## Image formation in CARS microscopy: effect of the Gouy phase shift |

Optics Express, Vol. 19, Issue 7, pp. 5902-5911 (2011)

http://dx.doi.org/10.1364/OE.19.005902

Acrobat PDF (904 KB)

### Abstract

Image formation in Coherent Anti-Stokes Raman Scattering (CARS) microscopy of sub-wavelength objects is investigated via a combined experimental, numerical and theoretical study. We consider a resonant spherical object in the presence of a nonresonant background, using tightly focused laser pulses. When the object is translated along the laser propagation axis, we find the CARS signal to be asymmetric about the laser focal plane. When the object is located before the focus, there is a distinct shadow within the image, whereas the brightest signal is obtained when the object is behind the focus. This behaviour is caused by interference between resonant and nonresonant signals, and the Gouy phase shift is responsible for the observed asymmetry within the image.

© 2011 Optical Society of America

## 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]

5. M. D. Duncan, J. Reintjes, and T. J. Manuccia, “Scanning coherent anti-Stokes Raman microscope,” Opt. Lett. **7**, 350–352 (1982). [CrossRef] [PubMed]

6. C. L. Evans, E. O. Potma, M. Puoris’haag, D. Côté, C. P. Lin, and X. S. Xie, “Chemical imaging of tissue in vivo with video-rate coherent anti-Stokes Raman scattering microscopy,” Proc. Natl. Acad. Sci. USA **102**, 16807–16812 (2005). [CrossRef] [PubMed]

*in-vivo*real-time studies of biological processes [7

7. 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,” Opt. Express **17**, 2984–2996 (2009). [CrossRef] [PubMed]

*i.e.*, the signal decreases below the level of the nonresonant background. Although a very simple system, this shadow formation and the apparent displacement and asymmetry of the bead indicates the need for a clear understanding of the image formation process.

## 2. Experimental observations

10. A. D. Slepkov, A. Ridsdale, A. F. Pegoraro, D. J. Moffatt, and A. Stolow, “Multimodal CARS microscopy of structured carbohydrate biopolymers,” Biomed. Opt. Express **1**, 1347–1357 (2010). [CrossRef]

*μ*m polystyrene fluorescent beads (Polysciences Inc.) suspended in an agarose gel. The two-photon excited fluorescence (TPEF) due to the pump was collected in the epi-direction. CARS was tuned to the resonance of the polystyrene beads (∼2850 cm

^{−1}) and was collected in the forward direction. The axial response of several beads were averaged. The location of each bead with respect to the best laser focus was determined by the TPEF signal, which is maximal for a bead at the best focus. The displacement of the bead relative to the focus was generated by moving the objective lens. The axial response as a function of the displacement from the focal point of the pump is shown in Fig. 1a, demonstrating that the peak of the CARS response is displaced from the peak of the TPEF. Furthermore, there is a clear decrease in the CARS signal for beads located on the other side of the focus.

*μ*m into the sample. The arrows indicate some dark spots (shadows) that are clearly visible in the image. In Fig 2b the focus position is moved to a depth of 8.8

*μ*m,

*i.e.*, closer to the laser source. The shadows of Fig 2a, have now turned into bright spots in Fig 2b.

## 3. Numerical simulations

*E⃗*= −(1/

*c*)

*∂B⃗*/

*∂t*, ∇ ×

*H⃗*= (1/

*c*)

*∂D⃗/∂t*, using the Finite-Difference-Time-Domain (FDTD) method [11] with the constitutive relations where

*χ*

^{(1)}and

*P⃗*is evaluated according to [12

_{R}12. 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. Quantum Electron. **40**, 175–182 (2004). [CrossRef]

*ω*the resonant frequency of molecular vibrations, and

_{R}*γ*the damping factor; ℱ

_{R}^{−1}denotes the inverse Fourier transformation. We use the auxiliary differential equation technique, described in detail in Ref. [12

12. 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. Quantum Electron. **40**, 175–182 (2004). [CrossRef]

*R*, tightly focused by a high-NA paraboloidal mirror with radius 2.5

_{G}*R*, are calculated using a technique described in a previous work [13

_{G}13. K. I. Popov, C. McElcheran, K. Briggs, S. Mack, and L. Ramunno, “Morphology of femtosecond laser modification of bulk dielectrics,” Opt. Express **19**, 271–282 (2011). [CrossRef] [PubMed]

15. G. Mur, “Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations,” IEEE Trans. Electromagn. Compat. **EMC-23**, 377–382 (1981). [CrossRef]

*n*

_{0}= 1.33 and a Kerr nonlinearity with

*R*and linear refractive index

*n*= 1.5 was situated at various locations along the laser propagation axis

*x̂*. Both the pump and Stokes pulses have a Full-Width-at-Half-Maximum (FWHM) of 300 fs, and are focused by a paraboloidal mirror with NA = 1.1. The value of

7. 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,” Opt. Express **17**, 2984–2996 (2009). [CrossRef] [PubMed]

*μ*m

^{3}box with 40

^{3}grid points per

*μ*m

^{3}. The scattered light was evaluated in the far field and integrated over a solid angle corresponding to a detector lens with NA = 0.6.

*R*= 0.3, 0.4, 0.5

*μ*m. We find results similar to the experimental observations plotted in Fig. 1a: there is a distinct shadow in the image when the sphere is located before the laser focus. The calculated signal is also asymmetric, with the maximum occurring when the scatterer lies after the best focus.

*R*= 0.4

*μ*m scatterer, where we 1) set the linear refraction index of the background equal to that of the scatterer by putting

*n*=

*n*

_{0}, and 2) removed the nonlinear nonresonant background by putting

## 4. Discussion

*ω*and

_{p}*ω*, respectively) have coinciding spot sizes

_{s}*w*

_{0}and Rayleigh lengths

*k*is the wave number (approximately equal for the pump and Stokes sources in this qualitative model). The propagation direction is

*x̂*and we take the laser focus to be the origin of our coordinate system. The resonant signal is generated by a Raman-active sphere located on the laser axis at

*x*=

*x*

_{0}. Both the resonant signal amplitude and phase thus depend on

*x*

_{0}. As we show in Appendix A, the forward-generated nonresonant and resonant signals in the continuous wave limit, as observed at the laser axis in the far-field zone at some point

*P*, can be written as where

*E*

_{0NR}is the amplitude of nonresonant signal,

*E*

_{0R}the amplitude of the resonant signal that would be generated by the sphere located at the origin,

*ω*= 2

_{as}*ω*–

_{p}*ω*is the anti-Stokes frequency,

_{S}*ϕ*

_{0}is an arbitrary phase constant that depends on the location of

*P*,

*δϕ*is the linear phase shift originating from unequal refractive indices of the scatterer and background medium, and

_{L}*ϕ*the Gouy phase shift. The latter depends on scatterer position, and is given by

_{G}*I*

_{0}is the intensity of the nonresonant signal and

*ρ*=

*E*

_{0R}/

*E*

_{0NR}. Eq. (6) is plotted in Fig. 4a. We estimated the linear phase shift by which gives values of ≈

*π*/10,

*π*/8,

*π*/7 for

*R*= 0.3, 0.4, 0.5

*μ*m, respectively. We have also set, correspondingly,

*ρ*= 0.4, 1, 2 in Eq. (6), that is consistent with Fig. 1b. We find that the shapes of our estimated signal in Fig. 4a match very well with our rigorously calculated signals in Fig. 1b. We thus conclude that Eq. (6) is a reasonable approximation and can be used for qualitative estimations.

*δϕ*is set to zero (Fig. 4b), we find that the asymmetry in the signal remains, however with an increased maximum signal and a smaller depth of shadow. This is in qualitative agreement with our numerical results in Fig. 1c. On the other hand, when

_{L}*ϕ*is set to zero (Fig. 4c), the patterns become symmetric, for both

_{G}*δϕ*= 0 and

_{L}*δϕ*≠ 0.

_{L}*x*

_{0}enters Eq. (6) via the amplitude of the resonant signal and Gouy phase shift

*ϕ*. The former is exactly symmetric about the laser focus (

_{G}*x*

_{0}= 0) whereas the latter has different signs for

*x*

_{0}> 0 and

*x*

_{0}< 0. Therefore, it is the Gouy phase shift that is responsible for the asymmetry of the signal. While shadows do appear even if the Gouy phase shift is zero (cf. Fig. 4c,

*δϕ*=

_{L}*π*/8), albeit with a lower contrast, the maximum of the intensity shifts from the origin only if the Gouy phase shift is nonzero.

*ρ*→ ∞ and

*I*

_{0}→ 0 in Eq.(6)), the dependence of the signal on the Gouy phase shifts disappears. It is thus important especially when the nonresonant and resonant signals are of comparable amplitude, which can occur for small Raman active scatters even in a weak nonresonant background.

## 5. Conclusion

## A. A Phenomenological Model of Interference Between the Resonant and Nonresonant Signals

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

*w*

_{0}and Rayleigh length

*x*:

_{R}*p*and

*s*correspond to pump and Stokes, respectively,

*E*

_{0}is the field amplitude,

*ϕ*

_{0}an arbitrary phase, and

*x̂*is the direction of the laser propagation;

*x*= 0 corresponds to the position of the laser focus. We also assume that the following condition is fulfilled everywhere in space (cf. [9

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

*k*is the wave vector length,

*as*stands for the anti-Stokes signal,

*n*is the constant linear refraction index in the medium and

*c*speed of light. Eq. (9) is a reasonable approximation for tightly focused lasers.

*A*located at position (

*x,r*). Interaction between the nonresonant medium and the pump and Stokes beams gives rise to the nonlinear polarization excited at the anti-Stokes frequency at point

*A*. Assuming an isotropic medium with weak nonlinearity and thus no coupling between different frequencies, this polarization is

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

*P*= (

*x*, 0) located on the laser axis in the far-field zone (

_{P}*i.e.*,

*x*≫

_{P}*x*)

_{R}*E*

_{0NR}is the overall amplitude of the nonresonant signal.

*R*located on the laser axis at point (

*x*

_{0}, 0). The resonant nonlinear susceptibility at the anti-Stokes frequency is purely imaginary; we denote it as

*R*≪

*w*

_{0},

*x*then the nonzero component of the nonlinear polarization is

_{R}**19**, 1363–1375 (2002). [CrossRef]

*P*

*E*

_{0R}is the amplitude of the resonant signal, generated by the scatterer located at the origin.

*δϕ*is the phase shift resulting from the linear index mismatch.

_{L}## 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. | J.-X. Cheng and X. S. Xie, “Coherent Anti-Stokes Raman Scattering: Instrumentation, Theory, and Applications,” J. Phys. Chem. B |

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

4. | E. O. Potma and X. S. Xie, “Theory of Spontaneous and Coherent Raman scattering,” in |

5. | M. D. Duncan, J. Reintjes, and T. J. Manuccia, “Scanning coherent anti-Stokes Raman microscope,” Opt. Lett. |

6. | C. L. Evans, E. O. Potma, M. Puoris’haag, D. Côté, C. P. Lin, and X. S. Xie, “Chemical imaging of tissue in vivo with video-rate coherent anti-Stokes Raman scattering microscopy,” Proc. Natl. Acad. Sci. USA |

7. | 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,” Opt. Express |

8. | M. Rivard, M. Laliberté, A. Bertrand-Grenier, C. Harnagea, C. P. Pfeffer, M. Valliéres, Y. St-Pierre, A. Pignolet, M. A. El Khakani, and F. Légaré, “The structural origin of second harmonic generation in fascia,” Biomed. Opt. Express |

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

10. | A. D. Slepkov, A. Ridsdale, A. F. Pegoraro, D. J. Moffatt, and A. Stolow, “Multimodal CARS microscopy of structured carbohydrate biopolymers,” Biomed. Opt. Express |

11. | A. Taflove and S. C. Hagness, |

12. | 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. Quantum Electron. |

13. | K. I. Popov, C. McElcheran, K. Briggs, S. Mack, and L. Ramunno, “Morphology of femtosecond laser modification of bulk dielectrics,” Opt. Express |

14. | A. Taflove and S. C. Hagness, |

15. | G. Mur, “Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations,” IEEE Trans. Electromagn. Compat. |

16. | A. Taflove and S. C. Hagness, |

17. | R. W. Boyd, |

**OCIS Codes**

(170.1650) Medical optics and biotechnology : Coherence imaging

(170.5660) Medical optics and biotechnology : Raman spectroscopy

(180.4315) Microscopy : Nonlinear microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: February 3, 2011

Revised Manuscript: February 28, 2011

Manuscript Accepted: March 1, 2011

Published: March 15, 2011

**Virtual Issues**

Vol. 6, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-5902

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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. 82, 4142–4145 (1999). [CrossRef]
- J.-X. Cheng and X. S. Xie, “Coherent Anti-Stokes Raman Scattering: Instrumentation, Theory, and Applications,” J. Phys. Chem. B 108, 827–840 (2004). [CrossRef]
- C. 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]
- E. O. Potma and X. S. Xie, “Theory of Spontaneous and Coherent Raman scattering,” in Handbook of Biological Nonlinear Optical Microscopy, B. R. Masters and P. T. C. So, eds. (Oxford University Press, 2008), pp. 164–185.
- M. D. Duncan, J. Reintjes, and T. J. Manuccia, “Scanning coherent anti-Stokes Raman microscope,” Opt. Lett. 7, 350–352 (1982). [CrossRef] [PubMed]
- C. L. Evans, E. O. Potma, M. Puoris’haag, D. Côté, C. P. Lin, and X. S. Xie, “Chemical imaging of tissue in vivo with video-rate coherent anti-Stokes Raman scattering microscopy,” Proc. Natl. Acad. Sci. USA 102, 16807–16812 (2005). [CrossRef] [PubMed]
- 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,” Opt. Express 17, 2984–2996 (2009). [CrossRef] [PubMed]
- M. Rivard, M. Laliberté, A. Bertrand-Grenier, C. Harnagea, C. P. Pfeffer, M. Valliéres, Y. St-Pierre, A. Pignolet, M. A. El Khakani, and F. Légaré, “The structural origin of second harmonic generation in fascia,” Biomed. Opt. Express 2, 26–36 (2011). [CrossRef] [PubMed]
- J.-X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterisation of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363–1375 (2002). [CrossRef]
- A. D. Slepkov, A. Ridsdale, A. F. Pegoraro, D. J. Moffatt, and A. Stolow, “Multimodal CARS microscopy of structured carbohydrate biopolymers,” Biomed. Opt. Express 1, 1347–1357 (2010). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics, 3rd ed. (Artech House, 2005), pp. 58–79.
- 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. Quantum Electron. 40, 175–182 (2004). [CrossRef]
- K. I. Popov, C. McElcheran, K. Briggs, S. Mack, and L. Ramunno, “Morphology of femtosecond laser modification of bulk dielectrics,” Opt. Express 19, 271–282 (2011). [CrossRef] [PubMed]
- A. Taflove and S. C. Hagness, Computational Electrodynamics, 3rd. ed. (Artech House, 2005), pp. 186–212.
- G. Mur, “Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics, 3rd ed. (Artech House, 2005), pp. 329–343.
- R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, 2003), p. 194.

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