Optical Polarization M\"obius Strips and Points of Purely Transverse Spin Density

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by  Thomas Bauer

Optical Polarization Mo¨bius Strips and Points of Purely Transverse Spin Density
Thomas Bauer,1,2 Martin Neugebauer,1,2 Gerd Leuchs,1,2,3 and Peter Banzer1,2,3,
∗
1Max Planck Institute for the Science of Light, Guenther-Scharowsky-Str. 1, D-91058 Erlangen, Germany
2Institute of Optics, Information and Photonics,
University Erlangen-Nuremberg, Staudtstr. 7/B2, D-91058 Erlangen, Germany
3Department of Physics, University of Ottawa, 25 Templeton, Ottawa, Ontario K1N 6N5 Canada
(Dated: January 25, 2016)
Tightlyfocusedlightbeamscanexhibitelectricfieldsspinningaroundanyaxisincludingtheone
transverse to the beams’ propagation direction. At certain focal positions, the corresponding local
polarizationellipsecandegenerateintoaperfectcircle,representingapointofcircularpolarization,
orC-point. WeconsiderthemostfundamentalcaseofalinearlypolarizedGaussianbeam,where–
6 upon tight focusing – those C-points created by transversely spinning fields can form the center of
1 3Dopticalpolarizationtopologieswhenchoosingtheplaneofobservationappropriately. Duetothe
0 high symmetry of the focal field, these polarization topologies exhibit non trivial structures similar
2 to Mo¨bius strips. We use a direct physical measure to find C-points with an arbitrarily oriented
spinning axis of the electric field and experimentally investigate the fully three-dimensional polar-
n
ization topologies surrounding these C-points by exploiting an amplitude and phase reconstruction
a
J technique.

2
PACSnumbers: 03.50.De,42.25.Ja,42.50.Tx
2

]
s Introduction.—Structuredlightfieldsrepresentimpor- plasmons [24–26], it was shown that tSAM is also natu-
c tant tools in modern optics due to their multitude of rallypresentinmanyfocusingscenarios[6,23,27–29]. As
i
t different applications, for example in microscopy [1–3], anexample,evenanordinarybeamsuchasalinearlypo-
p
nano-optics [4–7] and optical trapping [8, 9]. In general, larizedfundamentalGaussianbeamexhibitstransversely
o
. the spatial tailoring of amplitude, phase and polariza- spinning fields when tightly focused [6, 29, 30]. In this
s
c tion of paraxial or non-paraxial light beams can lead to case, the transverse (E ) and longitudinal (Ez) electric
i interestingandsometimescomplextopologicalstructures field components exhibi⊥t in the focal plane a π/2 phase
s
y of the electric field such as phase vortices [10, 11], opti- differencewithrespecttoeachother,andpositionswhere
h cal knots [12, 13] and optical polarization M¨obius strips both field components have the same amplitude repre-
p
[14–16]. Thelatterarelinkedtothepresenceofso-called sentC-pointsthatarelinkedtopurelytSAM.Thisraises
[
C-pointsorC-linesinthefielddistribution,i.e.positions, the question, whether those focal C-points of a Gaussian

1 wherethelocalpolarizationellipsedegeneratestoacircle beam also exhibit complex polarization topologies.
v and the field is circularly polarized [17]. While polariza- In this paper, we investigate the occurrence of M¨obius-
2
tionM¨obiusstripshavebeenpredictedadecadeagoand like optical polarization topologies in the focal field dis-
7
investigatedeversince[18],onlyrecentadvancesinnano- tributionofatightlyfocusedlinearlypolarizedGaussian
0
6 optics, inparticular3Damplitudeandphasereconstruc- beam. Furthermore, we show the generation of optical
0 tion techniques at the nanoscale [19], enabled their ex- polarization M¨obius strips around C-points with non-
1. perimentalverification[16]. Thiswasachievedbytightly zerotransversespinintheexperimentalrealizationofthe
0 focusing a light beam with spatially varying polarization mentioned field configuration using a nanointerferomet-
6 spanning the full Poincar´e sphere [20–22]. In the focal ric amplitude and phase reconstruction technique [19].
1
plane of such a full Poincar´e beam, this yields a complex C-points, transversespin and optical polarizationM¨obius
:
v fully vectorial field structure, including a C-point on the strips.—When examining polarization topologies in fully
i optical axis with the electric field spinning around said vectorial and highly confined structured light fields, the
X
(longitudinal) axis. Tracing the major axis of the polar- electric field E at each point r is in general oscillating in
r
a ization ellipse around this C-point allowed for revealing a plane oriented arbitrarily in space. This means, that
opticalpolarizationM¨obiusstripshiddeninthefocalpo- the local field can be described by a polarization ellipse
larization distribution [16]. with its major axis α(r), minor axis β(r) and normal
Inmoregeneralfielddistributions,C-pointswithanarbi- vector γ(r) defined in 3D-space by [31]
trarilyorientedspinningaxisoftheelectricfieldmightbe
observed. Thisincludesthespecialcaseofaspinningaxis 1
α= Re E √E E ,
perpendiculartotheopticalaxis[17],whichindicatesthe √E E ∗ ·
presence of purely transverse spin angular momentum | · | (cid:16) (cid:17)
1
(tSAM) [23]. While the rising interest in this intriguing β = Im E∗√E E , (1)
√E E ·
polarization component is strongly linked to its occur- | · | (cid:16) (cid:17)
rence in highly confined fields within guided modes and γ =Im(E∗ E),
×
2
with E denoting the complex conjugate of the field. that the longitudinal component is π/2 out of phase
∗
±
Thus, the field is oscillating in the plane spanned by α with respect to the in-phase transverse components,
andβ. Thenormalvectorγ is,inthisdefinition,directly indicating purely transversely spinning fields throughout
proportional to the electric part of the spin density [32– the focal plane. This can also be seen in the components
34] of the electric spin density (see Fig. 1(b) and Ref. [29])
and is a feature exhibited by many fundamental beams
(cid:15)
sE(x,y,z)= 0 Im(E∗ E), (2) after tight focusing. The strongest and, therefore,
4ω ×
dominant component of the electric spin density is sx,
E
with (cid:15) the permittivity of free space and ω the angu- with sy being weaker by one order of magnitude. As
0 E
lar frequency of the monochromatic light field. Since the expected, the longitudinal component sz is identical
E
electricspindensityspecifiestheorientationandsenseof zero, which implies that points of circular polarization
thespinningaxisofthelocalelectricfieldandrepresents in this case can only be linked to tSAM. In order to
aphysicallymeasurablequantity[29,33], anelegantand determine those C-points of purely transverse spin, the
straight forward method to determine points of arbitrar- ratio sE /wE is plotted in Fig. 1(c). The red solid
| |
ilyorientedcircularpolarization(C-points)inavectorial lines mark the C-lines, or equivalently, lines along which
light field is to normalize s with the energy density of Eq. (3) is fulfilled. It is worth mentioning, that in this
E
the electric field, w = (cid:15)0 E 2+ E 2+ E 2 [32]. theoretically calculated field distribution, C-points can
E 2 | x| | y| | z| only be found in the actual focal plane of the linearly
Thisamplitude-independent(cid:16)measureismaximized(cid:17)when
polarized tightly focused Gaussian beam. This is also
the polarization ellipse degenerates to a circle, and we
shown in Fig. 2(a), where the polarization ellipses are
thus can define a simple requirement for C-points:
plotted as solid white lines on a cross-section of the
s 1 electric energy density in the yz-plane for x = 0. Due
E
| | = . (3)
to the different phase velocities for the transverse and
w 2ω
E
longitudinal electric field components of the tightly
As discussed before, the polarization ellipse can show focused beam, all polarization ellipses outside the focal
intriguing topological patterns around such points in plane form ellipses, which in the far-field (z λ) have
(cid:29)
space. Bychoosingaplaneofobservationcontainingthe to transform to the initially linear polarization. It can
C-pointandwiththenormalvectorofthatplaneparallel be seen in Figs. 1(c) and 2(a) that along the y-axis in
tothespinningaxisγ oftheelectricfieldatthisC-point, the focal plane additional C-points can be found further
fundamental polarization topologies with a topological away from the optical axis.
index of 1/2 (in contrast to the integer numbered In the following, we concentrate on three specific
±
topological index of phase vortices) can be revealed C-points and investigate their polarization topologies in
[17, 35]. This half integer index is allowed, since the appropriately selected planes of observation. First, we
orientation of a (polarization) ellipse is indistinguishable examine a C-point on the y-axis (C1), approximately
under a rotation by π [15]. 275 nm away from the optical axis (see black crosses
Within the aforementioned formalism, we investigate in Figs. 1(a-c)), and look at the local polarization
the field distribution of a tightly focused y-polarized ellipses in its principal plane (here, the yz-plane).
Gaussian beam propagating along z. This scenario All electric field vectors and, therefore, also the local
corresponds to one of the most fundamental beam polarization ellipses in this plane are in-plane only, since
configurations in optics labs and is used in many exper- E (0,y,z) = 0 (see also Fig. 1(a)). Thus, we can use
x
imental studies. Figure 1(a) depicts the electric energy 2D polarization topologies to characterize the points
density and phase (insets) distributions of all three of circular polarization in the chosen meridional plane
Cartesian components in the focal plane [30, 36] for a of observation. With the terminology developed in
numerical aperture of 0.9, a wavelength of λ = 530 nm Refs. [17, 35], C1 represents a lemon-type polarization
and a fill-factor of the focusing aperture of w0/f =1.21. topology. The major axis of the polarization ellipse
The dominant field component E 2 has its maximum rotates clockwise when traced clockwise around the
y
| |
on the optical axis, while the longitudinal field E 2 central C-point and it performs a rotation of π (see
z
| |
shows a two-lobe structure, elongating the focal spot the upper part of Fig. 2(b)). Thus, the topological
along the y-axis [30, 37]. The crossed polarization index of C1 is +1/2. In contrast, the next C-point (C2)
component E 2, occurring analog to the longitudinal on the y-axis at a distance of approximately 455 nm
x
field compon|ent| due to a rotation of the polarization shows a star-type polarization topology (see the lower
components upon focusing, exhibits the characteris- part of Fig. 2(b)), associated with an index of 1/2.
−
tic four-lobe structure observed in cross-polarization This alternation of the two different planar topologies
measurements in many microscopy studies [38] and continues when moving further away from the optical
referred to as depolarization [39]. Considering the axis.
relative phases between the field components, we see Despite these sign changes of the topological index, the
3
π π π
(a) Φ Φ Φ (c)
0 x 0 y 0 z
x |E0.5x| 2 -π x3 10−3 |E0.5y| 2 -π 01.8 |E0.5z| 2 -π 0.12 0.5 |sE| / wE [11/2ω]
0.25 0.25 0.25
y y [µm] 0 2 0 00..46 0 0.08 0.25 0.8
z −0.25 1 −0.25 −0.25 0.04
0.2 0.6
−0.5 −0.5 −0.5 m]
(b) sx 0.5 0.25 0 −0.25−0.5 sy 0.5 0.25 0 −0.25−0.5 sz 0.5 0.25 0 −0.25−0.5 y [µ 0 0.4
E E E
1 0.06 −0.25
0.5 0.5 0.5
0.2
0.25 0.25 0.25
m] −0.5
y [µ−0.250 0 −0.250 0 −0.250 0 0.5 0.25 x [µ0m] −0.25 −0.5
−0.5 −0.5 −0.5
0.5 0.25 0 −0.25−0.5 −1 0.5 0.25 0 −0.25−0.5 −0.06 0.5 0.25 0 −0.25−0.5 C1 C1*
x [µm] x [µm] x [µm]
FIG. 1. (color). (a) The theoretically calculated components of the focal electric energy density distribution E 2, E 2 and
x y
E 2ofatightlyfocusedlinearlyy-polarizedGaussianbeam,normalizedtothemaximumelectricenergydensity.| Co|rre|spo|nding
z
| |
phase distributions are plotted as insets. (b) The distributions of the two non-zero components of the transverse spin density
sx andsy inthefocalplane(normalizedtothemaximumvalueofsx). Duetothesymmetryofthelightfield,sz isidentical
E E E E
zero across the whole focal plane. (c) Focal distribution of s normalized to the local electric energy density w . The red
E E
solidlinescorrespondtothemaximumvalue 1 . Theblacka|nd|graymarkersinalldistributionscorrespondtotheconsidered
2ω
C-pointsC1onthey-axisandC1∗ onthebisectorofthepositivex-andy-axis,whilethegraydashedlinesshowtheprincipal
plane of their polarization circle.
(a) |E|2 C1 1(b) (c) γ β α
−0.4 E(t)
x α x‘
y‘
y −0.2 C1 z‘
z
m]
z [µ0
0.2 -0.1
C1*
C2 α 0
0.4 z’ [µm]
0.1 0.1
0 0.2 0.4 0.6 C2 0 -0.1 0y’ [µm]
y [µm]
FIG. 2. (color). (a) The electric energy density distribution in the yz-plane, superimposed by the local polarization ellipses in
white. Details in the vicinity of the first two C-points C1 and C2 along the y-axis are shown as insets. (b) Trace of the major
axisofthepolarizationellipsearoundtheC-pointsmarkedinredintheinsetsin(a). (c)Arisingopticalpolarizationtopology
when tracing around C1∗ (in the principal plane of its polarization ellipse with local coordinates x0,y0,z0 = z) with a trace
radius of 100 nm. The occurring weak x0-component of the electric field is magnified 4 times to show the orientation of the
major axis of the polarization ellipse more distinctly. The definitions of the axes in an arbitrary polarization ellipse are shown
as inset. The magnified part of the trace shows the major axis rotating into the principal plane and pointing along the trace
direction.
general structure of the occurring polarization topology of its principal plane marked with a gray dashed line
and its local electric fields is limited to the yz-plane in Fig. 1(c), the electric field in the surrounding of
due to the missing out-of-plane electric field component. the C-point exhibits a 3D field configuration due to a
This means, that no 3D topologies can be observed in non-zero E -component, see also Fig. 1(a). This 3D
x0
the yz-plane, which is the plane of symmetry of the field causes the major axis of the polarization ellipses
overall focusing geometry. However, the symmetry can around the C-point to tilt out of plane. As a result,
be broken by investigating a C-point off the y-axis the previously two-dimensional ellipse field around C1
and choosing the principal plane of its polarization is transformed into a 3D structure around C1 with a
∗
ellipse as plane of observation. Since along a C-line, topology similar to an optical polarization M¨obius strip
the topological index of the polarization singularity is (see Fig. 2(c)). To highlight the out-of-plane orientation
conserved [17], we examine exemplarily the C-point on of the polarization ellipses, the x-components of the
0
the bisector of the positive x- and y-axis closest to the plotted major axes are magnified by a factor of 4. The
optical axis (C1 ). Observing the field structure around position of the discontinuity present in the trace of
∗
this point in the coordinates x,y ,z (with z = z) the major polarization axis is given by an arbitrary
0 0 0 0
4
(a) (b) [1/2ω] (c) β α
1 γ
Microscope 0.8 E(t)
objective NA=0.9 C1* y‘ 0.6
z‘
0.4
Nano-particle
0.2
100nm 3D-Piezo stage Glass 0.5
substrate
Microscope 0.25
objective NA=1.3
Ilemnasging x [µm] 0−0.25 0.25 0.5 -0.1 0 C1*
CCD-camera DMK 23G618 −0.5 −0.5 −0.25 y [µ0m] z’ [µm] 0.1 0 0.1
-0.1 y’ [µm]
FIG.3. (color). (a)Sketchoftheexperimentalsetupforthereconstructionofthegeneratedhighlyconfinedfielddistribution.
An SEM image of the employed gold nanoprobe is shown as inset. (b) The experimentally reconstructed focal distribution of
s /w andtheprojectionsofthetwoinnermostC-lines. ThelowerpartdepictstheprojectionoftheseC-linesontothefocal
E E
| |
planeandthecorrespondingdistributionof s /w . Theredlinesintheupperpartof(b)depictthe3Dtrajectoriesofboth
E E
| |
C-lines, which are crossing the focal plane (transparent white plane) repeatedly. The marked C-point C1∗ corresponds to the
one considered in (c). (c) Optical polarization Mo¨bius strip with one half-twist, generated by tracing the major axis of the
polarizationellipsearoundtheC-pointonthebisectorofthepositivex-andy-axisintheplanenormaltoitslocalspinvector,
marked in gray. The trace radius was chosen to be 100 nm. The magnified part of the trace shows that the major axis is, due
to slight phase aberrations, not at the same time parallel to the principal plane and pointing along the trace direction. The
occurringweakx0-componentoftheelectricfieldismagnified4timestoshowthehalf-twistofthemajoraxisofthepolarization
ellipse more distinctly.
choice of the offset phase of the field (as in the planar onaglasssubstrateandscannedthroughthefocalplane.
case, also seen in the projection onto the y z -plane A second MO (immersion type, NA = 1.3) collects the
0 0
in Fig. 2(c)). It is important to mention again, that transmitted field, including the light scattered off the
all components of the electromagnetic field are fully particleintheforwarddirection. Foreachpositionofthe
continuous at this point, only the major axis of the nanoprobe relative to the optical axis, back-focal-plane
polarization ellipse exhibits a discontinuity. In contrast images of the second MO are acquired, corresponding to
to the optical polarization Mo¨bius strips investigated in theangularspectrumtransmittedintotheforwarddirec-
Refs. [14–16], the major axis of the polarization ellipse tion. This experimental data can be used to reconstruct
coincides at one position along the chosen trace with the full vectorial focal field distribution following the
the direction of the trace around the C-point (for C1 : technique introduced in [19], since amplitude and phase
∗
z = 0, y = 0.1 µm, see magnified part of Fig. 2(c)) information are encoded in the angular interference
0 0
due to the purely tSAM in the whole focal plane. In between the transmitted beam and the scattered light.
other words, the major axis of the polarization ellipse More details about the experimentally measured scan
is tangential to the trace at this point. This special data and the resulting focal field distribution can be
orientation prevents the determination of the actual found in the Supplementary Material [41].
twist-number of the major polarization axis when traced Figure 3(b) illustrates the experimentally achieved dis-
around the C-point, rendering this case different from a tribution of s /w , calculated from the reconstructed
E E
| |
generic M¨obius strip. However, as will be shown later fully vectorial electric field distribution. Due to small
on, only slight aberrations in the focal field distribution, experimental deviations from the theoretically expected
for example due to experimentally unavoidable phase field distribution, the experimentally reconstructed
aberrations, might lead to a distribution exhibiting not C-lines are located not only in the focal plane, but
only purely tSAM. This implies, that the major axis cross it repeatedly. These deviations from the theoret-
of the polarization ellipse is not tangential to the trace ically expected field pattern can be explained by small
anymore, and even at points with locally purely tSAM, aberrations of the incoming beam and the microscope
the slight aberrations of the focal field will lead to a objective. Tracing the major axis of the polarization
M¨obius topology. ellipse around the C-point at the same bisector used in
Experimental approach and results.—To verify this Fig.2(c)(graycrossesinFig.1)intheprincipalplaneof
theoretical prediction, the experimental setup sketched its polarization ellipse results in the optical polarization
in Fig. 3(a) [40] was used. The incoming linearly M¨obius strip with one half-twist as depicted in Fig. 3(c).
y-polarized Gaussian beam is tightly focused by a first This C-point is not a point of purely tSAM in the ex-
microscope objective (MO) with a numerical aperture perimental case due to the experimentally present phase
(NA) of 0.9. A single spherical gold nanoparticle (radius aberrations, and thus leads to the shown M¨obius strip in
r = 42 nm), acting as a local nanoprobe, is immobilized agreement with the theoretical predictions. Also here,
5
the out-of-plane component of the field was magnified [11] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and
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[13] M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and
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M. J. Padgett, Nature Phys. 6, 118 (2010).
was additionally traced around such a C-point, and
[14] I. Freund, Opt. Commun. 249, 7 (2005).
the obtained polarization M¨obius strip can be seen
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Conclusion.—We investigated optical polarization
[18] I. Freund, Opt. Lett. 35, 148 (2010).
topologies in a tightly focused linearly polarized Gaus-
[19] T.Bauer,S.Orlov,U.Peschel,P.Banzer, andG.Leuchs,
sianbeamandshowedthataspecialcaseofaM¨obius-like
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[22] F. Cardano, E. Karimi, L. Marrucci, C. de Lisio, and
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E. Santamato, Opt. Express 21, 8815 (2013).
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[23] A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs,
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Nature Photonics 9, 789 (2015).
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Supplementary Materials to: Optical Polarization Mo¨bius Strips and Points of Purely
Transverse Spin Density
Thomas Bauer,1,2 Martin Neugebauer,1,2 Gerd Leuchs,1,2,3 and Peter Banzer1,2,3,
∗
1Max Planck Institute for the Science of Light, Guenther-Scharowsky-Str. 1, D-91058 Erlangen, Germany
2Institute of Optics, Information and Photonics,
University Erlangen-Nuremberg, Staudtstr. 7/B2, D-91058 Erlangen, Germany
3Department of Physics, University of Ottawa, 25 Templeton, Ottawa, Ontario K1N 6N5 Canada
(Dated: January 25, 2016)
Inthesesupplementarymaterials,weshowthedetailedexperimentallymeasureddataofalinearly
y-polarized tightly focused Gaussian beam and the corresponding focal field distribution, resulting
from a nanointerferometric reconstruction technique [1]. A short overview of the employed scheme
6 isgivenandthespindensitydistributionsderivedfromtheexperimentallyreconstructedfocalfield
1 areshown. TohighlighttheexperimentaloccurrenceofpolarizationMo¨biusstripsintightlyfocused
0
light fields with minute phase aberrations, even around C-points with purely transverse spin, the
2
experimentally realized trace of the major axis of the polarization ellipse is shown around such a

n C-point with purely transverse spin.
a
J PACSnumbers: 03.50.De,42.25.Ja,42.50.Tx

2
2
The experimental reconstruction of the focal field dis- symmetric scan image, as can be seen in Fig. 1(b). An

] tribution of a tightly focused y-polarized Gaussian beam integration of the transmitted intensity over an angu-
s
c is realized with the setup schematically shown in Fig- lar sector rather than a full 2π solid angle in the BFP
i ure 3(a) of the main manuscript and described in detail of the collection objective (see Figs. 1(c-f)) reveals the
t
p in Ref. [1]. A high numerical aperture (NA) objective interferometric information necessary to reconstruct the
o
with an NA of 0.9 and a focal length of f = 1.65 mm full three-dimensional amplitude and phase distribution
.
s focuses the beam (beam width w = 2 mm, wavelength of the focal field. Here, the sector integrated scan im-
0
c
λ=530 nm)toitsdiffractionlimitedfocalspot. Asingle ages show a two-lobe pattern each, with the power in
i
s spherical gold nanoprobe with a radius of 42 nm, immo- onelobedroppingbelowthecollectedbackgroundpower.
y
bilizedonaglasssubstrate,isscannedstep-wisethrough The orientation of the two lobes depends on the orienta-
h
p the focal plane with a lateral precision of approximately tion of the angular sector in the back focal plane. The
[ 2 nm. The light transmitted and scattered into the for- chosen sectors represent essentially four different obser-

ward direction is collected by an oil-immersion micro- vation directions of the interaction of the focal field with
1
v scope objective (NA = 1.3), and its back focal plane the nanoprobe.
2 (BFP) is imaged onto a camera. Integrating over dif- The experimental far-field intensity data presented in
7 ferent sectors of the BFP image for each position of the Figure 1 is used as input vector for the reconstruction
0
nanoprobe relative to the focal field distribution then al- algorithm described in detail in Ref. [1]. The basic con-
6
lowsforrecordingscanimagescorrespondingtodifferent cept of the algorithm is the decomposition of the fo-
0
. collectionangles. Theresultingscanimagesfortwointe- cal field distribution into vector spherical wave functions
1
gration sectors with an area spanning the full azimuthal (VSWFs)[2]andthesubsequentanalyticalexpressionof
0
angle around the optical axis and four angular sectors the scattering off the nanoprobe via generalized Lorenz-
6
1 each spanning an azimuthal angle of 1 rad are shown in Mie theory, including the interaction with the air-glass
: Fig. 1, with the chosen sectors displayed superimposed interface.
v
on a symbolic BFP image in the first row for each inte- The focal field distribution E can – in this VSWF
i in
X gration range. basis – be written as
r
n
a The integration over the intensity distribution in the ∞
E (r)= A M (r)+B N (r), (1)
BFP up to an NA of 0.8 results in a scan image whose in mn mn mn mn
n=1m= n
transmittancedistributionisslightlyelongatedalongthe X X−
y-direction. This elongation is an indication of the ex- with M and N representing regular VSWFs that
mn mn
pected longitudinal electric field component along the can be associated with magnetic and electric multipoles,
initial polarization direction due to the sensitivity of the respectively, and are expanded around the geometrical
collection angle to both the transverse and longitudinal focus of the beam [2]. With this decomposition the full
electric field components. Reducing the integration ra- focal field information is contained in the complex ex-
dius to an NA of 0.5 reduces the collection efficiency of pansion coefficients A and B . The interaction of
mn mn
the longitudinal electric field component with respect to the focal field distribution with the well characterized
thetransversecomponents,leadingtoanearlyrotational nanoprobe(here: sphericalgoldnanoparticlewithradius
2
e (a) (b) (c)
g 1.0 1.0 1.0
n
a
0.5 0.5 0.5
rP

onBF NA 0 0 0
atin
ri −0.5 −0.5 −0.5
g
e
nt −1.0 NA=0.8 −1.0 NA=0.5 −1.0 φ=1rad NA=0.8
I
−1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0
NA NA NA

ale 0.5 0.5 0.5 0.155
entag 0.25 0.94 0.25 0.44 0.25 0.15
mm m]
perian i y [µ 0 0.92 0 0.43 0 0.145
xc −0.25 −0.25 −0.25
Es 0.9
0.14
−0.5 −0.5 0.42 −0.5

−0.5 −0.25 0 0.25 0.5 −0.5 −0.25 0 0.25 0.5 −0.5 −0.25 0 0.25 0.5

d 0.5 0.5 0.5 0.155
ee
uctag 0.25 0.94 0.25 0.44 0.25 0.15
trm m]
onsan i y [µ 0 0.92 0 0.43 0 0.145
cc −0.25 −0.25 −0.25
es 0.9
R 0.14
−0.5 −0.5 0.42 −0.5

−0.5 −0.25 0 0.25 0.5 −0.5 −0.25 0 0.25 0.5 −0.5 −0.25 0 0.25 0.5
x [µm] x [µm] x [µm]

e (d) (e) (f)
g 1.0 1.0 1.0
n
a 0.5 0.5 0.5
rP

nF A
oB N 0 0 0
atin
ri −0.5 −0.5 −0.5
g
e
nt −1.0 φ=1rad NA=0.8 −1.0 φ=1rad NA=0.8 −1.0 φ=1rad NA=0.8
I
−1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0
NA NA NA
0.16
ale 0.5 0.15 0.5 0.155 0.5
entag 0.25 0.25 0.25
mm m] 0.145 0.15 0.15
perian i y [µ 0 0.14 0 0.145 0
xc −0.25 −0.25 −0.25
Es 0.135
−0.5 −0.5 0.14 −0.5 0.14

−0.5 −0.25 0 0.25 0.5 −0.5 −0.25 0 0.25 0.5 −0.5 −0.25 0 0.25 0.5
0.16
d 0.5 0.5 0.5
ee 0.15 0.155
uctag 0.25 0.25 0.25
trm m] 0.145 0.15 0.15
onsan i y [µ 0 0.14 0 0.145 0
cc −0.25 −0.25 −0.25
es
R −0.5 0.135 −0.5 0.14 −0.5 0.14

−0.5 −0.25 0 0.25 0.5 −0.5 −0.25 0 0.25 0.5 −0.5 −0.25 0 0.25 0.5
x [µm] x [µm] x [µm]
FIG.1. (color). Experimentallyachievedscanimagesofatightlyfocusedlinearly(y-polarized)Gaussianbeamfortheangular
ranges shown in the back focal plane of the collection microscope objective (first row), and comparison with the scan images
derived from the reconstructed focal field distribution (last row).
3
r =42 nmandrelativepermittivity(cid:15) = 3.4+2.4ıat These results also allow for determining the spin an-
Au
−
λ=530 nm)isthenfollowingclassicalscatteringtheory, gularmomentumdensityateachpointinthefocalplane
where the transmission and reflection at the air-glass in- using the relation
terface is taken into account using an effective scattering
(cid:15)
matrix Tˆ (see Ref. [1] for details of this matrix). The s = 0 (E E) , (4)
eff E ∗
4ω= ×
total field transmitted in the forward direction can thus
alreadydisplayedasEq.(2)inthemainmanuscript. The
be calculated by
resulting experimentally retrieved focal distributions are
in good agreement with the distributions calculated by
E (r)=E +E = 1+Tˆ E , (2) vectorial diffraction theory ([3]; see also Fig. 1 in the
t in,t sca,t eff in,t
mainmanuscript). Thenon-zerolongitudinalspindistri-
(cid:16) (cid:17)
bution present in the experimental field is caused by the
whereEin,t istheexaminedfocalfieldandEsca,t thefield non-planar phase profiles of the focal electric field com-
scatteredoffthenanoprobe,bothtransmittedtroughthe ponents, which can be seen in the slight deformations of
air-glass interface via Fresnel equations. the x-components of the reconstructed electric field.
With the obtained decomposition of the reconstructed
The angularly resolved optical power of the transmit-
focalfielddistributionintoVSWFs, theelectricfieldcan
ted light P (θ,φ) in the BFP of the collection objective
t
be retrieved not only in the focal plane but in a vol-
can be written as:
umearoundthefocus(seeFig.3inthemainmanuscript
and Fig. 3(a)). To investigate the optical polarization
Pt(θ,φ)=Pin+Psca+Pext (3) topologies present in this field around C-points associ-
1 ated with points of purely transverse spin density, the
P = Re(E H +E H ) ,
ext 2 ∗in× sca ∗sca× in fieldinthefocalvolumecanbeexaminedforpointswith
|sE| = 1 (see Eq. (3) in the main manuscript) and
with Pin, Psca and Pext representing the part of the op- swzEE= 0.2ωFor the focal distributions shown in Figure 2,
tical power originating from the initial light field, the this set of conditions is fulfilled at the point C1t with
scattered field, and the interference between both fields, coordinates x = 100 nm, y = 269 nm, and z = 3 nm
− −
called extinction in classical Lorenz-Mie-theory. The (see Fig. 3(a)). The normal vector of the principal plane
magnetic field components of the scattered and incom- of the polarization ellipse at this point is thus located
ing light are represented by H. in the transverse plane, confirming the point to exhibit
purely tSAM.
This angularly resolved detection of the optical far-
Tracing the major axis of the polarization ellipse α(r)
fieldpowerallowsforintegratingoversectorswithdiffer-
on a circle with radius 100 nm around said C-point C1
t
ent effective observation directions, preserving the inter-
results in the optical polarization M¨obius strip shown
ferenceinformationabouttherelativephasebetweenthe
in Fig. 3(b). Due to the non-planar phase front in all
different expansion coefficients and thus between the fo-
three electric field components in the transverse plane
calfieldcomponents. Byusingafixedcut-offofthemul-
of the tightly focused light beam through this chosen C-
tipolarexpansionorder(here: maximumorderofn=8),
point, the M¨obius-like polarization topology present in
itispossibletounambiguouslyinvertthesetofequations
thenumericallyconsideredfocalfielddistributionevolves
relating the far-field power collected in different angular
to a generic M¨obius strip with one half twist, as can be
sectors for all positions of the nanoprobe relative to the
seen by the inset in Fig. 3(b). The major axis of the
focal field to the complex VSWF expansion coefficients.
polarizationellipseatthetracepointwithz =0istilted
0
Ultimately, this permits the reconstruction of the ampli-
out of the principal plane of the C-point, allowing for
tudeandphasedistributionofallthreefocalelectricfield
unambiguously designating a twist index of +1/2 to this
components.
M¨obius strip.
With the experimental scan images shown in Figure
1 as input, the reconstruction algorithm leads to a set
of multipolar expansion coefficients that can be used to
calculate a set of reconstructed scan images using the
same integration range as in the experiment (see last ∗ [email protected]; http://www.mpl.mpg.de/
[1] T.Bauer,S.Orlov,U.Peschel,P.Banzer, andG.Leuchs,
row for each angular range in Fig. 1), exhibiting an ex-
Nature Photon. 8, 23 (2014).
cellent overlap with the experimental input data. The
[2] L.Tsang,J.A.Kong, andK.-H.Ding,Scatteringofelec-
experimentally determined focal field distribution of the tromagneticwaves,1sted.(JohnWiley&Sons,Inc.,New
electric field components shown in Figure 2 can then be York, 2000).
calculatedfromthesereconstructedmultipolarexpansion [3] L. Novotny and B. Hecht, Principles of Nano-Optics
coefficients. (Cambridge University Press, Cambridge, 2006).
4
π π π
(a) Φ Φ Φ
0 x 0 y 0 z
|E|2 |E|2 |E|2
d x -π ×10-3 y -π z -π
x cte 0.5 6 0.5 0.8 0.5 0.12
u 0.25 0.25 0.25
y tr m] 4 0.6 0.08
s µ 0 0 0
n y [ 0.4
z o −0.25 2 −0.25 −0.25 0.04
c 0.2
e
−0.5 −0.5 −0.5
r
y 0.5 0.25 0 −0.25−0.5 0.5 0.25 0 −0.25−0.5 0.5 0.25 0 −0.25−0.5
all (b) sx sy sz
nt E 1 E E 0.05
e
0.5 0.5 0.5
m 0.05
ri 0.25 0.25 0.25
e m]
p µ 0 0 0 0 0 0
x y [
E −0.25 −0.25 −0.25
-0.05
−0.5 −0.5 −0.5
0.5 0.25 0 −0.25−0.5 -1 0.5 0.25 0 −0.25−0.5 0.5 0.25 0 −0.25−0.5 -0.05
x [µm] x [µm] x [µm]
FIG. 2. (color). (a) Experimentally reconstructed electric energy density distributions E 2, E 2 and E 2, and their
x y z
| | | | | |
respectiverelativephasedistributionsΦ ,Φ andΦ (insets)inthefocalplane. Theenergydensitydistributionsarenormalized
x y z
to the maximum total electric energy density E2. (b) Components of the electric spin density sx, sy and sz, derived from
| | E E E
the reconstructed electric field. The spin density distributions are normalized to the maximum total spin density s .
E
| |
(a) [1/2ω] (b) β α
1 γ
0.8 E(t)
0.6
C1 0.4
t
y‘
z‘ 0.2
0.5
0.25
-0.1
x [µm] 0
−0.25 0.25 0.5 0 C1t
0 z‘ [µm]
−0.25
−0.5 −0.5 y [µm] 0.1 0.1
0
-0.1
y‘ [µm]
FIG.3. (color). (a)Theexperimentallyreconstructedfocaldistributionof s /w andtheprojectionsofthetwoinnermostC-
E E
| |
lines. ThelowerpartdepictstheprojectionoftheseC-linesontothefocalplaneandthecorrespondingdistributionof s /w .
E E
| |
Theredlinesintheupperpartof(a)depictthe3DtrajectoriesofbothC-lines,whicharecrossingthefocalplane(transparent
white plane) repeatedly. The marked C-point C1 corresponds to the one considered in (b). (b) Optical polarization Mo¨bius
t
stripwithonehalf-twist,generatedbytracingthemajoraxisofthepolarizationellipsearoundtheC-pointwithpurelytSAM
at the intersection of the plane of observation marked in gray. The trace radius was chosen to be 100 nm. The magnified part
of the trace shows that the major axis is, due to slight phase aberrations, at no time pointing along the trace direction. The
occurringweakx0-componentoftheelectricfieldismagnified4timestoshowthehalf-twistofthemajoraxisofthepolarization
ellipse more distinctly.