Use of multiple-wavelength and/or TE/TM effective= refractive-index measurements to reconstruct refractive- i n dex prof i I es

R.Oven S. Batchelor D. G. As hwo rth

Indexing terms: Refractive index, Optical waveguide

Abstract: A method is presented whereby the refractive-index profile of a planar, surface- dielectric optical waveguide may be reconstructed from sets of effective refractive indices, measured at different wavelengths and/or with sets of effective refractive indices measured with TE and TM polarisation. The index change and the substrate index may be wavelength dispersive and birefringent. The method is compared, both theoretically and experimentally, with a conventional single-wavelength method for a number of index profiles. This technique provides more information about the profile than can be obtained from one measured set and is used to analyse the index profiles of potassium ion exchange guides formed in soda-lime glass. The limitations of the technique are also discussed.

1 Introduction

The reconstruction of refractive-index profiles from a set of effective refractive indices { n e , } measured with a prism coupler is of great interest in the design and assessment of integrated optical waveguides [ 11. The reconstruction is conventionally performed, for planar surface guides, by numerical inversion of the Wentzel- Kramers-Brillouin (WKB) characteristic equation [2]:

0

In this equation n(x) is the refractive-index profile as a function of depth measured from the surface of the guide, n, is the cover refractive index ( = 1 for air), n(0) is the refractive index at the surface of the guide, xt is

0 IEE, 1997 IEE Proceeduzgs onhe no. 19971273 Paper first received 12th September 1996 and in remsed form 25th March 1997 The authors are with the Sohd State Electromcs Research Group, Elec- tromc Engneenng Laboratory, University of Kent, Canterbury, Kent CT2 7NT, UK

the mode turning point, ne, = n(xt), r = 1 for TE modes and r = {n(0)/n,}2 for TM modes, d is the free- space wavelength and m is the mode number (0, 1, 2 .. M-l), where M is the total number of modes. For a guide supporting M modes at the measurement wave- length there are M pieces of information with which to reconstruct the profile. A number of numerical algo- rithms have been devised in order to invert eqn. 1 and solve for n(x) [2-51. These methods prove successful in determining n(x) if the actual index profile varies smoothly with distance and supports a sufficiently large number of modes. However, the methods fail to give accurate reconstructions when either the actual index profile has many features in it or A4 is small, so there is little information. It has been shown [4] that, if the functional form of n(x) is known near the guide surface and/or deep inside the guide, then fewer modes are required to reconstruct accurate profiles. However, in most practical situations these dependencies are not known, so the technique is of limited use.

In this paper, methods are presented whereby the index profile may be reconstructed from sets of effective refractive indices measured at different wavelengths or, alternatively, from sets of effective refractive indices measured with TE and TM polarisation at the same wavelength. The use of at least two measurement wavelengths and/or measurements with both polarisations provides more information about the profile than can be obtained from one measured set. This method, using multiple wavelengths, has been presented previously [6]. However, in this paper it is discussed in more detail and extended to include the simultaneous analysis of TE and TM modes. A similar technique, capable of combining the modes measured at different wavelengths and polarisations, has also recently been published [7]. It is shown that our multiple-wavelength method is more general than that in [7] since it allows for the index change (as opposed to just the substrate index) to be wavelength dispersive. When applied to the simultaneous analysis of TE and TM modes, this technique is more general than [7] in that it allows for the guide to be birefringent. As with all profile reconstruction techniques, that described here has some limitations. These are discussed and compared with other techniques in this paper.

213 IEE Proc.-Optoelectron., Vol. 144, No. 4, August 1991

2 The continuous-effective-index function

The application of the technique to multiple- wavelength measurements is discussed first. The application of the technique to combining TE and TM Modes will be discussed in Section 6. The technique proposed is a generalisation of the single-wavelength reconstruction method devised by Chiang [2], with which familiarity is assumed. To reconstruct a smooth index profile and to assist in the numerical inversion of the WKB integral, Chiang used the concept of a continuous-effective-index function N(m), which replaces the discrete set {ne#} in eqn. 1, by making m a continuous variable. This function is generated by passing a monotonically decreasing polynomial through the points on the graph of m against nefl; hence n{x,(m)} = N(m). Chiang also showed that extrapolation of this polynomial to m = - 0.75 allows the surface index n(0) to be estimated. The continuous- effective-index function defined above cannot be used to combine sets of modes obtained at different wavelengths or polarisations. Hence, other plotting variables have to be determined. To determine these, it is first assumed that the index profile at the measurement wavelengths may be written in the form

45 , A,) = .S(XZ) + An(&)f(.) (2) where n,(h,) is the substrate index at the ith wavelength h,; An(h,) is the refractive-index change for the ith wavelength, and f ( x ) is a shape function [0 5 f ( x ) 5 11 which is wavelength independent. Although eqn. 2 is a reasonable functional form to assume for n(x, AI), and has been used on many occasions when comparing the theoretical modal structure of various index-profile shapes [8, 91, it is necessary to establish its validity experimentally. That Ax) is independent of wavelength can be confirmed experimentally by measuring the index profiles of guides which are highly multimode at the measurement wavelengths and then comparing their shape functions. If eqn. 2 is valid then, for A n ( Q << n,(h,) and f ( 0 ) = 1, eqn. 1 may be written in the form

7 Jf(4 - f ( x t )dx 0

L .. - - --.\/n(O, 27r Xz)2 - n s ( X i ) 2

( 3 ) The left-hand side of eqn. 3 is independent of wave- length; thus the right-hand side should also be inde- pendent of wavelength. This suggests that a smooth curve may be formed by plotting the right-hand side of eqn. 3, which for convenience we call F(x,), against f(x,). The data from all wavelengths should fall on this smooth curve. This will be confirmed in a subsequent section for deep multimode guides.

The determination of the shape function f ( x ) from the measured sets of effective refractive indices is a two-stage process. First, it is necessary to plot the experimental curve of F(x,) against f (x t ) using the measured sets of effective refractive indices. Secondly, an inversion algorithm, similar to Chiang's [2], is used to determine the spatial location xt of a number of values of fix,) obtained from interpolation of the curve

214

of F(x,) against f(x,). These two processes are described below and in the Appendix (Section 11).

3

Since ne# = n(x,, hi) then, from eqn. 2, f ( x J is given by

Experimental curve of F(xJ against fCxJ

(4)

for a particular mode. F(x,) for the same mode is determined by evaluating the right-hand side of eqn. 3 . Hence, to determine f ( x J and F(x,) and, thus, plot the curve of F(x,) against f(xt) , it is necessary to know ns(hi) and n(0, hi) for i = 1, 2, ..., iMAx. In the single- wavelength technique of Chiang [2], n(0) was determined by extrapolating the N(m) curve to m = - 0.75. In the multiple-wavelength technique this is equivalent to the requirement that the curve of F(xJ against Ax,) curve passes thought the point (F(x, = 0) = 0, f i x r = 0) = 1). The procedure for generating the required polynomial is, hence, to adjust n(0, hl) iteratively until the polynomial passes through the point (F(x, = 0) = O,f(x, = 0) = 1). According to eqn. 2, during the iteration, n(0, Aj) is calculated using

4 0 , Xi) = %(&) + R(n(0 , X l ) - %(A,)) (5)

(6)

where the ratio Ri is determined from

a%"Xi) - - %"O, Xi) - n s , m m ( G

A%"Xl) %"O, Xl) - %,mm(Al) R; =

where Anmm(hi), n,,(O, hi) and ns,mm(hi) are the index change, surface index and substrate index of a highly multimode guide measured at the ith wavelength. The highly multimode guide is formed by using the same manufacturing technique as the guide under analysis. Eqn. 6 is derived from eqn. 2 by assuming that the index change of the guide being analysed has the same dispersive nature as the highly multimode guide from which Ri was obtained. However, it does not assume that the two guides have the same absolute values for An(Al) and An(&). Since the guide used to determine R, is highly multimode, it is sufficient to use Chiang's single-wavelength technique to determine An,,(&) and nmm(O, hi). Since all data points should fall on a smooth curve of F(x,) against Ax,), an alternative technique which has been used to determine Ri is to adjust Rj until the mean-square deviation between the data points and the interpolating polynomial is minimised.

The substrate refractive indices may be measured with the prism coupler [lo] or by other methods [8]. Once the curve has been generated, a numerical-inver- sion technique may be used to solve eqn. 2 for f ( x ) . This is discussed in the Appendix (Section 11).

4 Validation of assumptions: wavelength

According to the above analysis, modes supported by a given guide but measured at different wavelengths should lie on the same curve of F(xJ against f(x,). This may be confirmed by using mode data generated from a Helmholtz-equation solver [l l] . Fig. 1 shows an example curve of F(x,) against f ( x J generated from theoretical data for an index profile which is Gaussian in shape. Parameters relevant to the simulation are given in the caption and are typical of those found when guides are produced in glass substrates by potassium-sodium-ion exchange. The fact that the

IEE Proc.-Optoelectuon., Vol. 144, No. 4, August 1997

F (Xt 1 Fig. 1 sian index profile 0 modes supported at A = 633nm + modes supported at A = 543nm .... fitted polynomial Parameters for the Gaussian function are: An(x) = O.Ol,f(x) = exp{-(x/5)*} at both wavelengths, n,(h) = 1.512 at A = 633nm and ns(A) = 1.515 at h = 543nm x is in microns

Theoretical curve of F(xr) againstf(xt) for a guide with a Gaus-

single-wavelength technique. Hence, strictly speaking, a check should be made to show that this further approximation does not significantly alter the shape of any recovered index profile. This check has been made using the theoretical mode data for the Gaussian discussed above. From the curve of F(xJ against f(xt) , the shape function was recovered using the above inversion technique and compared with the original Gaussian function. The comparison, made in Fig. 2, shows that the Gaussian function can indeed be reconstructed very accurately, thus justifying the approximation. Next, it is necessary to establish the validity of eqn. 2 experimentally. To do this, the index profile of deep multimode guides, produced by potassium-sodium-ion exchange at 395°C for 16h in a soda-lime glass, was measured with both red (633nm) and green (543nm) light separately. The shape function Ax) deduced from both wavelengths separately is shown in Fig. 3 . It can be seen that within reconstruction errors the profiles are identical. It was found experimentally that, for these types of waveguides, R, was close to unity. Fig. 4 shows the curve of F(xJ against Axt) for this guide. It can be seen that the modes obtained, using both red and green light, lie on a smooth curve which, experimentally, justifies the above analysis and the value of R. For ion- exchanged guides manufactured in glass, the substrate index for the actual guide under investigation should be used in the analysis since the substrate index can change during the processing due to the slight densification of the glass. Calculations using simulated data show that, if either Ri or a substrate index is incorrect, then the two sets of modes, when plotted on the F(x) against f(xJ graph, are displaced from one another. The validity of eqn. 2 has also been confirmed experimentally for silver-ion diffusion into glass. This is consistent with other studies [7, 81.

depth, pm Fig. 2 ..... Gaussian function

~ reconstructed profile

Comparison of Gaussian function and reconstructed profile

Fig.4 0 measured at h = 633nm + measured at h = 543nm _ _ - _ fitted polynomial

F(x,) against f i x , ) for the multimode guide shown in Fig. 3

depth, pm Shape function f ( x ) for a multimode guide produced by potas- Fig.3

sium-sodim-ion exchange (395°C for 16h) ~ reconstructed from modes measured at A = 633nm ____. reconstructed from modes measured at A = 543nm

function of F(x,) against f(xJ is smooth will also be confirmed below by using experimental data. To derive eqn. 3 , the approximation AH(&) << n,(Ai) has been made. This is an approximation not made in Chiang's

IEE Proc.-Optoelectron., Vol. 144, No. 4, August 1997

5 Results: dual wavelength

Fig. 5 shows the curve of F(xJ against Axt) for a guide where the combination of both sets of data provides significantly more profile information than the single- wavelength technique. In this example, a shallow guide has been produced by potassium-sodium-ion exchange for a diffusion time of 2h at 395°C in the same type of

215

0 1 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

F(x t ) Fig.5 ion exchange (39592 for 2h)

F(x,) against f i x , ) for a guide produced by potassim-sodim-

0 measured at h = 633nm + measured at h = 543nm _ _ _ _ fitted polynomial

1 o-2

I

' 10-5 L 0

1 o-6

IO-^ 0.90 0.95 1.00 1.05 1.10

Ri Fig.6 Variation of error with R,

Fig. 7 dotted apainst xr1I2

Shape function f i x ) produced by potassim-sodium-ion exchange

_ _ _ _ gand 16h exchanges reconstructed with h = 633nm modes -+- produced by a 2 hour exchange and reconstructed from both I = 543nm and h = 633nm modes

-0- produced by a 70 min exchange and reconstructed from both h = 543nm and h = 633nm modes

-0- produced from a 2h exchange and reconstructed from A = 543nm modes only

-x- produced by a 70 min exchange and reconstructed from h = 543nm modes only

216

soda-lime glass as the sample shown in Figs. 3 and 4. This produces a guide which supports two modes at a wavelength of 543nm and two modes at 633nm. With this guide, only a linear graph may be passed through the mode data for either red or green light but a higher-order polynomial may be passed through the combined data. To demonstrate that the correct value of R, was used when combining the red- and green- mode data, Fig. 6 shows R, plotted against the error between the data points and the interpolating polynomial. It can be seen that the error is indeed a minimum when R, is very close to 1. Fig. 7 shows the shape function f(x) for this guide, reconstructed using the dual-wavelength technique, plotted as a function of Xt-112 where t is the diffusion time. Also shown in Fig. 7 are the shape functions of multimode guides produced by 8 h and 16 h potassium-sodium-ion exchanges and reconstructed using measurements at 633 nm. Standard diffusion theory suggests that the profiles should be identical when plotted as a function of xt-Il2 provided that the potassium-ion surface concentration remains constant and that stress-relaxation effects do not occur. Also shown in Fig. 7 is the shape function for a guide produced by ion exchange for only 70 min which supports two modes at a wavelength of 543nm and only one mode at 633nm and again reconstructed using the dual-wavelength technique. It can be seen that there is good agreement between the four profiles. For comparison, Fig. 7 also shows the shape functions of the 2h and 70 min guides reconstructed using modes measured only with green light. Clearly, there is poor agreement between these two profiles and the previous four due to the fact that only a linear curve of F(xt) against f(xJ may be formed when the mode data from the green light are used alone. Surface index changes An of 0.010 and 0.0098 were measured for TM-mode excitation for the 8h and 16h diffusions. For the 2 hour and 70 min guides, An was 0.010 and 0.0096 (for TM-mode excitation), respectively, using the dual- wavelength technique while values determined from the mode data when using only green light were of the order of -20% in error.

6

The above technique may also be used to reconstruct the shape function, at a single wavelength, using both TE and TM modes. The assumption is made that the index profile measured with both polarisations may be written in the form

Reconstruction using TE and TM modes

n(2, T E ) = n,(TE) + An(TE) f ( z ) n ( 2 , T M ) = n , ( T M ) + A n ( r r M ) f ( x ) (7 )

where n,(TX) is the substrate index measured with the TX polarisation; An( TX) is the refractive-index change for the TX polarisation andf(x) is a shape function (0 2 Ax) 2 1) which is now polarisation-independent. The assumption that the shape function is independent of polarisation must be justified experimentally. It will be shown below that, for potassium-ion diffusion into soda-lime glass, this is a valid assumption. However, this is not true for all guide systems. For example, it has been shown that the index change An(x) for Ti- diffused LiNbO, varies with a power of the Ti concentration with a different exponent for TE- and TM-mode excitation [12]. Hence, for this system,f(x) is not polarisation independent and the technique cannot

IEE Proc -0ptoelectron , Vol 144, No 4, August 1557

be used. The independence off(x) on polarisation may be confirmed experimentally by measuring the index profiles of guides which are highly multimode at both polarisations independently. If eqn. 7 is valid, then a curve of F(x,) against f (x t ) may be plotted with

and F(x,) given by the right-hand side of eqn. .3. To determine f ( x J and F(x,), n(0, TE) is adjusted itera- tively untilf(x, = 0) = 1. During the iteration, n(0, TM) is calculated using

4 0 , T M ) = n,(TM) +R,(n(O, T E ) -ns (TE) ) (9) where the ratio Rp is determined from

A%nm ( T M ) - 72" (0, T M ) - ns," ( T M ) nTnm(0, T E ) - %,mm(TE)

(10)

R - - - h n m ( T E )

where An,,(TX), n,,(O, TX) and n,,,(TX) are the index change, surface index and substrate index of a highly multimode guide measured with TX polarisation. The highly multimode guide should again be formed by using the same manufacturing technique as the guide under analysis. Since all data points should fall on a smooth curve of F(x,) against f(x,), an alternative technique which has been used to determine Rp is to adjust Rp until the sum of the squares between the data points and the interpolating polynomial is minimised.

1

1

1 X

$ 1 - I I I

cl

1.516-

1.51L-

1.512-

I 0 2 r, 6 8 10 12

depth, pm Refractive-index profiles measured at A = 633nm for ran electric-

1.510- *

Fi .8 le d assisted potassium-sodiurr-ion exchanged guide

~ TM TE

f ? .....

7 Results: polarisation

For guide systems which show no birefrigence (Rp = l), the technique yields virtually identical results to those of Chiang's recent method [7]. The technique discribed here, however, is more general than that of Chiang [7] in that it may also be used to reconstruct f ( x ) for guides which do show birefringence. It is well known that guides produced by potassium-ion diffusion into glass produce birefringence due to planar stress built up in the glass [13]. Fig. 8 shows the index profiles reconstructed with TE and TM modes separately for an electric-field-assisted potassium-ion-diffused guide into a soda-lime glass. These guides are highly birefrin- gent with Rp = 1.21. Fig. 9 shows the curve of F(xJ against f ( x t ) for this case where the TE and TM modes are combined. It can be seen that TE and TM modes

IEE Proc -0ptoekctron , Vol 144, No 4, August 1997

lie on a smooth curve. Fig. 10 shows another guide produced by the same technique reconstructed using both TE- and TM-mode data. To reconstruct this func- tion it is assumed that Rp = 1.21 as obtained from the previous guide. Also shown in Fig. 10 is the index pro- file obtained from just the TM measurements. This particular guide only supports two TE modes and two TM modes; hence, if only a single set of either TE or TM modes is used then only a linear curve of F(xJ against Axt) may be generated. Fig. 11 shows F(xJ plotted against Axt) for this guide using both TE- and TM-mode data. A quadratic interpolating polynomial was used. Since the TE and TM modes fall on the same smooth curve the use of the particular value of Rp is validated. It can be seen that, by combining both TE and TM modes, an index profile with a more step-like character is produced. Also, the surface-index estimate is improved. However, it is not perfect since a surface index close to that in Fig. 8 is expected.

1 .o

0.8

0.6 - X - c

0 . L .

0 . 2 .

01 \ 0 1 2 3 ~ 5 6 7 8 9 '

F(x t ) Fig.9 0 TM data points 0 TE data points ~ fitted polynomial

F(xJ against f ( x J for the guide in Fig. 8

I

6 1.51 0

0 1 2 3 L 5 depth, pm

Refractive-index projZe for a guide produced by electric-$eld- Fig. 10 assisted potassium-sodium-ion exchange in glass (A = 633nm)

~ produced by combining both TE and TM modes _ _ _ _ _ produced with TM modes only

As a final illustration of the technique, Fig. 12 shows the TM refractive-index profile of a natural thermal diffused guide produced at 395°C for 70 min. When measured at 543nm it supports two TE and two TM modes. The solid line shows the index profile recon- structed using the dual wavelength measurements and the dashed curve shows the index profile reconstructed

211

using the TE and TM modes. For this guide Rp = 1.19. It can be seen that there is good agreement between the two profiles using the two techniques.

Fig. 11 F(x,) against f ix,) for the guide shown in Fig. 10 TM data points

0 TE data points ~ fitted polynomial

0 1 2 3 L 5 6 depth, pm

Index pro de (TM) of a potassium-sodium-ion-exchange guide Fig. 12 Droduced ut 395 “C t6 70 min ~ reconstructed using TM modes measured at h = 543nm and 633nm

and scaled to h = 543nm ..._. reconstructed using TE and TM modes measured at h = 543nm

Some reduction of Rp with increasing process time has been observed for potassium-sodium-ion exchanged guides in glass. It is believed that this is due to the onset of stress-relaxation effects which will reduce Rp with increasing process time [13] even at the low processing temperature used in the present study. If this is significant for a particular glass and process temperature, the use of an Rp determined from a deep multimode guide will be unsatisfactory and will not place the TE- and TM-data points of a shallow guide onto a smooth curve of F(xt) against Axt). In this case the procedure whereby Rp is adjusted to minimise the error between the mode data and the interpolating polynomial is to be preferred. Excessive stress relaxation can cause the TE and TM profiles to become significantly different in shape [13], ultimately invalidating eqn. 7 . In this respect, guides produced by potassium-sodium-ion exchange in soda-lime glass are not ideal for demonstration of the technique using TE and TM modes.

218

8 Discussion and conclusions

The methods proposed in this paper are extensions of the single-wavelength technique of Chiang [2] and are more general than the multiple-wavelength and polari- sation technique recently proposed [7]. If the combina- tion of mode indices at different wavelengths is considered first, then the present technique is more general in that it does not assume that the index changes at the wavelengths concerned are the same, i.e it does not assume that R = 1. Hence it can be used in waveguide systems where An is wavelength-dispersive. For the case where modes of different polarisations are combined, the method is more general in that birefrin- gence can be included provided that the shape function is polarisation-independent. Chiang’s proposed tech- nique [7] is not applicable to cases where the guide is birefringent. In principle, the technique described in this paper may be used to combine both TE- and TM- mode data with multiple-wavelength data. However, in this case more experimental values of R are required since the Rp linking the TE and TM modes will, in gen- eral, be wavelength dependant. So, for example, if two measurement wavelengths are used with both TE- and TM-mode data then, in general, three values of R are needed.

Both the technique described in this paper and Chi- ang’s new proposed technique [7] have limitations, some of which are also applicable to the single-wave- length method [2]. First, the order of the interpolating polynomial depends on a number of factors. These fac- tors are: the estimated accuracy of the mode data; whether the polynomial chosen is monotonically decreasing; and the number of modes available for its generation. Secondly, a similar extrapolation procedure is adopted to estimate surface-index values. The surface index can be sensitive to the polynomial order chosen. A very high-order polynomial is not recommended since it can vary rapidly and can result in gross errors when extrapolating. This is particularly so with tech- niques discussed in this and other papers [7] since neighbouring data points on the curve of F(x,) against fix,) can be very close. Notwithstanding these limita- tions, the methods proposed use more mode informa- tion than the single-wavelength technique and, thus, a higher-order interpolating polynomial may be chosen to capture what may be considered to be genuine mode structure and, thus, index profile information. The more data that are plotted on the curve of F(xt) against f(x,), the more accurately the shape function f ( x ) will be defined.

9 Acknowledgments

S. Batchelor is supported by a CASE studentship with Pilkington Technology Management Ltd.

10 References

1 RAMASWAMY, R.V., and SRIVASTAVA, R.: ‘Ion-exchanged glass waveguides: a review’, J. Lightwave Technol., 1988, 6, pp. 9861001 CHIANG, K.S.: ‘Construction of refractive index profiles of pla- nar dielectric waveguides from the distribution of effective indexes’, J. Lightwave Technol., 1985, 3, pp. 385-391 WHITE, J.W., and HEIDRICH, P.F.: ‘Optical waveguide refrac- tive index profiles determined from measurement of mode indices: a simple analysis’, Appl. Opt., 1976, 15, pp. 151-155

4 HERTEL, P., and MENZLER, H.P.: ‘Improved Inverse WKB procedure to reconstruct refractive index profiles of dielectric pla- nar waveguides’, Appl. Phys., 1987, B44, pp. 75-80

2

3

IEE Proc-Optoebctron., Vol. 144, No. 4, August 1997

OVEN, R., BATCHELOR, S., ASHWORTH, D.G., GELDER. D., and BRADSHAW. J.M.: ‘Iterative refinement technique for reconstructing refractive index profiles from mode indices’, Electron. Lett., 1995, 31, pp. 229-231 BATCHELOR, S., OVEN, R., and ASHWORTH, D.G.: ‘Recon- struction of refractive index profiles from multiple wavelength mode indices’, Optics Commun., 1996, 131, pp. 31-36 CHSANG, K.S., WONG, C.L., CHAN, H.P., and CHOW, Y.T.: ‘Refractive-index profiling of graded-index planar waveguides from effective indexes measured for both mode types and at dif- ferent wavelengths’, J. Lightwave Technol., 1996, 14, pp. 827-832 NAJAFI, S.I., SRSVASTAVA, R., and RAMASWAMY, R.V.: ‘Wavelength dependant propagation characteristics of Ag+-Na+ exchanged - - . - - - . - planar glass waveguides’, Appl. Opt., 1986, 25, pp. 1X4U-1x43 RAMASWAMY, R.V., and LAGU, R.K.: ‘Numerical field solu- tion for an arbitrary asymmetrical graded-index planar waveguide’, J. Lightwave Techno/., 1983, 1, pp. 408417

10 ZERNIKE. F.: ‘Fabrication and measurement of Dassive comDo-

Ax). First assume that Ax) may be represented by a piecewise linear function:

f(.) = f(12:,-1) + m& - 12:J-1) 211-1 < 12: < 2,

(11)

(121

where

f ( 2 3 ) - f(zJ-1) m3 = 2 3 - 2 3 - 1

This is in contrast to Chiang’s [2] algorithm where, between xJ and x,~, n ( ~ ) ~ is assumed to be a constant. Associated with these approximations are errors which vanish as the interval width (xJ - x,,) becomes arbitrar- ily small. Substituting eqns. 11 and 12 into eqn. 3 gives

nents’ in TAMIR, T. (Ed.): ‘Integrated optics. Tdpics in appfied physics’ (Springer-Verlag, 1975), vol, 7., pp. 223-231

1 1 KUESTER. E.F., and CHANG, D.C.: ‘Propagation, attenuation, and dispersion characteristics of inhomogeneous dielectric slab waveguides’, IEEE Trans., 1975, MTT-23, pp. 98-106

12 FOUCHET, S., CARENCO, A., DAGUET, C., GUGLIELMI, R., and RIVIERE, L.: ‘Wavelength dispersion of Ti induced refractive index change in LiNbO, as a function of diffusion parameters’, J. Lightwave Technol., 1987,5, pp. 700-708

ERA, A., and MORRA, M.: ‘Compositional and stress-optical effects in glass waveguides: comparison between K-Na and Ag- Na ion exchange’, J. Non-Cuyst. Solids, 1990, 119, pp. 195-204

13 DE BERNARDI, C., MORASCA, S., SCARANO, D., CARN-

11 Appendix: inversion technique

Once the experimental curve of F(xt) against Axt) has been generated, a numerical algorithm, similar to Chiang’s [2], may be used to find the shape function

X i

Integrating eqn. 13 and solving for xi gives

xi = xi-1 + {.f(X2-1) - f(.i))”2

(14) Eqn. 14 may be evaluated, sequentially, for i = 1,2, ... starting with the initial valuef(0) = 1. Once the xi have been determined, the shape function f ( x ) can be plotted.

IEE Proc -Optoelectron., Vol. 144, No. 4, August 1997 219