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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

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

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using a Bromwich formulation”, J. Opt. Soc. Am. A 5, 1427–1443 (1988).

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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).

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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).

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2003).

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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).

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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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