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Limits of light guidance in optical nanofibers

Alexander Hartung*, Sven Brueckner, and Hartmut Bartelt

Institute of Photonic Technology, Albert-Einstein-Str. 9, 07745 Jena, Germany

*alexander.hartung@ipht-jena.de

Abstract: Reducing the waist of an optical fiber taper to diameters below

1 µm can be interpreted as creating an optical nanofiber with propagation

properties different from conventional optical fibers. Although there is

theoretically no cutoff of the fundamental mode expected, a steep decline in

transmission can be observed when the fiber diameter is reduced below a

specific threshold diameter. A simple estimation of this threshold diameter

applicable to arbitrary taper profiles and based on the diameter variation

allowing adiabatic transmission behavior is introduced and experimentally

verified. In addition, this threshold behavior is supported by investigating

the variation of the power distribution of the nanofiber fundamental mode

as a function of the fiber diameter.

©2010 Optical Society of America

OCIS codes: (060.2310) Fiber optics; (060.2400) Fiber properties; (220.4241) Nanostructure

fabrication; (230.7370) Waveguides

References and Links

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Received 6 Jan 2010; revised 26 Jan 2010; accepted 29 Jan 2010; published 9 Feb 2010

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1. Introduction

In recent years, optical nanofibers have attracted much interest due to their remarkable

properties and the multitude of their possible applications. Optical couplers [1], micro ring

resonators [2], optical refractive index sensors [3], and nonlinear devices for super continuum

generation [4] have been fabricated successfully, and efficient atom trapping and guiding

schemes have been proposed [5].

Despite many different possibilities for producing freestanding nanofibers, their

fabrication is still challenging because of their low mechanical stability and high demands

concerning their diameter uniformity. The most common way is the preparation of optical

nanofibers by means of tapering existing optical fibers down to diameters in the range of 1

µm and below. In such a taper configuration, the submicron diameter waist is well applicable

for the investigation of the optical transmission properties of nanofibers. The transition

regions on both sides of the waist can serve for efficient input and output coupling of the light

and operate as a mechanical support simplifying the handling of the nanofiber. Various

mechanisms as, for instance, a traveling gas flame [1,4,6–8], an electric strip heater [9], a

fiber coupler production system [10], and an indirect CO2 laser heating method [11] have

been employed for fabricating nanofibers. Our experimental setup is based on a CO2 laser

heating principle as briefly described in section 2.

High transmission of fiber tapers is important for many possible applications. For this

purpose we will discuss in section 3 the transmission phenomena during the tapering process

when the fiber diameter is reduced to submicrometer dimensions. A threshold diameter with

strong increasing losses can be observed in the submicron diameter range. This threshold

diameter of a nanofiber for efficient light guiding is the main topic of the further discussions

in section 4 and 5. Experiments have already revealed a distinct increase in losses of the taper

when its waist diameter is decreased below the investigation wavelength [1,8,12]. According

to [1], these losses are due to the scattering of raising evanescent fields by surface

contaminants. Sumetsky traced these losses back to input and output losses, which therefore

occur mainly in the transition regions with varying diameter [12]. A formula for the

transmission power has been deduced depending on the Landau-Dykhne formula obtained in

quantum mechanics [13], the characteristic length of diameter variation and an assumptive

Lorentzian variation of the transverse propagation constant along the nanofiber [14].

Our estimation for the threshold diameter as introduced in section 4 relies on the main

coupling and energy transfer between the fundamental mode HE11 of the nanofiber with its

diameter-dependent propagation constant β(d) and the external radiation modes with

propagation constant β = 2π/λ. It is based on the local diameter variation along the nanofiber.

Discrimination between the constant diameter taper waist and the changing diameter taper

transitions becomes redundant from this point of view. Only the local taper length scale,

which might be significantly different or comparable at fixed points within the taper waist and

the taper transitions, decides between lossy and adiabatic guidance of light along the

submicron-diameter fiber taper regions. The threshold diameter obtained is verified by

experiments in section 5. In addition, the diameter-dependent distribution of power density of

the fundamental mode is numerically investigated in section 6 to achieve further clarification

concerning the threshold behavior of nanofibers.

2. Experimental setup

We have used an indirect laser heating principle for taper preparation due to its high

flexibility in controlling the heating impact on the fiber (Fig. 1). The beam of a CO2 laser

(10.6 µm wavelength) with a maximum optical power output of 30 W is split with a 50/50-

beam splitter into two identical beams and subsequently focused on two opposite sides of the

micro furnace (corundum - multi crystalline Al2O3, outer diameter 1.7 mm, inner diameter 1.1

mm) with the standard optical fiber to be tapered placed in its center. The two-sided heating

setup assures that the initial cylindrical symmetry of the fiber is maintained for all possible

taper diameters. This is achieved with a relatively low power of 6 W per beam. During the

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Received 6 Jan 2010; revised 26 Jan 2010; accepted 29 Jan 2010; published 9 Feb 2010

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drawing process, the fiber is pulled by two translation stages moving in opposite direction. In

addition to this pulling motion, the translation stages adopt an oscillatory movement to scan

the hot zone of the microfurnace along a fixed length w of the fiber and to achieve a waist of

constant diameter and of equal length w [6]. The setup allows preparation of silica tapers

down to diameters of about 500 nm. Examples of tapers made with this setup are shown in

Fig. 2.

Fig. 1. Sketch of the nanofiber fabrication setup using a bidirectional indirect laser heating

method.

Fig. 2. Examples of submicron tapers.

3. Investigation of transmission properties during taper drawing

The variation of the transmission intensity during the tapering process is not only helpful for

control purposes of the process but also gives an insight into the mode guiding and coupling

properties of fiber tapers. The coupling strengths of modes can be influenced by the slope of

the transition regions. The coupling effects are expected to be negligible for slowly varying,

adiabatic tapers [15]. If the changes in slope of the taper transitions are strong, higher order

modes of equal symmetry or even leaky modes are accessible due to mode coupling, and the

transmission observed will then be reduced.

In Fig. 3 characteristic transmission phenomena of the overall taper geometry during the

drawing process are shown. During the tapering process the waist was kept at a fixed length

of 4 mm and the taper transmission was modified by a gradually decreasing waist diameter

and by changing the length of the transition regions. As a light source, a tunable laser was

used at a wavelength of 1550 nm with a coherence length considerably longer than the overall

taper length, to assure that interference phenomena between excited modes become visible for

the complete process.

At the beginning of the process, only the fundamental mode HE11 of the standard single-

mode fiber is exited in the fiber core. A high effective index difference between the HE11 and

higher order cladding modes prevents mode coupling, and nearly no change in transmission is

observed. While the cross section is reduced, the influence of the core material on the guiding

properties of the fundamental mode diminishes, and the fundamental mode then is guided by

the silica-air interface as the higher order modes. Therefore the effective indices of the

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Received 6 Jan 2010; revised 26 Jan 2010; accepted 29 Jan 2010; published 9 Feb 2010

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fundamental and higher order modes approach each other. Calculations show that the smallest

effective index difference and therefore the highest mode coupling tendency can be found in

the diameter region around 60 µm. If the transition slope is too steep, energy is transferred

from the HE11 to the HE12, the nearest mode of the same azimuthal symmetry, and decreased

transmission is observed. Higher order modes of different azimuthal symmetry are not

expected, due to the cylindrical symmetry maintained by the two-sided heating setup.

Continued reduction and elongation leads to a beating phenomenon with constant

amplitude and increasing frequency between the two lowest excited modes [16]. The beating

is visible down to the cut-off diameter of the HE12 mode occurring at a value of V = 3.832,

corresponding to a fiber diameter of 1815 nm. For observing the HE12 cutoff it is important to

use a narrow-band light source fulfilling the coherency requirements. Otherwise the

oscillation amplitude will decay gradually with elongation of the taper, and reach zero at a

specific elongation length.

In the following single-mode diameter condition of the waist, transmission stays almost

fixed without strong oscillation up to a point where a steep decline in intensity can be

observed. The value of this threshold diameter will be discussed in the next section.

Fig. 3. Typical variation of transmission during the tapering process of a standard single-mode

fiber. The initial fiber diameter of 125 µm is reduced by a factor of 2 nearly every 250 s.

Fig. 4. Sketch for the definition of the local taper length scale l based on the local taper angle Ω

and the local taper radius r.

4. Estimation of the local taper length scale for adiabatic optical nanofibers

The following estimation of the threshold diameter for highly transmitting nanofibers is based

on the length scale criterion for adiabatic fiber tapers first introduced by Love et al [15]. Jung

et al successfully applied this criterion for the explanation of efficient inline fiber mode filters

based on taper transitions that are only adiabatic for the fundamental mode and non-adiabatic

for higher order modes [17]. It is derived from the fact that the local taper length scale has to

be much larger than the coupling length between the fundamental mode and the ruling

coupling mode for negligible power transfer. The local taper length scale l is defined as the

height of a circular cone with the base matching the local fiber cross-section and the apex

angle matching the local taper angle (Fig. 4). This yields a local taper angle of Ω =

arctan(dr/dz), resulting in Ω = r/l for small angles Ω << 1. In our case of a single-mode optical

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Received 6 Jan 2010; revised 26 Jan 2010; accepted 29 Jan 2010; published 9 Feb 2010

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nanofiber that only exhibits the fundamental mode with propagation constant β1(d), the

dominating coupling mode is the radiation mode with a propagation constant of β2 = 2π/λ,

assuming the refractive index of air being equal to 1. The beat length b equals 2π/(β1-β2). If l

>> b everywhere along the taper, coupling and loss are negligible, and the fundamental mode

can propagate adiabatically. On the other hand, if b >> l, coupling to radiation modes

becomes possible, and significant losses will occur. Therefore l = b offers an approximate

delineation between lossy and adiabatic diameter variations. This results in the condition of Ω

= r(β1-β2)/2π for a delineation concerning the local taper angle or l = 2π/(β1-β2) concerning the

local taper length scale as shown in Fig. 5.

Although the modeling has been done for a refractive index of n = 1.444 and a wavelength

of λ = 1550 nm, the obtained delineation curve is valid for different fiber diameter and

wavelength combinations as long as the ratio d/λ does not change. This is due to only small

changes ∆n/n = 0.02 of the refractive index of pure silica in a wide wavelength range between

400 nm and 1700 nm.

Due to the large difference in propagation constants, a relatively small taper length scale

down to 10 µm should be sufficient for fiber diameters above 0.6λ to assure approximately

adiabatic behavior. The conditions change strongly for smaller diameters. The requirement for

adiabatic behavior grows more than exponentially and leads to d = 0.29λ for a local taper

length scale of 1mm and only d = 0.16λ for a giant local taper length scale of 1 km. Increasing

the local taper length scale over 6 orders of magnitude results in a change of the threshold

diameter of only about a factor of 2.

For a typical taper with a monotonic change in diameter and local taper length scale, the

smallest diameter will define the necessary demands concerning uniformity and strongest

coupling to radiation modes, and therefore also the overall performance of the taper. The

actual loss of a specific nanofiber is not given by the model, since such a value would depend

on the specific profile of the taper region.

The results found by [13] support this behavior but exact comparisons are difficult due to

different intentions. While we estimate the transmission threshold diameter for a wide range

of possible local and global diameter variations, reference [13] deduced a formula for the

propagation losses caused by a specific variation of the transverse propagation constant. From

these results, changing the characteristic length from 10 mm to 10 km leads to a change of the

threshold diameter roughly by a factor of 2.

Fig. 5. Delineation curve between lossy and adiabatic local taper length scale l depending on

the nanofiber diameter d. The diameter d is normalized to the wavelength.

5. Experimental verification

For experimental verification we prepared submicron taper structures and measured their

diameter-dependent transmission characteristic in the critical range between adiabatic and

non-adiabatic behavior.

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Keeping the heated length w fixed in our taper drawing setup during the whole elongation

process results in an exponential taper profile of r(z) = r0 exp(-z/w) along the symmetry axis z

of the fiber [6]. Based on this known profile, the local taper angle is then tan(Ω) = dr(z)/dz = -

r0/w exp(-z/w) = -r(z)/w.

This equation corresponds to the definition of the local taper length scale l. Drawing the

fiber taper at constant waist length w results in a taper profile with constant local taper length

scale l = w.

Figure 6 shows the variation of transmission during tapering for two equivalent tapering

processes. The local fiber diameter of the 0.5 transmission level is taken as threshold diameter

dt between adiabatic and lossy propagation. For the nanofibers drawn with a waist length of l

= 6 mm, the threshold diameter lies at dt = 0.38 µm (dt/λ = 0.25), which agrees well with the

diameter predicted by the length scale criterion.

Fig. 6. Variation of transmission for two identical fiber tapers with a waist length of 6 mm.

Fig. 7. Axial component of the Poynting vector of nanofibers for different fiber diameters of a)

0.75λ, b) 0.3λ, and c) 0.2λ, respectively. The displayed area size is 2λ x 2λ. Only one of two

polarization modes is shown. Red signifies high power density and blue low power density.

6. Power density distribution of optical nanofibers

Further numerical simulations concerning the shape of the fundamental mode for small fiber

diameters have been performed to further understand the obtained threshold behavior of

optical nanofibers. All calculations have been done with a full vectorial finite element method

at a wavelength of λ = 1.55 µm, a refractive index of nnanofiber = 1.444 and nair = 1. In Fig. 7,

the power flow density along the fiber axis is simulated for different nanofiber diameters. For

a relatively big nanofiber diameter of d = 0.75λ, the single polarization mode has the expected

Gaussian-like shape. However, for fiber diameters near the experimentally observed

transmission threshold, the linearly polarized character and the evanescent fields becomes

dominant, and the common assumption of a Gaussian-like distribution is no longer valid [18].

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Received 6 Jan 2010; revised 26 Jan 2010; accepted 29 Jan 2010; published 9 Feb 2010

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Furthermore, the fraction of power propagating outside the fiber area changes dramatically

within a narrow diameter variation from only 0.18 at d = 0.75λ to 0.97 at d = 0.3λ, indicating

a threshold behavior [19]. It can be expected that coupling to outside propagating radiation

modes will be less critical for mode distributions concentrated within the fiber than for modes

mainly concentrated outside the fiber. Therefore, small deviations from the perfect cylindrical

symmetry are sufficient for diameter values around d = 0.2λ to couple light out.

In Fig. 8 the variation of the power density as a function of the fiber diameter is shown for

three selected points of the fiber cross-section. A maximum value for the center of the fiber

can be found at a fiber diameter of d = 0.60λ. Down to this diameter an increasing

concentration of the guided light within the fiber area is possible. For smaller fiber diameters

this center power density declines rapidly over several orders of magnitude, again indicating

threshold behavior below this fiber diameter. At the experimentally achievable diameter of d

= 0.25λ, the core power density declined over 2 orders of magnitude compared to the

maximum value at d = 0.6λ. The highest power density at the fiber-air interface is reached at a

diameter of d = 0.44λ [7].

Fig. 8. Variation of power density as a function of the fiber diameter to wavelength ratio in the

center and at the surface of the fiber. Due to an asymmetric power distribution of the polarized

fundamental mode, the direction of polarization has to be considered. The overall guided

power is normalized to 1 W.

Both maximum power density values lie at experimentally achievable local taper length

scales. Especially the diameter value for the highest power density at the surface could be

interesting for sensor applications. At this diameter, already 68 percent of the guided power is

propagating outside the fiber, which offers the possibility of high sensitivity in interaction

with external analytes. Furthermore, the guided light is still confined to a region close to the

fiber. A decline to a value of 1/e takes place within a distance of only 0.14λ apart from the

fiber surface as expected for evanescent waves, promising strongly localized sensing effects.

The power density increases due to the reduced cross section of the nanofiber in comparison

to an untapered fiber by up to a factor of 75 for the center of the fiber core, and in the same

order of magnitude for the core-cladding interface.

7. Summary

A length scale criterion for the delineation between adiabatic and lossy optical nanofibers

applicable to arbitrary taper profiles has been derived. For experimentally realizable local

taper length scales of about 10mm, the smallest adiabatic diameter turns out to be in the range

of about d = 0.24λ (d = 370 nm for λ = 1550 nm or d = 96 nm for λ = 400 nm). This threshold

value is consistent with a mode distribution where a power fraction of more than 97 percent is

propagating outside the solid taper material.

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Received 6 Jan 2010; revised 26 Jan 2010; accepted 29 Jan 2010; published 9 Feb 2010

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The observed threshold diameter for effective light guidance of nanofibers is caused by

geometric variations or nonuniformities. Since the difference in the propagation constant of

the fundamental mode and the radiation threshold vanishes for arbitrary small fiber diameters,

the demands on the geometric uniformity grow extremely, restricting the experimentally

usable diameters.

For fiber sensing applications that require a high power density at the fiber-air interface,

the threshold range of the diameter is attractive due to possible strong interaction with outside

media.

Acknowledgement

Funding by the Thuringian Ministry of Education, Science and Culture is gratefully

acknowledged.

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Received 6 Jan 2010; revised 26 Jan 2010; accepted 29 Jan 2010; published 9 Feb 2010

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