Optical nanofiber interferometer and resonator

Chengjie Ding,1, 2 Vivien Loo,1, 3 Simon Pigeon,1 Romain Gautier,1 Maxime

Joos,1 E Wu,2 Elisabeth Giacobino,1 Alberto Bramati,1 and Quentin Glorieux1, ∗

1Laboratoire Kastler Brossel, Sorbonne Universite, CNRS,ENS-PSL Research University, College de France2State Key Laboratory of Precision Spectroscopy,

East China Normal University, Shanghai 200062, China3ESPCI Paris, PSL Research University, CNRS, Institut Langevin

(Dated: March 26, 2019)

We report the fabrication and characterization of photonic structures using tapered opticalnanofibers. Thanks to the extension of the evanescent electromagnetic field outside of the nanofibertwo types of devices can be built: a ring interferometer and a knot resonator. We propose a generalapproach to predict the properties of these structures using the linear coupling theory. In addition,we describe a new source of birefringence due to the ovalization of a nanofiber under strong bending,known in mechanical engineering as the Brazier effect.

I. INTRODUCTION

Due to their low losses, optical fibers are undoubtedly amedium of choice to transport optical information, mak-ing them critical to current telecommunication networksand to the future quantum internet [1]. However inject-ing a specific state of light, for example a single photonin a fiber with a good coupling efficiency is not an easytask. A typical way to couple light into a fiber is to placethe emitter directly at one end of a fiber with or withoutadditional optical elements [2]. An alternative approachrecently raised significant interest, by injecting light fromthe side of a fiber [3, 4]. Indeed, stretching down thefiber diameter to the wavelength scale allows for a cou-pling between the fiber guided mode and an emitter in itsvicinity [5]. In such a nanofiber, the fundamental prop-agating mode has a significant evanescent component atthe glass/air interface, which allows for interacting withemitters on the surface [3, 4, 6–10].

Collection efficiency is limited so far to 22.0 ± 4.8% fora bare nanofiber [6]. Maximizing this coupling is a chal-lenging task as it requires simultaneously a fine-tuningof the fiber size and the largest possible cross-section forthe emitter. To render this ”injection by the side” tech-nique more attractive, the collection efficiency has to beincreased. One approach to do so is to enhance the ef-fective light-matter interaction. It is commonly done byreducing the mode volume using confined modes of theelectromagnetic field rather than propagating modes. Itleads, via the Purcell effect, to an increase of the sponta-neous emission within the nanofiber confined mode andtherefore to an increase of the emitter-fiber coupling [11].A detailed model predicts more than 90% collection effi-ciency if one adds an optical cavity of moderate finesseto the nanofiber [11]. Diverse strategies have been in-vestigated to do so. One is to fabricate two mirrors di-rectly in the fiber to add a Fabry–Perot cavity within

∗ Corresponding author: quentin.glorieux@lkb.upmc.fr

the nanofiber itself [12]. This strategy requires advancednanofabrication methods such as femtosecond laser abla-tion to modify the fiber index. Using a Talbot interfer-ometer, it has been possible to fabricate two fiber Bragggratings and form an optical cavity with a transmissionof 87% for a finesse of 39 [12]. A similar strategy, callednanofiber Bragg cavity, where a focused ion beam millsthe nanofiber to create mirrors has shown a Purcell fac-tor and coupling efficiency of 19.1 and 82% respectively[13, 14]. Another solution relies on coupling the nanofiberwith a whispering gallery mode resonator with very highquality factor up to 109 [15, 16]. With this strategy, atthe difference of previous ones, the cavity is exterior tothe nanofiber.

Beam splitter

Ain

Aout

Mirrora b

Beam splitter

Aout

AinMirror

dc

FIG. 1. Optical nanofiber structures. a- Nanofiber twistedloop: optical microscope image. b, Sagnac interferometerequivalent optical setup: the light emerging from the portAout consists of two reflections or two transmissions of thelight from incident port Ain through the beamsplitter. c-Nanofiber knotted loop: scanning electron microscope image.d- Fabry-Perot ring resonator equivalent optical setup: lightcoming from Ain that is not directly reflected to Aout by thebeam splitter, is trapped in the cavity formed by the beamsplitter and the mirrors.

In this article, we study an alternative approach, par-

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ticularly interesting because it does not involve nanofab-rication capabilities. The idea is to loop the nanofiberin order to directly create a cavity thanks to the evanes-cent coupling. Such cavities have been obtained in thetelecom range at 1.5 microns using microfiber with a fi-nesse up to 20 [17, 18]. Here we demonstrate the firstexperimental implementation of this approach in the vis-ible range using a fiber tapered to the nanoscale whilemaintaining a similar finesse. Moreover, using the linearcoupling theory, we present a generalized model for twocomplementary geometries: twisted and knotted loops,illustrating the crucial role of the topology of the loopformed. While the twisted loop is found to work as aSagnac interferometer, the nanofiber knot behaves as aFabry–Perot micro-resonator.

In addition, we report here a novel source of birefrin-gence for nanoscale tapered fiber. In such micron-sizestructures the nanofiber region is put under strong bend-ing constraints and therefore this induces an ovalizationof its transverse section, known as the Brazier effect inmechanical engineering [19]. We have estimated that thisovalization leads to a substantial difference in the effec-tive refractive index similar to the the well-known stressinduced birefringence [20].

II. EFFECTIVE COUPLING THEORYAPPROACH

The manufacturing of nanofibers is a well-controlledprocess, and it is possible to fabricate fibers with a di-ameter down to 200 nm [5, 21]. At this size, only thecore of the fiber remains, and the surrounding air actsas a cladding. Consequently, there is a strong evanes-cent field extending around the surface of the nanofiber.The fundamental mode does not correspond anymore tothe standard linearly polarized mode LP01. Neverthe-less, using Maxwell’s equations the correct propagatingmode profile can be precisely characterized [22]. We willconsider single mode air-cladding nanofibers only, thatis, nanofibers in which the fundamental mode HE11 isthe only propagating solution[22]. This is the case if the

normalized frequency V with V ≡ ka√n2 − 1 is lower

than the cutoff normalized frequency Vc = 2.405, wherek is the wavevector, a is the fiber radius, and n is thefiber index.

We have bent and twisted manually such nanofiberswith great care to realize two miniaturized optical setups:a fiber loop and a fiber knot (Fig. 1). The commonfeature of these two structures is that they both presenta section where the two parts of the nanofiber touch eachother as shown on Fig. 1. However, fiber knots andfiber loops are topologically distinct, as it will be detailedlater.

Given the strong evanescent field of the propagat-ing mode, the contact between different nanofiber re-gions leads to a coupling between the HE11 propagatingmodes. To model this coupling, let us consider two par-

A0

B0

E1

E2Z

A(z)

B(z)

FIG. 2. Schematic of codirectional couplers. A0 and B0 arethe amplitude of electric fields at the input of the two fibers.The outputs of each fiber are labelled A(z) and B(z).

allel nanofibers nearby to each other, as represented inFig. 2. In this scheme, the two fibers exchange energy inthe contact region of length z with coupling coefficientκ. Therefore, given the input amplitudes A0 and B0 theoutput amplitudes are:

A(z) = A0 cos(κz)− iB0 sin(κz) (1)

B(z) = B0 cos(κz)− iA0 sin(κz) . (2)

The coupling coefficient κ depends on the overlap ofthe coupled modes [20]:

κ =ωε0

∫∫(n2

air − n2silica)E∗1 ·E2 dxdy∫∫

uz · (E∗1 ×H1 + E1 ×H∗1) dxdy(3)

where Ei and Hi are respectively the electric andmagnetic components of the modes propagating in thenanofiber labeled i, uz is the unitary vector directed to-ward the propagation direction and ω the field frequency.

According to Eqs. (1) and (2), we can regard the lightgoing through the fiber 1 as transmitted, and the lightgoing from fiber 1 to fiber 2 as reflected. The system actsas a beam-splitter of transmission coefficient t = cos(κz),and reflection coefficient r = −i sin(κz). For example,when the system has an input A0 = 1 and B0 = 0, theoutput intensities appear to be |A(z)|2 = cos2(κz) and|B(z)|2 = sin2(κz), and complete power transfer occurswhen κz = (2p+1)π/2, p being an integer. Consequentlythe quantity π/2κ is equivalent to a coupling length.

Interestingly the orientation of the effective beam split-ter depends on the topology of the structure. In the caseof the twisted loop represented in Fig. 1-b, the beamsplitter is equivalent to a Sagnac interferometer allowingfor only one lap in the structure. Whereas in the case ofthe knot (Fig. 1-d), the effective beam splitter allows formultiple laps inside the setup and therefore is equivalentto a ring resonator.

As mentioned above, we place ourselves in conditionsunder which only one propagating mode exists: HE11.To estimate the coupling coefficient κ, we computed Eq.(3) using the exact profile of modes HE11 [22]. In orderto study the coupling coefficient dependence on the po-larization, we assume that nanofibers are identical and incontact at (0,0) on x−y plane, as shown in Fig. 3-a, andthat the light is linearly polarized in one fiber, whereasit is circularly polarized in the other one.

The field density in the transverse section in this con-figuration is presented in Fig. 3-b. We numerically calcu-lated the coupling strength as a function of the fiber di-ameter averaged over the polarization degree of freedom

3

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

10-4

10-2

100

x

y

z(μm)

|E2|2 |E1|

2 E1*• E2

[u.a.]

2a

y=0

x=0a

b

FIG. 3. Field distribution and overlap for two adjacentnanofibers (fiber radius a = 250 nm). a, The centers of twonanofibers are located on axis y=0. Fiber 1 is located at po-sition (a,0), and fiber 2 is located at (−a,0). b, The light inthe fiber 2 corresponds to an HE11 and linearly polarized asvisible in the left panel (upper left panel for the horizontallypolarized and lower left panel for vertically polarized). Thefield is circularly polarized in fiber 1 as visible in the cen-tral panels. Right panels represent the overlap between thetwo fields directly related to the coupling strength κ given inEq. (3).

κ = 〈κ〉ϕ, where ϕ is the angle of the polarization vectorwith the x axis. Results are presented in Fig. 4-a. We seethat κ decays exponentially with the fiber diameter. Thelarger the fibers are, the smaller their evanescent partof the field is. This decay is exponential so it is for theoverlap of the fields. This strong dependency illustrateswell the general interest to work with fiber at subwave-length scale rather than micrometric scale. Focusing nowon the polarization dependency of the coupling strengthwe show two cases in Fig. 3-b : (i) the polarization ofthe linearly polarized field in fiber 2 is along the x axis(along the direction that connects the centers of the twofibers) (ϕ = 0) (upper panels of Fig. 3-b) and (ii) thepolarization of the linearly polarized field in fiber 2 isnormal to the x axis (ϕ = π/2) (lower panels of Fig. 3-b). For both cases the field density is presented for thelinearly polarized field in fiber 2 (left panels), for thecircularly polarized field in fiber 1 (central panels) andfor their overlap E∗1 · E2 appearing in Eq. (3), whereE∗1 ·E2 = E∗1x ·E2x + E∗1y ·E2y + E∗1z ·E2z. The resultsshown are for a fiber diameter of 500 nm and wavelengthof 800 nm. We see that even if the overlap intensity dis-tribution is different from one case to the other, theiraverage magnitude and then the coupling coefficient aresimilar in the two cases as shown in Fig.3-b.

Actually, the coupling coefficient κ is found to be only

slightly dependent on the polarization. This variationdepends on the fiber diameter, as visible in the b panelof Fig. 4 but leads to a marginal relative change. In therealization of the nanofiber twisted loop below, we use afiber diameter of 500 nm for which the relative changeis estimated to be less than ±3%. This is illustrated inFig. 4-a by the colored region surrounding the mean cou-pling strength κ, corresponding to the amplitude of thevariation with respect to the polarization. Given that,we can reasonably neglect the effect due to polarizationand approximate κ ≈ κ.

0

400 nm480 nm560 nm640 nm720 nm800 nm

Polarization angleFiber diameter (

-(%

) 2

4

6

-2

-4

-6

a b

(μm

-1)

FIG. 4. a- Coupling coefficient κ (at 800 nm) betweennanofibers with varying diameter and averaged over the po-larization degree of freedom. The colored area refers to theamplitude of the variation with the polarization. b- Relativechange of the coupling coefficient as a function of the polar-ization angle with a fiber diameter varying from 500 nm to900 nm.

III. NANOFIBER INTERFEROMETER

We realized an optical nanofiber by pulling a commer-cial single-mode fiber to reach a 500 nm diameter overa length of 1 mm following [23]. The transition betweenthe commercial single mode fiber and the single modenanofiber is adiabatic and its transmission is over 95%[24].

To make a twisted loop structure presented in Fig.1-a,we first make a ring in the nanofiber region. Then, byfixing one side of the nanofiber, and rotating the otherside, we can slowly increase the length of the entwinedpart. This is a well-known mechanical phenomenon stud-ied in many contexts [25]. Increased torsion will reducethe size of the loop and bend it locally. At some pointthe bending exceeds the fiber tolerance and it breaks. Inthe experiment we carefully choose to remain below thisthreshold.

With this geometry, the system corresponds to aSagnac interferometer. As represented in Fig.1-b,light propagating towards the loop finds two counter-propagating optical paths. After the entwined region,part of the light is transferred into the clockwise path(red arrow) with a coefficient of r, whereas the remain-ing propagates along the anti-clockwise path with a co-

4

efficient of t. This two beams propagate separately ac-cumulating a phase of eiβLr , where β is the propagationconstant, and Lr is the length of the ring. Then, bothpaths interfere back into the entwined region, which actsas a beam-splitter, as represented in Fig.1.b. The ampli-tude of output electromagnetic field can be written as:

Aout = [r2 + t2]eiβLrA0, (4)

with t = cos(κz), and r = −i sin(κz), as shown above,which leads to the following transmittance for the device:

Tint =

∣∣∣∣Aout

A0

∣∣∣∣2 =∣∣r2 + t2

∣∣2 . (5)

As mentioned above, we control the length of the cou-pling region by varying the torsion applied on thenanofiber, which ultimately tunes the reflection andtransmission coefficients of our device. Applied mechani-cal stress will be translated into an optical response, fromreflective to transmissive.

0.6 0.7 0.8 0.9 1

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m)

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λ (μm) λ (μm)

a

b

Tin

t

75

87.5

100

10075

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75

87.5

100

75

87.5

10075

87.5

100

FIG. 5. Transmission spectrum through twisted loop struc-ture. a and b- Experimental data and numerical simulationof the transmission spectrum of broad light source throughthe nanofiber interferometer as a function of the wavelengthand coupling distance. c and d- Experimental data (blue) andnumerical simulation (red) of the transmission spectrum foran entwined length of 75 µm (panel c) and 100 µm (panel d)respectively. The experiment data shown are smoothed usingrloess methods with a span of 1%.

To probe the system transmittance we use a fiber-coupled linearly polarized super-continuous white laser(NKT Photonics SuperK COMPACT). The laser beamis coupled into the single mode fiber (SM800) with thetwisted nanofiber region in the middle. The output sig-nal is sent to a spectrometer. Fig. 5-a shows a map ofthe transmission spectrum through the nanofiber twisted

loop structure as a function of the entwined region size.To understand the spectrum obtained for a given en-twined region length, also presented in Figs. 5-c and din blue, one has to note that when the fiber diameter isfixed, the extension of the evanescent part of the field in-creases with the wavelength. Accordingly the couplingstrength κ and so the effective reflection coefficient rchange too as shown by the red curves in Fig. 5-c and d.In consequence, the spectrum for a fixed coupling lengthleads to the interference pattern visible in Fig. 5 andagrees with our description of the device as an interfer-ometer. Moreover, increasing the coupling length leadsto a shift of the interference to smaller wavelengths. InFig. 5-b we represent the same map as in Fig. 5-a, cal-culated from Eq. 3. Despite the variability of many ex-perimental parameters, our theoretical model shows goodagreement with the experimental data, This agreementcan be verified quantitatively in panel c and d, where weshow the measured (blue) and calculated (red) spectrumfor a coupling length of respectively 75 µm and 100 µmwith no adjustable parameters.

IV. NANOFIBER RESONATOR

Optical ring resonators can have many applications,such as optical add-drop filters, modulation, switchingand dispersion compensation devices. To fabricate theknot structure as presented in Fig. 1-c, we carefully madea large knot centered on the nanofiber region and by pre-cise control of the spacing between the two displacementplatforms used to pull the fiber, we can decrease the di-ameter of the fiber to tens of micrometers (≈ 30 µm). Asin the case of the twisted loop, the knotted loop inducedan important bending of the nanofiber. Accurate controlof the size of the knot allows us to avoid breaking it.

With this geometry, the system acts as a resonator. Asrepresented in Fig. 1-d, the light injected in the devicewill either be directly reflected to the output with a co-efficient r, or transmitted in the loop with a coefficientt. In contrast with the twisted loop, there is only oneoptical path within the loop. Moreover, the light circu-lating inside the knot will split again everytime it passesthrough the coupling region: some will go to the output,the rest will stay inside the knot. We represent in Fig. 1-d the corresponding optical setup. It is remarkable thatthe change of the topology of the loop formed, twisted orknotted, completely changes the behavior of the device.Schematically, passing from one device to the other oneis equivalent to rotating by 90◦ the beam splitter mim-icking the fiber coupling region as presented in Fig. 1-band Fig. 1-d.

Along the propagation in a loop, the field undergoeslosses with a rate ρ due to scattering. In Fig. 6-a wepresent an optical microscopy image of the knotted loopwhen light is propagating on the fiber. Bright spots onthe fiber are due to the scattering of imperfections on thefiber surface. Given the significant evanescent component

5

of the field, any defects located close to the surface willstrongly scatter the propagating light. However, most ofthe losses come from the knotted region itself as visiblein Fig. 6-a. They would drastically be reduced in a cleanroom environment. Indeed, when we fabricate the knotmanually, the knot was gradually tightened into smallsize. Thus, the entwined region sweeps several centime-ters of fiber. Impurities on the surface are blocked by theknot and inevitably accumulate there.

After one lap in the loop we have B′0 = (1 −ρ)eiβLkt2B0, where Lk is the length of the ring. Assum-ing the reflection coefficient r = −i sin(κz) and trans-mission coefficient t = cos(κz), then we get the equationgiving the amplitude of the electromagnetic field at theoutput:

Aout = A0[r + (1− ρ)t2eiβLk + (1− ρ)2t2re2iβLk

+(1− ρ)3t2r2e3iβLk + · · · ](6)

leading to the following transmittance Tres of the device:

Tres =

∣∣∣∣Aout

A0

∣∣∣∣2 =

∣∣∣∣r +(1− ρ)t2eiβLk

1− (1− ρ)reiβLk

∣∣∣∣2 (7)

In contrast to the nanofiber interferometer case, thesize of the loop plays a major role here as it dictates theamount of phase accumulated after one lap in the device.Optical resonances appear when the light traversing theloop accumulates a phase integer multiple of 2π.

To characterize the system we use the same setup as forthe twisted loop. Fig. 6-b is the transmission spectrumof the knot for a given diameter. Within the bandwithof the SM800 fiber (≈ 700 nm to ≈ 950 nm), many finepeaks are observed, revealing the resonant wavelengths.

In Fig. 6-c, we present the calculated transmissionspectrum, which agrees well with the experimental data,for its three main features. Firstly, the wide spectralGaussian envelop; this is measured beforehand by record-ing the spectrum of our laser transmitted through a fiberwithout the loop. Secondly, the fine peaks; they exhibitsmatched free spectral range (FSR) and amplitudes. Fi-nally, the larger scale contrast modulation ; its maximumaround 750 nm and 850 nm are faithfully reproduced. Toexplain this contrast modulation, we represent in Fig. 6-d the reflectance of the entwined region (i.e. withoutknot) calculated for the same condition. As observedpreviously, it depends strongly on the wavelength. Forinstance, at λ ≈ 800 nm, the reflectance |r|2 is zero; itcorresponds to a scenario without beamsplitter. As 100%of the light leaves the ring after one lap, there cannot beinterferences, and the resonance peaks contrast vanishesaccordingly. Similarly when r2 ∼ 1, around 900 nm and670 nm, the entwined region acts as a simple mirror in-stead of a beam-splitter, and the resonance peaks fade aswell, since no light gets inside the ring.

The length of the ring Lk determines the interval be-tween the peaks in the spectrum known as FSR and givenby ∆υFSR = c/(neffLk), where neff is the effective re-fractive index of air clad optical fiber. The analysis of

b

a

Exp.

0.6 0.7 0.8 0.9 1

Theory

Cts/ms

Normalized

r2 =si

n2 (κ

z)

50 �m

c

d1

50

100

150

0.2

0.4

0.6

0.7T

res

Contrast

00

FIG. 6. Transmission spectrum through a knot structure.a- Image of the nanofiber knot recorded by an optical mi-croscope. b- Experimental transmission spectrum of a broadlight source through the nanofiber resonator as a function ofthe wavelength. c- Calculated transmittance of the devicesas given in eq. (7). d- Left axis is the reflectance of the con-tact zone and right axis is the visibility of resonance betweenwavelength 0.68 µm and 0.9 µm. We measure ∆υFSR=2.214THz.

the spectrum gives ∆υFSR=2.21 THz, which correspondsto a cavity length of about 108 µm. The finesse variesslightly along the spectrum reaching 8 from 820 nm to860 nm. In this range, we measure a quality factor of1300. It agrees well with the calculation which predicteda ∆υFSR of 2.136 THz, a finesse of 7.5 and a qualityfactor of 1100. These calculations have been done withan estimate of 35% losses, extracted from experimentaldata. In the next section, we push forward the analysisof the spectrum and we identify an original birefringenceeffect.

Birefringence induced by ovalization under bending

A keener look at the data reveals that two differentresonance modes (see insert curve in Fig. 7) contribute

6

1

2

840 860 880λ (nm)

kCts.

40

160

1

2

FIG. 7. Blue: Fourier transform of experimental transmissionspectrum. Inset yellow: zoom of the Fig 6-b. Two differentsets of peaks can be seen. The corresponding ∆υFSR are 2.21THz and 2.28 THz (peaks location in the blue curve). Thissplitting corresponds to different polarization modes under-going birefringence along the propagation in the structure.

to the spectrum, otherwise we would observe regularlyspaced peaks. The FSR for the different modes is slightlyoffset. When the peaks positions of the two modes arestaggered, we can clearly see the two discrete peaks. Toidentify the two values of FSR, we took the Fourier trans-form of the spectrum, which is plotted in Fig.7. There, weclearly see two distinct resonant contributions labeled 1and 2. This shift between the FSR values corresponds toa relative variation of the indices between the two prop-agating modes of ∆ = (n1−n2)/n1 = 3.1%, with n1 andn2 the effective refractive indices of both modes.

To determine the origin of this lifting of degeneracy, wewill now focus on the mechanical properties of the knot.As for the twisted loop we have here a nanofiber understrong bending constraints. This bending implies stressand related mechanical effect that we assume at the ori-gin of the mode splitting. We investigate two possiblesources of birefringence: (i) stress-induced birefringence(Bs) and (ii) ovalization-induced birefringence (Bo) [20].Stress-induced birefringence is a well-known phenomenonin optical fiber [26] whereas ovalization-induced birefrin-gence is directly linked to the diameter of the fiber con-sidered here.

Ovalization of an elastic rod is a well-known phe-nomenon in mechanical engineering [27], and can be com-monly experienced when bending an elastic tube. Theperfect circular section of the tube will change due to thebending to an oval section with its short axis along thedirection of the bending. Such a mechanism is commonlyneglected in optical fibers given the rigidity of standardfibers in which a bending force will lead to stress-inducedbirefringence and break the fiber before leading to signifi-cant ovalization of the section. However, in our case, withfiber of nanometric diameter, the mechanical properties

are very different. Moreover, the electromagnetic field issignificantly less sensitive to stress, since it is largely lo-calized outside the fiber. The stress only affects the partof the mode confined inside the silica. We evaluate therelative change of indices due to stress to be of ∆ ≈ 1.7%.This is not enough to explain the different values of FSRwe observed, and leads us to consider ovalization, or Bra-zier effect, as a significant factor in the degeneracy lifting.

σy (GPa)

0

-0.03

-3 -2 -1 0 1 2 31.235

1.24

1.245

1.25

1.255

1.26

n eff

a b

Variation of fiber diameter in %

630 nm

y

xθnanofiber

y

x

FIG. 8. Brazier effect on a bent optical fiber. a- Schematicof the fiber cross-section (x − y plane) ovalization. σy is they component of the bending induced stress, represented withthe color scale within the fiber cross-section. θ is the angle ofrotation of the fiber ends. b- Relative refractive index vari-ation as a function of the fiber diameter change in percentcompared with an initial diameter of 630 nm.

We observe that ovalization or Brazier effect also par-ticipates in the birefringence. Brazier effect is a mechan-ical deformation that causes the ovalization of the cross-section of a bent tube. As shown in the Fig. 8-a, σy isdefined as the y component of the bending induced stress.The color in the fiber cross-section shows the value of σygiven by [26]:

σy = K2(E/2)(x2 − a2) (8)

where E is the Young’s modulus of silica, K = 1/R is thecurvature of the longitudinal axis, and a is the fiber ra-dius. The value of σy varies along the y axis from 0 GPaat the fiber surface to -0.03 GPa at y = 0 (correspondingto the compression stress), which gives an increased cir-cumferential strain at fiber surface along the x axis and areduced circumferential strain at fiber surface along they axis [28]. This effect, named as Brazier effect, causedthe ovalization in the plane perpendicular to the bend-ing axis. The relative displacement of two axes can beapproximated as [27]:

δ = 0.553aK (9)

where a is the initial fiber radius before bending. Tak-ing into account the measured loop radius (R ≈ 11µm)and the fiber diameter we found a reduction of the fiberdiameter of 1.5% parallel to the bending direction andan increase of 1.5% in the normal direction. In con-trast to standard fibers, this small change in the geom-etry will have strong impact given the transverse dis-tribution of the field. A difference of 3% of the ra-dius, leads to a difference of effective refractive index of

7

(n+1.5% − n−1.5%)/n = 0.8%, as shown in Fig.8-b. Thejoint effect of stress and ovalization give a birefringenceof 2.5%, which is in reasonable agreement with the ob-served value (∆ = 3.1%). The small discrepancy betweenthese two values is likely due to the fact that the loop isnot perfectly circular (as visible in Fig.6-a ) and thereforeleads to a non-homogeneous ovalization and stress effectalong the ring.

V. CONCLUSION

In this paper, we have reported the fabrication andcharacterization of two optical devices based on loopednanometrical optical fibers. The nature of the deviceschanges with the topology of the loop. The entwinedpart of the loop can be treated with the coupled modetheory and can be seen as a tunable beam splitter. Weshowed that a twisted loop creates a Sagnac interferom-eter, of which the dephasing is tuned by the torsion ap-plied on the nanofiber. The second device is a cavity,simply made out of a nanofiber knot. We analyzed itsspectral response and found a finesse of 8 and a quality

factor of 1100. Unlike in common resonators, the cou-pling efficency into the cavity strongly depends on thewavelength, which modulates the visibility of its reso-nances. It is reproduced accurately by our theoreticalmodel. Both setups, Sagnac interferometer and Fabry–Perot resonator are essential to photonics applicationsand optics in general. Their miniaturized versions pre-sented here pave the way toward their integration in pho-tonic circuits. A refined analysis of the cavity spectrumrevealed that birefringence of a bent nanofiber is alsoaffected by ovalization of its profile, and not only bystress as a normal fiber would be. With both devices,we showed how sensitive nanofibers are to mechanicalconstraints. In this regard, they could open an excitingplayground in optomechanics.

VI. ACKNOWLEDGMENTS

This work has been supported by the C-FLigHT ANRproject, Emergences Ville de Paris Nano2, CaiyuanpeiProgramm. C.D. is supported by the CSC scholarship.

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