynch, and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A **14**(5), 1748–1755 (1976). [CrossRef]

### 2.2 Coherent anti-Stokes Raman scattering

_{p}photons are indistinguishable. In fact, in 2002 Volkmer et al. demonstrated time-resolved CARS microscopy by taking advantage of non-degenerate frequency mixing mediated by a single Raman resonance [13

13. A. Volkmer, L. D. Book, and X. S. Xie, “Time-resolved coherent anti-Stokes Raman scattering microscopy: Imaging based on Raman free induction decay,” Appl. Phys. Lett. **80**(9), 1505–1507 (2002). [CrossRef]

_{p}−ω

_{s}=ω

_{p’}−ω

_{s}both possible combinations of the three incident fields probe the same Raman resonance and the third-order nonlinear susceptibility contains just a single Raman-resonant term.

_{p}−ω

_{s}approaches a Raman resonance the CARS intensity yields:

_{p}, ω

_{p’}, and ω

_{s}are all far from electronic resonances, a comparison of Eqs. (1) and 3 shows that there should be no difference in non-resonant background levels for DR-FWM and CARS regardless of the number of mixing processes that are coherently adding up to produce the resulting signal. Therefore DR-FWM microscopy should improve the signal to non-resonant background ratio when compared with CARS microscopy.

### 2.3 Double resonance enhancement

_{DR-FWM}, can be compared with the signal intensity of the singly resonant process, I

_{CARS}. If we ignore the non-resonant background terms in Eq. (1) (i.e. taking just the last line of the equation), χ

^{(3)}

_{R}can be factored out to give the expression:where |χ

^{(3)}

_{DR-FWM}|

^{2}is proportional to I

_{DR-FWM}and |χ

^{(3)}

_{R}|

^{2}is proportional to I

_{CARS}. From Eq. (4) it can be seen that a strong |χ

^{(3)}

_{R’}|

^{2}of one Raman resonance, based on e.g. high bond density or large Raman cross-section, could be used to enhance the much weaker χ

^{(3)}

_{R}of another Raman resonance. For example if |χ

^{(3)}

_{R}|/|χ

^{(3)}

_{R}|≈10 then |χ

^{(3)}

_{R}|

^{2}will be enhanced by approximately 2 orders of magnitude. Even a situation in which |χ

^{(3)}

_{R’}|

^{2}≈|χ

^{(3)}

_{R}|

^{2}will still lead to an approximately 4x increase in signal strength over the signal intensities of either singly resonant process.

## 3. Methodology and Results

### 3.1 Spectral dependence

*ω*) and (

_{p}−ω_{s}*ω*), determine the value of

_{p’}−ω_{s}*χ*[10

^{(3)}10. H. Lotem, R. T. Lynch, and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A **14**(5), 1748–1755 (1976). [CrossRef]

^{−1}C-H stretch and 2115 cm

^{−1}alkyne stretch vibrations (see the spectrum in Fig. 1(b)) was dried onto a glass coverslip and then submerged in water. Imaging was achieved with a modified CARS microscope, described in detail elsewhere, where a 1064 nm Stokes pulse and ~817 nm and ~869 nm pump pulses are used to match the appropriate Raman resonances [14

14. I. W. Schie, T. Weeks, G. P. McNerney, S. Fore, J. K. Sampson, S. Wachsmann-Hogiu, J. C. Rutledge, and T. Huser, “Simultaneous forward and epi-CARS microscopy with a single detector by time-correlated single photon counting,” Opt. Express **16**(3), 2168–2175 (2008). [CrossRef] [PubMed]

*ω*) and (

_{p}−ω_{s}*ω*) were tuned between 2845 and 2871 cm

_{p’}−ω_{s}^{−1}and 2101-2116 cm

^{−1}, respectively (see Fig. 2(a-i) ). It is important to note that resonant FWM peaks are often calculated for the ideal case in which the resonances involved are narrow and well isolated. This, however, is not always the case. Therefore, to illustrate this process and its favorable properties for molecularly specific imaging, we probed a single spectral quadrant within the 2D parameter space (see highlighted region in Fig. 3(a) ). Figure 1b shows that, while the 2115 cm

^{−1}peak is narrow and isolated, the 2845 cm

^{−1}line is clearly much more complex and includes multiple tightly packed spectral lines. CARS images were also acquired from 2846 to 2872 cm

^{−1}(Figs. 2(j-l)) and compared with the DR-FWM images. The images are normalized to the laser input power so that each image represents the magnitude of |

*χ*|

^{(3)}*at a particular spectral location.*

^{2}*ω*) and (

_{p}−ω_{s}*ω*) are both tuned away from their respective Raman resonances, the DR-FWM signal and the CARS signal are roughly equivalent in strength and dominated by non-resonant background contributions as can be seen in Figs. 2(a) and 2(j). As (

_{p’}−ω_{s}*ω*) and (

_{p}−ω_{s}*ω*) are simultaneously tuned to their respective Raman resonances a significant increase in the DR-FWM signal compared to the CARS signal can be observed, see Fig. 2(i and l). This increase in signal is consistent with the inclusion of an additional resonant

_{p’}−ω_{s}*χ*term. Substituting the average intensity over a small region of the image, <

^{(3)}*I*>, for both Figs. 2(i) and 2(l) into Eq. (4), we obtain

_{DR-FWM}*|χ*|/

^{(3)}_{R}*|χ*|≈

^{(3)}_{R}*0.5*. This value, along with the ratio of the Raman peaks, which we found to be approximately 6.5, suggest that the ratio of the line widths

*Γ*≈3.2 and the ratio of the differential Raman cross-sections (

_{R}/Γ_{R’}*∂α/∂Ω*)/(

*∂α/∂Ω*)’≈4.6. These relationships are valuable for comparison with theory.

### 3.2 Comparison with theory

*Γ*and

*∂α/∂Ω*, see Fig. 3a. In order to fit the theoretical surface to the data it was necessary to model the 2845 cm

^{−1}line as the superposition of two resonances at 2845 cm

^{−1}and at 2860 cm

^{−1}with a combined

*∂α/∂Ω*that maintained the necessary relationship, which is consistent with the convoluted nature of the aliphatic Raman peak. Using the average intensity over a small region of the image, <

*I*>, from Fig. 2(g), (h), and (i), Fig. 2(c), (f), and (i), and Fig. 2(a), (e), and (i) as data points and plotting them with corresponding cuts through the parameter space, Figs. 3(b-d) demonstrate excellent agreement between the theoretical DR-FWM line shapes and the collected data. For comparison, <

_{DR-FWM}*I*> from Fig. 2(j), (k), and (l) was fit with a CARS line probing the same 2845 cm

_{CARS}^{−1}resonance as in the DR-FWM case. This resulted in Fig. 3(e), which also shows excellent agreement with the measured values.

### 3.3 Signal enhancement

**14**(5), 1748–1755 (1976). [CrossRef]

15. R. Lynch, S. Kramer, H. Lotem, and N. Bloembergen, “Double Resonance Interference in Third-Order Light Mixing,” Opt. Commun. **16**(3), 372–375 (1976). [CrossRef]

17. H. Fei, Y. Zhang, L. Han, F. Zhao, and Z. Wei, “Raman-enhanced nondegenerate four-wave mixing,” Appl. Phys. B **52**(6), 395–399 (1991). [CrossRef]

1. J. X. Cheng and X. S. Xie, “Coherent anti-Stokes Raman scattering microscopy: Instrumentation, theory, and applications,” J. Phys. Chem. B **108**(3), 827–840 (2004). [CrossRef]

18. W. M. Tolles, J. W. Nibler, J. R. Mcdonald, and A. B. Harvey, “Review of Theory and Application of Coherent Anti-Stokes Raman-Spectroscopy (CARS),” Appl. Spectrosc. **31**(4), 253–271 (1977). [CrossRef]

19. G. Bjorklund, “Effects of focusing on third-order nonlinear processes in isotropic media,” IEEE J. Quantum Electron. **11**(6), 287–296 (1975). [CrossRef]

^{−1}vibration. Only the modified oleic acid, however, exhibits the 2115 cm

^{−1}Raman resonance (Fig. 1(b)). Figure 4(a) shows a DR-FWM image of these modified oleic acid crystal-like structures in regular oleic acid. The structures appear as bright features on the submicron to micron length scale and are visible throughout the field of view above the combined resonant and non-resonant background from the aliphatic CH vibration of all oleic acid species. Figure 4(b) shows the same region imaged at the 2845 cm

^{−1}CARS signal. Note the overall weaker signal obtained when imaging just a single resonance. Figure 4(c) again shows the same region imaged in CARS mode at 2115 cm

^{−1}. The signals were normalized consistently across all three images resulting in the apparent lack of background in Fig. 4(c). Lastly, by subtracting Fig. 4(b) from Fig. 4(a) the enhancement of the weak 2115 cm

^{−1}signal by DR-FWM can be visualized (see Fig. 4(d)). According to Eq. (1) and Eq. (3) this difference image should be free from non-resonant background and represent the 2115 cm

^{−1}resonance including an enhancement factor, as shown in Eq. (4).

## 4. Discussion and Conclusions

^{−1}mode by itself. It is also interesting to compare these results with Fig. 3(d) and Fig. 3(e). Although spontaneous Raman spectroscopy (Fig. 1(b)) shows that the intensity of the 2845 cm

^{−1}Raman peak is more than a factor of 5 times greater than the intensity of the 2115 cm

^{−1}peak (based on the area under the peak), the inclusion of the 2115 cm

^{−1}alkyne resonance still more than doubles the intensity of the signal without increasing the non-resonant background. In other words it can be seen from our results that not only does a strong Raman resonance enhance a weaker one but a weaker resonance can also be used to significantly enhance a much stronger one.

## Acknowledgements

## References and links

1. | J. X. Cheng and X. S. Xie, “Coherent anti-Stokes Raman scattering microscopy: Instrumentation, theory, and applications,” J. Phys. Chem. B |

2. | M. Müller and A. Zumbusch, “Coherent anti-stokes Raman scattering microscopy,” ChemPhysChem |

3. | J. Chan, S. Fore, S. Wachsman-Hogiu, and T. Huser, “Raman spectroscopy and microscopy of individual cells and cellular components,” Laser Photon. Rev. |

4. | C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. W. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science |

5. | S. Wachsmann-Hogiu, T. Weeks, and T. Huser, “Chemical analysis in vivo and in vitro by Raman spectroscopy--from single cells to humans,” Curr. Opin. Biotechnol. |

6. | C. L. Evans and X. S. Xie, “Coherent Anti-Stokes Raman Scattering Microscopy: Chemical Imaging for Biology and Medicine,” Ann. Rev. Anal. Chem. |

7. | F. Ganikhanov, C. L. Evans, B. G. Saar, and X. S. Xie, “High-sensitivity vibrational imaging with frequency modulation coherent anti-Stokes Raman scattering (FM CARS) microscopy,” Opt. Lett. |

8. | M. Jurna, J. P. Korterik, C. Otto, and H. L. Offerhaus, “Shot noise limited heterodyne detection of CARS signals,” Opt. Express |

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

10. | H. Lotem, R. T. Lynch, and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A |

11. | S. A. J. Druet, B. Attal, T. K. Gustafson, and J.-P. Taran, “Electronic resonance enhancement of coherent anti-Stokes Raman scattering,” Phys. Rev. A |

12. | Y. J. Lee, Y. Liu, and M. T. Cicerone, “Characterization of three-color CARS in a two-pulse broadband CARS spectrum,” Opt. Lett. |

13. | A. Volkmer, L. D. Book, and X. S. Xie, “Time-resolved coherent anti-Stokes Raman scattering microscopy: Imaging based on Raman free induction decay,” Appl. Phys. Lett. |

14. | I. W. Schie, T. Weeks, G. P. McNerney, S. Fore, J. K. Sampson, S. Wachsmann-Hogiu, J. C. Rutledge, and T. Huser, “Simultaneous forward and epi-CARS microscopy with a single detector by time-correlated single photon counting,” Opt. Express |

15. | R. Lynch, S. Kramer, H. Lotem, and N. Bloembergen, “Double Resonance Interference in Third-Order Light Mixing,” Opt. Commun. |

16. | S. Saha and R. Hellwarth, “Raman-Induced Phase Conjugation Spectroscopy,” Phys. Rev. A |

17. | H. Fei, Y. Zhang, L. Han, F. Zhao, and Z. Wei, “Raman-enhanced nondegenerate four-wave mixing,” Appl. Phys. B |

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

19. | G. Bjorklund, “Effects of focusing on third-order nonlinear processes in isotropic media,” IEEE J. Quantum Electron. |

**OCIS Codes**

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

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

(180.4315) Microscopy : Nonlinear microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: August 5, 2009

Revised Manuscript: September 1, 2009

Manuscript Accepted: September 2, 2009

Published: September 9, 2009

**Virtual Issues**

Vol. 4, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Tyler Weeks, Sebastian Wachsmann-Hogiu, and Thomas Huser, "Raman Microscopy based on Doubly-Resonant Four-Wave Mixing (DR-FWM)," Opt. Express **17**, 17044-17051 (2009)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-17-19-17044

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### References

- J. X. Cheng and X. S. Xie, “Coherent anti-Stokes Raman scattering microscopy: Instrumentation, theory, and applications,” J. Phys. Chem. B 108(3), 827–840 (2004). [CrossRef]
- M. Müller and A. Zumbusch, “Coherent anti-stokes Raman scattering microscopy,” ChemPhysChem 8(15), 2157–2170 (2007). [CrossRef]
- J. Chan, S. Fore, S. Wachsman-Hogiu, and T. Huser, “Raman spectroscopy and microscopy of individual cells and cellular components,” Laser Photon. Rev. 2(5), 325–349 (2008). [CrossRef]
- C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. W. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322(5909), 1857–1861 (2008). [CrossRef] [PubMed]
- S. Wachsmann-Hogiu, T. Weeks, and T. Huser, “Chemical analysis in vivo and in vitro by Raman spectroscopy--from single cells to humans,” Curr. Opin. Biotechnol. 20(1), 63–73 (2009). [CrossRef] [PubMed]
- C. L. Evans and X. S. Xie, “Coherent Anti-Stokes Raman Scattering Microscopy: Chemical Imaging for Biology and Medicine,” Ann. Rev. Anal. Chem. 1(1), 883–909 (2008). [CrossRef]
- F. Ganikhanov, C. L. Evans, B. G. Saar, and X. S. Xie, “High-sensitivity vibrational imaging with frequency modulation coherent anti-Stokes Raman scattering (FM CARS) microscopy,” Opt. Lett. 31(12), 1872–1874 (2006). [CrossRef] [PubMed]
- M. Jurna, J. P. Korterik, C. Otto, and H. L. Offerhaus, “Shot noise limited heterodyne detection of CARS signals,” Opt. Express 15(23), 15207–15213 (2007). [CrossRef] [PubMed]
- E. O. Potma, C. L. Evans, and X. S. Xie, “Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging,” Opt. Lett. 31(2), 241–243 (2006). [CrossRef] [PubMed]
- H. Lotem, R. T. Lynch, and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A 14(5), 1748–1755 (1976). [CrossRef]
- S. A. J. Druet, B. Attal, T. K. Gustafson, and J.-P. Taran, “Electronic resonance enhancement of coherent anti-Stokes Raman scattering,” Phys. Rev. A 18(4), 1529–1557 (1978). [CrossRef]
- Y. J. Lee, Y. Liu, and M. T. Cicerone, “Characterization of three-color CARS in a two-pulse broadband CARS spectrum,” Opt. Lett. 32(22), 3370–3372 (2007). [CrossRef] [PubMed]
- A. Volkmer, L. D. Book, and X. S. Xie, “Time-resolved coherent anti-Stokes Raman scattering microscopy: Imaging based on Raman free induction decay,” Appl. Phys. Lett. 80(9), 1505–1507 (2002). [CrossRef]
- I. W. Schie, T. Weeks, G. P. McNerney, S. Fore, J. K. Sampson, S. Wachsmann-Hogiu, J. C. Rutledge, and T. Huser, “Simultaneous forward and epi-CARS microscopy with a single detector by time-correlated single photon counting,” Opt. Express 16(3), 2168–2175 (2008). [CrossRef] [PubMed]
- R. Lynch, S. Kramer, H. Lotem, and N. Bloembergen, “Double Resonance Interference in Third-Order Light Mixing,” Opt. Commun. 16(3), 372–375 (1976). [CrossRef]
- S. Saha and R. Hellwarth, “Raman-Induced Phase Conjugation Spectroscopy,” Phys. Rev. A 27(2), 919–922 (1983). [CrossRef]
- H. Fei, Y. Zhang, L. Han, F. Zhao, and Z. Wei, “Raman-enhanced nondegenerate four-wave mixing,” Appl. Phys. B 52(6), 395–399 (1991). [CrossRef]
- W. M. Tolles, J. W. Nibler, J. R. Mcdonald, and A. B. Harvey, “Review of Theory and Application of Coherent Anti-Stokes Raman-Spectroscopy (CARS),” Appl. Spectrosc. 31(4), 253–271 (1977). [CrossRef]
- G. Bjorklund, “Effects of focusing on third-order nonlinear processes in isotropic media,” IEEE J. Quantum Electron. 11(6), 287–296 (1975). [CrossRef]

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