(PDF) Two-Mode Waveguide Characterization by Intensity Measurements from Exit Face Images

Two-Mode Waveguide Characterization by Intensity Measurements from Exit Face ImagesVolume 4, Number 1, February 2012

Xesús Prieto-BlancoJesús Liñares

Two-Mode Waveguide Characterizationby Intensity Measurements from

Exit Face ImagesXesus Prieto-Blanco and Jesus Linares

Departamento de Fısica Aplicada, Area de Optica, Escola Universitaria de Optica e Optometrıa,Campus Vida, Universidade de Santiago de Compostela, 15782 Galicia, Spain

DOI: 10.1109/JPHOT.2011.21775161943-0655/$26.00 �2011 IEEE

Manuscript received October 11, 2011; revised November 15, 2011; accepted November 18, 2011.Date of publication December 6, 2011; date of current version December 30, 2011. Correspondingauthor: X. Prieto-Blanco (e-mail: [emailprotected]).

Abstract: A method that characterizes two-mode waveguides whose modes cannot beselectively excited (such as buried waveguides) is presented and demonstrated. The theo-retical results are presented for N modes, although for the sake of simplicity, only the two-mode case is developed. The values of the optical mode fields are recovered from severalimages of the waveguide exit face, where both modes interfere with different relativeintensity in each image. From these mode fields, both the squared index distribution, exceptin an additive constant, and the effective index difference can be obtained by inverting theHelmholtz equation. As in the case of the standard monomode intensity method, the relativevalues of effective indices and index distributions become absolute if the modes are re-trieved in a point of known index, for instance, in the substrate. The method can also beapplied to multimode guides if only two modes are excited. In fact, an inexpensive setup isproposed to excite the first two modes of a multimode buried waveguide. This waveguidewas fabricated by ion exchange in glass and buried by electromigration. The shape of thesquared refractive index recovered by the proposed method agrees with that reported in theliterature.

Index Terms: Visible lasers, waveguide devices, waveguide characterization.

1. IntroductionOne of the main tasks in integrated optics is the characterization of waveguides. Optical characteri-zation spans a wide range of measurements of waveguide characteristics, such as, for example,refractive index profile, mode effective indices, modal intensities, modal losses, modal dispersion,and so on. Many different characterization methods implemented by diverse techniques have beenproposed ever since the beginnings of integrated optics, and they have set down the foundations ofmany other techniques. Thus, interferometry techniques are among the most used [1]–[6] to mea-sure mainly the refractive index profiles of integrated waveguides, however they used to be verylaborious and destructive. Another well-known method of slab waveguide characterization is theM-line one, based on the prism-guide coupling technique, to measure the effective indices, and,accordingly, to model the refractive index profile and the optical modes [7], [8]. This method is veryuseful and powerful when there are several guided modes; however it requires that the integratedguides be found on the surface of the substrate, otherwise (buried guides) the prism-guide couplingtechnique cannot be applied because the coupling between the guided modes and the prismradiation modes is negligible. An adaptation was also proposed for the measurement of effectiveindices of surface channel waveguides [9], but the recovery of the 2-D index profile is not obvious.

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IEEE Photonics Journal Two-Mode Waveguide Characterization

Another interesting technique of characterization is the measurement of the modal intensity,which allows us to recover the index distribution by inverting the Helmholtz equation. The mode isusually obtained in far-field regime, either from the diffracted modal fields [10] or by imaging the exitface of the guide on an image sensor [11]–[13], although the near field scanning was also used [14].This technique is particularly useful for monomode guides because their mode can be excited bythe end-fire coupling method. For multimode guides we should excite the modes in a selective way,as for example the modes of a surface (nonburied) slab guide which are excited by prism-guidecoupling. In other cases, such as optical fibers or buried (channel or slab) guides with two or moremodes, the end-fire coupling technique also fails because a selective modal excitation is notpossible. In this paper, we focus our attention on this last case, that is, on the buried guides withmore than one mode.

We must underline that although the transmission and/or interconnection in integrated optics isusually made in a monomode regime, a recent interest is arising in few-mode fibers to improve thedata throughput of monomode fibers [15]. Furthermore, the most of the integrated devices arebased on components working at a regime of a few modes. Basic examples of such componentsare the couplers of two monomode guides or even couplers of one guide with two modes. If thesecomponents were buried then the most direct and nondestructive technique of modal charac-terization would be the direct measurement of simultaneously excited modal fields. In particular, wewill analyze a slab buried guide obtained by electric field-assisted ion exchange in glass by usingimage intensity measurements at the exit face. Since the modes cannot be excited in a selectiveway, then we will have in each point of the image the interference of the excited modes in the guide.Our method is based on the capture of several images of the waveguide exit plane, each one for adifferent modal excitation. This allows us to determine the contribution of each mode at each imagepoint and therefore to obtain the values of the modal fields. From these modal fields, the differencebetween their effective indices and the index profile can also be obtained in the same way as themonomode intensity method. In that sense, our method can be considered as a generalization ofthe former one. The theoretical results are presented for the case of N modes, although we mustunderline that the method takes a simple form in the two-mode case N ¼ 2 which is developed.Likewise, the results of the two-mode case can be used in multimode guides under particular modalexcitation conditions, for instance, a modified end-fire coupling method, which primarily excites thefirst two modes, is proposed.

The plan of this paper is as follows: In Section 2, we present the main theoretical results for Nmodes concerning the method of characterization based on the multimode intensity measurements,as well as the main steps of the algorithm to be used in the data processing of the two-mode case.Section 3.1 is devoted to describe the fabrication of a slab buried waveguide by a field-assisted ion-exchange process while the optical setup to perform the intensity measurements is presented inSection 3.2. Section 4 deals with the image intensity data processing when only two modes areexcited, and accordingly, the retrieval of the modal amplitude, the relative modal phase, therefractive index profile, and the effective index difference are discussed. Conclusions are given inSection 5.

2. Coupling TheoryIn this section, we present the main theoretical results to be used throughout the work. First of all,we deal with the modal coupling problem of an input optical field Ei into an integrated N-modewaveguide in order to derive a consistent multimode optical intensity function. Next, we present ageneral method to recover the modal amplitude values (characterization) starting from intensitymeasurements. Finally, we apply this method to the two-mode case, leading to explicit and, aboveall, manageable results.

2.1. Multimode IntensityLet us consider a multimode optical waveguide represented by a refractive index profile nðx ; yÞ

and supporting monochromatic guided modes E�ðx ; y ; z; tÞ, H�ðx ; y ; z; tÞ (in which, for the sake of

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simplicity, we choose only a modal subindex �) propagating along z. These modes can be repre-sented by the following vectorial complex expressions:

E�ðx ; y ; z; tÞ ¼ E�ðx ; y ; zÞ e�i!t ¼ E�ðx ; yÞ ei½��z�!t � (1)

H�ðx ; y ; z; tÞ ¼H�ðx ; y ; zÞ e�i!t ¼ H�ðx ; yÞ ei½��z�!t � (2)

where fE�ðx ; yÞ;H�ðx ; yÞg and �� are the modal complex vectorial amplitude and the propagationconstant of the �-mode. It must be noted that the transverse and longitudinal modal components arereal and imaginary, respectively.

On the other hand, the quasi-complete orthonormalization condition of two modes: � and �0, on across section of an optical guide, is given by the following expression [16], [17]:

2 sgn�

ZZe�0 ^ h?��

uz dx dy ¼ �j�j; j�0 j (3)

where �j�j; j�0j is the Kronecker delta, the function sgn� is defined as þ1 if � 9 0 (forward modes) andas �1 if � G 0 (backward modes), and e�, h� are the normalized modes, that is, e� ¼ E�=kE�k andh� ¼ H�=kH�k, with [17]

kE�k � kH�k ¼ 2 sgn�

ZZE� ^ H?

�

� �uz dx dy

� �1=2

(4)

that is, the modal norm. Note that the orthonormalization condition (3) is determined only by thetransverse field components of the guided modes and it is a quasi-complete orthonormalizationcondition since for cases such as � ¼ ��0, (3) is not equal to zero; however, it is an exact expres-sion for copropagating (forward) modes which are considered throughout this work.

It is well known that any optical field can be expressed as a linear superposition of normalizedmodes, thus the spatial complex amplitude fE;Hg of a forward field at any plane z of the guide canbe written by means of forward modes as follows:

Eðx ; y ; zÞ ¼X�

a� e�ðx ; yÞ ei��z Hðx ; y ; zÞ ¼X�

a� h�ðx ; yÞ ei��z (5)

where

a� ¼ 2ZZ

Eðx ; y ; 0Þ ^ h?��

uz dx dy : (6)

Thus, starting from (5) and taking into account the modal normalization condition given by (3), thefollowing expression for the total power P of the optical field E is derived:

P ¼ 2ZZ

E ^ H?f guz dx dy ¼X�

ja�j2 þ Pr (7)

with ja�j2 the power coupled to each guided mode (guided mode power) and where, for the sake ofconsistency, we have added the power coupled to radiation modes Pr . It would even be necessaryto add other powers such as reflection power, absorption power, and so on; therefore, Pr can alsobe understood as the contribution of all of them. In Section 2.2, we will take into account thesepowers Pr in order to define the power coupled to each mode in a compatible way with theexperimental intensity measurements.

On the other hand, we can define the intensity I of the optical field E as a function of theamplitudes of the optical modes, that is, the multimode intensity

Iðx ; y ; zÞ ¼ E ^ H?f guz ¼X�;�0

a� e�ðx ; yÞ ei��z ^ a?�0 h?�0 ðx ; yÞ ei��0z

( )uz : (8)

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This equation can be simplified by means of very usual assumptions on modal fields. Thus, in thecase of modes of slab guides, that is, with refractive index profile nðxÞ, we can distinguishbetween TE and TM modes because the TE modes fulfill the relationship: ��Ey� ¼ �!�oHx� andEx� ¼ 0 and the TM modes fulfill the relationship: ��Hy� ¼ �!�oEx� and Ey� ¼ 0. These relation-ships are quasi-exact for most of the channel waveguides and in particular for those whose widthis larger than their height, and therefore these modes are called QTE (quasi-TE) and QTM (quasi-TM)modes.

Note that the expansion coefficients a� correspond to the case of an optical field E inside theguide, however the practical problem always involves the coupling of an input optical field E i from,for instance, the vacuum. Therefore, we must slightly change the expansion coefficients, that is,within the paraxial approximation and in a good approximation for an input optical field with planephase, that is, E i a real function at z ¼ 0, we can write

c� � 2ZZ

t� E iðx ; y ; 0Þ ^ h?��

uz dx dy (9)

where t� ¼ 2N�=ð1þ N�Þ are the transmission coefficients for each mode � which has an effectiveindex N�. Thus, taking into account the last relationship and that in most integrated waveguidesthe effective indices fulfill the relation N� � N 0� � N1, then (8) can be approximately rewritten asfollows:

Iðx ; y ; zÞ � N1

c�o

X�

jc�j2e2� ðx ; yÞ þ

X� 6¼�0

c�c�0ei��0 e�ðx ; yÞe�0 ðx ; yÞ( )

(10)

with ��0 ¼ ð�� 0 Þz, e� � ey� and, in a good approximation, for calculations concerning theintensity, we have chosen N� � N1, 8�. Note that all coefficients c� are real because t�, hy� , andE iðx ; y ; 0Þ are also real; therefore, the only complex numbers are the phases ei��0 .

2.2. Modal Amplitudes From Intensity MeasurementsBy taking into account the results of the above subsection, we describe a general procedure to

obtain the modal amplitudes starting from measurements of multimode intensity at a plane z of amultimode guide. Let us consider a slab guide, although the results can be extended to channelguides in a straightforward way. Let us consider N modes in a guide; then we perform a first set ofmeasurements of intensity at N points at the exit plane of the waveguide excited by an input realfield E i ðx ; y ;0Þ, that is, ðIð1Þe ð1Þ; . . . ; Ið1Þe ðNÞÞ, with subindex e indicating the experimental values;next, we modify the input field, for instance, by shifting it or by changing its size (by image system)but keeping the phase of the input field constant; thus, we obtain a new set of measurement, andwe continue until reaching the M -set, that is, ðIðMÞe ð1Þ; . . . ; IðMÞe ðNÞÞ.

On the other hand, from a theoretical point of view, we have N equations for the modal couplingintensity of the points j ¼ 1; . . . ;N , that is

IðjÞ � N1

c�o

X�

jc�j2e2� ðjÞ þ

X� 6¼�0

c�c�0ei��0e�ðjÞe�0 ðjÞ( )

: (11)

Note that we have N equations and N � 1 parameters c� because of the condition given by (7)applied to the modal coupling coefficients c�, that is, because of the power conservation. Never-theless, as commented above, there is coupling to the radiation modes, the reflective modes and soon, accordingly we must modify slightly the condition (7) to be compatible with both experimentalmeasurements and with (11), that is, we define new coefficients ~c� fulfilling

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2 u �

sin2 utan2 �12

s0@

vuuut ; e�v ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2 v �

sin2 vtan2 �12

s0@

vuuut :

We must stress that this procedure can also be applied to guides with more than two modes ifonly two modes are excited simultaneously. For instance, in the case of a three-mode symmetric

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guide, we could excite the first and the third mode by using symmetric input fields of variable widthor shape in order to achieve the M measurements.

3. Experimental Implementation

3.1. Slab Buried Waveguide FabricationA standard microscope slide (1-mm-thick piece of soda-lime glass) was immersed in a mixture of

molten salts. The selected salt composition was: 47.5 mol% NaNO3, 47.5 mol% KNO3, and 5 mol%AgNO3, since it presents a melting point of 220 �C [20], which allows us to use a diffusion tem-perature as low as 240 �C. Under these ion-exchange conditions only two modes are obtained after24 min of diffusion (see Table 1). As this waveguide is a surface slab one, its effective indices weremeasured by prism coupling method with a Model 2010 prism coupler from Metricon. A simplesodium/silver salt would require too short diffusion time for the same penetration depth of silvercations. We must note that the Kþ cations did not influence the waveguide formation because theyhave a much lower tendency to enter into the glass than Agþ cations. A second diffusion step wascarried out to bury this waveguide some micrometers under the glass surface. The sample wasplaced in a quartz support that allowed us to put in contact each sample side with its respectivemolten salt at a different electrical potential. In order to remove silver cations from the surface,a 50 mol% NaNO3 : 50 mol% KNO3 salt composition at 272 �Cwas used as anode in contact with theguiding surface. However, a 50 mol% AgNO3 : 50 mol% KNO3 salt mixture was used as cathode.This silver/potassium composition is nearly eutectic, with the melting point lower than 160 �C [20]; agood electric contact is achieved at the cathode when this salt melts. Initially, the sample wasimmersed for 10 min without current flow; next, a potential difference of 155 V was applied for30 min to perform a field-assisted diffusion. However, it is interesting to underline that, since theanode and cathode salts have different composition, an effective potential actually occurs, whichcan be different from the applied one by some Volt [21]. Once the field-assisted diffusion processwas finished, a 7.02-mm-long piece of sample was prepared by cutting and polishing two facesperpendicular to the waveguide and parallel to each other.

3.2. Optical SetupThe optical setup is prepared for illuminating one polished face of the sample (input face defined

by z ¼ 0) with a cylindrical focused beam whose focal line is parallel to the waveguide (Y axis). Inthis way, only two modes (as shown later) of the waveguide are excited, and they propagate to theother face (end face). Moreover, the setup contains a microscope objective which forms an image ofthe end face of the waveguide on a CCD camera to obtain the intensity profile resulting from theinterference between both modes.

As the input beam profile is a Gaussian one along the confinement direction of the waveguide(vertical or X axis), and it is collimated along the perpendicular direction, the phase is uniform at thewaist plane, where the input face is. It ensures that both modes have the same phase at the inputface (except in a � relative phase), irrespective of the height of the beam center or the size of thewaist, and therefore, their phase difference at the end face takes always the same value. A variationof the coupling coefficient to each mode is needed in order to apply the algorithm presented inSection 2. A relative vertical shift between the illumination system and the rest of the optical system(sample, imaging objective and CCD) could be an effective way to achieve such variations, but thisrequires a big stage with submicron resolution. A more cheap and convenient solution that performsthe same task is sketched in Fig. 1.

TABLE 1

Effective indices after first ion-exchange

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First, let us see Fig. 1(a), which shows a vertical section of the system. A He-Ne laser beam isrefracted through a low-power positive lens (0.5 diopter) that can be moved up and down to redirectthe beam with a negligible change in its original divergence. The following lens is a negativecylindrical one with its power meridian along the horizontal axis; therefore, it does not affect the raysof the vertical section. Next, a high-power positive lens (10 diopter) focuses the rays contained inthe vertical plane and projects an image of the low-power lens on the back focal plane of a finite-conjugated (DIN standard) 10 microscope objective. The distances between lenses were ar-ranged in such a way that this image was 160 mm apart from the beam focus. Thus, the objectivefocuses a high-quality beam again. To place just the input face on the beam waist, the back-reflected beam from this face must maintain the same transversal size as the incident beam alongthe illumination system. Besides, to ensure that the beam initially falls on the waveguide, its outputface is illuminated with a retractable white LED; therefore, both the laser back-reflection (suitablyattenuated) and the input face are simultaneously seen through the 10 microscope objectiveusing a cube beam-splitter placed between the high power positive lens and an eyepiece (notshown in Fig. 1). This illumination system allows us a great control of the beam height since amacroscopical rise (1 mm) of only one element (the first low-power lens) generates a microscopicalbeam drop at the input face (5 �m). Note that the central ray of this beam is kept parallel to theoptical axis; then, its phase is constant along the input face.

If the beam waist at the input face is fitted to the mode size, the beam matches the fundamentalmode very well for a particular beam height. In this way, the best coupling coefficient to the firstmode is achieved, while the excitation of the second and third mode becomes negligible. A smallbeam shift from this position increases mainly the coupling coefficient to the second mode (at theexpense of the fundamental one) but retains a negligible third mode excitation under moderateshifts. Consequently, only two modes are involved within an interval. As the required waist is widerthan the objective resolution, the Gaussian beam must illuminate only a portion of the objectiveaperture stop. The appropriate expansion of the beam was obtained by taking advantage of its owndivergence under a particular propagation distance from the laser to the first lens.

On the other hand, in Fig. 1(b), a projection on the horizontal plane of the beam as it propagatesthrough the system is shown. The main difference with respect to the vertical section is the pre-sence of the power meridian of the cylindrical lens whose axial position was fitted to obtain ahorizontally collimated beam at the waveguide input face.

In short, the illumination system generates a high-quality cylindrical beam with its focal line on theinput face of the waveguide and parallel to it. Moreover, the beam can be finely moved along theperpendicular direction to moderately change the coupling efficiencies to the first and second mode.

Fig. 1. Simplified diagram of the optical setup.

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The end face was imaged with a standard 40 microscope objective on to an analog PulnixTM-765 CCD camera, and the signal was digitized to a frame of 767 575 pixels and an 8-bit depthby an acquisition board. For each particular beam height, the intensity was adjusted by a set ofattenuators, both to achieve a good illumination and to prevent saturation. Then, 256 images werecaptured and averaged to improve the signal-to-noise ratio (SNR). Next, the beam was verticallyshifted 0.5 �m, the intensity was adjusted again, a new averaged image was captured, and so on.Moreover, an averaged final image was taken with the laser obstructed; this final image wassubtracted to the previous ones to remove both the background and the camera dark current. In thisway, a series of 37 irradiance distributions at the end face was obtained. Some representatives ofsuch distributions are shown in Fig. 2.

4. Image Processing and Results

4.1. Mode AmplitudesThe proprietary software MATLAB was used to apply the algorithm of Section 2 to these images,

although the free software Octave could have also been used. The first task is the selection of theregion of interest in the images which contains 100 60 pixels. Ideally, as the intensity profileshould not depend on the y coordinate, each image could have been collapsed to a function byaveraging it along this dimension, resulting in a new SNR improvement. Actually, a smooth variationcan be seen along the y dimension, probably due to some lack of uniformity in the waveguide (seeFig. 2). For this reason, groups of only five contiguous columns were collapsed into a new columnby applying a median filter. Thus, most defective pixels caused by dust in the cover plate of theCCD camera are discarded. Note that the discrepancy between columns gives us an idea of theaccuracy achieved, since each collapsed column will be processed independently. Next, the modeintensity is normalized along the x direction according to (12); that is, every column is divided by thesum of the values of its elements. It results in a series of compacted images of 20 60 pixels each.The intensity of a given pixel oscillates along the series. According to the general theory, an ellipsearc must be obtained when the intensity of two pixels is plotted one versus another (Fig. 3). Notethat this plot contains as many points as images are in the series. However, only 16 images in thecenter of the series have suitable values to be fitted to ellipses. Modes other than the first andsecond were noticeably excited in the rest of images that were discarded. From the five ellipseparameters, the absolute value of each mode amplitude at each pair of points, together with thesquared tangent of the relative phase, are calculated. This procedure has two inherent shortcom-ings: the sign of the second mode is lost and the lowest amplitude is always assigned to the firstmode, while the highest one is given to the second mode. A posteriori, each amplitude value mustbe assigned to the right mode and the sign of the second mode must be restored. When the secondmode at the point u ðvÞ is negative, u ðv Þ must also be considered negative in (23). The depths

Fig. 2. Some irradiance distributions at a selected region of the end face of the waveguide. The imagesin (a) and (f) were discarded for processing. In these images, the X axis is upwards, while the Y axis ishorizontal.

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(x values) where both amplitudes meet, or where the second mode cancels, remain the same forevery y value, which is an indicator of the consistency of this method.

As the intensities at any pair of points can be fitted to ellipses, all combinations within a givencolumn are checked. Those pairs of points that fit to ellipses which are not fully contained within thefirst quadrant are discarded, since they provide negative intensities. The modal amplitudes and thephase resulting from the resting pairs are averaged. Thus, 20 profiles (one per column) of the firstand second mode are obtainedVf�V. From their dispersion, we can estimate the uncertainty of theexperimental modal amplitudes of both modesVf� ðxÞVwhich are shown in Fig. 4. The absolutevalue of the relative phase between both modes is 1.06 0.17. Likewise, we can check that themodes are approximately orthonormal (see Table 2). However, the point of the second modeclosest to its node seems slightly deviated for most of profiles, that is, it contains a bias; therefore, itwas removed from subsequent calculations.

Fig. 4. Retrieved the first and second mode (a.u.) and j�12 � p�j in radians at the output face, where pan unknown integer. Each graph is an average along the slab guide; it is represented by two lines thatindicate one standard deviation below and above the mean.

Fig. 3. Experimental normalized intensity of some pairs of points at the waveguide exit face. Theintensity of one point ðIv Þ is graphed against the other ðIuÞ for several coupling heights of the inputbeam. The intensities from the same point pair were joined to show their elliptical-like form.

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The left tails of both modes are slightly higher than expected, probably due to a residual excitationof the third mode, which achieves large values in this region. The existence of the third mode isclear in Fig. 2(f), where two intensity minima can be seen.

4.2. Squared Refractive Index ProfileAs the experimental modal amplitudes are known, we can retrieve the squared index profile,

except an additive constant, starting from the scalar Helmholtz equation, that is

n2ðxÞ � N2� ¼ �

1k20e�

d2e�dx2 ; � ¼ 1; 2

by computing

n2ðxÞ � N2� � m�ðxÞ � �

f�ðx � h; yÞ þ f�ðx þ h; yÞ � 2f�ðx ; yÞf�ðx ; yÞh2k2

(24)

where k0 is the wavenumber, f�ðx ; yÞ are the experimental modal amplitudes, h is the step to obtainthe second derivative, and the symbol h iy means an average along y direction. This calculationmust be made carefully since the experimental modal amplitudes contain noticeable noise, that is

f�ðx ; yÞ ¼ e�ðxÞ þO f� ðxÞð Þ

where 2f� ðxÞ are the variances of f�ðx ; yÞ. On the one hand, since the second derivative [denoted asf 00� ðx ; yÞ] is computed by a typical finite difference scheme, two sources of uncertainty arise: onefrom the noise and another from the discretization; specifically

f 00� ðx ; yÞ �f�ðx � h; yÞ þ f�ðx þ h; yÞ � 2f�ðx ; yÞ

h2 ¼ e�ðx � hÞ þ e�ðx þ hÞ � 2e�ðxÞh2 þO

4f� ðxÞh2

� �

¼ d2e�dx2 þO h2 d

4e�dx4

� �þO

4f� ðxÞh2

� �:

The last line of this equation shows that very small or very high value of h gives rise to high errors.In our case, a good balance was achieved when h is equal to four pixels, although a three-pixel stepwas used sometimes to prevent the use of the discarded point. Moreover, a linear variation of thesecond derivative along x of f� was assumed at both the end points and its neighbors.

On the other hand, low values of f� have high relative uncertainty which leads to remarkableerrors in specific regions when computing n2ðxÞ � N2

� by (24). In other words, if the variance ofm�ðxÞ is calculated as

2m�ðxÞ ¼ ny

ny � 1f 00� ðx ; yÞf�ðx ; yÞk2

� �2* +

�m2� ðxÞ

where ny ¼ 20 is the number of compacted columns, the standard deviations m�ðxÞ are noticeably

dependent on x , as can be seen in Fig 5(a). Fortunately, regions of high values of 2m1and 2m2

onlyoverlap at the far tails since the mode orthogonality constrains the node of the second mode nearthe maximum of the first mode, and conversely, the left tail of the first mode starts where the secondmode still takes high values. This fact suggests to combine information from both modes to

TABLE 2

Orthonormality check of the retrieved modes. The values of the table are ð2ns=c�0ÞRf� f�0 dx dy , where

� and �0 are the row and column of the table. The ideal values are indicated in parenthesis

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construct the refractive index profile more accurately. However, the profiles obtained from eachmode differ in a constant N2

1 � N22 that must be calculated previously. This constant was computed

as a weighted mean difference between both profiles; the weight must be high only where bothprofiles have low variance. Therefore, we choose as weight 1=ð2m1

ðxÞ þ 2m2ðxÞÞ, which leads to

N21 � N2

2 ’ �m �

m2ðxÞ�m1ðxÞ2m1ðxÞþ2m2

ðxÞPx

12m1ðxÞþ2m2

ðxÞ:

A value of ð514 26Þ � 10�5 was obtained for N21 � N2

2 . Once �m is known, the final squared indexprofile can be obtained as a weighted mean of both profiles, but now, the weight of each one isinversely proportional to its variance

n2ðxÞ � N22 ’

m1ðxÞþ�m2m1ðxÞ þ

m2ðxÞ2m2ðxÞ

12m1ðxÞ þ

12m2ðxÞ

The resulting profile is shown in Fig. 5(b). As expected, the standard deviation of n2ðxÞ � N22 keeps

low or moderate for a wide central region and increases strongly in few points at both ends.Specifically, half of the points have a standard deviation of nðxÞ � N2 ’ ðn2ðxÞ � N2

2 Þ=ð2N1Þ be-tween 2 � 10�4 and 6 � 10�4; and it keeps below 2 � 10�3 in the 88% of the points. For comparison,the refractive index accuracies of the interferometric methods are 2 � 10�3 in [2] or 5 � 10�4 in [4] atevery point of the profile. In respect of the shape, a clear asymmetry of the squared index profile canbe seen in Fig. 5(b). The index decreases slowly toward the waveguide surface ðx ¼ 0Þ at the left ofthe maximum, while the reduction toward the substrate is more abrupt. This result is consistent withthe one previously reported for buried waveguides fabricated by field-assisted ion exchange inglass [13], [22].

4.3. Effective Index DifferenceA parameter of great importance in devices such as directional couplers or multimode interfer-

ence (MMI) couplers is the coupling length or alternatively the coupling coefficient. It is well known thatit* value is directly related to the effective index difference, which can be obtained from �m, that is

N1 � N2 ¼N2

1 � N22

N1 þ N2’ �m

2ns¼ ð170 13Þ � 10�5 (25)

where the effective indices of the denominator were approximated by the substrate indexns ¼ 1:5100. The inaccuracy from this approximation is low for the expected values of N1 and N2.

On the other hand, we can improve the above result since we know the propagation distance z,therefore we can relate the phase difference between both modes and the difference of effective

Fig. 5. (a) Squared refractive index profile retrieved from the first (blue) and second mode (green). Eachgraph is an average along the slab guide; it is represented with two lines that indicate one standarddeviation below and above the mean. (b) Combination of such profiles by a weighted average,represented in the same way; a histogram of its standard deviation is shown in the inset (number ofpoints against standard deviation).

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indices as follows:

�12 ¼2��ðN1 � N2Þz:

If the accuracy of N1 � N2 allows us to find which quadrant �12 is in, a more accurate value of thisphase may be derived from its squared tangent provided by the fitting to ellipses. Once �12 isimproved, N1 � N2 and the coupling length or the coupling coefficient can be also refined by againapplying the last equation. Unfortunately, the variance of �12 calculated from the effective indexdifference exceeds 2� in our case because of the large value of z, and hence, this improvementwas not made.

Finally, the values of N1 � ns and N2 � ns can be obtained from the squared index profile if thetails of each mode could be recovered in the substrate region. This does not happen in our sample,but the step index waveguides are good candidates for that. Provided that ns is known, we mayobtain the effective indices. In this case, the difference N1 � N2 could also be directly obtained fromthese values, but probably (25) will be more accurate, mainly if N1 and N2 are used to evaluate thedenominator.

5. Conclusion and PerspectivesWhen a waveguide is illuminated with a field with constant phase at the input face that excitesN modes, the normalized intensities at N points of its exit face are correlated, that is, there is aconstraint between these intensities on a hypersurface in an abstract N-dimensional space, whicharises from the interference between the excited modes. This interference is very stable since ittakes place after a common path along the waveguide. If only two modes are excited, such aconstraint is an ellipse. When the normalized intensity of a point is represented against the one ofany other point, then the ellipse is mapped by changing the coupling condition. From the ellipseparameters (center coordinates, height, width and tilt), we can retrieve the squared tangent of therelative phase between both modes as well as the absolute value of their amplitude at the consideredtwo points, although the equations do not indicate which value corresponds to each mode.

By repeating the procedure for different point pairs, it is possible to assign the amplitude values tothe right mode and fix the sign, that is, it is possible to reconstruct the amplitude of both modes. Theself-consistence of the results can be checked in several ways: ellipses fully contained in the firstquadrant, redundancy in both the relative phase and the amplitudes from the multiple way to selectthe point pairs, orthogonality of the final modes or fulfilling some symmetry of the waveguide, etc.Once the modes are obtained, the difference of effective indices and the squared refractive indexprofile, except an additive constant, can be also retrieved. Moreover, both the index profile and theeffective indices may be obtained without ambiguity in those cases in which the retrieved mode tailsreach the substrate region.

The proposed method was successfully applied to a buried slab waveguide fabricated by field-assisted ion exchange in glass. Since this guide is multimode (actually a few modes), an opticalsystem to excite almost exclusively the first two modes with a variable relative efficiency was alsoproposed. This optical system is simple, inexpensive, and accurate since it makes the most of thereduction that the focusing objective provides. The shape of the index profile agrees with thatobtained in similar waveguides from other methods previously reported.

Since each mode is extracted from two-mode interferences, the proposed method can be con-sidered as an extension (essentially a preliminary step) of the monomode intensity method tomultimode buried waveguides in which the prism-coupling method is not possible. Therefore, ourmethod is an alternative to interferometric methods, although a comparison is not simple. Interfe-rometric methods provide the index profile in a wide region with a uniform accuracy that dependsstrongly on the system geometry, but the sample preparation is destructive and very time-consuming.On the other hand, our two-mode intensity method is restricted to the waveguide region but has agood repeatability for the most part (between 2 � 10�4 and 6 � 10�4 in half of the points). Moreover,

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we should stress that it also provides the mode profiles and the mode effective indices or, at least,their difference ð13 � 10�5Þ, and this information is of great importance in many cases.

The coupling to the third mode was a limitation in recovering the first and second modes, andtherefore, we expect better results in two-mode guides. For instance, monomode dielectric guidesat 1.31 or 1.55 �m could be characterized as two-mode guides within the red wavelength range orat least below 1000 nm, provided that the dispersion is known. This could allow the use of commonsilicon-based cameras whose technology is more mature than that of infrared cameras. Similarly,optical elements for visible spectrum are much more common than infrared optics.

The diffraction is another limit of the proposed setup, which is especially relevant when the modesize is smaller than a wavelength, for instance, in some silicon waveguides. The diffraction limit canbe circumvented by a near-field scanning technique, in the same way that some versions of themonomode intensity method do.

The characterization of few-mode fibers, directional couplers or MMI couplers is a natural ap-plication of this method. In the last two cases, the difference of the effective indices is especiallyuseful since it sets the coupling length, which is an important parameter of these devices. Evenmore, if the propagation distance is moderate, the relative phase between the modes can beobtained without ambiguity from its squared tangent. This will lead to a more accurate value of thecoupling length.

In short, there are few methods to characterize multimode buried waveguides, and we haveproposed and checked a new method to solve this problem.

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