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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 20 — Oct. 1, 2007
  • pp: 13360–13374
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Rigorous analysis of spheres in Gauss-Laguerre beams

A.S. van de Nes and P. Torok  »View Author Affiliations


Optics Express, Vol. 15, Issue 20, pp. 13360-13374 (2007)
http://dx.doi.org/10.1364/OE.15.013360


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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

In recent years the experimental use and importance of Gauss-Laguerre beams have increased significantly [1–4

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]

]. The main benefit of Gauss-Laguerre beams is the helical phase front which allows for the transfer of angular momentum to the illuminated object, or alternatively a sensitive detection of small features due to its inherent differential field distribution. Although numerical tools, such as FDTD [5

05. A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 1998).

] and FEM [6

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

], exist to calculate the interaction of these beams with small objects rigorously, these techniques are often time consuming and do not always help in obtaining a physical understanding of the scattering process. A good alternative has been considered in Ref. [7

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]

] to treat the interaction of a spherical scatterer with the first order approximation of a vectorial Gaussian beam quasi-analytically.

In this paper we derive the expressions for fully vectorial, possibly focused, Gauss-Laguerre beams in terms of Mie modes. We also study the interaction of aluminium spheres of various sizes with the low order Gauss-Laguerre beams. We obtain the intensity distribution for spheres translated in the focal plane. The model presented here can for example be applied in the fields of optical detection and characterisation of small particles [8

08. C.F. Bohren and D.R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, 1983).

], or the manipulation of small particles using optical tweezers [9

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

].

2. Theory

In order to develop a rigorous model for calculating the electromagnetic field scattered by a spherical particle illuminated with a, possibly focused, Gauss-Laguerre beam we first briefly recall theories pertinent to both Gauss-Laguerre beams and Mie scattering. Therefore, we start in the first subsection with a discussion of the electromagnetic field distribution for a focused Gauss-Laguerre beam. This is followed by a subsection discussing Mie’s solution for light scattered by a sphere. As one of the major results of this paper we decompose a Gauss-Laguerre beam in terms of Mie modes which is discussed in the last subsection.

A schematic representation of our layout is shown in Fig. 1a, where the spherical scatterer, located in the focal plane and initially on the optic axis, is illuminated by a Gauss-Laguerre beam. Four detectors 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

We use the vectorial equivalent [10–12

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

] of the scalar Gauss-Laguerre beam [13

13. A.E. Siegman, Lasers (University Science Books, Sausalito, CA,1986).

] to illuminate the scatterer. We distinguish the unfocused from the focused vectorial Gauss-Laguerre beam, where the latter has been transformed by an imaging system. The description of the focused Gauss-Laguerre illumination given in this subsection is chiefly based on Ref. [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]

] and extends the formalism to the magnetic field.

The scalar Gauss-Laguerre modes [13

13. A.E. Siegman, Lasers (University Science Books, Sausalito, CA,1986).

] consist predominantly of a term describing propagation in the z-direction exp[ikz], with in the transversal plane a Gaussian beam profile exp [-ρ2/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 zr, the distance from the origin in which the beamwidth increases a factor √2, or equivalently, the Gaussian beamwidth w(0) = (2zr/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]

] as a linear combination of three eigenmodes of the vectorial Helmholtz equation,

Fig. 1. (a) Schematic of the optical system using a Gauss-Laguerre beam for illumination. The sphere is initially placed in focus but is later allowed to be translated along the x-axis. Four detectors are located around the sphere: detector Da measures the transmitted light, detector Db the reflected light, detector Dc the light reflected along the x-direction and detector Dd the light reflected along the y-direction. (b) Each detector consists of four segments with axes x 1 = x and x 2 = y for Da and Db, x 1 = z and x 2 = y for Dc, and x 1 = z and x 2 = x for Dd.
E(r)=12Epl,0(r;α,β)+14Epl,2(r;α+,β)+14Epl,2(r;αβ),
(1a)
H(r)=12Hpl,0(r;α,β)+14Hpl,2(r;α+,β)+14Hpl,2(r;αβ),
(1b)

with, using cylindrical coordinates r = (ρ,ϕ,z),

Epl,j(r;α,β)=0kNAEpl,j(kρ)ei(l+j)ϕ+ikzz((αx̂+βŷ)Jl+j(kρρ)+kρ2kzẑ×[(+β)eJl+j1(kρρ)(+β)eJl+j+1(kρρ)])dkρ,
(2a)
Hpl,j(r;α,β)=εμ0kNAEpl,j(kρ)2kkzei(l+j)ϕ+ikzz((βx̂+αŷ)[2k2kρ2]Jl+j(kρρ)+kρ22[(x̂+iŷ)(β)Jl+j2(kρρ)e2(x̂iŷ)(+β)Jl+j+2(kρρ)e2]kρkzẑ[(α+)Jl+j1(kρρ)e+(α)Jl+j+1(kρρ)e])dkρ,
(2b)

where kz = (k 2 - k 2ρ)1/2 and Jn(x) are the Bessel functions of the first kind [14

14. G.N. Watson, A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1966).

]. The freely chosen complex coefficients α and β determine the dominant state of polarisation, where α is associated with oscillation along x̂ and β along ŷ. The electric permittivity e and magnetic permeability μ are material properties of the medium in which the beam propagates. The function Epl,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]

], allowing us to write

Epl,j(kρ)=(1)pil1uplkρRfkzzr(1+(1j)kzk)(kρ2Rf2kzr)l2Lpl(kρ2Rf2kzr)exp[kρ2Rf22kzr],
(3)

with upl 2 =p!/[(1 + δ0l)π(p+ |l|)!], and Rf the focal length of the imaging system. The integration domain is bounded by kp ∊ [0, kNA〉. The ratio zr/Rf 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

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

,16

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

In a homogeneous medium it is possible to solve Maxwell’s equations analytically for a few well-known configurations with a particular shaped scattering object [17–19

17. A. Stratton, Electromagnetic Theory (McGraw-Hill book company, Inc., New York, 1941).

]. For a spherical scatterer, a separation of variables yields an analytical solution in terms of the Debye potentials, referred to as the Mie theory [20

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

].

The electromagnetic field can be resolved as an electric field component tangential to the surface of the sphere (TE) and a tangential magnetic field (TM) which can be described in terms of the Debye potentials Πe(r) and Πh(r), respectively. Any field distribution is fully determined by the set of modes

rΠe,h(r)=n=0m=nnanme,hrjn(kr)Pnm(cosθ)eimϕ,
(4)

where m and n are integers, P |m| n(x) the associated Legendre polynomials [21

21. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970).

] and jn(x) = (π/2x)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 hn(x) = jn(x) + iyn(x) to describe the scattered field outside the sphere. The radial component of the field distributions is given by

Er(r)=n(n+1)r2rΠh=Ex(r)cosϕsinθ+Ey(r)sinϕsinθ+Ez(r)cosθ,
(5a)
Hr(r)=n(n+1)r2rΠe=Hx(r)cosϕsinθ+Hy(r)sinϕsinθ+Hz(r)cosθ,
(5b)

where θ is the angle between the positive z-axis and the position vector r = (r, θ, ϕ) in spherical coordinates.

To obtain a solution which satisfies Maxwell’s equations the boundary conditions have to be matched for the incident and scattered field outside the sphere with the field inside the sphere, given in terms of the coefficients ae,hnm, be,hnm and ce,hnm, respectively. The induced field inside and outside the sphere only couples to modes of the incident field with the same mode number m,

cnme=iμ2(k1rs)μ1hn(k1rs)[k2rsjn1(k2rs)njn(k2rs)]μ2jn(k2rs)[k1rshn1(k1rs)nhn(k1rs)]anme,
(6a)
bnme=μ2jn(k2rs)[k1rsjn1(k1rs)njn(k1rs)]μ1jn(k1rs)[k2rsjn1(k2rs)njn(k2rs)]μ1hn(k1rs)[k2rsjn1(k2rs)njn(k2rs)]μ2jn(k2rs)[k1rshn1(k1rs)nhn(k1rs)]anme,
(6b)
cnmh=iμ2ε2(μ1k2rs)ε1hn(k1rs)[k2rsjn1(k2rs)njn(k2rs)]ε2jn(k2rs)[k1rshn1(k1rs)nhn(k1rs)]anmh,
(6c)
bnmh=ε2jn(k2rs)[k1rsjn1(k1rs)njn(k1rs)]ε1jn(k1rs)[k2rsjn1(k2rs)njn(k2rs)]ε1hn(k1rs)[k2rsjn1(k2rs)njn(k2rs)]ε2jn(k2rs)[k1rshn1(k1rs)nhn(k1rs)]anmh,
(6d)

where material parameters outside the sphere are indicated with the subscript 1 and inside the sphere with the subscript 2, and rs denotes the radius of the sphere.

2.3. Decomposition of Gauss-Laguerre beams in Mie modes

The Mie theory discussed above is of general validity and provides an analytic solution for the electromagnetic field in terms of a sum over an infinite number of modes. However, apart from cases with natural symmetry [22

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]

], the incident illumination cannot analytically be written in terms of the Mie modes. To obtain the coefficients corresponding to a general incident illumination, the inner-product of the right-hand side of Eqs. (5) with the basis-functions Eq. (4) has to be taken, resulting in

anmh=1SrSθSϕSaEinc,rrjn(kr)Pnm(cosθ)eimϕ,
(7a)
anme=1SrSθSϕSaHinc,rrjn(kr)Pnm(cosθ)eimϕ,
(7b)

where dσ is a surface element on the surface Sa enclosing the scattering sphere, and the normalisation constants Sr, Sθ and Sϕ are obtained by integration along that surface for all modes (n,m). For integration over a spherical shell with radius ra, we obtain

Sr=n(n+1)rajn(kra),Sθ=2(n+m)!(2n+1)(nm)!,Sϕ=2π.
(8)

Decomposing the illumination in Mie modes for a plane wave can be done analytically [23

23. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).

]. Without loss of generality we can assume an x-polarised plane wave propagating in the z-direction E inc,r = cos ϕ sin θ exp [ikr cos θ], which yields

anmh={in+1(2n+1)2kn(n+1)m=±10m±1,anme={i(m1)εμin+1(2n+1)2ikn(n+1)m=±10m±1.
(9)

For an unfocused vectorial Gauss-Laguerre beam, we substitute the field given by Eqs. (2) in Eqs. (5) with j = 0, but now using a different function Epl. 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

Er=0kEpl(kρ)2eilϕ+ikzz([(α+)e+(α)e]Jl(kρρ)sinθ+kρkz[(β)eJl1(kρρ)(+β)eJl+1(kρρ)]cosθ)dkρ,
(10a)
Hr=εμ0kEpl(kρ)2kkzeilϕ+ikzz[12((2k2kρ2)[(β)e(+β)e]Jl(kρρ)+kρ2[(+β)Jl2(kρρ)e(+β)Jl+2(kρρ)e])sinθkρkz[(α)eJl1(kρρ)+(α)eJl+1(kρρ)]cosθ]dkρ,
(10b)

with z = ra cosθ, ρ = ra sinθ, and the function Epl chosen such that it forms the vectorial equivalent of the ‘elegant’ form of the scalar Gauss-Laguerre solution [15

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

],

Epl(kρ)=(1)p(kzr2)(p+l+1)2kkz(kρ2k2kρ2)(2p+l+1)2exp[kzrkρ22(k2kρ2)].
(11)

This expression can be substituted into Eqs. (7) to obtain an expression for the incident field in terms of its coefficients ae,hmn. The integration over ϕ can be performed analytically by effectively replacing the exp [i(l ± 1)ϕ] terms by 2πδml±1. This yields only non-zero coefficients for two sets of modes with m = l ± and nm. 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 ra has to be large requiring an increased number of points for the integration over θ.

For the focused vectorial Gauss-Laguerre beam, we apply the same steps as above to obtain expressions for the radial component of the incident field but now we use all three terms in Eqs. (1). The dependence on the [1 + (1 - |j|)kz/k] term included in the function Epl,j (Eq. 3), is taken into account explicitly. After collecting the terms with the same ϕ and θ dependence this yields

Er=0kNAEpl(kρ)eilϕ+ikzz(12[(1+kzk)[(α+)e+(α+)e]Jl(kρρ)+(1+kzk)[(α+)eJl2(kρρ)+(α+)eJl+2(kρρ)]]sinθ+kρk[(+β)eJl2(kρρ)(+)eJl+1(kρρ)]cosθ)dkρ,
(12a)
Hr=εμ0kNAEpl(kρ)kkzeilϕ+ikzz[12((k2kρ2+kkz)[(α+)e(+β)e]Jl(kρρ)(k2kρ2+kkz)[(αβ)Jl2(kρρ)e(+)Jl+2(kρρ)e])sinθkρkz[(α+)eJl2(kρρ)+(α+)eJl+1(kρρ)]cosθ)dkρ,
(12b)

where the function Epl is given by the simplified expression

Epl(kρ)=(1)pil1uplkρRfkzzr(kρ2Rf2kzr)l2Lpl(kρ2Rf2kzr)exp[kρ2Rf22kzr].
(13)

Again, we substitute Eqs. (12) in Eqs. (7). Similarly, the integration over ϕ can be done analytically, by replacing exp [i(l ± 1)ϕ] with 2πδm,l±1, numerically over θ, and numerically over k ρ due to the focusing of the field.

Finally, when the focused beam has an offset with respect to the coordinate-system of the sphere, the ϕ coordinate of the sphere defined in Eqs. (5) is no longer equal to ϕ′ in Eqs. (2). The radial components of the electric and magnetic field are given by

Er=0kNAEpl(kρ)eilϕ+ikzz(12[(1+kzk)[(α+)e+(α+)e]Jl(kρρ)+(1+kzk)[(α+)eJl2(kρρ)+(α+)ei(2ϕϕ)Jl+2(kρρ)]]sinθ+kρk[(+β)eJl1(kρρ)(+β)eJl+1(kρρ)]cosθ)dkρ,
(14a)
Hr=εμ0kNAEpl(kρ)kkzeilϕ+ikzz[12((k2kρ2+kkz)[(+β)e(+β)e]Jl(kρρ)(k2kρ2+kkz)[(β)Jl2(kρρ)ei(2ϕϕ)(+β)Jl+2(kρρ)ei(2ϕϕ)])sinθkρkz[(α+)eJl2(kρρ)+(α+)eJl+1(kρρ)]cosθ)dkρ,
(14b)

with the same definition for Epl(k ρ) as Eq. (13). In addition we now have

ρ'=[(rasinθcosϕxoff)2+(rasinθsinϕyoff)2]12,
(15a)
tanϕ=rasinθsinϕyoffrasinθcosϕxoff.
(15b)

To obtain the coefficients ae,hnm 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 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

Using the tools described above we now study the field scattered by spherical particles when illuminated by various modes of a focused Gauss-Laguerre beam. We have considered three different modes of these beams choosing p = 0 and l = 0,1 or 2, denoted by GL00, GL01 and GL02, 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 θd = w(Rf)/Rf = 0.1 indicated in Fig. 1. This choice corresponds, for example, to an initial beamwidth of w(Rf) = 1 mm and lens with focal length Rf= 10 mm, resulting in a Rayleigh range of zr = 31.8λ and abeam waist of 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.

The polarisation state of the illumination, due to the symmetry of the configuration, has been chosen, without loss of generality, along the 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).

The dominant contribution is from the 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.

Fig. 2. The x-, y- and z-component of the electric field distribution in the focal plane for a Gauss-Laguerre beam with l = 0,1 and 2 are depicted in the first, second and last row, respectively. The phase is shown as an inset in each figure, where the colour scale changes linearly from blue to red corresponding to [-π, π⟩.

4. Spheres in Gauss-Laguerre beams

In Fig. 3 the logarithmic intensity distribution corresponding to a spherical scatterer with a radius of 1.82 μm and GL01 illumination is shown. The position of three of the four detectors are indicated by black circles. The detector placed in reflection (Db) is not visible because it is located at the south-pole of the sphere.

Fig. 3. Logarithmic intensity distribution due to a spherical particle of 1.82 μm radius, illuminated by a focused p = 0, l = 1 Gauss-Laguerre beam. Indicated by black circles are the detector for transmission Da, and the transversal detectors Dc and Dd corresponding to TM and TE, respectively.

4.1. Scattering as a function of sphere radius

Fig. 4. (1st col.) The total intensity as a function of sphere radius for the detector (Da) placed in transmission, (2nd col.) placed in reflection (Db), and (3rd col.) placed in the transversal plane with the blue line for TM (Dc) and red for TE (Dd). Each row corresponds with illumination of increasing order, GL00, GL01 and GL02, respectively.

Additional information on the spheres can be obtained from studying different combinations of the four segments of the detectors. Fig. 5 shows the differential intensity in a quadrant-detector configuration (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 GL00, GL01 and GL02. 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 GL00 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 (GL00) and a rotation of the field (GL01,GL02) due to a choice of an almost zero tilt angle.

Fig. 5. (1st col.) Difference intensity as a function of sphere radius for the detector (Da) placed in transmission, (2nd col.) placed in reflection (Db), and (3rd col.) placed in the transversal plane with the blue line for TM (Dc) and red for TE (Dd). Each row corresponds with illumination of increasing order, GL00, GL01 and GL02, respectively.

The split detector configuration yields additional information from detectors Dc and Dd. For detectors Da and Db these signals do not provide extra information due to the intrinsic symmetry of the configuration. The differential intensity in split-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 Dd (TE), and split-y configuration for detector Dc (TM). The detected signal with the split-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.

Fig. 6. Split-z (left) and split-x/y (right) intensity as a function of sphere radius for detector placed in the transversal plane with the blue line for TM (Dc) and red for TE (Dd). Each row corresponds to illumination with increasing order, GL00, GL01 and GL02, respectively.

To explain the oscillation frequency in the transversal plane we have to consider the actual shape of the contributing modes. The location of both detectors is in the same plane with θ = π/2. However, the associated Legendre polynomials 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.

Fig. 7. Absolute value of the bhnm coefficients for m = l -1 (left) and m = l + 1 (right) as a function of the sphere radius rs and the mode number n, obtained by illumination with the three Gauss-Laguerre beams GL00, GL01 and GL02.
Fig. 8. (a) Absolute value of the bhnm coefficients for m = ±1 of the plane wave illumination as a function of the sphere radius rs and the mode number n. (b) Absolute value of the bnm coefficient with n = 15 and m = 0 for GL01 illumination as a function of sphere radius, with blue corresponding to the TM coefficient bh and red to the TE coefficient be.

Note that increasing the beam divergence results in a stronger apodisation, i.e. less relevant modes, and therefore more pronounced oscillations occur

4.2. Off-axis illumination

In this section we discuss the signal at the four detectors when the scattering sphere is moved off the optic axis. The amplitude and phase of the doughnut beam along the 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 zr = 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 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. rs = 1.279 μm, rs = 1.818 μ and rs = 2.239 μm for GL00, GL01 and GL02, 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 rs = 1.82 μm and GL01 illumination is shown in Fig. 3.

In Fig. 9 the transmitted intensity is shown for the four different detector configurations. The signal has been normalised to the transmitted integrated intensity when there is no scattering sphere. When the sphere is positioned outside the illumination at 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 GL01 and GL02 modes.

Fig. 9. Transmitted integrated intensity for three different Gauss-Laguerre modes, GL00 blue, GL01 red and GL02 green. (a) The sum signal (b) the difference signal, (c) the split-x configuration and (d) the split-y configuration.

The reflected signals are shown in Fig. 10 for the four different detector configurations. The signal closely follows the maximum intensity of the illuminating Gauss-Laguerre beam. For higher order helical-phase distributions we observe additional oscillations due to a more complicated interference pattern. The dominant effect occurs now for the split-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 (Db) is ideal for probing the shape of the illumination.

In Fig. 11 the detected signal in the transversal plane is shown. The signal scattered in the direction of oscillation is depicted in the top row figures, and the signal scattered in the orthogonal direction is shown in the bottom row figures. The maximum amplitude is approximately equal compared to the maximum amplitude of the reflected signal. Interestingly, the sum signal for TE does not probe the field with a high enough resolution to notice the doughnut rings due to the size of the sphere.

Fig. 10. Reflected integrated intensity for three different Gauss-Laguerre modes, GL00 blue, GL01 red and GL02 green. (a) The sum signal (b) the difference signal, (c) the split-x configuration and (d) the split-y configuration.
Fig. 11. Integrated intensity scattered to the transversal plane for three different Gauss-Laguerre modes, GL00 blue, GL01 red and GL02 green. The top row corresponds to TM and the bottom to TE. (a) The sum signal (b) the difference signal, (c) the split-z configuration and (d) the split-y configuration for TM and split-x configuration for TE.

5. Conclusion

We have derived an expression for the field distribution of a, possibly focused, Gauss-Laguerre beam in terms of Mie modes. The usefulness of these expressions have been demonstrated by a thorough study of the light scattered by an aluminium sphere, illuminated by three modes of the Gauss-Laguerre beam. The intensity distribution has been obtained for four segments of the 4π solid angle as a function of the radius of the scattering sphere. The resulting detector signal exhibits oscillatory behaviour due to either the ellipticity of the Gauss-Laguerre beam with 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

This work is supported by the European Union within the 6th Framework Programme as part of NANOPRIM (contract number: NMP3-CT-2007-033310).

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 54, 1593–1596 (1996). [CrossRef] [PubMed]

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 64, 023603 (2001) [CrossRef]

03.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of orbital angular momentum states of photons”, Nature (London) , 412, 3123–3316 (2001). [CrossRef]

04.

K. OߣHolleran, M.R. Dennis, and M.J. Padgett, “Illustrations of optical vortices in three dimensions”, J. Europ. Opt. Soc. Rap. Public. 1, 06008 (2006). [CrossRef]

05.

A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 1998).

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]

08.

C.F. Bohren and D.R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, 1983).

09.

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

10.

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

11.

P. Török and P.R.T. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy”, Opt. Express 12, 3605–3617 (2004). [CrossRef] [PubMed]

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]

13.

A.E. Siegman, Lasers (University Science Books, Sausalito, CA,1986).

14.

G.N. Watson, A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1966).

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]

17.

A. Stratton, Electromagnetic Theory (McGraw-Hill book company, Inc., New York, 1941).

18.

P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill book company, Inc., New York, 1953).

19.

H.C. van de Hulst, Light scattering by small particles (Dover publications, New York, 1981).

20.

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

21.

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970).

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]

23.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).

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/oe/abstract.cfm?URI=oe-15-20-13360


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References

  1. 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]
  2. 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]
  3. A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of orbital angular momentum states of photons", Nature (London),  412, 3123-3316 (2001). [CrossRef]
  4. 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]
  5. A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 1998).
  6. P. Monk, Finite Element Methods for Maxwell’s equations (Oxford University Press, Oxford, 2003). [CrossRef]
  7. 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]
  8. C.F. Bohren and D.R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, 1983).
  9. J.E. Molloy and M.J. Padgett, "Light, action: optical tweezers", Contemp. Phys. 43, 241-258 (2002). [CrossRef]
  10. S.M. Barnett and L. Allen, "Orbital angular momentum and non-paraxial light beams", Opt. Commun. 110, 670-678 (1994). [CrossRef]
  11. 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]
  12. 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]
  13. A.E. Siegman, Lasers (University Science Books, Sausalito, CA, 1986).
  14. G.N. Watson, A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1966).
  15. S.M. Barnett, "Optical angular-momentum flux", J. Opt. B: Quantum and Semiclass. Opt. 4, S7-S16 (2002).Q3 [CrossRef]
  16. L. Allen, S.M. Barnett and M.J. Padgett, Optical Angular Momentum (Institute of Physics Publishing, Bristol, 2003). [CrossRef]
  17. A. Stratton, Electromagnetic Theory (McGraw-Hill book company, Inc., New York, 1941).
  18. P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill book company, Inc., New York, 1953).
  19. H.C. van de Hulst, Light scattering by small particles (Dover publications, New York, 1981).
  20. G. Mie, "Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen", Ann. Phys. 330, 377-445 (1908). [CrossRef]
  21. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970).
  22. 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]
  23. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).

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