1. INTRODUCTIONThe emerging field of metamaterials and the realization of ma-terials exhibiting a negative index of refraction (NIM) in thepast ten years has opened the way for various fascinating ap-plications. With these novel materials, physical applicationslike cloaking devices [1–3] and optical imaging beyond the dif-fraction limit [4–6] have become possible. A metamaterial con-sists of artificial structures that are much smaller than thewavelength of the operating incident radiation. NIMs havebeen realized by metal–dielectric structures, e.g., split-ring re-sonators deposited on dielectrics [7,8] or fishnet structures[9–11]. However, due to the electron-beam lithography fabri-cation process of these structures, the realization of three-dimensional metamaterials requires a sequential stacking ofsingle layers [12–14]. Three-dimensional metamaterials con-sisting of multistacks of metal–dielectric layers can also beobtained by rolling up strained layers into microrolls [15–17].Because of the rolling-up process, the metal–dielectric super-lattice formed by the wall of the microroll is curved. Further-more, it is well established to transform the curved layers intoplanar stacks by compressing the microrolls [18].
The efficiency of metamaterials suffers from the absorptionof radiation in the metal, caused by the finite imaginary part ofthe metals’ dielectric function. This causes strong energydissipation and a rapid field decay. A possible solution to over-come this problem would be the incorporation of gain materi-als. Different approaches in the recent literature include dyes[19], quantum dots [20], and optically pumped quantum wellsthat have been brought in close proximity to split-ring resona-tors [21]. Semiconductor quantum wells show no bleachingand electrical pumping is well established for these structures.It was recently demonstrated that the self-rolling process of-fers the possibility to easily include active quantum wells inthe semiconductor layer of three-dimensional metamaterials.It was found that such a novel structure shows enhanced lighttransmission upon optical pumping [22]. In the next step, the
concept of rolled-up nanotechnology allows the realization ofthree-dimensional metamaterials with sophisticated metallicnanostructures instead of planar metal layers, which enablesus to profit from plasmonic resonances.
In this paper we investigate alternating layers of amplifyingquantum wells and metal gratings supporting surface plasmonpolaritons (SPPs). Such structures can be realized in the wallsof rolled-up metamaterials, as sketched in Fig. 1. Experimen-tally, one can prepare different types of layered systems roll-ing along or perpendicular to the grating stripes. Here we areinterested in the fundamental effect of grating coupler en-hanced gain and concentrate on a microroll with its rollingdirection along the grating stripes, where we can preciselyalign the stripes on top of each other, whereas, in the caseof a perpendicular rolling direction, commensurability effectsbetween the microroll perimeter and the grating period canoccur. We present finite-difference time-domain (FDTD) si-mulations that show a transmission enhancement in the struc-tures with metallic gratings for incident angles between �5°.The transmission is much larger than observed for sampleswith flat silver surfaces.
2. STRUCTURES AND SIMULATIONMETHODIn Fig. 1, a sketch of a microroll is shown. The preparation ofthis structure is based on strain relaxation of pseudomorphi-cally grown (AlIn)GaAs heterostructures, which roll up into atube when released from the substrate by etching away anAlAs sacrificial layer (for details see [17]). The importantingredients for our investigation here are, in addition, (i) anInGaAs gain layer and (ii) a grating that is etched into thesemiconductor layer and covered with Ag before the rolling-up process. As a result, one obtains alternating layers of quan-tum wells and Ag gratings.
We performed FDTD simulations on the metamaterials byusing the commercial software Lumerical FDTD Solutions
2402 J. Opt. Soc. Am. B / Vol. 28, No. 10 / October 2011 Rottler et al.
[23]. For the simulations, we neglected the curvature of theactual structure and investigated flat structures in two dimen-sions. We simulated a grating unit cell (lattice constant a andfilling factor t ¼ b=a, where b denotes the width of the etchedregion) with three layers, where we used periodic boundaryconditions in the x direction and perfectly matched layerboundary conditions in the y direction (Fig. 2). We irradiatedthe investigated structures with a plane wave with zero inci-dent angle. To fulfill the criterion for SPP excitation, we as-sume that this plane wave has the H field ~H ¼ ð0; 0; HzÞand the E field ~E ¼ ðEx; 0; 0Þ. Because of the periodic corru-gation, y components of the electric field are also induced. Inthe frequency spectrum, the plane wave pulse has a Gaussian-shaped distribution with a maximum at 406THz and a fullwidth at half-maximum (FWHM) of 385THz. This correspondsto broadband excitation with a center wavelength of λ ¼950 nm and a FWHM of Δλ ¼ 900nm. Inside the simulationvolume, we used an appropriate nonuniform mesh that hasbeen optimized by convergence testing. The transmission
spectrum was recorded in a distance of dtrans ¼ 0:6 μm inthe y direction behind the structure. For the dielectric func-tions, we take polynomial fits on experimental data from Palik[24]. The dielectric functions of the AlGaAs, the InGaAs, andthe AlInGaAs used in the simulations are based on the dielec-tric function of GaAs and were adjusted as follows: thepercentage of In and Al in the semiconductor layer decreasesand increases the bandgap of the composite semiconductor,which we assume to be Al26Ga74As, In16Ga84As, andAl20In16Ga64As. In first approximation, this shift of the band-gap is linear, causing the dielectric function to change slightly.This effect was taken into account by dislocating the dielectricfunction of GaAs linearly:
ϵSCðℏωÞ ¼ ϵGaAs�ℏω −ΔEIn53
PIn
0:53þΔEAl30
PAl
0:3
�; ð1Þ
where PIn and PAl are the percentages of In and Al in the semi-conductor, respectively. The difference of the bandgap ofGaAs and In53Ga47As [25] is ΔEIn53 ¼ 0:7 eV and the differ-ence of the bandgap of Al30Ga70As and GaAs [26] isΔEAl30 ¼ 0:45 eV.
The amplification effect of the InGaAs quantum well canbe described by a negative imaginary part of the refractiveindex. In the simulations, the dielectric function ϵgain of thisgain layer was modeled qualitatively as a Lorentz oscillatoraccording to [27]
ϵgain ¼ ϵInGaAs −ω20ξ
ω20 − iωγ − ω2 ; ð2Þ
with ξ as the Lorentz oscillator strength, γ as the damping fre-quency, and ω0 as the resonant frequency of the quantumwell.
3. PARAMETER CALIBRATIONThe beneficial exploitation of the SPP excitations on the trans-mission requires that the lattice constant of the metallic grat-ing is chosen appropriately to adjust the energy of the SPPresonance to the active wavelength regime of the quantumwell. Furthermore, it is favorable to tune the SPP resonanceenergy to the Fabry–Perot transmission peak of the systemarising from the layered semiconductor/metal system withits high index of refraction. The optically active wavelengthregime of the quantum well can, in fact, be tailored by theamount of indium used during sample preparation. In the si-mulations presented in this paper, we consider a quantumwellemitting at λ ¼ 933nm. The Fabry–Perot resonance peaks canbe shifted by varying the individual layer thicknesses. Theblack curve in Fig. 3(a) shows the simulated transmissionspectrum for a three-layer structure, with each layer consist-ing of 10nm Ag, 26nm AlInGaAs, 9 nm InGaAs, and 25nmAlGaAs. We see that, for these dimensions, the structureexhibits a pronounced Fabry–Perot peak at the desired wave-length of λ ¼ 933nm. The appropriate grating was examinedby performing simulations with various lattice constants. InFig. 3(a) we display the transmission spectra for different grat-ings in comparison with the spectrum of unstructured layers.We first choose a grating filling factor t ¼ 0:5, a depthd ¼ 10nm, and an Ag thickness tAg ¼ 10nm. One can see that,for an a ¼ 600 nm grating, a deep SPP resonance is located atthe desired wavelength of λ ¼ 933nm. For smaller grating
AlInGaAs barrier layer
InGaAs gain layer
AlGaAs barrier layer
Ag metal layer
AlAs sacrificial layer
Fig. 1. (Color online) Sketch of a microroll that can be fabricatedby rolling up strained layers. The tube wall represents a three-dimensional metamaterial consisting of a metal–semiconductorsuperlattice containing quantum wells and metal gratings.
AlInGaAs barrier layer
InGaAs gain layer AlGaAs barrier layer
Ag metal layer
2D simulationvolume
sourceH field
E field x
y
ba
d
z
Ex
Ey
Hz
Fig. 2. (Color online) Scheme of the two-dimensional simulationvolume (orange box) with p-polarized fields. a is the lattice constant,b denotes the width of the etched region, and d is the grating depth.
Rottler et al. Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. B 2403
periods, the SPP resonance shifts to shorter wavelengths(λ ¼ 856nm for an a ¼ 400nm grating and λ ¼ 901nm foran a ¼ 500 nm grating), whereas, for larger grating periods,the SPP resonance wavelength is longer (λ ¼ 960nm for ana ¼ 700 nm grating and λ ¼ 980nm for an a ¼ 800nm grating).Variations in the lattice depth do not affect the position of theSPP resonance, but a smaller lattice depth causes less pro-nounced resonances. Moreover, several other SPP excitationscan be observed; however, they are considerably less pro-nounced. From the observed resonance positions, we canreconstruct a dispersion relation ωSP ¼ ωSPðkx ¼ n · 2π=aÞ,where n is the diffraction order. We find that the pronouncedminima, starting for a ¼ 800nm at λ ¼ 980nm, follow a disper-sion ωSP;a with n ¼ 1. The minima in the regime from λ ¼ 650to λ ¼ 750nm follow the same dispersion with n ¼ 2, thusthey represent a higher diffraction order. This dispersionωSP;a lies between the SPP dispersion of a flat-Ag/vacuum in-terface and a flat-Ag/GaAs interface. This is what we actuallyexpect for our complex semiconductor/metal structures in-cluding vacuum gaps. We further expect even different SPPbranches due to the asymmetric arrangement and couplingbetween the three metal layers. Indeed, the resonances start-ing for a ¼ 800nm at λ ¼ 1155nm follow a dispersion ωSP;b
that lies energetically below ωSP;a. Our studies showed thatthe variation of the lattice filling factor (we simulated fillingfactors between t ¼ 0:2 and t ¼ 0:8) does affect the position ofthe SPP resonance only slightly by about Δλ ¼ 20nm. InFig. 3(b), we independently show the transmission spectrafor the above structure without and with an a ¼ 600 nm grat-ing. It can be seen clearly that the SPP resonance and theFabry–Perot resonance peak are matched to the active wave-length regime of the quantum well (λ ¼ 933 nm). In Fig. 3(c)we plot the two-dimensional electric-field distribution ofthe λ ¼ 933nm SPP mode (at the left, we show the one-dimensional intensity profile corresponding to the white
dashed line). We recognize that the intensity profile exhibitsa large intensity inside the structure, in particular at thequantum well locations. We note that, in two-dimensional me-tamaterials, the gain material has to be brought in very closeproximity to the metallic structure (see, e.g., [21]), which cancause manufacturing problems. In our three-dimensionalcase, the mode pattern allows us to use thick barrier layers,which are necessary to maintain the optical quality of thequantum well and still have a strong field coupling to thequantum well.
4. RESULTS AND DISCUSSIONThe Lorentz oscillator strength in the gain layer was adjustedby investigating the transmission enhancement on a singlelayer: in previous experiments without a grating, it was shownthat, for single layers with and without a gain layer, which ex-hibit transmissions T and T þΔT , respectively, a transmis-sion enhancement of about ΔT=T ¼ 2% is easily realizable[22]. We performed simulations where we adjusted theLorentz oscillator strength ξ to ξ ¼ −0:03, −0:035, −0:037,and −0:04. This corresponds to single layer transmission en-hancements of ΔT=T ¼ 2%, 3%, 4%, and 5% and to imaginaryparts of the Lorentz oscillator refractive index of nQW
im ¼ −0:37,−0:43, −0:45, and −0:50 at λ ¼ 933nm, respectively. The re-sults are plotted in Fig. 4. Figure 4(e) shows the transmissionspectrumwith no gain as a reference. Figures 4(a)–4(d) depictthe transmission versus wavelength spectra for the above-mentioned values of ξ. It is observed that the transmissionat the SPP resonance is drastically enhanced with increasinggain. For ξ ¼ −0:03 (corresponding to nQW
im ¼ −0:37 andΔT=T ¼ 2% for an unstructured single layer film), we alreadyobtain a transmission of more than 1 at the wavelength of933nm. With increasing gain, the transmission at this wave-length increases up to 10 for ξ ¼ −0:04 (nQW
im ¼ −0:50,ΔT=T ¼ 5%). Figure 3(c) shows that there is a high intensity
933 nm
01
ytisnetni |E|
(b)400 nm grating(856 nm SPP)
500 nm grating(901 nm SPP)
600 nm grating(933 nm SPP)
700 nm grating(960 nm SPP)
800 nm grating(980 nm SPP)
(a)
no grating
100 nm
|E| intensity (a.U.)
01
intensitymonitor
x
(c)
933 nm
Fig. 3. (Color online) (a) Transmission spectrum of a three-layer structure of 10nm Ag, 26nm AlInGaAs, 9 nm InGaAs, and 25nm AlGaAs withdifferent metal grating periods and 10nm lattice depth in comparison to flat layers (black curve). (b) Transmission spectrum of an a ¼ 600nmgrating (green curve) together with the transmission spectrum of flat layers (black curve). The quantum well wavelength is marked with a reddashed line. (c) Two-dimensional electric-field distribution in the regime of the λ ¼ 933nm SPP mode. At the left, we plot the one-dimensionalintensity profile originating from the white dashed line (the intensity axes have the same linear scaling in both cases).
2404 J. Opt. Soc. Am. B / Vol. 28, No. 10 / October 2011 Rottler et al.
in all three layers. This indicates that the multilayered systemsthat we can realize with the microroll technique are indeedquite helpful. We have actually performed simulations inwhich we selectively turned on the gain in different layers,and it turned out that the effects on the transmission are con-siderably smaller. It is, therefore, advantageous to use a struc-ture with three or even more layers. We like to note that, asobserved in Fig. 4, the transmission enhancement actually hasa Fano-type shape with a minimum at the low wavelengthside. This indicates that the negative imaginary part of the gainmaterial shifts the SPP resonance position, i.e., there is an in-creasing interaction between the SPP and the gain layer.While the excitation of SPPs is linked to illumination withp-polarized light, we also performed for comparison simula-tions in s polarization. The transmission versus wavelengthspectra of this run are presented in Fig. 5. Figures 5(a)–5(d)show the spectra for Lorentz oscillator strengths ξ ¼ −0:03,−0:035, −0:037, and −0:04 corresponding to ΔT=T ¼ 2%,3%, 4%, and 5%, respectively. Figure 5(e) depicts the transmis-sion spectrum without gain. As expected, there are no SPPresonances. Turning on the gain leads to an increasing trans-mission at the resonant frequency of the quantum well.However, in the s-polarized case, a strong transmission en-hancement, as in Figs. 4(a)–4(d), is not observed. Our simula-tions show that the increased transmission for s polarization iscomparable to simulations with flat layers instead of a grating.This observation supports that the SPP resonances are crucialfor the extraordinary transmission. Further simulations showthat the strength of the transmission enhancement, as seen inFig. 4, is strongly dependent on the interplay of the Fabry–
Perot resonance, the SPP resonance, and the active wave-length regime of the quantum well.
The spectral position of the SPP resonance is not onlyaffected by the lattice constant a but also by the incident angle
(a)
(c)
(d)
(e)
(b)
= -0.04
= -0.035
= -0.03
= 0
= -0.037
|n | (a.U.)QWim
Fig. 4. (Color online) (a)–(d) Transmission versus wavelength spec-tra for different values of gain and a grating period of a ¼ 600nm forillumination with a p-polarized wave. The InGaAs gain layer has athickness of 9 nm. (d) also shows the maginary part of the refractiveindex of the gain layer. (e) Transmission spectrum without gain [com-pare with Fig. 3(b)] for illumination with a p-polarized wave.
(a)
(c)
(d)
(e)
(b)
600 800 1000 12000,0
0,5
1,0
1,50,0
0,5
1,0
1,50,0
0,5
1,0
1,50,0
0,5
1,0
1,50,0
0,5
1,0
1,5600 800 1000 1200
tran
smis
sion
wavelength (nm)
= -0.04
= -0.035
= -0.03
= 0
= -0.037
Fig. 5. (Color online) (a)–(d) Transmission versus wavelength spec-tra for different values of gain and a grating period of a ¼ 600nm forillumination with a s-polarized wave. The quantum well has a thick-ness of 9 nm. (e) Transmission spectrum without gain for illuminationwith a s-polarized wave. A pronounced SPP resonance at λ ¼ 933nmwavelength is not observed [compare with Fig. 4(e)].
= -0.04
24
68
10
Fig. 6. (Color online) Transmission versus wavelength spectra fordifferent angles of incidence α and a grating period of a ¼ 600nm.The structure is illuminated with a p-polarized wave. The Lorentzoscillator strength is always set to ξ ¼ −0:04. The transmissiondecreases considerably for angles larger than α ¼ 5°.
Rottler et al. Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. B 2405
α of the incident plane wave. In Fig. 6 we display the transmis-sion versus wavelength spectra for different angles of inci-dence α and a grating period of a ¼ 600nm. The planewave is p polarized and the Lorentz oscillator strength is al-ways set to ξ ¼ −0:04, i.e., the spectrum for zero angle is thesame as the spectrum in Fig. 4(a). The angle α was changed insteps of 0:5°. It is observed that the enhancement of the trans-mission is of the same dimension for angles of incidence up toα ¼ 5°. The slight increase of enhancement reflects an opti-mized coupling of the three resonances. For angles betweenα ¼ 5° and α ¼ 10°, the transmission decreases considerablyuntil, for angles larger than α ¼ 10°, it cannot be distinguishedfrom illumination with s polarization. These findings can beexplained by the additional kx component that is providedby the nonzero incident angle, thus causing the SPP resonanceshifting to shorter wavelengths. Therefore, the previouslymentioned matching of the Fabry–Perot resonance, the SPPresonance, and the active wavelength regime of the quantumwell is no longer given. This distinct angular dependence lim-its the usage of the structure for subwavelength resolution.For this, the enhanced transmission of all kx componentswould be required (see, e.g., [17,28,29]). As a possible furtherdevelopment, this might be realized by using spherical metal-lic nanoparticles incorporated into a host medium instead of agrating. Intuitively, spherical particles would exhibit angularindependent plasmonic resonances, although their close as-sembly in a host medium can lead to collective modes alteringtheir angular independence. The effective dielectric functionsof these structures and their applications in metamaterialshave been extensively studied in the recent literature [30,31].
5. CONCLUSIONSIn conclusion, we demonstrated that SPP resonances on me-tallic gratings embedded into three-dimensional metamater-ials containing quantum structures can lead to a strongtransmission enhancement. The dimensions of the structuresand the grating period have to be chosen such that the wave-length of the SPP resonance spectrally and spatially matchesthe emission of the quantum well. Moreover, our simulationsshow that it is desirable to also shift the Fabry–Perot reso-nance peak to the SPP resonance.
ACKNOWLEDGMENTSThe authors thank Markus Bröll and Jens Ehlermann for fruit-ful discussions and gratefully acknowledge financial supportof the Deutsche Forschungsgemeinschaft (DFG) via theGraduiertenkolleg 1286 “Functional Metal-SemiconductorHybrid Systems.”
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