## Rigorous analysis of spheres in Gauss-Laguerre beams

Optics Express, Vol. 15, Issue 20, pp. 13360-13374 (2007)

http://dx.doi.org/10.1364/OE.15.013360

Acrobat PDF (1726 KB)

### Abstract

In this paper we develop a rigorous formulation of Gauss-Laguerre beams in terms of Mie scattering coefficients which permits us to quasi-analytically treat the interaction of a spherical particle located in the focal region of a possibly high numerical aperture lens illuminated by a Gauss-Laguerre beam. This formalism is used to study the scattered field as a function of the radius of a spherical scatterer, as well as the translation of a spherical scatterer through the Gauss-Laguerre illumination in the focal plane. Knowledge of the Mie coefficients provides a deeper insight to understanding the scattering process and explaining the oscillatory behaviour of the scattered intensity distribution.

© 2007 Optical Society of America

## 1. Introduction

01. M.E.J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N.R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particles”, Phys. Rev. A **54**, 1593–1596 (1996). [CrossRef] [PubMed]

06. P. Monk, *Finite Element Methods for Maxwell’s equations* (Oxford University Press, Oxford, 2003). [CrossRef]

07. G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation”, J. Opt. Soc. Am. A **5**, 1427–1443 (1988). [CrossRef]

09. J.E. Molloy and M.J. Padgett, “Light, action: optical tweezers”, Contemp. Phys. **43**, 241–258 (2002). [CrossRef]

## 2. Theory

*D*

_{{a,b,c,d }}have been placed around the sphere in order to study characteristics of the scattered field. Each detector consists of four segments

*S*

_{{1,2,3,4}}, as shown in Fig. 1b.

### 2.1. Gauss-Laguerre illumination

10. S.M. Barnett and L. Allen, “Orbital angular momentum and non-paraxial light beams”, Opt. Commun. **110**, 670–678 (1994). [CrossRef]

12. A.S. van de Nes, S.F. Pereira, and J.J.M. Braat, “On the conservation of fundamental optical quantities in non-paraxial imaging systems”, J. Mod. Opt. **53**, 677–687 (2006). [CrossRef]

*z*-direction exp[

*ikz*], with in the transversal plane a Gaussian beam profile exp [-ρ

^{2}/

*w*(

*z*)

^{2}] and a helical phase front exp [

*ilϕ*]. The Gauss-Laguerre modes at wavelength γ are fully determined in terms of the mode numbers (

*p,l*) and the Rayleigh range

*z*, the distance from the origin in which the beamwidth increases a factor √2, or equivalently, the Gaussian beamwidth

_{r}*w*(0) = (2

*z*/

_{r}*k*)

^{1/2}with

*k*the wavenumber. An imaging system with numerical aperture NA which obeys Abbe’s sine condition is illuminated by a scalar Gauss-Laguerre mode. The electromagnetic field in the focal region of the imaging system can be written [12

12. A.S. van de Nes, S.F. Pereira, and J.J.M. Braat, “On the conservation of fundamental optical quantities in non-paraxial imaging systems”, J. Mod. Opt. **53**, 677–687 (2006). [CrossRef]

**r**= (ρ,ϕ,

*z*),

*k*= (

_{z}*k*

^{2}-

*k*

^{2}

_{ρ})

^{1/2}and

*J*(

_{n}*x*) are the Bessel functions of the first kind [14]. The freely chosen complex coefficients α and β determine the dominant state of polarisation, where α is associated with oscillation along

**x**̂ and β along

**y**̂. The electric permittivity e and magnetic permeability μ are material properties of the medium in which the beam propagates. The function

*E*(

_{pl,j}*k*

_{ρ}) can also be chosen freely as long as it tends to zero sufficiently quickly to keep the energy associated to the field finite [10

10. S.M. Barnett and L. Allen, “Orbital angular momentum and non-paraxial light beams”, Opt. Commun. **110**, 670–678 (1994). [CrossRef]

*u*

_{pl}^{2}=

*p*!/[(1 + δ

_{0l})π(

*p*+ |

*l*|)!], and

*R*the focal length of the imaging system. The integration domain is bounded by

_{f}*k*∊ [0,

_{p}*k*NA〉. The ratio

*z*/

_{r}*R*determines how the divergence of the beam is scaled before and after the imaging system. In the limit of NA = 0 the solution reduces to the scalar Gauss-Laguerre solution. This type of light beam with a helical phase front carries an amount of orbital angular momentum which is a conserved quantity [15

_{f}15. S.M. Barnett, “Optical angular-momentum flux”, J. Opt. B: Quantum and Semiclass. Opt. **4**, S7–S16 (2002). [CrossRef]

16. L. Allen, S.M. Barnett, and M.J. Padgett, *Optical Angular Momentum* (Institute of Physics Publishing, Bristol, 2003). [CrossRef]

### 2.2. Interaction with a scattering sphere

20. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallüsungen”, Ann. Phys. **330**, 377–445 (1908). [CrossRef]

^{e}(

**r**) and Π

^{h}(

**r**), respectively. Any field distribution is fully determined by the set of modes

*m*and

*n*are integers,

*P*

^{|m|}

_{n}(

*x*) the associated Legendre polynomials [21] and

*j*(

_{n}*x*) = (π/2

*x*)

^{1/2}

*J*

_{n+1/2}(

*x*) the spherical Bessel function of the first kind that needs to be replaced by the spherical Hankel function

*h*(

_{n}*x*) =

*j*(

_{n}*x*) +

*iy*(

_{n}*x*) to describe the scattered field outside the sphere. The radial component of the field distributions is given by

**r**= (

*r*, θ, ϕ) in spherical coordinates.

*a*,

^{e,h}_{nm}*b*and

^{e,h}_{nm}*c*, respectively. The induced field inside and outside the sphere only couples to modes of the incident field with the same mode number

^{e,h}_{nm}*m*,

*r*denotes the radius of the sphere.

_{s}### 2.3. Decomposition of Gauss-Laguerre beams in Mie modes

22. P. Török, P.D. Higdon, R. Juškaitis, and T. Wilson, “Optimising the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers”, Opt. Commun. **155**, 335–341 (1998). [CrossRef]

*S*enclosing the scattering sphere, and the normalisation constants

_{a}*S*,

_{r}*S*and

_{θ}*S*are obtained by integration along that surface for all modes (

_{ϕ}*n,m*). For integration over a spherical shell with radius

*r*, we obtain

_{a}*x*-polarised plane wave propagating in the

*z*-direction

*E*

_{inc,r}= cos ϕ sin θ exp [

*ikr*cos θ], which yields

*j*= 0, but now using a different function

*E*. Replacing cosϕ and sinϕ by the equivalent expressions in terms of the exponential function permits us to collect the terms with the same ϕ and θ dependence. After some straightforward algebra we obtain the radial component of the incident field

_{pl}*z*=

*r*cosθ, ρ =

_{a}*r*sinθ, and the function

_{a}*E*chosen such that it forms the vectorial equivalent of the ‘elegant’ form of the scalar Gauss-Laguerre solution [15

_{pl}15. S.M. Barnett, “Optical angular-momentum flux”, J. Opt. B: Quantum and Semiclass. Opt. **4**, S7–S16 (2002). [CrossRef]

*a*. The integration over ϕ can be performed analytically by effectively replacing the exp [

^{e,h}_{mn}*i*(

*l*± 1)ϕ] terms by 2πδ

_{ml±1}. This yields only non-zero coefficients for two sets of modes with

*m*=

*l*± and

*n*≥

*m*. However, the integration over θ needs to be carried out numerically. Note that an additional numerical integration over

*k*

_{ρ}is required due to the definition of the field. To obtain the coefficients for higher order modes accurately, either the accuracy with which the associated Legendre polynomials are determined has to be high, or alternatively the radius of the integration sphere

*r*has to be large requiring an increased number of points for the integration over θ.

_{a}*j*|)

*k*/

_{z}*k*] term included in the function

*E*(Eq. 3), is taken into account explicitly. After collecting the terms with the same ϕ and θ dependence this yields

_{pl,j}*E*is given by the simplified expression

_{pl}*i*(

*l*± 1)ϕ] with 2πδ

_{m,l±1}, numerically over θ, and numerically over

*k*

_{ρ}due to the focusing of the field.

*a*the decomposition of the field in terms of Mie modes using Eqs. (7) has to be done numerically over both ϕ and θ, also including an additional integration over

^{e,h}_{nm}*k*

_{ρ}as required for focusing the Gauss-Laguerre beam. However, although this integration can be time consuming, the expression yields the incident field distribution in terms of its Mie coefficients, so extending the computations to spheres of different compositions or radii is straightforward.

## 3. Optical model configuration

*p*= 0 and

*l*= 0,1 or 2, denoted by GL

_{00}, GL

_{01}and GL

_{02}, respectively. Although typical distances can be expressed in wavelength-units, the electric permittivity depends on the wavelength and therefore we elect to use λ = 405 nm. In order to have a reasonably large focal field distribution such that any size-dependent effects are clearly separated from each other over a range of radii up to 10 μm, we choose a beam divergence for the imaging system of sin θ

_{d}=

*w*(

*R*)/

_{f}*R*= 0.1 indicated in Fig. 1. This choice corresponds, for example, to an initial beamwidth of

_{f}*w*(

*R*) = 1 mm and lens with focal length

_{f}*R*= 10 mm, resulting in a Rayleigh range of

_{f}*z*= 31.8λ and abeam waist of

_{r}*w*(0)= 3.2λ.Note that the beam waist corresponds to the 1/

*e*-radius of the field strength for the

*l*= 0 beam. As long as the divergence angle of the beam is maintained the focal length and initial beamwidth can be scaled freely in order to match experimental conditions.

*x*-axis. The focal field distribution shown in Fig. 2 forthe three different Gauss-Laguerre modes is obtained using the expression given in Eqs. (1).

*x*-component; and it strongly resembles the scalar Gauss-Laguerre mode. Typically for all three modes the

*y*- and

*z*-components are two and one order of magnitude smaller than the

*x*-component, respectively. The phase distribution is shown as an inset of the various amplitude distributions.

*r*in the focus. The layout of the system is shown in Fig. 1a where a Gauss-Laguerre beam is focused onto the sphere and the scattered field is studied at four different detector planes

_{s}*D*

_{{a,b,c,d}}. Each detector with a 1.0 mm radius is placed 10 mm away from the sphere, and consists of 4 segments defined as

*S*

_{{1,2,3,4}}(Fig. 1b). The location of the detectors are chosen to maximise the obtained information since the contributions of the different Mie-modes can easily be identified. To obtain a high contrast we assume an aluminium sphere with refractive index

*n*

_{al}= 0.503 +

*i*4.923 corresponding to the λ = 405 nm illumination. In the following section we translate the spherical scatterer along the

*x*-axis by a distance varying from –12.5λ to 12.5λ. The number of modes that have to be taken into account depends strongly on the radius of the sphere but due to the fast calculation speed of the model we fixed

*N*

_{max}= 115, which is more than sufficient for the largest radius scatterer. For spheres located in focus we only need to consider the modes with

*m*=

*l*± 1, however for the off-axis spheres all

*m*-modes should be taken into account with -

*n*≤

*m*≤

*n*. Typically, an on-axis sphere requires a calculation time of less than a minute, while an off-axis sphere requires approximately 25 minutes due to the additional numerical integration over ϕ. Note, using a plane-wave decomposition and the Mie-solution for a plane-wave takes approximately 25 minutes as well, but contrary to our modal decomposition, this has to be repeated for each sphere-radius and composition. A typical calculation with a rigorous EM-solver takes several hours and yields a much lower accuracy.

## 4. Spheres in Gauss-Laguerre beams

_{01}illumination is shown. The position of three of the four detectors are indicated by black circles. The detector placed in reflection (

*D*) is not visible because it is located at the south-pole of the sphere.

_{b}### 4.1. Scattering as a function of sphere radius

*S*

_{1}-

*S*

_{2}+

*S*

_{3}-

*S*

_{4}). Again the columns correspond to the detectors placed in transmission, reflection and the transversal plane, and the rows correspond to the different illumination modes GL

_{00}, GL

_{01}and GL

_{02}. The signals are normalised to the total integrated intensity obtained without a scattering sphere for each Gauss-Laguerre mode. The quadrant configuration is sensitive to an elliptical intensity distribution when the axes are not aligned with the detector-segments, and therefore, to any rotation of an elliptical intensity distribution. We have chosen the segments to have a very small tilt with respect to the Cartesian coordinate-system such that the preferential symmetry of the system is broken. Considering the GL

_{00}illumination, we expect no obvious asymmetries in the intensity distribution due to the intrinsic symmetry of the illumination. Note that an initial non-zero value for the transmitted beam is present which corresponds to detecting an ellipticity of the intensity distribution of the illumination. This ellipticity changes as a function of sphere radius as clearly observed in the reflected signal. A rapid change of the major axis along the

*x*- to

*y*-axis, and vice versa, is responsible for the observed oscillations. In the transverse plane this behaviour can also be seen but here the major axis changes with a different oscillation frequency. For higher order Gauss-Laguerre modes the asymmetry of the illumination causes the elliptical intensity distribution to rotate as a function of the sphere radius. However, the oscillation frequency is for all modes almost identical, 77 nm or 0.19λ for the reflected light, and 138 nm or 0.34λ for the light scattered in the transverse plane. Note the difference in strength between the effect of a change in ellipticity (GL

_{00}) and a rotation of the field (GL

_{01},GL

_{02}) due to a choice of an almost zero tilt angle.

*D*and

_{c}*D*. For detectors

_{d}*D*and

_{a}*D*these signals do not provide extra information due to the intrinsic symmetry of the configuration. The differential intensity in split-

_{b}*z*detector configuration (

*S*

_{1}-

*S*

_{2}-

*S*

_{3}+

*S*

_{4}) is shown in Fig. 6 in the left column and in split-

*x*/

*y*detector configuration (

*S*

_{1}+

*S*

_{2}-

*S*

_{3}-

*S*

_{4}) in the right column. The notation split-

*z*indicates that the split detector is used to compare the +

*z*segment with the -

*z*segment. The notation split-

*x*/

*y*indicates an split-

*x*configuration for detector

*D*(TE), and split-

_{d}*y*configuration for detector

*D*(TM). The detected signal with the split-

_{c}*z*detector is considerably stronger than the asymmetry observed with the split-

*x*/

*y*configuration. The oscillation frequency of the rotational features are similar to the 134 nm observed previously.

*P*

^{|m|}

_{n}(cos θ) only contribute for

*n*is odd or

*n*is even depending on the mode number

*m*. Effectively for the particular modes that contribute the frequencies obtained above have to be doubled, which explains the observed oscillation frequency of 0.34λ.

### 4.2. Off-axis illumination

*z*-axis is predominantly determined by the exp [

*ikz*] term due to the small divergence of the beam. The width of the beam at a Rayleigh distance

*z*= 12.9 μm from focus is increased by a factor of 2 as compared to the in focus width. It is relatively straightforward to correct for an increase in width of the illuminating beam provided an estimate of the distance of the sphere to focus is available. Therefore, we only consider spheres moving off-axis in the focal plane. Due to the symmetry of the illumination it is sufficient to study translations along the

_{r}*x*-axis only. The sphere radius has been fixed at which a maximum amount of light is scattered in the

*y*-direction (TE), which is the radius corresponding to the maximum value of the red line in the third column of Fig. 4, i.e.

*r*= 1.279 μm,

_{s}*r*= 1.818 μ and

_{s}*r*= 2.239 μm for GL

_{s}_{00}, GL

_{01}and GL

_{02}, respectively. The choice of sphere radius for a particular Gauss-Laguerre mode provides a very basic match of the scattering pattern, however that does not allow a direct comparison of the obtained signals. The spheres are moved along the

*x*-direction for -12.5λ ≤

*x*≤ 12.5λ. The logarithmic intensity distribution for an on-axis sphere with

*r*= 1.82 μm and GL

_{s}_{01}illumination is shown in Fig. 3.

*x*= ±12.5λ, the detected intensity is, as can be expected, equal to that without a scattering sphere. More surprising is that the strongest signal is observed with the split-

*y*configuration, as opposed to the split-x configuration. The reason for this is that interference of the light scattered by the sphere with the unaffected transmitted beam causes a strong anti-symmetric intensity distribution with respect to the

*y*-axis for the GL

_{01}and GL

_{02}modes.

*x*configuration. However the split-

*y*configuration detects a substantial amount of asymmetry and the difference signal a significant rotation of the intensity distribution when the sphere is moved off-axis. The detector placed in reflection (

*D*) is ideal for probing the shape of the illumination.

_{b}## 5. Conclusion

*p*= 0 and

*l*= 0, or the rotation of a similar beam with higher mode index

*l*. This oscillation was explained by a careful study of the Mie coefficients. A translation of a spherical scatterer with a fixed radius through the light distribution in the focal plane gives more insight to the behaviour of the detector signals. Beside a dramatic improvement in speed compared to more general solvers, our solution also provides additional information in terms of the contribution of individual Mie coefficients resulting in a deeper physical insight of the scattering process.

## Acknowledgements

## References and links

01. | M.E.J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N.R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particles”, Phys. Rev. A |

02. | J. Tempere, J.T. Devreese, and E.R.I. Abraham, “Vortices in Bose-Einstein condensates confined in a multiply connected Laguerre-Gaussian optical trap”, Phys. Rev. A |

03. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of orbital angular momentum states of photons”, Nature (London) , |

04. | K. OߣHolleran, M.R. Dennis, and M.J. Padgett, “Illustrations of optical vortices in three dimensions”, J. Europ. Opt. Soc. Rap. Public. |

05. | A. Taflove, |

06. | P. Monk, |

07. | G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation”, J. Opt. Soc. Am. A |

08. | C.F. Bohren and D.R. Huffman, |

09. | J.E. Molloy and M.J. Padgett, “Light, action: optical tweezers”, Contemp. Phys. |

10. | S.M. Barnett and L. Allen, “Orbital angular momentum and non-paraxial light beams”, Opt. Commun. |

11. | P. Török and P.R.T. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy”, Opt. Express |

12. | A.S. van de Nes, S.F. Pereira, and J.J.M. Braat, “On the conservation of fundamental optical quantities in non-paraxial imaging systems”, J. Mod. Opt. |

13. | A.E. Siegman, |

14. | G.N. Watson, |

15. | S.M. Barnett, “Optical angular-momentum flux”, J. Opt. B: Quantum and Semiclass. Opt. |

16. | L. Allen, S.M. Barnett, and M.J. Padgett, |

17. | A. Stratton, |

18. | P.M. Morse and H. Feshbach, |

19. | H.C. van de Hulst, |

20. | G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallüsungen”, Ann. Phys. |

21. | M. Abramowitz and I.A. Stegun, |

22. | P. Török, P.D. Higdon, R. Juškaitis, and T. Wilson, “Optimising the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers”, Opt. Commun. |

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

**OCIS Codes**

(260.1960) Physical optics : Diffraction theory

(260.5740) Physical optics : Resonance

(290.4020) Scattering : Mie theory

**ToC Category:**

Scattering

**History**

Original Manuscript: July 25, 2007

Revised Manuscript: September 9, 2007

Manuscript Accepted: September 27, 2007

Published: September 28, 2007

**Virtual Issues**

Vol. 2, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

A. S. van de Nes and P. Török, "Rigorous analysis of spheres in Gauss-Laguerre beams," Opt. Express **15**, 13360-13374 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-20-13360

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### References

- M.E.J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N.R. Heckenberg, "Optical angular-momentum transfer to trapped absorbing particles", Phys. Rev. A 54, 1593-1596 (1996). [CrossRef] [PubMed]
- J. Tempere, J.T. Devreese and E.R.I. Abraham, "Vortices in Bose-Einstein condensates confined in a multiply connected Laguerre-Gaussian optical trap", Phys. Rev. A 64, 023603 (2001) [CrossRef]
- A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of orbital angular momentum states of photons", Nature (London), 412, 3123-3316 (2001). [CrossRef]
- K. O’Holleran, M.R. Dennis and M.J. Padgett, "Illustrations of optical vortices in three dimensions", J. Europ. Opt. Soc. Rap. Public. 1, 06008 (2006).Q1 [CrossRef]
- A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 1998).
- P. Monk, Finite Element Methods for Maxwell’s equations (Oxford University Press, Oxford, 2003). [CrossRef]
- G. Gouesbet, B. Maheu and G. Gr’ehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation", J. Opt. Soc. Am. A 5, 1427-1443 (1988). [CrossRef]
- C.F. Bohren and D.R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, 1983).
- J.E. Molloy and M.J. Padgett, "Light, action: optical tweezers", Contemp. Phys. 43, 241-258 (2002). [CrossRef]
- S.M. Barnett and L. Allen, "Orbital angular momentum and non-paraxial light beams", Opt. Commun. 110, 670-678 (1994). [CrossRef]
- P. Török and P.R.T. Munro, "The use of Gauss-Laguerre vector beams in STED microscopy", Opt. Express 12, 3605-3617 (2004).Q2 [CrossRef] [PubMed]
- A.S. van de Nes, S.F. Pereira and J.J.M. Braat, "On the conservation of fundamental optical quantities in nonparaxial imaging systems", J. Mod. Opt. 53, 677-687 (2006). [CrossRef]
- A.E. Siegman, Lasers (University Science Books, Sausalito, CA, 1986).
- G.N. Watson, A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1966).
- S.M. Barnett, "Optical angular-momentum flux", J. Opt. B: Quantum and Semiclass. Opt. 4, S7-S16 (2002).Q3 [CrossRef]
- L. Allen, S.M. Barnett and M.J. Padgett, Optical Angular Momentum (Institute of Physics Publishing, Bristol, 2003). [CrossRef]
- A. Stratton, Electromagnetic Theory (McGraw-Hill book company, Inc., New York, 1941).
- P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill book company, Inc., New York, 1953).
- H.C. van de Hulst, Light scattering by small particles (Dover publications, New York, 1981).
- G. Mie, "Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen", Ann. Phys. 330, 377-445 (1908). [CrossRef]
- M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970).
- P. Török, P.D. Higdon, R. Ju¡skaitis and T.Wilson, "Optimising the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers", Opt. Commun. 155, 335-341 (1998).Q4 [CrossRef]
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).

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