Rigorous analysis of spheres in Gauss-Laguerre beams

A.S. van de Nes; P. Torok

doi:10.1364/OE.15.013360

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 sensitive 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 been considered in Ref. [7] 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], 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 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] 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 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 z_r, the distance from the origin in which the beamwidth increases a factor √2, or equivalently, the Gaussian beamwidth w(0) = (2z_r/k)^1/2 with k the wavenumber. An imaging system with numerical aperture NA which obeys Abbe’s sine condition is illuminated by a scalar Gauss-Laguerre mode. The electromagnetic field in the focal region of the imaging system can be written [12] 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 D_a measures the transmitted light, detector D_b the reflected light, detector D_c the light reflected along the x-direction and detector D_d the light reflected along the y-direction. (b) Each detector consists of four segments with axes x ₁ = x and x ₂ = y for D_a and D_b, x ₁ = z and x ₂ = y for D_c, and x ₁ = z and x ₂ = x for D_d.

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E (r) = \frac{1}{2} E_{pl, 0} (r; α, β) + \frac{1}{4} E_{pl, - 2} (r; α + iβ, iα - β) + \frac{1}{4} E_{pl, 2} (r; α - iβ - iα - β),

H (r) = \frac{1}{2} H_{pl, 0} (r; α, β) + \frac{1}{4} H_{pl, - 2} (r; α + iβ, iα - β) + \frac{1}{4} H_{pl, 2} (r; α - iβ - iα - β),

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

E_{pl, j} (r; α, β) = \int_{0}^{k NA} E_{pl, j} (k_{ρ}) e^{i (l + j) ϕ + i k_{z} z} ((α \hat{x} + β \hat{y}) J_{l + j} (k_{ρ} ρ) + \frac{k_{ρ}}{2 k_{z}} \hat{z} \times [(iα + β) e^{- iϕ} J_{l + j - 1} (k_{ρ} ρ) - (iα + β) e^{iϕ} J_{l + j + 1} (k_{ρ} ρ)]) d k_{ρ},

H_{pl, j} (r; α, β) = \sqrt{\frac{ε}{μ}} \int_{0}^{k NA} \frac{E_{pl, j} (k_{ρ})}{2 k k_{z}} e^{i (l + j) ϕ + i k_{z} z} ((- β \hat{x} + α \hat{y}) [2 k^{2} - k_{ρ}^{2}] J_{l + j} (k_{ρ} ρ) + \frac{k_{ρ}^{2}}{2} [(\hat{x} + i \hat{y}) (iα - β) J_{l + j - 2} (k_{ρ} ρ) e^{- 2 iϕ} - (\hat{x} - i \hat{y}) (iα + β) J_{l + j + 2} (k_{ρ} ρ) e^{2 iϕ}] - k_{ρ} k_{z} \hat{z} [(α + iβ) J_{l + j - 1} (k_{ρ} ρ) e^{- iϕ} + (α - iβ) J_{l + j + 1} (k_{ρ} ρ) e^{iϕ}]) d k_{ρ},

where k_z = (k ² - k ²_ρ)^1/2 and J_n(x) are the Bessel functions of the first kind [14]. The freely chosen complex coefficients α and β determine the dominant state of polarisation, where α is associated with oscillation along 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 E_pl,j(k _ρ) can also be chosen freely as long as it tends to zero sufficiently quickly to keep the energy associated to the field finite [10], allowing us to write

E_{pl, j} (k_{ρ}) = {(- 1)}^{p} i^{l - 1} u_{pl} \frac{k_{ρ} R_{f}}{\sqrt{k_{z} z_{r}}} (1 + (1 - ∣ j ∣) \frac{k_{z}}{k}) {(\frac{k_{ρ}^{2} R_{f}^{2}}{k z_{r}})}^{\frac{∣ l ∣}{2}} L_{p}^{∣ l ∣} (\frac{k_{ρ}^{2} R_{f}^{2}}{k z_{r}}) \exp [- \frac{k_{ρ}^{2} R_{f}^{2}}{2 k z_{r}}],

with u_pl ² =p!/[(1 + δ_0l)π(p+ |l|)!], and R_f the focal length of the imaging system. The integration domain is bounded by k_p ∊ [0, kNA〉. The ratio z_r/R_f 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 [15,16].

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 separation of variables yields an analytical solution in terms of the Debye potentials, 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 Π^e(r) and Π^h(r), respectively. Any field distribution is fully determined by the set of modes

r Π^{e, h} (r) = \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} a_{nm}^{e, h} r j_{n} (kr) P_{n}^{∣ m ∣} (\cos θ) e^{imϕ},

where m and n are integers, P ^|m| _n(x) the associated Legendre polynomials [21] and j_n(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 h_n(x) = j_n(x) + iy_n(x) to describe the scattered field outside the sphere. The radial component of the field distributions is given by

E_{r} (r) = \frac{n (n + 1)}{r^{2}} r Π^{h} = E_{x} (r) \cos ϕ \sin θ + E_{y} (r) \sin ϕ \sin θ + E_{z} (r) \cos θ,

H_{r} (r) = \frac{n (n + 1)}{r^{2}} r Π^{e} = H_{x} (r) \cos ϕ \sin θ + H_{y} (r) \sin ϕ \sin θ + H_{z} (r) \cos θ,

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 a^e,h_nm, b^e,h_nm and c^e,h_nm, respectively. The induced field inside and outside the sphere only couples to modes of the incident field with the same mode number m,

c_{nm}^{e} = \frac{- \frac{i μ_{2}}{(k_{1} r_{s})}}{μ_{1} h_{n} (k_{1} r_{s}) [k_{2} r_{s} j_{n - 1} (k_{2} r_{s}) - n j_{n} (k_{2} r_{s})] - μ_{2} j_{n} (k_{2} r_{s}) [k_{1} r_{s} h_{n - 1} (k_{1} r_{s}) - n h_{n} (k_{1} r_{s})]} a_{nm}^{e},

b_{nm}^{e} = \frac{μ_{2} j_{n} (k_{2} r_{s}) [k_{1} r_{s} j_{n - 1} (k_{1} r_{s}) - n j_{n} (k_{1} r_{s})] - μ_{1} j_{n} (k_{1} r_{s}) [k_{2} r_{s} j_{n - 1} (k_{2} r_{s}) - n j_{n} (k_{2} r_{s})]}{μ_{1} h_{n} (k_{1} r_{s}) [k_{2} r_{s} j_{n - 1} (k_{2} r_{s}) - n j_{n} (k_{2} r_{s})] - μ_{2} j_{n} (k_{2} r_{s}) [k_{1} r_{s} h_{n - 1} (k_{1} r_{s}) - n h_{n} (k_{1} r_{s})]} a_{nm}^{e},

c_{nm}^{h} = \frac{- \frac{i μ_{2} ε_{2}}{(μ_{1} k_{2} r_{s})}}{ε_{1} h_{n} (k_{1} r_{s}) [k_{2} r_{s} j_{n - 1} (k_{2} r_{s}) - n j_{n} (k_{2} r_{s})] - ε_{2} j_{n} (k_{2} r_{s}) [k_{1} r_{s} h_{n - 1} (k_{1} r_{s}) - n h_{n} (k_{1} r_{s})]} a_{nm}^{h},

b_{nm}^{h} = \frac{ε_{2} j_{n} (k_{2} r_{s}) [k_{1} r_{s} j_{n - 1} (k_{1} r_{s}) - n j_{n} (k_{1} r_{s})] - ε_{1} j_{n} (k_{1} r_{s}) [k_{2} r_{s} j_{n - 1} (k_{2} r_{s}) - n j_{n} (k_{2} r_{s})]}{ε_{1} h_{n} (k_{1} r_{s}) [k_{2} r_{s} j_{n - 1} (k_{2} r_{s}) - n j_{n} (k_{2} r_{s})] - ε_{2} j_{n} (k_{2} r_{s}) [k_{1} r_{s} h_{n - 1} (k_{1} r_{s}) - n h_{n} (k_{1} r_{s})]} a_{nm}^{h},

where material parameters outside the sphere are indicated with the subscript 1 and inside the sphere with the subscript 2, and r_s 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], 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

a_{nm}^{h} = \frac{1}{S_{r} S_{θ} S_{ϕ}} \int \int_{S_{a}} E_{inc, r} r j_{n} (kr) P_{n}^{∣ m ∣} (\cos θ) e^{- imϕ} dσ,

a_{nm}^{e} = \frac{1}{S_{r} S_{θ} S_{ϕ}} \int \int_{S_{a}} H_{inc, r} r j_{n} (kr) P_{n}^{∣ m ∣} (\cos θ) e^{- imϕ} dσ,

where dσ is a surface element on the surface S_a enclosing the scattering sphere, and the normalisation constants S_r, S_θ and S_ϕ are obtained by integration along that surface for all modes (n,m). For integration over a spherical shell with radius r_a, we obtain

S_{r} = \frac{n (n + 1)}{r_{a}} j_{n} (k r_{a}), S_{θ} = \frac{2 (n + ∣ m ∣)!}{(2 n + 1) (n - ∣ m ∣)!}, S_{ϕ} = 2 π .

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 E _inc,r = cos ϕ sin θ exp [ikr cos θ], which yields

a_{nm}^{h} = {\begin{matrix} \frac{i^{n + 1} (2 n + 1)}{2 kn (n + 1)} m = \pm 1 \\ 0 m \neq \pm 1 \end{matrix}, a_{nm}^{e} = {\begin{matrix} i^{(m - 1)} \sqrt{\frac{ε}{μ}} \frac{i^{n + 1} (2 n + 1)}{2 ikn (n + 1)} m = \pm 1 \\ 0 m \neq \pm 1 \end{matrix} .

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 E_pl. 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

E_{r} = \int_{0}^{k} \frac{E_{pl} (k_{ρ})}{2} e^{il ϕ + i k_{z} z} ([(α + iβ) e^{- iϕ} + (α - iβ) e^{iϕ}] J_{l} (k_{ρ} ρ) \sin θ + \frac{k_{ρ}}{k_{z}} [(iα - β) e^{- iϕ} J_{l - 1} (k_{ρ} ρ) - (iα + β) e^{iϕ} J_{l + 1} (k_{ρ} ρ)] \cos θ) d k_{ρ},

H_{r} = \sqrt{\frac{ε}{μ}} \int_{0}^{k} \frac{E_{pl} (k_{ρ})}{2 k k_{z}} e^{il ϕ + i k_{z} z} [\frac{1}{2} ((2 k^{2} - k_{ρ}^{2}) [(iα - β) e^{- iϕ} - (iα + β) e^{- iϕ}] J_{l} (k_{ρ} ρ) + k_{ρ}^{2} {[(iα + β) J}_{l - 2} (k_{ρ} ρ) e^{- iϕ} - (iα + β) J_{l + 2} (k_{ρ} ρ) e^{iϕ}]) \sin θ - k_{ρ} k_{z} [(α - iβ) {e^{iϕ} J}_{l - 1} (k_{ρ} ρ) {+ (α - iβ) e}^{iϕ} J_{l + 1} (k_{ρ} ρ)] \cos θ] d k_{ρ},

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

E_{pl} (k_{ρ}) = {(- 1)}^{p} {(\frac{k z_{r}}{2})}^{\frac{(p + ∣ l ∣ + 1)}{2}} \frac{k}{k_{z}} {(\frac{k_{ρ}^{2}}{k^{2} - k_{ρ}^{2}})}^{\frac{(2 p + ∣ l ∣ + 1)}{2}} \exp [- \frac{k z_{r} k_{ρ}^{2}}{2 (k^{2} - k_{ρ}^{2})}] .

This expression can be substituted into Eqs. (7) to obtain an expression for the incident field in terms of its coefficients a^e,h_mn. The integration over ϕ can be performed analytically by effectively replacing the exp [i(l ± 1)ϕ] terms by 2πδ_ml±1. This yields only non-zero coefficients for two sets of modes with m = l ± and n ≥ m. However, the integration over θ needs to be carried out numerically. Note that an additional numerical integration over k _ρ is required due to the definition of the field. To obtain the coefficients for higher order modes accurately, either the accuracy with which the associated Legendre polynomials are determined has to be high, or alternatively the radius of the integration sphere r_a 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|)k_z/k] term included in the function E_pl,j (Eq. 3), is taken into account explicitly. After collecting the terms with the same ϕ and θ dependence this yields

E_{r} = \int_{0}^{k NA} E_{pl} (k_{ρ}) e^{ilϕ + i k_{z} z} (\frac{1}{2} [(1 + \frac{k_{z}}{k}) [(α + iβ) e^{- iϕ} + (α + iβ) e^{iϕ}] J_{l} (k_{ρ} ρ) + (1 + \frac{k_{z}}{k}) [(α + iβ) e^{- iϕ} J_{l - 2} (k_{ρ} ρ) + (α + iβ) e^{iϕ} J_{l + 2} (k_{ρ} ρ)]] \sin θ + \frac{k_{ρ}}{k} [(iα + β) e^{- iϕ} J_{l - 2} (k_{ρ} ρ) - (iα + iβ) e^{iϕ} J_{l + 1} (k_{ρ} ρ)] \cos θ) d k_{ρ},

H_{r} = {\sqrt{\frac{ε}{μ}} \int}_{0}^{k NA} {\frac{E_{pl} (k_{ρ})}{k k_{z}} e}^{ilϕ + i k_{z} z} [\frac{1}{2} ((k^{2} - k_{ρ}^{2} + {kk}_{z}) [(α + iβ) e^{- iϕ} - (iα + β) e^{iϕ}] J_{l} (k_{ρ} ρ) - (k^{2} - k_{ρ}^{2} + k k_{z}) [(α - β) {J_{l - 2} (k_{ρ} ρ) e}^{- iϕ} - (iα + iβ) {J_{l + 2} (k_{ρ} ρ) e}^{iϕ}]) \sin θ - k_{ρ} k_{z} [(α + iβ) e^{- iϕ} J_{l - 2} (k_{ρ} ρ) + (α + iβ) e^{iϕ} J_{l + 1} (k_{ρ} ρ)] \cos θ) d k_{ρ},

where the function E_pl is given by the simplified expression

E_{pl} (k_{ρ}) = {(- 1)}^{p} i^{l - 1} u_{pl} \frac{k_{ρ} R_{f}}{\sqrt{k_{z}} z_{r}} {(\frac{k_{ρ}^{2} R_{f}^{2}}{k z_{r}})}^{\frac{∣ l ∣}{2}} L_{p}^{∣ l ∣} (\frac{k_{ρ}^{2} R_{f}^{2}}{k z_{r}}) \exp [- \frac{k_{ρ}^{2} R_{f}^{2}}{2 k z_{r}}] .

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 2πδ_m,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

E_{r} = \int_{0}^{k NA} E_{pl} (k_{ρ}) e^{ilϕ' + i k_{z} z} (\frac{1}{2} [(1 + \frac{k_{z}}{k}) [(α + iβ) e^{- iϕ} + (α + iβ) e^{iϕ}] J_{l} (k_{ρ} ρ') + (1 + \frac{k_{z}}{k}) [(α + iβ) e^{- iϕ} J_{l - 2} (k_{ρ} ρ') + (α + iβ) e^{i (2 ϕ' - ϕ)} J_{l + 2} (k_{ρ} ρ')]] \sin θ + \frac{k_{ρ}}{k} [(iα + β) e^{- iϕ} J_{l - 1} (k_{ρ} ρ') - (iα + β) e^{iϕ'} J_{l + 1} (k_{ρ} ρ')] \cos θ) d k_{ρ},

H_{r} = {\sqrt{\frac{ε}{μ}} \int}_{0}^{k NA} {\frac{E_{pl} (k_{ρ})}{k k_{z}} e}^{ilϕ + i k_{z} z} [\frac{1}{2} ((k^{2} - k_{ρ}^{2} + {kk}_{z}) [(iα + β) e^{- iϕ} - (iα + β) e^{iϕ}] J_{l} (k_{ρ} ρ') - (k^{2} - k_{ρ}^{2} + k k_{z}) [(iα - β) {J_{l - 2} (k_{ρ} ρ') e}^{- i (2 ϕ' - ϕ)} - (iα + β) {J_{l + 2} (k_{ρ} ρ') e}^{i (2 ϕ' - ϕ)}]) \sin θ - k_{ρ} k_{z} [(α + iβ) e^{- iϕ'} J_{l - 2} (k_{ρ} ρ') + (α + iβ) e^{iϕ'} J_{l + 1} (k_{ρ} ρ')] \cos θ) d k_{ρ},

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

ρ' = {[{(r_{a} \sin θ \cos ϕ - x_{off})}^{2} + {(r_{a} \sin θ \sin ϕ - y_{off})}^{2}]}^{\frac{1}{2}},

\tan ϕ' = \frac{r_{a} \sin θ \sin ϕ - y_{off}}{r_{a} \sin θ \cos ϕ - x_{off}} .

To obtain the coefficients a^e,h_nm 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 θ_d = w(R_f)/R_f = 0.1 indicated in Fig. 1. This choice corresponds, for example, to an initial beamwidth of w(R_f) = 1 mm and lens with focal length R_f= 10 mm, resulting in a Rayleigh range of z_r = 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⟩.

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Initially we place a spherical scatterer of radius r_s in the focus. The layout of the system is shown in Fig. 1a where a Gauss-Laguerre beam is focused onto the sphere and the scattered field is studied at four different detector planes D _{a,b,c,d}. Each detector with a 1.0 mm radius is placed 10 mm away from the sphere, and consists of 4 segments defined as S _{1,2,3,4} (Fig. 1b). The location of the detectors are chosen to maximise the obtained information since the contributions of the different Mie-modes can easily be identified. To obtain a high contrast we assume an aluminium sphere with refractive index n _al = 0.503 + 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 varying from –12.5λ to 12.5λ. The number of modes that have to be taken into account depends strongly on the radius of the sphere but due to the fast calculation speed of the model we fixed N _max = 115, which is more than sufficient for the largest radius scatterer. For spheres located in focus we only need to consider the modes with m = l ± 1, however for the off-axis spheres all m-modes should be taken into account with -n ≤ m ≤ n. Typically, an on-axis sphere requires a calculation time of less than a minute, while an off-axis sphere requires approximately 25 minutes due to the additional numerical integration over ϕ. Note, using a plane-wave decomposition and the Mie-solution for a plane-wave takes approximately 25 minutes as well, but contrary to our modal decomposition, this has to be repeated for each sphere-radius and composition. A typical calculation with a rigorous EM-solver takes several hours and yields a much lower accuracy.

4. Spheres in Gauss-Laguerre beams

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 (D_b) 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 D_a, and the transversal detectors D_c and D_d corresponding to TM and TE, respectively.

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4.1. Scattering as a function of sphere radius

Interaction of the illumination with a sphere located in focus results in a field distribution spread 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 (S ₁ + S ₂ + S ₃ + S ₄) 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, GL₀₁ and GL₀₂, respectively. The signal of the detector placed in transmission (D_a), 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 in the column with transmitted intensity plots. The reflected intensity, obtained from detector D_b, 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 GL₀₁ and 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 D _c and D_d 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 D_c is placed on the x-axis which is the direction of oscillation of the illumination (TM) and D_d on the y-axis (TE), which is apparent from the difference in oscillatory behaviour of the respective intensity distributions.

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

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

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The split detector configuration yields additional information from detectors D_c and D_d. For detectors D_a and D_b 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 D_d (TE), and split-y configuration for detector D_c (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, a^e,h_nm has to be determined only once and any radius-dependent effects are purely related to matching the boundary conditions. The plotted coefficients b^h_nm correspond to the TM-contribution of the field outside the spherical scatterer as obtained from Eqs. (6). For each mode number n, an oscillation of the coefficient is observed as a function of the sphere radius. These oscillations are directly related to the argument of the functions j_n(kr_s) yielding a frequency of 202.5 nmor0.5A for kr_s ≫ 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.

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 (D_c) and red for TE (D_d). Each row corresponds to illumination with increasing order, GL₀₀, GL₀₁ and GL₀₂, respectively.

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The oscillation frequency associated with the ellipticity of the intensity distribution is related to the amount and shape of the modes involved in the scattering process. Adding an additional mode to the intensity distribution changes the shape of the total intensity distribution dramatically. The sphere radius for the q ^th-order peak as a function of the mode-number can 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 0.19 λ 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 = 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 b^h_nm coefficients for m = l -1 (left) and m = l + 1 (right) as a function of the sphere radius r_s and the mode number n, obtained by illumination with the three Gauss-Laguerre beams GL₀₀, GL₀₁ and GL₀₂.

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Fig. 8. (a) Absolute value of the b^h_nm coefficients for m = ±1 of the plane wave illumination as a function of the sphere radius r_s and the mode number n. (b) Absolute value of the b_nm coefficient with n = 15 and m = 0 for GL₀₁ illumination as a function of sphere radius, with blue corresponding to the TM coefficient b^h and red to the TE coefficient b^e.

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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 z_r = 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. r_s = 1.279 μm, r_s = 1.818 μ and r_s = 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 r_s = 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.

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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 (D_b) 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.

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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.

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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).

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Rigorous analysis of spheres in Gauss-Laguerre beams

Abstract

1. Introduction

2. Theory

2.1. Gauss-Laguerre illumination

2.2. Interaction with a scattering sphere

2.3. Decomposition of Gauss-Laguerre beams in Mie modes

3. Optical model configuration

4. Spheres in Gauss-Laguerre beams

4.1. Scattering as a function of sphere radius

4.2. Off-axis illumination

5. Conclusion

Acknowledgements

References and links

Cited By

Figures (11)

Equations (26)

Optics Express