Polarization-controlled directional scattering for nanoscopic position sensing

Abstract

Controlling the propagation and coupling of light to sub-wavelength antennas is a crucial prerequisite for many nanoscale optical devices. Recently, the main focus of attention has been directed towards high-refractive-index materials such as silicon as an integral part of the antenna design. This development is motivated by the rich spectral properties of individual high-refractive-index nanoparticles. Here we take advantage of the interference of their magnetic and electric resonances to achieve strong lateral directionality. For controlled excitation of a spherical silicon nanoantenna, we use tightly focused radially polarized light. The resultant directional emission depends on the antenna's position relative to the focus. This approach finds application as a novel position sensing technique, which might be implemented in modern nanometrology and super-resolution microscopy set-ups. We demonstrate in a proof-of-concept experiment that a lateral resolution in the Ångström regime can be achieved.

ARTICLE

Received 11 Dec 2015 | Accepted 8 Mar 2016 | Published 20 Apr 2016

Controlling the propagation and coupling of light to sub-wavelength antennas is a crucial prerequisite for many nanoscale optical devices. Recently, the main focus of attention has been directed towards high-refractive-index materials such as silicon as an integral part of the antenna design. This development is motivated by the rich spectral properties of individual high-refractive-index nanoparticles. Here we take advantage of the interference of their magnetic and electric resonances to achieve strong lateral directionality. For controlled excitation of a spherical silicon nanoantenna, we use tightly focused radially polarized light. The resultant directional emission depends on the antennas position relative to the focus. This approach nds application as a novel position sensing technique, which might be implemented in modern nanometrology and super-resolution microscopy set-ups. We demonstrate in a proof-of-concept experiment that a lateral resolution in the ngstrm regime can be achieved.

DOI: 10.1038/ncomms11286 OPEN

Polarization-controlled directional scattering for nanoscopic position sensing

Martin Neugebauer1,2, Pawe Woniak1,2, Ankan Bag1,2, Gerd Leuchs1,2,3 & Peter Banzer1,2,3

1 Max Planck Institute for the Science of Light, Gnther-Scharowsky-St. 1, D-91058 Erlangen, Germany. 2 Institute of Optics, Information and Photonics, Department of Physics, Friedrich-Alexander-University Erlangen-Nuremberg, Staudtstrasse 7/B2, D-91058 Erlangen, Germany. 3 Department of Physics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario K1N 6N5, Canada. Correspondence and requests for materials should be addressed to P.B. (email: mailto:[email protected]

Web End [email protected] ).

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Cylindrical vector beams are well-established tools in modern microscopy, ranging from scanning microscopy, where a reduced focal spot size can be achieved with a

radially polarized beam14, to more sophisticated techniques such as stimulated-emission depletion5,6 and multi-photon microscopy7. In addition, those polarization-tailored beams have also paved the way towards versatile applications in recent nanophotonic experiments by enabling selective excitation of nanoparticle eigenmodes8,9 or controllable directional emission and waveguide-coupling of single plasmonic nanoantennas10.

In this work, we combine several aspects of both research elds to present a novel approach towards high-precision position sensing, a discipline that is of paramount importance in modern nanometrology1118, because of its special role in super-resolution microscopy1922. Our all-optical technique for localization of a single nanoantenna is thereby based on encoding the position of the antenna in its laterally directional scattering pattern. For that purpose, we take advantage of the resonance properties of a high-refractive-index silicon nanoantenna featuring electric and magnetic resonances2326.

ResultsExcitation scheme. It was shown that the simultaneous excitation of transverse electric and magnetic resonances of a high-refractive-index dielectric nanoparticle may yield enhanced or suppressed forward/backward scattering due to their interference2730. However, by carefully structuring the excitation eld three-dimensionally (3D) and thus also exciting longitudinal particle modes9, the scattering pattern can be tailored to achieve lateral directivity in the far eld. For example, a tightly focused radially polarized beam features a promising 3D focal eld with cylindrical symmetry1,2. Figure 1ac shows its electric and magnetic eld intensity distributions and the corresponding phases, calculated by vectorial diffraction theory31,32, while taking into account the experimental parameters. Apart from the transverse (in-plane) radially polarized electric eld E? Ex; Ey

,

a strong longitudinal component Ez is formed, reaching its maximum amplitude on the optical axis. In contrast, the magnetic eld H? Hx; Hy

is purely transverse and azimuthally polarized. In close vicinity to the optical axis, Ez exhibits a phase delay of Dfz p/2 with respect to the

transverse eld components, and the electric and magnetic elds can be approximated by:

E x; y

/ xE0?^x yE0?^y iE0z^z; 1 H x; y

/ yH0?^x xH0?^y: 2 Here, E0?, E0z and H0? are real valued amplitudes of the transverse

electric, longitudinal electric and transverse magnetic eld components, respectively, and (x, y) are Cartesian coordinates in the focal plane. Without loss of generality, the point in time is chosen such that the transverse eld components Ex, Ey, Hx and

Hy are real, and the longitudinal component Ez is imaginary, owing to the aforementioned phase delay of p/2. For the chosen beam parameters (see Methods section), we estimate equations(1) and (2) to be valid within the region up to 50 nm away from the optical axis (see grey area in Fig. 1d). In this limited range, the transverse electric and magnetic elds are linearly dependent on the coordinates x and y, while Ez is assumed to be approximately constant. In order to adopt this linear position dependence of the transverse electromagnetic eld for position sensing, a sub-wavelength antenna, capable of incorporating the local eld in its far-eld emission pattern, is required to localize the antenna unambiguously by its (directional) scattering pattern recorded in the far eld. In the following, we discuss a silicon nanosphere (radius r 92 nm), whose spectrum in the visible range was

experimentally investigated previously9, and we explain how its

far-eld emission pattern is governed by its position.

Tailored directional scattering. Figure 2a shows the scattering cross-section of the antenna sitting on a glass substrate, simulated using the nite-difference time-domain method (similar to ref. 9). Here, only the forward scattering efciency into the angular region within the numerical aperture (NA)A[0.95, 1.3] is considered to match the experimental detection scheme described below (see also Fig. 2be). In the visible spectral range, the silicon antenna supports the following three pronounced resonances:9 the magnetic dipole (lMDE670 nm), the electric dipole (lEDE540 nm) and the magnetic quadrupole (lMQE515 nm). For wavelengths above 600 nm, the weak contribution of the magnetic quadrupole can be completely neglected9, and the antenna can be approximated by a point-like dipole (electric and magnetic). Assuming that the dipole moments are proportional to the respective local eld vectors, ppE and mpH, we yield the position-dependent dipole moments p / xE0?^x yE0?^y iE0z^z and m / yH0?^x xH0?^y.

The aim of our experimental concept is to achieve highly position-sensitive far-eld directivity caused by the interference of the, in rst approximation, constant z-oriented electric dipole pz and the position-dependent transverse components of the magnetic dipoles mx and my. The inuence of the transverse electric dipole components px and py will be proven to be negligible later on.

In Fig. 2b, the far-eld intensities of a z-oriented electric dipole (see dashed red line) and a y-oriented magnetic dipole (see black line) emitted into the glass substrate are depicted. Here, we consider the electric and magnetic dipole moments to exhibit the same strength. If the dipole moments are in phase, the interference of both far elds yields a remarkably strong lateral directivity (Fig. 2c). Figure 2d shows the corresponding calculated k-spectrum in the experimentally accessible region within NAA[0.95, 1.3]. At this point, the relative phase between the longitudinal and the transverse eld components (Dfz p/2,

see insets in Fig. 1) of the excitation beam needs to be considered. If the electric and magnetic dipoles oscillate p/2 out of phase, no directivity would be observed in the far eld because their symmetric far-eld intensity distributions add up. Hence, an additional phase of p/2 is required to compensate for Dfz. Since the relative phase between a dipole moment and its respective excitation eld (DfMD for the magnetic, DfED for the electric eld) depends on the wavelength, we can compensate for Dfz by

carefully choosing the wavelength of the incoming light with respect to the spectral positions of the electric and magnetic dipole resonances. From simulation, we retrieve the relative phase between the electric and magnetic dipole moment to be Df DfMD DfED p/2 for a wavelength of l 652 nm

(Supplementary Fig. 1 and Supplementary Note 1). Using this wavelength for excitation, we expect to achieve strongly directional scattering at antenna positions where the longitudinal electric and transverse magnetic elds overlap. For experimental verication, a measured far-eld image is plotted in Fig. 2e. The image was retrieved by placing the antenna on the x axis B140 nm away from the centre of the beam in the focal plane, effectively obtaining longitudinal electric and transverse magnetic dipole moments of equal strength. The strong directivity proves that the compensation of the relative phase has been successful, and the very good overlap of 94% with the theoretical pattern suggests that, in rst-order approximation, the transverse electric dipole moments, not taken into account here, can indeed be neglected (details can be found in Supplementary Fig. 2 and Supplementary Note 2).

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a b

0

0.2

0.6

0

0.8

0.8

0.2

0

0 0.4

E2

H2

E

Ez2

zE

Lateral position y(m)

0

Lateral position y(m)

Lateral position x (m) Lateral position x (m)

0

0.8

0 0.8

0.8

0 0.8

c d

0

0.2

0.4

Re[Ex]

Re[Hy]

Im[Ez]

1

0

0.8

0

H

Lateral position y(m)

1 0.8

0 0.8

Lateral position x (m)

0

0.2

0.2

Re[Ex]

Re[Hy]

Im[Ez]

1.0

0.6

0

0.8

0.8

0 0.8

0 80

Lateral position x (m)

80 Lateral position x (nm)

Figure 1 | Theoretical eld intensity distributions and relative phases of a tightly focused radially polarized beam. The wavelength l 652 nm and

experimental parameters are taken into account (see Methods section). (a) The transverse (radial) electric eld intensity E?

j j2 Ex

j j2 Ey

2, (b)

the

longitudinal electric eld intensity Ez

j j2 and (c) the transverse (azimuthal) magnetic eld H?

j j2 Hx

j j2 Hy

2

are all normalized to the maximum value of

j j2 (Gaussian units). The corresponding phase distributions fE?, fEz and fH? are plotted as insets. (d) Cross-sections of

the focal elds along the x axis. Close to the centre (grey area in the lower image), the transverse eld amplitudes Re[Ex] and Re[Hy] are linearly dependent on the position, while the longitudinal eld Im[Ez] is approximately constant.

the total eld intensity, Itot E

j j2 H

In short, we optimized the polarization distribution and the wavelength of our excitation beam to achieve strongly directional emission depending on the position of a single silicon nanoantenna relative to the beams optical axis. The underlying principle causing the directivity is the simultaneous and in-phase excitation of a longitudinal electric and a transverse magnetic dipole moment.

Experimental implementation and calibration. A sketch of the experimental set-up is depicted in Fig. 3a (for more details see ref. 33). The collimated incoming radially polarized beam (l 652 nm) was tightly focused by a microscope objective

with an NA of 0.9 onto the silicon nanosphere sitting on a glass substrate (see electron micrograph in Fig. 3a), which was positioned precisely within the focal plane by a 3D piezo-stage. A second microscope objective (oil-immersion type, NA 1.3)

below the substrate collected both the transmitted beam and the forward scattered light. Imaging the back-focal plane of the second microscope objective onto a CCD camera enabled acquisition of the intensity distribution emitted into the far eld and grants access to the angular spectrum of the scattered eld

(see examples in Figs 2e and 3b,c). Similar to ref. 10, only the region of NAA[0.95, 1.3] was considered, where the scattered light can be detected without interfering with the transmitted beam.

To retrieve the position of the antenna from the back-focal plane images, we averaged the measured intensity over four small regions in k-space (black dotted lines in Fig. 3b,c)34, resulting in four averaged intensity values I1, I2, I3 and I4. The size and position of these four regions was chosen to include only the strongest change of the far-eld intensity for an antenna shift along the x or y axis. The normalized intensity differences Dx I3 I1

=

~I and Dy I2 I4

=

~I, with ~I I1 I2 I3 I4

=2,

represent directivity parameters, which are linear functions of the antenna position (Supplementary Note 3).

In order to compensate for experimental imperfections such as beam aberrations or deviations from the ideal antenna shape, the measurement approach requires initial calibration, for which we placed the antenna centrally in the focus. At this position, the far-eld distribution of the scattered light is expected to be cylindrically symmetric, since only a longitudinal electric dipole moment can be excited (Fig. 3b)9,10. From this reference point,

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a

Scattering (a.u.)

1

=652

ED

MD

MQ

0 480 520 560 600 640 680 720

Wavelength (nm)

b c

my, pz my + pz

y

x

Air Glass

z

NA=1.3

NA=0.95

d e

1.3

1.3

k y/k 0

0

0 Max

Critical angle

Intensity I (a.u.) Intensity I (a.u.)

Theory

k y/k 0

0

0 Max

Experiment

1.3 0 1.3

1.3 0

1.3

kx /k0 kx /k0

Figure 2 | Scattering of a silicon nanoantenna sitting on a dielectric interface. (a) Simulated scattering cross-section (linearly polarized Gaussian beam used for excitation) of a silicon sphere with radius r 92 nm; only the forward scattering efciency into the angular region within the NAA[0.95, 1.3]

is considered to match the experimental detection scheme (see grey arcs in b,c, and far-eld patterns in d,e). In the visible range, the nanosphere supports magnetic dipole (lMDE670 nm), electric dipole (lEDE550 nm) and magnetic quadrupole (lMQE520 nm) resonances. At the excitation wavelength of l 652 nm, the magnetic and electric dipole moments are p/2 out of phase with respect to each other. (b) Emission of a longitudinal electric dipole pz

(see dashed red line) and a transverse magnetic dipole my (see black line) into the glass substrate. (c) In-phase far-eld interference of pz and my results in strong directivity. Comparison of (d) a calculated far-eld pattern (interference of pz and my) and (e) a measured back-focal-plane image, retrieved at an antenna position on the x axis 140 nm away from the centre of the beam.

the antenna was scanned across the focal plane (100 100 nm),

with a step-size of 10 nm. For each position, an image of the back-focal plane was acquired and the corresponding values of Dx and

Dy were determined. The whole procedure was repeated 40 times and the measured directivity parameters were averaged in order to decrease the inuence of the instability of our set-up (position uncertainty of 5 nm). Thereupon, linear equations were tted to the averaged directivity parameters Dx(x, y) and Dy(x, y) (see equation (8) in the Methods section), which allow for retrieving the antenna position from individual back-focal plane images. As an example, Fig. 3d shows the averaged directivity parameter Dx plotted against the x coordinate and the corresponding linear t.

Lateral resolution. In order to demonstrate the accuracy in the measurement of the antenna position, which can be achieved with a single camera shot, far-eld images for different antenna positions are analysed. To this end, we normalize the intensity maps recorded in each back-focal plane to the intensity ~I and calculate the difference to a reference image, which corresponds to the antenna sitting on the optical axis (Fig. 3b). For the demonstration of this inherently 2D localization technique, we show results for x displacements only. In Fig. 3e, we depict four

post-selected difference images for the antenna being placed on the x axis, for which our calibration measurement indicated relative positions of DxE40, 20, 10 and 5 nm (DyE0 nm). For a relatively large displacement of DxE40 nm, the difference image corresponding to the difference between Fig. 3b,c yields a very good signal-to-noise ratio. Even for a small displacement of only DxE5 nm, the difference image reveals predominately negative values on the left side (kxo0) and positive values on the right side (kx40). However, the signal-to-noise ratio decreases with shorter distances Dx. The theoretical limit of our resolution is determined by the derivatives (slopes) of Dx(x, y) and Dy(x, y), the intensity noise of an individual camera pixel, and the actual number of pixels in each integration region. Our calculations yield that a position uncertainty below 2 could be achieved. More details and an actual experimental example can be found in Supplementary Fig. 3 and Supplementary Note 4. However, a direct proof of this accuracy would require a highly stabilized set-up including a piezo-stage with ngstrm precision.

DiscussionIn summary, we experimentally demonstrated that the simultaneous and phase-adapted excitation of longitudinal electric and

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a b c

1.3

1.3

NA0.9

NA1.3

CCD

I4

I2

Intensity I (a.u.)

0 0.6

0.4

0.2

I3 I1

x 0 nm

I4

I2

Intensity I (a.u.)

0 0.8

0.4

I3 I1

x 40 nm

k y/k 0

0

k y/k 0

0

200 nm

1.3

0

1.3

1.3

0

1.3

kx/k0 kx/k0

d e

0.5

0.5

Directivity x

0.2 0

0.2

I (a.u.)

0.1 0

0.1

I (a.u.)

0.1 0

0.1

I (a.u.)

0.05 0

0.05

I (a.u.)

0

0

50 50

x 40 nm

x 20 nm

x 10 nm

x 5 nm

Lateral position x (nm)

Figure 3 | Set-up and experimental results. (a) A radially polarized collimated beam is tightly focused by a microscope objective (NA 0.9) onto a

sub-wavelength silicon antenna sitting on a glass substrate (see inset). The light emitted into the angular regime with NAA[0.95, 1.3] is collected by an oil-immersion type microscope objective (NA 1.3). The back-focal plane is imaged onto a CCD camera. (b) Acquired far-eld intensity distribution

I(kx, ky) for the antenna placed on the optical axis and (c) for the antenna displaced laterally by xE40 nm (off-axis). The dashed black lines and the magnied insets indicate four regions with averaged intensities I1, I2, I3 and I4. (d) Directivity parameter Dx versus the antennas position along the x axis; the slope of the curve (0.01 nm 1) denes the sensitivity of the measurement to the antenna displacement. (e) Far-eld intensity difference images for four antenna positions along the x axis (left to right: DxE40, 20, 10 and 5 nm away from the optical axis). The left image corresponds to the intensity difference DI(kx, ky) between b,c.

transverse magnetic dipole modes of a high-refractive-index nanosphere yields extraordinarily strong directionality. Especially the spectral tuning of the relative phase between both dipole modes in combination with the appropriate choice of a 3D focal eld pattern enabled highly position-sensitive transverse scattering directionality. We utilized the approach as a novel technique for single-shot lateral position sensing, achieving localization accuracies down to a few ngstrm, which is comparable to other state-of-the-art localization methods presented in literature12,13,18. Our technique could be applied for the stabilization of samples, for instance, in super-resolution microscopy. Furthermore, since the directionality is also present in the super-critical regime (NA41), evanescent coupling to waveguide modes will allow for on-chip detection of the directional scattering and, hence, of the lateral position of the sample. Finally, future studies might demonstrate that antenna design and size, as well as the excitation eld can be optimized to achieve an even stronger dependence of the directionality on the particle position, which would allow for sub-ngstrm localization accuracies.

Methods

Experimental set-up. A tunable light source (NKT Photonics SuperK Extreme & SpectraK Dual) emits a linearly polarized Gaussian beam at a wavelength of652 nm, which is converted into a radially polarized beam by a liquid-crystal polarization converter (q-plate)35,36. The beam with radius w0 1.26 mm is then

guided into a microscope objective with NA 0.9 and an entrance aperture radius

of 1.8 mm (Leica HCX PL FLUOTAR 100 /0.90 POL 0/D). A single spherical

silicon nanoparticle with radius r 92 nm on a glass substrate is scanned through

the focal plane by a high-precision 3D piezo-stage (PI P-527), and the transmitted light is collected with an oil-immersion objective with NA 1.3 (Leica HCX PL

FLUOTAR 100 /1.30 OIL). The angular intensity distribution of the transmitted

light is detected by imaging the back-focal plane of the oil-immersion objective onto a CCD camera (The Imaging Source DMK 23U618). The four solid angles

corresponding to I1, I2, I3 and I4 (see dashed lines in Fig. 3b,c) are dened by an azimuthal angular range of DF 45 and by NAA[0.98, 1.02].

Calculation of the far-eld distribution. We make use of the cylindrical symmetry of the beam and, without loss of generality, only consider antenna positions along the x axis. Therefore, only the longitudinal electric (pz) and transverse magnetic (my) dipole moments need to be considered. The transverse electric (s-polarized) and transverse magnetic (p-polarized) far-eld distributions r l

emitted into the dielectric substrate (refractive index n 1.5) are expressed

in ref. 32.

EED;zp Ctp

k?

k0 pz; 3

EED;zs 0; 4

EMD;yp

Cc0 tp

kx k?

my; 5

EMD;ys

Cc0 ts

p

ky k0k?

k20 k2?

my; 6

with

p d; 7

the Fresnel coefcients for transmission tp and ts, the wavenumber in vacuum

k0 2p/l, the transverse component of the k-vector k? k2x k2y

1=2

C eik nr

k20

p

4prE0

k20n2 k2?

ei

k k

p

k20 k2?

and the

vacuum speed of light c0. Comparison between theoretical and experimental far-eld patterns enables estimating the distance between the effective point-like dipole and the interface, d 70 nm. The emission patterns in Fig. 2b,c are

calculated using equations (3)(6), whereby we considered a similar strength for both dipole moments pz my/c0 to achieve maximum directivity. Taking into

account the aplanatic microscope objective, an additional energy conservation factor proportional to k20n2 k2?

1=2

is introduced for Fig. 2d (ref. 32).

Calibration measurement. The directivity parameters Dx and Dy are linear functions of the lateral antenna position x and y, respectively. Thus,

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we t a system of two linear equations to the averaged calibration measurement data, resulting in

x

y

103:4 nm 4:8 nm

2:5 nm 94:8 nm

3:5 nm

12:7 nm

: 8

Ideally, the matrix has non-zero values on its diagonal only. The small off-diagonal elements indicate a minor rotation of the coordinate system and, in addition, not entirely orthogonal directivity parameters Dx and Dy. The rotation of the coordinates might stem from a misalignment of our camera with respect to the coordinate frame of the piezo-stage, while the non-orthogonal basis can be related to aberrations of the beam and asymmetries of the antenna (see electron micrograph in Fig. 3a). The derivatives (slopes) of Dx and Dy dene the sensitivity of the directivity to a displacement of the antenna. At the rim of the region of linearity, 50 nm away from the centre, we already achieve a directivity Dx 48%

(Dy 52%) if the antenna is shifted in x direction (y direction).

Post-selection of difference images. The instability of our experimental set-up causes an uncertainty of 5 nm regarding the position of the particle relative to the beam. For this reason, we took 40 individual images for each position set by the piezo-stage (DxE40, 20, 10 and 5 nm), and then post-selected the far-eld images of which the directivity parameters Dx and Dy best-represented the position set by the piezo-stage according to the calibration measurement (equation (8)). Finally, we calculated the difference images depicted in Fig. 3e.

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Acknowledgements

We thank A. Rubano and L. Marrucci for the fabrication of the liquid-crystal polarization converter (q-plate) utilized for the generation of the radially polarized beam, andT. Bauer and U. Mick for inspiring discussions. P.B. acknowledges nancial support by the Alexander von Humboldt Foundation and the Canada Excellence Research Chair (CERC) in Quantum Nonlinear Optics.

Author contributions

P.B. and M.N. conceived the idea and the experiment; P.W. prepared the sample; P.W. and A.B. performed the measurements; A.B. and M.N. analysed the data; M.N., P.W. and P.B. wrote the manuscript; P.B. and G.L. supervised the project; and all authors discussed the results and commented on the nal manuscript.

Additional information

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Competing nancial interests: The authors declare no competing nancial interests.

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How to cite this article: Neugebauer, M. et al. Polarization-controlled directional scattering for nanoscopic position sensing. Nat. Commun. 7:11286doi: 10.1038/ncomms11286 (2016).

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