## Abstract

Fourier domain optical coherence tomography (FDOCT) is used to non-invasively measure properties of the hole pattern in microstructured fibers. Features in the FDOCT data are interpreted and related to the hole diameter and spacing. Measurement examples are demonstrated for three different fibers with one hole, three holes at the vertices of an equilateral triangle, and a full triangular lattice. These studies provide the first path to real time monitoring of microstructured fibers during their draw.

© 2005 Optical Society of America

## 1. Introduction

Microstructured optical fibers (MOF) are of scientific interest because of their unique guiding mechanisms and associated modal and dispersion properties [1, 2]. They greatly extend the realm of possibilities for waveguides in the fiber geometry beyond what is possible with conventional optical fibers. For example, with photonic bandgap guidance it is possible to propagate light through an air core with negligible nonlinearity and theoretical loss limits an order of magnitude lower than those of conventional fibers. Effective index guided MOF have been designed to be single moded at all wavelengths, and to have extremely high nonlinearity for octave spanning continuum generation. These unique properties of MOF depend critically on their hole size and distribution. Pressure and temperature of the preform can be varied to control hole size and separation during fiber draw. Currently fiber samples are taken by interrupting the draw and checked under a microscope to optimize draw parameters. However drawing a MOF is a nonequilibrium process where the balance between viscosity, surface tension, and pressure in the holes can change with length of the fiber drawn resulting in a drift of hole parameters with length [3]. Successful manufacture of long MOF lengths requires non-invasive on-line measurement of hole parameters so that real time corrections can be made to the draw conditions – this cannot be achieved by instruments used for monitoring standard fibers. To our knowledge such a technique does not exist and this paper is the first attempt to address the problem.

In this paper Fourier Domain Optical Coherence Tomography [4] (FDOCT), which is widely used for medical imaging, is used to measure MOF. The technique was previously used to simultaneously measure the dimensions of multiple coatings applied to fibers and the uncoated fiber diameter [5, 6].

## 2. Theory and experimental setup

The FDOCT technique images refractive index distributions in samples by analyzing their reflectivity as a function of wavelength. Figure 1 depicts our FDOCT setup. The fiber sample to be characterized is placed in one arm of a Michelson interferometer and illuminated with >300 nm wide continuum generated by pumping nonlinear fiber with high power cw light [6]. The orientation of the fiber hole pattern with respect to the incident beam is monitored with a camera looking at a fiber end. The interferometer output is coupled into a spectrometer where reflections from each interface in the sample and reference arms interfere to produce modulations in the spectrum with periods corresponding to the optical path length difference between them.

The spectrogram *I(ν)* is described mathematically by,

where *I _{r}* is the spectral intensity reflected by the reference arm mirror,

*I*is the spectral intensity reflected by the

_{i}*i*interface in the fiber that is an

^{th}*optical*distance

*d*from the reference,

_{i}*ν*is the frequency and

*c*is the velocity of light in space. The third term is the interference of the reference with each interface of the fiber. The fourth term is the interference between various fiber interfaces and constitutes the autocorrelation of the fiber structure. The Fourier transform of

*I(ν)*contains peaks at 2

*d*/

_{i}*c*and 2(

*d*-

_{i}*d*)/

_{j}*c*, corresponding to the optical separation between the various reflecting surfaces. Hence the location of various interfaces of the object can be determined from the FFT and the object cross section can be reconstructed.

Our spectrometer recorded a 277 nm bandwidth (from 1418 nm to 1695 nm) with a 0.6 nm resolution. During data processing, the spectrograms were normalized with the source spectrum and a linear fit to it was subtracted in order to eliminate the d.c. component before the FFT was taken. The measured free space axial resolution (FWHM of the system point spread function) was *F _{R}*=5.5

*µ*m which implies a resolution of ~4.0

*µ*m in glass of index

*n*=1.444. By peak fitting the FFT, the reflecting surfaces could be located with a~1

*µ*m accuracy.

## 3. Fiber with a single hole at center

We begin by studying a fiber with a single hole which is the basic element that constitutes a MOF. Figure 2 is the FFT recorded for an uncoated fiber with a single hole of diameter *BC*=8*µ*m at its center (see caption for peak locations). Peaks A and D arise from reflections off the external fiber surfaces, and peaks B and C arise from reflections off the near and far surfaces of the hole. Half the difference between B and C yields a hole size of ~9.3*µ*m after accounting for the *π* phase shift on reflection at interface C. Thus both front and back surfaces of holes contribute to the spectrogram and with a high resolution system they can be resolved.

## 4. Fiber with three holes arranged in a concentric equilateral triangle

In order to understand the FFTs of a MOF with a full triangular lattice, we next examine a MOF containing just three holes of diameter *d*=8*µ*m at the vertices of an equilateral triangle with side of length Λ=16*µ*m, which is concentric with the fiber. This structure represents the unit cell of a triangular lattice. The FFTs in Figs. 3(a) and (b) were recorded with the fiber in orientation *M* and *K* respectively. Orientation *M* is defined as when light is incident perpendicular to a side of the triangle, whereas orientation *K* is defined as when light is incident parallel to a side of the triangle. The reflecting planes of the holes and their orientation with respect to the beam are shown in the insets. Peak S arises from the reflection at the fiber surface. Peaks from various reflecting planes of the holes (shown in the insets) are labeled accordingly on the FFT graphs and their exact locations are given in the figure caption.

In order to interpret the FFT pattern, we assume that (i) light incident on a hole surface has not passed through a hole other than itself, which is reasonable since after propagation through a hole light diverges strongly, and (ii) only light that is reflected exactly back on itself is recorded in the spectrogram.

In the *M* orientation (cf. Fig. 3(a)), for light that scatters off just one surface, the round trip optical distance of surfaces from plane *A* are *AB*=2*d*=16*µ*m, *AC*=2*n*Λ√3/2=40*µ*m, and *AD*=2[*n*Λ√3/2+*d*]=56*µ*m. The measured distances of *AB*=16.2*µ*m (⇒*d*=8.1*µ*m), *AC*=37.4*µ*m (⇒Λ=15*µ*m), and *AD*=54.8*µ*m compare well with these expected values.

In addition, light incident on a hole can suffer total reflection if the angle of incidence on the hole exceeds *θ _{c}*=sin

^{-1}(1/

*n*)=43.85°. This totally reflected light can suffer retro reflection at another hole if it is incident along its diagonal. For our studies we assume that the light recorded in the spectrogram has scattered off a maximum of two holes and the two holes involved are the nearest neighbors. Path

*X*(shown in the inset of Fig. 3(a)) represents such a two hole scattered ray. The round trip optical path difference of

*X*with respect to the ray reflected at surface

*A*is,

where,

In the FFT of Fig. 3(a), the measured distance *AX*=28.2µm agrees well with the calculated value of *T*=30.6*µ*m for such a ray that is scattered off two holes.

Figure 3(b) is the FFT recorded in the *K* orientation arrived at by rotating the fiber by thirty degrees from orientation*M*. The round-trip optical distance of the various surfaces from surface *A* are, *AB*=2*d*=16*µ*m, *AC*=2[*n*Λ/2]=23*µ*m, *AD*=2[*n*Λ/2+*d*]=39*µ*m, *AE*=4[*n*Λ/2]=46*µ*m, and *AF*=4[*n*Λ/2]+2*d*=62*µ*m. From the measured peaks we get *AC*=20.3*µ*m *AD*=34.6*µ*m, *AE*=45.7*µ*m, and *AF*=58.9*µ*m. Except for *AD*, the error in the *one way geometrical* distance between the reflecting planes lies within the 1*µ*m accuracy of the system. The surfaces *B&C*, and *D&E* have round trip optical separations of 7*µ*mfrom each other which is less than the FWHM, 2*F _{R}*, of the FFT peaks. Hence peak

*B*is not resolved, and the error in peak

*D*is higher than the stated 1

*µ*m accuracy.

## 5. Fiber with concentric triangular lattice of holes

Having understood the FFT patterns of a fiber consisting of a single three hole unit cell along two cardinal directions, we now examine the FFTs of a MOF with a full triangular lattice concentric with the fiber. Our MOF had a diameter *D*=245µm, with six rings of holes of diameter *d*=5.5±0.5*µ*m, and pitch Λ=10.8*µ*m surrounding a core formed by the absence of a single hole at the center. Figure 4(a) shows the FFT recorded for orientation *M* (light incident perpendicular to a side of the hexagonal hole pattern where the holes are oriented as shown in inset of Fig. 3(a) with respect to the incident beam) with the reference arm blocked. The FFT (which, with the reference arm blocked, is an autocorrelation of the fiber cross section) shows very distinct peaks. The spectrogram (shown in inset of (a)) consists of slow modulations due to interference amongst the holes (resulting in peaks at short distances in the FFT), and fast modulations due to the interference of holes with the fiber surfaces (resulting in peaks at larger distances in the FFT). The early peaks therefore contain direct information about the lattice structure. However, in an autocorrelation the origin of the FFT peaks can be ambiguous and more complicated to associate with real crystal features and therefore we will stick to examining the cross correlation of the crystal features with the reference arm.

Figures 4(b) and (c) show the FFTs recorded with the reference arm along diagonals *M*, and *K* (light incident on vertex of hexagonal hole pattern where the holes are oriented as shown in inset of Fig. 3(b) with respect to the incident beam) respectively. The distances 2*nX _{1}* and 2

*nX*(for orientations

_{2}*K*and

*M*respectively; see Figs. 4(b) and (c)) between peak

*S*(due to the fiber surface) and the first peak from the hole pattern

*A*, give the round trip distance of the first hole surface from the fiber surface. If the crystal pattern is uniform and concentric with the fiber surface (as with our fiber), then the hole pitch Λ can be derived from measurements along

*K*and

*M*directions as Λ=(

*X*

_{2}-

*X*

_{1})/

*N*(1-√3/2) where

*N*(

*N*=6 in our case) is the number of lattice periods from the center of the fiber to the center of the hole in the outermost ring. From the FFTs we get

*X*

_{1}=55.8

*µ*m, and

*X*

_{2}=64.5

*µ*m, which gives Λ=10.8

*µ*m in agreement with measurements under the microscope.

In the *M* direction, light sees the three hole structure of Fig. 3(a) repeated several times. The round trip optical distance of hole surfaces from the first reflecting hole surface *A* are given by *U _{m}*=2

*m*(

*n*Λ√3/2), and

*V*=

_{m}*U*+2

_{m}*d*(

*m*=0,1,2, …). Figure 4(d) shows that in Fig. 4(b) the distance of most measured peaks from peak

*A*lie within 3

*µ*m of the predicted

*U*, and

_{m}*V*values. The average hole diameter

_{m}*d*=(

*V*-

_{m}*U*)/2 from the measured FFT is 5.9

_{m}*µ*m which agrees well with microscope measurements. In addition, we observe a peak at 18.8

*µ*m from peak

*A*which agrees well with the expected value of 20.5

*µ*m for a ray that suffers total reflection at one hole surface and is retro reflected by another hole (like

*X*in Fig. 3(a)).

Similarly, following the analysis for the three hole fiber in the *K* orientation (Fig. 3(b)), the peak distances from peak *A* in Fig. 4(c) are *P _{m}*=2

*m*(

*n*Λ/2), and

*Q*=

_{m}*P*+2

_{m}*d*. The peaks are closer in the

*K*orientation than the

*M*orientation and are therefore more difficult to resolve. With a limited resolution system, the error due to overlapping of peaks is therefore larger in the

*K*orientation.

The fibers we have presented have relatively large feature sizes so that they can be resolved by our system. Usually the desired accuracy is ~5%, which is ~100 nm for typical MOF. Today’s state-of-the-art FDOCT systems have sub-micron resolutions [7] with which fibers with smaller feature sizes can possibly be measured with a 100 nm scale accuracy.

On the draw tower, where the fiber spins and vibrates so that the orientation of the crystal with respect to the beam changes constantly, clear images can still be obtained with high speed FDOCT systems capable of kHz data acquisition rates [8]. To implement this technique on the draw tower two FDOCT systems spaced angularly by thirty degrees can be used to make simultaneous measurements on the fiber. The *K* orientation is identified unambiguously when one device measures the shortest distance between the fiber surface peak *S* and the first peak from the hole structure *A* - the second device then records data for the*M* orientation. From these measurements Λ can be retrieved. In addition, if *F _{R}*≤

*d*, the

*M*orientation can be analyzed to retrieve

*d*.

In conclusion, FDOCT can measure the hole diameter and pitch in MOF non-invasively. The MOF features could, in principle, be monitored in real time during draw thereby enabling fabrication of long lengths of uniform fiber through the active control of draw conditions.

## Acknowledgments

The author thanks R.Windeler, and R. Bise for fiber samples, S.Wielandy for the spectrometer, and E. Monberg, F. Dimarcello, J. Fini, and D. Trevor for useful discussions.

## References and links

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