JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 20, NO. 4, APRIL 2002 651

Phase-Matching Engineering in Birefringent AlGaAsWaveguides for Difference Frequency Generation

J. C. G. de Sande, G. Leo, and G. Assanto, Member, IEEE, Member, OSA

Abstract—We investigate refractive-index engineering in form-birefringent AlGaAs–AlAs multilayer waveguides aiming at themaximization of the phase-matching bandwidth for difference fre-quency generation. Acting on layer thickness and sequence, we pre-dict a threefold improvement, as compared to previously reportedresults.

Index Terms—Nonhomogeneous media, nonlinear optics, opticalwaveguides, optimization methods, phase matching.

I. INTRODUCTION

I N THE last two decades, quadratic optical interactions havebeen intensively studied both toward frequency generation

at wavelengths not accessible by established laser transitions,and toward applications in areas such as parametric amplifi-cation and oscillation, signal processing, quantum optics, andcascading [1], [2]. In the presence of fields at frequenciesand , the -induced polarization waves generally includefrequency-sum, frequency-difference, and second-harmoniccomponents. These interactions are governed by two mainconditions, energy conservation (relating the frequencies ofthe three interacting fields) and momentum conservation orphase-velocity matching, required for constructive interferenceof the generated field amplitudes during propagation. In orderto obtain efficient quadratic effects, it is, therefore, necessaryto employ materials with large susceptibilities where itis also possible to implement phase-matching schemes. Thelatter can be grouped in two main families, birefringent phasematching (BPM) and quasi-phase matching (QPM), with theaddition of temperature tuning via the thermooptic response.In bulk media, the former approach is not viable in opticallyisotropic crystals such as cubic semiconductors, whereas QPMhas recently proven successful through wafer bonding ofseveral materials [3], [4] and ferroelectric poling [5], [6]. In fer-roelectric waveguides, conversely, QPM is well established indielectric substrates, [7], [8] while its implementation in III–Vsemiconductors such as GaAs requires complex technologiesand faces severe material problems [9], [10].

More recently, stemming from the availability of molecularbeam epitaxy (MBE), design and fabrication of semiconductor

Manuscript received March 27, 2001; revised November 26, 2001. This workwas supported by the European Union (Project ESPRIT 28.202: OFCORSE II).

J. C. G. de Sande was with the National Institute for the Physics of MatterINFM-RM3 and Department of Electronic Engineering, University Roma Tre,Rome 00146, Italy. He is now with the Departamento de Ingeniería de Circuitosy Sistemas, EUITT, Universidad Politécnica de Madrid, Madrid 28031, Spain.

G. Leo and G. Assanto are with the National Institute for the Physics of MatterINFM-RM3 and Department of Electronic Engineering, University Roma Tre,Rome 00146, Italy.

Publisher Item Identifier S 0733-8724(02)02855-4.

waveguides and photonic structures are also benefitting fromthe so-called refractive-index engineering [11]. The latter, rem-iniscent of band-gap engineering in terms of added flexibilityand potentials [12], allows the realization of complex multi-layer optical structures from integrated mirrors to vertical cavitysurface emitting lasers (VCSEL) in material systems such asAl Ga As, owing to a large refractive index reduction withincreasing Al fraction (from 3.48 to 2.94 at m).With regards to nonlinear optics, refractive-index engineeringhas been successfully applied to second order interactions incubic AlGaAs, with BPM achieved through form birefringencein multilayer waveguides [13]. For near-infrared (NIR) lightgeneration, because BPM stems from the balance of dispersionand birefringence between orthogonally polarized modes, mul-tilayers with higher index contrast proved necessary and wereobtained through selective AlAs oxidation, with refractive indexvarying from 2.94 to 1.61 [14].

The use of AlAs oxide (AlOx), recently employed also inVCSEL technology [15], provides further degrees of freedomin AlGaAs refractive index engineering. Such fabricationflexibility paves the way to a true designer’s approach forthe realization of semiconductor nonlinear integrated devices.The optimum design and reliable fabrication of guiding struc-tures with specified linear–nonlinear susceptibility profiles,in fact, can lend itself to devices with a prescribed opticalfunctionality. In this framework, a desirable requirement is therelaxation of the phase-matching condition in guided-wavefrequency conversion. This issue, analogous in many respectsto angle-bandwidth maximization in noncritical BPM in bulk[16], has been addressed in terms of backward parametricinteractions [17], [18], quadratic solitons [19], and laser am-plification by the insertion of quantum wells in AlGaAs–AlOxwaveguides [20]. While the practical importance of BPM re-laxation is rather general, AlGaAs–AlOx nonlinear waveguidesalready led to successful demonstrations of refractive-indexengineering for second-harmonic and difference-frequencygeneration (DFG) [21]–[23], and parametric fluorescence [24].However, the reported peak efficiencies became negligible forwavelength deviations of less than 1 nm from phase matching.

In this paper, we investigate the optimized design of Al-GaAs–AlOx waveguides for DFG aiming at broadening thephase matching condition while retaining a substantially largeconversion efficiency. In terms of practical relevance, however,as a cornerstone for this study we consider the specificationsof the sample experimentally analyzed in [23], and hereafterindicated as REF. The rest of the paper is organized as fol-lows. In Section II, we analyze the DFG phase matching inAlGaAs–AlOx waveguides, introducing a phase-matching

0733-8724/02$17.00 © 2002 IEEE

652 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 20, NO. 4, APRIL 2002

relaxation (PMR) figure that will prove useful in comparingthe performance of various devices. In Section III, we describeour optimization approach and discuss technical–practicallimitations to be accounted for in the design. In Section IV, wediscuss the expected performances. Finally, after exploring,in Section V, a few alternative options, in Section VI, wesummarize the main results and conclude.

II. PHASE MATCHING

In this section, after defining the phase-matching conditionfor DFG, we study the phase mismatch versus the frequency ofone of the two pump fields and introduce a figure of merit forthe BPM bandwidth. Then, with reference to the AlGaAs–AlOxwaveguides under investigation, we evaluate numerically itsmaximum value.

A. Figure of Merit for Phase-Matching Relaxation

Let us consider the collinear interaction of three plane wavesof frequencies . For DFG, energy conserva-tion implies

(1)

and phase matching

(2)

with being the angular frequency (wave vector) of theth beam. Assuming (2) to be fulfilled for a particular set of

frequencies , the phase-matching condition can be expressedas

(3)

with being the refractive index of the nonlinear medium atand being the speed of light in a vacuum. This condition,

which would be automatically satisfied for a dispersion-free ma-terial, can only be met in birefringent media by noncopolar-ized fields of specific frequencies; slight changes in one of thepump frequencies make the process completely inefficient. Theplane-wave conversion efficiency of collinear DFG in absorp-tion-free media, neglecting boundary effects and pump deple-tion, is given by with

, ( ) being the power (in-tensity) of the th field, being the device length, beingthe effective nonlinear coefficient, being the vacuum permit-tivity, and being the wavelength of the generated signal atphase matching.

Efficiency is maximum when the phase mismatch, goes to zero when , and remains very low

for , i.e., the generated power is only appreciablearound phase matching within a small frequency bandwidth,usually defined as the efficiency full-width at half-maximum(FWHM) and corresponding to the . Ourgoal is to maximize such bandwidth. To this extent, adopting astandard approach, let us perturb the pump frequency,while

keeping fixed the pump frequency . Assuming the followingchanges in frequencies and refractive indexes

(4)

neglecting second-order terms and approximating, the phase mismatch can be written as

(5)

Within the limits of this linearized approach, the phase mis-match grows linearly with both the frequency shift aroundand the device length. However, because we aim at maximizing

while considering the device long enough to grant high effi-ciency at BPM, in order to have a small mismatch, the factor insquare brackets should be minimized. Let us introduce a mea-sure of the BPM relaxation through the figure of merit

PMR (6)

where is the frequency shift at which the efficiency fallsat . This positively defined figure provides a quantitative es-timate of the bandwidth within which one of the pump frequen-cies can vary without a significant reduction in. Moreover, thenormalization allows a direct comparison between devices op-erating at different pump wavelengths. The greater the PMR,the broader the band for efficient frequency conversion. Struc-tures with larger PMR not only tolerate larger errors in pumpwavelengths, but also permit a wider tunability of the generatedsignal. From (5) and (6)

PMR

(7)

Clearly, to have a large PMR for a given length, the term insquare brackets should be as small as possible.

B. PMR in AlGaAs–AlOx Multilayer Waveguides

Let us apply the previous expressions to partially ox-idized, multilayer AlGaAs waveguides, with referenceto sample REF in [23]. To this extent, in a one-dimen-sional (1-D) treatment, the efficiency at phase matching is

, where, is a modal effective index

(8)

DE SANDEet al.: BIREFRINGENT AlGaAs WAVEGUIDES FOR DIFFERENCE FREQUENCY GENERATION 653

Fig. 1. TE transverse field profiles at 1.312�m (dashes) and 5.47�m (dots),and TM field profile at 1.058�m (solid line) in sample REF. The refractiveindex distribution is also graphed at 1.312�m. The inset shows a typicalREF-like structure.

the nonlinear overlap integral, the eigenmodal elec-tric-field amplitude at , and runs perpendicular to thelayers. Waveguide REF was grown by MBE on 100 GaAs,with the structure: AlAs substrate / 1300 nm Al Ga As/ 3 [40 nm AlOx/325 nm GaAs] 40 nm AlOx / 1300 nmAl Ga As. Its refractive index profile is sketched inFig. 1, along with the distributions of the interacting eigen-modes in DFG, transverse magnetic (TM)at m( ), transverse electric (TE)at m ( ), TEat m ( ). The large TM field density in theAlOx layers, due to the unusual index contrast betweencrystalline and oxidized films, considerably lowers the TMdispersion curve with respect to the unoxidized case, thusendowing a large TE–TM modal birefringence. In addition(not shown in the figure), is piece-wise constant along

, AlOx , Al Ga As , andGaAs pm/V. [25] This also allows the

nonlinear overlap integral (8) to be rewritten in terms of anonlinear effective thickness , as (in the caseof REF, m).

In the following, we develop the investigation as a sequenceof perturbations on the structure REF, for the DFG interactionTM TE TE . Unless otherwise stated, wekeep constant the waveguide length ( mm) and the ap-proximate values of the pump wavelengths ( m,

m), as in REF.For photon energies below the gap, AlGaAs undergoes

normal dispersion (see Fig. 2 for GaAs and AlAs). In addition,modal dispersion in planar waveguides gives modal effectiveindexes which monotonically increase with frequency,from the substrate lowest refractive index (cutoff) to the corehighest value. Therefore, as illustrated in Fig. 2 for TEandTM modes of REF, for each eigenvalue it is .Conversely, it can be readily shown that, in DFG processes,one of the following must occur at phase matching

(9a)

or

(9b)

Fig. 2. TE and TM effective indexes vsersus frequency (solid lines), foran ideal REF-like structure, with maximum relaxation of phase matching. Thecorresponding effective indexes in sample REF are also shown for comparison(dashes), as well as the material dispersion of GaAs and AlAs (dots).

Because , however, neither one of (9) can be at-tained with a single mode, thus requiring at least two distinctmodes. Moreover, for guided-wave interactions in 100 GaAs,the structure of the tensor couples orthogonally polarizedpumps and a TE-generated signal. Because in DFG,

, dispersion makes (9b) more demanding than (9a) interms of the required birefringence; therefore, we will focus on(9a), with TM and TE polarizations at and , respectively.The modal orders of the inputs are dictated by practical rea-sons; while is larger for pump eigenmodes of the same order,end-fire input coupling is more efficient with fundamental pumpmodes. Finally, there are several reasons to generate a signalin the TE rather than in higher order modes. TEis the leastdemanding in terms of the necessary birefringence for phasematching; a fundamental order output is better coupled to cas-caded devices; if TEis the only eigensolution at , the numberof modes at the pump frequencies is minimized, thus limitingunwanted input coupling to phase-mismatched fields.

It stems from these considerations that in the PMR (rewrittenhere for guided waves)

PMR

(10)

the first two terms in square brackets are positive and the thirdone is negative. Therefore, in order to relax the BPM condition,

should be large enough to compensate for bothand the TE dispersion from to

, weighed by the factor .Let us derive a PMR upper limit in REF-like waveguides, con-

strained to have the same claddings (as in REF), and a multilayerAlGaAs–AlOx composite core to be specified. With reference toFig. 2, we can linearize the TEdispersion curve in a frequencyinterval around , with a slope that zeros the square-bracketterm in (10). The optimum would occur for the largest exten-sion of such frequency range, limited by the maximumpossible variation of the TE effective index around ,

654 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 20, NO. 4, APRIL 2002

i.e., by the material dispersion of AlAs (substrate) and GaAs(core), . In general,

. This slope must balance the positive contri-butions of the first two terms in the square brackets of (10);the smaller terms are the higher is, which, in a givenwaveguide, would be highest for .Furthermore, by assuming the TEminimum possible disper-sion for (i.e., requiring the second por-tion of the TE dispersion in Fig. 2 to be parallel to the dis-persion of AlAs, the least dispersive within the REF-like struc-ture), the minimum value of the difference is

. Analogously, the min-imization of TM dispersion around is limited by the con-dition , and gives the lowersolid line in Fig. 2. With these limits, within the signal interval

, the PMR upper limit is

PMR

% (11)

However, the range , where we applied a linear perturba-tion of BPM, has a considerable extension. As a consequence,higher order terms become important, such that this upper limitis far from being practically reachable. This occurs because thesecond derivatives of TEand TM dispersion have oppositesigns around and , respectively (in REF, for example,PMR 0.09%). Nevertheless, as shown in the following, thelinear approach provides effective and simple design guidelinesfor BPM relaxation.

Notice that, from a physical standpoint, PMR maximizationcan be pursued by either minimizing slopes of the effective in-dexes around and or increasing the TEslope at .The first option implies an increase of Al content in the wave-guide such that dispersion at pump wavelengths tends to thatof bulk AlAs. The second option, conversely, requires a highercutoff frequency. Although both options lead to a reduction inthe overlap integral , we discarded the second because the gen-erated signal would only be weakly guided, thus undergoingsubstantial scattering losses in a real device.

III. OPTIMIZATION PROCEDURE

Hereafter, we design a series of REF-like waveguides,varying their parameters in order to relax the phase-matchingcondition. In this section, specifically, we address the utilizedstrategy.

To evaluate the guided field profiles and effective indexes,we adopted a standard 1-D transfer matrix method. Amongthe AlGaAs dispersion models reported in literature, weadopted that of Afromowitz [26], and for the oxide, we used

[27], [28]. The examined planar waveguideshave the following structure: AlAs (substrate) AlGaAs–

(AlOx–AlGaAs)-AlOx–AlGaAs–Air, with being aninteger (see inset of Fig. 1). Both the AlAs substrate andthe top–bottom AlGa As layers (usually thicker and with

higher Al content than the inner AlGa As ones) are selectedto ensure sufficient confinement of the generated signal at.We varied both thickness and Al fraction for each AlGa Aslayer, but we assumed equal AlOx layers, because this does notlimit design flexibility, although it reduces the number of pa-rameters. For the same reason, the (AlOx–AlGaAs)–AlOxwaveguide core, allowing the confinement of pump modes, wastaken to be symmetric.

In order to design structures with high PMR, we used amultivariable function minimization routine based on theNelder–Mead simplex method [29]. Due to the several costfunctions (CFs) needed to take into account technical andpractical limitations, however, the function to minimize is notsimply 1/PMR. The cost functions relate to the requirementsof a large overlap integral ( ), an appropriate AlOx thickness( ), a low Al fraction ( ) in each Al Ga As layer, a lowoverall Al content ( ), and a given pump-frequency rangewhere BPM condition is nearly satisfied ( ). The functionto minimized is then

CFPMR

(12)

where

(13)

The lowest acceptable values of nonlinear overlap integral,AlOx layer thickness , and pump frequency are , ,and , respectively. Correspondingly, , , , andare the maximum allowed values for, , Al fraction ineach Al Ga As layer, and overall Al content . Theweight factors make CF grow more or less smoothly as eachgiven parameter falls out of the desired range. Consistent withthe first DFG demonstration in REF, we let the range [, ]approximately correspond to a Ti : sapphire tunable pump laserof wavelengths between 0.95 and 1.06m. For the AlOx thick-ness, due to the peculiarity of the lateral oxidation processestypically employed [14], both a lower and an upper limit exist;the former ( nm) is set by the out diffusion of As atomsduring oxidation, whereas the latter ( nm) cannot beexceeded without incurring strain-induced delamination withinthe multilayer [14]. Furthermore, the maximum Al fraction ineach Al Ga As layer ( ) is to prevent their oxida-tion, whereas the upper limit on the overall Al content (

nm) is to prevent cross-hatching effects during the MBEgrowth of thick multilayers. Finally, we selected the minimumoverlap integral to be one third than in REF, i.e., .

These constraints also result in an upper limit for the numberof AlOx–AlGaAs periods in REF-like structures. Moreover,

DE SANDEet al.: BIREFRINGENT AlGaAs WAVEGUIDES FOR DIFFERENCE FREQUENCY GENERATION 655

TABLE ITHICKNESSt (nm), COMPOSITIONy OF Al Ga As LAYERS, AND RECIPROCALNONLINEAR EFFECTIVE THICKNESS�=d (�m ), FOR OPTIMIZED

STRUCTURES. THE (PERCENT) VALUES OF THEFIGURES OFMERIT PMR AND � d PMR ARE ALSO INCLUDED IN THE CASE OFFREQUENCYTUNING ON

EITHER! (WITH ! = ! ) OR! (WITH ! = ! ). THE TYPE ISEXPRESSED BY THENUMBERN OF AlGaAs LAYERS IN THE CORE AND THEINDEX PROFILE

SHAPE (p: PERIODIC, �: HIGHEST REFRACTIVE INDEX IN THE CENTRAL LAYER AND LOWESTINDEX IN THE OUTERMOSTLAYERS; V : LOWESTINDEX IN THE

CENTRAL LAYER AND HIGHEST IN THE OUTERMOSTLAYERS; W : ALTERNATING REFRACTIVE INDEXES). FOR ALL OF THE SAMPLES, THE FOLLOWING VALUES

HAVE BEEN SELECTED: h = 35 nm; 2�=! = 1062 nm; 2�=! = 980 nm; 2�=! = 1312 nm; � = � =3; WAVEGUIDE

LENGTHL = 1:2 nm. THE CASE OFREF IS REPORTED FORCOMPARISON

a lower limit for ensures a minimum amount of form birefrin-gence to satisfy BPM in the sought frequency range. Hereafter,we consider waveguides with .

IV. RESULTS

Because we deal with the minimization of a complex mul-tivariable function with specific constraints for the parameters,the solution depends on the initial trial values. In general, thefunction described in (12) and (13) has several local minima de-pending on the selected parameters, , , and . We alwaysstart with values for which BPM is satisfied in the permitted fre-quency range.

In order to keep the number of parameters within limits, weinvestigate structures with a symmetric core consisting of AlOxlayers of the same thickness, alternating with one of the fol-lowing structures:

a) identical AlGaAs layers (structure);b) central Al Ga As layer with highest Al-content, i.e.,

lowest refractive index, and surrounding layers withmonotonically decreasing( index profile);

c) central Al Ga As layer with lowest Al content, i.e.,highest refractive index, and surrounding layers with mo-notonously increasing ( index profile);

d) Al Ga As layer with alternating values of( indexprofile).

With regard to such structures, the parameters and the calculatednonlinear overlap integral are summarized in Table I, whichrefers to designs where corresponds to m and

to m (throughout the paper, each struc-ture is labeled as , with and referring to Table and to the

th row within such table, respectively). The related values ofPMR shown in Table I ( ), are almost twice as high asin REF. The detuning curve of waveguide (which is the onewith highest PMR in Table I as is tuned) is drawn in Fig. 3,where the efficiency is also shown.

The considerable relaxation of BPM with respect to REFappears counteracted by a lower phase-matched efficiency,but higher values of could be achieved by operating withlonger propagation distances. By doing so, becauseand PMR , the effects of a larger in improving theefficiency prevail on the PMR-driven reduction. Therefore, inorder to ascertain whether the bandwidth increase associatedto a given design is more useful than simply making a shorterversion of another design, we also introduce the dimensionlessfigure of merit PMR ).

Finally, Fig. 4 shows TEand TM dispersion curves for ,compared with REF. Clearly, the optimization leads to consider-able reductions of TMdispersion around and TE between

656 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 20, NO. 4, APRIL 2002

TABLE IIDESCRIPTIONS ANDPERFORMANCES FOROPTIMIZED STRUCTURES OF THESAME TYPE AS IN TABLE I, WITH � = � =5 AND FREQUENCYTUNING ON !

Fig. 3. Conversion efficiency versus� for samplesI:5 (solid line), II:5(dotted line), and REF (dashed line). Curves are normalized to the peakefficiency of REF; the same waveguide lengthL = 1.2 mm and pump powerP have been used in all cases.

Fig. 4. TE and TM effective indices versus frequency (solid lines) forsample I:5. The corresponding effective indexes in REF are shown forcomparison (dashes), as well as the material dispersion for GaAs and AlAs(dots).

and . Although this is consistent with the strategy of in-creasing Al content rather than the TEcutoff frequency, theslope of TE dispersion around is also reduced.

V. VARIOUS DESIGNS

In the previous sections, we defined a set of design require-ments (such as AlOx layer thickness and pump sources). In thissection, we explore the possibility of enlarging PMR when someof them are modified.

A. Lower Bound in the Nonlinear Overlap Integral

So far, we set . In order to increase PMR, itseems appropriate to relax such conditions, such as by requiring

. With this new lower bound for the overlap in-tegral (8), we explored further design options, summarized inTable II. Regretfully, although BPM is only moderately relaxed(with a slight increase of PMR compared to Table I), a sig-nificant reduction occurs, typically larger than twofold (seeFig. 3).

B. Lower Thickness of AlOx Layers

If AlOx thickness is left as a free parameter, the minimizationconsistently converges to solutions with . Therefore, tospeed up computation, we set in the previous geome-tries. Nevertheless, it is interesting to study PMR in structureswith thinner AlOx layers. In fact, in spite of the limits discussedin Section III in the framework of a conservative design, Al-GaAs–AlOx waveguides with AlOx-layers as thin as 21 nm hasbeen reported [27]. Taking nm, we launched a fewoptimization routines with , which converged to the

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TABLE IIIDESCRIPTIONS ANDPERFORMANCES OFOPTIMIZED, PERIODIC-CORE WAVEGUIDES WITH FREQUENCY TUNING ON ! , h = 21 nm, AND

OTHER PARAMETERS, AS IN TABLE I

TABLE IVDESCRIPTIONS ANDPERFORMANCES OFOPTIMIZED PERIODIC-CORE WAVEGUIDES WITH FREQUENCY TUNING ON ! , h = 0, AND

OTHER PARAMETERS, AS IN TABLE I

Fig. 5. Interleaved histograms of PMR (odd columns, left axis) and� d PMR (even columns, right axis), for severalp structures withsmall AlOx thickness. The case of REF is reported for comparison. In thePMR histogram, white, gray, and black columns correspond to Tables I,III, and IV. In the � d PMR histogram, up-right-down-left-hatched,up-left-down-right-hatched, and cross-hatched columns correspond to Tables I,III, and IV, respectively.

results listed in Table III. Albeit moderate, the observed PMRimprovements suggest an exploration of AlOx thickness reduc-tion, regardless of the present technological limitations. The nu-merical study without lower bound onprovides an additionalincrease on both PMR and PMR, detailed in Table IValong with the relevant parameters. Fig. 5 displays such figuresof merit, comparing them with those obtained in previous cases.Note that the best case refers to an AlOx thickness as low as4 nm, which at present appears beyond realization by wet selec-tive oxidation.

C. Tunable Pump at

The choice of the tunable source as the high-frequencypump around , adopted so far, stems from the availability ofTi : sapphire lasers, as employed in DFG demonstrations withREF [23]. However, a PMR amelioration can be achieved bykeeping the high-frequency source at a fixed frequency and

tuning the low-frequency pump. This can be appreciated byrewriting (4) for this case

(14)

Linearizing the phase mismatch, the corresponding PMR is

PMR

(15)

Therefore, for a fixed and a tunable , large PMR valuesare obtained if the difference and the derivative

are low while is high. Expressions(15) and (10) are similar, except for the first two terms in squarebrackets. However, because TEdispersion around is usu-ally much lower than for TM modes around (see Fig. 4), thephase mismatch increases more slowly whenis tuned with

kept constant.Let us then reconsider the structures described Table I, and

calculate the corresponding PMRs for fixed and tun-able, i.e., for m and ranging from 1.317to 1.330 m, depending on design. As can be seen in Fig. 6,a considerable PMR increment is obtained throughout. As anexample, in Fig. 7, we compare the efficiencies for, for atuning in or in , respectively. It is apparent that, for a givensample, as the tunable source greatly relaxes the BPM con-dition, proving to be an approach more effective than those ex-amined in Sections V-A and V-B.

D. Other Pump Sources

From the previous sections, it appears that a relevant relax-ation of BPM (without unacceptable penalties on the nonlinearoverlap) is more readily achieved through a reduction of the first

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TABLE VDESCRIPTIONS ANDPERFORMANCES OFOPTIMIZED PERIODIC-COREWAVEGUIDES WITH N = 3 FORDIFFERENTWAVELENGTHS� , � , and� (�m) OF THE

THREE INTERACTING FIELDS, TUNING ON ! , AND OTHER PARAMETERS, AS IN TABLE I

Fig. 6. Interleaved histograms of PMR (odd columns, left axis) and� d PMR (even columns, right axis) relative to the samples of Table I,for two different choices of the tunable frequency:! (PMR white columns;� d PMR up-right-down-left-hatched columns) and! (PMR graycolumns;� d PMR up-left-down-right-hatched columns).

Fig. 7. Conversion efficiency versus wavelength of the generated signalaround! for sampleI:5, with the tuning on either! (solid line) or!(dotted line). The detuning is also shown for REF (dashed line). Curves arenormalized to the peak efficiency of REF, and the same pump powerP hasbeen used in all cases.

two terms in square brackets in (10) and (15), rather than an in-crease of the third term. It is noteworthy that, in order to increasethe third term, the frequency should be close to cutoff, wherehigher losses would occur for the generated signal. The maincontribution to the first two terms is due to the high materialdispersion in the near infrared region. Therefore, an approachto generate midinfrared signals through relaxed-BPM DFG be-tween two NIR pumps is to use the lowest possible pump fre-quencies. Fig. 8 shows the figures of merit relative to four-corestructures, for several pairs of pump wavelengths. Such simplewaveguides ( , nm, , and further pa-rameters listed in Table V) can be tuned via the low-frequencypump. The first pair of wavelengths in Table V corresponds to

Fig. 8. Interleaved histograms of PMR (odd columns, left axis) and� d PMR (even columns, right axis) for REF and samples of Table V,where phase matching is satisfied at different sets of frequencies (frequencytuning on the pump! ).

Fig. 9. Conversion efficiency versus wavelength of the generated signalaround! for waveguides of various lengthsL: sampleV:4 (L = 1:96

mm, solid line); sampleIII:1 (L = 1:96 mm, dash-dot-dot line); sampleI:5 and tuning! (L = 1:99 mm, dash-dot line); sampleI:5 and tuning!(L = 1:99 mm, dotted line); and REF sample (L = 1:2 mm, dashed line).Curves are normalized to the peak efficiency of REF, and the same pump powerP has been used in all cases.

Nd : YAG and a commercial laser diode, whereas the second andthird combine the He–Ne emission at 1.15m with NaCl-centerlaser wavelengths. For these pump wavelengths, signals are gen-erated at m (the higher the PMR, the largerthe signal bandwidth). Finally, DFG pumped by the fourth pairof pumps ( , m, available by commercial laserdiodes) provides a signal at m: in this case, PMR ismore than six times larger than in REF. Fig. 9 shows the calcu-lated conversion efficiencies for some of the samples illustratedhere, for signals about 5.5m. In plotting such detuning curves,at variance with Fig. 3 and Fig. 7, we have adjusted the wave-guide length in order to maintain . In such a way,

DE SANDEet al.: BIREFRINGENT AlGaAs WAVEGUIDES FOR DIFFERENCE FREQUENCY GENERATION 659

although keeping the same phase-matched efficiency as in REF,the BPM relaxation afforded by various designs is readily ap-preciable.

VI. CONCLUSION

We have carried out an optimization study for DFG inAlGaAs–AlOx multilayer waveguides, a versatile scenariofor refractive-index engineering in semiconductor integratedoptics.

Whereas the flexibility offered by growth–oxidation tech-nologies has only been exploited to date in a punctual fashion(i.e., by conceiving structures for DFG at a specific wavelength),here we have focused on broadening the phase-matching wave-length acceptance in these form-birefringent waveguides.Clearly, the motivation stems from the practical difficultyof experimentally achieving momentum conservation inintegrated frequency generators with high dispersion and,therefore, narrow tolerances. With reference to the parametersof the initial DFG experiments in AlGaAs–AlOx waveguides[23], we have examined several optimization options aimingat increasing the energy (or frequency) difference betweenthe band gap of the materials and the interacting waves. Thisenables the lowering of the dispersion and, thus, enhances thefigure of merit PMR suitably in (10) in order to quantify therelaxation of the phase-matching condition in different devices.

Choosing the wavelength as the detuning parameter, thestrategy consists of reducing the overall dispersion on thewavelength-tunable input. Assuming that phase matching ismaintained, this can be pursued through the reduction of AlOxcontent in the waveguide. Such an approach, however, wouldbe severely hampered by technologic limits on the lowestachievable thickness of oxide layers. Bandwidth enhancementcan, therefore, be improved through wavelength tuning on thelow-frequency input. In this case, constraining, for instance,the pump lasers to be exactly as in [23], PMR can be doubledat the expenses of a 60% reduction in peak conversion effi-ciency. Moreover, operating in the same range of generatedwavelengths but with different and currently available lasers, asubstantially larger improvement of PMR (more than threefold)can be expected at equally high conversion levels. Finally, bymonitoring the figure of merit PMR, which followsthe same trends as PMR versus design parameters, we couldascertain that our optimized structures are preferable to shorterversions of other designs.

The present study is a thorough example of refractive indexengineering in a complex multilayer structure for opticalparametric generation, and can be readily generalized to otherquadratic interactions such as second-harmonic and sum-fre-quency generation. Such capability, with the extreme flexibilitybuilt-in a numerical approach, is expected to play a significantrole in the development of semiconductor nonlinear elements inintegrated optics circuitry for tunable light generation, sensing,and communications.

ACKNOWLEDGMENT

The authors wish to thank Prof. V. Berger for stimulatingdiscussions and for critical reading of the manuscript, and

the Spanish MEC (Ref. PR2000-0048 0 042 072 826) for anextended visit at the department of electrical engineering atUniversity Roma Tre, Rome, Italy.

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J. C. G. de Sande, photograph and biography not available at the time of pub-lication.

G. Leo, photograph and biography not available at the time of publication.

G. Assanto(M’99), photograph and biography not available at the time of pub-lication.