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Rigorous analysis of spheres in

Gauss-Laguerre beams

A.S. van de Nes and P. T¨ or¨ ok

Department of Physics, Imperial College London

Prince Consort Road, London SW7 2BW, UK.

arthur.vandenes@imperial.ac.uk

Abstract:

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.

In this paper we develop a rigorous formulation of Gauss-

© 2007 Optical Society of America

OCIS codes: (290.4020) Mie theory; (260.1960) Diffraction theory; (260.5740) Resonance.

References and links

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).

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)

3. A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, “Entanglement of orbital angular momentum states of photons”,

Nature (London), 412, 3123–3316 (2001).

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).

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).

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).

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).

10. S.M. Barnett and L. Allen, “Orbital angular momentum and non-paraxial light beams”, Opt. Commun. 110,

670–678 (1994).

11. P. T¨ or¨ ok and P.R.T. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy”, Opt. Express 12,

3605–3617 (2004).

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).

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).

16. L. Allen, S.M. Barnett and M.J. Padgett, Optical Angular Momentum (Institute of Physics Publishing, Bristol,

2003).

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).

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1 October 2007 / Vol. 15, No. 20 / OPTICS EXPRESS 13360

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

20. G.Mie, “Beitr¨ age zur Optik tr¨ uber Medien, speziell kolloidaler Metall¨ osungen”, Ann.Phys.330, 377-445 (1908).

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

22. P. T¨ or¨ ok, P.D. Higdon, R. Juˇ 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).

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

1.Introduction

In recent years the experimental use and importance of Gauss-Laguerre beams have increased

significantly [1–4]. 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 sen-

sitive detection of small features due to its inherent differential field distribution. Although

numerical tools, such as FDTD [5] and FEM [6], 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 beenconsideredin Ref. [7]to treat the interactionofa sphericalscatterer 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 orderGauss-Laguerrebeams. We obtainthe 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], or the manipulation of small

particles using optical tweezers [9].

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 oneof the major results of this paperwe decomposea 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 charac-

teristics 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] of the scalar Gauss-Laguerre beam [13] 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] and extends

the formalism to the magnetic field.

The scalar Gauss-Laguerre modes [13] consist predominantly of a term describing prop-

agation 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

λ 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

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x

z

Dc

Db

Da

10 mm

1 mm

sin θd = 0.1

λ = 405 nm

Ex,GL

S1

S4

S2

S3

(a)(b)

Dd

D{a,b,c,d}

x1

x2

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 Dameasures the transmitted

light, detector Dbthe reflected light, detector Dcthe light reflected along the x-direction

and detector Ddthe light reflected along the y-direction. (b) Each detector consists of four

segments with axes x1= x and x2= y for Daand Db, x1= z and x2= y for Dc, and x1= z

and x2= x for Dd.

Gaussian beamwidth w(0) = (2zr/k)1/2with 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] as a linear combination of three eigenmodes of the vectorial Helmholtz equation,

E(r) =1

2Epl,0(r;α,β)+1

H(r) =1

4Epl,−2(r;α +iβ,iα −β)+1

4Hpl,−2(r;α +iβ,iα −β)+1

4Epl,2(r;α −iβ,−iα −β) ,

4Hpl,2(r;α −iβ,−iα −β) ,

(1a)

2Hpl,0(r;α,β)+1

(1b)

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

Epl,j(r;α,β) =

?kNA

×?(iα −β)e−iφJl+j−1(kρρ)−(iα +β)eiφJl+j+1(kρρ)??

?ε

+k2

2

−kρkzˆ z?(α +iβ)Jl+j−1(kρρ)e−iφ+(α −iβ)Jl+j+1(kρρ)eiφ??

where kz= (k2−k2

chosen complex coefficients α and β determine the dominant state of polarisation, where α is

associated with oscillation along ˆ x and β along ˆ y. The electric permittivity ε and magnetic per-

meability μ are material properties of the medium in which the beam propagates. The function

Epl,j(kρ) can also be chosen freely as long as it tends to zero sufficiently quickly to keep the

0

Epl,j(kρ)ei(l+j)φ+ikzz

?

(αˆ x+β ˆ y)Jl+j(kρρ)+kρ

2kzˆ z

dkρ,

(2a)

Hpl,j(r;α,β) =

μ

?(ˆ x+iˆ y)(iα −β)Jl+j−2(kρρ)e−2iφ−(ˆ x−iˆ y)(iα +β)Jl+j+2(kρρ)e2iφ?

?kNA

0

Epl,j(kρ)

2kkz

ei(l+j)φ+ikzz

?

(−βˆ x+αˆ y)

?

2k2−k2

ρ

?

Jl+j(kρρ)

ρ

dkρ,

(2b)

ρ)1/2and Jn(x) are the Bessel functions of the first kind [14]. The freely

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energy associated to the field finite [10], allowing us to write

Epl,j(kρ) = (−1)pil−1upl

kρRf

√kzzr

?

1+(1−|j|)kz

k

??k2

ρR2

kzr

f

?|l|/2

L|l|

p

?k2

ρR2

kzr

f

?

exp

?

−k2

ρR2

2kzr

f

?

(3)

,

with u2

tion domainis boundedby kρ∈ [0,kNA?. The ratio zr/Rfdetermineshow the divergenceof 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 [15,16].

pl= p!/[(1+δ0l)π(p+|l|)!], and Rfthe focal length of the imaging system. The integra-

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]. For a spherical

scatterer, a separationof variablesyields an analytical solutionin terms ofthe Debyepotentials,

referred to as the Mie theory [20].

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=0

n

∑

m=−n

ae,h

nmrjn(kr)P|m|

n (cosθ)eimφ,

(4)

where m and n are integers, P|m|

(π/2x)1/2Jn+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

n (x) the associated Legendre polynomials [21] and jn(x) =

Er(r) =n(n+1)

r2

rΠh= Ex(r)cosφ sinθ +Ey(r)sinφ sinθ +Ez(r)cosθ ,

(5a)

Hr(r) =n(n+1)

r2

rΠe= Hx(r)cosφ sinθ +Hy(r)sinφ sinθ +Hz(r)cosθ ,

(5b)

whereθ is the anglebetweenthe positivez-axisandthe positionvectorr=(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,h

outside the sphere only couples to modes of the incident field with the same mode number m,

nm, be,h

nmand ce,h

nm, respectively. The induced field inside and

ce

nm=

−iμ2/(k1rs)

μ1hn(k1rs)[k2rsjn−1(k2rs)−njn(k2rs)]−μ2jn(k2rs)[k1rshn−1(k1rs)−nhn(k1rs)]ae

nm,

(6a)

be

nm=μ2jn(k2rs)[k1rsjn−1(k1rs)−njn(k1rs)]−μ1jn(k1rs)[k2rsjn−1(k2rs)−njn(k2rs)]

μ1hn(k1rs)[k2rsjn−1(k2rs)−njn(k2rs)]−μ2jn(k2rs)[k1rshn−1(k1rs)−nhn(k1rs)]ae

nm,

(6b)

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ch

nm=

−iμ2ε2/(μ1k2rs)

ε1hn(k1rs)[k2rsjn−1(k2rs)−njn(k2rs)]−ε2jn(k2rs)[k1rshn−1(k1rs)−nhn(k1rs)]ah

nm,

(6c)

bh

nm=ε2jn(k2rs)[k1rsjn−1(k1rs)−njn(k1rs)]−ε1jn(k1rs)[k2rsjn−1(k2rs)−njn(k2rs)]

ε1hn(k1rs)[k2rsjn−1(k2rs)−njn(k2rs)]−ε2jn(k2rs)[k1rshn−1(k1rs)−nhn(k1rs)]ah

nm,

(6d)

where material parameters outside the sphere are indicated with the subscript 1 and inside the

sphere with the subscript 2, and rsdenotes 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], the incident illumination cannot analytically be written in

terms of the Mie modes. To obtain the coefficients corresponding to a general incident illumi-

nation, the inner-product of the right-hand side of Eqs. (5) with the basis-functions Eq. (4) has

to be taken, resulting in

ah

nm=

1

SrSθSφ

1

SrSθSφ

??

??

Sa

Einc,rrjn(kr)P|m|

n (cosθ)e−imφdσ ,

(7a)

ae

nm=

Sa

Hinc,rrjn(kr)P|m|

n (cosθ)e−imφdσ ,

(7b)

where dσ is a surface element on the surface Saenclosing the scattering sphere, and the nor-

malisation constants Sr, Sθand Sφare 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)

ra

jn(kra) ,

Sθ=

2(n+|m|)!

(2n+1)(n−|m|)!,

Sφ= 2π .

(8)

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

Without loss of generality we can assume an x-polarised plane wave propagating in the z-

direction Einc,r= cosφ sinθ exp[ikrcosθ], which yields

ah

nm=

?in+1(2n+1)

0

2kn(n+1)

m = ±1

m ?= ±1

,

ae

nm=

⎧

⎩

⎨

i(m−1)?ε

0

μ

in+1(2n+1)

2ikn(n+1)

m = ±1

m ?= ±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

??(α +iβ)e−iφ+(α −iβ)eiφ?Jl(kρρ)sinθ

+kρ

kz

Er=

?k

0

Epl(kρ)

2

?(iα −β)e−iφJl−1(kρρ)−(iα +β)eiφJl+1(kρρ)?cosθ

eilφ+ikzz

?

dkρ,

(10a)

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Hr=

?ε

+k2

μ

?k

?(iα −β)Jl−2(kρρ)e−iφ−(iα +β)Jl+2(kρρ)eiφ??

−kρkz

0

Epl(kρ)

2kkz

eilφ+ikzz

?1

2

??

2k2−k2

ρ

??(iα −β)e−iφ−(iα +β)eiφ?Jl(kρρ)

sinθ

ρ

?(α +iβ)e−iφJl−1(kρρ)+(α −iβ)eiφJl+1(kρρ)?cosθ

with z = racosθ, ρ = rasinθ, and the function Eplchosen such that it forms the vectorial

equivalent of the ‘elegant’ form of the scalar Gauss-Laguerre solution [15],

?

?

dkρ,

(10b)

Epl(kρ) = (−1)p

?kzr

2

?(p+|l|+1)/2k

kz

k2

ρ

k2−k2ρ

?(2p+|l|+1)/2

exp

?

−

kzrk2

2(k2−k2ρ)

ρ

?

.

(11)

This expression can be substituted into Eqs. (7) to obtain an expression for the incident field in

terms ofits coefficientsae,h

replacing the exp[i(l±1)φ] terms by 2πδm,l±1. This yields only non-zero coefficients for two

sets of modes with m = l ±1 and n ≥ m. However, the integration over θ needs to be carried

out numerically. Note that an additional numerical integration over kρ 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

alternativelytheradiusoftheintegrationsphererahastobelargerequiringanincreasednumber

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

?kNA

+

k

+kρ

k

?ε

−(k2−k2

?(α +iβ)e−iφJl−1(kρρ)+(α −iβ)eiφJl+1(kρρ)?cosθ

where the function Eplis given by the simplified expression

?k2

kzr

Again, we substitute Eqs. (12) in Eqs. (7). Similarly, the integration over φ can be done analyt-

ically, by replacing exp[i(l±1)φ] with 2πδm,l±1, numerically over θ, and numerically over kρ

due to the focusing of the field.

mn. Theintegrationoverφ canbeperformedanalyticallybyeffectively

Er=

0

Epl(kρ)eilφ+ikzz

??(α +iβ)e−iφJl−2(kρρ)+(α −iβ)eiφJl+2(kρρ)??

?(iα −β)e−iφJl−1(kρρ)−(iα +β)eiφJl+1(kρρ)?cosθ

?kNA

ρ−kkz)?(iα −β)Jl−2(kρρ)e−iφ−(iα +β)Jl+2(kρρ)eiφ??

−kρkz

?1

2

??

1+kz

k

??(α +iβ)e−iφ+(α −iβ)eiφ?Jl(kρρ)

?

1−kz

sinθ

?

dkρ,

(12a)

Hr=

μ

0

Epl(kρ)

kkz

eilφ+ikzz

?1

2

?

(k2−k2

ρ+kkz)?(iα −β)e−iφ−(iα +β)eiφ?Jl(kρρ)

sinθ

?

dkρ,

(12b)

Epl(kρ) = (−1)pil−1upl

kρRf

√kzzr

ρR2

f

?|l|/2

L|l|

p

?k2

ρR2

kzr

f

?

exp

?

−k2

ρR2

2kzr

f

?

.

(13)

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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

?1

+

k

+kρ

k

?ε

−(k2−k2

?

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

Er=

?kNA

?

0

Epl(kρ)eilφ?+ikzz

??

?

?kNA

2

??

1+kz

k

??(α +iβ)e−iφ+(α −iβ)eiφ?Jl(kρρ?)

1−kz

(α +iβ)e−i(2φ?−φ)Jl−2(kρρ?)+(α −iβ)ei(2φ?−φ)Jl+2(kρρ?)

??

sinθ

(iα −β)e−iφ?Jl−1(kρρ?)−(iα +β)eiφ?Jl+1(kρρ?)

Epl(kρ)

kkz

2

?

(α +iβ)e−iφ?Jl−1(kρρ?)+(α −iβ)eiφ?Jl+1(kρρ?)

?

cosθ

?

dkρ,

(14a)

Hr=

μ

0

eilφ+ikzz

(iα −β)Jl−2(kρρ?)e−i(2φ?−φ)−(iα +β)Jl+2(kρρ?)ei(2φ?−φ)??

?1

?

(k2−k2

ρ+kkz)?(iα −β)e−iφ−(iα +β)eiφ?Jl(kρρ?)

ρ−kkz)

sinθ

−kρkz

?

cosθ

?

dkρ,

(14b)

ρ?=?(rasinθ cosφ −xoff)2+(rasinθ sinφ −yoff)2?1/2,

rasinθ cosφ −xoff

To obtain the coefficients ae,h

Eqs. (7) has to be done numerically over both φ and θ, also including an additional integration

over kρas 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.

(15a)

tanφ?=rasinθ sinφ −yoff

.

(15b)

nmthe decomposition of the field in terms of Mie modes using

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 sinθd= 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=10mm,resultinginaRayleighrangeofzr=31.8λ andabeamwaist ofw(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

for the three differentGauss-Laguerremodes is obtainedusing the expression givenin Eqs. (1).

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Thedominantcontributionis fromthex-component;andit stronglyresemblesthescalar 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.

10

8

6

4

2

0

-10

-2

-4

-6

-8

10 -100 -55

0.1

0.05

0.2

0.15

0

0.4

0.35

1

0.5

0

x 10-3

2

1.5

0.02

0.03

0.01

0

0.04

10

8

6

4

2

0

-10

-2

-4

-6

-8

10-100 -55

10

8

6

4

2

0

-10

-2

-4

-6

-8

10-100-55

2.5

0.3

0.25

0.05

10

8

6

4

2

0

-10

-2

-4

-6

-8

10-100 -55

0.1

0.3

0.2

0

0.4

0.4

0.2

0

x 10

1.2

-3

0.8

0.6

0.02

0.03

0.01

0

0.04

10

8

6

4

2

0

-10

-2

-4

-6

-8

10-100 -55

10

8

6

4

2

0

-10

-2

-4

-6

-8

10 -100 -55

0.05

1.0

10

8

6

4

2

0

-10

-2

-4

-6

-8

10-100 -55

0.1

0.3

0.2

0

0.5

2

1

0

x 10

-4

4

3

0.01

0.015

0.005

0

0.02

|Ex|

10

8

6

4

2

0

-10

-2

-4

-6

-8

10-100-55

10

8

6

4

2

0

-10

-2

-4

-6

-8

10-100 -55

5

0.4

GL00

GL01

GL02

y [λ]

y [λ]

x [λ]

y [λ]

x [λ]

x [λ]

|Ey||Ez|

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?.

Initially we place a spherical scatterer of radius rsin 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 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 con-

tributions of the different Mie-modes can easily be identified. To obtain a high contrast we

assume an aluminium sphere with refractive index nal= 0.503+i4.923 corresponding to the

λ = 405 nm illumination. In the following section we translate the spherical scatterer along the

x-axis by a distance varyingfrom −12.5λ to 12.5λ.The numberof 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 Nmax=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

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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

In Fig. 3 the logarithmic intensity distribution corresponding to a spherical scatterer with a

radius of 1.82 µm and GL01illumination 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.

10

-10

-10

5

-5

0

z [mm]

10

5

-5

0

y [mm]

x [mm]

10

-10

5

-5

0

-9.5

-9

-8.5

-8

-7.5

-10

Dc

Dd

Da

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 Dcand Ddcorresponding to

TM and TE, respectively.

4.1.Scattering as a function of sphere radius

Interactionoftheilluminationwitha spherelocatedin focusresults in afield distributionspread

over the full 4π solid angle, as is shown on a logarithmic scale in Fig. 3. Depending on the size

of the sphere, different detectors detect a different amount of light. Using the signal of all four

detectors we acquire information about the sphere in order to identify size related effects. The

total intensity (S1+S2+S3+S4) for the various detectors is shown in Fig. 4. The first, second

and third row of figures correspond to the focused Gauss-Laguerre beam illumination with

GL00, GL01and GL02, respectively. The signal of the detector placed in transmission (Da),

presented in the first column of Fig. 4, shows maximum transmission for small spheres and,

as the radius of the sphere increases, the transmitted light intensity drops and eventually is

completely blocked by the scatterer. Although we calculated the response for a sphere radius

up to 10 µm, for clarity we limit the range of the plots to a radius of 5 µm, since for larger radii

the trend of the signal does not change substantially. The presence of a larger hole at the centre

of the illumination for increasing order l is clearly observed as a widening of the initial plateau

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012345

0

0.2

0.4

0.6

0.8

1

norm. total intensity

012345

0

4x 10−3

2

4

6

8

012345

0

5x 10−3

0.5

1

1.5

2

2.5x 10−3

TM

TE

012345

0

0.2

0.4

0.6

0.8

1

norm. total intensity

012345

0

2x 10−4

1

2

3

012345

0

1

2

3

4

TM

TE

0

12345

0

0.2

0.4

0.6

0.8

1

sphere radius [μm]

norm. total intensity

012345

0

0.5

1

1.5

sphere radius [μm]

TM

TE

Reflection

Transmission

Transverse plane

GL00

GL01

GL02

012345

0

0.5

1

1.5

sphere radius [μm]

x 10−2

x 10−2

Fig.4. (1stcol.)The totalintensity asa function of sphere radius for thedetector (Da)placed

in transmission, (2ndcol.) placed in reflection (Db), and (3rdcol.) 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, GL01and GL02, respectively.

in the column with transmitted intensity plots. The reflected intensity, obtained from detector

Db, is shown in the second column. Initially, there is almost no reflection for small spheres and

as the radius of the sphere increases the amplitude of the reflected light increases. For a radius

of 10 µm, the total reflected intensity reaches only 23.6% for GL00, 4.2% for GL01and 0.65%

for GL02, as compared to a perfect reflecting plane in focus. Obviously, the reflected light is

still scattered outside the detector due to the curvature of the sphere. For detectors Dcand Dd

it is not easy to estimate what the optimum radius of the scatterer should be in order to achieve

maximum scattered intensity, but as expected, this optimum radius increases with increasing

order l of the Gauss-Laguerre beam since the doughnut increases in width. Note that Dcis

placed on the x-axis which is the direction of oscillation of the illumination (TM) and Ddon

the y-axis (TE), which is apparent from the difference in oscillatory behaviour of the respective

intensity distributions.

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-

detectorconfiguration(S1−S2+S3−S4).Againthe columnscorrespondto thedetectorsplaced

in transmission, reflection and the transversal plane, and the rows correspond to the different

illumination modes GL00, GL01and 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 illumi-

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012345

−15

−10

−5

0

5

sphere radius [μm]

norm. difference intensity

012345

−2

−1

0

1

2x 10-7

norm. difference intensity

012345

0

2

4

6

norm. difference intensity

012345

−2

4x 10-6

−1

0

1

2x 10-5

012345

−4

1.2x 10-5

−2

0

2

4x 10-6

TM

TE

012345

−6

−4

−2

0

2

sphere radius [μm]

Reflection

Transmission

Transverse plane

GL00

GL01

GL02

012345

−5

0

5

10x 10-9

012345

−5

−2.5

0

2.5

5x 10-8

TM

TE

012345

−1.2

−0.6

0

0.6

sphere radius [μm]

TM

TE

x 10

−2

x 10

−2

Fig. 5. (1stcol.) Difference intensity as a function of sphere radius for the detector (Da)

placed in transmission, (2ndcol.) placed in reflection (Db), and (3rdcol.) 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, GL01and GL02, respectively.

nation.This ellipticitychangesas a functionofsphereradius as clearlyobservedin 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 differencein 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.

The split detector configuration yields additional information from detectors Dcand Dd. For

detectorsDaandDbthesesignalsdonotprovideextrainformationduetotheintrinsicsymmetry

of the configuration. The differential intensity in split-z detector configuration (S1−S2−S3+

S4) is shown in Fig. 6 in the left column and in split-x/y detector configuration (S1+S2−S3−

S4) 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.

To explain the origin of the observed oscillations, we consider the modal distribution as a

function of scattering sphere radius, shown in Fig. 7. Since the incident illumination is known

and does not change as a function of radius of the spherical scatterer, ae,h

only once and any radius-dependent effects are purely related to matching the boundary con-

ditions. The plotted coefficients bh

spherical scatterer as obtained from Eqs. (6). For each mode number n, an oscillation of the

nmhas to be determined

nmcorrespond to the TM-contribution of the field outside the

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012345

−5

−2.5

0

2.5

5x 10

−9

norm. split-x/y intensity

5

TM

TE

01234

−5

−2.5

0

2.5

5x 10

−4

norm. split-z intensity

TM

TE

012345

−4

−2

0

2

4x 10

−6

norm. split-x/y intensity

5

TM

TE

01234

−1

−0.5

0

0.5

1x 10

−3

norm. split-z intensity

TM

TE

012345

−6

−3

0

3

6x 10

−6

sphere radius [μm]

norm. split-x/y intensity

5

TM

TE

01234

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

sphere radius [μm]

norm. split-z intensity

TM

TE

Transverse plane

GL00

GL01

GL02

Transverse plane

GL00

GL01

GL02

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, GL01and GL02, respectively.

coefficient is observed as a function of the sphere radius. These oscillations are directly related

to the argument of the functions jn(krs) yielding a frequency of 202.5 nm or 0.5λ for krs? n.

In comparison with the coefficients for the plane wave, as shown in Fig. 8a over a much larger

range n, it is clear that the finite extent of the Gaussian beam is responsible for an apodisation

effect to limit the amount of relevant coefficients. The plane wave illumination does not exhibit

a similar oscillatory behaviour for the signals at the various detectors. We concentrate on the

coefficients for the TM-modes only, since the coefficients for the TE-modes are quite similar,

except for a π-phase difference of the oscillation, as shown in Fig. 8b.

The oscillation frequencyassociated with the ellipticity of the intensity distribution is related

to the amount and shape of the modes involved in the scattering process. Adding an addi-

tional mode to the intensity distribution changes the shape of the total intensity distribution

dramatically. The sphere radius for the qth-order peak as a function of the mode-numbercan be

estimated with a linear fit, as indicated by the white lines drawn for the 5 lowest order peaks

shown in Fig. 8a. The derivative of these linear estimates are found to be 0.159λ, 0.178λ,

0.187λ, 0.193λ and 0.198λ, which values are close to the observed oscillation frequency of

0.19λ for the reflected light. Careful study of the figure reveals that each line q deviates from

the peak value for the higher mode-numbers.Since only a small range of q-modes contribute to

the intensity distribution for the Gauss-Laguerre illumination due to the Gaussian apodisation,

the linear fit represents the modal dependence closely and we observe well defined oscillations

of the intensity distribution. For plane wave illumination, these effects are averaged out due to

the high number of relevant q-modes, as well as the deviation from the linear fit.

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 =

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Page 13

30

10

20

0

5

0

3

4

21

0.4

0.1

0.2

0.3

GL00 : m = -1

n - mode number

30

10

20

0

5

0

3

4

21

0.4

0.1

0.2

0.3

GL00 : m = 1

30

10

20

0

5

0

3

4

21

0.4

0.1

0.2

0.3

GL01 : m = 0

n - mode number

0

30

10

20

0

5

0

3

4

21

0.02

0.005

0.01

0.015

GL02 : m = 1

n - mode number

0

30

10

20

0

5

0

3

4

21

0.02

0.005

0.01

0.015

GL01 : m = 2

0

30

10

20

0

5

0

3

4

21

8

2

4

6

GL02 : m = 3

0

x 10-4

sphere radius [μm]sphere radius [μm]

Fig. 7. Absolute value of the bh

function of the sphere radius rsand the mode number n, obtained by illumination with the

three Gauss-Laguerre beams GL00, GL01and GL02.

nmcoefficients for m = l−1 (left) and m = l +1 (right) as a

0 0.51 1.522.5

0

0.1

0.2

0.3

0.4

0.5

sphere radius [μm]

|bn=15,m=0|

TM

TE

80

60

40

20

00

1

2

34

5

sphere radius [μm]

n - mode number

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

(a) (b)

70

50

30

10

Fig. 8. (a) Absolute value of the bh

as a function of the sphere radius rsand the mode number n. (b) Absolute value of the bnm

coefficient with n = 15 and m = 0 for GL01illumination as a function of sphere radius,

with blue corresponding to the TM coefficient bhand red to the TE coefficient be.

nmcoefficients for m =±1 of the plane wave illumination

π/2. However, the associated Legendre polynomials P|m|

n is even dependingon 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.

Note that increasing the beam divergence results in a stronger apodisation, i.e. less relevant

modes, and therefore more pronounced oscillations occur.

n (cosθ) only contribute for n is odd or

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4.2.Off-axis illumination

Inthis sectionwediscuss thesignalat thefourdetectorswhenthescatteringsphereis movedoff

the optic axis. Theamplitudeandphase of the doughnutbeamalongthe z-axis is predominantly

determinedby theexp[ikz] term dueto the small divergenceof thebeam.The widthof 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 beenfixedat whicha maximumamountoflightis scatteredinthey-direction(TE),whichis

the radius correspondingto the maximum value of the red line in the third column of Fig. 4, i.e.

rs= 1.279 µm, rs= 1.818 µm and rs= 2.239 µm for GL00, GL01and GL02, respectively. The

choiceofsphereradiusfora particularGauss-Laguerremodeprovidesa verybasicmatchofthe

scattering pattern, howeverthat 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 GL01illumination 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.Thereasonforthis is thatinterferenceofthe lightscatteredbythespherewiththe

unaffected transmitted beam causes a strong anti-symmetric intensity distribution with respect

to the y-axis for the GL01and GL02modes.

−10 −505 10

0

0.2

0.4

0.6

0.8

1

x [λ]

norm. total intensity

−10−50510

−4

−2

0

2

4

6

x [λ]

norm. difference intensity

−10 −505 10

−5

−2.5

0

2.5

5

x [λ]

norm. split-x intensity

−10 −505 10

−0.5

−0.25

0

0.25

0.5

x [λ]

norm. split-y intensity

GL00

GL01

GL02

x 10

-2

x 10

-2

(a)(b)(c)(d)

Fig. 9. Transmitted integrated intensity for three different Gauss-Laguerre modes, GL00

blue, GL01red and GL02green. (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 configurationdetects 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 orthog-

onal direction is shown in the bottom row figures. The maximum amplitude is approximately

equal compared to the maximumamplitude 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.

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−10−505 10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x [λ]

norm. total intensity

−10−505 10

−1

0

1

2

3

4

5x 10

norm. difference intensity

-4

x [λ]

−10 −505 10

−1.5

−1

−0.5

0

0.5

1

1.5x 10

-3

x [λ]

norm. split-x intensity

−10−505 10

−1.5

−1

−0.5

0

0.5

1

1.5x 10

-4

x [λ]

norm. split-y intensity

GL00

GL01

GL02

x 10

-2

(a)(b) (c)(d)

Fig.10. Reflected integrated intensity forthree different Gauss-Laguerre modes, GL00blue,

GL01red and GL02green. (a) The sum signal (b) the difference signal, (c) the split-x

configuration and (d) the split-y configuration.

−10

x 10

−505 10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x [λ]

norm. total intensity

−10−505 10

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

-5

x [λ]

norm. difference intensity

−10−505 10

−8

−6

−4

−2

0

2

4

6

8x 10

-4

x [λ]

norm. split-z intensity

−10 −505 10

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

-4

x [λ]

norm. split-y intensity

GL00

GL01

GL02

−10−505 10

0

0.2

0.4

0.6

0.8

1

1.2

x [λ]

norm. total intensity

−10 −505 10

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

-5

x [λ]

norm. difference intensity

−10−505 10

−5

−4

−3

−2

−1

0

1x 10

-4

x [λ]

norm. split-z intensity

−10 −505 10

−6

−4

−2

0

2

4

6

8x 10

-4

x [λ]

norm. split-x intensity

GL00

GL01

GL02

TE

TM

x 10

-2

-2

(a)(b) (c)(d)

Fig. 11. Integrated intensity scattered to the transversal plane for three different Gauss-

Laguerre modes, GL00blue, GL01red and GL02green. 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 andl =0,or therotationofa similar beamwithhighermodeindexl. 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).

#85671 - $15.00 USD

(C) 2007 OSA

Received 25 Jul 2007; revised 9 Sep 2007; accepted 27 Sep 2007; published 28 Sep 2007

1 October 2007 / Vol. 15, No. 20 / OPTICS EXPRESS 13374