Complete polarization control for a nanofiber waveguide using directional coupling

Fuchuan Lei,1 Georgiy Tkachenko,1, ∗ Jonathan M. Ward,1 and Sıle Nic Chormaic1, †

1Light-Matter Interactions Unit, Okinawa Institute of Science andTechnology Graduate University, Onna, Okinawa 904-0495, Japan

(Dated: April 12, 2019)

Optical nanofiber waveguides are widely used for near-field delivery and measurement of light.Despite their versatility and efficiency, nanofibers have a critical drawback - their inability to main-tain light’s polarization state on propagation. Here, we design a directional coupler consisting oftwo crossed nanofibers to probe the polarization state at the waist region. Directionality of couplingoccurs due to asymmetric dipolar emission or spin-locking when the evanescent field pattern breaksthe mirror symmetry of the crossed-nanofiber system. We demonstrate that, by monitoring theoutputs from the directional coupler, two non-orthogonal polarization states can be prepared at thenanofiber waist with a fidelity higher than 99%. Based on these states, we devise a simple andreliable method for complete control of the polarization along a nanofiber waveguide.

I. INTRODUCTION

Tapered optical fibers are unique because they allowfor the smooth transition of light from macro- to micro-or nanoscale systems. Owing to the strong evanescentfield around the ultrathin waist region, such fibers areefficient and versatile tools for optical manipulation [1–3], sensing [4, 5], nonlinear optics [6, 7], microcavities [8],atomic physics [9–13], and various studies of light-matterinteractions in both classical and quantum regimes [14–16]. As light propagates from the fiber pigtail to the waistregion (Fig. 1(a)), the mode volume reduces, and theintensity of the evanescent field increases [17]. Opticalpower can be transferred to the waist without significantlosses, even for ultrathin fibers. However, the other im-portant parameter of the field—its polarization—is usu-ally impossible to control. The polarization of the evanes-cent field is crucial for many nanofiber-based studies in-cluding microcavity mode excitation [18] and interactionswith nanoparticles [19] or atomic ensembles [20, 21].

As simple as it may seem, the problem of polarizationuncertainty in tapered fibers has challenged the scientificcommunity for decades. To solve it, one must establish adirect link between the near field at the ultrathin waistand the far field where macroscopic detectors and con-trollers are typically placed. In this work, we explore di-rectional coupling between two crossed nanofibers, and,consequently, develop a reliable method for achievingan arbitrary polarization state in the evanescent fieldof a single-mode nanofiber. The physics behind thisdirectionality features the interplay of two phenomena:the well-known spin-momentum locking, and the newlyfound mirror-symmetry breaking of a light-matter systemby an induced electric dipole.

We begin by introducing optical nanofibers and theirfabrication in Sec. II, and discuss how the polarizationof light propagates in adiabatically tapered fibers. Weemphasize that the polarization transformation in suchfibers, as in any optical elements free of depolarizationand dichroism, is equivalent to rotations of the Poincare

x

y

z

down-taper

up-taper

waist

pigtail pigtail

MM

corein out

wcladding

V

L

A

R

H D

y

x

(a)

(b) (c)

FIG. 1. (a) A tapered optical fiber with an ultrathin waistdoes not, generally, maintain the polarization of guided light.Transformation of the input polarization state, sin, to that atthe waist, sw, is described by the Mueller matrix M−. An-other matrix, M+ 6= M−, describes the transformation intothe output state, sout. (b) The trajectories traced on thePoincare sphere, P, by the input (sin, diamonds and dottedline) and the output (sout, circles and solid line) polarizationstates are different, even for short and straight nanofibers.(c) Polarization ellipse defined by the angles ψ and χ, whichcorrespond to half the azimuthal and polar angles on P.

sphere. In Sec. III, we devise a simple, two-step pro-cedure, which allows for reversing the above rotationaltransformations via consecutive mapping of two non-orthogonal polarization states. By compensating an ar-bitrary transformation, one achieves complete controlover the polarization state. Section IV is dedicated tothe directional coupling between two single-mode opticalnanofibers crossed at right angles. We present a detailedexperimental and numerical study of the directional cou-pler’s operation. The results indicate how to securelyidentify a pair of non-orthogonal states required for thetwo-step compensation procedure to be applicable in thecase of a nanofiber waveguide. Section V presents a prac-tical demonstration of the polarization control. We dis-cuss its precision and accuracy, as well as the experimen-tal evidence related to polarization evolution in a tapered

arX

iv:1

904.

0551

3v1

[ph

ysic

s.op

tics]

11

Apr

201

9

2

optical fiber.

II. POLARIZATION OF LIGHT IN ANANOFIBER WAVEGUIDE

Nanofibers are generally produced from conventionaloptical fibers by controlled heating and pulling [22]. Atypical, single-mode nanofiber consists of a cylindricalsubmicron-diameter waist connected to fiber pigtails bytwo taper regions, see Fig. 1(a), where the Cartesian co-ordinate system (x, y, z) originates in the middle of thewaist and z is parallel to the fiber axis. For this work, thetapered fibers were prepared from a step-index cylindri-cal optical fiber with a cut-off wavelength of 920±50 nm.The cylindrical waist regions were about 2 mm long withradii of 159 ± 3 nm, as measured by a scanning elec-tron microscope. The input pigtail was coupled to acollimated Gaussian beam from a continuous wave laser(980 nm wavelength). Each fiber was kept as short,straight, and strain-free as possible. Such precautionsare common practice in nanofiber experiments where po-larization (or mode) transformations are undesirable.

The key questions to be addressed are (i) how doesthe polarization of guided light change upon propaga-tion through a tapered fiber? and—more importantly—(ii) how can this change be controlled? A polarizationstate can be treated as a unit vector s = (1, S1, S2, S3),where S1,2,3 are the Stokes parameters which define apoint on the Poincare sphere, P, see Fig. 1(b). The az-imuthal (2ψ) and polar (2χ) angles on the sphere aredirectly linked to the orientation, ellipticity, and hand-edness (through the sign of χ) of the polarization el-lipse traced in the (x, y) plane by the tip of the elec-tric field vector, E(t), see Fig. 1(c), in the following way:s = (1, cos 2ψ cos 2χ, sin 2ψ cos 2χ, sin 2χ).

First, we compared the polarization state at the input,sin, and the output, sout, of our tapered fibers. In orderto probe a large part of P, sin was driven around thePoincare sphere in a figure-of-eight trajectory by pass-ing the horizontally-polarized (H) input beam through arotating quarter-wave plate (QWP). The trajectory wasrecorded by means of a commercial polarization analyzer.For every fiber under test, the output trajectory repeatedthe shape of the input, but always exhibited a significantshift on the sphere (see a typical example in Fig. 1(b)),regardless of our efforts to maintain the polarization us-ing the fore-mentioned precautions.

To understand how the shape of the trajectory on Pis maintained, let us note that the fibers were designedaccording to the adiabatic condition which implies energytransfer with minimum loss. Adiabaticity was ensuredby keeping the taper angle below the critical value [23,24] and the transmission at 980 nm wavelength above97% throughout the pulling process. Both pigtails andthe cylindrical waist region are single-mode, hence losses

can only occur in the tapers via coupling to radiationmodes and higher-order modes which cannot propagatealong the single-mode fiber. The guided electric field,E(z), in adiabatic single-mode fibers can be described asa combination of two orthogonal, quasi-linearly polarizedfundamental eigenmodes, HEx11 and HEy11 [17]:

E(z) = α(z)HEx11 + β(z)HEy11 , (1)

where α and β are variable complex amplitudes. Thepolarization ellipse is associated with the Jones vector

j =1√

|α|2 + |β|2

[αβ

]. (2)

Its evolution upon propagation from z0 to z through thefiber can be written as

j(z) = ufiber j(z0) , (3)

and, given that adiabatic fibres have transmission closeto unity, ufibre must be a 2× 2 unitary matrix:

u†fibreufibre = I , (4)

where I is the identity matrix. Therefore, transforma-tions of the Stokes vectors in adiabatically-tapered fibersare restricted to the 3D rotation (SO(3)) group [25]. Ro-tations of the Poincare sphere (P → P ′) preserve anglesbetween the Stokes vectors, and, consequently, the shapeof the trajectories for sin and sout in Fig. 1(b). Thispreservation of angles is the key condition for the polar-ization control that we achieve.

III. TWO-STEP POLARIZATIONCOMPENSATION

The polarization state at the nanofiber waist, sw, canbe linked to the input and output states in terms ofMueller calculus [26]: sw = M−sin, and sout = M+sw,where M− and M+ are 4 × 4 matrices describing theStokes vector evolution before (z < 0) and after (z > 0)the waist. These two matrices do not correlate and can-not be determined if one only measures sin and sout.Consequently, in order to control sw, one has to probethe evanescent field.

Theoretically, M− can be found by measuring severalsets of (sin, sw) [26]. However, this procedure is not al-ways realistic. Instead of controlling the target polar-ization state by deducing M−, in this work we followed adifferent strategy, which is more practical and, as will be-come clear, is the only option for nanofiber waveguides.Namely, we reverse the transformation of the Poincaresphere, P → P ′, thus achieving P → P and sw = sin.

From a geometrical point of view, the orientation of asphere can be completely defined by two independent an-gles, i. e., latitude and longitude. It is, therefore, logicalto realize the P → P mapping in two steps:

3

in out

(a) (b)

(c)

WP1

WP2

MPC

MM

Fiberpaddles

Polarizationcompensator

12

1 2

2

VR

FIG. 2. (a),(b) An unitary transformation of the Poincaresphere P → P ′ can be decomposed into two independent ro-tations by angles ϕ1 and ϕ2. To reverse the transformation,we adjust ϕ1 in order to achieve one state (here H or S1 = 1)maintained (that is H → H, S′1 = S1 = 1). Then, whilekeeping ϕ1 fixed, ϕ2 is adjusted until S′2,3 = S2,3. (c) Tocontrol polarization at the nanofiber waist, we compensatefor the unknown matrix M− by sequentially picking ϕ1 andϕ2 using two quarter-wave plates (WP1,2) and a variable re-tarder (VR). Fiber paddles allow randomization of M− whenstudying the precision of the control.

1. tilting of one axis by an angle ϕ1 (Fig. 2(a));

2. rolling of the two other axes about the first one byan angle ϕ2 (Fig. 2(b)).

Importantly, this two-step procedure for reversing un-known polarization transformations is not restricted tonanofibers, but can be applied to any optical elementfree of depolarization and dichroism. In practice, we re-alized the procedure by means of a free-space polarizationcompensator (PC) consisting of a variable retarder (VR,with the fast axis parallel to x) and a pair of quarter-wave plates (WP1,2), see Fig. 2(c). The compensatoris characterized by an unitary Jones matrix, uPC, or aMueller matrix, MPC. In step (1), WP1 and WP2 areindependently rotated until the input horizontal polar-ization (sin = H; S1 = 1) is mapped onto itself at thenanofiber waist (sw = H; S′1 = 1). Next, in step (2),an input state with |S1| 6= 1 is selected, and the re-tardance of VR is adjusted. This drives sw along thecircle in the plane parallel to (S2, S3) until eventuallyS′2,3 = S2,3, and thus sw = sin due to MPC = M−1

− . Infact, P → P mapping can be performed with any pair ofnon-orthogonal states, i. e. such states that do not lie onthe same diameter of the Poincare sphere, or—in math-ematical terms—have a non-zero inner product (see theproof in Appendix).

(a)

50µm

outin

out

xy

z

in

Outputfiber

x

y

z

(b) (c)

Inputfiber

Linearpolarization

Circularpolarization

out

(t)

Inputfiber

z

FIG. 3. (a) Crossed-nanofiber optical coupler and the imag-ing system consisting of a ×20 objective lens, a linear po-larizer, and a video camera. The camera image (top-left)shows nanofibers and the scattering spot from the crossing.(b) Tilted linear polarization states of guided light break themirror symmetry of the system, thus leading to power imbal-ance between the output channels. (c) Circular polarizationbreaks the symmetry too. Directional coupling occurs due tolocking of the transverse spin angular momentum, Jz, to theoutput wave vector, here k+.

IV. CROSSED-NANOFIBER DIRECTIONALCOUPLER

In practice, to identify two non-orthogonal polariza-tion states at the waist of a nanofiber, we cross it witha near-identical nanofiber at right angles, as sketched inFig. 3(a). Near-field probing one (input) ultrathin fiberwith another (output) one is not new, see for instanceits application for the purpose of profilometry [27, 28].In this work, for the first time, we consider symmetryof such a system (hence the importance of the right-angle crossing) and use output signals from both endsof the probe fiber. The efficiency of near-field couplingto guided modes which produce these signals is very low,typically under 0.1%, see Fig. 6(d),(e). However, it is suf-ficient for unambiguous identification of non-orthogonalH and RM (see below) polarization states at the waist ofthe input fiber.

The system of two crossed fibers has a symmetry plane,(x, z), shown as the dash-dotted line in Figs. 3(b),(c).When the input beam is H- or V-polarized, the overallsymmetry of the light-matter system is preserved and theoptical power values, P+ and P−, measured at the endsof the output fiber, are equal: P+ = P−. Otherwise, thesymmetry is broken and in general P+ 6= P−. Interest-ingly, this directional coupling of light can be separatedinto two independent effects associated with orientation(ψ) and shape (χ) of the polarization ellipse. Let us nowconsider the two special cases of linear (ψ 6= const; χ = 0)and circular (χ = ±π/4) polarizations.

For a tilted linear polarization (Fig. 3(b)), the mirrorsymmetry is broken by the emission pattern of an elec-tric dipole induced at the crossing point. As a result, twocounter-propagating modes of non-equal power are gener-ated in the output fiber. A wave with reflection angle, θ,carries energy proportional to sin2(θ+ψ). Assuming that

4

all angles allowed in this nanofiber equally contribute tothe energy transfer, the net power radiated along the ±direction is proportional to

∫ π/2θc

sin2(θ ± ψ)dθ, where θc

is the critical angle. Therefore, the output power sum,

PΣ = (P+ + P−) ∝ (cos 2ψ + const) , (5)

which has a maximum (minimum) for H (V) polariza-tion. This effect, which we dubbed “asymmetric dipo-lar emission” in order to emphasize its geometrical ori-gin, provides an alternative way to achieve directionalitywithout spin-momentum locking [29]. This possibilityhas been overlooked in earlier works on directional cou-pling in similar systems [30, 31].

For circular polarization, the electric field vector, E(t),traces a circle about the longitudinal axis, z, see Fig. 3(c).In free space, such a field only contains the longitudinalcomponent of the photon spin, Jz. The evanescent fieldmay also contain a significant transverse spin component,Jtrans, which appears due to interaction between the real,Re(k), and imaginary, Im(k), parts of the wave vector,k, at the interface between two different media [32]. In-terestingly, Re(k), Im(k), and Jtrans must form a right-handed system [33]. As a result, the direction of Jtrans de-fines the direction of the propagating wave [30, 34]. Thisphenomenon is known as “spin locking” or the “quan-tum spin Hall effect” of light [35]. With regard to ourcrossed nanofibers, spin locking causes P− 6= P+, sincethe vector Jz is simultaneously the longitudinal spin forthe input nanofiber and the transverse spin for one ofthe counter-propagating, x-polarized modes of the out-put nanofiber. Simple trigonometric considerations yieldthat the output power difference produced by the spinlocking effect,

P∆ = (P+ − P−) ∝ sin 2χ , (6)

with maximum and minimum values at χ = π/4 (R po-larization state) and χ = −π/4 (L state), respectively.

For elliptical polarization (ψ, χ 6= 0), both asymmetricdipolar emission and spin locking effects may influencethe output power balance. In order to confirm the aboveanalytical predictions for the special cases and generalizethe understanding of the directional coupler’s operation,finite-element numerical simulations of P± for sw cover-ing the whole Poincare sphere were performed. The re-sults for variable ψ and χ are shown in Figs. 4(a),(b) [36].The insets in Fig. 4(a) demonstrate that H and V cor-respond to the global extrema of PΣ and each of thesetwo states can be identified and used in the first step ofthe polarization compensation. Experimentally, ψ wasvaried while keeping S3 = 0 by sending the H-polarizedinput beam through a rotating half-wave plate (HWP).The data (diamonds) agree with the simulation (solidcurve). Thus, PΣ can be readily used for H → H map-ping, which can be further verified by monitoring theintensity of scattering from the fiber crossing, I, see the

A

D

V HR

V H

A

D

L

RM

LM

0

0.2

0.4

0.6

0.8

1

1.2

0-90 -45 45 90

Output power sum, (a. u.)

10.5 1-1

Output power difference, (a. u.)

0

( )o ( )o0-45 45

-0.5

0.5

1

-1

(a) (b)RM

0

0.2

0.4

0.6

0.8

1

1.2

HA D

R

LV

D A

R

Sca

tterin

g in

tens

ity,

(a.

u.)

L

FIG. 4. Operation of the crossed-nanofiber coupler. (a) Mea-sured (diamonds) and simulated (solid curve) values of theoutput power sum, PΣ, versus orientation of the linear polar-ization, ψ. The numerical simulation over the Poincare sphere(insets) shows that PΣ has the global maximum for the hor-izontal and the global minimum for the vertical polarizationstates. The intensity of scattering from the fiber crossing(dots—measured, dashed curve—simulated) is exactly oppo-site in phase with respect to PΣ. (b) Measured (circles) andsimulated (dashed curve) values of the output power differ-ence, P∆, versus the polarization ellipticity. The simulatedglobal maximum corresponds to the state RM, which is shiftedfrom R towards D due to the interplay between the two mech-anisms of mirror symmetry breaking.

experimental (dots) and theoretical (dashed curve) re-sults in Fig. 4(a). We measured I as the total brightnessof the camera image captured through a linear polarizerparallel to the y axis [11].

The simulations of P∆ (see insets in Fig. 4(b)) revealthat, due to the interplay between the asymmetric dipo-lar emission and the spin locking effects, the global max-imum (minimum) of P∆ is shifted from the expected R(L) to RM (LM) by ∆χ ≈ 8◦. This value depends on theradii of the nanofibers and was under 10◦ for the regionwhere significant coupling can be achieved (see Fig. 5(b)and Figs. 6(d),(e)). Experimentally, χ was varied whilekeeping S1 = 0 by rotation of a HWP in front of a QWPfixed at 45◦ to the x axis, as depicted in Fig. 8(g). Inorder to avoid systematic errors associated with ∆χ, inthe second step of the polarization control, we mappedRM → RM instead of R→ R.

All data presented in Fig. 4 were collected while keep-ing the fiber crossing point fixed. Now let us checkwhether the choice of the crossing point makes a dif-ference. For instance, if the polarization at the inputfiber waist depends on the longitudinal position, z, thecurves for PΣ,∆ versus the HWP orientation, ϕHWP, willhave variable phase shifts. Alterations to PΣ,∆ will alsoappear if the directional coupling depends on the verti-cal position of the output fiber, or, effectively, its radiusat the crossing. We tested both possibilities using oneperiod of the HRVL trajectory on the Poincare sphere(Figs. 8(d),(f)). First, we displaced the output fiber alongits axis, thus varying ∆y = (y − y0) while keeping thelongitudinal position of the crossing point fixed, so that

5

-2-2-1

0 00

1

1 12

2

2

2

3

3

3

3

33

4

4

4

4

4

5

5

5

6

6

7

7

8

8

910

200

250

300

350

200 250

r

(nm

)ou

t

r (nm)in

300 350

A D

(a)R

L

RM

(b) shift

FIG. 5. Shift of the maximum P∆ for sin tracing the circle inthe S1 = 0 plane of the Poincare sphere. (a) On the sphere,the shift appears as 2∆χ = 4∆ϕ, where ∆ϕ is the corre-sponding orientation of the HWP in the polarization genera-tor. (b) Simulated map of ∆ϕ versus radii of the input (rin)and output (rout) nanofibers. Noteworthy, the shift can benegative, in which case RM is located between R and A. How-ever, in practice, the coupling efficiency of the crossed fibersin this regime is too weak to be detected, see Figs. 6(d),(e)

∆z = (z − z0) = const, where y0, z0 are random ini-tial coordinates. As shown in Fig. 6(a), the measuredP∆ is independent of ∆y, up to an amplitude scalingfactor. The same behavior was observed when the cross-ing point was displaced along z with ∆y = const, seeFig. 6(b). These results indicate that sw is maintainedthroughout the coupling region, which we define as they×z area where light can still couple to the output fiber.We found that the amplitudes of P+ and P− are nonzerowithin 4 mm×4 mm area covering the waist and thinnerparts of the tapers, see the radius profile in Fig. 6(c).

According to our numerical simulations presented inFig. 6(d), the coupling efficiency, η = PΣ/PT (wherePT is the optical power transmitted through the in-put fiber) is maximum for an input nanofiber radius,rmaxin ≈ 210 nm, and an output nanofiber radius, rmax

out ≈220 nm. We have measured η over the whole couplingrange with a step of 0.25 mm in y and z. The re-sulting efficiency map shown in Fig. 6(e) agrees reason-ably with the simulations. Notably, the whole exploredrange for the nanofiber radii, r, corresponds to the single-mode regime, since the normalized frequency parameter,V = (2πr/λ)

√1.452 − 1 < 2.356, is below the cut-off

value of 2.405 [37]. In fact, the polarization compensa-tion is valid only for single-mode input nanofibers, witha minimum wavelength-to-diameter ratio of πNA/2.405(about 1.372 for glass in air).

V. POLARIZATION CONTROL

Figure 7 illustrates the precision of the polarizationcontrol we achieved. In order to estimate the precision,the input pigtail was spliced to fiber paddles (Fig. 2(c))that allowed us to produce a random M−. For each ran-

0( )o

45 90

-0.1

0.1

0

-0.1

0.1

0

(a)

(a. u

.)

00.50.91.1

�y (mm)

02

2.43

�z (mm)

�z = const

�y = const

160

200

250

300

350

200 250

r

(nm

)ou

t

r (nm)in

y,z (mm)

r (n

m)

0

400

800

-2 -1 0 1 2

waist radius: 159 nm

0 45 90

300 200 250r (nm)in

300

Coupling efficiency, (‰)

0 1 0.5 1

Simulation Experiment

2 0 1.5

(a. u

.)

(b)

(c)

(d) (e)

FIG. 6. The role of crossing-point position and fiber thick-ness. (a) Output power difference, P∆, measured at a fixedpoint (0, 0,∆z) on the input fiber with the output fiber beingdisplaced vertically by ∆y. Although the amplitude of P∆

depends on ∆y, its shape and phase are preserved. There-fore, polarization information gathered by the output fiberdoes not depend on its radius. (b) The input fiber is probedat various ∆z by the output fiber of a fixed radius. Thebehaviour of P∆ indicates that the polarization state is main-tained throughout the whole coupling region. (c) A typicalnanofiber radius profile measured by a scanning electron mi-croscope. (d),(e) Simulated and measured efficiency of thedirectional coupler, η = PΣ/PT, dependent on the radii ofthe nanofibers.

dom setting of the paddles, the two-step compensationprocedure was performed by H → H and RM → RM

mapping. Then, sout was measured for the three princi-pal states: H, D, and R. The resulting statistics over26 sets (see Fig. 7(a)) gives the following deviations fromthe mean: 1.16 ± 1.43◦, 6.03 ± 3.82◦, and 4.29 ± 2.37◦

for H, D, and R, respectively. The fidelities (defined asthe cosine of the mean angular distances) for these statesare 0.9998, 0.9945, and 0.9972, respectively. The offsetsfrom the target states (about 10◦ for this fiber) are dueto the unknown, constant matrix M+.

(a) (b)

uncompensated

V

L

A

R

H

D

V

L

A

R

H D

Rcompensated,with

RM RM

compensated,with

input

output:

Polarization statetrajectory:

RM

FIG. 7. Precision of the polarization control. (a) Statisticsfor the principal states measured at the end of the input fiberafter compensation for a random M−. (b) Recovering an ar-bitrary input trajectory (black diamonds) for sw using com-pensation with RM → RM (solid red squares) or R → RM

(empty green squares) mapping.

6

H D V A H

0( )o

45 90

-0.2

0

0.2

0.4

0

0.2

0.4

0.6

0.8

1

,

(a. u

.),

(a

. u.)

0

0.1

0.2

0.3(b)

H R V L H D R A L D

( )o

-0.2

0

0.2

0.4

0

0.2

0.4

0.6

0.8

1

,

(a. u

.),

(a

. u.)

0

0.2

0.1

0.3

( )o

-0.2

0

0.2

0.4

0

0.2

0.4

0.6

0.8

1

,

(a. u

.),

(a

. u.)

0

0.2

0.1

0.3

Simulation

Experiment

Simulation

Experiment(a. u

.)

(a. u

.)

(a. u

.)

135 180 0 45 90 135 180 0 45 90 135 180

D V A H R V L H R A L D

(e) (h)

(c) (f) (i)

o22.5

FAHx

y

z

HWP QWP

HWP

o22.5

o45 QWP

HWP

in

(a) (d) (g)

FIG. 8. Testing the crossed-nanofiber directional coupler after the polarization compensation. (a),(d),(g) The input polarizationstate, sin, traces a circle in one of the principal planes of the Poincare sphere, driven by the polarization generator representedby a rotating HWP and a static QWP. (b) Simulated (solid and dashed lines) and measured (diamonds and circles) opticalpowers at the ‘+’ and ‘−’ ends of the output fiber for sin lying in the S3 = 0 plane. (c) Simulated (lines) and measured(diamonds and circles) sum and difference of the output powers, along with the scattering intensity (dots). (e),(f) Same as(b),(c), but for the S2 = 0 plane. The simulated P∆ is maximum for the R state, and minimum for the L state. (h),(i) Samefor the S1 = 0 plane. The simulated maximum of P∆ is slightly shifted from R towards D (by ∆ϕ, see Fig. 5) due to theinterplay between the asymmetric dipolar emission and the spin locking effects. Here the experimental P−,+,Σ,∆ are in units ofvoltage from photodetectors; simulated P−,+ are normalized by the experimental maxima; scattering intensity I is normalizedby its maximum. The grey bands are added as a guide to the eye.

Besides the principal states, the method was testedfor a figure-of-eight trajectory (black diamonds and dot-ted line in Fig. 1(b) and Fig. 7(b)). The randomly ar-ranged paddles could move the uncompensated trajec-tory to any part of the sphere (see the blue solid curvein Fig. 7(b) as an example). Compensation with H→ Hand RM → RM brings the trajectory (red filled squares)close to the initial one. This result indicates that the sec-ond (straight) part of the tapered fiber (correspondingto M+) contributed only a minor transformation to sw.The case where the correction to ∆χ is ignored was alsochecked. In this case, the initial state is R, and R→ RM

mapping is achieved by locating the maximum of P∆.This mapping causes systematic errors for ϕ2, see thecompensated trajectories in Fig. 7(b).

In order to demonstrate the accuracy of the polariza-tion control, we have performed a detailed study sum-marized in Fig. 8. In this study, the input polarizationstate, sin, was driven along a circular trajectory in oneof the principal planes of the Poincare sphere: S3 = 0,

S2 = 0, or S1 = 0. This was achieved by the free-spacepolarization generator depicted in the top panels (thatis, (a), (d), and (g)) for each trajectory. The exper-imental data for the directional coupler’s outputs andthe scattering intensity were collected after the polariza-tion compensation procedure. We note that spin-lockingand asymmetric dipolar emission effects produce compa-rable modulations of the directionality characterized byP∆, see Figs. 8(c),(f). The higher noise in the ‘−’ chan-nel (especially noticeable in Fig. 8(b)) was repeatableover numerous experimental attempts and independentof the detectors. The noise levels in the two channelswere closer to each other for thicker nanofibers (withradii over 250 nm). Therefore, we attribute this effectto higher sensitivity of thinner nanofibers to bends dueto vibrations or sagging, and, perhaps, to unknown devi-ations of the generated trajectory from the target one.

Our results contradict the general belief that shortlengths of straight tapered fibers do not change the po-larization by much and, even if they do, the shifts of the

7

input state, sin, by the down- and up-tapers are equal,so that the state at the waist, sw, lies in the center be-tween sin and sout. According to our findings, presentedin Fig. 1(b) and Fig. 7(b), the transformation matricesM− and M+ lead to oppositely-directed, non-equal shiftsof significant magnitudes. Therefore, it is clear that ex-periments with high levels of precision (like those involv-ing quantum emitters) do require near-field polarizationcontrol, which can now be achieved using the reportedmethod.

VI. CONCLUSION

We demonstrated the first method for the completecontrol of polarization of light in a single-mode opti-cal nanofiber waveguide, using directional coupling be-tween two such nanofibers. Although based on complexphysical phenomena (the spin-momentum locking andthe newly discovered asymmetric dipolar emission), ourmethod is surprisingly simple and highly reliable. We be-lieve that it will have significant impact on the vast rangeof experimental systems based on optical nanofibers andevanescently coupled elements, in general. The demon-strated directional coupler itself is a promising platformfor developing integrated photonic circuits.

ACKNOWLEDGMENTS

The authors would like to thank S. P. Mekhail for helpwith automation of the experiment, K. Karlsson for assis-tance with nanofiber fabrication, and F. Le Kien for help-ful discussions. This study was partially supported by theOkinawa Institute of Science and Technology GraduateUniversity. G. T. was supported by the Japan Societyfor the Promotion of Science (JSPS) as an InternationalResearch Fellow (Standard, ID No.P18367).

Appendix: polarization compensation requiresnon-orthogonal states

Our polarization compensation is based on sequentialmapping of two input polarization states, A and B, ontothemselves at the waist of an optical nanofiber. As aresult, {

uA = eiδAA

uB = eiδBB, (7)

where the unknown phase factors, δA and δB , do notchange the state, and the Jones matrix, u = ufiberuPC,describes the polarization transformation in the compen-sator (uPC) and the tapered fiber until the waist (ufiber).

Since u is unitary, the inner product of these states

〈A|u†u |B〉 = 〈A|B〉 = ei(δA−δB) 〈A|B〉 . (8)

If these two states are not orthogonal, that is

〈A|B〉 6= 0 , (9)

then Eq. 8 requires δA = δB = δ. Under this condition,an arbitrary input state, C, can be defined in the basis(A, B): {

C = aA + bB

uC = eiδC, (10)

where a and b are complex coefficients. Such a stateC will be maintained in the polarization-compensatednanofiber.

∗ Corresponding author: georgiy.tkachenko@oist.jp; thefirst two authors contributed equally to this work.

† Also at Institut Neel, Universite Grenoble Alpes, F-38042Grenoble, France

[1] G. Brambilla, G. Senthil Murugan, J. S. Wilkinson, andD. J. Richardson, “Optical manipulation of microspheresalong a subwavelength optical wire,” Opt. Lett. 32, 3041–3043 (2007).

[2] A. Maimaiti, V. G. Truong, M. Sergides, I. Gusachenko,and S. Nic Chormaic, “Higher order microfibre modes fordielectric particle trapping and propulsion,” Sci. Rep. 5,9077 (2015).

[3] Y. Ren, R. Zhang, C. Ti, and Y. Liu, “Tapered opti-cal fiber loops and helices for integrated photonic devicecharacterization and microfluidic roller coasters,” Optica3, 1205–1208 (2016).

[4] T. Yoshie, L. Tang, and S.-Y. Su, “Optical microcavity:sensing down to single molecules and atoms,” Sensors 11,1972–1991 (2011).

[5] X.-C. Yu, B.-B. Li, P. Wang, L. Tong, X.-F. Jiang, Y. Li,Q. Gong, and Y.-F. Xiao, “Single nanoparticle detectionand sizing using a nanofiber pair in an aqueous environ-ment,” Adv. Mater. 26, 7462–7467 (2014).

[6] J.-C. Beugnot, S. Lebrun, G. Pauliat, H. Maillotte,V. Laude, and T. Sylvestre, “Brillouin light scatteringfrom surface acoustic waves in a subwavelength-diameteroptical fibre,” Nat. Commun. 5, 5242 (2014).

[7] H. Zoubi and K. Hammerer, “Quantum nonlinear opticsin optomechanical nanoscale waveguides,” Phys. Rev.Lett. 119, 123602 (2017).

[8] M. Cai, O. Painter, and K. J. Vahala, “Observation ofcritical coupling in a fiber taper to a silica-microspherewhispering-gallery mode system,” Phys. Rev. Lett. 85,74 (2000).

[9] F. Le Kien, V. I. Balykin, and K. Hakuta, “Atom trapand waveguide using a two-color evanescent light fieldaround a subwavelength-diameter optical fiber,” Phys.Rev. A 70, 063403 (2004).

[10] S. M. Hendrickson, M. M. Lai, T. B. Pittman, and J. D.Franson, “Observation of two-photon absorption at lowpower levels using tapered optical fibers in rubidium va-por,” Phys. Rev. Lett. 105, 173602 (2010).

8

[11] E. Vetsch, S. T. Dawkins, R. Mitsch, D. Reitz,P. Schneeweiss, and A. Rauschenbeutel, “Nanofiber-based optical trapping of cold neutral atoms,” IEEE J.Sel. Top. Quantum Electron. 18, 1763–1771 (2012).

[12] R. Kumar, V. Gokhroo, and S. Nic Chormaic, “Multi-level cascaded electromagnetically induced transparencyin cold atoms using an optical nanofibre interface,” NewJ. Phys. 17, 123012 (2015).

[13] D. F. Kornovan, M. I. Petrov, and I. V. Iorsh, “Transportand collective radiance in a basic quantum chiral opticalmodel,” Phys. Rev. B 96, 115162 (2017).

[14] X. Guo, M. Qiu, J. Bao, B. J. Wiley, Q. Yang, X. Zhang,Y. Ma, H. Yu, and L. Tong, “Direct coupling ofplasmonic and photonic nanowires for hybrid nanopho-tonic components and circuits,” Nano Lett. 9, 4515–4519(2009).

[15] L. Tong, F. Zi, X. Guo, and J. Lou, “Optical microfibersand nanofibers: a tutorial,” Opt. Commun. 285, 4641–47(2012).

[16] P. Solano, J. A. Grover, J. E. Hoffman, S. Ravets, F. K.Fatemi, L. A. Orozco, and S. L. Rolston, Advances inatomic, molecular, and optical physics (Elsevier, 2017)pp. 439–505.

[17] F. Le Kien, J. Q. Liang, K. Hakuta, and V. I. Balykin,“Field intensity distributions and polarization orienta-tions in a vacuum-clad subwavelength-diameter opticalfiber,” Opt. Commun. 242, 445–455 (2004).

[18] J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks,“Phase-matched excitation of whispering-gallery-moderesonances by a fiber taper,” Opt. Lett. 22, 1129–1131(1997).

[19] S. Wang, X. Pan, and L. Tong, “Modelingof nanoparticle-induced rayleigh-gans scattering fornanofiber optical sensing,” Opt. Commun. 276, 293–297(2007).

[20] T. Nieddu, V. Gokhroo, and S. Nic Chormaic, “Opti-cal nanofibres and neutral atoms,” J. Opt. 18, 053001(2016).

[21] M. Sadgrove, S. Wimberger, and S. Nic Chormaic,“Quantum coherent tractor beam effect for atomstrapped near a nanowaveguide,” Sci. Rep. 6, 28905(2016).

[22] J. M. Ward, A. Maimaiti, Vu H. Le, and S. Nic Chor-maic, “Contributed review: Optical micro- and nanofiberpulling rig,” Rev. Sci. Instrum. 85, 111501 (2014).

[23] J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black,S. Lacroix, and F. Gonthier, “Tapered single-mode fibres

and devices, part 1: Adiabaticity criteria,” IEE Proceed-ings J - Optoelectronics 138, 343–354 (1991).

[24] Y. Jung, G. Brambilla, and D. J. Richardson, “Broad-band single-mode operation of standard optical fibers byusing a sub-wavelength optical wire filter,” Opt. Express16, 14661–14667 (2008).

[25] J. J. Sakurai, Quantum mechanics (Addison-WesleyReading, 1994).

[26] R. A. Chipman, Handbook of optics: Ch. 22 Polarimetry(McGraw-Hill, 2010).

[27] L. S. Madsen, C. Baker, H. Rubinsztein-Dunlop, andW. P. Bowen, “Nondestructive profilometry of opticalnanofibers,” Nano Lett. 16, 7333–7337 (2016).

[28] F. K. Fatemi, J. E. Hoffman, P. Solano, E. F. Fenton,G. Beadie, S. L. Rolston, and L. A. Orozco, “Modalinterference in optical nanofibers for sub-angstrom radiussensitivity,” Optica 4, 157–162 (2017).

[29] M. F. Picardi, A. V. Zayats, and F. J. Rodrıguez-Fortuno, “Janus and huygens dipoles: near-field direc-tionality beyond spin-momentum locking,” Phys. Rev.Lett. 120, 117402 (2018).

[30] J. Petersen, J. Volz, and A. Rauschenbeutel, “Chiralnanophotonic waveguide interface based on spin-orbit in-teraction of light,” Science 346, 67–71 (2014).

[31] M. Sadgrove, M. Sugawara, Y. Mitsumori, andK. Edamatsu, “Polarization response and scaling law ofchirality for a nanofibre optical interface,” Sci. Rep. 7,1–9 (2017).

[32] K. Y. Bliokh, F. J. Rodrıguez-Fortuno, F. Nori, andA. V. Zayats, “Spin-orbit interactions of light,” Nat. Pho-tonics 9, 796–808 (2015).

[33] T. Van Mechelen and J. Zubin, “Universal spin-momentum locking of evanescent waves,” Optica 3, 118–126 (2016).

[34] D. O’Connor, P. Ginzburg, F. J. Rodrıguez-Fortuno,G. A. Wurtz, and A. V. Zayats, “Spin–orbit couplingin surface plasmon scattering by nanostructures,” Nat.Commun. 5, 5327 (2014).

[35] K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spinhall effect of light,” Science 348, 1448–1451 (2015).

[36] For the ease of presentation, this figure shows experimen-tal data collected after the polarization compensation.

[37] A. W. Snyder and J. D. Love, Optical waveguide theory(Chapman and Hall, 1983).