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Determination of refractive index profiles of planar buried waveguides on the basis of a set of modal propagation constants
Optics Communications 219 (2003) 199–214 www.elsevier.com/locate/optcom
Determination of refractive index profiles of planar buried waveguides on the basis of a set of modal propagation constants ski * Roman Rogozin Optoelectronics Department, Institute of Physics, Silesian University of Technology, ul. Krzywoustego 2, 44-100 Gliwice, Poland Received 14 May 2002; received in revised form 6 December 2002; accepted 5 February 2003
Abstract In the work the algorithm involving the reconstruction of refractive index profiles of planar buried waveguides based on modal equation was proposed. The analysis testing the algorithm with the help of functions simulating buried profiles was carried out. Using ion-exchange technique in glass substrate, buried waveguide structures were manufactured. Propagation constants of all modes of those structures were determined by a prism coupling method with the measurements of synchronic angles. On the basis of the measured propagation constants the reconstruction of their refractive index distribution was carried out with the help of the proposed algorithm. The obtained results were compared with the results of interferometric investigations. 2003 Elsevier Science B.V. All rights reserved. Keywords: Planar buried waveguides; Refractive index profiles
1. Introduction Planar optical waveguide structures produced in glass substrate can have a variously shaped distribution of the refractive index. The ion-exchange technique applied for the production of these types of structures in the superficial area of glass allows a relatively high degree of freedom when forming their refractive profiles [1]. Depending on the requirements for an optical structure, it can have a monotonic or unmonotonic form of refractive index. The latter type of refractive profile is effected in technological processes, in which the already produced form of refractive index is purposefully reduced by the introduction of a suitable admixture from the side of glass surface. As a result, the maximum value of refractive profile is shifted far into the substrate. It results from this that in glass exists an area where the value of refractive
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0030-4018/03/$ - see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01286-0
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index is increasing with depth. Waveguide structures characterized by such refractive profiles are referred to as buried waveguides. In the technique of ion-exchange applied for the production of optical structures in glass, it is extremely important that we know the refractive profile of the structure being produced, both in view of cognitive aspect with respect to the processes of theoretical modelling of ion-exchange process, and also with respect to the description of transmission properties of the waveguide [2]. In accepted theoretical models of the phenomenon of ion-exchange in glass it challenges often the dependence of parameters of diffusion on the concentration of introduced into glass ions of the admixture [3–5]. The concentration profile of this admixture in the ion-exchange process is then the solution of the non-linear equation of diffusion. The form of this solution is here dependent on a certain number of parameters accepted in the model assumptions. The determination of these parameters is effected by the adjustment of the diffusion solution to the real refractive profile obtained from the measurements. The knowledge of the parameters determined in such a way makes it possible to predict the shape of refractive profiles obtained in other technological processes (secondary diffusion, electrodiffusion). There are a number of measurement techniques applied in the metrology of refractive profiles of planar optical structures. Considering the results obtained in effect of their application, they can be divided into two groups: direct and indirect. In direct methods the refractive profile is obtained immediately as a result of the measurement. This group of methods is represented by the interference method [6,7] (where the information about the refractive profile is enclosed in the perturbation of interference field), method of near field analysis [8,9] (where the analyzed distribution of power at the waveguideÕs exit is reflecting the profile), or the method of isotopic loftsmen [10], or X-ray microanalysis (where the distribution of concentration of the admixture introduced to the glass is defined directly). In indirect methods the refractive profile of the waveguide is obtained as a result of an additional analysis of the obtained measurement data with the application of more or less complex mathematical models which approximate the searched profile. In this group of methods the most commonly applied is a so called inverted WKB method [11], which is based on the modal equation and is using a set of propagation constants of the waveguide for the reconstruction of ðiÞ ðiÞ refractive profile. The refractive index distribution received with this method is a set of points fxt ; Neff g, ðiÞ ðiÞ where xt is the turning point of the ith mode having the effective index Neff . The method has the following advantages: high accuracy involving the acquisition of input data (involving the set of propagation constants which are determined using prism coupling measurements), and easy realization of measurements – without any special preparation of waveguide sample (nondestructive method). There is one disadvantage of this method – its application is limited to multimode waveguide structures (the reliability of the reconstruction of refractive profile is increasing with the number of modes). This method was proposed for the first time in 1976 by White and Heidrich [12]. In the modal equation the searched function describing the profile was replaced by the authors with a stepping function, which yielded a recurrent formula for the ð0Þ calculation of individual modal turning points. In this method, in order to determine the turning point xt of the zero mode, it is necessary to know the refractive index on the surface of waveguide. Its value is selected in the way ensuring that it meets a specific criterion. The smoothness of the obtained profile is accepted here as a feature characteristic for diffusive processes. In 1985 Chiang [13] introduced the notion of effective refractive index as a continuous function of modal order. Basing on the set of measured values of effective indexes, this procedure makes it possible to reconstruct the refractive profile defined in any number of points. The criterion of smoothness of the received profile does not occur here as it was in the procedure of White and Heidrich. The problem of uniqueness is shifted here towards the selection of a suitable kind of interpolation polynomial. In the work [14] the authors proposed the reconstruction of the monotonic refractive index profile of planar waveguide basing on the measurements of effective indexes of waveguide modes measured for different wavelengths. As in ChiangÕs method, it is possible here to define with arbitrary accuracy (the number of testing points) the shape of the function describing the profile. This function is not dependent on wavelength. Therefore, it can be determined basing on effective refractive indexes of
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only one mode, determined for many wavelengths. Due to this, the method enables the reconstruction of waveguide profiles, even single-mode ones. There is, however, a necessity to know the dispersion properties of the base glass. The specified above methods basing on modal equation are applied for the reconstruction of only monotonic distribution of refractive index. The present work introduces a proposal of algorithm making use of modal equation for the reconstruction of unmonotonic distribution of refractive indexes of buried waveguides. 2. Buried waveguide – own equation of buried modes A typical refractive profile of planar buried waveguide is presented in Fig. 1. Such a profile has the following characteristic features: the value of refractive index on the surface of waveguide nð0Þ and the maximum value of refractive index within the area of waveguide nmax shifted far into the base of glass (xmax > 0). Such a shape of the refractive profile results in the division of the set of propagation constants (effective indexes) of waveguide modes into two groups: • Buried modes, characterized by the occurrence of two turning points in the area of glass: xa > 0 and xb > 0, for which the value of effective indexes satisfies the inequality: nð0Þ < Nm0 < nmax (Fig. 1). • Unburied modes, for which always one turning point occurs on the surface of glass, and the values of effective indexes satisfy the relation: ng < Nm < nð0Þ, where ng stands for the refractive index of base glass. The form of modal equation allowing for the change of phase in turning points will therefore be different for each group of modes. For the TE polarization this equation assumes the following form [15,16]: Buried modes rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 8 Nm2 n2 ð0Þ > > > > Z xb qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 = <1 xa qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nm2 n20 1 2 2 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 2k0 : k0 n ðxÞ Nm dx ¼ p m þ Nm n ðxÞ dx þ arctg > > 2 2 2 2 ð0Þ xa 0 > > ; : 1 þ Nm n 2 2 Nm n0
ð1Þ Unburied modes Z k0 0
xc
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n2 ðxÞ Nm2 dx ¼ p m þ þ arctg 4
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Nm2 n20 : 2 n ð0Þ Nm2
Fig. 1. Refractive index profile of the planar buried waveguide.
ð2Þ
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In above mentioned equations, k0 ¼ 2p=k, nðxÞ is the function describing the refractive index profile of the waveguide, m is the order of mode, Nm is the effective index of the mth mode, n0 is the refractive index of surroundings, nð0Þ ¼ ns is the refractive index on the surface of the waveguide, ng is the refractive index of the glass base, xi is the turning points of mode (i ¼ a; b; c; see Fig. 1). When we know the set of effective indexes fNm g of waveguide modes we can reconstruct the profile function nðxÞ using the above-mentioned equations.
3. Reconstruction of the refractive index profile of the buried waveguide In the reconstruction procedures of monotonic refractive index profile nðxÞ based on the modal equation, the approximation of this profile by stepping function is most commonly applied [12] nðxÞ ¼ 0:5ðNk1 þ Nk Þ
ðk1Þ
for xt
ðkÞ
6 x < xt ;
ð3Þ
where k ¼ 0; 1; 2; . . . ; M 1; M is the total number of waveguide modes. By replacing in the modal equation the searched function nðxÞ with the approximation (3), as well as the integral with summation, we obtain the approximate form of modal equation, from which we can derive a recurrent formula which allows one to determine the next turning points of modes [12]. In the case of the monotonic profile, in order to determine the turning point x0 of the zero order mode, it is necessary to know the refractive index on the surface of waveguide (N1 ¼ ns ). As it was mentioned before, it is not measured directly, and its value is selected here in the way ensuring that the obtained profile fxk ; Nk g satisfies the criterion of smoothness as the feature characteristic for diffusive processes. A similar approach in the case of buried profile necessitates that the number of buried modes (z) is defined and that the initial value of turning points closer to glass surface is accepted for buried modes. The idea involving the approximation of the searched refractive index profile with a stepping function is presented in Fig. 2. This function has now the following form: nðxÞ ¼ 0:5ðNk1 þ Nk Þ
ðk1Þ
for xt
ðkÞ
6 x < xt ;
ð4Þ
where k ¼ z; z þ 1; . . . ; 1; 0; 1; . . . ; M; M is the number of all modes of the waveguide, z is the assumed quantity of buried modes (z < M).
Fig. 2. Idea of the reconstruction of buried refractive index profile.
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The values of effective indexes Nm corresponding to the successive orders of modes are numbered from 1 to M (where M denotes the number of all modes of the waveguide). In Fig. 2 the exemplary value M ¼ 8. The value of index m ¼ 1 is attributed to the zero order mode. Since the number of buried modes (z) is unknown, therefore in the successive approximations it is changed. This value, being a natural number, can assume values from the range: 1 6 z < M. It results from the assumed value of refractive index ns on the glass surface. For each fixed value (z) (in Fig. 2 the exemplary value z ¼ 3) negative indexes mð1 6 =m= 6 zÞ are introduced for the accepted buried effective indexes Nk ¼ Nk for k 6 z (values N1 ; N2 ; N3 in Fig. 2). Also negative indexes are attributed to the corresponding turning points from the side of glass surface (values x1 ; x2 ; x3 in Fig. 2). Another preset quantity is the hypothetic value N0 of the maximum value of refractive index in the waveguide, which has the coordinate x0 on the axis x. This value satisfies the inequality: N1 < N0 . As it was mentioned above, the algorithm is based on the approximate form of modal equation, separately for buried and unburied modes. Eq. (1), after the profile function nðxÞ has been replaced with the approximation (4) and the integration with summation, assumes the following form: k0
k1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2 0:25ðNiþ1 þ Ni Þ Nk2 ðxiþ1 xi Þ ¼ pðk 0:5Þ i¼k
(
"
þ arctg Ck exp 2k0
ðkþ1Þ X
#) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Nk2 0:25 Njþ1 þ Nj xjþ1 xj ;
ð5Þ
j¼ðzþ1Þ
where z is the quantity of buried modes, whereas rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ck ¼
1 2
1
2 Nk2 Nðzþ1Þ
Nk2 n20
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1þ
2 Nk2 Nðzþ1Þ
Nk2 n20
For the assumed value ns and the resulting from this value z, the turning point x1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 P2 p 2 N x þ arctg C exp 2k 0:25 N þ N x 1 0 jþ1 j jþ1 j 1 j¼ðzþ1Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x1 ¼ x0 þ 2 k0 0:25ðN1 þ N0 Þ N12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 k0 0:25ðN0 þ N1 Þ N1 ðx0 x1 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 k0 0:25ðN1 þ N0 Þ N12 The remaining turning points (for 1 < k 6 z) of buried modes are defined recurrently qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pðkþ1Þ p k 12 þ arctg Ck exp 2k0 j¼ðzþ1Þ Nk2 0:25 Njþ1 þ Nj xjþ1 xj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xk ¼ xk1 þ 2 k0 0:25ðNk þ Nk1 Þ Nk2 Pk2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k0 i¼k 0:25ðNiþ1 þ Ni Þ Nk2 ðxiþ1 xi Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 k0 0:25ðNk þ Nk1 Þ Nk2
ð6Þ
ð7Þ
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x Fig. 3. Criterion of smoothness of reconstructed refractive index profile.
Having completed the values of turning points of buried modes, we calculate the turning points of the remaining modes using Eq. (2). As before, taking into consideration (4), it assumes the following form (for m > z): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 Pm2 Nm2 n2a 2 x 0:25 N p m þ 14 þ arctg þ N N k 0 jþ1 j 2 j¼ðzþ1Þ m jþ1 xj Nðzþ1Þ Nm2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xm ¼ xm1 þ : ð8Þ 2 k0 0:25ðNm þ Nm1 Þ Nm2 Making use of (8) we can recurrently determine the remaining turning points of unburied modes for ðiÞ ðiÞ z < m 6 M. The obtained set of points fxt ; Neff g for i ¼ 1; . . . ; M is the approximation of the searched ðiÞ refractive index profile of the buried waveguide. Since the obtained values of turning points xt for i ¼ z þ 1; . . . ; M, and in consequence of profile points, are the function of the preset initial values ns ; N0 ; xðzþ1Þ ; . . . ; x1 ; x0 , therefore its final shape is the function of these variables. Not all final profiles obtained in this way have the shape corresponding with physical conditions they are produced in. Therefore, we should accept a certain criterion of its final shape. Due to the character of the production process of waveguides, we assume here the criterion of smoothness as characteristic for diffusive processes. Since the obtained profile has the form of a broken line – the criterion of smoothness has been established here as the sum of triangles area created by each of its three neighboring points. The pictorial view is presented in Fig. 3. Through the change of initial values we aim to minimize this sum. In further this sum as a function of the number of buried modes z will be called as criterionðzÞ. 4. Reconstruction algorithm of refractive index profile The described algorithm relies on the search of minimum function of many variables with the help of the nongradient method by Hooke–Jeeves [21]. The demand is here the minimum of function ‘‘criterion’’ of variables: ns ; N0 ; x0 ; x1 ; . . . ; xz . The initial values of these variables are given in certain intervals. For receive number of buried modes – z, initial value ns is from interval: Nzþ1 < ns < Nz (see Fig. 4). Thus with a change in the value of z this interval will be changed. However, for variables x0 and N0 intervals of admissible changes their values are given once. For variable x0 one should to qualify the admissible interval of changes. For variable N0 the maximum admissible value N0 max poses as only. Besides the initial value, for which it begins a search of minimum of function, it defines the preliminary step of variable changes – s. In the introduced here algorithm the step of changes (s) for every variable is equal. During calculations its initial value is changed. Constant coefficients (a, b and c see Fig. 5) decide about the quantity of this change.
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Fig. 4. Initial values: ns ; N0 and x0 in case z ¼ 3 (M ¼ 8).
Thus the input data of the calculation algorithm are as follows: the set of effective indexes fN1 ; . . . ; NM g, the maximum admissible value of N0 max , the admissible values x0 min and x0 max (see Fig. 4). Additionally, we introduced a coefficient max defining the maximum number of iterations. The quantity e defines the minimum admissible value of a particular step of investigation. From among the input data of the algorithm, preliminary numerical value of the criterion is also defined. As it was mentioned, there are an unknown number of buried modes – z in the set of measured values of effective indexes of modes of the waveguide. In the proposed here reconstruction algorithm of the refractive index profile this quantity is changed. Assuming M as the number of all modes of the waveguide – z assumes the successive integral values from the set: 1; 2; . . . ; M 1. For each value z the following is accepted: initial value ns ¼ 0:5ðNz þ Nzþ1 Þ, the initial value N0 ¼ 0:5ðN1 þ N0 max Þ and the initial value x0 ¼ 0:5ðx0 min þ x0 max Þ. Basing on x0 the initial values of turning points of buried modes being below x0 are assumed. They have to satisfy inequalities: xz < < x1 < x0 . During further calculations (at a settle magnitude of z) these values are independently changed about the current value of step s. Then on the basis of Eqs. (6) and (7) the value of the turning points (x1 ; . . . ; xz ) of buried modes is calculated, and, on the basis of Eq. (8), the value of the turning points of the remaining unburied modes (xzþ1 ; . . . ; xM ) is calculated. For the fixed value z, the mentioned initial values are exchanged with the steps of changes declared at the beginning, aiming to minimize the value of criterionðzÞ (Fig. 3). This value corresponding to z is retained in memory (as well as corresponding with its refractive index profile) and the whole course of calculations is repeated for z ¼ z þ 1. The computational procedure is terminated when the value z ¼ M 1 is reached. As output result, we obtain a set of reconstructed refractive index profiles corresponding with the values 1 6 z 6 M 1, as well as a set of values of criterionðzÞ. The smallest value of criterionðzÞ shows the final profile of waveguide determined with the algorithm. Detailed block diagram of the discussed algorithm is presented in Fig. 5.
5. Testing of algorithm In order to test the algorithm discussed above with respect to its efficiency to reconstruct refractive profiles, a few nðxÞ functions having the characteristics of buried profiles were generated. The Gauss function was then selected as the most appropriate testing function
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Fig. 5. Block diagram of the proposed algorithm reconstructing the refractive index profile of the buried waveguide.
" nðxÞ ¼ ng þ dn exp
xd a
2 # :
ð9Þ
In the above equation it was assumed that: ng ¼ 1:5, dn ¼ 0:1. The values of parameters d and a were changed in the way ensuring that the function (9) described the profiles of different degrees of burying. For ðiÞ every function of the profile nðxÞ, with the help of modal Eqs. (1) and (2), a set of effective indexes fNeff g ðiÞ was determined. Basing on each of the sets determined in this way fNeff g, using the described algorithm, the
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Table 1 Counted values of criterionðzÞ Criterion(z) z
1
2
3
4
5
7
1 2 3 4 5 6 7 8 9 10 11 12
0:03128 0.18847 1.76011 2.52622 0.46032 0.28631 3.37928 2.05148 4.25871 3.78945 5.57618 –
1.85071 0:06996 0.20201 0.41448 0.85455 0.44762 3.61139 0.72275 1.52374 3.91754 6.24971 –
0.18315 0.09226 0:04712 0.65232 0.31433 3.52259 0.76537 2.61585 2.47135 5.91421 4.28463 –
0.04658 0.10666 0.11849 0:04583 0.23589 0.53661 0.95841 1.25137 1.59831 2.69728 2.78194 –
0.10808 0.12108 0.39215 0.73833 0:07493 3.20917 0.31550 0.42054 4.35814 3.87994 4.55741 –
0.12325 0.14614 0.18586 0.29226 0.45655 0.39013 0:10334 0.14252 0.36422 0.25129 0.21699 0.22405
function of a respective input profile was reconstructed. Counted values of criterionðzÞ indicate the quantity of buried modes is collected in Table 1. Figs. 6 and 7 present functions simulating buried profiles having a suitable number of buried modes (value z) as well as the obtained reconstruction of refractive profiles. For every profile the course of changes of criterionðzÞ was presented with the smallest value being indicated.
6. Production of buried waveguides For the measurements of refractive profiles, the structures of planar buried waveguides were produced, using the ion-exchange technique. As substrate, soda-lime glass of refractive index n ¼ 1:5111 for k ¼ 677 nm was used. The technological processes were realized in four samples. With respect to all samples, preliminary diffusion of Agþ ions from molten pure salts AgNO3 was realized. The temperature of the process was Td ¼ 303 C, and its duration td ¼ 10 h. For planar waveguides produced in such a way, propagation constants of modes for k ¼ 677 nm were measured, and refractive index profile with the use of modal equation for monotonic profile was determined. Then, for individual waveguides, the processes of burying were carried out through the diffusion of Naþ ions in NaNO3 bath of the temperature Tb ¼ 320 C. The duration times of burying processes were, respectively, 1, 2, 4 and 8 h. A part of the waveguide samples produced in that way were prepared for comparative interference investigations (see Section 8.2).
7. Measurements of propagation constants of modes of buried waveguides The measurements of propagation constants of modes in planar waveguides having a monotonic profile are carried out using prism couplers. Refractive index of the prism has here a higher value than the maximum refractive index of the waveguideÕs. The light wave propagating inside the prism can then penetrate the area of waveguide, and when the components of wave vectors in the prism and in the waveguide are coordinated along the propagation direction – it can stimulate individual modes [17,18]. The penetration depth of the wave field in the prism into the waveguide area will be the higher the lower is the difference between refrective indexes of the prism and waveguide [19]. The coupling obtained in this way depends also on the width of the gap at the contact area between the prism and glass. In the case of buried
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Fig. 6. Refractive profiles (continuous line) and their reconstructions (intermittent lines) on the basis of the set of effective indexes ðiÞ fNeff g, as well as courses of the changes of the value of criterionðzÞ. Quantity of buried modes: (a) z ¼ 1, (b) z ¼ 2, (c) z ¼ 3.
waveguides, the maximum of refractive index distribution is shifted far into glass base, and hence the distributions of modal fields take place also further from the glass surface. In that case, if we want to effect the excitation of modes in the waveguide with the help of a prism coupler, its refractive index should have the value a little higher than the maximum refractive index of the waveguide. It requires a suitable selection of coupling prisms [20]. Fig. 8 presents values of the refractive index of the applied prism couplers (glass of Schott firm) for wavelength k ¼ 677 nm against the background of a typical refractive index profile produced in soda-lime glass (ng677 ¼ 1:5111), with ion-exchange technique Agþ () Naþ . Table 2 presents the parameters of prism couplers applied in the measurements. Table 3 contains the collected results involving the measurements of effective indices of waveguides produced in burying processes in times: 1, 2, 4 and 8 h. The left-hand part of the Table 3 contains the measurement results for buried waveguide in time 4 h, accomplished with the use of different prism couplers. We can see here the influence of the refractive index of the prism on the possibility to stimulate a number of modes in the buried waveguide. The right-hand part of the Table 3 contains the measurement results of effective indexes for all buried waveguides carried out with the help of a prism having the lowest refractive index.
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Fig. 7. Refractive profiles (continuous line) and their reconstructions (intermittent lines) on the basis of the set of effective indexes ðiÞ fNeff g, as well as courses of changes of the value of criterionðzÞ. Quantity of buried modes: (a) z ¼ 4, (b) z ¼ 5, (c) z ¼ 7.
8. Determination of the refractive index profiles of buried waveguides 8.1. Reconstruction algorithm of refractive index profile To determine the refractive index profiles of the produced buried waveguides, the algorithm described in Section 4 was applied. The values of effective indexes of modes of buried structures gathered in Table 3 were applied here as input data. Fig. 9 presents the obtained reconstructions of buried profiles as well the refractive index profile of the waveguide after preliminary diffusion obtained using the reconstruction procedure for monotonic profiles. 8.2. Interference method The interference method described in work [7] was used to determine comparative profiles of the produced waveguide. For that purpose, three samples were prepared from the produced waveguides (buried in
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Fig. 8. Refractive indexes of applied prism couplers for k ¼ 677 nm, on the background of refractive index profile of planar waveguide. Table 2 Parameters of prism couplers Prism couplers Glass (Schott) Prism angle d(deg) Uncertainty of prism angle Dd (deg) n677
SF-14 49.9821 0.0002 1.7516
SK-10 65.0117 0.0011 1.6186
SK-2 65.0222 0.0011 1.6033
SK-12 70.0787 0.0008 1.5794
Table 3 Measurement results of effective indexes of buried waveguides Value of effective indexes Prism
SF-14
Number of mode
Soda-lime glass diffusion AgNO3 : T ¼ 303 C; tdiff ¼ 10 h þ diffusion ðburyingÞ NaNO3 : T ¼ 320 C
18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
SK-10
SK-2
SK-12
tb ¼ 4 h
1.5332 1.5319 1.5306 1.5290 1.5277 1.5262 1.5247 1.5233 1.5219 1.5204 1.5189 1.5174 1.5160 1.5145 1.5132 1.5118
1.5338 1.5324 1.5311 1.5297 1.5283 1.5269 1.5255 1.5241 1.5226 1.5211 1.5197 1.5182 1.5167 1.5153 1.5138 1.5125 1.5113
1.5345 1.5332 1.5319 1.5305 1.5291 1.5277 1.5263 1.5248 1.5233 1.5218 1.5204 1.5189 1.5174 1.5160 1.5147 1.5134 1.5125
1.5360 1.5347 1.5334 1.5321 1.5307 1.5293 1.5278 1.5264 1.5250 1.5235 1.5220 1.5205 1.5190 1.5175 1.5161 1.5147 1.5134 1.5123
SK-12
tb ¼ 1 h
tb ¼ 2 h
tb ¼ 4 h
tb ¼ 8 h
1.5451 1.5438 1.5419 1.5399 1.5379 1.5358 1.5337 1.5316 1.5294 1.5273 1.5251 1.5228 1.5207 1.5185 1.5164 1.5143 1.5125 1.5116
1.5411 1.5399 1.5384 1.5368 1.5351 1.5334 1.5317 1.5300 1.5282 1.5264 1.5245 1.5227 1.5209 1.5191 1.5173 1.5155 1.5138 1.5124
1.5360 1.5347 1.5334 1.5321 1.5307 1.5293 1.5278 1.5264 1.5250 1.5235 1.5220 1.5205 1.5190 1.5175 1.5161 1.5147 1.5134 1.5123
1.5309 1.5301 1.5290 1.5280 1.5269 1.5258 1.5247 1.5236 1.5225 1.5214 1.5202 1.5191 1.5180 1.5169 1.5158 1.5147 1.5137 1.5128
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Fig. 9. Refractive index profiles of buried waveguides reconstructed on the basis of the value of effective indices from Table 3.
times: 1, 4 and 8 h, respectively) by cutting them out and grinding them suitably to ensure a definite thickness. These thickness values were subjected to interference measurements, and for particular samples the following results were obtained, respectively, 77 lm (for sample buried in time 1 h), 91 lm (for sample buried in time 4 h), and 143 lm (for sample buried in time 8 h), with uncertainty of measure of about 1 lm. Polarization microinterferometer BIOLAR-PI was applied for interferometric investigations. Fringe interferograms of the studied samples were registered with a CCD camera. In order to make it possible to compare the refractive index profiles obtained with both methods, the same wavelength was applied (k ¼ 677 nm) with the registration of interference patterns as with the measurements of propagation constants. Fig. 10 presents fringe interferograms for the samples of waveguides buried in times 1, 4 and 8 h, as well profiles of refractive index changes dnðxÞ reconstructed on the basis of the shape of interference fringes [7]. Broken lines were used to mark measure uncertainties resulting from the measurement error involving the thickness of samples. The profile of absolute value of refractive index in this measurement method is nðxÞ ¼ dnðxÞ þ ng . 9. Comparison of reconstructed refractive index profiles Fig. 11 presents refractive index profiles obtained using both measurement methods. The obtained refractive profiles display a certain degree of similarity, yet the differences are larger than the measure uncertainty. When comparing the results, one should remember that the applied measurement methods are very different from each other. The interference method, however, can bring about a relatively high measurement error resulting from the uncertainty to qualify the run of interference fringes. In the microinterferometer type BIOLAR-PI applied for the measurements, two-beam interference is used. In the case of interference of that type, wide interference fringes are created. Yet the form of interference fringe contains direct information about the refractive profile [7]. The method basing on the described algorithm makes use, however, of the values of effective indexes, whereof measurements are performed with high accuracy with the help of a goniometer. With the application of prism couplers specified in Table 2, taking into account the uncertainty of measured synchronic angles at the level of 0.05, the uncertainty determination of effective index is DNeff 0:0003. But in this case we obtain from the measurements only a set of exact values of the function fNi g without the information involving the respective values of arguments fxi g. The error involving the profile shape results here from the calculations carried out with the help of the proposed algorithm values of these arguments (turning points). Although the interference method requires time-consuming preparations of sample waveguide, it provides essential information on the range of refractive index profile. On the basis of Eqs. (1) and (2) it is possible, as in Section 5, to determine the values
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Fig. 10. Registered fringe interferograms of buried waveguides and reconstructed profiles of refractive index on the basis of the interference fringe course, for waveguides buried in times: (a) tb ¼ 1 h, (b) tb ¼ 4 h, (c) tb ¼ 8 h. Intermittent lines were marked measure uncertainties resulting from a mistake of the measurement thickness of samples [4].
Fig. 11. Comparison of refractive index profiles obtained with the help of the described algorithm with that obtained by the interference method, for waveguides buried in times: (a) tb ¼ 1 h, (b) tb ¼ 4 h, (c) tb ¼ 8 h.
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Table 4 Values of effective indices defined with the prism coupler in comparison with those counted on the basis of Eqs. (1) and (2) for refractive profiles measured by the interference method Order of mode
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Value of effective indexes tb ¼ 4 h
tb ¼ 1 h
tb ¼ 8 h
Interference method
Prism coupler
Interference method
Prism coupler
Interference method
Prism coupler
1.5429 1.5419 1.5405 1.5390 1.5372 1.5354 1.5336 1.5316 1.5298 1.5279 1.5261 1.5243 1.5224 1.5204 1.5185 1.5165 1.5146 1.5128 1.5113 – – –
1.5451 1.5438 1.5419 1.5399 1.5379 1.5358 1.5337 1.5316 1.5294 1.5273 1.5251 1.5228 1.5207 1.5185 1.5164 1.5143 1.5125 1.5116 – – – –
1.5332 1.5323 1.5314 1.5304 1.5295 1.5286 1.5275 1.5264 1.5253 1.5242 1.5231 1.5218 1.5205 1.5192 1.5179 1.5166 1.5154 1.5142 1.5129 1.5117 – –
1.5360 1.5347 1.5334 1.5321 1.5307 1.5293 1.5278 1.5264 1.5250 1.5235 1.5220 1.5205 1.5190 1.5175 1.5161 1.5147 1.5134 1.5123 – – – –
1.5293 1.5286 1.5279 1.5271 1.5264 1.5256 1.5248 1.5239 1.5231 1.5223 1.5213 1.5204 1.5195 1.5185 1.5176 1.5166 1.5156 1.5146 1.5136 1.5127 1.5118 1.5112
1.5309 1.5301 1.5290 1.5280 1.5269 1.5258 1.5247 1.5236 1.5225 1.5214 1.5202 1.5191 1.5180 1.5169 1.5158 1.5147 1.5137 1.5128 – – – –
of effective indices for the refractive profiles obtained with the interference method. The compared values of the of refractive profiles obtained in this way by means of a prism coupler are presented in Table 4.
10. Conclusions The proposed in the present work algorithm of the reconstruction of refractive index profiles of planar buried waveguides, basing on the measurements of propagation constants of modes, shows good reproduction effectiveness of the profiles of testing functions (see Section 5). What is the limitation involving the applicability of the algorithm described above is the number of waveguide modes – as in the case of the reconstruction of monotonic profiles – resulting from the depth of profile nðxÞ. In the refractive index profiles applied to test the algorithm (Section 5), the total number of modes was higher than 10 in each case. For the reconstruction of refractive index profiles of the real structures of buried waveguides, a complete set of propagation constants of modes should be obtained. This necessitates the application of suitable prism couplers (see Table 3, Section 7). The application of smoothness criterion in the proposed algorithm (see Fig. 3) is not the only possible criterion which can be used to qualify the obtained profiles. The interference method applied as a supplement allows one to verify the obtained refractive index profiles. Despite the time-consuming preparation of the sample, this method provides essential information about the depth of the profile. As it was already mentioned in the Introduction, the credibility of the obtained results is most essential in order to determine the refractive index profiles of buried waveguides. Therefore, it seems advisable, in the cases where it is possible, to correlate the results obtained with both methods.
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Acknowledgements This work was performed within the research project KBN 8 T11B 05 218.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
S.I. Najafi, Introduction to Glass Integrated Optics, Artech House, 1992. P. Karasi nski, Proc. SPIE 4516 (2001) 15. J.M. Inman, J.L. Bentley, S.N. Houde-Water, J. Non-Cryst. Solids 191 (1995) 209. A. Lupascu, A. Kevorkian, T. Boudet, F. Saint-Andre, D. Persegol, M. Levy, Opt. Eng. 35 (6) (1996) 1603. B.M. Schulz, S. K€ urbitz, J. Porstendorfer, K.-J. Berg, M. Schulz, J. Non-Cryst. Solids 248 (1999) 99. M. Sochacka, E.L. Lago, Z. Jaroszewicz, Appl. Opt. 33 (16) (1994) 3342. R. Rogozi nski, A. Opilski, Opt. Appl. 26 (2) (1996) 71. K. Morishita, J. Lightwave Technol. 4 (8) (1986) 1120. G.L. Yip, P.C. Noutsios, L. Chen, Appl. Opt. 35 (12) (1996) 2060. S.A. Poling, S.N. Houde-Walter, J. Non-Cryst. Solids 255 (1999) 78. M.J. Adams, in: John Wiley and Sons (Ed.), An Introduction to Optical Waveguides, Wiley, Chichester, 1981, p. 191. J.M. White, P.F. Heidrich, Appl. Opt. 15 (1) (1976) 151. K.S. Chiang, J. Lightwave Technol. 3 (2) (1985) 385. S. Batchelor, R. Owen, D.G. Ashworth, Opt. Commun. 131 (10) (1996) 31. tyrok^ J. Janta, J. C y, Opt. Commun. 25 (1) (1978) 49. F. Xiang, G.L. Yip, J. Lightwave Technol. 12 (3) (1994) 443. H.G. Unger, Optische Nachrichtentechnik, Elitera–Verlag, Berlin, 1976. R.G. Hunsperger, Integrated Optics: Theory and Technology, Springer, Berlin, 1984. R. Ulrich, R. Torge, Appl. Opt. 12 (12) (1973) 2901. J. Seligson, Appl. Opt. 26 (13) (1987) 2609. B. Baron, Metody Numeryczne w Turbo Pascalu, Helion, Gliwice, 1994, p. 129 (in Polish).
Determination of refractive index profiles of planar buried waveguides on the basis of a set of modal propagation constants ski * Roman Rogozin Optoelectronics Department, Institute of Physics, Silesian University of Technology, ul. Krzywoustego 2, 44-100 Gliwice, Poland Received 14 May 2002; received in revised form 6 December 2002; accepted 5 February 2003
Abstract In the work the algorithm involving the reconstruction of refractive index profiles of planar buried waveguides based on modal equation was proposed. The analysis testing the algorithm with the help of functions simulating buried profiles was carried out. Using ion-exchange technique in glass substrate, buried waveguide structures were manufactured. Propagation constants of all modes of those structures were determined by a prism coupling method with the measurements of synchronic angles. On the basis of the measured propagation constants the reconstruction of their refractive index distribution was carried out with the help of the proposed algorithm. The obtained results were compared with the results of interferometric investigations. 2003 Elsevier Science B.V. All rights reserved. Keywords: Planar buried waveguides; Refractive index profiles
1. Introduction Planar optical waveguide structures produced in glass substrate can have a variously shaped distribution of the refractive index. The ion-exchange technique applied for the production of these types of structures in the superficial area of glass allows a relatively high degree of freedom when forming their refractive profiles [1]. Depending on the requirements for an optical structure, it can have a monotonic or unmonotonic form of refractive index. The latter type of refractive profile is effected in technological processes, in which the already produced form of refractive index is purposefully reduced by the introduction of a suitable admixture from the side of glass surface. As a result, the maximum value of refractive profile is shifted far into the substrate. It results from this that in glass exists an area where the value of refractive
*
Fax: +48-32-237-2216. E-mail address: [email protected].
0030-4018/03/$ - see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01286-0
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index is increasing with depth. Waveguide structures characterized by such refractive profiles are referred to as buried waveguides. In the technique of ion-exchange applied for the production of optical structures in glass, it is extremely important that we know the refractive profile of the structure being produced, both in view of cognitive aspect with respect to the processes of theoretical modelling of ion-exchange process, and also with respect to the description of transmission properties of the waveguide [2]. In accepted theoretical models of the phenomenon of ion-exchange in glass it challenges often the dependence of parameters of diffusion on the concentration of introduced into glass ions of the admixture [3–5]. The concentration profile of this admixture in the ion-exchange process is then the solution of the non-linear equation of diffusion. The form of this solution is here dependent on a certain number of parameters accepted in the model assumptions. The determination of these parameters is effected by the adjustment of the diffusion solution to the real refractive profile obtained from the measurements. The knowledge of the parameters determined in such a way makes it possible to predict the shape of refractive profiles obtained in other technological processes (secondary diffusion, electrodiffusion). There are a number of measurement techniques applied in the metrology of refractive profiles of planar optical structures. Considering the results obtained in effect of their application, they can be divided into two groups: direct and indirect. In direct methods the refractive profile is obtained immediately as a result of the measurement. This group of methods is represented by the interference method [6,7] (where the information about the refractive profile is enclosed in the perturbation of interference field), method of near field analysis [8,9] (where the analyzed distribution of power at the waveguideÕs exit is reflecting the profile), or the method of isotopic loftsmen [10], or X-ray microanalysis (where the distribution of concentration of the admixture introduced to the glass is defined directly). In indirect methods the refractive profile of the waveguide is obtained as a result of an additional analysis of the obtained measurement data with the application of more or less complex mathematical models which approximate the searched profile. In this group of methods the most commonly applied is a so called inverted WKB method [11], which is based on the modal equation and is using a set of propagation constants of the waveguide for the reconstruction of ðiÞ ðiÞ refractive profile. The refractive index distribution received with this method is a set of points fxt ; Neff g, ðiÞ ðiÞ where xt is the turning point of the ith mode having the effective index Neff . The method has the following advantages: high accuracy involving the acquisition of input data (involving the set of propagation constants which are determined using prism coupling measurements), and easy realization of measurements – without any special preparation of waveguide sample (nondestructive method). There is one disadvantage of this method – its application is limited to multimode waveguide structures (the reliability of the reconstruction of refractive profile is increasing with the number of modes). This method was proposed for the first time in 1976 by White and Heidrich [12]. In the modal equation the searched function describing the profile was replaced by the authors with a stepping function, which yielded a recurrent formula for the ð0Þ calculation of individual modal turning points. In this method, in order to determine the turning point xt of the zero mode, it is necessary to know the refractive index on the surface of waveguide. Its value is selected in the way ensuring that it meets a specific criterion. The smoothness of the obtained profile is accepted here as a feature characteristic for diffusive processes. In 1985 Chiang [13] introduced the notion of effective refractive index as a continuous function of modal order. Basing on the set of measured values of effective indexes, this procedure makes it possible to reconstruct the refractive profile defined in any number of points. The criterion of smoothness of the received profile does not occur here as it was in the procedure of White and Heidrich. The problem of uniqueness is shifted here towards the selection of a suitable kind of interpolation polynomial. In the work [14] the authors proposed the reconstruction of the monotonic refractive index profile of planar waveguide basing on the measurements of effective indexes of waveguide modes measured for different wavelengths. As in ChiangÕs method, it is possible here to define with arbitrary accuracy (the number of testing points) the shape of the function describing the profile. This function is not dependent on wavelength. Therefore, it can be determined basing on effective refractive indexes of
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only one mode, determined for many wavelengths. Due to this, the method enables the reconstruction of waveguide profiles, even single-mode ones. There is, however, a necessity to know the dispersion properties of the base glass. The specified above methods basing on modal equation are applied for the reconstruction of only monotonic distribution of refractive index. The present work introduces a proposal of algorithm making use of modal equation for the reconstruction of unmonotonic distribution of refractive indexes of buried waveguides. 2. Buried waveguide – own equation of buried modes A typical refractive profile of planar buried waveguide is presented in Fig. 1. Such a profile has the following characteristic features: the value of refractive index on the surface of waveguide nð0Þ and the maximum value of refractive index within the area of waveguide nmax shifted far into the base of glass (xmax > 0). Such a shape of the refractive profile results in the division of the set of propagation constants (effective indexes) of waveguide modes into two groups: • Buried modes, characterized by the occurrence of two turning points in the area of glass: xa > 0 and xb > 0, for which the value of effective indexes satisfies the inequality: nð0Þ < Nm0 < nmax (Fig. 1). • Unburied modes, for which always one turning point occurs on the surface of glass, and the values of effective indexes satisfy the relation: ng < Nm < nð0Þ, where ng stands for the refractive index of base glass. The form of modal equation allowing for the change of phase in turning points will therefore be different for each group of modes. For the TE polarization this equation assumes the following form [15,16]: Buried modes rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 8 Nm2 n2 ð0Þ > > > > Z xb qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 = <1 xa qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nm2 n20 1 2 2 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 2k0 : k0 n ðxÞ Nm dx ¼ p m þ Nm n ðxÞ dx þ arctg > > 2 2 2 2 ð0Þ xa 0 > > ; : 1 þ Nm n 2 2 Nm n0
ð1Þ Unburied modes Z k0 0
xc
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n2 ðxÞ Nm2 dx ¼ p m þ þ arctg 4
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Nm2 n20 : 2 n ð0Þ Nm2
Fig. 1. Refractive index profile of the planar buried waveguide.
ð2Þ
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In above mentioned equations, k0 ¼ 2p=k, nðxÞ is the function describing the refractive index profile of the waveguide, m is the order of mode, Nm is the effective index of the mth mode, n0 is the refractive index of surroundings, nð0Þ ¼ ns is the refractive index on the surface of the waveguide, ng is the refractive index of the glass base, xi is the turning points of mode (i ¼ a; b; c; see Fig. 1). When we know the set of effective indexes fNm g of waveguide modes we can reconstruct the profile function nðxÞ using the above-mentioned equations.
3. Reconstruction of the refractive index profile of the buried waveguide In the reconstruction procedures of monotonic refractive index profile nðxÞ based on the modal equation, the approximation of this profile by stepping function is most commonly applied [12] nðxÞ ¼ 0:5ðNk1 þ Nk Þ
ðk1Þ
for xt
ðkÞ
6 x < xt ;
ð3Þ
where k ¼ 0; 1; 2; . . . ; M 1; M is the total number of waveguide modes. By replacing in the modal equation the searched function nðxÞ with the approximation (3), as well as the integral with summation, we obtain the approximate form of modal equation, from which we can derive a recurrent formula which allows one to determine the next turning points of modes [12]. In the case of the monotonic profile, in order to determine the turning point x0 of the zero order mode, it is necessary to know the refractive index on the surface of waveguide (N1 ¼ ns ). As it was mentioned before, it is not measured directly, and its value is selected here in the way ensuring that the obtained profile fxk ; Nk g satisfies the criterion of smoothness as the feature characteristic for diffusive processes. A similar approach in the case of buried profile necessitates that the number of buried modes (z) is defined and that the initial value of turning points closer to glass surface is accepted for buried modes. The idea involving the approximation of the searched refractive index profile with a stepping function is presented in Fig. 2. This function has now the following form: nðxÞ ¼ 0:5ðNk1 þ Nk Þ
ðk1Þ
for xt
ðkÞ
6 x < xt ;
ð4Þ
where k ¼ z; z þ 1; . . . ; 1; 0; 1; . . . ; M; M is the number of all modes of the waveguide, z is the assumed quantity of buried modes (z < M).
Fig. 2. Idea of the reconstruction of buried refractive index profile.
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The values of effective indexes Nm corresponding to the successive orders of modes are numbered from 1 to M (where M denotes the number of all modes of the waveguide). In Fig. 2 the exemplary value M ¼ 8. The value of index m ¼ 1 is attributed to the zero order mode. Since the number of buried modes (z) is unknown, therefore in the successive approximations it is changed. This value, being a natural number, can assume values from the range: 1 6 z < M. It results from the assumed value of refractive index ns on the glass surface. For each fixed value (z) (in Fig. 2 the exemplary value z ¼ 3) negative indexes mð1 6 =m= 6 zÞ are introduced for the accepted buried effective indexes Nk ¼ Nk for k 6 z (values N1 ; N2 ; N3 in Fig. 2). Also negative indexes are attributed to the corresponding turning points from the side of glass surface (values x1 ; x2 ; x3 in Fig. 2). Another preset quantity is the hypothetic value N0 of the maximum value of refractive index in the waveguide, which has the coordinate x0 on the axis x. This value satisfies the inequality: N1 < N0 . As it was mentioned above, the algorithm is based on the approximate form of modal equation, separately for buried and unburied modes. Eq. (1), after the profile function nðxÞ has been replaced with the approximation (4) and the integration with summation, assumes the following form: k0
k1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2 0:25ðNiþ1 þ Ni Þ Nk2 ðxiþ1 xi Þ ¼ pðk 0:5Þ i¼k
(
"
þ arctg Ck exp 2k0
ðkþ1Þ X
#) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Nk2 0:25 Njþ1 þ Nj xjþ1 xj ;
ð5Þ
j¼ðzþ1Þ
where z is the quantity of buried modes, whereas rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ck ¼
1 2
1
2 Nk2 Nðzþ1Þ
Nk2 n20
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1þ
2 Nk2 Nðzþ1Þ
Nk2 n20
For the assumed value ns and the resulting from this value z, the turning point x1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 P2 p 2 N x þ arctg C exp 2k 0:25 N þ N x 1 0 jþ1 j jþ1 j 1 j¼ðzþ1Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x1 ¼ x0 þ 2 k0 0:25ðN1 þ N0 Þ N12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 k0 0:25ðN0 þ N1 Þ N1 ðx0 x1 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 k0 0:25ðN1 þ N0 Þ N12 The remaining turning points (for 1 < k 6 z) of buried modes are defined recurrently qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pðkþ1Þ p k 12 þ arctg Ck exp 2k0 j¼ðzþ1Þ Nk2 0:25 Njþ1 þ Nj xjþ1 xj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xk ¼ xk1 þ 2 k0 0:25ðNk þ Nk1 Þ Nk2 Pk2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k0 i¼k 0:25ðNiþ1 þ Ni Þ Nk2 ðxiþ1 xi Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 k0 0:25ðNk þ Nk1 Þ Nk2
ð6Þ
ð7Þ
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x Fig. 3. Criterion of smoothness of reconstructed refractive index profile.
Having completed the values of turning points of buried modes, we calculate the turning points of the remaining modes using Eq. (2). As before, taking into consideration (4), it assumes the following form (for m > z): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 Pm2 Nm2 n2a 2 x 0:25 N p m þ 14 þ arctg þ N N k 0 jþ1 j 2 j¼ðzþ1Þ m jþ1 xj Nðzþ1Þ Nm2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xm ¼ xm1 þ : ð8Þ 2 k0 0:25ðNm þ Nm1 Þ Nm2 Making use of (8) we can recurrently determine the remaining turning points of unburied modes for ðiÞ ðiÞ z < m 6 M. The obtained set of points fxt ; Neff g for i ¼ 1; . . . ; M is the approximation of the searched ðiÞ refractive index profile of the buried waveguide. Since the obtained values of turning points xt for i ¼ z þ 1; . . . ; M, and in consequence of profile points, are the function of the preset initial values ns ; N0 ; xðzþ1Þ ; . . . ; x1 ; x0 , therefore its final shape is the function of these variables. Not all final profiles obtained in this way have the shape corresponding with physical conditions they are produced in. Therefore, we should accept a certain criterion of its final shape. Due to the character of the production process of waveguides, we assume here the criterion of smoothness as characteristic for diffusive processes. Since the obtained profile has the form of a broken line – the criterion of smoothness has been established here as the sum of triangles area created by each of its three neighboring points. The pictorial view is presented in Fig. 3. Through the change of initial values we aim to minimize this sum. In further this sum as a function of the number of buried modes z will be called as criterionðzÞ. 4. Reconstruction algorithm of refractive index profile The described algorithm relies on the search of minimum function of many variables with the help of the nongradient method by Hooke–Jeeves [21]. The demand is here the minimum of function ‘‘criterion’’ of variables: ns ; N0 ; x0 ; x1 ; . . . ; xz . The initial values of these variables are given in certain intervals. For receive number of buried modes – z, initial value ns is from interval: Nzþ1 < ns < Nz (see Fig. 4). Thus with a change in the value of z this interval will be changed. However, for variables x0 and N0 intervals of admissible changes their values are given once. For variable x0 one should to qualify the admissible interval of changes. For variable N0 the maximum admissible value N0 max poses as only. Besides the initial value, for which it begins a search of minimum of function, it defines the preliminary step of variable changes – s. In the introduced here algorithm the step of changes (s) for every variable is equal. During calculations its initial value is changed. Constant coefficients (a, b and c see Fig. 5) decide about the quantity of this change.
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Fig. 4. Initial values: ns ; N0 and x0 in case z ¼ 3 (M ¼ 8).
Thus the input data of the calculation algorithm are as follows: the set of effective indexes fN1 ; . . . ; NM g, the maximum admissible value of N0 max , the admissible values x0 min and x0 max (see Fig. 4). Additionally, we introduced a coefficient max defining the maximum number of iterations. The quantity e defines the minimum admissible value of a particular step of investigation. From among the input data of the algorithm, preliminary numerical value of the criterion is also defined. As it was mentioned, there are an unknown number of buried modes – z in the set of measured values of effective indexes of modes of the waveguide. In the proposed here reconstruction algorithm of the refractive index profile this quantity is changed. Assuming M as the number of all modes of the waveguide – z assumes the successive integral values from the set: 1; 2; . . . ; M 1. For each value z the following is accepted: initial value ns ¼ 0:5ðNz þ Nzþ1 Þ, the initial value N0 ¼ 0:5ðN1 þ N0 max Þ and the initial value x0 ¼ 0:5ðx0 min þ x0 max Þ. Basing on x0 the initial values of turning points of buried modes being below x0 are assumed. They have to satisfy inequalities: xz < < x1 < x0 . During further calculations (at a settle magnitude of z) these values are independently changed about the current value of step s. Then on the basis of Eqs. (6) and (7) the value of the turning points (x1 ; . . . ; xz ) of buried modes is calculated, and, on the basis of Eq. (8), the value of the turning points of the remaining unburied modes (xzþ1 ; . . . ; xM ) is calculated. For the fixed value z, the mentioned initial values are exchanged with the steps of changes declared at the beginning, aiming to minimize the value of criterionðzÞ (Fig. 3). This value corresponding to z is retained in memory (as well as corresponding with its refractive index profile) and the whole course of calculations is repeated for z ¼ z þ 1. The computational procedure is terminated when the value z ¼ M 1 is reached. As output result, we obtain a set of reconstructed refractive index profiles corresponding with the values 1 6 z 6 M 1, as well as a set of values of criterionðzÞ. The smallest value of criterionðzÞ shows the final profile of waveguide determined with the algorithm. Detailed block diagram of the discussed algorithm is presented in Fig. 5.
5. Testing of algorithm In order to test the algorithm discussed above with respect to its efficiency to reconstruct refractive profiles, a few nðxÞ functions having the characteristics of buried profiles were generated. The Gauss function was then selected as the most appropriate testing function
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Fig. 5. Block diagram of the proposed algorithm reconstructing the refractive index profile of the buried waveguide.
" nðxÞ ¼ ng þ dn exp
xd a
2 # :
ð9Þ
In the above equation it was assumed that: ng ¼ 1:5, dn ¼ 0:1. The values of parameters d and a were changed in the way ensuring that the function (9) described the profiles of different degrees of burying. For ðiÞ every function of the profile nðxÞ, with the help of modal Eqs. (1) and (2), a set of effective indexes fNeff g ðiÞ was determined. Basing on each of the sets determined in this way fNeff g, using the described algorithm, the
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Table 1 Counted values of criterionðzÞ Criterion(z) z
1
2
3
4
5
7
1 2 3 4 5 6 7 8 9 10 11 12
0:03128 0.18847 1.76011 2.52622 0.46032 0.28631 3.37928 2.05148 4.25871 3.78945 5.57618 –
1.85071 0:06996 0.20201 0.41448 0.85455 0.44762 3.61139 0.72275 1.52374 3.91754 6.24971 –
0.18315 0.09226 0:04712 0.65232 0.31433 3.52259 0.76537 2.61585 2.47135 5.91421 4.28463 –
0.04658 0.10666 0.11849 0:04583 0.23589 0.53661 0.95841 1.25137 1.59831 2.69728 2.78194 –
0.10808 0.12108 0.39215 0.73833 0:07493 3.20917 0.31550 0.42054 4.35814 3.87994 4.55741 –
0.12325 0.14614 0.18586 0.29226 0.45655 0.39013 0:10334 0.14252 0.36422 0.25129 0.21699 0.22405
function of a respective input profile was reconstructed. Counted values of criterionðzÞ indicate the quantity of buried modes is collected in Table 1. Figs. 6 and 7 present functions simulating buried profiles having a suitable number of buried modes (value z) as well as the obtained reconstruction of refractive profiles. For every profile the course of changes of criterionðzÞ was presented with the smallest value being indicated.
6. Production of buried waveguides For the measurements of refractive profiles, the structures of planar buried waveguides were produced, using the ion-exchange technique. As substrate, soda-lime glass of refractive index n ¼ 1:5111 for k ¼ 677 nm was used. The technological processes were realized in four samples. With respect to all samples, preliminary diffusion of Agþ ions from molten pure salts AgNO3 was realized. The temperature of the process was Td ¼ 303 C, and its duration td ¼ 10 h. For planar waveguides produced in such a way, propagation constants of modes for k ¼ 677 nm were measured, and refractive index profile with the use of modal equation for monotonic profile was determined. Then, for individual waveguides, the processes of burying were carried out through the diffusion of Naþ ions in NaNO3 bath of the temperature Tb ¼ 320 C. The duration times of burying processes were, respectively, 1, 2, 4 and 8 h. A part of the waveguide samples produced in that way were prepared for comparative interference investigations (see Section 8.2).
7. Measurements of propagation constants of modes of buried waveguides The measurements of propagation constants of modes in planar waveguides having a monotonic profile are carried out using prism couplers. Refractive index of the prism has here a higher value than the maximum refractive index of the waveguideÕs. The light wave propagating inside the prism can then penetrate the area of waveguide, and when the components of wave vectors in the prism and in the waveguide are coordinated along the propagation direction – it can stimulate individual modes [17,18]. The penetration depth of the wave field in the prism into the waveguide area will be the higher the lower is the difference between refrective indexes of the prism and waveguide [19]. The coupling obtained in this way depends also on the width of the gap at the contact area between the prism and glass. In the case of buried
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Fig. 6. Refractive profiles (continuous line) and their reconstructions (intermittent lines) on the basis of the set of effective indexes ðiÞ fNeff g, as well as courses of the changes of the value of criterionðzÞ. Quantity of buried modes: (a) z ¼ 1, (b) z ¼ 2, (c) z ¼ 3.
waveguides, the maximum of refractive index distribution is shifted far into glass base, and hence the distributions of modal fields take place also further from the glass surface. In that case, if we want to effect the excitation of modes in the waveguide with the help of a prism coupler, its refractive index should have the value a little higher than the maximum refractive index of the waveguide. It requires a suitable selection of coupling prisms [20]. Fig. 8 presents values of the refractive index of the applied prism couplers (glass of Schott firm) for wavelength k ¼ 677 nm against the background of a typical refractive index profile produced in soda-lime glass (ng677 ¼ 1:5111), with ion-exchange technique Agþ () Naþ . Table 2 presents the parameters of prism couplers applied in the measurements. Table 3 contains the collected results involving the measurements of effective indices of waveguides produced in burying processes in times: 1, 2, 4 and 8 h. The left-hand part of the Table 3 contains the measurement results for buried waveguide in time 4 h, accomplished with the use of different prism couplers. We can see here the influence of the refractive index of the prism on the possibility to stimulate a number of modes in the buried waveguide. The right-hand part of the Table 3 contains the measurement results of effective indexes for all buried waveguides carried out with the help of a prism having the lowest refractive index.
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Fig. 7. Refractive profiles (continuous line) and their reconstructions (intermittent lines) on the basis of the set of effective indexes ðiÞ fNeff g, as well as courses of changes of the value of criterionðzÞ. Quantity of buried modes: (a) z ¼ 4, (b) z ¼ 5, (c) z ¼ 7.
8. Determination of the refractive index profiles of buried waveguides 8.1. Reconstruction algorithm of refractive index profile To determine the refractive index profiles of the produced buried waveguides, the algorithm described in Section 4 was applied. The values of effective indexes of modes of buried structures gathered in Table 3 were applied here as input data. Fig. 9 presents the obtained reconstructions of buried profiles as well the refractive index profile of the waveguide after preliminary diffusion obtained using the reconstruction procedure for monotonic profiles. 8.2. Interference method The interference method described in work [7] was used to determine comparative profiles of the produced waveguide. For that purpose, three samples were prepared from the produced waveguides (buried in
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Fig. 8. Refractive indexes of applied prism couplers for k ¼ 677 nm, on the background of refractive index profile of planar waveguide. Table 2 Parameters of prism couplers Prism couplers Glass (Schott) Prism angle d(deg) Uncertainty of prism angle Dd (deg) n677
SF-14 49.9821 0.0002 1.7516
SK-10 65.0117 0.0011 1.6186
SK-2 65.0222 0.0011 1.6033
SK-12 70.0787 0.0008 1.5794
Table 3 Measurement results of effective indexes of buried waveguides Value of effective indexes Prism
SF-14
Number of mode
Soda-lime glass diffusion AgNO3 : T ¼ 303 C; tdiff ¼ 10 h þ diffusion ðburyingÞ NaNO3 : T ¼ 320 C
18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
SK-10
SK-2
SK-12
tb ¼ 4 h
1.5332 1.5319 1.5306 1.5290 1.5277 1.5262 1.5247 1.5233 1.5219 1.5204 1.5189 1.5174 1.5160 1.5145 1.5132 1.5118
1.5338 1.5324 1.5311 1.5297 1.5283 1.5269 1.5255 1.5241 1.5226 1.5211 1.5197 1.5182 1.5167 1.5153 1.5138 1.5125 1.5113
1.5345 1.5332 1.5319 1.5305 1.5291 1.5277 1.5263 1.5248 1.5233 1.5218 1.5204 1.5189 1.5174 1.5160 1.5147 1.5134 1.5125
1.5360 1.5347 1.5334 1.5321 1.5307 1.5293 1.5278 1.5264 1.5250 1.5235 1.5220 1.5205 1.5190 1.5175 1.5161 1.5147 1.5134 1.5123
SK-12
tb ¼ 1 h
tb ¼ 2 h
tb ¼ 4 h
tb ¼ 8 h
1.5451 1.5438 1.5419 1.5399 1.5379 1.5358 1.5337 1.5316 1.5294 1.5273 1.5251 1.5228 1.5207 1.5185 1.5164 1.5143 1.5125 1.5116
1.5411 1.5399 1.5384 1.5368 1.5351 1.5334 1.5317 1.5300 1.5282 1.5264 1.5245 1.5227 1.5209 1.5191 1.5173 1.5155 1.5138 1.5124
1.5360 1.5347 1.5334 1.5321 1.5307 1.5293 1.5278 1.5264 1.5250 1.5235 1.5220 1.5205 1.5190 1.5175 1.5161 1.5147 1.5134 1.5123
1.5309 1.5301 1.5290 1.5280 1.5269 1.5258 1.5247 1.5236 1.5225 1.5214 1.5202 1.5191 1.5180 1.5169 1.5158 1.5147 1.5137 1.5128
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Fig. 9. Refractive index profiles of buried waveguides reconstructed on the basis of the value of effective indices from Table 3.
times: 1, 4 and 8 h, respectively) by cutting them out and grinding them suitably to ensure a definite thickness. These thickness values were subjected to interference measurements, and for particular samples the following results were obtained, respectively, 77 lm (for sample buried in time 1 h), 91 lm (for sample buried in time 4 h), and 143 lm (for sample buried in time 8 h), with uncertainty of measure of about 1 lm. Polarization microinterferometer BIOLAR-PI was applied for interferometric investigations. Fringe interferograms of the studied samples were registered with a CCD camera. In order to make it possible to compare the refractive index profiles obtained with both methods, the same wavelength was applied (k ¼ 677 nm) with the registration of interference patterns as with the measurements of propagation constants. Fig. 10 presents fringe interferograms for the samples of waveguides buried in times 1, 4 and 8 h, as well profiles of refractive index changes dnðxÞ reconstructed on the basis of the shape of interference fringes [7]. Broken lines were used to mark measure uncertainties resulting from the measurement error involving the thickness of samples. The profile of absolute value of refractive index in this measurement method is nðxÞ ¼ dnðxÞ þ ng . 9. Comparison of reconstructed refractive index profiles Fig. 11 presents refractive index profiles obtained using both measurement methods. The obtained refractive profiles display a certain degree of similarity, yet the differences are larger than the measure uncertainty. When comparing the results, one should remember that the applied measurement methods are very different from each other. The interference method, however, can bring about a relatively high measurement error resulting from the uncertainty to qualify the run of interference fringes. In the microinterferometer type BIOLAR-PI applied for the measurements, two-beam interference is used. In the case of interference of that type, wide interference fringes are created. Yet the form of interference fringe contains direct information about the refractive profile [7]. The method basing on the described algorithm makes use, however, of the values of effective indexes, whereof measurements are performed with high accuracy with the help of a goniometer. With the application of prism couplers specified in Table 2, taking into account the uncertainty of measured synchronic angles at the level of 0.05, the uncertainty determination of effective index is DNeff 0:0003. But in this case we obtain from the measurements only a set of exact values of the function fNi g without the information involving the respective values of arguments fxi g. The error involving the profile shape results here from the calculations carried out with the help of the proposed algorithm values of these arguments (turning points). Although the interference method requires time-consuming preparations of sample waveguide, it provides essential information on the range of refractive index profile. On the basis of Eqs. (1) and (2) it is possible, as in Section 5, to determine the values
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Fig. 10. Registered fringe interferograms of buried waveguides and reconstructed profiles of refractive index on the basis of the interference fringe course, for waveguides buried in times: (a) tb ¼ 1 h, (b) tb ¼ 4 h, (c) tb ¼ 8 h. Intermittent lines were marked measure uncertainties resulting from a mistake of the measurement thickness of samples [4].
Fig. 11. Comparison of refractive index profiles obtained with the help of the described algorithm with that obtained by the interference method, for waveguides buried in times: (a) tb ¼ 1 h, (b) tb ¼ 4 h, (c) tb ¼ 8 h.
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Table 4 Values of effective indices defined with the prism coupler in comparison with those counted on the basis of Eqs. (1) and (2) for refractive profiles measured by the interference method Order of mode
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Value of effective indexes tb ¼ 4 h
tb ¼ 1 h
tb ¼ 8 h
Interference method
Prism coupler
Interference method
Prism coupler
Interference method
Prism coupler
1.5429 1.5419 1.5405 1.5390 1.5372 1.5354 1.5336 1.5316 1.5298 1.5279 1.5261 1.5243 1.5224 1.5204 1.5185 1.5165 1.5146 1.5128 1.5113 – – –
1.5451 1.5438 1.5419 1.5399 1.5379 1.5358 1.5337 1.5316 1.5294 1.5273 1.5251 1.5228 1.5207 1.5185 1.5164 1.5143 1.5125 1.5116 – – – –
1.5332 1.5323 1.5314 1.5304 1.5295 1.5286 1.5275 1.5264 1.5253 1.5242 1.5231 1.5218 1.5205 1.5192 1.5179 1.5166 1.5154 1.5142 1.5129 1.5117 – –
1.5360 1.5347 1.5334 1.5321 1.5307 1.5293 1.5278 1.5264 1.5250 1.5235 1.5220 1.5205 1.5190 1.5175 1.5161 1.5147 1.5134 1.5123 – – – –
1.5293 1.5286 1.5279 1.5271 1.5264 1.5256 1.5248 1.5239 1.5231 1.5223 1.5213 1.5204 1.5195 1.5185 1.5176 1.5166 1.5156 1.5146 1.5136 1.5127 1.5118 1.5112
1.5309 1.5301 1.5290 1.5280 1.5269 1.5258 1.5247 1.5236 1.5225 1.5214 1.5202 1.5191 1.5180 1.5169 1.5158 1.5147 1.5137 1.5128 – – – –
of effective indices for the refractive profiles obtained with the interference method. The compared values of the of refractive profiles obtained in this way by means of a prism coupler are presented in Table 4.
10. Conclusions The proposed in the present work algorithm of the reconstruction of refractive index profiles of planar buried waveguides, basing on the measurements of propagation constants of modes, shows good reproduction effectiveness of the profiles of testing functions (see Section 5). What is the limitation involving the applicability of the algorithm described above is the number of waveguide modes – as in the case of the reconstruction of monotonic profiles – resulting from the depth of profile nðxÞ. In the refractive index profiles applied to test the algorithm (Section 5), the total number of modes was higher than 10 in each case. For the reconstruction of refractive index profiles of the real structures of buried waveguides, a complete set of propagation constants of modes should be obtained. This necessitates the application of suitable prism couplers (see Table 3, Section 7). The application of smoothness criterion in the proposed algorithm (see Fig. 3) is not the only possible criterion which can be used to qualify the obtained profiles. The interference method applied as a supplement allows one to verify the obtained refractive index profiles. Despite the time-consuming preparation of the sample, this method provides essential information about the depth of the profile. As it was already mentioned in the Introduction, the credibility of the obtained results is most essential in order to determine the refractive index profiles of buried waveguides. Therefore, it seems advisable, in the cases where it is possible, to correlate the results obtained with both methods.
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Acknowledgements This work was performed within the research project KBN 8 T11B 05 218.
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