time-frequency analysis for an efficient detection and localization of side-coupled cavities in real...

816 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 5, MARCH 1, 2010

Time-Frequency Analysis for an Efficient Detectionand Localization of Side-Coupled Cavities in

Real Photonic CrystalsYaneck Gottesman, Sylvain Combrié, Alfredo DeRossi, Anne Talneau, Philippe Hamel, Alberto Parini,

Renaud Gabet, Yves Jaouen, Badr-Eddine Benkelfat, and Elchuri V. K. Rao

Abstract—We propose and demonstrate here the high efficiencyof concurrent time and frequency analysis to detect and unambigu-ously identify the coupled cavities in real photonic crystals con-taining imperfections and/or process-induced disorder. This pro-cedure when applied to reflectograms recorded using phase-sensi-tive optical low-coherence reflectometry allows a straightforwardand complete assessment of cavities (spectral and spatial localiza-tion in addition to photon lifetime) over a wide spectral range. Con-sidering such a reflectogram (recorded in 2 s), we show that thisprocedure greatly eases the evaluation of cavities under guidingconditions in real photonic crystals by discriminating their signa-ture from the in-plane scattering induced by disorder.

Index Terms—Cavities, interferometry, optical low-coherencereflectometry, photonic crystals, time-frequency analysis.

I. INTRODUCTION

T HE ability of photonic crystals (PhCs) for light confine-ment in sub- scale offers the unique opportunity for

photonic crystal cavities to play a stimulating role in the de-velopment of innovations both in the fields of basic and appliedphysics. As examples of recent innovations implying PhC cavi-ties of small volumes, one can mention the control of light emis-sion in the cavity [1] and the observation of bio-molecules insmall quantities [2], [3] when combined to optofluidic devices.Obviously, the latter applications warrant a thorough evaluationof the properties of cavities such as their location and spectralpositions in addition to the photon lifetime (or Q-factors). Sofar such an assessment is a delicate task since the properties ofreal (or actual) crystals substantially differ from those of idealones. Fabrication imperfections such as, pattern definition andalso those induced during etching can screen the cavity signa-ture and render delicate the identification of cavity resonant fre-

Manuscript received December 22, 2008; revised August 11, 2009 andOctober 06, 2009. First published October 30, 2009; current version publishedMarch 05, 2010. This work was supported by the Agence Nationale de laRecherche (ANR) (Project L2CP).

Y. Gottesman, A. Parini, B-E. Benkelfat are with Institut TELECOM, T&MSudParis, Laboratoire SAMOVAR UMR INT-CNRS 5157, 91011 Evry, France(e-mail: [email protected]).

S. Combrié and A. DeRossi are with Thalès Research and Technology, 91767Palaiseau, France.

A. Talneau and E.V.K. Rao are with Laboratoire de Photonique et Nano-Structures CNRS-LPN, 91460 Marcoussis, France.

P. Hamel, R. Gabet, and Y. Jaouen are with Institut Telecom ParisTech, 75013Paris, France.

Digital Object Identifier 10.1109/JLT.2009.2034988

quency under guiding conditions. This task is far more difficultwhile dealing with newly elaborated coupled-cavities since ourunderstanding of their performance relies only on simulationsand design compromises. For example, for a coupled cavity, theresonant frequency is critically dependent on its distance withrespect to the coupling waveguide (WG). Too close to the WGresults in a shift of the cavity resonant frequency (as comparedto its frequency when isolated). On the other hand, too far fromthe WG inhibits coupling and hence the cavity excitation.

At present two methods are principally employed to char-acterize coupled-cavities. The first, performed in time domain,consists in measuring the power decay of out of plane light.Although direct, this method however necessitates an experi-mental setup with an adequate (fine and broadband) tuning ofthe light source [4], [5] and rapid detection electronics. Thesecond method, as detailed in [6] is realized in spectral domainunder guiding conditions. It consists in exploiting the specificinterference signatures of the cavity in the midst of Fabry–Perot(FP) fringes originating from the reflections at the end-facets ofthe coupling WG. Nevertheless, the identification of cavity reso-nant frequency can be difficult because of possible multiple par-asitic abnormal fringes and/or parasitic out-of-plane scattering.This method is time consuming since it necessitates high (spec-tral) resolution measurements as dictated by the cavity finesse.Furthermore, the -factor is indirectly extracted after fittingmodal parameters with the experimental results. To summarize,the above methods are not always suitable for a direct study ofcoupled cavities, particularly those with time-dependant char-acteristics (e.g., bio-photonic applications).

In the above context, a novel procedure based on concurrenttime (or spatial) and frequency (or spectral) analysis (CTFA) ex-ploiting the reflection properties of guiding structure, i.e., ampli-tude and phase information, is proposed here. It permits a directand complete assessment (spatial and spectral positions in addi-tion to Q-factors or photon lifetimes) of coupled cavities in real(or actual) PhCs under guiding conditions. The CTFA treatmentis applied here on the experimental data intentionally acquiredusing a complex optical low-coherence reflectometry (c-OLCR)[7]. This setup permits to record both module and phase infor-mation of the reflection originating from the coupled cavity overa broad spectral domain within a couple of seconds ( 2 s).

The paper is organized as follows. The Section II and its sub-sections are entirely devoted to simulations by considering a 1-Dlaterally coupled cavity in a PhC-WG (denoted hereafter as sidecoupled cavity, SCC). We introduce the basics of simulations

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GOTTESMAN et al.: TIME-FREQUENCY ANALYSIS 817

based on a known physical model of the coupled cavity whichis then incorporated in a Transfer Matrix Formalism (TMF) tocompute coupled cavity reflections (Section II.A). The simula-tions are then undertaken in time-domain to study the depen-dence of coupled cavity signature in an ideal WG under dif-ferent conditions namely, with and without reflections at the WGend facets and varied probe characteristics (Section II.B). Usingthe same formalism, a phenomenological model is utilized fora qualitative understanding of the parasitic scattering influencein a WG with no cavity (Section II.C). In light of these re-sults, the CTFA is proposed to filter cavity related information inthe presence of disorder induced scattering (Section II.D). TheSection III is devoted to experimental results and discussion.Here, after a brief description of the sample and the c-OLCRsetup employed to probe it, the benefits of CTFA applicationto assess coupled cavity are described. Finally, the Section IVsummarizes the results of this work by illustrating the impor-tance of CFTA to investigate coupled cavities.

II. TIME AND FREQUENCY ANALYSIS OF PHCS : SIMULATIONS

A. Principle of Simulations

By definition, the photon lifetime in a cavity is asso-ciated to a generic 1st order differential equation whose gen-eral solution has a temporal dependence on . Obviously,the differential equation depends on the physical model chosento describe the coupled cavity. After testing different existingmodels [8], [9], we chose to work in the framework of the onedescribed in [9] for the following reasons. Firstly, the results ofsimulations described here are independent of the model. Sec-ondly, the reflection and transmission properties of a coupledcavity are derived using coupled mode theory and have beenexperimentally validated by applying it to the case of SCC [6]under guiding conditions.

Thus, the reflectivity of the coupled cavity can be analyticallyexpressed as

(1)

where represents its resonant pulsation, the cavity in-trinsic linewidth and the coupling factor between the wave-guide and cavity.

The coupled cavity structure considered here can be dividedinto 3 different sections: a PhC-WG followed by a SCC andanother PhC-WG. As this structure is embedded in air, onewould expect reflections from the PhC-WG end facets (A andB). This structure is modelled using Transfer Matrix Formalism[10] (TMF) to calculate the complex reflectivity both in thespectral and time domains. This is carried out by multiplyingthe matrix corresponding to each section (see inset of Fig. 1)to generate a global matrix representative of the wholestructure. In particular, the elementary matrices correspondingto propagation in the two PhC-WG sections , SCC section

Fig. 1. Time-domain simulations of a SCC inserted in an ideal PhC-WG (A-B)(see inset of 1(a)). (a) Data simulated considering both without (thick line) andwith (thin line) antireflection coatings at the WG facets (A and B). (b) Datasimulated using different temporal resolutions ��.

and the reflections at the WG end facets can bewritten as

(2)

Here, represents the additional phase due to light propaga-tion in the WG, the reflection coefficient and the identitymatrix. The construction of the matrix [9] is directly de-duced from (1). This approach differs from the one proposed byPendry [11] which takes into consideration the geometry of thestructure and the super-cell method recently applied to model1-D PhC cavity [12] wherein the saturation value correspondsto .

In spectral domain, the complex reflectivity of the struc-ture when probed by a light source can be expressed as

. We emphasize here that is deduced from theelements of the global matrix . In time domain, the reflectedsignal can now be expressed as

(3)

where FT represents the Fourier Transform. As is obvious fromthe convolution product in (3), the temporal resolution avail-able to monitor the reflectivity in time-domain is only limited bythe width of the probing light source [13]. This means, thewider the probe spectrum, the higher is the temporal resolutionas dictated by inequalities related to Fourier transformations.


B. Time-Domain Probing of SCC

As is illustrated in Fig. 1, the detection, spatial localizationand a direct determination of photon lifetime can bereadily achieved in time domain. This figure depicts in timedomain the reflected signal amplitudes of different PhCWGs with a SCC . All simu-lations are carried using a probe whose spectral distributionis a Gaussian with normalized intensity at its maximum andcentered at the SCC resonant pulsation . The Fig. 1(a) iscomputed using a high resolution probe ( 50 nm) andcorresponds to a SCC placed in the middle of a PhC-WG. Here,the thin and thick curves represent respectively the case withand without reflections at the WG facets (A and B). As seen inthis figure, the cavity is distinctly localized spatially at pointC by a sharp rise in the reflection level followed by a lineardecay (in log scale). The latter decay precisely corresponds to

. This is verified by carrying out simulations on cavitieswith various Q values in the range (usingvarious values of and ) and comparing the resulting decayswith the analytical expressions of extracted from (1). Forsuch assessment the probe spectrum must contain the resonantwavelength of the cavity (a spectrally selective reflector). Inthe opposite case, the cavity becomes invisible since no en-ergy is reflected by it, as is the case of wavelength selectiveBragg gratings [14]. Also, while analyzing structures withoutantireflection coatings at the WG facets, the can only beextracted from the information contained in the 1st round trip(see Fig. 1(a)) because of the superimposition of cavity echoesin subsequent round-trips.

We thus conclude from the above simulations that SCCs inideal PhC WGs can be completely assessed by performing mea-surements in time domain using a suitably chosen probe to studyreflections.

Since the SCC dimensions are often much smaller than theresolution accessible to optical probes, we have further exam-ined the dependence of its signature using a probe same as abovebut with different half-widths , 1.5, 2.5, and 50 nm. Thesesimulations shown in Fig. 1(b) reveal that the slope of reflec-tivity is invariant with . On the other hand, the rise time atC exhibits an increase with a drop in since the temporalresolution degrades with decreasing , as mentionedbefore. Let us remark that a probe of nm (low-reso-lution probe) which corresponds to 0.8 ps is adequate toextract the photon lifetime of commonly realized PhC cavities( ps). Another interesting feature observable from Fig. 1(b)is the invariance of SCC reflectivity with . This is not sur-prising since the power spectral density of probes with varying

is invariant at (normalized). On the other hand, as seen inFig. 1(b) and unlike the case of SCC, the achromatic reflectionsat the end-facets (A and B) depend on the integral of the powerspectral density of the probe.

C. Influence of Scattering Centers

As is already mentioned, the observation of SCCs insertedin real (or actual) PhC WGs is undoubtedly more delicate be-cause of the presence of (diffusion or) scattering centers intro-duced during fabrication. In what follows, the influence of scat-tering centers is investigated under guiding conditions by car-

Fig. 2. Simulated data showing the dependence of scattering signature in timedomain on temporal resolution.

rying out simulations as before for different probe spectra (of, and 50 nm). Here, we have examined the case of

a PhC-WG without a SCC but containing several deliberately in-troduced achromatic scattering centers all along the WG length.To this end, the properties of scatterers, like their cross sections

and phases have been generated randomly. Also, eachscattering center and their multiple interactions are taken intoaccount in TMF calculations by attributing a reflection matrix

, a propagation matrix that includes the losses encoun-tered between centers and the out of plane radiation. We furtherassigned two uniform (or ‘white’) and uncorrelated probabilitydensities for and where represents the physical sepa-ration between successive centers (e.g., centers i and ). Thelatter is chosen to be about the PhC lattice dimension which isshorter than the spatial resolution of an optical probe defined as

where represents the group index.For example, for a high resolution source probing a PhC

under guiding conditions is about 3 m.This means, the influence of scatterers is observable withoutresolving them spatially.

The results of simulation are presented in Fig. 2. Here, the in-fluence of scatterers in time domain is clearly visible by a con-tinuous speckle all along the length of the WG. Interestingly,one can notice from Fig. 2 that the intensity (or power) reflectedby the scattering centers is strongly dependent on . For ex-ample a variation as high as 20 dB can be witnessed for probesof nm (thin curve) and nm (thick curve).This result is not unexpected since the temporal signature ofachromatic reflectors (scattering centers) depends on . An-other specificity of diffusion by scatterers is its evolution afterthe WG end-facet B. For example, the noise exhibits a markeddrop after B due to additional reflection losses at B on accountof continued probing.

D. Benefits of CTFA Application

We have seen from the simulations presented in Section II(Sections II.B and II.C) that a SSC incorporated in PhC WGcan be identified in the midst of disorder induced scattering. Ac-cordingly, the assessment of SCC (detection, identification anddetermination ) can be readily achieved by sliding the low


Fig. 3. Flowchart showing the numerical processing of phase-sensitive OLCRdata used for concurrent time-frequency analysis (CTFA). The arrows withdotted lines represent the repeated operations by varying � , the central filterpulsation.

resolution probe across a broad spectral domain. This definesprecisely the CTFA.

From an experimental point of view, the use of CTFA can behighly beneficial when the reflection amplitude and phase infor-mation is available across a broad spectral domain. This helpsan easy conversion from time-domain to spectral domain andvice-versa using Fast Fourier Transform (FFT) and its inverse(iFFT), respectively. This also renders possible a further numer-ical filtering in spectral domain carried out by sliding a Gaussianfilter which replaces the use of a tunable optical probe, asindicated in flowchart of Fig. 3.

Additionally, the temporal resolution of the numerical filtercan be adapted to the practical case under study. In this regard,we turned to data recorded using c-OLCR which operates intime domain by providing information on reflection module andphase over a broad spectral domain. This is discussed in thefollowing section.

III. RESULTS AND DISCUSSION

In what follows, we first give a brief description of the sampleand the experimental setup employed in this work. Later on, the

results describing the identification of coupled cavity are pre-sented and discussed. Finally, the data relative to the determi-nation resonant wavelength of the cavity and are pre-sented to show the efficiency of CFTA to achieve a completeassessment of coupled cavities.

A. Sample and Experimental Setup

The sample investigated here is a 1 mm-long 2-D PhC-WGwith a single-line-defect containing a donor-type SCC locatedapproximately in the middle of the WG similar to the oneswhose fabrication is detailed in [6]. It consists of a perforatedGaAs membrane in an aluminum-free material and exhibitspropagation losses of about 5 dB/cm.

The data exploited is recorded using a phase sensitive OLCRsetup (i.e., c-OLCR). This setup is basically a Michelson inter-ferometer fitted with a broadband source that has a flatspectral power density distribution between 1520–1610 nm.Under this condition the estimated temporal resolution is below100 fs. One arm of the interferometer contains the sampleunder investigation while the other is equipped with a movabledelay-line fitted with a reference mirror. The position of thedelay-line is measured with high precision to obtain phase in-formation [7]. The resulting reflection signal is mathematicallydescribed as

(4)

Here t, unlike in (3) where it is simply a time variable, repre-sents the delay between the two-arms. A reflectogram consistsbasically of a series of fringe patterns whose envelop and posi-tion correspond respectively to the reflection module and phasein time-domain. In our case, the interferometer is completelyfibered and the probe light is injected into the sample usinga single-mode lensed fiber. The reflectograms are recorded byscanning the delay-line between 10 ps to 60 ps over a period of2 s.

B. Identification of Coupled Cavity

A typical reflectogram recorded on the sample is presentedFig. 4(a). It shows two important peaks (A and B) which weidentified to the reflections originating at the end-facets (en-trance and exit) from their position. Interestingly one can fur-ther note three important features in this figure. First, a strongbackscattering amounting to dB all along the WG (in thefirst round trip between A and B). Second, an unusually largewidth for peak B at the exit end (designated here as and

) as compared to the one at the entrance-end (A). Lastlyand most importantly, we noticed no apparent optical signa-ture of the SCC which indeed was surprising. For a further un-derstanding of this observation we relied on the simulationspresented in Section II. They indicate that SCC can be dis-tinguished from disorder induced scattering using low resolu-tion probe conditions. To accomplish this, we applied CTFA inthe following manner. The c-OLCR data recorded in time do-main in a wide spectral range is spectrally filtered (as said, in apurely numerical way, i.e., applying a filter to the Fourier trans-form of the interferogram) by sliding a 2 nm (wavelength-wise)Gaussian-type filter . The latter spectral width is precisely


Fig. 4. Phase-sensitive OLCR data recorded on a PhC-WG inserted with aSCC. (a) Experimental reflectogram. (b) A 3-D representation of ConcurrentTime-Frequency Analysis applied to the data of Fig. 4(a). The first and secondround-trips (r.t.) of light (defined as the time needed for the light to propagateback and forth in the WG) are indicated. Note the decrease in the intensity ofbackscattering after B.

chosen as a trade-off to identify the SCC and to suppress the in-fluence of scattering induced by disorder.

These results in the form of 3-D representation are shown inFig. 4(b). Here, the ordinate corresponds to the central wave-length of the Gaussian filter, the abscissa to time-delay and acoded color to the reflection intensity. At nm, onecan notice in this figure a thin trace starting from C and ex-tending beyond B. We assign it to the signature of SCC forthe following reasons: i) the narrowness of its spectral width(comparable to the filter width) suggests that it is associated toa highly wavelength selective reflector ii) it extends beyond the1st round trip indicating a large photon lifetime around 1553nm (as expected from the discussion in relation to Fig. 1). iii) itcommences in the middle of PhC WG where the cavity is knownto be located physically from technological realization. Finally,the abnormally large width of peak B (observed in Fig. 4(a)) issimply a consequence of PhC-WG dispersion (see wavelengthdependant position of peak B in Fig. 4(b)) that is readily pre-dicted from band structure calculations (performed with planewave expansion method).

C. Determination and

To determine the and resonant frequency of thecavity, we have further examined the temporal evolution ofthe cavity signal around nm and nm,previously estimated resonant wavelength. To this end, thehorizontal sections at these wavelengths are extracted fromFig. 4(b) and plotted as Fig. 5(b) and (a), respectively. A

Fig. 5. Time and spectral signatures of a SCC in a PhC-WG. (a) A cross sec-tion of Fig. 4(b) at the cavity resonant wavelength (� � �� nm). (b) A crosssection of Fig. 4(b) close to the cavity resonant wavelength (� � �� nm).(c) Representation of the data of Fig. 4(b) in spectral domain without (uppercurve) and with filtering (lower curve). The filtering operation consists in re-taining only the information that lies between dashed lines of Fig. 4. Note theabsence of clear evidence of SCC without time-domain filtering.

simple comparison between these two figures clearly reveal thefollowing interesting features: i) at 1553 nm, one can easilylocalize spatially the cavity C and a linear decrease of its signalwith time ii) at 1550 nm, the cavity is not visible but instead,one can only notice a noise ceiling around dB whoseappearance is much like speckle interference effects from thescattering or diffusion signals, iii) the reflection signal from thecavity is far superior to the one due to scattering (12 dB). Inagreement with the above observations, the photon lifetime ofthe cavity is evaluated as ps from Fig. 5(a).

A more precise determination of the cavity resonant wave-length is now undertaken from the examination of the reflec-togram data in spectral domain. The upper curve of Fig. 5(c)is readily obtained after a FT of the reflectogram (of Fig. 4(a))data. It exhibits no clear information on the cavity resonant fre-quency but instead an interference fringe pattern that we as-signed to PhC-WG FP reflections (in relation to multiple roundtrips of light indicated in Fig. 4). Since FP reflections most likelyscreens the observation of cavity signature, the data containedin a selected time-domain region ( ps ps, seedashes in Fig. 4) are further examined in spectral domain. Thisselected time-domain region contains exclusively the informa-tion from the cavity and a single reflection at end-facet B inthe first round-trip. The corresponding data in spectral domain(see lower curve in Fig. 5(c) does not reveal FP fringes pattern


but now exhibits a wavelength selective change in the reflec-tion level. The cavity resonant frequency is consequently deter-mined as nm. The spectral resolution associatedto this measure is estimated 0.2 nm from the width of the se-lected time-domain region and the number of sampled values ofFig. 4(a).

At last, from the and values determined in this work,a complete assessment of the SCC in PhC WG is achieved byestimating a Q factor of 5000.

IV. CONCLUSION

We have proposed and demonstrated the application of con-current time-frequency analysis (CTFA) to achieve a completeassessment of coupled cavities in real PhC WG. Its applicationto the data recorded using c-OLCR measurements fitted with abroad band source ( nm) is shown to suppress selec-tively the signal from disorder-induced scattering often encoun-tered under guiding conditions in real (or actual) PhC-WGs. Theavailability of phase information and the subsequent exploita-tion of CTFA with straightforward linear numerical filtering en-able a rapid assessment of coupled cavities over a broad spectraldomain.

ACKNOWLEDGMENT

Y. Gottesman gratefully acknowledges Dr. L.Nice for en-lightening discussions.

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[14] Y. Gottesman, E. V. K. Rao, H. Sillard, and J. Jacquet, “Modelingof optical low coherence reflectometry recorded Bragg reflectograms:Evidence to a decisive role of Bragg spectral selectivity,” J. Lightw.Technol., vol. 20, no. 3, pp. 489–493, Mar. 2002.

Yaneck Gottesman graduated from the Ecole Supérieure d’Ingénieurs de Mar-seille, France, in 1997, and received the M.S. degree in optics from the EcoleNationale Supérieure de Physique de Marseille, France, and the Ph.D. degreefrom the University of Marseille, France, in 1997 and 2001, respectively.

He is an Associate Professor of optical engineering at the Institut TELECOM,T&M SudParis, France. His current research interests include integrated optics,nano-photonics and Erbium amplification.

Sylvain Combrié, photograph and biography not available at the time ofpublication.

Alfredo DeRossi, photograph and biography not available at the time ofpublication.

Anne Talneau, photograph and biography not available at the time ofpublication.

Philippe Hamel, photograph and biography not available at the time ofpublication.

Alberto Parini was born in Milan, Italy, in 1977. He received the M.S. degreein electronic engineering and the Ph.D. degree in information technology engi-neering from the University of Ferrara, Italy, in 2001 and 2005, respectively.

From 2007 to 2009 he worked on slow-light effects in photonic crystal wave-guides at the Institut TELECOM, T&M SudParis, France, His interests covernonlinear optics, electromagnetic propagation and antennas, design of photonicdevices and photonic crystals.

Renaud Gabet, photograph and biography not available at the time ofpublication.

Yves Jaouen, photograph and biography not available at the time of publication.

Badr-Eddine Benkelfat, photograph and biography not available at the time ofpublication.

Elchuri V. K. Rao, photograph and biography not available at the time ofpublication.

time-frequency analysis for an efficient detection and localization of side-coupled cavities in real...

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