## Differential imaging in coherent anti-Stokes Raman scattering microscopy with Laguerre-Gaussian excitation beams

Optics Express, Vol. 15, Issue 16, pp. 10123-10134 (2007)

http://dx.doi.org/10.1364/OE.15.010123

Acrobat PDF (327 KB)

### Abstract

We propose a new differential imaging technique to visualize the fine structures and the edges of a sample in coherent anti-Stokes Raman Scattering (CARS) microscopy. Both the pump and Stokes excitation fields are modulated simultaneously with a spiral phase mask which transforms them from Gaussian modes into Laguerre-Gaussian modes of LG01 for CARS excitation. With an accurate three dimensional finite-difference time-domain (FDTD) method, the intensity and phase distributions of focused input fields, the scattering pattern of generated CARS signal as well as the formation of differential images are studied detailedly, and by simulating the sensitivity range and reliability of this method, we have verified that it is much suitable for visualizing structures with a scale comparable to the excitation wavelength and has higher reliability in retrieving chemical structural information of the sample compared to common CARS microscopy.

© 2007 Optical Society of America

## 1. Introduction

_{p}, a Stokes beam of frequency ω

_{s}, and a CARS signal at the anti-Stokes frequency of ω

_{as}=2ω

_{p}-ω

_{s}generated in the phase matching direction [1]. The vibrational contrast in CARS microscopy is created when the frequency difference ω

_{p}-ω

_{s}between the pump and the Stokes beams is tuned to be resonant with a Raman-active molecular vibration of samples. Recently, CARS technique has received great interest in imaging live cells [2

2. P.D. Maker and R.W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys, Rev. **137**, A801–A818 (1965). [CrossRef]

10. E. R. Andresen, H. N. Paulsen, and V. Birkedl “Broadband multiplex coherent anti-Stokes Raman scattering microscopy employing photonic-crystal fibers,” J. Opt. Am. B. **22**, 1934–1938 (2005). [CrossRef]

4. J. X. Cheng and X. S. Xie, “Coherent Anti-Stokes Scattering Microscopy: Instrumentation, Theory, and Application,” J. Phys. Chem. B. **108**,827–840 (2004). [CrossRef]

4. J. X. Cheng and X. S. Xie, “Coherent Anti-Stokes Scattering Microscopy: Instrumentation, Theory, and Application,” J. Phys. Chem. B. **108**,827–840 (2004). [CrossRef]

5. J. X. 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]

11. J. X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B. **19**, 1604–1610 (2002). [CrossRef]

12. V. V. Krishnamachari and E. O. Potma, “Focus-engineered coherent anti-Stokes Raman scattering microscopy: a numerical investigation,” J. Opt. Soc. Am. A **24**, 1138–1147 (2007). [CrossRef]

12. V. V. Krishnamachari and E. O. Potma, “Focus-engineered coherent anti-Stokes Raman scattering microscopy: a numerical investigation,” J. Opt. Soc. Am. A **24**, 1138–1147 (2007). [CrossRef]

13. K. S. Yee. “Numerical solution of initial boundary value problem involving Maxwell equations in isotropic media,”.IEEE Trans. Antennas Propagat. **14**, 302–307 (1966). [CrossRef]

## 2. Geometry and parameters of the simulated setup

*P*

^{(3)}(

*r*,

*t*),

*E*(

*r*,

*t*) are third-order nonlinear polarization and generated electric field, respectively, n is the refractive index of a medium, and c is the speed of light in vacuum. The third-order polarization at the anti- Stokes frequency of ω

_{as}=2ω

_{p}-ω

_{s}can be written as

*χ*

^{(3)}

*is the third-order nonlinear coefficient of a medium, and*

_{ijkl}*E*

^{P}*(*

_{j}*r*,

*ω*

*,*

_{p}*t*),

*E*

^{P}*(*

_{k}*r*,

*ω*

*,*

_{p}*t*) and

*E*

^{S*}l(

*r*,

*ω*

*,*

_{s}*t*) are the time-dependant amplitudes of pumping and Stokes electric fields in

*j*,

*k*, and

*l*directions, respectively. If the distributions of the pumping and the Stokes electric fields as well as the third-order nonlinear coefficient are known, the third-order nonlinear polarization at any direction can be determined by Eq. (2), and then the generated electric field of a CARS signal can be obtained by solving Eq.(1).

## 3. Finite-difference time-domain (FDTD) method for simulations

13. K. S. Yee. “Numerical solution of initial boundary value problem involving Maxwell equations in isotropic media,”.IEEE Trans. Antennas Propagat. **14**, 302–307 (1966). [CrossRef]

*CA*(

*m*)=[2

*ε*(

*m*)-

*σ*(

*m*)Δ

*t*]/[2

*ε*(

*m*)+

*σ*(

*m*)Δ

*t*], and

*CB*(

*m*)=2Δ

*t*/[2

*ε*(

*m*)+

*σ*(

*m*)Δ

*t*]. The discretized equations for the magnetic fields in the FDTD simulation can be written similarly to the electric fields in Eq. (4). After a few leapfrog iterations with Eq. (4), the electromagnetic field will converge to a stable value determined by given boundary conditions. Based on this leapfrog approach, the FDTD simulation will determine not only the steady-state parameters (e.g., intensity, phase and polarization distribution) of light fields, but also the temporal evolution of these parameters with respect to time. For the analysis of the generation and the propagation of CARS signal, the FDTD method was firstly used to determine the local excitation light fields surrounding scatterers. And then Eq. (2) was used to calculate the induced nonlinear polarizations. Finally the scattering pattern of the generated CARS signal was calculated in Eq. (1) by taking the induced nonlinear polarization into account as an additional radiation source.

## 4. CARS generation and scattering

### 4.1 Excitation field and scattering pattern of uniform bulk material

^{6}=279 times stronger than those along the y or z axes. Therefore, we may only consider the CARS signal polarized along the x-axis for simplicity in calculation. The Stokes beam shows a similar distribution as those given in Fig. 2, and those results are not shown here.

### 4.2 Excitation field and CARS scattering pattern of tiny scatterer

*um*and comparable to the wavelength used in CARS microscopy. Thus generated CARS signal as well as both of pump and stokes lights will be remarkable scattered when some organelles are included inside the focal volume of a confocal CARS setup. To investigate the effects of the spiral phase excitation beam on CARS signal generation and propagation around these organelles, a polystyrene bead with a diameter of 800nm assumed to be located inside the focal spot, and the distribution of the excitation field and the scattering of the induced CARS signal are studied with an FDTD method. The position of the polystyrene bead is indicated in Fig. 4 with a circle. In the calculation, the refractive indices of the polystyrene bead and surrounding water are assumed to be 1.56 and 1.33, respectively.

*P*

^{(3)}

*(*

_{i}*r*,

*ω*,

_{as}*t*) into Eq. (1). Fig. 6(a) is a calculated scattering patterning, which is intensity distribution on a sphere centered on the geometric focal point with a diameter of 200 um. The position of the poly-bead is same as that of Fig. 4 or Fig. 5. We have assumed the nonlinear coefficient and the refractive index of the poly-bead to be 0.6(1+0.5i) and 1.56, respectively [1, 19

19. K. Takeda, Y. Ito, and C. Munakata, “Simultaneous measurement of size and refractive index of a fine particle in flowing liquid,” Meas. Sci. Technol. **3**, 27–32 (1992). [CrossRef]

## 5. Differential image with spiral phase excitation beams

20. M. Hashimoto and T. Araki, “Three-dimensional transfer functions of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. A **18**, 771–776 (2001). [CrossRef]

_{1}, y

_{1}) is the amplitude point spread function of the collecting objective lens, which always is circularly symmetric and can be simplified with a two-dimensional Bessel function.

*P*

^{(3)}(

*x*,

*y*,

*ω*

*) is the induced nonlinear CARS polarization in the focal plan. Δ*

_{as}*x*and Δ

*y*are the sizes of the detecting pinhole in x- and y- directions respectively. For common CARS system with Gaussian excitation beams,

*P*

^{(3)}(

*x*,

*y*,

*ω*

*) is circularly symmetric in a uniform specimen. Then the detected signal determined by Eq. (5) will be relatively very strong, because Eq. (5) essentially indicates constructive interferences between induced nonlinear polarizations from different points on the focal plan. For our case, both the pumping and Stokes lights are LG01 Laguerre-Gaussian beams, and the induced nonlinear CARS polarization*

_{as}*P*

^{(3)}(

*x*,

*y*,

*ω*) has a radially anti-symmetric phase distribution and a symmetric intensity distribution for a uniform specimen. Then Eq. (5) indicates an overall cancellation in the detected CARS signal due to destructive interferences of

_{as}*P*

^{(3)}(

*x*,

*y*,

*ω*) on the focal plan, resulting in a very weak detected signal of I

_{as}_{dec}. When a tiny scatterer is included in the focal volume, the excitation and induced CARS fields inside the focal volume will be redistributed due to the index-mismatch, scattering, diffraction, and near-field effects. The symmetry in the intensity and the anti-symmetry in the phase of generated nonlinear polarization

*P*

^{(3)}(

*x*,

*y*,

*ω*) would disappear in this case. Thus there should be an increase in the intensity of the CARS signal IDec determined by Eq. (5). This suggests that the CARS microscopy has the ability to suppress the background from uniform bulk material and highlight the fine structures of a specimen by using LG01 Laguerre-Gaussian beams for excitation and a confocal system for signal detection. For imaging a bio-sample, this means that the strong non-resonant background from the bulk scattering of the water could be effectively eliminated, and the fine structures dimmed by the background can be highlighted accordingly. Therefore, the capability of the CARS microscopy on identifying fine structures is improved in our scheme.

_{as}*um*in length and width with 3

*um*in thickness. The numerical aperture of the illuminating and the collector lenses are both assumed to be 0.75. The nonlinear coefficient of the water and the polystyrene are assumed to be 0.6 and 0.6(1+0.5i), respectively. The refractive index of water is assumed to be 1.33 while that of the polystyrene is assumed to be 1.56. Fig. 7 (a) shows the image under common Gaussian excitation beams. There is a ringing effect around the border of the polystyrene square, and its degree is much larger in vertical edges at the right and left sides of the square than horizontal edges at upper and lower sides. This asymmetric intensity ringing along the horizontal and vertical directions can be explained with the combining effect of the diffraction and the polarization of input fields. Due to the scattering of input fields at the boundary of the sample, the intensities of exciting fields decrease near the boundary. Hence the generated CARS signal decreases there as well, and this generates a dark ring around the edge of a specimen. We have assumed linearly polarized input fields in our simulation along the vertical direction. On the upper and lower horizontal boundaries in Fig. 7(a) the continuity condition of the electric displacement D leads to a larger intensity of electric field [21

21. L. Cheng, H. Zhiwei, L. Fake, Z. Wei, H. W. Dietmar, and S. Colin“Near-field effects on coherent anti-Stokes Raman scattering microscopy imaging,” Opt. Express **15**, 4118–4131(2007). [CrossRef]

^{6}, and the intensity of the water is about 6.5×10

^{6}. In Fig. 7(b) the edges of the polystyrene slice are remarkably highlighted while uniform CARS intensity inside the square of the polystyrene sample shown in Fig. 7(a) is quite much depressed. The contrast of the image is about 50 in this case; the peak intensity of the polystyrene slice edge is about 3.7×10

^{6}, and the background intensity is about 7.4×10

^{4}. From the analysis in section 2.4 we know that the edge highlighting in Fig. 7 (b) originates from the asymmetry of the optical parameters at the boundary, and it includes the effects of both the index-mismatch and the nonlinear coefficient difference between the surrounding water and the polystyrene sample. Thus our edge detection scheme in CARS microscopy is due to the combined effect of both linear and the nonlinear refractive index differences between two media at boundaries. It has been shown in Ref [12

12. V. V. Krishnamachari and E. O. Potma, “Focus-engineered coherent anti-Stokes Raman scattering microscopy: a numerical investigation,” J. Opt. Soc. Am. A **24**, 1138–1147 (2007). [CrossRef]

_{scatter}to that of the total signal I

_{total}is given in Fig. 8(c). A curve marked with hollow circles corresponds to the method with Laguerre-Gaussian excitation beams, and another one with small solid dots corresponds to the case with common Gaussian excitation beams. For the case of Laguerre-Gaussian beams, the CARS signal from the scatterers is dominant when the size of the scatterer is small, and the ratio of I

_{scatter}to I

_{total}decreases with the increase of scatterer size. On the contrary, for the case with common Gaussian excitation beams the ratio of I

_{scatter}to I

_{total}increases with the scatterer size and approaches a constant of 0.3 when the scatterer size becomes larger than 400 nm. It shows that the I

_{scatter}to I

_{total}ratio with Laguerre-Gaussian excitation beams is larger than that of common Gaussian excitation beams when the size a scatterer ranges from 400 nm to 1000 nm, which corresponds to the case of enhanced signal intensity with Laguerre-Gaussian excitation beams. This means that CARS confocal microscopy based on Laguerre-Gaussian excitation beams not only can highlight the fine structures of a sample by suppressing the non-resonant CARS signal from bulk water but also has enhanced its ability in revealing the chemical structure information of a sample in comparison with common CARS microscopy based on Gaussian excitation beams. Based on these results, we can conclude that CARS microscopy based on Laguerre-Gaussian excitation beams is much suitable for highlighting the structures with scale comparable to the excitation wavelength. When the size of the structure is much larger or smaller than the excitation wavelength used, the sensitivity of this method decreases.

## 6. Conclusion

## Acknowledgments

## References:

1. | R. J.H. Clark and R. E. Hester, Advances in Nonlinear Spectroscopy, (Wiley, New York, 1988) 15. |

2. | P.D. Maker and R.W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys, Rev. |

3. | Y. R. Shen, |

4. | J. X. Cheng and X. S. Xie, “Coherent Anti-Stokes Scattering Microscopy: Instrumentation, Theory, and Application,” J. Phys. Chem. B. |

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

6. | H. Wang, Y. Fu, P. Zickmund, R. Shi, and J. X. Cheng, “Coherent Anti-Stokes Raman Scattering Imaging of Axonal Myelin in Live Spinal Tissues,” Biophysical Journal |

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

8. | J. P. Ogilvie, E. Beaurepire, A. Alexandrou, and M. Joffre, “Fourier-transform coherent anti-Stokes Raman scattering microscopy,” Opt. Lett. |

9. | D. Oron, N. Dudovich, and Y. Silberberg, “Single-Pulse Phase-contrast nonlinear Raman Spectroscopy,” Phys. Rev. Lett. |

10. | E. R. Andresen, H. N. Paulsen, and V. Birkedl “Broadband multiplex coherent anti-Stokes Raman scattering microscopy employing photonic-crystal fibers,” J. Opt. Am. B. |

11. | J. X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B. |

12. | V. V. Krishnamachari and E. O. Potma, “Focus-engineered coherent anti-Stokes Raman scattering microscopy: a numerical investigation,” J. Opt. Soc. Am. A |

13. | K. S. Yee. “Numerical solution of initial boundary value problem involving Maxwell equations in isotropic media,”.IEEE Trans. Antennas Propagat. |

14. | A. Taflove, |

15. | R. W. Boyd, |

16. | S. Mukamel, |

17. | M.W. Beijersbergen, R.P.C. Coewinkel, M. Kristensen, and J.P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Comm. |

18. | K. Crabtree, J. A. Davis, and I. Moreno, “Optical processing with votex-producing lenses,” Appl. Opt. |

19. | K. Takeda, Y. Ito, and C. Munakata, “Simultaneous measurement of size and refractive index of a fine particle in flowing liquid,” Meas. Sci. Technol. |

20. | M. Hashimoto and T. Araki, “Three-dimensional transfer functions of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. A |

21. | L. Cheng, H. Zhiwei, L. Fake, Z. Wei, H. W. Dietmar, and S. Colin“Near-field effects on coherent anti-Stokes Raman scattering microscopy imaging,” Opt. Express |

**OCIS Codes**

(180.1790) Microscopy : Confocal microscopy

(290.5860) Scattering : Scattering, Raman

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

(330.6130) Vision, color, and visual optics : Spatial resolution

**ToC Category:**

Microscopy

**History**

Original Manuscript: June 21, 2007

Revised Manuscript: July 23, 2007

Manuscript Accepted: July 24, 2007

Published: July 26, 2007

**Virtual Issues**

Vol. 2, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Liu Cheng and Dug Y. Kim, "Differential imaging in coherent anti-Stokes Raman scattering microscopy with Laguerre- Gaussian excitation beams," Opt. Express **15**, 10123-10134 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-16-10123

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### References

- R. J.H. Clark and R. E. Hester, Advances in Nonlinear Spectroscopy, (Wiley, New York, 1988) 15.
- P.D. Maker and R.W. Terhune, "Study of optical effects due to an induced polarization third order in the electric field strength," Phys, Rev. 137, A801-A818 (1965). [CrossRef]
- Y. R. Shen, The principles of nonlinear optics (Wiley, New York, 1984).
- J. X. Cheng and X. S. Xie, "Coherent Anti-Stokes Scattering Microscopy: Instrumentation, Theory, and Application," J. Phys. Chem. B. 108,827-840 (2004). [CrossRef]
- J. X. 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]
- H. Wang, Y. Fu, P. Zickmund, R. Shi, and J. X. Cheng, "Coherent Anti-Stokes Raman Scattering Imaging of Axonal Myelin in Live Spinal Tissues," Biophysical Journal 89, 581-591 (2005). [CrossRef] [PubMed]
- 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]
- J. P. Ogilvie, E. Beaurepire, A. Alexandrou, and M. Joffre, "Fourier-transform coherent anti-Stokes Raman scattering microscopy," Opt. Lett. 31, 480- 482 (2006). [CrossRef] [PubMed]
- D. Oron, N. Dudovich, and Y. Silberberg, "Single-Pulse Phase-contrast nonlinear Raman Spectroscopy," Phys. Rev. Lett. 89, 273001-273004 (2002). [CrossRef]
- E. R. Andresen, H. N. Paulsen, V. Birkedl "Broadband multiplex coherent anti-Stokes Raman scattering microscopy employing photonic-crystal fibers," J. Opt. Am. B. 22, 1934-1938 (2005). [CrossRef]
- J. X. Cheng and X. S. Xie, "Green’s function formulation for third-harmonic generation microscopy," J. Opt. Soc. Am. B. 19, 1604-1610 (2002). [CrossRef]
- V. V. Krishnamachari, E. O. Potma, "Focus-engineered coherent anti-Stokes Raman scattering microscopy: a numerical investigation," J. Opt. Soc. Am. A 24, 1138-1147 (2007). [CrossRef]
- K. S. Yee. "Numerical solution of initial boundary value problem involving Maxwell equations in isotropic media,".IEEE Trans. Antennas Propagat. 14, 302-307 (1966). [CrossRef]
- A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).
- R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992).
- S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford U. Press New York, 1995).
- M.W. Beijersbergen, R.P.C. Coewinkel, M. Kristensen, J.P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Opt. Comm. 112, 321-327 (1994) [CrossRef]
- K. Crabtree, J. A. Davis, and I. Moreno, "Optical processing with votex-producing lenses," Appl. Opt. 43, 1360-1367(2004). [CrossRef] [PubMed]
- K. Takeda, Y. Ito and C. Munakata, "Simultaneous measurement of size and refractive index of a fine particle in flowing liquid," Meas. Sci. Technol. 3, 27-32 (1992). [CrossRef]
- M. Hashimoto and T. Araki, "Three-dimensional transfer functions of coherent anti-Stokes Raman scattering microscopy," J. Opt. Soc. Am. A 18,771-776 (2001). [CrossRef]
- L. Cheng, H. Zhiwei, L. Fake, Z. Wei, H. W. Dietmar, S. Colin "Near-field effects on coherent anti-Stokes Raman scattering microscopy imaging," Opt. Express 15, 4118-4131(2007). [CrossRef]

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