1. INTRODUCTIONPlanar photonic crystals (PCs) are now widely recognizedas promising platforms for the realization of ultracompactphotonic circuits on a submicrometer scale. Their three-dimensional counterparts, which would provide optimalconfinement of light,1 are still extremely difficult tofabricate.2 Based on ideal two-dimensional PCs consist-ing of infinite-depth air holes, planar PCs further confinethe beam in the vertical direction by conventional indexguiding: For the results presented in this paper, this ver-tical index guiding consists of a heterostructure [typically,AlxGa12xAs/GaAs/AlyGa12yAs (Ref. 3)]. The reflectivityof such planar photonic crystals was found to be morethan 90% at wavelengths within the photonic bandgap(PBG), even for PCs limited to a few periods.3–5 By per-turbing the PC lattice in a controlled manner, however,we can create a particular set of modes that have theirwavelengths in the PBG.4 In this paper we focus on theline defects that result from the removal of one or morerows of air holes.
As there are no propagating modes in the PBG, a beamguided along such a linear defect at wavelengths withinthe PBG cannot be scattered in the surrounding PC by de-
fects or fluctuations. Therefore, with the same fabrica-tion tolerances, propagation losses a in PC waveguidesshould be smaller than in conventional waveguides de-fined by a deeply etched ridge, in which propagationmodes exist in the surrounding media. However, whenthe vertical confinement is a heterostructure rather thana membrane, such planar PC devices suffer from out-of-plane losses in a quite systematic fashion, as all modes lieabove the so-called light line in a dispersion diagram.6–8
This penalty has been suggested nevertheless to be mod-est and controllable because vanishing losses have beenpredicted for systems with low De.9 For such less con-fined systems, however, the mode swells vertically, so thefinite depth of the holes is an additional cause of out-of-plane losses. As a consequence in a planar PC channelwaveguide the resultant value of the propagation lossesdepends on the outcome of this competition between theout-of-plane losses and the loss inhibition mechanism de-scribed above.
To compare the performance, in terms of propagationlosses, of conventional index-guiding channel waveguideswith that of planar PC waveguides, their counterparts, itis thus necessary to estimate such propagation losses for
2002 Optical Society of America
2404 J. Opt. Soc. Am. B/Vol. 19, No. 10 /October 2002 Schwoob et al.
both kinds of waveguide. One way to measure theselosses is to quantify the attenuation encountered by a sig-nal propagating along the waveguides.
For conventional index-guiding channel waveguides,quantitative measurements of propagation losses havebeen carried out for decades, and steady progress has ledto scattering losses in such ridges in the rangea , 1 cm21 for optimized, InP-based, 1.5-mm-wide guidesat l 5 1.55 mm.10 Measurements of propagation lossesare more critical and still scarce for planar PCwaveguides. Note that all data reported in this paperuse as a photonic crystal a triangular lattice of air holes,which provides the largest PBG.11 The first quantitativeexperimental loss results were presented in Ref. 12; byuse of the internal probe method it was demonstratedthat, provided that the etching is deep enough, propaga-tion losses of PC waveguides at l 5 1 mm (GaAs-basedstructures) are lower than those of ridge waveguidesetched in the same run. A transmission measurementled to the conclusion that the out-of-plane losses of thefundamental transverse mode are indistinguishable fromthe material absorption, setting an upper limit of 50 cm21
(2 dB/100 mm).12 Recently, by the end-fire method,13
propagation losses of planar PC waveguides at l5 1.55 mm (InP-based structures) were also measured,
yielding a value of 25 cm21 (1 dB/100 mm). It is thus ofinterest to determine losses more accurately in GaAs-based structures and, more generally, to find a way to in-crease the sensitivity of the convenient internal probemethod.
Among all methods aimed at measuring the propaga-tion losses in waveguides, the use of a waveguide Fabry–Perot structure was pioneered by Kaminow and Stulz in1978.14 They obtained quantitative measurements of thepropagation losses by measuring the Fabry–Perot fringecontrast. Here we introduce a variant of this method,without genuine facets however. Specifically, we discussthe results obtained from straight channel waveguideswith two-dimensional PC boundaries modified at bothends by constrictions instead of facets (see Fig. 1) with theinitial intention of producing a somewhat comparableFabry–Perot resonator, and we present an analysis of thesubsequent fringe contrast. Whereas these Fabry–Perotfringes clearly appeared on the shortest waveguide trans-mission spectra, analysis of the longest waveguide spec-
Fig. 1. (a) Scanning-electron microscope picture of a waveguideof length Lc 5 160 rows; period a is 260 nm. Arrows indicatethe constrictions at both ends. (b) Constrictions at one end ofthe waveguide.
tra revealed an additional phenomenon, which led tofringes with a largely different fringe spacing. We at-tribute the observed fringes to a two-mode interferencebetween the fundamental transverse mode and the thirdtransverse mode of the PC waveguide. We report hereour analysis of this phenomenon and explain how it canbe used to extract propagation losses with sensitivity en-hanced by an order of magnitude.
In Section 2 we briefly outline the relevant experimen-tal issues, especially the design of these waveguides withconstrictions. Section 3 is devoted to a discussion of bothexperimental and theoretical results. In Subsection 3.Awe analyze the Fabry–Perot fringes, in Subsection 3.Bwe turn to two-mode interference in the PC waveguideswith constrictions, and in Subsection 3.C we extract aquantitative value for out-of-plane losses, namely, 25cm21 for the fundamental transverse mode.
2. EXPERIMENTIn the research reported in Ref. 12 the propagation losseswere deduced from a comparison of the attenuation en-countered by the fundamental transverse mode in planarPC waveguides of various lengths. However, this methodwas limited by the emitters’ reabsorption, typically 50cm21 in the internal probe method with quantum dots asemitters, which tends to hide genuine out-of-plane losses.As it had limited sensitivity, this method could set only anupper limit for these losses. To increase the sensitivity,we thought about a well-documented technique that con-sists of measuring fringe contrast. In optics, this inter-ference technique, which requires a Fabry–Perot resona-tor with some variable parameters for creation of thefringes, has been used for decades to measure preciselythe reflection and transmission coefficients of mirrors.In common guided optics this method has also beenwidely used to measure waveguide propagation losses: Aprobe beam, produced by an external light source, propa-gates along a Fabry–Perot waveguide structure obtainedby polishing or cleaving of the sample end faces. Thetransmitted beam is collected and spectrally analyzed.The fringe contrast is measured and gives an estimate ofthe attenuation coefficient, the facet reflectivity beingtaken into account in this contrast.14,15 As our internalprobe technique16 cannot fit into the former method (thewaveguide does not even reach the cleaved facet), we haveto implement special reflecting structures. In Section 3we explain the design and fabrication of a Fabry–Perotresonator in our planar PC waveguides.
Electron-beam lithography and reactive-ion etching areused to define the PC channel waveguides in a GaAs-based heterostructure identical to that of Ref. 16 (400-nm-thick Al0.8Ga0.2As/220-nm GaAs core/310-nmAl0.8Ga0.2As top cladding, with an effective index of 3.38at 955 nm). We recall that three layers of InAs quantumdots (QDs) are embedded in the GaAs core.16,17 TheseQDs have a large size distribution, which leads to a broadphotoluminescence spectrum and makes the QDs a versa-tile broadband internal light source (960–1050 nm, lim-ited by a Si detector), with reabsorption ranging from 50to 100 cm21 in the spectra of interest. Various periods(a 5 240, 260, 280 nm) of the PC triangular lattice of airholes are used to scan the TE polarization photonic band-
Schwoob et al. Vol. 19, No. 10 /October 2002 /J. Opt. Soc. Am. B 2405
gap in the GK direction at normalized frequencies,u 5 a/l, ranging from 0.23 to 0.32. The air-filling factoris ;f 5 37%. Holes have a depth/diameter aspect ratioof ;5. With f 5 37%, a typical value for the diameter is180 nm for a 5 280 nm, and the typical depth is 0.9 mm.As out-of-plane losses depend on the shape and the depthof the holes,9 the value of the propagation losses that weestimate here relates to the particular hole shape andhole depth in the structure discussed here.
The PC waveguide, W3, is defined as three missingrows in the GK direction in the triangular planar PC lat-tice [Fig. 2(a)]. The lengths range from Lc 5 10a to Lc5 640a in a geometric series of ratio 1.414 [Fig. 2(b)]. Tobuild up a Fabry–Perot cavity without cleaved mirrors,and hence to apply the classic guided optics loss measure-ment method (see above), we add constrictions at bothends of each waveguide. The constrictions consist of twoholes added five rows from the end of the waveguide, suchthat the waveguide is, locally, one row (W1) wide (Fig. 1).The lengths quoted above are actually the distances be-tween two constrictions.
The characterization technique is similar in all re-spects to that described in Ref. 12: The photolumines-cence of the layers of QDs is excited by a 678-nm red laserdiode focused to a spot on the surface of the sample 20 mmfrom the entrance of the waveguide. The QDs emit bothTE and TM polarization, but we select only TE polariza-tion, as an omnidirectionnal photonic bandgap exists onlyfor this polarization. As there were many closely spacedpatterns, to check the absence of spurious light scatteredby these patterns it proved practical to focus the collectionlens inside the sample to a point corresponding to the PCwaveguide exit’s virtual image at a locus determined byrefraction at the facet.18 This approach ensured that thecollected light had indeed traveled along the waveguidebeing probed.
The intensity of the collected beam, I2(l), is the inten-sity of the fundamental transverse mode of thewaveguide.12 Other modes with higher orders can alsopropagate along waveguide W3, which is multimode inthe PBG. However, at the entrance of the waveguide,coupling to the fundamental mode is much more efficient
Fig. 2. Schematic representations of (a) a W3 waveguide and (b)the multiwaveguide sample. The lengths of the waveguides,Lc , range from 10 to 640 rows, with a scale factor of A2. Notethe constrictions at both ends of each waveguide. The variouscollection configurations of interest are also sketched.
than to higher-order modes. The slow group velocity ofhigher-order modes, combined with their larger penetra-tion into the PC barriers compared with the fundamentalmode, makes then the higher-order modes much moresensitive to out-of-plane losses: Their decay length isthus shorter. Finally, as the collected light correspondsto a maximum internal angle of 6.5° (Ref. 16) and becausethe divergence of the beam at the output of the waveguideis much greater for the third mode than for the funda-mental mode, the amount of latter mode that is collectedis greater, by a large factor.
For a single-mode waveguide the intensity I2(l) is ofthe form (I0 /Lc)exp(2aLc) f (l). The function f(l) takesinto account all wavelength dependences (source, cou-pling...). Coefficient a measures the total extinction ofthe guided beam: It contains, in addition to out-of-planelosses aPC that are directly caused by out-of-plane scatter-ing, absorptive contribution aQD that is due to the layersof QDs. Intensity I1(l) of a beam propagating the samedistance Lc along the heterostructure without beingguided in a PC waveguide is then of the form(I0 /Lc)exp(2aQD Lc) with an almost identical value ofaQD , as the fundamental mode group velocity is close toits planar counterpart. To compensate for most of theQDs absorption we normalize I2(l) by I1(l).
With this sole normalization, all transmission spectrashow additional fringes at the same spectral interval,whatever the waveguide: They are due to the Fabry–Perot cavity formed by the cleaved edge and the edge ofthe etched part of the sample that is near the cleavededge [Fig. 2(b)]. To eliminate them, we first devized anextra normalization based on a third intensity, I3(l), forwhich the photoluminescence is excited in the nonetchedpart of the sample near the cleaved edge. However, aswe know the optical thickness of this extra cavity, a betterway to eliminate these fringes is to numerically filterthem out in the Fourier-transformed I2 by means of anotch filter. This filter degrades other fringe patternswhenever their free spectral range is of the same order,but this degradation, as we shall see below, causes littleinconvenience. Moreover, this method was found betterin terms of signal/noise ratio. Finally, each intensity isnormalized by its corresponding frontal spectrum,I1,2
front(l), to take into account possible inhomogeneitiesof the local photoluminescence efficiency.3 Finally, whatwe call, for convenience, the transmission of each wave-guide is given by the following formula, where I2(l) is theFourier-filtered intensity, which contains only fringesthat are due to the presence of the constrictions:
T~l! 5I2~l!
I1~l!
I1front~l!
I2front~l!
. (1)
Note that Eq. (1) is not, strictly speaking, a transmission,as guided intensities I1 and I2 are collected from twobeams with quite different geometrical divergence. Theresults, consisting of transmission spectra T(u), u5 a/l, are presented for some selected guides (Lc rangesfrom 20a to 456a), for TE polarized light only, in Fig. 3.Section 3 is devoted to analysis of these spectra.
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3. RESULTS AND DISCUSSIONA. Qualitative ResultsThe first significant feature of our measurements is thatthe average transmission values are roughly of the sameorder for all waveguides, from 0.2 for the shortest to 0.1for the longest (Fig. 3). As a first consequence, a beamcan propagate with only moderate losses in a 150-mm-long photonic crystal waveguide. The second feature isthe transmission extinction from u 5 0.258 to u 5 0.263
Fig. 3. Experimental transmission spectra of five planar PCwaveguides with constrictions as a function of normalized fre-quency u 5 a/l for the lengths (a) Lc 5 452a, (b) Lc 5 226a, (c)Lc 5 80a, (d) Lc 5 56a, and (e) Lc 5 20a. Solid arrows, ex-pected Fabry–Perot fringes; dashed arrows, two-mode fringes,which are discussed below.
on all transmission spectra, which corresponds to a ministopband of the dispersion relation of waveguide W3.19
Outside this mini stopband, and of direct relevance forthe scope of this study, we find that two different kinds offringe can be identified: (i) for Lc < 80a, rather low-contrast fringes are observed from u 5 0.28 to u 5 0.31[Figs. 3(c)–3(e)] and (ii) for Lc > 56a, rather high-contrast fringes are observed for all values of u outsidethe mini stop band of the waveguide [Figs. 3(a)–3(d)]. Itis also interesting to note that, in both cases, the fringecontrasts decrease as length increases, as expected be-cause of damping on propagation.
B. Fabry–Perot FringesConstrictions at both ends of each waveguide were ini-tially added to create a Fabry–Perot resonator in the lon-gitudinal direction of the waveguide. Our aim was to ob-serve the fringes and deduce the propagation losses justas for cleaved ridge guides, that is, by monitoring fringecontrast versus guide length. For clarity, notation andbasic concepts are described in Appendix A. Figure 4shows calculated and experimental values for fringe spac-ing for each waveguide versus the waveguide length, forboth kinds of fringe and for both regions corresponding tofrequencies outside the W3 mini stop band: The first re-
Fig. 4. Solid curves, plots of theoretical calculations for Fabry–Perot fringe spacing versus length of the waveguide. Errorbars, Fabry–Perot (dashed) and two-mode (solid) experimentallymeasured fringes. Regions both below and above the mini stop-band, regions A and B, respectively, are represented in terms ofnormalized frequency u.
Schwoob et al. Vol. 19, No. 10 /October 2002 /J. Opt. Soc. Am. B 2407
gion, region A, corresponds to the values of normalizedfrequency u below the mini stop band (0.24 , u, 0.258), and the second region, region B, corresponds tothe values of normalized frequency u above the mini stopband (0.28 , u , 0.30).
The narrowest set of fringes, which are visible only onwaveguides with length verifying Lc < 80a, can clearlybe identified as Fabry–Perot fringes. The measuredfringe spacing of the fringes identified on longerwaveguides (Lc > 56a), however, does not correspond atall to Fabry–Perot fringe spacing. Moreover, experimen-tally these fringes are no longer observed when photolu-minescence is excited in the waveguide between the twoconstrictions.20 As a conclusion, these fringes are not ofthe Fabry–Perot type; that is, they are not due to roundtrips of the fundamental mode between constrictions.One could wonder why the Fabry–Perot fringes are nolonger visible on the longer waveguides, as they shouldhave longer decay lengths than the second kind of fringe,as we shall see below. There are two reasons for the dis-appearance of the Fabry–Perot fringes: (i) The notch fil-ter discussed above turns out to suppress Fabry–Perotfringes for waveguides with length near 100 rows and (ii)for the narrowest Fabry–Perot fringes of the longestwaveguides we are limited by the sensitivity of the setup.
C. Two-Mode FringesTo explain the role of higher modes in fringe measure-ment, we follow the path of the guided probe light. Asthe designs of the entrances are identical for PCwaveguides without and with constrictions (constrictionsare located five rows away from the ends of thewaveguides), the coupling between the guided photolumi-nescence and the fundamental transverse mode is stillpredominant. However, when this fundamental modeimpinges on the input constrictions, part of its amplitudeis converted into higher-order even modes owing to themirror symmetry of the constrictions (Fig. 5). Loss con-
Fig. 5. Schematic of the input constrictions of a planar PCwaveguide. Schematics of the magnetic-field profiles for thefundamental mode and the third mode are also presented.
siderations similar to those of Section 2 favor the funda-mental and the third modes, leading principally to a two-wave phenomenon. Each mode propagates separatelyalong the waveguide, with its own wave vector.
At the output constrictions, a converse mode-mixingmechanism occurs: Part of the third mode is convertedinto the fundamental mode and vice versa. According towhether either mode is in phase or out of phase, the re-sultant amplitude of the fundamental mode is maximal orminimal. For reasons similar to those mentioned above,the collection geometry is in fact quite selective with re-spect to the fundamental mode and is only weakly sensi-tive to the third mode: The fundamental mode is thus bya large factor the predominant one collected. Conversely,for a waveguide of fixed length, because of the dependenceof propagation constant difference Db on the excitationwavelength (Fig. 6), fringes with wavelength-dependentfrequency and relative phase appear on the transmissionspectra of the waveguides. We believe that the observedfringes of long waveguides stem from this phenomenon.
What are the expected characteristics of such two-modefringes? Let a1 be the amplitude of the fundamental modeat the entrance of the waveguide before the constrictions,a i be the extinction coefficient that is due to propagationlosses of mode i, and b i be the propagation constant alongthe longitudinal direction of the waveguide for mode i (i5 1, 3). When the beam impinges on the input constric-tions, we define the following amplitude coefficients: t11for the transmission of the fundamental mode throughthe constrictions, t13 for the conversion of the fundamen-tal mode into the third mode, and t31 for the conversion ofthe third mode into the fundamental mode (Fig. 7). Theamplitude of the fundamental (third) mode just after theinput constrictions is then of the form a18 5 a1t11 (a385 a1t13). Just before the output constrictions,these amplitudes are affected by phase factorsexp(2a1L)exp(ib1L) and exp(2a3L)exp(ib3L), respectively.Finally, the output constrictions recombine the twomodes, so the fundamental mode amplitude is
Fig. 6. Band diagram v 5 v(b) of a W3 waveguide of type Aand with an air-filling factor of 37%. We also show the width ofthe PBG of an infinite PC and regions A and B. Note the differ-ence in the slopes of modes 1 and 3, which is responsible for thefringe spacing (here, Db decreases with increasing u).
2408 J. Opt. Soc. Am. B/Vol. 19, No. 10 /October 2002 Schwoob et al.
as 5 a1t11t11 exp~2a1Lc!exp~ib1Lc!
1 a1t13t31 exp~2a3Lc!exp~ib3Lc!. (2)
The intensity is then
Is 5 uA0u2@1 1 uCu2 exp~2DaLc!
1 2C exp~2DaLc!cos~DbLc!#, (3)
where A0 5 a1t11t11 exp(2a1Lc/2)exp(ib1Lc), C 5 (t13t31)/(t11t11), Da 5 a3 2 a1 , and Db 5 b3 2 b1 . Equation(3) is typical of a two-wave interference phenomenon. Itis interesting to note that the modulation prefactor, C, islinked only to the initial amplitude division process andto the reciprocal combination process. The relativephase (Dw 5 DbLc) depends spatially on the length ofthe waveguide and spectrally on the variation of Db withl (Fig. 6). For each waveguide, as the length is fixed,fringes on the transmission spectra are the spectral sig-natures of this two-mode phenomenon. These fringes ap-pear clearly in Fig. 8, where we have plotted the calcu-lated values of intensity Is(u) for waveguides withlengths that range from Lc 5 80a to Lc 5 452a (we dis-carded the most uncertain data). The propagation con-stants, b1 and b3 , were taken directly from the band dia-gram of a W3 waveguide obtained by supercellcalculations based on a two-dimensional plane-waveexpansion.21 As Db diverges in the mini stop band
Fig. 7. Schematic explanation of the mode-mixing process thatresults from the interaction between the launched mode and theconstrictions at both ends of the waveguide.
Fig. 8. Calculated transmission spectra for waveguide lengths(a) Lc 5 452a, (b) Lc 5 320a, (c) Lc 5 226a, (d) Lc 5 160a, (e)Lc 5 113a, and (f) Lc 5 80a as a function of normalized fre-quency u 5 a/l.
(MSB), the two relevant regions for u are 0.24 , u, 0.258 (region A) and 0.28 , u , 0.30 (region B). Weare now in a position to compare the model with our ex-periment. Figure 9 presents the spectral fringe spacingdeduced from the Is(u) graph in both relevant regions foru, compared with the experimental fringe spacing for allwaveguides with lengths ranging from Lc 5 80a to Lc5 452a. There is perfect agreement between the experi-mental fringe spacing and the theoretical spacing de-duced from the two-mode interference assumption.
It is, however, somewhat frustrating to require numeri-cal computation of guided modes, whereas only two rathersimple modes are involved. For this reason we show be-low that the previous results can also be approximatelydeduced from a model as simple as the scalar classicwaveguide theory for a slab. Let us assume that the PCwalls of our waveguides are perfect walls, at frequenciesin the PBG. Field amplitudes are of the form am(x, y)5 am sin(mpy/w)exp(ibmx), where w is the width of thewaveguide and m is the mode order (m 5 1 for the fun-damental mode and m 5 3 for the third mode). We ig-nore z, as this degree of freedom is frozen by the hetero-structure and results in the use of an effective index,neff 5 3.38 here. As a consequence, propagation con-stants bm are given by bm
2 1 (mp/w)2 5 k2, where k isequal to 2pneff /l. From this approach we may estimateboth properties of the two-mode phenomenon. The first
Fig. 9. Solid curves, calculated beat mode fringe spacing versuslength of planar PC waveguides in units of rows, for regions Aand B. Experimental fringe spacing is indicated by correspond-ing error bars. Dashed–dotted curves, fringe spacing deducedfrom the scalar classic waveguide theory for a slab.
Schwoob et al. Vol. 19, No. 10 /October 2002 /J. Opt. Soc. Am. B 2409
property is beat length, obviously 2p/Db. An expansionto first order in 1/(kw) gives the relation Db5 2(4kp2)/(kw)2, leading to the characteristic beatlength: Lcbeat 5 2p/uDbu 5 neffw
2/l; hence Lcbeat 5 9afor region A and Lcbeat 5 11a for region B [spacing w istaken as w 5 4(A3/2)a to fit the typical width of theguide, taken as the distance between the axes of the firstconfining rows, l 5 1000 nm, and neff 5 3.38]. Figure10 presents a simulated profile of the magnetic field inwaveguide W3 for the TE polarization and for both re-gions (hence uA 5 0.24 and uB 5 0.28). These valuesare consistent with our experimental results: They indi-cate that there is enough room to develop this two-wavephenomenon in our waveguides, as their lengths rangefrom 10 to 640 rows.
The second property is the fringe spacing, which is thespectral period of the cos(DbLc) term and is due to the un-derlying frequency dependence of Db. Expressed in di-mensionless units of u, spacing Du is given by Du5 2p/Lc 3 1/(b38 2 b18), where b i8 5 db i /du (i5 1, 3). An expansion to first order in 1/(kw) leads tothe following expression for fringe spacing: Du5 a/12Lcneff 3 (kw/p)2. We have added in Fig. 9(dashed–dotted curves) this analytical estimate of thefringe spacing for waveguides with lengths that rangefrom 80a to 452a for both regions [region A and region B,in Figs. 9(a) and 9(b), respectively] at l 5 1000 nm(hence uA 5 0.24 and uB 5 0.28). This estimate is con-sistent with both the exact bidimensional calculation andour experimental results. Estimations of both character-istics of the two-mode phenomenon are quite consistentwith experimental results, justifying a posteriori the as-sumption made about the walls of the PC waveguides. Itis also interesting to note that the two-mode phenomenonreported here is not specific to photonic bandgap materi-als; but the existence of multimode waveguides at submi-crometer width is not common in other fields of guided op-tics.
D. Propagation LossesOur first aim was to estimate propagation losses, exploit-ing the Fabry–Perot fringes in the spectrum of the col-lected beam. From an analysis not detailed here, wefound that the Fabry–Perot fringe contrast variation wastoo weak and that the noise level was too high to enableus to extract propagation loss values from this kind of
Fig. 10. Simulation of mode beating between the fundamentalmode and the third mode over a distance of 40 rows in a PC pla-nar waveguide for both region A and region B. Distances aregiven in units of rows.
fringe. If we now consider the second kind of fringe thatis due to two-mode interference, Eq. (3) shows that extinc-tion coefficients for modes 1 and 3 appear not individuallybut through their difference Da 5 a3 2 a1 in the attenu-ation factor of Is(u). In a first step, we proceed by ex-tracting the value of Da from Is(u) [Eq. (3)]. In a secondstep, each extinction coefficient is extracted for each modeseparately. The fringe visibility can be written as fol-lows:
V 52C exp@2~Da/2!Lc#
1 1 uCu2 exp~2DaLc!. (4)
As V is weak (uCu , 0.3; see Fig. 3), we may use the sim-plified relation V 5 2C exp@2(Da/2)Lc#. In Fig. 11 weplot ln(V) as a function of Lc for both relevant regions ofthe values of the normalized frequency, u (region A andregion B). The slope of the curve gives a value for Da:
DaA 5 235 cm21 6 15%, DaB 5 205 cm21 6 15%.
We now proceed to extract the value of each extinction co-efficient. Two phenomena contribute to the extinction co-efficients a i (i 5 1, 3) and thus to their difference Da.The first phenomenon is material absorption of the wavein the core of the heterostructure. Let us denote by a i
m
(Dam) this contribution to a i (Da). At these wavelengthsthe layers of GaAs and QDs are not highly absorbent.Moreover, as the group velocity and the mode confinementare greater for the fundamental mode than for the third
Fig. 11. Ln(V) plotted as a function of Lc , where V is the fringevisibility and Lc is the length of the cavity in centimeters. Theslope of these curves is the loss difference Da between the funda-mental mode and the third mode.
2410 J. Opt. Soc. Am. B/Vol. 19, No. 10 /October 2002 Schwoob et al.
mode, we believe that the material absorption encoun-tered by each mode is roughly the same: a1
m > a3m.
Subsequently we neglect the material absorption contri-bution to Da. The second phenomenon is due to genuineout-of-plane losses to the substrate or to the air. It hasbeen shown that these losses are essentially dictated bythe amount of overlap of the D field with the air holes,leading to an estimation of the ratio of the two extinctioncoefficients.9 From a qualitative point of view, the thirdmode is less transversally confined than the fundamentalmode, leading to more overlap and higher losses for thethird mode than for the fundamental mode. Quantita-tively, one uses supercell methods to obtain the eigen-modes and eigenfrequencies in planar PC waveguideW3.21 We exploit the results as follows: from the com-puted eigenmode in k-space, $Hk(G)%, we first go back toreal space and obtain Hz(r). Then we deduce from thecurl Maxwell equations the excitation field, D(r)5 @Dx(r),Dy(r)#. The square of the excitation field am-plitude is plotted in Fig. 12. Parameters used here aree 5 11 and an air-filling factor of 37%. To calculate theratio a3 /a1 we use the approach presented in Ref. 9,which we adapt here to a waveguide: Extinction coeffi-cient a i is proportional in this case to horizontal overlapfactor G i of the guided mode with the air holes, defined as
G i 5
EEholes
iDii2dS
EEallspace
iDii2dS
. (5)
Fig. 12. Representation of the square of the amplitude of the di-electric vector in a planar PC waveguide for (a) the fundamentalmode and (b) the third mode. The vertical scale is logarithmic.
We eventually estimate the ratio a3 /a1 : G3 /G15 a3 /a1 5 9 6 15% (Ref. 22); hence Da 5 a3 2 a15 8a1 . Finally, from this analysis, the extinction coef-ficient a1 for the fundamental mode is taken as Da/8;hence a1
A 5 26 cm21 6 30% and a1B 5 25 cm21
6 30%. Considering the uncertainties, one can safelyretain as a result of this study a single estimate of thefundamental mode propagation losses:
a1 5 25 cm21 6 30% ~1 dB/100 mm!.
This value is consistent with the upper limit of 50 cm21
set in Ref. 12 for GaAs-based structures. Compared tothe value of 25 cm21 found in Ref. 13 for InP-based struc-tures, it validates the assumption that in InP structuresone can substantially diminish propagation losses by im-proving the etching technique.23 Finally, for future ap-plications it is a promising value, as it shows that eventhe third mode can be easily detected after propagationalong a 125-mm-long planar PC waveguide.
4. CONCLUSION AND PERSPECTIVESAs a conclusion, the inclusion of the simple constrictionsof Fig. 1 within planar PC straight multimodewaveguides has been found to affect the system much be-yond the initially desired creation of detectable Fabry–Perot fringes for the fundamental mode. For experimen-tal reasons, such Fabry–Perot fringes can be observedonly for waveguides with lengths of fewer than 80 rows,and the contrast of these fringes is always weak, giving atbest an upper bound of the propagation losses. However,we found that the interaction of the excited fundamentalmode with the constrictions leads to a relatively simpletwo-mode beating phenomenon, which appears to be evenmore sensitive to out-of-plane losses, here by a factor of 9.In essence, the next even mode was found to be excited atthe input constrictions and mixed back with the funda-mental mode at the output constrictions, leading to acos(DbLc) term in the detected signal. As Db depends onfrequency, the detected spectrum displays novel fringes.The contrast of this new kind of fringe depends essen-tially on the larger extinction coefficient of the highermode, leading to a new and more sensitive measurementof propagation losses. It can be hoped that such a sensi-tive method will help in assessment of the lower losses ex-pected from better etching.23
Quantitatively, for the fundamental mode of planar PCwaveguide W3, propagation losses are found here to be25 cm21 6 30%, that is, 1 dB/100 mm, with a reactive ionetching technology. As a consequence of the perspectivesopened by a level of losses near 30 cm21, a wide range ofPC-based elements can be cascaded in a 100-mm lengthwith losses lower than 3 dB. Moreover, as it is likely thateven more-diminished losses will be obtained for betteretching technologies and optimized choice ofheterostructures,23 the number of cascadable elementswill plausibly not be limited by propagation losses.
Finally, one could go to single-missing-row waveguideswith losses not larger than those of the present larger W3waveguides. Compared with W3 waveguides, the so-called W1 waveguides present a monomode transmissionover a broader range of frequencies owing to the absence
Schwoob et al. Vol. 19, No. 10 /October 2002 /J. Opt. Soc. Am. B 2411
of mini stop bands in their dispersion relations.24 Theycould be useful in many integrated optics applicationsthat require single-mode propagation. Experimentally,such broadband transmission in W1 waveguides has al-ready been demonstrated.25 Moreover, recently a low-loss sharp turn in a PC was achieved for a single-missing-row waveguide.25 One can therefore envisage moreplausibly that PC devices will be able to shrink the sizesof all the basic functions of integrated optics circuits.
APPENDIX A: FABRY–PEROT RESONATORLet us recall some background knowledge of Fabry–Perotresonators. The following analysis still holds for the PCdevices (mirrors and waveguides) that we consider in thepresent study in the absence of diffraction and thus for asingle mode. We consider a symmetric cavity made fromtwo identical parallel planar mirrors separated by aspacer of width Lc and characterized by their reflectionand transmission amplitude coefficients, r and t, respec-tively. Standard analysis gives the total intensity trans-mission as the following function:
TFP 5 U t
1 2 r2 exp~2ij!exp@2a~l!Lc#U2
, (A1)
where 2j 5 2k iLc 5 4pLcneff /l is related to the compo-nent of the wave vector in the cavity that is orthogonal tothe mirrors and where exp@2a(l)Lc# is the amplitude at-tenuation for a round trip of the light in the cavity. Therelevant index in our cavity is the effective index of thenonpatterned planar structure neff , whose typical value is3.38 for the vertical fundamental mode of the heterostruc-ture. Equation (A1) can also be written as
TFP
5utu2
1 1 r4 exp@22a~l!Lc# 2 2r2 exp@2a~l!Lc#cos~2j!.
(A2)
The spacing between fringes is then given by the follow-ing relation: 2j 5 2p ⇒ Dl 5 l2/(2neff Lc) ⇒ Du 5 1/(2neff Lc), with u 5 a/l. We also have to take into ac-count the material dispersion, which is by far the leadingfactor of ]neff /]l that reduces fringe spacing accordingto26 Du 5 1/(2ngLc), where ng is the group index,ng 5 n 2 l]n/]l (Ref. 27), with n 5 neff and ]neff /]l5 28 3 1024 nm21 (Ref. 18); hence ng 5 4.2 here. Thevisibility of the fringes that are due to the Fabry–Perotresonator is defined by the following relation: V5 (TFP
max 2 TFPmin)/(TFP
max 1 TFPmin); it relates to the
attenuation coefficient according to V 5 r2 exp@2a(l)Lc#.
*Present address: Intense Photonics, 4 Stanley Boule-vard, High Blantyre, G72 OUX, Scotland.
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