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Journal of Quantitative Spectroscopy &Radiative Transfer

Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 122–132

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journal homepage: www.elsevier.com/locate/jqsrt

Photonic nanojets in optical tweezers

Antonio Alvaro Ranha NevesCentro de Ciências Naturais e Humanas, Universidade Federal do ABC (UFABC), Santo André – São Paulo 09.210-170, Brazil

a r t i c l e i n f o

Article history:Received 4 February 2015Received in revised form12 March 2015Accepted 17 March 2015Available online 25 March 2015

Keywords:Photonic nanojetsGeneralized Lorenz–Mie theoryBeam shape coefficientsMie scatteringOptical tweezers or optical manipulationDielectric particles

x.doi.org/10.1016/j.jqsrt.2015.03.01973/& 2015 Elsevier Ltd. All rights reserved.

ail address: antonio.neves@ufabc.edu.br

a b s t r a c t

Photonic nanojets have been brought into attention ten years ago for potential applicationin ultramicroscopy, because of its sub-wavelength resolution that can enhance detectionand interaction with matter. For these novel applications under development, the opticaltrapping of a sphere acts as an ideal framework to employ photonic nanojets. In thepresent study, we generated nanojets by using a highly focused incident beam, in contrastto traditional plane waves. The method inherits the advantage of optical trapping,especially for intracellular applications, with the microsphere in equilibrium on the beampropagation axis and positioned arbitrarily in space. Moreover, owing to optical scatteringforces, when the sphere is in equilibrium, its center shifts with respect to the focal point ofthe incident beam. However, when the system is in stable equilibrium with a configura-tion involving optical tweezers, photonic nanojets cannot be formed. To overcome thisissue, we employed double optical tweezers in an unorthodox configuration involving twocollinear and co-propagating beams, the precise positioning of which would turn on/offthe photonic nanojets, thereby improving the applicability of photonic nanojets.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

A photonic nanojet (PNJ) is a narrow (subwavelength)and elongated region with high intensity located at theshadow-side surface of an illuminated loss-less dielectricmicrocylinder or microsphere [1,2]. In the present study, wefocus on PNJs generated by a dielectric microsphere acting asa focusing lens, which results in the build up of sphericallyaberrated rays in the focal region. This build up of rays in thespatially localized high-energy-density region constitute theexternal caustic [3,4]. Therefore, PNJs are not a new phe-nomenon, but they have attracted renewed interest becauseof recent technological advances allowing the exploitation ofthis high-energy-density region. Nonetheless, contrary to apossible understanding via geometrical optics or catastrophetheory, we aim to obtain an exact solution of Maxwell'sequations by employing the generalized Lorenz–Mie theory

(GLMT) [5], as in the study by Devilez et al. [6]. Such focusingof light using a microsphere has also been investigated byKofler et al. [7] using the uniform caustic asymptotic method,to obtain analytical expressions for the intensity by matchingthe geometrical-optic solutions with Bessoid integrals. How-ever, for optical trapping applications, the scattering object isa microsphere of size comparable to the illuminating wave-length and consequently outside the regime of geometricaloptics.

Optical trapping results from the change in linearmomentum of a beam scattered by a microsphere, produ-cing a resultant restoring force. The most common setupfor laser trapping is of optical tweezers [8,9]. PNJs andoptical tweezers (OT) have attracted attention because, incombination, they can be highly useful in applicationsincluding nanoscale processing [10–15], high-resolutionmicroscopy [3,16–19], and enhanced inelastic spectro-scopy, such as Raman scattering [20–23], coherent anti-Stokes Raman scattering [24,25] and fluorescence [26,27],or elastic enhancement through backscattering from

A.A.R. Neves / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 122–132 123

nanoparticles [28]. A recent study that aimed to positionPNJs in a controlled manner involved a microsphereattached to a movable micropipette [29] and magneticallycontrolled Janus spheres [30]. In addition, an opticallytrapped non-spherical dielectric particle in a Besselbeam was employed to generate PNJs from anotherpulsed laser for direct laser writing [31]. The disadvan-tage of this approach is in the engineering of particularnon-spherical particles, each exhibiting a characteristicPNJ. Other studies on optical forces and PNJs are relatedto the radiation pressure effect produced by a PNJ onnearby nanoparticles [32–34]. In contrast, the scope ofthis study is different; we investigate the PNJs generatedfrom an optically trapped microsphere in terms of theincoming beam, trading the complexity of structuringspecial shaped particles for the ease of manipulating thetrapping beam.

It is widely known that the main features of PNJs arewaists smaller than the diffraction limit and propagationover several wavelengths without significant diffraction.These features lead to the potential application of PNJs fordeveloping spectroscopic methods with high spatial resolu-tion and high detection sensitivity through backscatteringenhancement. Therefore, the characteristics of a nanojethave been described with a few parameters: the location ofthe intensity maximum, the distance from this maximum tothe sphere surface (radial shift), the distance from theintensity peak to the point where the intensity decays toe�2 times the initial intensity in the direction of the beam(decay length), and the PNJ width, which is e�2 waist of thejet [35]. The behavior of PNJs for a given wavelength is asfollows: the PNJ waist widening is proportional to therelative refractive index, and the PNJ lengthening is due tothe increasing sphere size. This transverse confinement (PNJwaist) has been approximated as a Gaussian, while a part ofthe longitudinal confinement (decay length) is approxi-mated as a Lorentzian profile [36]. In the present work, itis shown that a Bessoidal-type surface best describes theshape of light confined by these PNJs originating from theinterference between the scattered field and the incidentbeam [6] and therefore the extinction component of thePoynting vector.

The modeling of PNJs originating from plane-waveincidence involves only four parameters: the refractiveindices of the particle and its surroundings, wavelength ofthe incident wave, and radius of the sphere. However, withan incident focused beam, which is required for an opticaltrap, the complexity of the modeling increases with addi-tional parameters: the numerical aperture and location ofthe waist with respect to the scatterer. The positioning ofthe incident focused beam with respect to the scatterer isvery important, and we demonstrate that it is responsiblefor switching the PNJs on and off for an optically trappedmicrosphere.

2. Theoretical model of photonic nanojets

Any Maxwellian beam can be expressed by its electro-magnetic field in terms of partial waves. We start with thegeneral expression for the electromagnetic fields of inter-est: the incident and scattered fields. Later, we simplify

this to a two-dimensional problem, which is useful for a2D plot and subsequently for a 1D profile such as thelongitudinal intensity of the PNJ.

2.1. Generalized Lorenz–Mie theory

The basic idea of GLMT is that a beam, which isexpressed by a solution to Maxwell's equations (i.e.,Maxwellian beam), can be written as an infinite series ofspherical functions and spherical harmonics, each mul-tiplied by a coefficient called a beam shape coefficient(BSC). These BSCs completely describe the incident,internal, and scattered electromagnetic fields in termsof partial-waves series. The correct determination ofthese radially independent BSCs allows for the precisedetermination of the observed electromagnetic phe-nomena [5].

The present notation follows that of previous works[37–39], expressing the electromagnetic fields in terms ofspherical vector wave functions even though the originalformulation of GLMT was in the framework of the Brom-wich scalar functions [40,41]. The microsphere scattererhas a radius a and real refractive index n, and the incidentbeam is directed towards the positive z-axis of the rectan-gular coordinate system. The incident wave has a timedependence expð� iωtÞ (omitted for clarity), wavelength λ,and wave number k¼ 2π=λ. The scatterer and the sur-rounding medium are homogeneous, isotropic, and non-magnetic. The incident and scattered electromagneticfields can be written as follows:

Einc ¼ E0X1p ¼ 1

Xpq ¼ �p

GTMpq NpqðrÞþGTE

pqMpqðrÞ; ð1Þ

Hinc ¼H0

X1p ¼ 1

Xpq ¼ �p

GTMpq MpqðrÞ�GTE

pqNpqðrÞ; ð2Þ

Esca ¼ E0X1p ¼ 1

Xpq ¼ �p

apqNpqðrÞþbpqMpqðrÞ; ð3Þ

Hsca ¼H0

X1p ¼ 1

Xpq ¼ �p

apqMpqðrÞ�bpqNpqðrÞ; ð4Þ

where the terms involving the spherical functions andvectorial spherical harmonics can be abbreviated as

Npq rð Þ ¼ ik∇�Mpq rð Þ; ð5Þ

MpqðrÞ ¼ zpqðkrÞXpqðθ;ϕÞ; ð6Þ

where zpq(kr) denotes spherical Bessel or sphericalHankel functions, depending on whether it expressesthe incident or scattered field, respectively. The vec-torial spherical harmonics are defined as XpqðrÞ ¼LYlmðrÞ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffilðlþ1Þ

p, where YlmðrÞ denotes the scalar sphe-

rical harmonics and L¼ � ir� d=dr is the angularmomentum operator in direct space. The scatteringpartial-wave coefficients apq and bpq are related to theMie scattering coefficients by

A.A.R. Neves / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 122–132124

�ap ¼apqGTMpq

¼mψpðmxÞψ 0pðxÞ�ψ 0

pðmxÞψpðxÞψ 0

pðmxÞξpðxÞ�mψpðmxÞξ0pðxÞ

; ð7Þ

�bp ¼bpqGTEpq

¼mψ 0pðmxÞψpðxÞ�ψpðmxÞψ 0

pðxÞψpðmxÞξ0

pðxÞ�mψ 0pðmxÞξpðxÞ

: ð8Þ

The minus sign was included to adopt the morefrequent notation for the Mie coefficients ap and bp. Thecoefficients Gpq

TMand Gpq

TErepresent the expansion coeffi-

cients (BSCs) of the incident fields. The BSCs are defined asfollows:

GTMpq ¼ �gp

E0

Z π

0dθ sin θ

Z 2π

0dϕYn

pq θ;ϕ� �

E � r ; ð9Þ

GTEpq ¼

gpH0

Z π

0dθ sin θ

Z 2π

0dϕYn

pq θ;ϕ� �

H � r ; ð10Þ

where gp ¼ kr=ðjpðkrÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinðnþ1Þ

pÞ was introduced to shorten

the equation. Note that the explicit cancelation of theradial dependence in gp has been the basis of variousapproximation techniques for the BSC [5,37]. Recently, theanalytical solution for any type of Maxwellian beam hasbeen demonstrated [42].

For all the incident beams described henceforth, the axisof symmetry coincides with that of the sphere, and thescenario is termed on-axis. This is due to the fact that theequilibrium position for a homogeneous sphere lies onthe axis of symmetry. For this particular configuration, thedouble sum in the partial wave expansion, Eqs. (1)–(4),simplifies because of contributions exclusively from q¼ 71[43,39]. For this case, it would be best to introduce a newvectorial function to describe our electromagnetic fields:

N7p ðrÞ ¼Np;1ðrÞ7Np;�1ðrÞ; ð11Þ

M7p ðrÞ ¼Mp;1ðrÞ7Mp;�1ðrÞ; ð12Þ

thereby simplifying the analytical solution up to this pointusing the symmetrical relations for q¼ 71. Further simpli-fication of the BSC would require knowledge of the exactvectorial electromagnetic fields near the focus, as will beshown in Section 2.4.

2.2. Plane-wave incidence

Any incident field can be expressed as an infinite seriesof vectorial spherical harmonics. As an initial approach, anarbitrary polarized plane wave traveling along the z-axis(k¼ kz) is used to validate the results and compare themwith known results. The electric field is

E¼ E0eik�rðpx; py;0Þ; ð13Þwhere the polarization is described by the components ofthe Jones vector ðpx; pyÞ. The magnetic field can be easilydetermined from the relation

H¼ kωμ

� E¼H0eik�r �py; px;0� �

: ð14Þ

The radial components of the fields in spherical coordi-nates are

E � r ¼ E0eikr cosθ sinθðpx cosϕþpy sinϕÞ; ð15Þ

H � r ¼H0eikr cosθ sinθðpx sinϕ�py cosϕÞ; ð16Þ

where we used r ¼ sinθ cosϕxþ sinθ sinϕyþ cosθz. Theintegrals over the solid angle are presented in Appendix A,and it can be seen that only the q¼ 71 components arepresent. The plane-wave (pw) BSCs are

GTM;pwp;71 ¼ Gpw

p ð8 ipx�pyÞ; GTE;pwp;71 ¼ Gpw

p ðpx8 ipyÞ; ð17Þ

Gpwp ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπð2pþ1Þ

pip: ð18Þ

2.3. Description of highly focused beam

The commonly employed Davis description of Gaussianbeams, a perturbation method that introduces higher-order corrections to the paraxial approximation, yields acondition that is rarely met for strong focusing (as inoptical tweezers). Consequently there is a need to gobeyond a paraxial-expansion description. To achieve thisaim, we resort to an early description based on the angularspectrum representation, which is later applied to opticaltrapping to result in a solution for a completely arbitraryvectorial Maxwellian beam [37]. This new approach hasthe benefit of providing an analytical expression for theBSCs as a function of beam position and polarizationdirectly with respect to the sphere without the need torely on vector translation theorems requiring high com-putational cost. Results obtained by using this approachhave recently been confirmed independently through anaberrated photothermal microscopy measurement to be inexcellent agreement with the theory [44].

To present a complete study of PNJs in optical tweezers,we start with a description of the highly focused incidentbeam in the framework of GLMT. The on-axis case isconsidered here, since the equilibrium trap position for ahomogeneous sphere lies on the propagation axis. Theexact BSCs for an on-axis highly focused beam (fb) posi-tioned at z0 with respect to the sphere center are [39]

GTM;fbp;71 ¼ Gfb

p ðipx7pyÞ; GTE;fbp;71 ¼ Gfb

p ð8pxþ ipyÞ; ð19Þ

Gfbp ¼ ikfeikf

Gpwp

pðpþ1Þ

�Z αmax

0dα sinα

ffiffiffiffiffiffiffiffiffiffiffifficosα

peikz0 cosαe�ðf sinα=ωÞ2 ½π1

pðαÞþτ1pðαÞ�:

ð20ÞIn terms of the two angular functions, which involves

associated Legendre functions,

πqp θ� �¼ Pq

pð cosθÞsinθ

; τqp θ� �¼ d

dθPqp cosθ� �

: ð21Þ

2.4. Approximations for fields along the incident plane (2D)

When plotting the fields in 2D, in the x–z plane, we willtake ϕ¼ 0 but θa0 in Eqs. (11) and (12). The vectorialincident and scattered fields from Eqs. (1)–(4) become forthe on-axis case,

Einc

E0¼ �2i

X1p ¼ 1

Gp

pðpþ1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2pþ1Þ

4π

rzpðkrÞkr

1sinθ

τ1p θ� �

cosθ�π1p θ� �� �

�∂τ1pðθÞ∂θ

" #pxr

(

� pþ1ð ÞzpðkrÞkr

�zpþ1

� �τ1p θ� �� izp krð Þπ1

p θ� �

pxθ� pþ1ð ÞzpðkrÞkr

�zpþ1

� �π1p θ� �� izp krð Þτ1p θ

� � pyϕ

�; ð22Þ

Hinc

H0¼ 2i

X1p ¼ 1

Gp

pðpþ1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2pþ1Þ

4π

rzpðkrÞkr

1sinθ

τ1p θ� �

cosθ�π1p θ� �� �

�∂τ1pðθÞ∂θ

" #pyr

(

� pþ1ð ÞzpðkrÞkr

�zpþ1 krð Þ� �

τ1p θ� �þ izp krð Þπ1

p θ� �

pyθþ pþ1ð ÞzpðkrÞkr

�zpþ1 krð Þ� �

π1p θ� �þ izp krð Þτ1p θ

� � pxϕ

�; ð23Þ

Esca

E0¼ 2i

X1p ¼ 1

Gp

pðpþ1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2pþ1Þ

4π

rapzpðkrÞkr

1sinθ

τ1pðθÞ cosθ�π1pðθÞ

� ��∂τ1pðθÞ

∂θ

" #pxr

(

� ap pþ1ð ÞzpðkrÞkr

�zpþ1 krð Þ� �

τ1p θ� �þ ibpzp krð Þπ1

p

pxθ� ap pþ1ð ÞzpðkrÞ

kr�zpþ1 krð Þ

� �π1 θ� �þ ibpzp krð Þτ1p

pyϕ

�; ð24Þ

Hsca

H0¼ 2i

X1p ¼ 1

Gp

pðpþ1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2pþ1Þ

4π

r�bp

zpðkrÞkr

1sinθ

τ1p θ� �

cosθ�π1p θ� �� �

�∂τ1pðθÞ∂θ

" #pyr

(

þ bp pþ1ð ÞzpðkrÞkr

�zpþ1 krð Þ� �

τ1p θ� �þ iapzp krð Þπ1

p

pyθ� bp pþ1ð ÞzpðkrÞ

kr�zpþ1 krð Þ

� �π1 θ� �þ iapzp krð Þτ1p

pxϕ

�: ð25Þ

For the plane-wave BSCs (i.e., Gp ¼ Gpwp ), results in

Einc

E0¼ � i

X1p ¼ 1

ð2pþ1Þpðpþ1Þi

p jpðkrÞkr

1sinθ

τ1p θ� �

cosθ�π1p θ� �� �

�∂τ1pðθÞ∂θ

" #pxr

(� pþ1ð ÞjpðkrÞ

kr� jpþ1

� �τ1p θ� �� ijp krð Þπ1

p θ� �

�pxθ� pþ1ð ÞjpðkrÞkr

� jpþ1

� �π1p θ� �� ijp krð Þτ1p θ

� � pyϕ

�; ð26Þ

Hinc

H0¼ i

X1p ¼ 1

ð2pþ1Þpðpþ1Þi

p jpðkrÞkr

1sinθ

τ1p θ� �

cosθ�π1p θ� �� �

�∂τ1pðθÞ∂θ

" #pyr

(� pþ1ð ÞjpðkrÞ

kr� jpþ1 krð Þ

� �τ1p θ� �þ ijp krð Þπ1

p θ� �

pyθ

þ pþ1ð ÞjpðkrÞkr

� jpþ1 krð Þ� �

π1p θ� �þ ijp krð Þτ1p θ

� � pxϕ

�; ð27Þ

Esca

E0¼ i

X1p ¼ 1

ð2pþ1Þpðpþ1Þi

p aphð1Þp ðkrÞ

kr1

sin θτ1p θ� �

cosθ�π1p θ� �� �

�∂τ1pðθÞ∂θ

" #pxr

(� ap pþ1ð Þh

ð1Þp ðkrÞkr

�hð1Þpþ1 krð Þ !

τ1p θ� �"

þ ibphð1Þp ðkrÞπ1

p

ipxθ� ap pþ1ð Þh

ð1Þp ðkrÞkr

�hð1Þpþ1 krð Þ !

π1 θ� �þ ibph

ð1Þp krð Þτ1p

#pyϕ

" ); ð28Þ

Hsca

H0¼ i

X1p ¼ 1

ð2pþ1Þpðpþ1Þi

p �bphð1Þp ðkrÞ

kr1

sin θτ1p θ� �

cosθ�π1p θ� �� �

�∂τ1pðθÞ∂θ

" #pyr

(þ bp pþ1ð Þh

ð1Þp ðkrÞkr

�hð1Þpþ1 krð Þ

!τ1p θ� �"

þ iaphð1Þp ðkrÞπ1

p

ipyθ� bp pþ1ð Þh

ð1Þp ðkrÞkr

�hð1Þpþ1 krð Þ !

π1 θ� �þ iaph

ð1Þp krð Þτ1p

#pxϕ

" ): ð29Þ

A.A.R. Neves / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 122–132 125

Note that for the Poynting vector in the x–z plane, weonly need the r and θ components. We are thus interestedin the following products from the vectorial fields above:

SrrþSθθ ¼ rðEθHϕ�EϕHθÞþ θðEϕHr�ErHϕÞ: ð30Þ

For the particular 2D case, since the photonic nanojet isextremely close to the z-axis immediately outside thespherical scatterer, the small-angle approximation forsmall θ is justified:

Pqp cosθ� �� ð�1Þq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðpþqÞ!ðp�qÞ!

s ffiffiffiffiffiffiffiffiffiffiffiθ

sinθ

sJq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipðpþ1Þ

pθ

� �; ð31Þ

P1p cosθ� �¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipðpþ1Þ

p ffiffiffiffiffiffiffiffiffiffiffiθ

sinθ

sJ1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipðpþ1Þ

pθ

� �: ð32Þ

This small-angle approximation yields numerically fas-ter 2D plots that are not noticeably different from the plotobtained through the use of the exact expression of theassociated Legendre functions.

2.5. Approximations for fields along the z-axis (1D)

When plotting only the values of the longitudinalintensity profile of the PNJ, we are interested only in thefields along the z-axis (i.e., θ¼ 0); the angular functions

A.A.R. Neves / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 122–132126

(Eqs. (5)–(6)) for θ-0 are

limθ-0

1sinθ

τ1p θ� �

cosθ�π1p θ� �� �

¼ 0; ð33Þ

limθ-0

∂τ1pðθÞ∂θ

¼ 0; ð34Þ

limθ-0

τ1p θ� �¼ �1

2p pþ1ð Þ; ð35Þ

limθ-0

π1 θ� �¼ �1

2p pþ1ð Þ: ð36Þ

This greatly simplifies the field, especially for a fastcalculation of the longitudinal profile using the Poynt-ing vector of Section 2.6, resulting in fields with no r-component.

2.6. Time-averaged Poynting vector

By examining the time-averaged Poynting vector, weaim to elucidate the vectorial characteristics of the PNJ andits dependence on the incident polarization. Previousresults [45] considered the total field for scattering by amicrosphere to represent the PNJ. Here, emphasis is givento the extinction component of the Poynting vector [46] asthe origin of the PNJ, as suggested in [6]. We can nowdetermine the Poynting vector for the incident, scattered,and interference fields. The electromagnetic fields outsidethe spherical scatterer are

Etot ¼ EincþEsca; Htot ¼HincþHsca: ð37ÞThe time-averaged Poynting vectors are as follows:

⟨Sinc⟩¼12R Einc �Hn

inc

� �; ð38Þ

⟨Ssca⟩¼12R Esca �Hn

sca

� �; ð39Þ

⟨Sext⟩¼12R Etot �Hn

tot

� �¼ 12R Einc �Hn

sca

� �þ12R Esca �Hn

inc

� �;

ð40Þwhere Sinc, Ssca, and Sext are, respectively, the incident,scattered, and interference (between the scattered andincident beams) Poynting vectors. We now carefully exam-ine each Poynting vector by using Gp ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπð2pþ1Þ

pip for the

case of plane waves. Since by definition, jpxj2þjpyj2 ¼ 1, werecover an on-axis intensity profile independent of thechosen incident polarization as follows:

⟨Sinc � z⟩¼18

X1p ¼ 1

2pþ1ð Þip jpðkzÞkz

þdjpðkzÞdðkzÞ � ijp kzð Þ

� �" #2: ð41Þ

By using the relations and identities of spherical func-tions, we recover the Poynting vector for the incident field as

⟨Sinc � z⟩¼12Rjieikzj2 ¼ 1

2: ð42Þ

This is expected, as the sum term must converge inEq. (41) to a particular value because the incident plane

wave is properly defined as

⟨Sinc⟩¼12R E0H

n

0 pxxþpyy� �

� �pn

yxþpn

x y� �� �

¼ 12R E0H

n

0z� �

:

ð43ÞThis plane-wave analysis validates the present equa-

tions. For Ssca, we start with

⟨Ssca � z⟩¼18R

X1p ¼ 1

2pþ1ð Þip ap pþ1ð Þhð1Þp ðkzÞkz

�hð1Þpþ1 kzð Þ

!"

þ ibphð1Þp ðkzÞ

#X1p ¼ 1

2pþ1ð Þð� iÞp

� bn

p pþ1ð Þhð1Þnp ðkzÞkz

�hð1Þnpþ1 kzð Þ

!� ian

phð1Þnp kzð Þ

" #!:

ð44ÞIt is tempting to approximate the spherical Hankel func-tion in its asymptotic form, since the fields are in theregion outside the spherical scatterer, i.e., kr4ka. For alow p-index, kr≫p. However, when p increases, the Miecoefficient decreases; in such a case, we expect theasymptotic form to be a very good approximation. Unfor-tunately the asymptotic expansion does not represent thePoynting vector up to twice the size factor.

Finally, we are interested in determining the Poyntingvector related to the interference between the incidentand scattered beams. From Eq. (40), this would be the realpart of a linear combination of the spherical Hankelfunction with a plane wave, which is described in theform of the equation below:

⟨Sext � z⟩¼ 14R

ieikzX1p ¼ 0

ð� iÞphð1Þnp ðkzÞBp� ie� ikzX1p ¼ 0

iphð1Þp ðkzÞAp

" #: ð45Þ

Note that Ap and Bp are linear combinations of thecomplex Mie coefficients ap and bp, and the positiondependence of the Poynting vector originates from theterms containing (kz) in Eq. (45).

2.7. Optical forces

Once the BSCs are calculated, the force components onan optically trapped microsphere are determined via theMaxwell stress tensor formalism, providing new insightsowing to the analytical nature of the solution. Finally, torepresent the PNJ due to an optically trapped microsphere,we first determine the location along the propagation axiswhere stable equilibrium is verified. The position is deter-mined from an optical-force curve with respect to the trapposition from the microsphere center. This type of forceprofile has been applied previously in the investigation ofmorphology dependent resonances of an optically trappedmicrosphere [38], as well as locations along the axis [39].At this stable position, an energy-density plot can bedrawn to identify the existence/location of PNJs for differ-ent optical trapping parameters (wavelength, refractiveindexes, polarization, and numerical aperture).

Fig. 1. Contour and vector plot of the intensity and time-averagedPoynting vector, respectively for the interference between the incidentand scattered fields. The result is for a 2�μm polystyrene microsphere(n¼1.59) in water (n¼1.33) under plane-wave incidence at λ¼633 nm.

Fig. 2. Contour and vector plot of the intensity and time-averagedPoynting vector, respectively for the interference between the incidentand scattered fields. The result is for a 4�μm polystyrene microsphere(n¼1.59) in water (n¼1.33) under plane-wave incidence at λ¼633 nm.

A.A.R. Neves / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 122–132 127

We know that the optical force along the z-axis is

Fz ¼ �ϵjE0j22k2

RX1p ¼ 1

Xþp

q ¼ �p

ipþ1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipðpþ2Þðpþqþ1Þðp�qþ1Þ

ð2pþ3Þð2pþ1Þ

s(

� ðapþ1þan

p�2apþ1an

pÞGTMpþ1;qG

TMn

p;q þðbpþ1þbn

p�2bpþ1

h

bn

pÞGTEpþ1;qG

TEnp;q

i�qp

ðapþbn

p�2apbn

pÞGTMp;qG

TEnp;q

h i�: ð46Þ

For the on-axis case, only p¼ 71 remains

Fz ¼ �ϵjE0j22k2

RX1p ¼ 1

ipþ1

pðpþ2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2pþ3Þð2pþ1Þ

p(

� ðapþ1þan

p�2apþ1an

pÞ GTMpþ1;1G

TMn

p;1 þGTMpþ1;�1G

TMn

p;�1

� �hþðbpþ1þbn

p�2bpþ1bn

pÞðGTEpþ1;1G

TEnp;1 þGTE

pþ1;�1GTEnp;�1Þ

i�1p

ðapþbn

p�2apbn

pÞðGTMp;1G

TEnp;1 �GTM

p;�1GTEnp;�1Þ

h i�: ð47Þ

The cross products of BSCs for the focused beam can besimplified to

ðGTMpþ1;1G

TMn

p;1 þGTMpþ1;�1G

TMn

p;�1Þ ¼ 2Gpþ1Gn

p; ð48Þ

ðGTEpþ1;1G

TEnp;1 þGTE

pþ1;�1GTEnp;�1Þ ¼ 2Gpþ1G

n

p; ð49Þ

ðGTMp;1G

TEnp;1 �GTM

p;�1GTEnp;�1Þ ¼ �2iGpG

n

p: ð50Þ

Therefore, the axial force becomes

Fz ¼ �ϵjE0j2k2

RX1p ¼ 1

ipþ1

pðpþ2ÞGpþ1Gn

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2pþ3Þð2pþ1Þ

p apþ1�(

þan

p�2apþ1an

pþbpþ1þbn

p�2bpþ1bn

pÞ

� iGpGn

p

papþbn

p�2apbn

p

� �): ð51Þ

This expression for the axial component of the forcemust be determined for each position of the beam focuswith respect to the sphere center, which involves deter-mining the BSCs for each of these positions. Owing to thelinearity of the electromagnetic fields, we can add anotherfocused field simply by adding BSCs. Let us denote z1 andz2 as the focal positions of the two independent focusedbeams, instead of the previous notation of z0 for the singlebeam in Eq. (20). For the double beam the correspondingBSC is

Gfbp ¼ ikfeikf

Gpwp

pðpþ1ÞZ αmax

0dα sinα

ffiffiffiffiffiffiffiffiffiffiffifficosα

peikz1 cosαþeikz2 cosα� �

�e�ðf sinα=ωÞ2 ½π1pðαÞþτ1pðαÞ�; ð52Þ

which can be simplified if the beam positions aresymmetric (i.e., when z1 ¼ z0þd=2 and z2 ¼ z0�d=2,where d represents the spacing between both focus) asfollows:

Gfbp ¼ 2ikfeikf

Gpwp

pðpþ1ÞZ αmax

0dα sinα

ffiffiffiffiffiffiffiffiffiffiffifficosα

peikz0 cosα

� cos ðkd=2 cosαÞe�ðf sinα=ωÞ2 ½π1pðαÞþτ1pðαÞ�: ð53Þ

3. Results and discussion

Mathematica (version 10, Wolfram Inc.) was chosen asthe computational software for performing the numericalsimulation presented in this section because of its arbitrary-precision computation, error tracking, and numericallibraries. A suitable truncation number is chosen to termi-nate the infinite sum, according to [47], and the floating-point machine error is chosen. Note that this is correct onlyfor the total cross section, as has been emphasized recently[48]. However, for specific fields instead of cross sections, asimilar rule is applicable in which the maximum number ofterms greater than the largest kr in the plot region shouldbe chosen.

3.1. PNJ as a Bessoidal-type surface

Starting with conventional plane-wave incidence, wedetermine the Poynting vector of the interference betweenthe incident and scattered fields in a region immediatelyafter the scatterer for a 2�μm (Fig. 1) and 4�μm micro-sphere (Fig. 2).

The fitting function for the transverse and longitudinalintensity profiles of a PNJ has been described as a Gaussianand a Lorentzian, respectively [36], noting that the inten-sity distribution of photonic jets along the z-axis (long-itudinal direction) is not symmetric with respect to themaximum intensity. From the results presented here(Fig. 3), we can obtain a dominant function that adequatelyrepresents the profiles of the PNJ. The transverse profile isbetter described by spherical (from Eq. (45)) and cylind-rical Bessel functions (from the small-angle approxima-tion, Eq. (32)), as can be observed by the fitting (Fig. 4).

1.0

0.8

0.6

0.4

0.2

0.0

1.0

1.5

0.5

0.0

-0.5

-1.0

-1.5

Normalized |Sext| (arb.unit)

Nor

mal

ized

|Sex

t| (a

rb.u

nit)

0.2 0.4 0.6 0.8 1.01 2 3 4 56 7 8

Tran

sver

se p

ositi

on, x

(μm

)

Longitudinal position, z (μm)

Fig. 3. PNJ intensity profile for plane-wave incidence. The result is for the 2�μm sphere of Fig. 1 at the maximum intensity. (a) Longitudinal intensityprofile. (b) Transverse intensity profile.

GaussianBessel

1.0

0.8

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8 1.0 1.2

Transverse position, x (μm)

Nor

mal

ized

|Sex

t| (a

rb.u

nit)

Fig. 4. Fitting of the PNJ transverse profile using a cylindrical Bessel typeversus that using a Gaussian function. The result is for the 2�μm sphereof Fig. 3(b), clearly showing a better fitting with the Bessel-type function.

Fig. 5. Contour and vector plot of the intensity and time-averagedPoynting vector, respectively for the interference between the incidentand scattered fields. The result is for the 2�μm sphere of Fig. 3(b), butwith a highly focused (NA¼1.25) Gaussian beam (waist¼2.5 mm beforeat the objective back aperture) placed at the origin (z0 ¼ 0 μm).

Fig. 6. Contour and vector plot of the intensity and time-averagedPoynting vector for the interference between the incident and scatteredfields. The result is for the 2�μm sphere of Fig. 3(b), but with a highlyfocused (NA¼1.25) Gaussian beam (waist¼2.5 mm before at the objec-tive back aperture) located outside the sphere (z0 ¼ 2 μm).

1.0

0.8

0.6

0.4

0.2

0.01 2 3 4 5 6

Longitudinal position, z (μm)

Nor

mal

ized

|Sex

t| (a

rb.u

nit)

Zfocus=0.5μmZfocus=1.0μmZfocus=1.5μmZfocus=2.0μmZfocus=2.5μmZfocus=3.0μmZfocus=3.5μmZfocus=4.0μmZfocus=4.5μmZfocus=5.0μm

Fig. 7. Overlapping plot of the on-axis intensity of the time-averagedPoynting vector for the interference between the incident and scatteredfields. The result is for the 2�μm sphere of Fig. 3(b), but with a highlyfocused (NA¼1.25) Gaussian beam (waist¼2.5 mm before at the objec-tive back aperture) located at specific positions (z0¼zfocus), i.e., differentradial shifts. The dashed line is an r�2 fitting though the maximumprofile points.

-6 -4 -2 0 2 4 6-1.0

-0.5

0.0

0.5

1.0

z (μm)

Nor

mal

ized

Forc

e,F z

(arb

.uni

t)

Fig. 8. Axial optical force as a function of microsphere displacementz ðμmÞ from the focus. The inset illustrates equilibrium configuration, thebeam propagation direction (arrow), the focus (plus) and microsphere(dot).

A.A.R. Neves / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 122–132128

3.2. PNJ from a focused beam

A PNJ is only observed when the focus is close to thesphere, as has been pointed out by Lecler et al. [45], andthe behavior of the PNJ with respect to the focal position ofthe beam was investigated by Devilez et al. [6]. Therefore,for a highly focused beam at the sphere center, which isclose to the stable trapping configuration of an opticaltweezer, we obtain no PNJ as illustrated in Fig. 5.

To observe any appreciable PNJ, we have to shift theincident-beam focus close to the surface. By placing it2 μm from the sphere center, we observed the PNJ of Fig. 6.

-6 -4 -2 0 2 4 6-1.0

-0.5

0.0

0.5

1.0

z (μm)

Nor

mal

ized

Forc

e,F z(

arb.

unit)

Nor

mal

ized

Forc

e,F z(

arb.

unit)

-6 -4 -2 0 2 4 6-1.0

-0.5

0.0

0.5

1.0

z (μm)

Fig. 9. Double co-propagating optical traps: one located at z¼ �1:0 μm and z¼ 1:0 μm, corresponding to (a)–(b), and the other located at z¼ �1:5 μm andz¼ 1:5 μm, shown by (c)–(d). The inset illustrates equilibrium configuration, the beam propagation direction (arrow), the focus (plus) and microsphere(dot). (a) Axial optical-force profile. (b) Contour intensity plot. (c) Axial optical-force profile. (d) Contour intensity plot.

A.A.R. Neves / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 122–132 129

Moreover, we examined the PNJ transverse profile as afunction of beam position, as shown in Fig. 7. It can beobserved that the PNJ due to the interference of theincident and scattered fields, inherits the r�2 intensity-decay characteristic of the scattered field and localizationfrom the highest-intensity position of the incident field.

The challenge here is to obtain a PNJ from an opticallytrapped sphere. In an optical tweezer, we should nothave a PNJ, since the microsphere center is brought nearthe focus of the beam, in a configuration similar to thatof Fig. 5.

3.3. Optical forces in a microsphere

The optical force profile of conventional optical twee-zers is presented in Fig. 8. Note that the equilibriumposition for the microsphere is located immediately after(� 0:2 μm) the highly focused beam position, which is dueto the scattering forces as the focused beam close to thesphere center yields no PNJ.

To circumvent the absence of a PNJ in conventionaloptical tweezers, we adopt double optical tweezers: not theconfiguration commonly employed in which each beamaxis is parallel to each other [49], but in a configuration inwhich both beams are on the same axis. This leads to theassumption of the original counter-propagating traps of1970 [50], but here, the two traps are co-propagating. Inthis double co-propagating optical-tweezers configuration,

an equilibrium position will exist if the gradient force fromone tweezer exceeds the scattering force from the other.

For the 2�μm sphere and two equal-intensity trapslocated 2 μm apart (Fig. 9a), because of the scattering forcefrom the first trap, the equilibrium position is displaced toabout 1:1 μm in the beam-propagation direction. There is acost associated with using these co-propagating traps: thereduction of the trap stiffness. The loss of stiffness for thepresent case is approximately 40% with respect to a singleoptical tweezer. Consequently, the second focused beam ispositioned within the trapped sphere, and no PNJ is gener-ated (Fig. 9b). On increasing the displacement between thetwo traps to 3 μm, we obtain two equilibrium positions, asillustrated in Fig. 9c. For the microsphere located in the firstequilibrium position of z¼�1.2 μm, a focused beam ispresent immediately outside the microsphere, resulting ina PNJ (Fig. 9d). For the second equilibrium position, there isno PNJ, as expected. Therefore depending on where themicrosphere is located within the two possible trap loca-tions, the PNJ can easily be switched on or off by modulatingthe intensity of both traps.

4. Conclusion

In summary, we presented a detailed theoretical inves-tigation of the formal relationship between the opticalforce on a microsphere and its photonic nanojet, whichresults from the interference between the scattered andincident fields. This approach is general, and a complex

A.A.R. Neves / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 122–132130

refractive index can be implemented to take into accountlight absorption by the scatterer. Even though doubleoptical tweezers were employed, a similar principle holdsas for other beam configurations, especially those gener-ated by holographic traps.

Consequently, an understanding between the interactionof an optically trapped microsphere and its PNJ field struc-ture is of fundamental importance in optical physics and haspractical significance in applications such as imaging, nano-lithography, detection, metrology, biophotonics, and spectro-scopy. Such an understanding also provides an opportunityto manipulate the intensities of two traps differently in orderto achieve a controlled PNJ near an optically trapped micro-sphere. Moreover, the loss in trap stiffness can be easilyovercome using anti-reflection coatings to enhance thetrapping force [51]. In the case of trapping soft spheres, suchas water drops, as has been recently reported [52], laser-induced surface stress would break the spherical symmetryof the scatterer leading to spheroidal particles. The PNJsgenerated by such scatterers have recently been reported in[53]. Finally when whispering-gallery modes or pulsedbeams are of interest, they can also be detected in an opticaltrap [49], thereby validating recent simulations on photonic-jet shaping [54] and temporal dynamics of the jet [55].

Acknowledgments

The present work received support from CNPq(308627/2012-1), Conselho Nacional de DesenvolvimentoCientífico e Tecnológico and FAPESP (2014/07191-9), Fun-dação de Amparo à Pesquisa do Estado de São Paulo, Brazil.

Appendix A. Shape-coefficient integrals of the beam

In this appendix, the analytical integrals for the inte-gration of the BSCs are presented. The first integral, interms of the azimuthal angle ϕ, is of the typeZ 2π

0dϕe� imϕ

cosϕsinϕ

" #¼ π

δq;1þδq;�1

� iðδq;1�δq;�1Þ

" #; ðA:1Þ

which does not depend on θ. Therefore the remainingintegral, in terms of the polar angle θ for the plane-wavecase, is of the typeZ π

0dθ sin 2θPq

pð cosθÞeikr cosθ : ðA:2Þ

This integral resembles a formerly reported one [56],the solution of which isZ π

0dθ sinθPq

pð cosθÞeikr cosα cos θJqðkr sinα sinθÞ

¼ 2ip�qPqpð cosαÞjpðkrÞ: ðA:3Þ

Now, we are interested in taking the limit α-0 of theabove integral. Because of the properties of the Besselfunction and associated Legendre functions, we obtain, forq¼ þ1 and q¼ �1,Z π

0dθ sin 2θP1

p cosθ� �

eikr cosθ ¼ 2p pþ1ð Þipþ1jpðkrÞkr

; ðA:4Þ

Z π

0dθ sin 2θP�1

p cosθ� �

eikr cosθ ¼ �2ipþ1jpðkrÞkr

; ðA:5Þ

thereby eliminating the radial-dependency term, jpðkrÞ=kr,from the BSCs.

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