## Interaction of spherical nanoparticles with a highly focused beam of light

Optics Express, Vol. 16, Issue 5, pp. 2874-2886 (2008)

http://dx.doi.org/10.1364/OE.16.002874

Acrobat PDF (176 KB)

### Abstract

The interaction of a highly focused beam of light with spherical nanoparticles is investigated for linear and radial polarizations. An analytical solution is obtained to calculate this interaction. The Richards-Wolf theory is used to express the incident electric field near the focus of an aplanatic lens. The incident beam is expressed as an integral where the integrand is separated into transverse-electric (TE) and transverse-magnetic (TM) waves. The interaction of each TE and TM wave with a spherical nanoparticle is calculated using the Mie theory. The resulting analytical solution is then obtained by integrating the scattered waves over the entire angular spectrum. A finite element method solution is also obtained for comparison.

© 2008 Optical Society of America

## 1. Introduction

1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. **24**, 156–159 (1970). [CrossRef]

2. C. Godefroy and M. Adjouadi, “Particle sizing in a flow environment using light scattering patterns,” Part. Part. Syst. Charact. **17**, 47–55 (2000). [CrossRef]

3. A. C. Eckbreth, “Effects of laser-modulated particulate incandescence on Raman scattering diagnostics,” J. Appl. Phys. **48**, 4473–4479 (1977). [CrossRef]

15. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. Roy. Soc. London Ser. A **253**, 349–357 (1959). [CrossRef]

16. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London Ser. A **253**, 358–379 (1959). [CrossRef]

15. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. Roy. Soc. London Ser. A **253**, 349–357 (1959). [CrossRef]

16. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London Ser. A **253**, 358–379 (1959). [CrossRef]

17. A. Hartschuh, E. J. Sánchez, X. S. Xie, and L. Novotny, “High-resolution near-field Raman microscopy of singlewalled carbon nanotubes,” Phys. Rev. Lett. **90**, 095503 (2003). [CrossRef] [PubMed]

18. W. A. Challener, I. K. Sendur, and C. Peng, “Scattered field formulation of finite difference time domain for a focused light beam in dense media with lossy materials,” Opt. Express **11**, 3160–3170 (2003). [CrossRef] [PubMed]

19. K. Sendur, W. Challener, and C. Peng, “Ridge waveguide as a near field aperture for high density data storage,” J. Appl. Phys. **96**, 2743–2752 (2004). [CrossRef]

## 2. Focused field formulation

15. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. Roy. Soc. London Ser. A **253**, 349–357 (1959). [CrossRef]

16. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London Ser. A **253**, 358–379 (1959). [CrossRef]

*α*is the half angle of the beam,

**r**

_{p}is the observation point

*λ*is the wavelength in the medium,

*ϕ*= arctan(

_{p}*y*). In Eq. (1),

_{p}/x_{p}**a**(

*θ,ϕ*) is the weighting vector for a plane wave incident from the (

*θ*,

*ϕ*) direction. Here it should be noted that

**a**(

*θ*,

*ϕ*) is a polarization dependent quantity.

**a**(

*θ*,

_{i}*ϕ*) is given as

_{j}*ω*are the numerical quadrature coefficients,

_{ij}**k**

_{ij}direction with an amplitude of

**a**(

*θ*,

_{i}*ϕ*) of the plane wave in the

_{j}**k**

_{ij}direction.

*x*-component of the electric field is much stronger than the other two components as shown in Fig. 1. The radially polarized wave has a strong

*z*-component in the focal region as shown in Fig. 2.

## 3. Analytical treatment of the interaction of focused light with spherical nanoparticles

20. G. Mie, “Beiträge zur optik truber medien, speziell kolloida ler metallösungen” Ann. d. Physik **25**, 377- (1908). [CrossRef]

21. M. Born and E. Wolf, *Principles of Optics 5 ^{th}* ed. (Pergamon Press, Oxford, 1975), section 13.5. [PubMed]

**E**

^{x}

_{tot}(

**r**). Explicit expressions for the

**E**

^{x}

_{tot}(

**r**) are given in the literature [21

21. M. Born and E. Wolf, *Principles of Optics 5 ^{th}* ed. (Pergamon Press, Oxford, 1975), section 13.5. [PubMed]

**E**

*(*

^{TE}_{inc}**r**) and

**E**

*(*

^{TM}_{inc}**r**) polarized plane waves in Eqs. (11) and (12) can be obtained from

**E**

*(*

^{x}_{inc}**r**) in Eq. (10) by simple coordinate transformations.

**E**

*(*

^{TM}_{inc}**r**) is obtained from

**E**

*(*

^{x}_{inc}**r**) by subsequent

*θ*=

*θ*, and

_{inc}*ϕ*=

*ϕ*transformations. Similarly,

_{inc}**E**

*(*

^{TE}_{inc}**r**) is obtained from

**E**

*(*

^{x}_{inc}**r**) by subsequent

*ϕ*=-

*π*/2,

*θ*=

*θ*, and

_{inc}*ϕ*=

*ϕ*transformations. Since the incident fields

_{inc}**E**

*(*

^{TM}_{inc}**r**) and

**E**

*(*

^{TE}_{inc}**r**) can be obtained from linear transformations of

**E**

*(*

^{x}_{inc}**r**), the total fields

**E**

*(*

^{TM}_{tot}**r**) and

**E**

*(*

^{TE}_{tot}**r**) can be obtained from

**E**

*(*

^{x}_{tot}**r**) by the same transformations due to the linearity of the system. In summary, the total electric fields due to the presence of the sphere

**E**

*(*

^{TM}_{tot}**r**) and

**E**

*(*

^{TE}_{tot}**r**) are obtained as

**E**

*(*

^{TM}_{tot}**r**) and

**E**

*(*

^{TE}_{inc}**r**) starting from the expression in the literature. Equations (13) and (14) will be utilized to find the solution scattering from spherical particles when the incident field is given by Richards-Wolf theory. In Eq. (1), a highly focused incident beam of light is expressed as an integral where the integrand is a plane wave propagating in the

**k**direction. A linearly polarized incident focused wave

*and TE*

_{inc}*incident plane waves, respectively. We can write the total (incident + scattered) electric field due to scattering from spherical particles when the incident field is given by Eq. (15) as*

_{inc}*θ*cos

_{inc}*ϕ*+cos

_{inc}◯*θ*sin

_{inc}*ϕ*+sin

_{inc}ŷ*θ*)exp(

_{inc}ẑ*i*

**k · r**) is a representation of the TM polarized incident plane waves. Using the linearity of the integration operation, we can write the total field due to the radially polarized focused light as

## 4. Results

*i*×4.523. The wavelength is selected around the plasmonic resonances of larger nanoparticles. However, no particular attempt is made to optimize the response of the nanoparticles as a function of wavelength when they are illuminated with a focused beam of light.

*ẑ*direction. In Figs. 4 and 5, the results of a silver sphere with a 50 nm radius are presented for radial and linear polarizations, respectively. The half beam angle of the optical lens is 60°. In Figs. 4 and 5, |

*E*|

_{x}^{2}and |

*E*|

_{z}^{2}are plotted. The |

*E*|

_{y}^{2}component was negligible for both linear and radial polarization. The Mie series solution agrees well with the FEM results. For the 50 nm sphere the results for radial polarization and linear polarization are very similar, except for a 90° rotation, which is consistent with the direction of the incident field at the focus, as shown in Figs. 1 and 2. In this case the sphere is too small to interact with field components in other directions.

*◯*-direction and the

*ẑ*-direction for radially polarized light. Although the amplitude of the electromagnetic wave in

*ẑ*-direction is similar for linear polarization, there is some difference in the amplitude of the

*◯*-component for linear polarization.

**k**vectors. The larger sphere feels the effect of various

**k**vectors of the incident field outside the focal point, producing different electric field distributions in the scattered field. In the area around the focus, the main contribution for the linear and radial polarized light comes from the

*x*-component and z-components, respectively. As shown in Figs. 1 and 2, the

*x*-component for the linear polarization and the

*z*-component of the radial polarization are similar. The other polarization components are small at the focus, therefore, a very small sphere does not feel the impact of these components. However, as the sphere becomes larger, it starts to feel the effect of the other polarization components. As shown in Figs. 1 and 2, the other polarization components have differences for linear and radial polarizations, which result in different response of the sphere for linear and radial polarizations. The amplitude of the |

*E*(

_{z}*x*,

*z*)|

^{2}component is stronger than the |

*E*(

_{x}*x*,

*z*)|

^{2}component.

*x̂-ẑ*plane for radial and linear polarized incident light, respectively. The results are in agreement.

## 5. Conclusion

## References and links

1. | A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. |

2. | C. Godefroy and M. Adjouadi, “Particle sizing in a flow environment using light scattering patterns,” Part. Part. Syst. Charact. |

3. | A. C. Eckbreth, “Effects of laser-modulated particulate incandescence on Raman scattering diagnostics,” J. Appl. Phys. |

4. | N. Morita, T. Tanaka, T. Yamasaki, and Y. Yakanishi, “Scattering of a beam by a spherical object,” IEEE Trans. Antennas Propag. |

5. | W.-C. Tsai and R. J. Pogorzelski, “Eigenfunction solution of the scattering of beam radiation fields by spherical objects,” J. Opt. Soc. Am. A |

6. | W. G. Tam and R. Corriveau, “Scattering of electromagnetic beams by spherical objects,” J. Opt. Soc. Am. |

7. | J. S. Kim and S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. |

8. | J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. |

9. | L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A |

10. | J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: Focal point positioning effects at resonance,” J. Appl. Phys. |

11. | J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundemental Gaussian beam,” J. Appl. Phys. |

12. | J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. |

13. | J. P. Barton, “Electromagnetic-field calculations for a sphere illuminated by a higher-order Gaussian beam. I. Internal and near-field effects,” Appl. Opt. |

14. | J. P. Barton, “Electromagnetic-field calculations for a sphere illuminated by a higher-order Gaussian beam. II. Far-field scattering,” Appl. Opt. |

15. | E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. Roy. Soc. London Ser. A |

16. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London Ser. A |

17. | A. Hartschuh, E. J. Sánchez, X. S. Xie, and L. Novotny, “High-resolution near-field Raman microscopy of singlewalled carbon nanotubes,” Phys. Rev. Lett. |

18. | W. A. Challener, I. K. Sendur, and C. Peng, “Scattered field formulation of finite difference time domain for a focused light beam in dense media with lossy materials,” Opt. Express |

19. | K. Sendur, W. Challener, and C. Peng, “Ridge waveguide as a near field aperture for high density data storage,” J. Appl. Phys. |

20. | G. Mie, “Beiträge zur optik truber medien, speziell kolloida ler metallösungen” Ann. d. Physik |

21. | M. Born and E. Wolf, |

22. | J. M. Jin, |

23. | E. D. Palik, |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(240.6680) Optics at surfaces : Surface plasmons

(290.4020) Scattering : Mie theory

**ToC Category:**

Scattering

**History**

Original Manuscript: January 7, 2008

Revised Manuscript: February 11, 2008

Manuscript Accepted: February 12, 2008

Published: February 15, 2008

**Virtual Issues**

Vol. 3, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Kursat Sendur, William Challener, and Oleg Mryasov, "Interaction of spherical nanoparticles with a highly focused beam of light," Opt. Express **16**, 2874-2886 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-5-2874

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### References

- A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970). [CrossRef]
- C. Godefroy and M. Adjouadi, "Particle sizing in a flow environment using light scattering patterns," Part. Part. Syst. Charact. 17, 47-55 (2000). [CrossRef]
- A. C. Eckbreth, "Effects of laser-modulated particulate incandescence on Raman scattering diagnostics," J. Appl. Phys. 48, 4473-4479 (1977). [CrossRef]
- N. Morita, T. Tanaka, T. Yamasaki, and Y. Yakanishi, "Scattering of a beam by a spherical object," IEEE Trans. Antennas Propag. 16, 724-727 (1968). [CrossRef]
- W.-C. Tsai and R. J. Pogorzelski, "Eigenfunction solution of the scattering of beam radiation fields by spherical objects," J. Opt. Soc. Am. A 65, 1457-1463 (1975). [CrossRef]
- W. G. Tam and R. Corriveau, "Scattering of electromagnetic beams by spherical objects," J. Opt. Soc. Am. 68, 763-767 (1978). [CrossRef]
- J. S. Kim and S. S. Lee, "Scattering of laser beams and the optical potential well for a homogeneous sphere," J. Opt. Soc. Am. 73, 303-312 (1983). [CrossRef]
- J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988). [CrossRef]
- L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979). [CrossRef]
- J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal fields of a spherical particle illuminated by a tightly focused laser beam: Focal point positioning effects at resonance," J. Appl. Phys. 65, 2900-2906 (1989). [CrossRef]
- J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989). [CrossRef]
- J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989). [CrossRef]
- J. P. Barton, "Electromagnetic-field calculations for a sphere illuminated by a higher-order Gaussian beam. I. Internal and near-field effects," Appl. Opt. 36, 1303-1311 (1997). [CrossRef] [PubMed]
- J. P. Barton, "Electromagnetic-field calculations for a sphere illuminated by a higher-order Gaussian beam. II. Far-field scattering," Appl. Opt. 37, 3339-3344 (1998). [CrossRef]
- E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. Roy. Soc. London Ser. A 253, 349-357 (1959). [CrossRef]
- B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. London Ser. A 253, 358-379 (1959). [CrossRef]
- A. Hartschuh, E. J. S’anchez, X. S. Xie, and L. Novotny, "High-resolution near-field Raman microscopy of singlewalled carbon nanotubes," Phys. Rev. Lett. 90, 095503 (2003). [CrossRef] [PubMed]
- W. A. Challener, I. K. Sendur, and C. Peng, "Scattered field formulation of finite difference time domain for a focused light beam in dense media with lossy materials," Opt. Express 11, 3160-3170 (2003). [CrossRef] [PubMed]
- K. Sendur, W. Challener, and C. Peng, "Ridge waveguide as a near field aperture for high density data storage," J. Appl. Phys. 96, 2743-2752 (2004). [CrossRef]
- G. Mie, "Beitr¨age zur optik truber medien, speziell kolloida ler metallosungen," Ann. d. Physik 25, 377 (1908). [CrossRef]
- M. Born and E. Wolf, Principles of Optics 5th ed., section 13.5 (Pergamon Press, Oxford, 1975). [PubMed]
- J. M. Jin, The Finite Element Method in Electomagnetics (John Wiley & Sons, New York, NY, 2000).
- E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, San Diego, CA, 1998).

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