Local supermode analysis of tapered fibre sensor

J.A. Besley J. D. Love G.B.Scelsi

Indexing terms: Supermode analysis, Tapered fibre sensor, Optical fibre

Abstract: The authors investigate the performance of an evanescent-field fibre sensor made from standard single-mode optical fibre with a tapered cladding region, upon which a thin attenuating material coating is deposited. The authors' method involves calculating the supennodes of the complete core-cladding- coating-air structure locally at each point along the taper and studying the evolution of the supermode guided power along the sensor. Results are compared with those from a numerical finite difference scheme; and also used to interpret experimental results obtained for a taper with a thin layer of sulphur deposited by electrical discharge in an atmosphere of SF,.

1 Introduction

An important class of sensors is based on evanescent field changes due to chemical substances deposited near the core of a single-mode optical fibre [l]. In this paper we consider both theoretically and experimentally the properties of such an evanescent field sensor on which an absorbing material coating is deposited in a thin layer. This sensor is fabricated from standard single mode fibre. The protective jacket is stripped to expose the cladding, which is then etched sufficiently using hydrofluoric acid to produce a slowly-varying, approxi- mately adiabatic taper along its length, with a mini- mum cladding radius approaching that of the core. Intuitively, the most sensitive region of this device is where the cladding is thinnest and hence the evanescent field of the fibre-core modal fields are strongest. Fig. 1 shows schematically the structure of the tapered sensor before deposition of the coating. The tapered fibre is very fragile, but if appropriately mounted it can be made into a fairly robust sensor.

The sensor was originally devised to detect the deposition of a dielectric film in a low-pressure environment. We consider the effect of a thin uniform coating of sulphur deposited around the tapered fibre. The properties of this film depend on the structural 0 IEE, 1997 IEE Proceedings online no. 19971609 Paper frst received 18th February and in revised form 13th August 1997 J.A. Besley and J.D. Love are with the Optical Sciences Centre, The Australian National University, ACT 0200, Australia G.B. Scelsi is with the Division of Physics and Electronics Engineering, University of New England, Armidale, NSW 2351, Australia

IEE Proc.-Optoelectron., Vol. 144, No. 6, December I997

form of the sulphur, and therefore, on the method of deposition. In general, though, sulphur has a higher refractive index than the fibre and a relatively high absorption coefficient. Accordingly, we model the coated fibre with the piecewise step refractive index profile illustrated in Fig. 2.

Fig. 1 Schematic offibre taper before deposition of ohemical layer

refractive ,.> ,., index, n nde=riBJde+in'l'de

1

radial distance, units of p

Fig. 2 Refactive index profile of taper with deposited chemical layer , refrac$:fx:n , "Cl

i 1 T(Z)

radial distance, units of p Fig. 3 Refactive index profile of core structure for local mode analysis

We can understand qualitatively the effect of the sul- phur coating on propagation of the fundamental mode along the tapered fibre in terms of mode coupling. It is well known that, when analysing a hybrid waveguide

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structure, such as a coupler, it is possible to find the modes of the individual guiding subprofiles (two fibres cores in the case of a coupler) and then apply a pertur- bation approach to determine the coupling between them [2]. In the tapered sensor described above, there are two guiding subprofiles, shown in Figs. 3 and 4, which consist of the core-cladding-air and the cladding- sulphur-air, respectively. The fundamental mode, which is localised in the core, can couple power across to the annular sulphur coating and thus to the 'coating' modes which are then attenuated by the absorption of the sulphur. The different cross-sections and profile of the core and the coating layer mean that strong cou- pling will occur at certain resonant wavelengths. In terms of the transmission spectrum for the fundamental core mode, this is equivalent to dips appearing in the spectral response of the sensor.

refractive ,.\ I

index, n nde=n"'de+in''de

radial distance, units of p

Fig. 4 sis

Repactive index profile of coating structure for local mode analy-

This coupled mode analysis would be accurate in the limit of greatly separated guiding substructures. Our tapered sensor is not in this regime, however, as the inner radius of the sulphur layer approaches the outer radius of the core at the midpoint of the taper. The coupled mode analysis accordingly becomes less accurate, and we therefore consider the modes of the complete core-cladding-coating-air structure which we call supermodes. We assume weak guidance, so that the fields and propagation constants of the supermodes are determined by solving the scalar wave equation across the whole structure, as described in [3]. We assume that each region of the taper is of uniform refractive index, in which case it is equivalent in structure to the stepwise matched-cladding fibre in [3]. We can then determine the local supermode fields and eigenvalue equations analytically, from which follow the propagation constants. The tapering introduces coupling between these supermodes, which we can quantify following the analysis in [2].

In the following Sections, we set up the mathematical model for the problem and present the method of solu- tion of the local coupled supermode equations. For comparison, we also use a numerical simulation of propagation through the complete tapered fibre using the beam propagation method (BPM). We then com- pare the results of both methods with the experimen- tally measured results.

2 Model

In the absence of an accurate measurement of the com- plete refractive index profile of the fibre and the precise

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taper shape, we adopt, for simplicity, a refractive index profile which is a piecewise step function across the core-cladding-coating-air structure, as depicted in Fig. 2. In this model, the core has uniform refractive index n,, and constant radius p, and the cladding layer has index n,l and outer radius e=o(z)p, where Nz) is a dimensionless linear function of the distance z along the fibre, imitating the taper shape. The deposited sul- phur coating has index ndep and an outer radius of z(z)p, where z(z) is a second linear function of z, beyond which is free space with unit refractive index. The coating has a constant thickness of d=(z(z) - o(z))p. The core and cladding materials are assumed to be nonabsorbing with real refractive indices, while the coating is assumed absorbing, with complex refractive index ndep = n$& + i n$&,, where superscripts (i) and ( r ) denote real and imaginary parts, respectively. The imaginary part of nd,, accounts for both absorption and scattering in the coating.

2. I Source representation In the following applications, we model a focussed laser source on the endface of the slab or fibre by assuming a coherent source with a transverse electric field distribution, E,. We also assume that E, includes the effect of reflection from the endface and has an approximately Gaussian distribution:

where s is the spot size, and A($) is the amplitude as a function of the azimuthal co-ordinate 4, which is con- stant for excitation of the axisymmetric modes, and has the form:

A($) = Ccos($) or Csin(4) (2) for second-order antisymmetric modes, where C is a constant.

2.2 Parameter values The parameter values are based on the data from the experimental results. The experiment used standard sin- gle-mode optical fibre with a nominal core diameter of 3 . 8 ~ and cladding diameter of 125p-1, which was immersed in hydrofluoric acid and tapered down to a diameter of about 6pm and back up again, with the core radius remaining constant over a 6mm length of fibre. The core index, n,, = 1.4616, and the cladding index nC1 = 1.4571. A thin layer of sulphur (of the order of a micron) was deposited around the outside of the cladding during several electrical discharges in an atmosphere of SF6. We ignore any asymmetry intro- duced by the deposition process, assume a coating thickness of 0 . 6 ~ and taper down to l p from the outer radius of the core.

Sulphur in the solid phase can take many forms [4], so it is difficult to estimate accurately values of the real and, in particular, the imaginary components of the refractive index. It may not be accurate enough to assume a single form of sulphur during deposition. Relaxation, and other chemical processes can occur. In general, though, the real and imaginary parts of the refractive index are higher in the blue than in the red part of the spectrum, the variation of the imaginary part being much more marked than the real part. Fol- lowing the results of Sasson et al. [4, 51, it is possible to approximate the real part of the refractive index by a curve which gives the best fit to the experimental data

IEE Proc.-Optoelectron., Vol. 144, No. 6, December 1997

over the range of wavelengths of interest. The index would be in the range 1.85-2.0 for the wavelengths we study. For simplicity, we take nominal values for the real n$;? -1.95 and the imaginary n$iP = 0.001 parts of the refractive index of the sulphur, and ignore disper- sion.

3 Modal analysis

The variation between the core and cladding indices is small, so we can use the weak-guidance approximation to determine the modal fields and propagation con- stants [2], Ch. 13. Although the coating index is some- what larger than that of the cladding, for the purposes of our approximate model, it is adequate to use the same approximation for the coating region.

3.1 Local modes As explained above, it is possible to calculate the local modes of the guiding subprofiles of our structure and then predict resonances using the results of mode cou- pling. These subprofiles are shown in Figs. 3 and 4. Details of the methods involved in these calculations can be found in [3].

Fig. 5 shows the spectral variation of the effective index of the fundamental ‘core’ mode and of the adja- cent coating-layer modes of the taper structure for a range of values of o, the normalised outer radius of the cladding. We note that the effective indices of the coat- ing modes are insensitive to the parameter o on the scale of this plot. In other words, the effective index values are determined predominantly by the normalised thickness of the coating layer (z - 0) rather than its normalised inner radius (o), down to very small radii.

frequency, THz Fi .5 Spectral variation of effective indices,offundumentaI mode (bold) an2 of adjacent symmetric ‘coating modes’ of taper structure for range of values of o, normalised distance to inside layer of coating Horizontal dashed lines show refractive indices of core (&) and cladding (n,,)

Fig. 6 shows the spectral variation of the effective index of the second ‘core’ mode and of adjacent antisymmetric coating-layer modes. We see a similar trend to the fundamental mode plot above the cladding index. The second mode can be followed beyond the traditional cutoff at the cladding index, however, since we have considered a guiding subprofile which extends out to air (Fig. 3). Below the cladding index value, the effective indices for different values of o separate. This is due to the fact that the continuation of the coating mode effective indices below the cladding index cross ‘cladding’ modes which are sensitive to the normalised cladding radius o. At the intersection of these modes we see a process analogous to that described in [3] for

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the modes of a depressed-cladding fibre below the clad- ding index. The continuation of the ‘coating’ modes can be found by following the mode of the next lowest ‘coating’ mode to the point where it effectively swaps over from the previous mode. The resonances in local mode coupling are therefore still expected to occur, but if the effective index of the second mode decreases below the cladding index (i.e. it is cutoff), it is possible for there to be no effective resonance for some values of the parameter 0.

0.75 0.60 ”W 0.50

1.458 0 E * - $ 1.457 c ._ .- J 4-

a, 5 1.456

400 500 600 700

frequency, THz

Fi .6 Spectral variation of effective indices of second core mode (bold) an? o adjacent antisymmetric ‘coating modes’ of ta er structure for range

Horizontal dashed line shows refractive indices of claddin,g (n,J 0 = 6.0. ~ ~ . 0 = 4.0. ... ... 0 = 3.0. ~ 0 = 2.0

of vu f ues qf o, normalised distance to inside layer ofcoating ~ . ~

It is noted in Figs. 5 and 6 that the curves for the axisymmetric and antisymmetric ‘coating’ modes coin- cide very closely. This can be explained iin terms of the asymptotic behaviour of the Bessel functions. For large parameter values, the first- and second-order Bessel functions approximate to the same sinusoidal function, and so we would expect the effective indices of the cor- responding modes of each symmetry to coincide. Since the ‘coating’ modes cross the ‘core’ modes at a large angle, we see that resonances are expected at approxi- mately the same frequency for the fundamental and second modes. Closer inspection reveals that the peak for the second ’core’ mode is slightly to the left of that for the fundamental.

The results obtained in this Section lead to the conclusion that we expect sharp resonances in the transmission spectrum of the sensor, even though it has a tapered profile and higher-order core modes may be present. Put another way, the resonance is insensitive to the cladding radius (0) and to the azimuthal symmetry.

3.2 Local supermodes For reasons given in the Introduction, we now put aside the modes of the subprofiles of the sensor in favour of the supermodes, which are the modes of the complete waveguide structure. In the case of longitudi- nally invariant fibre, these modes are orthogonal and no coupling will occur between them. As we quantify later, the tapering of the cladding leads to some cou- pling between these supermodes.

First, we give the derivation of the supermodes of the

the scalar transverse electric field of each supermode is expressible in the separable form:

uniform Lbre. Ifi the weak-guidance !approximation,

E(T, z ) = $(r, 4 ) exp(iP2:) (3) 385

where r is the radial co-ordinate relative to the fibre axis, 4 is the azimuthal angle, b is the supermode prop- agation constant, and W(r, y) is a bound solution of the scalar wave eqn. 4.

(4)

d 2 I d l a 2 2 2 - + - - + - - + k n ( r ) - P 2 dr2 r dr r2 dqP

where the wavenumber k=2 nlA, A is the source wave- length, and n(r) denotes the radially symmetric refrac- tive index profile of the complete structure.

The supermode fields are constructed from the bound solutions of this equation in each region of the profile, and continuity of the field and its first deriva- tives at every interface across the fibre, including the jacket-air interface, give the eigenvalue equation for the propagation constant. The detailed derivation for each supermode is very similar to that given in [3]. Within the coating region, the refractive index is assumed real when solving the eigenvalue equation, and so the calculated propagation constants are also real. The effect of the small imaginary part of the index in the coating on each supermode is regarded as a per- turbation, introducing a small imaginary part to its propagation constant, as described in [3]. To generalise these results to local supermodes, the representation of the transverse electric field in eqn. 3 is modified to:

where y ( r , 4) denotes the orthonormal form of the transverse field, i.e. with unit normalisation, and the integral denotes the accumulated phase along the length of the taper, since the local propagation con- stant now varies with the cross-section.

3.3 Coupling of local supermodes When the core-cladding region of the fibre is single- moded, the fundamental mode is axisymmetric, (i.e. y ( r , 4) = y ( r ) ) , and hence the corresponding local supermode is also axisymmetric, with an effective index n,ff = blk lying between the core and cladding index values, with the supermode field confined predomi- nantly over the core and cladding. The tapering of the cladding results in coupling between the fundamental supermode and other supermodes (which have fields spread over the cladding-coating-air regions). Since the fibre taper is azimuthally symmetric, however, coupling is only possible between supermodes of the same azi- muthal symmetry.

If the core-cladding region of the fibre becomes two moded, then the second core-guided supermode can propagate. This supermode is azimuthally antisymmet- ric (i.e. y ( r , 4) = @(r)sin@ or @(r, 4) = y(r)cos$), and coupling will occur only with the corresponding antisymmetric coating-guided supermodes.

supermodes are derived in [2], Ch. 28. If there are N bound supermodes, the amplitude 6, = b,(z) of the nth local supermode at a distance z along the fibre is cou- pled to the amplitudes of the other supermodes by the following relation:

The equationa governing the coupling between local

where P, = p,(z) is the complex propagation constant of the nth supermode, and Cm, = Cm,(z) is the coupling

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coefficient between the mth and nth local supermodes as defined by [2], Ch. 28:

where k is the wavenumber, is the permittivity of free space, p0 is the permeability of free space, rimed is the refractive index profile of the taper, and the inte- gration is over the infinite cross-section (A,) of the tapered fibre and coating. yn,m = yn,m(r, 4, z ) repre- sents the local supermode field. For the piecewise step- profile taper, this equation reduces to [2], Ch. 28.7 to:

( 8 ) where 0 = o(z) and z = z(z) are the normalised trans- verse distances to the cladding-coating interface and the coating-air interface, respectively, as shown in Fig. 2, and:

The amplitudes of each of the N bound supermodes will vary with z according to eqn. 6 , and the system is described by a set of N coupled local supermode equa- tions. These equations can be integrated along the length of the taper.

For the present case, tapering of the fibre sensor is very gradual and approximately adiabatic. Thus none of the power in any supermode couples to other super- modes. This allows us to make the simplifying approxi- mation that C,, = 0 for all m and n, hence decoupling the supermodes. The simplified equation for the ampli- tude of each supermode becomes:

Note that eqn. 5 is just the integrated form of this equation. The taper is essentially adiabatic, and the sys- tem can be considered as an analogue of a highly asymmetric single to multimode core coupler.

Our method of analysis is now as follows. The local supermode fields, yn, and their (complex) propagation constants, Oa, are determined at the start of the taper, z = 0. An overlap integral with the Gaussian source described in Section 2.1 is then used to calculate the excitation of each supermode b, (0). Eqn. 10 is inte- grated numerically along the length of the sensor for each supermode amplitude, and the total power at dis- tance z along the device P(z) can be obtained by sum- ming the power in all supermodes:

n

To obtain the total power guided in core region, we can include a term related to the field distribution of each local supermode. The total number of local super- modes at the input, where the cladding region is thick- est, is larger than at the midpoint of the taper. The effective index of each supermode decreases as the clad- ding becomes thinner, and, if it falls to the refractive index of air (l.O), the supermode is no longer guided

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and its power radiates away. Supermodes which are cutoff in this manner, however, carry negligible power. The transmission characteristics of the device are there- fore determined from eqn. 11, but with a summation only over the supermodes which are not cutoff in the central region of the sensor. Typically, for the device described here, there are 5-10 such supermodes.

3.4 Numerical solution Propagation along the tapered fibre can also be deter- mined numerically using a finite difference (FD) scheme, as described in [6]. This procedure integrates the scalar wave equation along the complete length of the taper for an input condition of the form described in Section 2.1. In this analysis, the transverse field is set to zero beyond a small, fixed distance from the coat- ing-air boundary.

4 Results

Here we present numerical results for the single-mode, tapered fibre using the parameter values in Section 2.2. The results obtained are plotted as a function of fre- quency, rather than of wavelength, as successive reso- nances were found to be equidistant in frequency (similar to the peaks in a Fabry-Perot interferometer).

400 500 600 700 frequency, THz

Fi .7 Spectral variation of transmitted power in axisymmetric (bold) an% antisymmetric (dashed) supermodes, modelled by local supermode analysis, with 0 6 p n coating of sulphur

I 0 " . . J " . , . , . . . . " ' ' . . J . . . 1 I

400 500 600 700

frequency, THz

Fi 8 Spectral variation of transmitted ower for axisymmetric (bold) an?ktisymmetric (dashed) excitation, moci)elled by FD scheme with 0.6pn coating of sulphur

Fig. 7 shows the results of the local supermode anal- ysis for the core-guided supermodes, and Fig. 8 shows the results of the FD scheme analysis for both axisym- metric and antisymmetric excitation.

IEE Proc-Optoelectron., Vol. 144, No. 6, December 1997

Several trends can be noted from these plots. Firstly, the peaks in the attenuation of the antisymmetric power coincide very closely with those of the axisym- metric core-guided supermode (the peak for the antisymmetric supermode being slightly displaced to the left). This is as one would expect from the discus- sion of Section 3.1.

Secondly, the agreement between the local super- mode analysis and the finite differences scheme is very good throughout the frequency range of the results, with a slight broadening of peak being attributable to the discretisation in the FD scheme.

Finally, at lower frequencies, the antisymmetric supermode effective index is below that o F the cladding. In classical waveguide optics the mode would be cutoff, and the peak in attenuation of the antisyimmetric mode at 460THz is much broader than at 660THz as a result of the loss of guidance associated with this cutoff. It is worth noting that we can still model the supermode in this regime because the taper structure is surrounded by air, and so can guide at effective indices below the cladding index, but above that of air. In this regime, the superposition of the supermodes provides an ana- logue of a leaky mode [3].

h,pm 0.75 0.60 0.50

1 . 2 ~ . " " ' " ' I ' " ' ' " ' ' I

0 400 500 600 700

frequency, THz Experimental results of spectral variation of transmitted power Fig.9

during discharge process in SF,

We can compare these results with the experimental results. Fig. 9 shows the spectral response of the taper at a fixed time in the deposition process. The broadness of the peaks is attributed to a lack of uniformity in the sulphur film. Deposition is likely to have occurred pref- erentially on one side of the taper due to the geometry of the discharge apparatus. On the other hand, the positions of the absorption peaks in Fyig. 9 coincide well with those for the theoretical analysis. Sharp selec- tive behaviour has been verified experimentally on bent fibres polished to the core on the convex side of the bend, on which a ZnS film is deposited [7]. We expect that the theoretically predicted sharp absorption peaks could be observed in this experiment with a more even deposition process. A calibrated tapered fibre could then be used to monitor the thickness of the film to a high degree of accuracy.

5 Conclusions

We have developed a simple local supermode analysis for studying propagation in tapered single mode fibres and the coupling of core modal power to higher-index, lossy coatings. The method was applied to a tapered single-mode fibre sensor with a thin coating of sulphur,

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and the results were found to be compatible with exDerimenta1 measurements. The taDered structure

7 References

sh’ows a wavelength-selective behaviou; which could be exploited in optical fibre sensing. The position of the sharp absorption peaks depends primarily on the thick- ness of the sulphur layer, while being relatively insensi- tive to the taper geometry. This last characteristic would result in a high manufacturing tolerance.

6 Acknowledgments

James Besley is the recipient of a Commonwealth Scholarship and Fellowship Plan scholarship. This pro- gram is administered in conjunction with the Depart- ment of Employment, Education, Training and Youth Affairs representing the Commonwealth of Australia.

James Besley and John Love are members of the Australian Photonics Co-operative Research Centre.

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3 BESLEY, J.A., and LOVE, J.D.: ‘Supermode analysis of fibre transmission’, IEE Proc. J, Optoelectron., (in press)

4 SASSON, R., WRIGHT, R., ARAKAWA, E.T., KHARE, B.N., and SAGAN, C.: ‘Optical properties of solid and liquid sulphur at visible and infrared wavelengths’, ICARUS, 1985, 64, pp. 368- 374 SASSON, R., and ARAKAWA, E.T.: ‘Temperature dependence of index of refraction, reflection. and extinction coefficient of lia-

5

uid sulphur in the 0.4-2.0 pm’wavelengths range’, Appl. O p i , 1986, 25, (16), pp. 2675

6 SCARMOZZINO. R.. and OSGOOD. R.M.. JR.: ‘Comoarison of finite-difference and Fourier-transform solutions of thi para- bolic wave equation with emphasis on integrated-optics applica- tions’, J. Opt. Soc. Am. A, 1991, 8, (5), pp. 724 CREANEY, S., JOHNSTONE, W., and HUA, N.P.: ‘Low loss fibre optic polarisers using differential coupling to dielectric waveguide overlays’, Electron. Lett., 1994, 30, (4), pp. 349-351

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