## Coupled and uncoupled dipole models of nonlinear scattering |

Optics Express, Vol. 20, Issue 23, pp. 25834-25842 (2012)

http://dx.doi.org/10.1364/OE.20.025834

Acrobat PDF (947 KB)

### Abstract

Dipole models are one of the simplest numerical models to understand nonlinear scattering. Existing dipole model for second harmonic generation, third harmonic generation and coherent anti-Stokes Raman scattering assume that the dipoles which make up a scatterer do not interact with one another. Thus, this dipole model can be called the uncoupled dipole model. This dipole model is not sufficient to describe the effects of refractive index of a scatterer or to describe scattering at the edges of a scatterer. Taking into account the interaction between dipoles overcomes these short comings of the uncoupled dipole model. Coupled dipole model has been primarily used for linear scattering studies but it can be extended to predict nonlinear scattering. The coupled and uncoupled dipole models have been compared to highlight their differences. Results of nonlinear scattering predicted by coupled dipole model agree well with previously reported experimental results.

© 2012 OSA

## Introduction

1. J. N. Gannaway and C. J. R. Sheppard, “Second-harmonic imaging in the scanning optical microscope,” Opt. Quantum Electron. **10**(5), 435–439 (1978). [CrossRef]

2. W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science **248**(4951), 73–76 (1990). [CrossRef] [PubMed]

3. P. J. Campagnola, M. D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. **77**(6), 3341–3349 (1999). [CrossRef] [PubMed]

4. A. Zoumi, A. Yeh, and B. J. Tromberg, “Imaging cells and extracellular matrix in vivo by using second-harmonic generation and two-photon excited fluorescence,” Proc. Natl. Acad. Sci. U.S.A. **99**(17), 11014–11019 (2002). [CrossRef] [PubMed]

5. S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. **90**(2), 693–703 (2006). [CrossRef] [PubMed]

6. P. J. Campagnola, A. C. Millard, M. Terasaki, P. E. Hoppe, C. J. Malone, and W. A. Mohler, “Three-dimensional high-resolution second-harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J. **82**(1), 493–508 (2002). [CrossRef] [PubMed]

8. W. R. Zipfel, R. M. Williams, R. Christie, A. Y. Nikitin, B. T. Hyman, and W. W. Webb, “Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation,” Proc. Natl. Acad. Sci. U.S.A. **100**(12), 7075–7080 (2003). [CrossRef] [PubMed]

9. J. Squier, M. Muller, G. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express **3**(9), 315–324 (1998). [CrossRef] [PubMed]

10. D. Yelin and Y. Silberberg, “Laser scanning third-harmonic-generation microscopy in biology,” Opt. Express **5**(8), 169–175 (1999). [CrossRef] [PubMed]

11. J.-X. Cheng, Y. K. Jia, G. Zheng, and X. S. Xie, “Laser-scanning coherent Anti-Stokes Raman scattering microscopy and applications to cell biology,” Biophys. J. **83**(1), 502–509 (2002). [CrossRef] [PubMed]

12. H. A. Rinia, K. N. J. Burger, M. Bonn, and M. Müller, “Quantitative label-free imaging of lipid composition and packing of individual cellular lipid droplets using multiplex CARS microscopy,” Biophys. J. **95**(10), 4908–4914 (2008). [CrossRef] [PubMed]

13. J. Lin, H. Wang, W. Zheng, F. Lu, C. Sheppard, and Z. Huang, “Numerical study of effects of light polarization, scatterer sizes and orientations on near-field coherent anti-Stokes Raman scattering microscopy,” Opt. Express **17**(4), 2423–2434 (2009). [CrossRef] [PubMed]

14. G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P.-F. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” J. Opt. Soc. Am. B **25**(6), 955–960 (2008). [CrossRef]

15. J. Mäkitalo, S. Suuriniemi, and M. Kauranen, “Boundary element method for surface nonlinear optics of nanoparticles,” Opt. Express **19**(23), 23386–23399 (2011). [CrossRef] [PubMed]

16. L. Moreaux, O. Sandre, and J. Mertz, “Membrane imaging by second-harmonic generation microscopy,” J. Opt. Soc. Am. B **17**(10), 1685–1694 (2000). [CrossRef]

17. J.-X. Cheng and X. S. Xie, “Green's function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B **19**(7), 1604–1610 (2002). [CrossRef]

18. 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**(6), 1363–1375 (2002). [CrossRef]

19. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. **186**, 705–714 (1973). [CrossRef]

20. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. **333**, 848–872 (1988). [CrossRef]

22. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A **11**(4), 1491–1499 (1994). [CrossRef]

23. N. K. Balla, P. T. C. So, and C. J. R. Sheppard, “Second harmonic scattering from small particles using Discrete Dipole Approximation,” Opt. Express **18**(21), 21603–21611 (2010). [CrossRef] [PubMed]

## Theory

23. N. K. Balla, P. T. C. So, and C. J. R. Sheppard, “Second harmonic scattering from small particles using Discrete Dipole Approximation,” Opt. Express **18**(21), 21603–21611 (2010). [CrossRef] [PubMed]

## Results and discussion

24. E. Yew and C. Sheppard, “Effects of axial field components on second harmonic generation microscopy,” Opt. Express **14**(3), 1167–1174 (2006). [CrossRef] [PubMed]

*x*-polarized, is focused through a high numerical aperture (NA) lens, the polarization of light at the focus is not entirely along the

*x*-axis and there is significant power in

*y*- and

*z*- polarized components as well [25

25. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. ii. structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. **253**(1274), 358–379 (1959). [CrossRef]

*x*-polarised light is focussed onto a thin layer of actin fiber bundles oriented along

*x*-axis. The wavelength of the excitation source is 800 nm and the NA of the objective is 0.87 ( = sin60°). The vectorial distribution of electric field at the focus of a high NA lens was calculated [25

25. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. ii. structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. **253**(1274), 358–379 (1959). [CrossRef]

*E*, followed by

_{x}*E*and

_{z}*E*. The driving field for second order dipoles in the actin fibers was calculated (Eq. (13)) from the nonlinear susceptibility (

_{y}**χ**

^{(2)}) of actin fiber bundle [26

26. S.-W. Chu, S.-Y. Chen, G.-W. Chern, T.-H. Tsai, Y.-C. Chen, B.-L. Lin, and C.-K. Sun, “Studies of X^{(2)}/X^{(3)} tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. **86**(6), 3914–3922 (2004). [CrossRef] [PubMed]

*x*-component of the second order polarization (

*P*

_{x}^{(2)}) in the focal plane shows three different lobes (Fig. 2 ). The two side lobes are contributions from the

*E*component of the local field. The central lobe comes from the

_{z}*E*component of the local field.

_{x}*E*being the weakest component, its contribution to

_{y}*P*

_{x}^{(2)}is very small. Results calculated using UDM (Fig. 2(a)) show that the central lobe is very weak as compared to the side lobes. On the other hand, results predicted using CDM (Fig. 2(b) and 2(c)) show the central lobe becomes stronger with increase in refractive index of the actin fibers. The results show that with increase in refractive index, the

*E*component of the local field becomes stronger as compared to the

_{x}*E*and

_{y}*E*components. Any change in refractive index of the sample should affect the scattering from the sample and CDM predicts how it happens in this case.

_{z}*P*and

_{y}^{(2)}*P*depend on the products

_{z}^{(2)}*E*and

_{x}E_{y}*E*respectively. Even if the

_{x}E_{z}*E*component of local field becomes stronger with increasing magnitude of refractive index, the spatial distribution of the products

_{x}*E*and

_{x}E_{y}*E*does not change significantly (Fig. 3 ). Here it should be noted that each plot in Figs. 2 and 3 has been normalized to have a maximum value of unity. This helps us to show the distribution of induced second order polarizations but not their relative magnitudes.

_{x}E_{z}27. 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**(1), 27–32 (1992). [CrossRef]

**χ**

^{(3)}) of polystyrene was assumed to be the same as that of an isotropic material [28]. Forward scattered THG was calculated using UDM and CDM as the bead was scanned axially through the focus of the excitation beam. Since THG is strong at interfaces, axial scanning should give strong signals at the front and back surface of the bead. From the center of the sphere, the surface in the direction of incident light is referred to as the front surface. In the case of UDM, both these surfaces are symmetric and so is the signal generated by them (Fig. 4 ). In reality, the bead interacts with the excitation field and therefore the field distribution is different when light is focused at the front and back surfaces of the bead. CDM is able to capture this effect because it takes into account the presence of sample dipoles when calculating the field distributions. Therefore signal from the two interfaces is different (Fig. 4). CDM results agree well with experimental observations reported in literature [29

29. D. Débarre, W. Supatto, and E. Beaurepaire, “Structure sensitivity in third-harmonic generation microscopy,” Opt. Lett. **30**(16), 2134–2136 (2005). [CrossRef] [PubMed]

*), the Stokes wavelength (λ*

_{p}*) and the CARS wavelength (λ*

_{s}*) in this study are 750 nm, 852 nm and 670 nm respectively. This corresponds to a Raman shift of 1600 cm*

_{c}^{−1}in polystyrene beads [30

30. C. Liu, Z. Huang, F. Lu, W. Zheng, D. W. Hutmacher, and C. Sheppard, “Near-field effects on coherent anti-Stokes Raman scattering microscopy imaging,” Opt. Express **15**(7), 4118–4131 (2007). [CrossRef] [PubMed]

**γ**of polystyrene should have resonant and a non-resonant component whereas the

**γ**of water should have only the non-resonant component. The non-resonant component of

**γ**was taken to be of the same form as that in THG calculations. The

**γ**of water, which contributes to background here, is assumed to be 60% in magnitude of the non-resonant

**γ**of polystyrene. For the resonant

**γ**, the ratios of non-zero elements are γ

*/ γ*

_{xyyx}*= 3/4, γ*

_{xxxx}*/ γ*

_{xxyy}*= 1/8 and γ*

_{xxxx}*/ γ*

_{xyxy}*= 1/8. Furthermore, the resonance of*

_{xxxx}**γ**was taken to be 2.5 times the magnitude of non-resonant

**γ**

*Being a third order scattering process, CARS is present in all materials. The effect of the Gouy phase shift at the focus is partially compensated by the interaction between the pump field and the conjugate Stokes field [31*

_{.}31. 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]

32. N. Djaker, D. Gachet, N. Sandeau, P.-F. Lenne, and H. Rigneault, “Refractive effects in coherent anti-Stokes Raman scattering microscopy,” Appl. Opt. **45**(27), 7005–7011 (2006). [CrossRef] [PubMed]

32. N. Djaker, D. Gachet, N. Sandeau, P.-F. Lenne, and H. Rigneault, “Refractive effects in coherent anti-Stokes Raman scattering microscopy,” Appl. Opt. **45**(27), 7005–7011 (2006). [CrossRef] [PubMed]

## Conclusions

22. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A **11**(4), 1491–1499 (1994). [CrossRef]

## Acknowledgments

## References and links

1. | J. N. Gannaway and C. J. R. Sheppard, “Second-harmonic imaging in the scanning optical microscope,” Opt. Quantum Electron. |

2. | W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science |

3. | P. J. Campagnola, M. D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. |

4. | A. Zoumi, A. Yeh, and B. J. Tromberg, “Imaging cells and extracellular matrix in vivo by using second-harmonic generation and two-photon excited fluorescence,” Proc. Natl. Acad. Sci. U.S.A. |

5. | S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. |

6. | P. J. Campagnola, A. C. Millard, M. Terasaki, P. E. Hoppe, C. J. Malone, and W. A. Mohler, “Three-dimensional high-resolution second-harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J. |

7. | E. Brown, T. McKee, E. diTomaso, A. Pluen, B. Seed, Y. Boucher, and R. K. Jain, “Dynamic imaging of collagen and its modulation in tumors in vivo using second-harmonic generation,” Nat. Med. |

8. | W. R. Zipfel, R. M. Williams, R. Christie, A. Y. Nikitin, B. T. Hyman, and W. W. Webb, “Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation,” Proc. Natl. Acad. Sci. U.S.A. |

9. | J. Squier, M. Muller, G. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express |

10. | D. Yelin and Y. Silberberg, “Laser scanning third-harmonic-generation microscopy in biology,” Opt. Express |

11. | J.-X. Cheng, Y. K. Jia, G. Zheng, and X. S. Xie, “Laser-scanning coherent Anti-Stokes Raman scattering microscopy and applications to cell biology,” Biophys. J. |

12. | H. A. Rinia, K. N. J. Burger, M. Bonn, and M. Müller, “Quantitative label-free imaging of lipid composition and packing of individual cellular lipid droplets using multiplex CARS microscopy,” Biophys. J. |

13. | J. Lin, H. Wang, W. Zheng, F. Lu, C. Sheppard, and Z. Huang, “Numerical study of effects of light polarization, scatterer sizes and orientations on near-field coherent anti-Stokes Raman scattering microscopy,” Opt. Express |

14. | G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P.-F. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” J. Opt. Soc. Am. B |

15. | J. Mäkitalo, S. Suuriniemi, and M. Kauranen, “Boundary element method for surface nonlinear optics of nanoparticles,” Opt. Express |

16. | L. Moreaux, O. Sandre, and J. Mertz, “Membrane imaging by second-harmonic generation microscopy,” J. Opt. Soc. Am. B |

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

18. | 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. | E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. |

20. | B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. |

21. | J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett. |

22. | B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A |

23. | N. K. Balla, P. T. C. So, and C. J. R. Sheppard, “Second harmonic scattering from small particles using Discrete Dipole Approximation,” Opt. Express |

24. | E. Yew and C. Sheppard, “Effects of axial field components on second harmonic generation microscopy,” Opt. Express |

25. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. ii. structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

26. | S.-W. Chu, S.-Y. Chen, G.-W. Chern, T.-H. Tsai, Y.-C. Chen, B.-L. Lin, and C.-K. Sun, “Studies of X |

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

28. | R. W. Boyd, “The Nonlinear Optical Susceptibility,” in |

29. | D. Débarre, W. Supatto, and E. Beaurepaire, “Structure sensitivity in third-harmonic generation microscopy,” Opt. Lett. |

30. | C. Liu, Z. Huang, F. Lu, W. Zheng, D. W. Hutmacher, and C. Sheppard, “Near-field effects on coherent anti-Stokes Raman scattering microscopy imaging,” Opt. Express |

31. | J.-X. Cheng and X. S. Xie, “Coherent Anti-Stokes Raman Scattering microscopy: instrumentation, theory, and applications,” J. Phys. Chem. B |

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

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(190.3970) Nonlinear optics : Microparticle nonlinear optics

(190.4720) Nonlinear optics : Optical nonlinearities of condensed matter

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 24, 2012

Revised Manuscript: September 10, 2012

Manuscript Accepted: September 21, 2012

Published: November 1, 2012

**Citation**

Naveen K. Balla, Elijah Y. S. Yew, Colin J. R. Sheppard, and Peter T. C. So, "Coupled and uncoupled dipole models of nonlinear scattering," Opt. Express **20**, 25834-25842 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-23-25834

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### References

- J. N. Gannaway and C. J. R. Sheppard, “Second-harmonic imaging in the scanning optical microscope,” Opt. Quantum Electron.10(5), 435–439 (1978). [CrossRef]
- W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science248(4951), 73–76 (1990). [CrossRef] [PubMed]
- P. J. Campagnola, M. D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J.77(6), 3341–3349 (1999). [CrossRef] [PubMed]
- A. Zoumi, A. Yeh, and B. J. Tromberg, “Imaging cells and extracellular matrix in vivo by using second-harmonic generation and two-photon excited fluorescence,” Proc. Natl. Acad. Sci. U.S.A.99(17), 11014–11019 (2002). [CrossRef] [PubMed]
- S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J.90(2), 693–703 (2006). [CrossRef] [PubMed]
- P. J. Campagnola, A. C. Millard, M. Terasaki, P. E. Hoppe, C. J. Malone, and W. A. Mohler, “Three-dimensional high-resolution second-harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J.82(1), 493–508 (2002). [CrossRef] [PubMed]
- E. Brown, T. McKee, E. diTomaso, A. Pluen, B. Seed, Y. Boucher, and R. K. Jain, “Dynamic imaging of collagen and its modulation in tumors in vivo using second-harmonic generation,” Nat. Med.9(6), 796–801 (2003). [CrossRef] [PubMed]
- W. R. Zipfel, R. M. Williams, R. Christie, A. Y. Nikitin, B. T. Hyman, and W. W. Webb, “Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation,” Proc. Natl. Acad. Sci. U.S.A.100(12), 7075–7080 (2003). [CrossRef] [PubMed]
- J. Squier, M. Muller, G. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express3(9), 315–324 (1998). [CrossRef] [PubMed]
- D. Yelin and Y. Silberberg, “Laser scanning third-harmonic-generation microscopy in biology,” Opt. Express5(8), 169–175 (1999). [CrossRef] [PubMed]
- J.-X. Cheng, Y. K. Jia, G. Zheng, and X. S. Xie, “Laser-scanning coherent Anti-Stokes Raman scattering microscopy and applications to cell biology,” Biophys. J.83(1), 502–509 (2002). [CrossRef] [PubMed]
- H. A. Rinia, K. N. J. Burger, M. Bonn, and M. Müller, “Quantitative label-free imaging of lipid composition and packing of individual cellular lipid droplets using multiplex CARS microscopy,” Biophys. J.95(10), 4908–4914 (2008). [CrossRef] [PubMed]
- J. Lin, H. Wang, W. Zheng, F. Lu, C. Sheppard, and Z. Huang, “Numerical study of effects of light polarization, scatterer sizes and orientations on near-field coherent anti-Stokes Raman scattering microscopy,” Opt. Express17(4), 2423–2434 (2009). [CrossRef] [PubMed]
- G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P.-F. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” J. Opt. Soc. Am. B25(6), 955–960 (2008). [CrossRef]
- J. Mäkitalo, S. Suuriniemi, and M. Kauranen, “Boundary element method for surface nonlinear optics of nanoparticles,” Opt. Express19(23), 23386–23399 (2011). [CrossRef] [PubMed]
- L. Moreaux, O. Sandre, and J. Mertz, “Membrane imaging by second-harmonic generation microscopy,” J. Opt. Soc. Am. B17(10), 1685–1694 (2000). [CrossRef]
- J.-X. Cheng and X. S. Xie, “Green's function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B19(7), 1604–1610 (2002). [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. B19(6), 1363–1375 (2002). [CrossRef]
- E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J.186, 705–714 (1973). [CrossRef]
- B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J.333, 848–872 (1988). [CrossRef]
- J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett.16(15), 1198–1200 (1991). [CrossRef] [PubMed]
- B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A11(4), 1491–1499 (1994). [CrossRef]
- N. K. Balla, P. T. C. So, and C. J. R. Sheppard, “Second harmonic scattering from small particles using Discrete Dipole Approximation,” Opt. Express18(21), 21603–21611 (2010). [CrossRef] [PubMed]
- E. Yew and C. Sheppard, “Effects of axial field components on second harmonic generation microscopy,” Opt. Express14(3), 1167–1174 (2006). [CrossRef] [PubMed]
- B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. ii. structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci.253(1274), 358–379 (1959). [CrossRef]
- S.-W. Chu, S.-Y. Chen, G.-W. Chern, T.-H. Tsai, Y.-C. Chen, B.-L. Lin, and C.-K. Sun, “Studies of X(2)/X(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J.86(6), 3914–3922 (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(1), 27–32 (1992). [CrossRef]
- R. W. Boyd, “The Nonlinear Optical Susceptibility,” in Nonlinear Optics, 3rd ed. (Academic Press, 2008), pp. 1–67.
- D. Débarre, W. Supatto, and E. Beaurepaire, “Structure sensitivity in third-harmonic generation microscopy,” Opt. Lett.30(16), 2134–2136 (2005). [CrossRef] [PubMed]
- C. Liu, Z. Huang, F. Lu, W. Zheng, D. W. Hutmacher, and C. Sheppard, “Near-field effects on coherent anti-Stokes Raman scattering microscopy imaging,” Opt. Express15(7), 4118–4131 (2007). [CrossRef] [PubMed]
- J.-X. Cheng and X. S. Xie, “Coherent Anti-Stokes Raman Scattering microscopy: instrumentation, theory, and applications,” J. Phys. Chem. B108(3), 827–840 (2004). [CrossRef]
- N. Djaker, D. Gachet, N. Sandeau, P.-F. Lenne, and H. Rigneault, “Refractive effects in coherent anti-Stokes Raman scattering microscopy,” Appl. Opt.45(27), 7005–7011 (2006). [CrossRef] [PubMed]

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