Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group

Discrete photonics resonator in coupled waveguide arrays

Open Access Open Access

Abstract

We demonstrate both theoretically and experimentally that discrete diffraction resonance can be designed, fabricated, and successfully probed in functionalized – guidonic – coupled waveguide arrays. We evidence that double-barrier patterning of the coupling creates wavelength-independent angular tunnel resonance in the transmitted and the reflected intensity of light beams freely propagating in the plane of the array. Transmission peaks obtained are associated with resonant excitation of the engineered array bound supermodes of the functionalized array, in agreement with accurate and practical numerical modeling based on extended coupled-mode theory. The linear operation of the guidonic resonant tunneling double barrier makes up an original resonator for discrete photonics, suitable for all-optical control of light.

© 2014 Optical Society of America

1. Discrete resonant tunneling double barrier: modulated transmission for active control

1.1 Light control in structured arrays: guidonics

Coupled waveguide arrays open new ways to control light [1,2]. Due to the evanescent coupling of light from a waveguide to the next, the propagation behavior of beams – i.e. superpositions of Floquet-Bloch waves of limited extension over several waveguides – has no equivalent in continuous optical media. Engineering beam propagation in the guided regime is thus a promising route for all optical signal processing. We have proposed built-in patterning of the coupling constant C between pairs of neighboring waveguides for efficient and original manipulation of 2D Gaussian beams in discrete functionalized waveguide arrays. We use the phase shift between successive waveguides, i.e. the beam propagation direction, as an angular engineering parameter [3,4]. In analogy with electronics where built-in bandgap engineering is put to good use, we call “guidonics” this approach of discrete photonics [3]. Beyond the conventional guided regime, guidonic beams can indeed be stirred, redirected, or imaged by patterns of evanescently coupled waveguides while remaining in the guided regime. Several patterns have been demonstrated so far, based on a local defect [5] or on interfaces [4,68]. Guidonics thus offers basic propagation schemes implementing alternative lines of approach for manipulation of guided light in the context of all-optical light control [9]. Advanced active functions such as optical switching or routing within the guidonic approach have already been proposed [3]. Those functions require the implementation of a discrete photonics resonator.

The resonator is indeed a key photonic component which makes it possible to build up high intensities, enhance light-matter interactions both in free and guided wave propagation, and permit resonant operation. In numerous existing approaches, the wavelength is the control-parameter which is swept over the resonance, as in distributed feedback lasers [10], horizontally or vertically evanescently coupled lasers [11,12], micro-disk or photonic crystal optical memories [1315]. The incidence angle may also be a control-parameter together with the wavelength, e.g. in sub-wavelength metallic or dielectric arrays [16,17]. In strong coupling effects in semiconductor heterostructures, in-plane invariance also gives rise to dispersive angular behavior, but angle and wavelength are not independent [18,19].

In contrast, guidonic systems are periodic and highly anisotropic systems, so a strong angular dependence of propagation is expected within a Brillouin zone, whereas propagation dependence on wavelength remains small and non-resonant. Propagation is indeed essentially governed by the coupling constant pattern and coupling constants are only weakly dependent on wavelength. More precisely, propagation is dictated by the diffraction relationship between components of wavevectors, which can be seen as an angular dispersion. As a result, management of the propagation angle with respect to waveguide direction provides an independent control-parameter for all-optical functions [20,21], in particular for the operation of the resonator. In the following we show that such angle-controlled resonators can be designed, fabricated and probed in the guided wave regime using patterning of the coupling constant and guidonic beam dephasing rather than the conventional path length dephasing of standard plane waves.

1.2 Resonant leaky modes addressed by oblique beams - the double-barrier scheme

In guidonics, a resonator has to be devised around a cavity arising from interfaces between different coupling constant regions with strong reflection coefficients. An isolated channel – i.e. a CLCHCL pattern with two parallel interfaces between zones with low (CL) and high (CH) coupling constants between pairs of neighboring waveguides – indeed gives rise to one or several resonant bound super-modes and thus enables light confinement [3,7]. But these confined modes are accessible only through head-on i.e. inline injection of light in the high-C zone [8]. This is not desired in the context of routing and commuting beams, since the super-modes should be addressed with variable oblique beams. We thus need to design leaky quasi-modes that can be accessed resonantly by an incoming beam with an adequate angle. These bound quasi-modes of a discrete resonator, whose resonant excitation leads to angular transmission peaks, are a major ingredient for active discrete photonics.

Starting from the CLCHCL channel super-modes, we design those leaky modes via a new CHCLCHCLCH structure with four interfaces. We call our scheme a Resonant Tunneling Double Barrier (RTDB) as it is the discrete photonics equivalent of the double-barrier diode in electronics [22,23]. The term “barrier” for the low coupling constant zones will be used in reference to those former developments. In Section 2 we explain how this structure provides the desired leaky modes and modulated transmission, first using an analogy and then with quantitative arguments and accurate simulations. In Section 3 we take into account realistic conditions. Section 4 reports the experimental demonstration and Section 5 the discussion of the results.

2. Modeling the discrete guidonic resonator

2.1 Principle of operation

Let the straight waveguides lie in the {X,Z} plane with Z the waveguides' axis. Taking the origin of angles along Z, several variables can trace the angular dependence of the propagation of a beam: the angle of the central direction of energy propagation, either internal or external, the angle of the normal to the phase fronts, or more straightforwardly the Brillouin-zone wavevector Kx. In the following we indeed use Kx, which can easily be related to the other angles owing to the diffraction relations. Uppercase K refer to true wavevectors and lowercase k to reduced dimensionless wavevectors, i.e. kx = KxSH and kz = Kz/CH, where SH is the array period in the high-C regions. Beam propagation in the array is solely governed by its diffraction relation Kz(Kx) which is a succession of bands and bandgaps. In the guidonic approach, this relation is engineered by patterning the coupling coefficient [3].

An intuitive picture of the RTDB scheme operation in a simple ideal case is given in Fig. 1. A guidonic RTDB is a homogenous space of identical waveguides identically coupled everywhere – thus providing the reference Brillouin zone which we adopt henceforth – except within two barriers where the coupling constant is lowered (see Fig. 1, barriers in purple). The analogy of guidonics with optics or electronics has been developed elsewhere [3]. As can be expected from this analogy [22,23] and is demonstrated below, the two barriers create bound quasi-modes nearby specific kx values in the high-coupling region in-between the barriers. Those modes are connected with the truly confined modes of the isolated channel CLCHCL, but they are not strictly confined [24]. Therefore, due to conservation of Kz along the interfaces, they can be addressed by oblique incoming beams. At specific angles of incidence matched to a mode, the beam is transmitted, while it is reflected at other angles. This modulated transmission across the barriers can also be swept via an external parameter, e.g. optical non linearity for all-optical signal processing purposes.

 figure: Fig. 1

Fig. 1 Top left: the RTDB, a set of identical waveguides identically coupled almost everywhere (blue) except in two barriers (purple) where the coupling constant is lowered by a factor of 2 i.e. CH/CL = 2. A guidonic beam is sent across the array with an angle of incidence characterized by its wavevector component kx. Bottom middle: confinement in-between the barriers results in a set of bound quasi-modes at the band edges (red solid line) and at mid-band (red dashed line). Bottom right: CMT diffraction relations in the barriers and elsewhere (solid line), with evanescent continuations (dash-dotted line) plotted vs. ikx since they correspond to pure imaginary values of kx ; all zones here have the same array period. Top right (resp. bottom left): kx (resp. Kz) controls the transmission of the RTDB.

Download Full Size | PPT Slide | PDF

In the next subsections we develop the design and the probing of the RTDB on an accurate basis. In addition we give insightful simple interpretations in terms of plane waves or eigenmodes. We model RTDBs with M array periods in either barrier and N periods in the central channel, summarized as BMCNBM. The ratio of coupling coefficients in the channel and in the barriers is δ = CH/CL > 1. Our typical structure throughout Section 2 is the B1C6B1 structure where barriers involve a single reduced coupling with δ = 2 and the channel six normal couplings. This arrangement has been found to be a good test structure by simulations and is also implemented among others as real arrays (Section 4). The array period is 5.5 µm and the high coupling constant 1 mm−1 at the wavelength of 1.55 µm as in the experiments.

2.2 Simple analytical model - the coupled-mode theory

The above picture can be supported by a simple analytical model based on coupled-mode theory (CMT) and a transfer matrix approach (Appendix). In an homogeneous array with coupling constant C obeying CMT, plane waves use only the upper band of the diffraction relation approximated as a cosine curve Kz = 2Ccoskx with the origin of Kz at the propagation coefficient of the isolated waveguide. For the RTDB obeying CMT, the behavior of plane waves can be exactly established (Fig. 2).The main result is that reflectivity vanishes at discrete input kx values (see Fig. 1 and Fig. 2), solutions of Eq. (11) which writes for M = 1:

tanNkx=fM=1(kx)δ2sin2kx1δ2cos2kx
As expected the latter are in close connection with the wavevectors of the supermodes of an isolated channel (see Eq. (13)), an analogue for coupled waveguides of the quantum well equation for electrons). Those modes evanescently vanish on either side of the channel, within about one array period inside either barrier for large enough coupling ratios δ>1.5. Consequently, they extend over ~N + 1 waveguides and N + 2 array periods, so that their kx are approximately separated by π/(N + 2). The transmission peaks of the RTDB exhibit such regular arrangement in the Brillouin zone as well. Their width is given by Eq. (14). For the solutions kx = 0 and π the resonance width cancels, so only N + 1 transmission peaks are expected. Equation (14) also predicts that peaks nearby band extrema can be made as narrow as desired by tuning M and δ.

 figure: Fig. 2

Fig. 2 Left: incident (blue), reflected (red) and transmitted (green) plane waves in the typical RTDB, with phase fronts (dashed lines) and wavevectors ( ± Kx,Kz). Center: field modulus as a function of kx and waveguide index m, calculated by CMT for an incident wave of unit amplitude. Right: same as center but m is considered as a continuous variable. This distorted view displays more clearly the bound supermodes and the transmitted and stationary waves.

Download Full Size | PPT Slide | PDF

Concerning now the array quasi-modes, the conservation rule of Kz at interfaces δ.coskx = coskxB shows that the wavevector in the barriers kxB is imaginary when coskx>1/δ i.e. in the vicinity of the extrema of the widest band, and real between those of the narrowest band. Hence, there are two different kinds of unconfined modes bound to the RTDB. The modes at the band edges (Fig. 1, solid red lines) arise from the confined modes of the isolated channel and can be referred to as quasi-modes of the RTDB structure legitimately, since they are weakly coupled to the semi-infinite regions on either side of the double barrier through evanescent coupling. Their field is enhanced between the barriers. Conversely, the modes around the band center (dashed red lines) arise from the continuum of unconfined modes and are strongly coupled to the homogeneous semi-infinite zones through pure dephasing relations which do not exhibit strong constructive interference e.g. of the Fabry-Pérot type. They are expected to be less efficient for optical control than the first ones since the field between the barriers is less enhanced, despite their resonant behavior. This shows in the lesser finesse of the corresponding resonance peaks (Fig. 2 center).

CMT gives a rough but analytical description of guidonic structures. However it is not sufficient to describe correctly real arrays where modes of isolated waveguides are most often notably non-orthogonal and where second-neighbor coupling arises at large coupling [25]. Besides, CMT encompasses only the upper band of the diffraction relation. The following methods we developed do provide a correct description.

2.3 Accurate numerical model - the extended coupled-mode theory

A good approach to the full description of the RTDB transmission is the two-dimensional effective index (2DEI) method, which has been applied so far to periodic strip-loaded waveguide arrays [8]. The pure plane-wave scheme outlined in Appendix within CMT can also be implemented in 2DEI, and applied to non-periodic structures. This supplies their transmission curves, not only in the upper band but also incorporating the lower bands either confined or unconfined by the strips. Resonances in the lower bands are found to broaden and merge into a continuum as the effective index decreases. 2DEI indeed provides a comprehensive and realistic picture of simple guidonic systems. However its implementation in real cases with complex structures and limited-width beams is cumbersome. We use instead a lighter tool to simulate propagation in the upper band we actually consider in experiments, namely an extended CMT (eCMT) that we developed earlier [25].

eCMT provides the realistic framework that CMT cannot, at the expense of a slightly greater complexity, and is accurately validated by experiment [8,25]. In addition to the CMT coupling coefficient, eCMT involves three additional parameters describing the still small but significant deviations to CMT, obtained from the mode of an isolated waveguide or from universal equations [25]. It requires the numerical solution of the modified propagation equations [da/dz] = [[P]].[a], where [a] is the vector of modal amplitudes [am(z)] and [[P]] is the propagation matrix. Its coefficients for homogeneous zones are combinations of their eCMT parameters. Around interfaces, they are complex combinations of the parameters of the adjacent zones [8].

The eigenmodes of [[P]] strongly link with the CMT plane-wave calculation (Fig. 2) since they are stationary combinations of plane waves in a limited space, with reflected waves in both outer regions. With field cancellation at both ends of the structure as boundary conditions, the eigenvalues which sample the Brillouin zone can be labeled by wavevector kx (Fig. 3). As in CMT, N + 1 bound supermodes appear, approximately equally spaced in the Brillouin zone. The move from CMT to eCMT shows in the diffraction relation which has no longer a true cosine shape (Fig. 3 left). The profiles of the field at resonance kx clearly show exaltation within the resonator, all the greater for modes nearer to the band edges. Figure 3 also displays the outside lateral oscillations of the modes, i.e. the stationary waves in semi-infinite regions, which line up as hyperboles in the Brillouin zone with asymptotes along the interfaces with the barriers.

 figure: Fig. 3

Fig. 3 Eigenmodes of the eCMT propagation matrix for the typical RTDB in a 400-waveguide array. Left: eigenvalues Kz labeled by kx (line) nearly perfectly overlap the diffraction relation of the high-C regions. Selected values (dots) correspond to the 7 bound modes. Middle: map of eigenvectors I(kx,X/S), zoomed in X around the RTDB; eigenvector profiles have been convoluted with the single-waveguide mode profile to smooth the picture. Right: selected eigenvector profiles I(X/S) corresponding to the 7 bound modes, also zoomed; symbol colors are matched with those on the diffraction relation (left) and the eigenvector map (middle).

Download Full Size | PPT Slide | PDF

2.4 eCMT and Fabry-Pérot scheme - the shortest way to transmission curves

Within this simple eCMT scheme, the transmission curve is obtained by solving numerically the propagation equations for finite-width input beams (e.g. Figure 4, blue dots). Mimicking the ideal plane-wave system with a large enough input beam is more challenging. However since the curve for non-resonant structures can be accurately calculated using adequate beams, we can treat RTDBs by combining the accurate eCMT with a plane-wave analysis in the loss-less Fabry-Pérot interference scheme. To this end we use the mirror reflectivity of a single barrier CHCLCH obtained by the accurate eCMT calculation, and a resonator thickness of a N + 2 period array. As expected transmission curves obtained (e.g. Figure 4, green curve) are similar to the CMT result (Fig. 1) but are no longer symmetric with respect to kx = π/2, because of the asymmetry of the diffraction band and hence of the barrier reflectivity curve. In particular, the peaks near kx = π are significantly broadened. The transmission for finite beam widths is obtained readily by convolution in Kx space. As shown in Fig. 4 (blue line) this simple approach is validated by the full eCMT calculation; the same agreement is obtained for various RTDBs with high enough coupling ratio δ and various beam widths. The small discrepancies point to the limits of both approaches.

 figure: Fig. 4

Fig. 4 Transmission of the typical RTDB. Orange: eCMT reflectivity of CHCL interface. Red: eCMT reflectivity of single barrier CHCLCH. Green: plane wave Fabry-Pérot model for the CHCLCHCLCH double barrier using red curve as mirror reflectivity. Blue line: same as green with broadening by the finite beam width W = 100µm. Blue dots: eCMT simulation for the same RTDB and width. Vertical dashed lines: kx of eCMT eigenmodes of the double barrier.

Download Full Size | PPT Slide | PDF

This alternative view provides very fast assessment of RTDBs: different structures with the same barriers are evaluated with only one tedious calculation, the barrier reflectivity. The influence of beam width can also be readily explored. We now examine in more detail this problem, which is central in assessing experiments, before reporting them.

3. Probing realistic RTDB structures

3.1 The guidonic signature

First, as a reference we show in Fig. 5 left a generic eCMT propagation map across the typical RTDB which displays all features of such maps. The calculation involves injection in a large array of a wide beam and a long propagation path; all these notions will be specified in the following. The input beam is partly reflected and partly transmitted, quite satisfactorily preserving the beam shape even though the beam profile is not as regular in the reflected beam as in the incident beam. As expected, stationary waves between incident and reflected beams are observed near the first interface. Even though the selected kx does not correspond exactly to a resonance, a bound mode of the RTDB is excited locally, here the third mode with three maxima along X. It lasts until reflected and transmitted beams have been extracted.

 figure: Fig. 5

Fig. 5 Left: propagation of a guidonic beam (W = 100µm, kx = 0.35π, L = 50 mm) across the typical RTDB. Middle: guidonic signature. The input beam is rotated around mid-length and mid-structure to scan kx and the output intensity profile along X is monitored. Right: model signature I(kx,X/L). The vertical line indicates the 0.35 π abscissa used in the left-hand figure.

Download Full Size | PPT Slide | PDF

The best way – and the experimental one – to summarize such propagation patterns for beams injected at all angles is to map the output intensity profile as a function of kx. The injection point and angle are simultaneously adjusted so that the input beam rotates around the center of the RTDB at mid-propagation (Fig. 5 center). Figure 5 right shows the corresponding dimensionless map I(kx,X/L) involving the deviation X/L, that we call the signature of the RTDB, a modified version of the usual one where the input beam rotates around a point at the input facet [8]. This signature exhibits modulated alternate transmission and reflection spots with nearly zero minima. These spots lie on trajectories which merely reflect the normal beam behavior in either homogeneous semi-infinite region [8].

3.2 Expected experimental limitations

We first note that no intensity is observed in the signature within the central channel, except near kx = 0. This is an important though extrinsic point. At those grazing incidences, the propagation path required for the beams to clear out of the interface becomes larger than the one used in the calculation. Therefore at the output facet one still observes the first bound supermodes (#1 clearly and #2 dimly in Fig. 5 right). Decreased paths reveal more modes (left sequence of Fig. 6). This specifies what a “long enough” propagation path is, when one wants to observe clean output beams; the number of waveguides in the array must also be large enough to avoid beam reflections at its edges. Conversely, if short propagation lengths make output beams less distinct, they have the beneficial effect of revealing the resonant modes.

 figure: Fig. 6

Fig. 6 Influence of W and L parameters on the model guidonic signature of Fig. 5. Leftmost sequence: decreasing propagation length; due to the choice of a X/L deviation ordinate, decreasing the length increases the apparent size of the RTDB. Middle sequence: influence of beam waist. Rightmost signature: combination corresponding to the experimental case.

Download Full Size | PPT Slide | PDF

The other limiting parameter is the input beam waist W (radius at 1/e2 of the maximum intensity). First it must be large enough to limit divergence at the output and hence maintain well-defined beams. More important, a finite W corresponds to a wavevector spread Δkx = 2SH/W which defines the angular probe size. In order to display clearly resonances, this spread should be less than their separation π/(N + 2). The middle sequence of Fig. 6 shows the evolution of model signatures with decreasing W, from Δkx = 0.3 π/(N + 2) to 1.4 π/(N + 2), about the actual experimental case. Transmission and reflection remain strongly modulated, but most features are gradually distorted and dampened. Nevertheless the sequence of signatures makes it possible to identify the relevant features even at high Δkx. Figure 6 right displays the signature when both distortions are present, which is actually the experimental case we now describe in Section 4. Despite its limitations this realistic scheme for excitation and characterization of the resonator simultaneously displays the bound modes and the related modulations of the transmitted and reflected intensities.

4. Experimental demonstration: bound quasi-modes and resonant behavior

4.1 Test waveguide arrays and experimental setup

In the actual implementation of the RTDB scheme, we use strip-loaded waveguide arrays with an InP/InGaAsP shallow-ridge design [8]. Barriers are obtained by increasing locally the array period [26], from SH = 5.5 µm for the quasi homogenous array to SL = 6.5/7.0/8.0 µm for the barriers, corresponding to coupling constants of respectively 2.1 and 1.35/1.10/0.88 mm-1 in the TE (X) polarization. The typical structure B1C6B1 involves two barriers with a single low coupling between neighboring waveguides, enclosing a central channel with six high couplings. B1C4B1 and B1C2B1 have also been measured. Propagation length is L ~10 mm.

The experimental setup has been described in detail elsewhere [8]. Here polarized 1.55 µm light is input by the cleaved end of an air-silica microstructured optical fiber [27], specially designed to deliver a single mode of large waist W = 23 µm while remaining flexible (pitch = 28 µm, hole diameter = 13 µm). This fiber is rotated and displaced so as to maintain the guidonic beam passing at mid-length of the sample at the center of the RTDB. Injection by a single-mode fiber has many advantages: position and angle of injection are readily available and the input wave is unambiguously planar. Nonetheless the angular resolution dictated by the input waist is rather low. Even with the exceptionally large waist of our single mode fiber, Δkx is only ~1.2 π/8, i.e. similar to the separation between expected resonance peaks.

4.2 Experimental signature - transmission curve

A typical experimental signature of the RTDB together with the corresponding simulation using eCMT is shown in Fig. 7. Its symmetry and its agreement with the calculation are excellent, without any adjustable parameter. Note that the first 3 bound modes are observed. Similar results are also observed for the RTDBs with other barrier heights or widths and for the TM polarization.

 figure: Fig. 7

Fig. 7 Left: experimental signature of the B1C6B1 RTDB with an injected beam of waist W = 23µm and TE polarization. SB = 8.0µm and L ~10 mm. Middle: eCMT simulation. Right: superposition of the experimental signature with the model signature shown as contours lines.

Download Full Size | PPT Slide | PDF

In spite of those experimental distortions, an unambiguous transmission curve can be obtained in a straightforward manner. Even though transmitted and reflected beams are not clearly seen, their average deviation X/Z is well known since at each kx it is dictated by the derivative of the diffraction relation X/Z = -dKz/dKx. This is more easily seen in long samples (Fig. 5) and has been checked experimentally using homogeneous arrays [8]. In Fig. 8 the corresponding traces R and T are superimposed on the experimental signature in the range [0,π]. We can then extract from the signature the respective intensities r* and t* along those traces. r* and t* are precursors of what large beams would yield upon longer propagation. This procedure corresponds to using the output of selected waveguides as transmitted and reflected signals rather than trying to get the full transmission or reflection. The transmission curve of Fig. 8 is then obtained in a classical way as the variation of t*/(t* + r*) with kx.

 figure: Fig. 8

Fig. 8 Left: part of experimental signature of B1C6B1 (Fig. 7) between 0 and π. Traces of the center deviation of transmitted and reflected beams T and R are shown respectively as green and red lines. Right: transmission curves, calculated for plane waves (dashed line), and obtained along traces from the eCMT model (full line) and experimental signatures (dots).

Download Full Size | PPT Slide | PDF

As expected from the agreement between the complete signatures, the agreement between the experimental curve and the model one is good. More important, both curves display most resonance peaks expected from the plane wave calculation with a significant contrast, in spite of the distortions created by the limited length of the sample and the finite waist of the input beam. In our particular experimental case, a major part of the contrast originates in the reflected beam, but again this is in no way intrinsic to the RTDB scheme. The resonant character of RTDBs is then firmly demonstrated. In the next Section we discuss in detail two important points for future advances, the wavelength dependence and the finesse.

5. Potentialities and limitations of the double-barrier guidonic structure

5.1 Wavelength dependence

As stated in Section 1, the general properties of guidonic structures do not depend strongly or resonantly on wavelength, as they follow the coupling itself. For the experimental waveguides and a wide spectral region 1.55 ± 0.1 µm, coupling constants vary about linearly with λ, with a slope decreasing exponentially with S. The variation with λ of the relevant parameter – the relative barrier height δ – is shown in Fig. 9 left. This moderate variation has a still smaller impact on the RTDB transmission. As an example we show in Fig. 9 right the transmissions of the typical single and double barrier for wavelengths varying from 1.5 to 1.6µm, i.e. twice the telecom C band. The variation of the curves is very small, confirming the wavelength independence of the resonator.

 figure: Fig. 9

Fig. 9 Left: variation of relative barrier height δ with the wavelength λ for various barrier periods SB, for 9, 8, and 7 µm top to bottom. Right: plane-wave transmission of the typical single (dashed line) and double (solid line) barrier for SB = 8 µm and λ = 1.5 (blue), 1.55 (green), and 1.6 (red) µm.

Download Full Size | PPT Slide | PDF

5.2 Specifications for all-optical light control

We now address the potential efficiency of this resonator in functional configurations. We have proposed previously using it for instance in all-optical gates [3] based on the nonlinear shift of the transmission peaks by a control beam in the channel, the analogue of the Stark shift in the original electronic RTDB device. Such devices are inherently superior to solitonic devices operating in non-patterned arrays, especially since they have no threshold. The control power they require is obviously all the smaller as the peaks are steeper, i.e. have a narrower width and a higher contrast. This in turn requires a high barrier reflectivity, which can in principle be obtained at will by lowering the coupling in the barrier and increasing its width (Eq. (14)). However small injection waists and propagation paths lead to peak dampening (Fig. 4). A good compromise between resonance steepness (finesse ~3), resonance contrast (5dB), and realistic test configuration (L<50mm, W<100µm) is obtained for precisely the type of barrier we designed for the experiments, and yields the expected gating specifications with only slightly longer samples and wider beams. Furthermore preliminary calculations show that elaborate solutions are at hand for future improvements using oblique interfaces or highly-coupled waveguides. Deep-ridge waveguides could be used to that effect, at the expense of a higher sensitivity to defects. Since their coupling is intrinsically higher, more compact devices could be designed, or conversely devices with the same size but obeying CMT instead of the more complex eCMT. Due to their shorter periods, more waveguides could be injected by a given input beam, thus also allowing to come closer to the planar-wave case.

6. Conclusion

In agreement with our eCMT simulation, our experimental results directly demonstrate excitation of channel supermodes in a RTDB structure by discrete photonics beams, using incident beam wavevector – equivalently, angle or direction – as a control-parameter. Moreover, the transmitted and reflected beams are definitely connected with excitation of the bound supermodes and exhibit resonant angular behavior: the beam is transmitted when a super-mode is excited, otherwise it is reflected. The connection is made clear thanks to simulations with wider beams and longer samples than in experiments, which sample the narrow transmission peaks of the RTDB more efficiently.

To our knowledge, this is the first experimental evidence of resonant states in waveguide arrays. Furthermore, excitation of such states can be controlled by a wavelength-independent angular variable and, in a reflection set-up, angular resonance could be used to route a wide enough beam. A more general configuration of achromatic directional resonance for guidonic circuitry would only require reshaping and reamplification of the transmitted or reflected beams broadened by diffraction. This RTDB structure thus holds its promise for discrete photonics and nonlinear control of the resonant features in the transmission and reflection can now be expected to achieve all-optical gating. Along the way, we have designed, fabricated and characterized the first discrete photonics resonator. Additionally this resonator possesses a built-in promising advantage inherited from discreteness and coupling patterning, namely independence of wavelength.

Appendix

In an homogeneous region of a waveguide array obeying CMT, the projections an of the field amplitudes onto the modes of the isolated waveguides are solutions of the equations

anz=iC(an-1+an+1),
where C is the waveguide to waveguide coupling coefficient. Propagation eigenmodes can be cast as discrete waves
aneiKzZeinkx.
In a waveguide array pattern comprising two generic homogeneous regions p-1 and p characterized by constant coupling coefficients Cp-1 and Cp respectively - for instance CH and CL in the leftmost part of the RTDB - and separated by an interface waveguide at index np, conservation of the longitudinal component Kz of the wavevector implies that
KZ=2Cp-1coskxp1=2Cpcoskxp.
As two waves, ± kxp, have longitudinal component Kz in region p, a field propagating with that longitudinal component can be written as a superposition of two discrete waves
Apeinkxp+Bpe-inkxp=Ap(n)+Bp(n).
CMT at interface np,
anpz=i(Cp1anp1+Cpanp+1),
and continuity of the field at that interface yield two relations between (Ap(np)+Bp(np)) and(Ap1(np)+Bp1(np)), which can be written using transfer matrix
12isinkxp[e-ikxpCp1Cpeikxp-1e-ikxpCp1Cpe-ikxp-1(eikxpCp1Cpeikxp-1)(eikxpCp1Cpe-ikxp-1)].
Another matrix accounting for propagation between waveguides np and np + 1, according to
[Ap(np+1)Bp(np+1)]=[ei(np+1np)kxp00e-i(np+1np)kxp][Ap(np)Bp(np)],
allows for CMT transfer matrix treatment of any structure made up of any number of stripes of coupled waveguides parallel to the waveguides. In particular, if the transfer matrix of RTDB with N and M inter-waveguide separation distances in the channel and in the barriers respectively is denoted by TNM, the reflection coefficient is given by
rNM=TNM21TNM22,=2iαβ*sinMkxB(|α|2eiMkxB|β|2eiMkxB)+(|α|2eiMkxB|β|2eiMkxB)e2iNkx(|α|2eiMkxB|β|2eiMkxB)2+4|α|2|β|2sin2MkxBe2iNkx,
where
α=eikxBδeikxβ=eikxBδeikx,
and δ is the ratio of the coupling coefficients in the channel – as well as in the incidence and transmission regions – and the barriers. It is easy to see that the numerator of the fraction in Eq. (8) cancels when
tanNkx=cotMkxB|α|2|β|2|α|2+|β|2
or
tanNkx=TM(δcoskx)UM-1(δcoskx)2δsinkx1δ2cos2kxfM(kx),
where TM and UM-1 are Chebyshev polynomials, the ratio of which is δ.coskx when M = 1. Then:
tanNkx=fM=1(kx)δ2sin2kx1δ2cos2kx.
The roots of Eq. (11) are in close connection with the wavevectors of the supermodes of an isolated channel given by:
tanNkx=fM(kx)2δsinkxδ2cos2kx11δ2cos2kx
In addition, as the argument of rNM/tNM is ± π/2 in any symmetric system [28], the width of each transmission resonance at half maximum can be assessed from
tanNkxfM(kx)=±δ21δ21UM12(δcoskx)cosNkx1cos2kx1δ2cos2kx
if the resonance is narrow enough to reach the condition |r|2 = |t|2 = 0.5.

The transmission coefficient can be derived similarly using tNM=(TNM11+rNMTNM12)ei(Nkx+2MkxB), and the relation |rNM|2+|tNM|2=1 holds since the model is lossless.

Acknowledgments

The authors thank Isabelle Sagnes, Anne Talneau, Sophie Bouchoule, and Edmond Cambril for fruitful discussions and key participation in the fabrication of waveguide arrays, as well as Rémi Habert for the optical fiber characterization. This work was partly supported by French Ministry of Higher Education and Research, the Nord-Pas de Calais Regional Council and FEDER through the “Contrat de Projets Etat Region (CPER) 2007-2013”, the “Campus Intelligence Ambiante” (CIA), and the FLUX Equipex Project (“Programme Investissement d’Avenir”).

References and links

1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003). [CrossRef]   [PubMed]  

2. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1–3), 1–126 (2008). [CrossRef]  

3. J. M. Moison, N. Belabas, C. Minot, and J. A. Levenson, “Discrete photonics in waveguide arrays,” Opt. Lett. 34(16), 2462–2464 (2009). [CrossRef]   [PubMed]  

4. A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, “Fresnel’s laws in discrete optical media,” New J. Phys. 10(10), 103020 (2008). [CrossRef]  

5. H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, and A. Bräuer, “Tailoring guided modes in waveguide arrays,” Opt. Express 11(25), 3404–3411 (2003). [CrossRef]   [PubMed]  

6. R. A. Syms, “Approximate solution of eigenmode problems for layered coupled arrays,” IEEE J. Quantum Electron. 23(5), 525–532 (1987). [CrossRef]  

7. N. Belabas, S. Bouchoule, I. Sagnes, J. A. Levenson, C. Minot, and J. M. Moison, “Confining light flow in weakly coupled waveguide arrays by structuring the coupling constant: towards discrete diffractive optics,” Opt. Express 17(5), 3148–3156 (2009). [CrossRef]   [PubMed]  

8. J. M. Moison, N. Belabas, J. A. Levenson, and C. Minot, “Light-propagation management in coupled waveguide arrays: quantitative experimental and theoretical assessment from band structures to functional patterns,” Phys. Rev. A 86(3), 033811 (2012). [CrossRef]  

9. D. N. Christodoulides and E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87(23), 233901 (2001). [CrossRef]   [PubMed]  

10. R. C. Alferness, C. H. Joyner, M. D. Divino, M. J. R. Martyak, and L. L. Buhl, “Narrowband grating resonator filters in InGaAsP/InP waveguides,” Appl. Phys. Lett. 49(3), 125–127 (1986). [CrossRef]  

11. D. Liang and J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4(8), 511–517 (2010). [CrossRef]  

12. Y. Halioua, A. Bazin, P. Monnier, T. J. Karle, G. Roelkens, I. Sagnes, R. Raj, and F. Raineri, “Hybrid III-V semiconductor/silicon nanolaser,” Opt. Express 19(10), 9221–9231 (2011). [CrossRef]   [PubMed]  

13. L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010). [CrossRef]  

14. K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics 6(4), 248–252 (2012). [CrossRef]  

15. A. Shinya, S. Mitsugi, T. Tanabe, M. Notomi, I. Yokohama, H. Takara, and S. Kawanishi, “All-optical flip-flop circuit composed of coupled two-port resonant tunneling filter in two-dimensional photonic crystal slab,” Opt. Express 14(3), 1230–1235 (2006). [CrossRef]   [PubMed]  

16. E. Sakat, S. Héron, P. Bouchon, G. Vincent, F. Pardo, S. Collin, J.-L. Pelouard, and R. Haïdar, “Metal-dielectric bi-atomic structure for angular-tolerant spectral filtering,” Opt. Lett. 38(4), 425–427 (2013). [CrossRef]   [PubMed]  

17. P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, and S. Collin, “Optical extinction in a single layer of nanorods,” Phys. Rev. Lett. 109(14), 143903 (2012). [CrossRef]   [PubMed]  

18. E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, and J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010). [CrossRef]  

19. H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D. Solnyshkov, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, G. Malpuech, and J. Bloch, “Realization of a double-barrier resonant tunneling diode for cavity polaritons,” Phys. Rev. Lett. 110(23), 236601 (2013). [CrossRef]  

20. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90(5), 053902 (2003). [CrossRef]   [PubMed]  

21. C. E. Rüter, J. Wisniewski, and D. Kip, “Prism coupling method to excite and analyze Floquet-Bloch modes in linear and nonlinear waveguide arrays,” Opt. Lett. 31(18), 2768–2770 (2006). [CrossRef]   [PubMed]  

22. T. C. L. G. Sollner, W. D. Goodhue, P. E. Tannenwald, C. D. Parker, and D. D. Peck, “Resonant tunneling through quantum wells at frequencies up to 2.5 THz,” Appl. Phys. Lett. 43(6), 588–600 (1983). [CrossRef]  

23. B. Rico and M. Y. Azbel, “Physics of resonant tunneling. The one-dimensional double-barrier case,” Phys. Rev. B 29(4), 1970–1981 (1984). [CrossRef]  

24. A. M. Childs and D. Gosset, “Levinson's theorem for graphs II,” J. Math. Phys. 53, 102207 (2012).

25. Ch. Minot, N. Belabas, J. A. Levenson, and J. M. Moison, “Analytical first-order extension of coupled-mode theory for waveguide arrays,” Opt. Express 18(7), 7157–7172 (2010). [CrossRef]   [PubMed]  

26. N. Belabas, Ch. Minot, J. A. Levenson, and J. M. Moison, “Ab initio design, experimental validation, and scope of coupling coefficients in waveguide arrays and discrete photonics patterns,” J. Lightwave Technol. 29(19), 3009–3014 (2011). [CrossRef]  

27. P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006). [CrossRef]  

28. A. Zeilinger, “General properties of lossless beamsplitters in interferometry,” Am. J. Phys. 49(9), 882–883 (1981). [CrossRef]  

References

  • View by:

  1. D. N. Christodoulides, F. Lederer, Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
    [Crossref] [PubMed]
  2. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1–3), 1–126 (2008).
    [Crossref]
  3. J. M. Moison, N. Belabas, C. Minot, J. A. Levenson, “Discrete photonics in waveguide arrays,” Opt. Lett. 34(16), 2462–2464 (2009).
    [Crossref] [PubMed]
  4. A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, A. Tünnermann, “Fresnel’s laws in discrete optical media,” New J. Phys. 10(10), 103020 (2008).
    [Crossref]
  5. H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, A. Bräuer, “Tailoring guided modes in waveguide arrays,” Opt. Express 11(25), 3404–3411 (2003).
    [Crossref] [PubMed]
  6. R. A. Syms, “Approximate solution of eigenmode problems for layered coupled arrays,” IEEE J. Quantum Electron. 23(5), 525–532 (1987).
    [Crossref]
  7. N. Belabas, S. Bouchoule, I. Sagnes, J. A. Levenson, C. Minot, J. M. Moison, “Confining light flow in weakly coupled waveguide arrays by structuring the coupling constant: towards discrete diffractive optics,” Opt. Express 17(5), 3148–3156 (2009).
    [Crossref] [PubMed]
  8. J. M. Moison, N. Belabas, J. A. Levenson, C. Minot, “Light-propagation management in coupled waveguide arrays: quantitative experimental and theoretical assessment from band structures to functional patterns,” Phys. Rev. A 86(3), 033811 (2012).
    [Crossref]
  9. D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87(23), 233901 (2001).
    [Crossref] [PubMed]
  10. R. C. Alferness, C. H. Joyner, M. D. Divino, M. J. R. Martyak, L. L. Buhl, “Narrowband grating resonator filters in InGaAsP/InP waveguides,” Appl. Phys. Lett. 49(3), 125–127 (1986).
    [Crossref]
  11. D. Liang, J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4(8), 511–517 (2010).
    [Crossref]
  12. Y. Halioua, A. Bazin, P. Monnier, T. J. Karle, G. Roelkens, I. Sagnes, R. Raj, F. Raineri, “Hybrid III-V semiconductor/silicon nanolaser,” Opt. Express 19(10), 9221–9231 (2011).
    [Crossref] [PubMed]
  13. L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
    [Crossref]
  14. K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics 6(4), 248–252 (2012).
    [Crossref]
  15. A. Shinya, S. Mitsugi, T. Tanabe, M. Notomi, I. Yokohama, H. Takara, S. Kawanishi, “All-optical flip-flop circuit composed of coupled two-port resonant tunneling filter in two-dimensional photonic crystal slab,” Opt. Express 14(3), 1230–1235 (2006).
    [Crossref] [PubMed]
  16. E. Sakat, S. Héron, P. Bouchon, G. Vincent, F. Pardo, S. Collin, J.-L. Pelouard, R. Haïdar, “Metal-dielectric bi-atomic structure for angular-tolerant spectral filtering,” Opt. Lett. 38(4), 425–427 (2013).
    [Crossref] [PubMed]
  17. P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, S. Collin, “Optical extinction in a single layer of nanorods,” Phys. Rev. Lett. 109(14), 143903 (2012).
    [Crossref] [PubMed]
  18. E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010).
    [Crossref]
  19. H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D. Solnyshkov, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, G. Malpuech, J. Bloch, “Realization of a double-barrier resonant tunneling diode for cavity polaritons,” Phys. Rev. Lett. 110(23), 236601 (2013).
    [Crossref]
  20. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90(5), 053902 (2003).
    [Crossref] [PubMed]
  21. C. E. Rüter, J. Wisniewski, D. Kip, “Prism coupling method to excite and analyze Floquet-Bloch modes in linear and nonlinear waveguide arrays,” Opt. Lett. 31(18), 2768–2770 (2006).
    [Crossref] [PubMed]
  22. T. C. L. G. Sollner, W. D. Goodhue, P. E. Tannenwald, C. D. Parker, D. D. Peck, “Resonant tunneling through quantum wells at frequencies up to 2.5 THz,” Appl. Phys. Lett. 43(6), 588–600 (1983).
    [Crossref]
  23. B. Rico, M. Y. Azbel, “Physics of resonant tunneling. The one-dimensional double-barrier case,” Phys. Rev. B 29(4), 1970–1981 (1984).
    [Crossref]
  24. A. M. Childs, D. Gosset, “Levinson's theorem for graphs II,” J. Math. Phys. 53, 102207 (2012).
  25. Ch. Minot, N. Belabas, J. A. Levenson, J. M. Moison, “Analytical first-order extension of coupled-mode theory for waveguide arrays,” Opt. Express 18(7), 7157–7172 (2010).
    [Crossref] [PubMed]
  26. N. Belabas, Ch. Minot, J. A. Levenson, J. M. Moison, “Ab initio design, experimental validation, and scope of coupling coefficients in waveguide arrays and discrete photonics patterns,” J. Lightwave Technol. 29(19), 3009–3014 (2011).
    [Crossref]
  27. P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006).
    [Crossref]
  28. A. Zeilinger, “General properties of lossless beamsplitters in interferometry,” Am. J. Phys. 49(9), 882–883 (1981).
    [Crossref]

2013 (2)

E. Sakat, S. Héron, P. Bouchon, G. Vincent, F. Pardo, S. Collin, J.-L. Pelouard, R. Haïdar, “Metal-dielectric bi-atomic structure for angular-tolerant spectral filtering,” Opt. Lett. 38(4), 425–427 (2013).
[Crossref] [PubMed]

H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D. Solnyshkov, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, G. Malpuech, J. Bloch, “Realization of a double-barrier resonant tunneling diode for cavity polaritons,” Phys. Rev. Lett. 110(23), 236601 (2013).
[Crossref]

2012 (4)

A. M. Childs, D. Gosset, “Levinson's theorem for graphs II,” J. Math. Phys. 53, 102207 (2012).

P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, S. Collin, “Optical extinction in a single layer of nanorods,” Phys. Rev. Lett. 109(14), 143903 (2012).
[Crossref] [PubMed]

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics 6(4), 248–252 (2012).
[Crossref]

J. M. Moison, N. Belabas, J. A. Levenson, C. Minot, “Light-propagation management in coupled waveguide arrays: quantitative experimental and theoretical assessment from band structures to functional patterns,” Phys. Rev. A 86(3), 033811 (2012).
[Crossref]

2011 (2)

2010 (4)

Ch. Minot, N. Belabas, J. A. Levenson, J. M. Moison, “Analytical first-order extension of coupled-mode theory for waveguide arrays,” Opt. Express 18(7), 7157–7172 (2010).
[Crossref] [PubMed]

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
[Crossref]

D. Liang, J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4(8), 511–517 (2010).
[Crossref]

E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010).
[Crossref]

2009 (2)

2008 (2)

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, A. Tünnermann, “Fresnel’s laws in discrete optical media,” New J. Phys. 10(10), 103020 (2008).
[Crossref]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1–3), 1–126 (2008).
[Crossref]

2006 (3)

2003 (3)

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90(5), 053902 (2003).
[Crossref] [PubMed]

D. N. Christodoulides, F. Lederer, Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, A. Bräuer, “Tailoring guided modes in waveguide arrays,” Opt. Express 11(25), 3404–3411 (2003).
[Crossref] [PubMed]

2001 (1)

D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87(23), 233901 (2001).
[Crossref] [PubMed]

1987 (1)

R. A. Syms, “Approximate solution of eigenmode problems for layered coupled arrays,” IEEE J. Quantum Electron. 23(5), 525–532 (1987).
[Crossref]

1986 (1)

R. C. Alferness, C. H. Joyner, M. D. Divino, M. J. R. Martyak, L. L. Buhl, “Narrowband grating resonator filters in InGaAsP/InP waveguides,” Appl. Phys. Lett. 49(3), 125–127 (1986).
[Crossref]

1984 (1)

B. Rico, M. Y. Azbel, “Physics of resonant tunneling. The one-dimensional double-barrier case,” Phys. Rev. B 29(4), 1970–1981 (1984).
[Crossref]

1983 (1)

T. C. L. G. Sollner, W. D. Goodhue, P. E. Tannenwald, C. D. Parker, D. D. Peck, “Resonant tunneling through quantum wells at frequencies up to 2.5 THz,” Appl. Phys. Lett. 43(6), 588–600 (1983).
[Crossref]

1981 (1)

A. Zeilinger, “General properties of lossless beamsplitters in interferometry,” Am. J. Phys. 49(9), 882–883 (1981).
[Crossref]

Aitchison, J. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90(5), 053902 (2003).
[Crossref] [PubMed]

Alferness, R. C.

R. C. Alferness, C. H. Joyner, M. D. Divino, M. J. R. Martyak, L. L. Buhl, “Narrowband grating resonator filters in InGaAsP/InP waveguides,” Appl. Phys. Lett. 49(3), 125–127 (1986).
[Crossref]

Amo, A.

H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D. Solnyshkov, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, G. Malpuech, J. Bloch, “Realization of a double-barrier resonant tunneling diode for cavity polaritons,” Phys. Rev. Lett. 110(23), 236601 (2013).
[Crossref]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1–3), 1–126 (2008).
[Crossref]

Azbel, M. Y.

B. Rico, M. Y. Azbel, “Physics of resonant tunneling. The one-dimensional double-barrier case,” Phys. Rev. B 29(4), 1970–1981 (1984).
[Crossref]

Baets, R.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
[Crossref]

Bardou, N.

P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, S. Collin, “Optical extinction in a single layer of nanorods,” Phys. Rev. Lett. 109(14), 143903 (2012).
[Crossref] [PubMed]

Bazin, A.

Belabas, N.

Bloch, J.

H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D. Solnyshkov, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, G. Malpuech, J. Bloch, “Realization of a double-barrier resonant tunneling diode for cavity polaritons,” Phys. Rev. Lett. 110(23), 236601 (2013).
[Crossref]

E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010).
[Crossref]

Bouchon, P.

Bouchoule, S.

Bowers, J. E.

D. Liang, J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4(8), 511–517 (2010).
[Crossref]

Bräuer, A.

Buhl, L. L.

R. C. Alferness, C. H. Joyner, M. D. Divino, M. J. R. Martyak, L. L. Buhl, “Narrowband grating resonator filters in InGaAsP/InP waveguides,” Appl. Phys. Lett. 49(3), 125–127 (1986).
[Crossref]

Childs, A. M.

A. M. Childs, D. Gosset, “Levinson's theorem for graphs II,” J. Math. Phys. 53, 102207 (2012).

Christodoulides, D. N.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1–3), 1–126 (2008).
[Crossref]

D. N. Christodoulides, F. Lederer, Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87(23), 233901 (2001).
[Crossref] [PubMed]

Collin, S.

E. Sakat, S. Héron, P. Bouchon, G. Vincent, F. Pardo, S. Collin, J.-L. Pelouard, R. Haïdar, “Metal-dielectric bi-atomic structure for angular-tolerant spectral filtering,” Opt. Lett. 38(4), 425–427 (2013).
[Crossref] [PubMed]

P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, S. Collin, “Optical extinction in a single layer of nanorods,” Phys. Rev. Lett. 109(14), 143903 (2012).
[Crossref] [PubMed]

de Vries, T.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
[Crossref]

Divino, M. D.

R. C. Alferness, C. H. Joyner, M. D. Divino, M. J. R. Martyak, L. L. Buhl, “Narrowband grating resonator filters in InGaAsP/InP waveguides,” Appl. Phys. Lett. 49(3), 125–127 (1986).
[Crossref]

Dreisow, F.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, A. Tünnermann, “Fresnel’s laws in discrete optical media,” New J. Phys. 10(10), 103020 (2008).
[Crossref]

Eisenberg, H. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90(5), 053902 (2003).
[Crossref] [PubMed]

Eugenieva, E. D.

D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87(23), 233901 (2001).
[Crossref] [PubMed]

Ferrier, L.

E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010).
[Crossref]

Galopin, E.

H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D. Solnyshkov, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, G. Malpuech, J. Bloch, “Realization of a double-barrier resonant tunneling diode for cavity polaritons,” Phys. Rev. Lett. 110(23), 236601 (2013).
[Crossref]

Geluk, E.-J.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
[Crossref]

Ghenuche, P.

P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, S. Collin, “Optical extinction in a single layer of nanorods,” Phys. Rev. Lett. 109(14), 143903 (2012).
[Crossref] [PubMed]

Goodhue, W. D.

T. C. L. G. Sollner, W. D. Goodhue, P. E. Tannenwald, C. D. Parker, D. D. Peck, “Resonant tunneling through quantum wells at frequencies up to 2.5 THz,” Appl. Phys. Lett. 43(6), 588–600 (1983).
[Crossref]

Gosset, D.

A. M. Childs, D. Gosset, “Levinson's theorem for graphs II,” J. Math. Phys. 53, 102207 (2012).

Grousson, R.

E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010).
[Crossref]

Haïdar, R.

E. Sakat, S. Héron, P. Bouchon, G. Vincent, F. Pardo, S. Collin, J.-L. Pelouard, R. Haïdar, “Metal-dielectric bi-atomic structure for angular-tolerant spectral filtering,” Opt. Lett. 38(4), 425–427 (2013).
[Crossref] [PubMed]

P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, S. Collin, “Optical extinction in a single layer of nanorods,” Phys. Rev. Lett. 109(14), 143903 (2012).
[Crossref] [PubMed]

Halioua, Y.

Heinrich, M.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, A. Tünnermann, “Fresnel’s laws in discrete optical media,” New J. Phys. 10(10), 103020 (2008).
[Crossref]

Héron, S.

Huybrechts, K.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
[Crossref]

Johne, R.

E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010).
[Crossref]

Joyner, C. H.

R. C. Alferness, C. H. Joyner, M. D. Divino, M. J. R. Martyak, L. L. Buhl, “Narrowband grating resonator filters in InGaAsP/InP waveguides,” Appl. Phys. Lett. 49(3), 125–127 (1986).
[Crossref]

Karle, T. J.

Kavokin, A. V.

E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010).
[Crossref]

Kawaguchi, Y.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics 6(4), 248–252 (2012).
[Crossref]

Kawanishi, S.

Kip, D.

Kumar, R.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
[Crossref]

Laroche, M.

P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, S. Collin, “Optical extinction in a single layer of nanorods,” Phys. Rev. Lett. 109(14), 143903 (2012).
[Crossref] [PubMed]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1–3), 1–126 (2008).
[Crossref]

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, A. Tünnermann, “Fresnel’s laws in discrete optical media,” New J. Phys. 10(10), 103020 (2008).
[Crossref]

H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, A. Bräuer, “Tailoring guided modes in waveguide arrays,” Opt. Express 11(25), 3404–3411 (2003).
[Crossref] [PubMed]

D. N. Christodoulides, F. Lederer, Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

Lemaître, A.

H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D. Solnyshkov, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, G. Malpuech, J. Bloch, “Realization of a double-barrier resonant tunneling diode for cavity polaritons,” Phys. Rev. Lett. 110(23), 236601 (2013).
[Crossref]

E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010).
[Crossref]

Levenson, J. A.

Liang, D.

D. Liang, J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4(8), 511–517 (2010).
[Crossref]

Liu, L.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
[Crossref]

Malpuech, G.

H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D. Solnyshkov, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, G. Malpuech, J. Bloch, “Realization of a double-barrier resonant tunneling diode for cavity polaritons,” Phys. Rev. Lett. 110(23), 236601 (2013).
[Crossref]

E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010).
[Crossref]

Mandelik, D.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90(5), 053902 (2003).
[Crossref] [PubMed]

Martyak, M. J. R.

R. C. Alferness, C. H. Joyner, M. D. Divino, M. J. R. Martyak, L. L. Buhl, “Narrowband grating resonator filters in InGaAsP/InP waveguides,” Appl. Phys. Lett. 49(3), 125–127 (1986).
[Crossref]

Matsuo, S.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics 6(4), 248–252 (2012).
[Crossref]

Michaelis, D.

Minot, C.

Minot, Ch.

Mitsugi, S.

Moison, J. M.

Monnier, P.

Morandotti, R.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90(5), 053902 (2003).
[Crossref] [PubMed]

Morthier, G.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
[Crossref]

Nguyen, H. S.

H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D. Solnyshkov, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, G. Malpuech, J. Bloch, “Realization of a double-barrier resonant tunneling diode for cavity polaritons,” Phys. Rev. Lett. 110(23), 236601 (2013).
[Crossref]

Nolte, S.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, A. Tünnermann, “Fresnel’s laws in discrete optical media,” New J. Phys. 10(10), 103020 (2008).
[Crossref]

Notomi, M.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics 6(4), 248–252 (2012).
[Crossref]

A. Shinya, S. Mitsugi, T. Tanabe, M. Notomi, I. Yokohama, H. Takara, S. Kawanishi, “All-optical flip-flop circuit composed of coupled two-port resonant tunneling filter in two-dimensional photonic crystal slab,” Opt. Express 14(3), 1230–1235 (2006).
[Crossref] [PubMed]

Nozaki, K.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics 6(4), 248–252 (2012).
[Crossref]

Pardo, F.

Parker, C. D.

T. C. L. G. Sollner, W. D. Goodhue, P. E. Tannenwald, C. D. Parker, D. D. Peck, “Resonant tunneling through quantum wells at frequencies up to 2.5 THz,” Appl. Phys. Lett. 43(6), 588–600 (1983).
[Crossref]

Peck, D. D.

T. C. L. G. Sollner, W. D. Goodhue, P. E. Tannenwald, C. D. Parker, D. D. Peck, “Resonant tunneling through quantum wells at frequencies up to 2.5 THz,” Appl. Phys. Lett. 43(6), 588–600 (1983).
[Crossref]

Pelouard, J.-L.

E. Sakat, S. Héron, P. Bouchon, G. Vincent, F. Pardo, S. Collin, J.-L. Pelouard, R. Haïdar, “Metal-dielectric bi-atomic structure for angular-tolerant spectral filtering,” Opt. Lett. 38(4), 425–427 (2013).
[Crossref] [PubMed]

P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, S. Collin, “Optical extinction in a single layer of nanorods,” Phys. Rev. Lett. 109(14), 143903 (2012).
[Crossref] [PubMed]

Pertsch, T.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, A. Tünnermann, “Fresnel’s laws in discrete optical media,” New J. Phys. 10(10), 103020 (2008).
[Crossref]

H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, A. Bräuer, “Tailoring guided modes in waveguide arrays,” Opt. Express 11(25), 3404–3411 (2003).
[Crossref] [PubMed]

Peschel, U.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, A. Tünnermann, “Fresnel’s laws in discrete optical media,” New J. Phys. 10(10), 103020 (2008).
[Crossref]

H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, A. Bräuer, “Tailoring guided modes in waveguide arrays,” Opt. Express 11(25), 3404–3411 (2003).
[Crossref] [PubMed]

Raineri, F.

Raj, R.

Regreny, P.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
[Crossref]

Rico, B.

B. Rico, M. Y. Azbel, “Physics of resonant tunneling. The one-dimensional double-barrier case,” Phys. Rev. B 29(4), 1970–1981 (1984).
[Crossref]

Roelkens, G.

Y. Halioua, A. Bazin, P. Monnier, T. J. Karle, G. Roelkens, I. Sagnes, R. Raj, F. Raineri, “Hybrid III-V semiconductor/silicon nanolaser,” Opt. Express 19(10), 9221–9231 (2011).
[Crossref] [PubMed]

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
[Crossref]

Russell, P. St. J.

Rüter, C. E.

Sagnes, I.

H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D. Solnyshkov, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, G. Malpuech, J. Bloch, “Realization of a double-barrier resonant tunneling diode for cavity polaritons,” Phys. Rev. Lett. 110(23), 236601 (2013).
[Crossref]

Y. Halioua, A. Bazin, P. Monnier, T. J. Karle, G. Roelkens, I. Sagnes, R. Raj, F. Raineri, “Hybrid III-V semiconductor/silicon nanolaser,” Opt. Express 19(10), 9221–9231 (2011).
[Crossref] [PubMed]

E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010).
[Crossref]

N. Belabas, S. Bouchoule, I. Sagnes, J. A. Levenson, C. Minot, J. M. Moison, “Confining light flow in weakly coupled waveguide arrays by structuring the coupling constant: towards discrete diffractive optics,” Opt. Express 17(5), 3148–3156 (2009).
[Crossref] [PubMed]

Sakat, E.

Sanvitto, D.

E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010).
[Crossref]

Sato, T.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics 6(4), 248–252 (2012).
[Crossref]

Segawa, T.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics 6(4), 248–252 (2012).
[Crossref]

Segev, M.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1–3), 1–126 (2008).
[Crossref]

Senellart, P.

E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010).
[Crossref]

Shinya, A.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics 6(4), 248–252 (2012).
[Crossref]

A. Shinya, S. Mitsugi, T. Tanabe, M. Notomi, I. Yokohama, H. Takara, S. Kawanishi, “All-optical flip-flop circuit composed of coupled two-port resonant tunneling filter in two-dimensional photonic crystal slab,” Opt. Express 14(3), 1230–1235 (2006).
[Crossref] [PubMed]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1–3), 1–126 (2008).
[Crossref]

D. N. Christodoulides, F. Lederer, Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90(5), 053902 (2003).
[Crossref] [PubMed]

Sollner, T. C. L. G.

T. C. L. G. Sollner, W. D. Goodhue, P. E. Tannenwald, C. D. Parker, D. D. Peck, “Resonant tunneling through quantum wells at frequencies up to 2.5 THz,” Appl. Phys. Lett. 43(6), 588–600 (1983).
[Crossref]

Solnyshkov, D.

H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D. Solnyshkov, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, G. Malpuech, J. Bloch, “Realization of a double-barrier resonant tunneling diode for cavity polaritons,” Phys. Rev. Lett. 110(23), 236601 (2013).
[Crossref]

Solnyshkov, D. D.

E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010).
[Crossref]

Spuesens, T.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
[Crossref]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1–3), 1–126 (2008).
[Crossref]

Streppel, U.

Sturm, C.

H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D. Solnyshkov, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, G. Malpuech, J. Bloch, “Realization of a double-barrier resonant tunneling diode for cavity polaritons,” Phys. Rev. Lett. 110(23), 236601 (2013).
[Crossref]

Suzaki, Y.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics 6(4), 248–252 (2012).
[Crossref]

Syms, R. A.

R. A. Syms, “Approximate solution of eigenmode problems for layered coupled arrays,” IEEE J. Quantum Electron. 23(5), 525–532 (1987).
[Crossref]

Szameit, A.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, A. Tünnermann, “Fresnel’s laws in discrete optical media,” New J. Phys. 10(10), 103020 (2008).
[Crossref]

Takahashi, R.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics 6(4), 248–252 (2012).
[Crossref]

Takara, H.

Tanabe, T.

Tanese, D.

H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D. Solnyshkov, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, G. Malpuech, J. Bloch, “Realization of a double-barrier resonant tunneling diode for cavity polaritons,” Phys. Rev. Lett. 110(23), 236601 (2013).
[Crossref]

Tannenwald, P. E.

T. C. L. G. Sollner, W. D. Goodhue, P. E. Tannenwald, C. D. Parker, D. D. Peck, “Resonant tunneling through quantum wells at frequencies up to 2.5 THz,” Appl. Phys. Lett. 43(6), 588–600 (1983).
[Crossref]

Trompeter, H.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, A. Tünnermann, “Fresnel’s laws in discrete optical media,” New J. Phys. 10(10), 103020 (2008).
[Crossref]

H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, A. Bräuer, “Tailoring guided modes in waveguide arrays,” Opt. Express 11(25), 3404–3411 (2003).
[Crossref] [PubMed]

Tünnermann, A.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, A. Tünnermann, “Fresnel’s laws in discrete optical media,” New J. Phys. 10(10), 103020 (2008).
[Crossref]

Van Thourhout, D.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
[Crossref]

Vincent, G.

E. Sakat, S. Héron, P. Bouchon, G. Vincent, F. Pardo, S. Collin, J.-L. Pelouard, R. Haïdar, “Metal-dielectric bi-atomic structure for angular-tolerant spectral filtering,” Opt. Lett. 38(4), 425–427 (2013).
[Crossref] [PubMed]

P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, S. Collin, “Optical extinction in a single layer of nanorods,” Phys. Rev. Lett. 109(14), 143903 (2012).
[Crossref] [PubMed]

Vishnevsky, D.

H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D. Solnyshkov, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, G. Malpuech, J. Bloch, “Realization of a double-barrier resonant tunneling diode for cavity polaritons,” Phys. Rev. Lett. 110(23), 236601 (2013).
[Crossref]

Wertz, E.

E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010).
[Crossref]

Wisniewski, J.

Yokohama, I.

Zeilinger, A.

A. Zeilinger, “General properties of lossless beamsplitters in interferometry,” Am. J. Phys. 49(9), 882–883 (1981).
[Crossref]

Am. J. Phys. (1)

A. Zeilinger, “General properties of lossless beamsplitters in interferometry,” Am. J. Phys. 49(9), 882–883 (1981).
[Crossref]

Appl. Phys. Lett. (2)

T. C. L. G. Sollner, W. D. Goodhue, P. E. Tannenwald, C. D. Parker, D. D. Peck, “Resonant tunneling through quantum wells at frequencies up to 2.5 THz,” Appl. Phys. Lett. 43(6), 588–600 (1983).
[Crossref]

R. C. Alferness, C. H. Joyner, M. D. Divino, M. J. R. Martyak, L. L. Buhl, “Narrowband grating resonator filters in InGaAsP/InP waveguides,” Appl. Phys. Lett. 49(3), 125–127 (1986).
[Crossref]

IEEE J. Quantum Electron. (1)

R. A. Syms, “Approximate solution of eigenmode problems for layered coupled arrays,” IEEE J. Quantum Electron. 23(5), 525–532 (1987).
[Crossref]

J. Lightwave Technol. (2)

J. Math. Phys. (1)

A. M. Childs, D. Gosset, “Levinson's theorem for graphs II,” J. Math. Phys. 53, 102207 (2012).

Nat. Photonics (3)

D. Liang, J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4(8), 511–517 (2010).
[Crossref]

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
[Crossref]

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics 6(4), 248–252 (2012).
[Crossref]

Nat. Phys. (1)

E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010).
[Crossref]

Nature (1)

D. N. Christodoulides, F. Lederer, Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

New J. Phys. (1)

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, A. Tünnermann, “Fresnel’s laws in discrete optical media,” New J. Phys. 10(10), 103020 (2008).
[Crossref]

Opt. Express (5)

Opt. Lett. (3)

Phys. Rep. (1)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1–3), 1–126 (2008).
[Crossref]

Phys. Rev. A (1)

J. M. Moison, N. Belabas, J. A. Levenson, C. Minot, “Light-propagation management in coupled waveguide arrays: quantitative experimental and theoretical assessment from band structures to functional patterns,” Phys. Rev. A 86(3), 033811 (2012).
[Crossref]

Phys. Rev. B (1)

B. Rico, M. Y. Azbel, “Physics of resonant tunneling. The one-dimensional double-barrier case,” Phys. Rev. B 29(4), 1970–1981 (1984).
[Crossref]

Phys. Rev. Lett. (4)

D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87(23), 233901 (2001).
[Crossref] [PubMed]

P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, S. Collin, “Optical extinction in a single layer of nanorods,” Phys. Rev. Lett. 109(14), 143903 (2012).
[Crossref] [PubMed]

H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D. Solnyshkov, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, G. Malpuech, J. Bloch, “Realization of a double-barrier resonant tunneling diode for cavity polaritons,” Phys. Rev. Lett. 110(23), 236601 (2013).
[Crossref]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90(5), 053902 (2003).
[Crossref] [PubMed]

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1 Top left: the RTDB, a set of identical waveguides identically coupled almost everywhere (blue) except in two barriers (purple) where the coupling constant is lowered by a factor of 2 i.e. CH/CL = 2. A guidonic beam is sent across the array with an angle of incidence characterized by its wavevector component kx. Bottom middle: confinement in-between the barriers results in a set of bound quasi-modes at the band edges (red solid line) and at mid-band (red dashed line). Bottom right: CMT diffraction relations in the barriers and elsewhere (solid line), with evanescent continuations (dash-dotted line) plotted vs. ikx since they correspond to pure imaginary values of kx ; all zones here have the same array period. Top right (resp. bottom left): kx (resp. Kz) controls the transmission of the RTDB.
Fig. 2
Fig. 2 Left: incident (blue), reflected (red) and transmitted (green) plane waves in the typical RTDB, with phase fronts (dashed lines) and wavevectors ( ± Kx,Kz). Center: field modulus as a function of kx and waveguide index m, calculated by CMT for an incident wave of unit amplitude. Right: same as center but m is considered as a continuous variable. This distorted view displays more clearly the bound supermodes and the transmitted and stationary waves.
Fig. 3
Fig. 3 Eigenmodes of the eCMT propagation matrix for the typical RTDB in a 400-waveguide array. Left: eigenvalues Kz labeled by kx (line) nearly perfectly overlap the diffraction relation of the high-C regions. Selected values (dots) correspond to the 7 bound modes. Middle: map of eigenvectors I(kx,X/S), zoomed in X around the RTDB; eigenvector profiles have been convoluted with the single-waveguide mode profile to smooth the picture. Right: selected eigenvector profiles I(X/S) corresponding to the 7 bound modes, also zoomed; symbol colors are matched with those on the diffraction relation (left) and the eigenvector map (middle).
Fig. 4
Fig. 4 Transmission of the typical RTDB. Orange: eCMT reflectivity of CHCL interface. Red: eCMT reflectivity of single barrier CHCLCH. Green: plane wave Fabry-Pérot model for the CHCLCHCLCH double barrier using red curve as mirror reflectivity. Blue line: same as green with broadening by the finite beam width W = 100µm. Blue dots: eCMT simulation for the same RTDB and width. Vertical dashed lines: kx of eCMT eigenmodes of the double barrier.
Fig. 5
Fig. 5 Left: propagation of a guidonic beam (W = 100µm, kx = 0.35π, L = 50 mm) across the typical RTDB. Middle: guidonic signature. The input beam is rotated around mid-length and mid-structure to scan kx and the output intensity profile along X is monitored. Right: model signature I(kx,X/L). The vertical line indicates the 0.35 π abscissa used in the left-hand figure.
Fig. 6
Fig. 6 Influence of W and L parameters on the model guidonic signature of Fig. 5. Leftmost sequence: decreasing propagation length; due to the choice of a X/L deviation ordinate, decreasing the length increases the apparent size of the RTDB. Middle sequence: influence of beam waist. Rightmost signature: combination corresponding to the experimental case.
Fig. 7
Fig. 7 Left: experimental signature of the B1C6B1 RTDB with an injected beam of waist W = 23µm and TE polarization. SB = 8.0µm and L ~10 mm. Middle: eCMT simulation. Right: superposition of the experimental signature with the model signature shown as contours lines.
Fig. 8
Fig. 8 Left: part of experimental signature of B1C6B1 (Fig. 7) between 0 and π. Traces of the center deviation of transmitted and reflected beams T and R are shown respectively as green and red lines. Right: transmission curves, calculated for plane waves (dashed line), and obtained along traces from the eCMT model (full line) and experimental signatures (dots).
Fig. 9
Fig. 9 Left: variation of relative barrier height δ with the wavelength λ for various barrier periods SB, for 9, 8, and 7 µm top to bottom. Right: plane-wave transmission of the typical single (dashed line) and double (solid line) barrier for SB = 8 µm and λ = 1.5 (blue), 1.55 (green), and 1.6 (red) µm.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

tan N k x = f M = 1 ( k x ) δ 2 sin 2 k x 1 δ 2 cos 2 k x
a n z =iC( a n-1 + a n+1 ),
a n e iK z Z e in k x .
K Z =2 C p-1 cos k x p1 =2 C p cos k x p .
A p e in k x p + B p e -in k x p = A p (n)+ B p (n).
a n p z =i( C p1 a n p 1 + C p a n p +1 ),
1 2isin k x p [ e -i k x p C p1 C p e i k x p-1 e -i k x p C p1 C p e -i k x p-1 ( e i k x p C p1 C p e i k x p-1 ) ( e i k x p C p1 C p e -i k x p-1 ) ].
[ A p ( n p+1 ) B p ( n p+1 ) ]=[ e i( n p+1 n p ) k x p 0 0 e -i( n p+1 n p ) k x p ][ A p ( n p ) B p ( n p ) ],
r NM = T NM 21 T NM 22 , =2iα β * sinM k x B ( | α | 2 e iM k x B | β | 2 e iM k x B )+( | α | 2 e iM k x B | β | 2 e iM k x B ) e 2iN k x ( | α | 2 e iM k x B | β | 2 e iM k x B ) 2 +4 | α | 2 | β | 2 sin 2 M k x B e 2iN k x ,
α= e i k x B δ e i k x β= e i k x B δ e i k x ,
tanN k x =cotM k x B | α | 2 | β | 2 | α | 2 + | β | 2
tanN k x = T M ( δcos k x ) U M-1 ( δcos k x ) 2δsin k x 1 δ 2 cos2 k x f M ( k x ),
tanN k x = f M=1 ( k x ) δ 2 sin2 k x 1 δ 2 cos2 k x .
tanN k x = f M ( k x ) 2δsin k x δ 2 cos 2 k x 1 1 δ 2 cos2 k x
tanN k x f M ( k x )=± δ 2 1 δ 2 1 U M1 2 ( δcos k x )cosN k x 1cos2 k x 1 δ 2 cos2 k x

Metrics

Select as filters


Select Topics Cancel
© Copyright 2022 | Optica Publishing Group. All Rights Reserved