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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 9 — Apr. 25, 2011
  • pp: 8735–8749
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Silicon-based optical leaky wave antenna with narrow beam radiation

Qi Song, Salvatore Campione, Ozdal Boyraz, and Filippo Capolino  »View Author Affiliations


Optics Express, Vol. 19, Issue 9, pp. 8735-8749 (2011)
http://dx.doi.org/10.1364/OE.19.008735


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Abstract

We propose a design of a dielectric (silicon nitride) optical leaky wave antenna (OLWA) with periodic semiconductor (silicon) corrugations, capable of producing narrow beam radiation. The optical antenna radiates a narrow beam because a leaky wave (LW) with low attenuation constant is excited at one end of the corrugated dielectric waveguide. We show that pointing angle, beam-width, and operational frequency are all related to the LW complex wavenumber, whose value depends on the amount of silicon perturbations in the waveguide. In this paper, the propagation constant and the attenuation coefficient of the LW in the periodic structure are extracted from full-wave simulations. The far-field radiation patterns in both glass and air environments predicted by LW theory agree well with the ones obtained by full-wave simulations. We achieve a directive radiation pattern in glass environment with about 17.5 dB directivity and 1.05 degree beam-width at the operative free space wavelength of 1.55 μm, pointing at a direction orthogonal to the waveguide (broadside direction). We also show that the use of semiconductor corrugations facilitate electronic tuning of the radiation pattern via carrier injection.

© 2011 OSA

1. Introduction

An optical leaky wave antenna (OLWA) is a device that radiates a light wave into the surrounding space from a leaky wave (LW) guided mode or, vice-versa, it couples receiving optical power from the surrounding space into a guided optical mode. Optical antennas have the capabilities to enhance the interaction between light and matter, and thus have the potential to boost the efficiency of optoelectronic devices such as light-emitting diodes, lasers and solar cells, and bio-chemical detection capabilities [1

1. P. Ghenuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopic mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett. 101(11), 116805 (2008). [CrossRef] [PubMed]

3

3. R. L. Olmon, P. M. Krenz, A. C. Jones, G. D. Boreman, and M. B. Raschke, “Near-field imaging of optical antenna modes in the mid-infrared,” Opt. Express 16(25), 20295–20305 (2008). [CrossRef] [PubMed]

].

The optical antenna cases studied up to date mostly focus on the local field distribution control of the device. Those kinds of antennas are fabricated in a sub-wavelength dimension and some exceptional physical phenomena, such as super-resolution effect and near-field enhancement [4

4. T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13(3), 429–432 (2002). [CrossRef]

], are realizable due to the resonant modes. Very directive near-IR optical antennas with electronically controlled beam steering and radiation pattern are the subject of great interest for applications such as planar imaging [5

5. Q. Song, F. Qian, E. K. Tien, I. Tomov, J. Meyer, X. Z. Sang, and O. Boyraz, “Imaging by silicon on insulator waveguides,” Appl. Phys. Lett. 94(23), 231101 (2009). [CrossRef]

] and LIDAR [6

6. C. K. Toth, “R&D of mobile LIDAR mapping and future trends,” in Proceeding of ASPRS 2009 Annual Conference (Baltimore, Maryland, 2009).

]. The dielectric near-IR antenna consisting of silicon perturbations designed here can be used to transform a guided mode into a leaky mode, thus radiating in a region of space. This radiation phenomenon can be precisely described by using the concept of LWs. Accordingly, the periodicity of the silicon perturbations of the waveguide produces a spatial harmonic of the guided mode in the visible region. Recently, LWs parameterization has been used to explain enhanced directivity of a corrugated thin silver film with a sub-wavelength hole [7

7. D. R. Jackson, J. Chen, R. Qiang, F. Capolino, and A. A. Oliner, “The role of leaky plasmon waves in the directive beaming of light through a subwavelength aperture,” Opt. Express 16(26), 21271–21281 (2008). [CrossRef] [PubMed]

]. LWs have also been used to parameterize directive radiation from a quasi-crystal waveguide [8

8. A. Micco, V. Galdi, F. Capolino, A. Della Villa, V. Pierro, S. Enoch, and G. Tayeb, “Directive emission from defect-free dodecagonal photonic quasicrystals: a leaky wave characterization,” Phys. Rev. B 79(7), 075110 (2009). [CrossRef]

], or from a photonic crystal interface layer [9

9. E. Colak, H. Caglayan, A. O. Cakmak, A. D. Villa, F. Capolino, and E. Ozbay, “Frequency dependent steering with backward leaky waves via photonic crystal interface layer,” Opt. Express 17(12), 9879–9890 (2009). [CrossRef] [PubMed]

].

In the optoelectronics domain, silicon-on-insulator (SOI) devices have been widely utilized recently to deliver chip scale active and passive photonic devices such as amplifiers, switches and modulators with potential optoelectronic integration on the same platform [10

10. R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678–1687 (2006). [CrossRef]

,11

11. Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]

]. The SOI platform provides tight mode confinement due to a large index contrast, which is essential for miniaturization, and simultaneous low loss operation [12

12. L. Friedman, R. A. Soref, and J. P. Lorenzo, “Silicon double-injection electro-optic modulator with junction gate control,” J. Appl. Phys. 63(6), 1831–1839 (1988). [CrossRef]

]. In particular, the electronic tunability of the optical parameters of silicon (such as refractive index and absorption coefficient) via current injection renders itself the ideal platform for optical antennas that can facilitate electronic beam control [13

13. C. K. Tang and G. T. Reed, “Highly efficient optical-phase modulator in SOI waveguides,” Electron. Lett. 31(6), 451–452 (1995). [CrossRef]

15

15. S. M. Csutak, S. Dakshina-Murthy, and J. C. Campbell, “CMOS-compatible planar silicon waveguide-grating-coupler photodetectors fabricated on silicon-on-insulator (SOI) substrates,” IEEE J. Quantum Electron. 38(5), 477–480 (2002). [CrossRef]

]. The OLWA can function as a passive transmitter or receiver as well as beam steering device. Hence, the integration of a SOI waveguide and an optical antenna is a powerful tool for optical wave guiding and electronically controlled emitters. Previously, directional couplers and sensors devices with periodic perturbations have been presented [14

14. K. Van Acoleyen, W. Bogaerts, J. Jágerská, N. Le Thomas, R. Houdré, and R. Baets, “Off-chip beam steering with a one-dimensional optical phased array on silicon-on-insulator,” Opt. Lett. 34(9), 1477–1479 (2009). [CrossRef] [PubMed]

16

16. P. Cheben, S. Janz, D. X. Xu, B. Lamontagne, A. Delage, and S. Tanev, “A broad-band waveguide grating coupler with a subwavelength grating mirror,” IEEE Photon. Technol. Lett. 18(1), 13–15 (2006). [CrossRef]

]. However, the proposed device in this article includes semiconductor periodic perturbations inside a dielectric waveguide (novel by itself) that facilitate miscellaneous optoelectronic tunability. Also, up to date, far field radiation pattern has not been utilized in the similar sense.

In this paper, we provide results for a novel CMOS compatible OLWA with electronic tuning capability, and most of the investigation is relative to a structure invariant along one direction (say z, Fig. 1
Fig. 1 Optical antenna based on SOI. The Si periodic perturbation (nSi=3.48) is inside the Si3N4 waveguide (nw=1.67). The waveguide has a width equal to tw=1μm, is upon a SiO2 substrate (nh=1.45) and is covered by a SiO2 layer. The Si perturbation is characterized by an optimized period d=970nm, a width w=300nm and a length l=d/2=485 nm, which are found after a parametric study. The number of elements of the perturbation is equal to 60. The red contour represents the boundary at which the field has been extracted for far-field calculations.
). Indeed, though in general waves can also be guided in wide slabs (in the z direction), the results shown provide insight into how a waveguide with rectangular cross section could radiate. The radiation pattern of this latter case, a linear OLWA in a three dimensional (3D) space, is also presented at the end of the paper. The proposed OLWA consists of a silicon nitride (Si3N4) waveguide comprising periodic silicon perturbations and it provides directive radiation patterns at communication wavelengths. Modeling of such an antenna indicates that CMOS compatible OLWAs are feasible and produce very directive radiation patterns with about 17.5 dB directivity and 1.05 degree beam-width pointing at a direction orthogonal to the waveguide direction (transverse radiation, also called broadside radiation). We show that the use of semiconductor corrugations facilitates electronic tuning of the radiation pattern via carrier injection. The Floquet theorem is adopted here to describe the periodic properties of the field inside the waveguide and LW theory is provided to give the physical explanation for the narrow beam radiation behavior. The propagation constant and the attenuation coefficient of the leaky wave propagating in the periodic structure are extracted from full-wave simulations and utilized to evaluate the far-field radiation patterns in both SiO2 and air environments by using the theoretical formulas provided. Results demonstrate that the theoretical predictions are in excellent agreement with radiation patterns obtained by full-wave simulations.

2. Antenna geometry

The two dimensional (2D) structure of the OLWA is presented in Fig. 1, assumed invariant along the z direction. The optical guided mode is excited from the left side. The same mode is used to feed the OLWA made of 60 periodic silicon perturbations positioned on the bottom side of the silicon nitride waveguide. This number has been observed to be large enough to provide a long radiating section, with length L, required to achieve high directivity. The physical dimensions of the silicon perturbation are parametrically studied to obtain high directivity around the bottom transverse direction (radiation along –y in Fig. 1). Power radiation takes place towards ± y directions. However, the radiation is stronger in the bottom transverse direction due to the fact that the perturbation is placed on the bottom side of the waveguide. Each silicon perturbation is characterized by a width and a length equal to w=300 nmand l=d/2=485 nm, respectively, where d=970 nm is the period of the perturbation. The waveguide is placed on a silica glass (SiO2) substrate and is covered by a SiO2 cover layer, and has a width tw=1 μm. The total length of the radiating section is L=Nd, where N is the number of the perturbation, chosen equal to 60 in this study.

3. Local field representations and far-field pattern theoretical derivations

In this section, Floquet theory specialized in one dimensional domain is introduced as a powerful tool to characterize the behavior of a SOI waveguide comprising a periodic perturbation. The equivalent aperture (EA) method and the array factor (AF) method are the two theoretical approaches proposed to predict far-field radiation patterns from a leaky wave produced by the presence of the silicon perturbation inside the waveguide. In this analysis, electromagnetic fields are assumed to be time harmonic with a exp(iωt) time variation.

3.1 Quasi-periodic field distribution along the periodic structure

The guided mode excited by the incoming field on the left side of the waveguide becomes leaky in correspondence of the periodic silicon perturbations. Such a mode decays exponentially along the structure, even for a lossless structure, and a leakage phenomenon (radiation) takes place.

In the 2D structure presented here, the electric field is polarized along z and travels along the x direction. Indeed, the problem is invariant in the z-direction and the z-component of the guided electric field satisfies the quasi-periodic property E(x+d,y)=E(x,y)exp(ikx,0d), where d is the period of the periodic structure and kx,0=β+iα is the wavenumber along the propagation direction. Here, β and α are the phase and attenuation constants, respectively, of the LW. Note that, before introducing the perturbations, the original guided mode has a phase propagation constant very close to β, whereas the attenuation constant of the waveguide would be much smaller than α (without perturbation the attenuation is caused only by material losses and scattering by irregularities of the structure). Instead, with perturbations, the value of α can be significantly large because of radiation, though to have a directive OLWA this value has to be maintained low. Accordingly, in terms of a Fourier series expansion, the electric field at any place along the periodic structure can be represented as the superposition of Floquet spatial harmonics
E(x,y)=n=En(y)eikx,nx,    kx,n=kx,0+2nπ/d,
(1)
where kx,n is the Floquet wavenumber, n is the order of the Floquet spatial harmonic, andEn(y) is the weight of the n-th harmonic [17

17. A. A. Oliner and D. R. Jackson, “Leaky-wave antennas,” in Antenna Engineering Handbook, J. Volakis, ed. (McGraw Hill, 2007), pp. 11.11–11.56.

19

19. S. Campione and F. Capolino, “Linear and Planar Periodic Arrays of Metallic Nanospheres: Fabrication, Optical Properties and Applications,” in Selected Topics in Metamaterials and Photonic Crystals, A. Andreone, A. Cusano, A. Cutolo, and V. Galdi, eds. (World Scientific Publishing, 2011).

]. Each Floquet wave number kx,n=βn+iα has the same attenuation constant α [7

7. D. R. Jackson, J. Chen, R. Qiang, F. Capolino, and A. A. Oliner, “The role of leaky plasmon waves in the directive beaming of light through a subwavelength aperture,” Opt. Express 16(26), 21271–21281 (2008). [CrossRef] [PubMed]

,20

20. D. R. Jackson and A. A. Oliner, “Leaky-wave antennas,” in Modern Antenna Handbook, C. A. Balanis, ed. (Wiley, 2008), pp. 325–367.

]. The purpose of the periodic perturbations is to create a radiating n=1 harmonic. Its wavenumber is
kx,1=β1+iα,  with  β1=β2π/d,
(2)
such that kh<β1<kh, where kh=nhk0, k0is the free space wavenumber, and nh is the refractive index of the surrounding material, that in our case is either SiO2 (nh=1.45) or free space (nh=1).

Therefore, in the periodic structure under analysis, all the n-indexed Floquet harmonics but one are evanescent waves (contrarily to what may happen in gratings where several harmonics have βn in the visible region, i.e., the wavenumber region kh<βn<kh relative to waves that can propagate away from the plane). The radiating harmonic, the one that falls in the interval (kh,kh) , is usually defined as the n=1 one since in slightly perturbed structures the 0th fundamental propagation constant ββ0>khhas a value very close to the one of the bound mode in the unperturbed waveguide, and thus it is not radiating.

3.2 The equivalent aperture method

Based on the Floquet field expansion [Eq. (1)], the electric field along the periodic structure can be described as a superposition of field terms, and only the n=1 is relevant for radiation. Therefore, its expression evaluated on the “aperture” (i.e, a section from where we assume radiation is propagating away [21

21. R. E. Collin, Antennas and Radiowave Propagation (McGraw Hill, 1985), p. 164.

]) is
E(x)=E1eiβ1xeαx      (0<x<L),
(3)
where L is the total length of the silicon perturbation, and E1is the amplitude of the n=1 harmonic of the electric field at the beginning of the silicon perturbation.

The far-field radiation pattern EFF(ϕ) is obtained by integrating Eq. (3) across an “equivalent aperture” (0<x<L)
EFF(ϕ)=E10Leiβ1xeαxeikhcosϕxdx=E1ei(khcosϕβ1iα)L1i(khcosϕβ1iα)
(4)
which leads to a normalized far-field pattern magnitude
F(ϕ)=|EFF(ϕ)||E1|=(1+e2αL2eαLcos[(khcosϕβ1)L](khcosϕβ1)2+α2)12,
(5)
that in the case of large αL (either when the antenna is long, or when the attenuation constant is large) can be further simplified.

The maximum radiation direction can be computed using Eq. (5). In the case of large αL (i.e. when exp(αL)<<1), the maximum value of F(ϕmax)=1/α occurs at ϕ=ϕmax such that cosϕmax=β1/kh. In a similar way, using Eq. (5), it is also possible to determine the 3 dB beam-width, defined as Δϕ3dB=|ϕ3dB+ϕ3dB|, where F(ϕ3dB±)=F(ϕmax)/2, which happens when cosϕ3dB±(β1±α)/kh. This leads to

Δϕ3dB2αkh
(6)

3.3 The array factor method

The periodic perturbation in the waveguide can be considered as an array of identical scatterers from which the far-field pattern is determined by the pattern multiplication, i.e., by multiplying the pattern of the single element and the array factor (AF) of the array [22

22. C. A. Balanis, Antenna Theory: Analysis and Design (Wiley, 2005), p. 286.

]. The pattern of the single perturbation element is quasi isotropic, whereas the array factor is very angle-selective. Therefore, as usual for directive antennas, the total far-field radiation pattern is approximated as EFF(ϕ)AF(ϕ), and thus AF(φ) represents the normalized far-field. For the OLWA under consideration, the array factorAF(φ) is defined as
AF(ϕ)=n=0N1ei(β1+iα)ndei(khcosϕ)nd=1ei(khcosϕβ1iα)Nd1ei(khcosϕβ1iα)d,
(7)
whose magnitude is
|AF(ϕ)|=(1+e2αL2eαLcos[(khcosϕβ1)L]1+e2αd2eαdcos[(khcosϕβ1)d])12,
(8)
where we have assumed that L=Nd. An analogous procedure was applied in [7

7. D. R. Jackson, J. Chen, R. Qiang, F. Capolino, and A. A. Oliner, “The role of leaky plasmon waves in the directive beaming of light through a subwavelength aperture,” Opt. Express 16(26), 21271–21281 (2008). [CrossRef] [PubMed]

] to find the directive radiation pattern of a narrow slit in a corrugated silver film.

4. Analysis of the radiation properties

In this section, in order to validate the theoretical results, full-wave simulations are implemented by using two commercial software, COMSOL and HFSS. Explanations of particular phenomena will be provided.

The input light used in these analyses is characterized by a free space wavelength equal to λ0=1.55 μm, and is polarized along z. The actual simulated structure presents a calculation domain in the x-y plane of 200 μm × 40 μm. Therefore, there are two waveports, 200 μm apart, one as input (marked as Port 1 in Fig. 1) whereas the other one (marked as Port 2 in Fig. 1) is used to monitor the power remaining after the OLWA section. The width of the SiO2 layers is 19.5 μm, above and below.

4.1 Preliminary analyses

Figure 2
Fig. 2 Comparison of the S-parameters obtained by COMSOL and HFSS with respect to the free space wavelength λ0 varying from 1.4 µm to 1.7 µm.
shows the comparison of the S-parameters obtained by COMSOL and HFSS, which are in agreement in all the analyzed wavelength range. It can be observed that S11 is lower than −10 dB at λ0=1.55 μmand almost in the entire swept range. Therefore, just a very small portion of the power injected to the antenna will be reflected back by the silicon perturbations. In other words, the power accepted by the antenna will be close in value to the incident power furnished by the incoming wavemode at the input. The transmission coefficient S21 is smaller than −20 dB at λ0=1.55 μmand in almost all the analyzed range. Therefore, very small percentage of power travelling along the waveguide will pass the silicon perturbation section. All the incident power is either lost by dissipation (material losses) or by radiation. Since material losses are much smaller than radiation losses, as stated in Sec. 3.1, we can consider that all the power has been radiated away. Note that there is a mismatch between COMSOL and HFSS only below −35 dB mostly due to different numerical precisions of the two software packages.

4.2 Local field across the aperture

Having assured that the proposed structure radiates, we then proceed on the analysis by computing the parameters characterizing the LW from the results obtained by full-wave simulations.

From Eq. (3), it is clear that the description of the aperture field is described by means of two parameters: the propagation constant β1 and the attenuation constant α.After having computed these parameters, the far-field radiation pattern can be easily constructed using the two theoretical methods presented in Sec. 3. The field along the silicon-perturbed waveguide contains the information regarding the two parameters stated above. Indeed, we know that the local field magnitude decays exponentially along the perturbed waveguide with attenuation constant α. Moreover, its phase shift is linearly dependent to the propagation constant, as can be seen looking at the phase term β1x.

Figures 3(a)
Fig. 3 Comparison between COMSOL and HFSS full-wave simulation results for the electric field along the waveguide, sampled in the silicon perturbations (one point per cell). (a) The magnitude and (b) the phase of the electric field along the aperture.
and 3(b) show the field magnitude and phase along the silicon-perturbed waveguide, sampled at one point per cell, positioned at the center of each Si perturbation, along x, and at the bottom of each Si perturbation, along y. The magnitude of the field E(x)in Fig. 3(a) exhibits a linear trend using a logarithmic scale, confirming its exponential decay along the waveguide longitudinal direction as expected from |E(x)/E1|dB=20logeαx=20αxloge. As can be easily inferred, the attenuation constant α can be measured from the slope of the line in Fig. 3(a). Figure 3(b) shows, instead, the unwrapped phase β1x of the leaky wave along the waveguide: the slope of the line provides the propagation constantβ1. Observing one sample of the field per period gives us only the phase propagation constant in the main Brillouin zone.

We used the least mean square interpolation to extract the above mentioned LW parameters. The wavenumber is equal to β1=2.46×105m-1. The normalized propagation constant with respect to the free space wavenumber is β¯1=β1/k00.06. In the same way, the extracted value of the attenuation constant is α=4.52×104m-1, which corresponds to the normalized value α¯=α/k00.01.

4.3 Far-field radiation pattern in free space and SiO2 environments

As we have demonstrated in Sec. 3, the far-field radiation pattern can be obtained theoretically by applying two different approaches: the equivalent aperture method, leading to Eq. (5), and the array factor method, leading to Eq. (8). The far-field radiation patterns are then analytically computed by using the previously extracted values for α and β1 into Eq. (5) and Eq. (8) for both free space and SiO2 environments.

The free space far-field radiation patterns, assuming free space outside the SiO2 region, are shown in Fig. 4
Fig. 4 Comparison of the normalized far-field radiation patterns in free space, obtained by COMSOL (red line), HFSS (black line), EA method (green line) and AF method (blue line). The inset shows an enlargement of the maximum radiation region.
. The red and black lines are the results obtained by near-field integration of fields obtained by COMSOL and HFSS, respectively. Whereas the green and blue lines are the EA method and the AF method results, respectively, considering 60 silicon perturbations.

The far-field radiation patterns of the main directive beam obtained with the LW theory are in good agreement with the full-wave simulation results, down to −20 dB. The full-wave simulation results present a very directive main-beam around 93.2°. This peak position is in agreement with the approximate value computed in Sec. 3.2, ϕmaxarccos(β1/kh)=93.5°. We have computed the directivity by
D=2π|EmaxFF|202π|EFF(ϕ)|2dϕ,
(9)
where EmaxFF represents the electric far-field along the direction ϕmax of maximum radiation obtained by either HFSS or COMSOL. This leads to 15.7 dB directivity. The full-wave simulation 3 dB beam-width is Δϕ3dB1.4°, in agreement with the approximate formula (6) which gives Δϕ3dB1.3°.

The side-lobe at around 51° in Fig. 4 is not predicted by LW theory based on the values α and β1 found in Sec. 4.2. Indeed, a different phase behavior is observed in Fig. 3 for a few elements at the beginning of the silicon perturbations. The side-lobe is explained by noticing that the first two/three elements of the perturbation form a small array. Indeed, if we interpolate the first two (three) points in Fig. 3(b), we are able to extract the value β1=3.02×106m-1 (β1=1.95×106m-1). According to the array factor for these two sub-arrays made of 2 (3) elements, respectively, the maximum radiation φM is obtained when khcosϕMβ1=0, as inferred from Eq. (8). These two different β1 values result in a maximum at ϕM42° (ϕM61°), which are close to the one obtained by full-wave simulations at around ϕ51°, observed in Fig. 4.

Figure 5
Fig. 5 COMSOL results of the normalized far-field radiation pattern in free space by varying the free space wavelength λ0varying from 1.49 µm to1.56 µm.
shows COMSOL results of the far-field radiation pattern around the main beam, assuming free space outside the SiO2 region, when operating at different free space wavelengths. The maximum radiation angle has a clockwise shift from 88.8°to 94° with 70 nm wavelength increase from λ0=1.49 μm to λ0=1.56 μm. The 3 dB beam-width becomes narrower from Δϕ3dB2.7° to Δϕ3dB1.4°.

Figure 6
Fig. 6 Comparison of the normalized far-field radiation patterns in SiO2, obtained by COMSOL (red line), HFSS (black line), EA method (green line) and AF method (blue line). The inset shows an enlargement of the maximum radiation region.
shows the far-field patterns assuming that the SiO2 region extends beyond the dashed red line shown in Fig. 1. The red and black lines are the results obtained by COMSOL and HFSS, respectively. Whereas the green and blue lines are the EA method and the AF method results, respectively, considering 60 silicon perturbations. The full-wave simulation results present a very directive main-lobe at 92.5°. Again, this peak position is in agreement with the approximate value that, adopting the formula provided in Sec. 3.2, is ϕmaxarccos(β1/kh)=92.4°.

The side-lobe at 65° in Fig. 6 is not predicted by LW theory based on the values α and β found in Sec. 4.2. The same approach adopted in the case of free space environment can be used here to explain the side-lobe in Fig. 6, estimating the maximum radiation of the sub-array of 2 or 3 elements radiating at ϕM60° or ϕM71.3°, respectively, close to the observed one at 65°.

The directive radiation has been obtained by either HFSS or COMSOL, resulting into 17.5 dB directivity by using Eq. (9). The full-wave simulation 3 dB beam-width is Δϕ3dB1.05° , in agreement with the approximate formula (6) which gives Δϕ3dB0.9°.

Figure 7
Fig. 7 COMSOL results of the normalized far-field radiation pattern in SiO2, by varying the free space wavelength λ0from 1.49 µm to1.56 µm.
illustrates the COMSOL results of the far-field radiation pattern when operating at different free space wavelengths of the incoming wave mode. It is noted that as far as the wavelength increases (frequency decreases), the maximum radiation angle turns in a clockwise way. In particular, with 70 nm wavelength increase from λ0=1.49 μm to λ0=1.56 μm, the radiation peak has shifted from 88.8°to 92.9°. It can be observed that not only the peak moves by varying the wavelength, but also the 3 dB beam-width becomes narrower, from Δϕ3dB2.2° to Δϕ3dB1°.

To conclude the analysis on the radiation pattern of the OLWA, we mention that power is also radiated in the region 0°<ϕ<180°, Fig. 1, with a maximum at ϕ=93.2°in free space and at ϕ=92.5°in silica glass, though radiation toward the bottom side of the waveguide is stronger. The correspondent far-field maxima in the top side of the waveguide are 2.5 dB (free space environment) or 5 dB (silica glass environment) lower than the maxima in the correspondent bottom side direction. Radiation in the region 0°<ϕ<180° can be avoided by adding reflectors.

5. Modulation by carrier injection

Linear and non-linear optical properties of silicon have attracted many researchers to develop chip scale planar optical devices on the same optoelectronic platform that is compatible with high density electronic integration [25

25. K. Preston, S. Manipatruni, A. Gondarenko, C. B. Poitras, and M. Lipson, “Deposited silicon high-speed integrated electro-optic modulator,” Opt. Express 17(7), 5118–5124 (2009). [CrossRef] [PubMed]

]. In particular, integration of electronic control to manipulate the optical properties of silicon led to development of novel switches and modulators that can provide solution for telecom applications. The plasma dispersion effect is one of effective phenomena to realize high-speed modulation in silicon waveguides [26

26. Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s carrier-injection-based silicon micro-ring silicon modulators,” Opt. Express 15(2), 430–436 (2007). [CrossRef] [PubMed]

]. The basic tuning parameter in these devices is the carrier density that can alter both the Si refractive index (unitless) and the attenuation coefficient (m1) as described by the Drude’s model [27

27. O. Boyraz, X. Sang, E.-K. Tien, Q. Song, F. Qian, and M. Akdas, “Silicon based optical pulse shaping and characterization,” Proc. SPIE 7212, 72120U (2009). [CrossRef]

]
ΔnSi(Ne,Nh)=(8.8×104Ne+8.5Nh0.8)×1018,
(10)
ΔαSi(Ne,Nh)=(8.5Ne+6.0Nh)×1016,
(11)
whereNe and Nhare the concentrations of electrons and holes (expressed in cm3) in Si, and k0 is the free space wavenumber (expressed inm1). Practically, the density of electrons and holes in Si can be increased up to 1019cm3 in optoelectronic devices via current injection or photo generation before Auger process starts to mitigate the device performance [28

28. W. P. Dumke, “Minority-carrier injection and storage into a heavily doped emitter—approximate solution for Auger recombination,” Solid-State Electron. 24(2), 155–157 (1981). [CrossRef]

]. Hence the variation in both the refractive index and the attenuation coefficient due to carrier injection may be significant to facilitate the electronic control of the radiation pattern of OLWAs.

To assess the true variations in the radiation pattern in presence of carrier injection into silicon perturbations, we may assume a simple model where we have equal concentration of holes and electrons, or Nh = Ne. The presence of Nh=Ne=1019cm3free carriers reduces the refractive index of silicon, ΔnSi, by 0.0223, as illustrated in Fig. 8(a)
Fig. 8 The variation of (a) the silicon refractive index and (b) the normalized attenuation constant in silicon versus the injected carrier density. The carrier density Nh=Ne varies from 1016cm3 to 1019 cm3.
, whereas it increases the normalized attenuation constant, Δα¯Si=ΔαSi/k0, by 3.6×103 as shown in Fig. 8(b).

In Sec. 4.2 we have extracted β1=2.46×105m-1, whereas the attenuation constant α will be considered here as a varying parameter. Also, as stated at the beginning of this paper, the period of the silicon perturbations is d=970 nmand the width of each perturbation is w=300 nm, thus the filling factor is ff=15%. The total length of the perturbed waveguide is L=Nd=58.2 μm, with N=60.

In Fig. 9
Fig. 9 The variation of Qat broadside versus the injected carrier density. The carrier density Nh=Ne varies from 1016cm3 to 1019 cm3.
we show the variation of Q in Eq. (12) for some examples of different values of the LW attenuation constant (before carrier injection) equal toα=0.01k0, α=0.005k0 and α=0.001k0. Figure 9 indicates that Q0.52 dB at the carrier injection of Nh=Ne=1019cm3when α=0.01k0. The value changes to Q0.65 dB and Q0.75 dBwhen α=0.005k0and α=0.001k0, respectively.

The preliminary results presented in Fig. 9 indicate that the effect of carrier injection on the power radiated at a certain direction is rather limited (0.52 dB with a LW attenuation constant α=0.01k0 which is similar to the one extracted by full-wave simulations in Sec. 4.2). One way of amplifying weak perturbations in the optical domain is achieved by placing perturbations into optical resonators so that these perturbations will be experienced by the optical field during multiple roundtrips [29

29. C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol. 24(3), 1433–1439 (2006). [CrossRef]

]. Indeed, it has been already demonstrated theoretically and experimentally that weak perturbations created via injection of carriers into silicon optical resonators alter the resonance wavelength and the quality factor of the resonator and provide modulation of the optical signals [29

29. C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol. 24(3), 1433–1439 (2006). [CrossRef]

]. Similarly, by integrating OLWAs into a racetrack or Fabry-Perot integrated cavities, it is expected to amplify the 0.52 dBperturbation in the radiation pattern of an amount which is proportional to the quality factor of the cavity. Future research should be dedicated to quantify the perturbations in the radiation pattern in resonators with practical geometries.

6. Three dimensional design of a linear OLWA

The 3D geometry reported in Fig. 10
Fig. 10 3D model for a linear OLWA. Notice that the lateral view is as in the 2D model.
consists of the Si3N4 waveguide with square cross section w×w, with w=1 μm and 39.285 μmlong. It is embedded in a SiO2 volume with cross section equal to 6 μm×6 μm. The number N of perturbations is now chosen equal to 35 (because of the high numerical simulation burden of 3D structures), and each perturbation extends to the edge of the silica glass domain (characteristic that will be adopted to perform carrier injection in future works, as discussed in Sec. 5). Each silicon perturbation has the following dimensions: wp=6 μm, h=300 nm, l=d/2, whered=1.015 μm.The input light used in these analyses is as the one described in Sec. 4. Adopted materials are as in Sec. 2.

The radiation pattern, assuming free space outside the SiO2 volume, of the linear OLWA in Fig. 10, is obtained through an HFSS simulation and shown in Fig. 11
Fig. 11 Comparison of the far-field radiation patterns in the xy plane of 2D and 3D optical leaky wave antennas. In both cases a narrow radiating beam is obtained.
. It has been compared with the HFSS result obtained for the 2D OLWA in Fig. 1. The result in Fig. 11 shows that the radiation pattern carried out for the 2D OLWA is also representative for the description of directive radiation in the xy plane, for the 3D case. However, some small difference is noticeable, and that is due to the different total length of the radiating section L=Nd, where N is the number of the perturbation, chosen equal to 60 in the 2D model and equal to 35 in the 3D model to limit the computational requirements.

7. Conclusion

In this work, a highly directive OLWA based on SOI waveguide (made of a Si3N4 waveguide comprising a planar periodic structure of silicon strips) has been proposed, whose working principle has been explained by using leaky wave and antenna array factor theory. A two dimensional (2D) implementation has been analyzed in details, and the radiation pattern has been compared to that of a linear OLWA in a 3D space. The highly directive radiation has been designed to be around the bottom transverse direction with 17.5 dBdirectivity and a 3 dB beam-width of Δϕ3dB1.05° in SiO2 environment and with 15.7 dBdirectivity and a 3 dB beam-width of Δϕ3dB1.4° in free space environment, when leaky wave phase and attenuation constants were β1=2.46×105m-1, and α=4.52×104m-1, as extracted by full-wave simulations. Far-field radiation pattern, beam angle and beam-width have been all parameterized in simple terms, based on the complex LW wavenumber β1+iα in the main Brillouin zone. Results have shown that the theoretical predictions agree well with both COMSOL and HFSS full-wave results. For the sake of further investigation, we have observed the relationship of excess index change and attenuation constant with various carrier injection densities in the Si perturbations. Results showed that up to 0.75 dB variation was achievable in the electric far-field radiation magnitude in presence of carrier injection of 1019cm-3 with respect to the one in absence of carrier injection, at ϕmax, with β10.06k0, α=0.001k0 when the perturbation length is 58.2 µm. Based on this investigation, the wave beaming of the optical radiator at a fixed direction could be varied by combining the electronic and optical control to the intrinsic quality factor of the SOI device. Based on reciprocity, the interesting radiation performance exactly holds also for the receiving case, showing very narrow angular selectivity. Our results made the high speed modulation and integration more feasible and promising.

Acknowledgments

This work is supported by National Science Foundation (NSF) Award # ECCS-1028727. The authors also thank Ansys and COMSOL Multiphysics for providing them their simulation tools (HFSS and COMSOL) that were instrumental in this analysis.

References and links

1.

P. Ghenuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopic mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett. 101(11), 116805 (2008). [CrossRef] [PubMed]

2.

S. Wedge, J. A. E. Wasey, W. L. Barnes, and I. Sage, “Coupled surface plasmon-polariton mediated photoluminescence from a top-emitting organic light-emitting structure,” Appl. Phys. Lett. 85(2), 182–184 (2004). [CrossRef]

3.

R. L. Olmon, P. M. Krenz, A. C. Jones, G. D. Boreman, and M. B. Raschke, “Near-field imaging of optical antenna modes in the mid-infrared,” Opt. Express 16(25), 20295–20305 (2008). [CrossRef] [PubMed]

4.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13(3), 429–432 (2002). [CrossRef]

5.

Q. Song, F. Qian, E. K. Tien, I. Tomov, J. Meyer, X. Z. Sang, and O. Boyraz, “Imaging by silicon on insulator waveguides,” Appl. Phys. Lett. 94(23), 231101 (2009). [CrossRef]

6.

C. K. Toth, “R&D of mobile LIDAR mapping and future trends,” in Proceeding of ASPRS 2009 Annual Conference (Baltimore, Maryland, 2009).

7.

D. R. Jackson, J. Chen, R. Qiang, F. Capolino, and A. A. Oliner, “The role of leaky plasmon waves in the directive beaming of light through a subwavelength aperture,” Opt. Express 16(26), 21271–21281 (2008). [CrossRef] [PubMed]

8.

A. Micco, V. Galdi, F. Capolino, A. Della Villa, V. Pierro, S. Enoch, and G. Tayeb, “Directive emission from defect-free dodecagonal photonic quasicrystals: a leaky wave characterization,” Phys. Rev. B 79(7), 075110 (2009). [CrossRef]

9.

E. Colak, H. Caglayan, A. O. Cakmak, A. D. Villa, F. Capolino, and E. Ozbay, “Frequency dependent steering with backward leaky waves via photonic crystal interface layer,” Opt. Express 17(12), 9879–9890 (2009). [CrossRef] [PubMed]

10.

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678–1687 (2006). [CrossRef]

11.

Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]

12.

L. Friedman, R. A. Soref, and J. P. Lorenzo, “Silicon double-injection electro-optic modulator with junction gate control,” J. Appl. Phys. 63(6), 1831–1839 (1988). [CrossRef]

13.

C. K. Tang and G. T. Reed, “Highly efficient optical-phase modulator in SOI waveguides,” Electron. Lett. 31(6), 451–452 (1995). [CrossRef]

14.

K. Van Acoleyen, W. Bogaerts, J. Jágerská, N. Le Thomas, R. Houdré, and R. Baets, “Off-chip beam steering with a one-dimensional optical phased array on silicon-on-insulator,” Opt. Lett. 34(9), 1477–1479 (2009). [CrossRef] [PubMed]

15.

S. M. Csutak, S. Dakshina-Murthy, and J. C. Campbell, “CMOS-compatible planar silicon waveguide-grating-coupler photodetectors fabricated on silicon-on-insulator (SOI) substrates,” IEEE J. Quantum Electron. 38(5), 477–480 (2002). [CrossRef]

16.

P. Cheben, S. Janz, D. X. Xu, B. Lamontagne, A. Delage, and S. Tanev, “A broad-band waveguide grating coupler with a subwavelength grating mirror,” IEEE Photon. Technol. Lett. 18(1), 13–15 (2006). [CrossRef]

17.

A. A. Oliner and D. R. Jackson, “Leaky-wave antennas,” in Antenna Engineering Handbook, J. Volakis, ed. (McGraw Hill, 2007), pp. 11.11–11.56.

18.

F. Capolino, D. R. Jackson, and D. R. Wilton, “Field representation in periodic artificial materials excited by a source,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 12.11.

19.

S. Campione and F. Capolino, “Linear and Planar Periodic Arrays of Metallic Nanospheres: Fabrication, Optical Properties and Applications,” in Selected Topics in Metamaterials and Photonic Crystals, A. Andreone, A. Cusano, A. Cutolo, and V. Galdi, eds. (World Scientific Publishing, 2011).

20.

D. R. Jackson and A. A. Oliner, “Leaky-wave antennas,” in Modern Antenna Handbook, C. A. Balanis, ed. (Wiley, 2008), pp. 325–367.

21.

R. E. Collin, Antennas and Radiowave Propagation (McGraw Hill, 1985), p. 164.

22.

C. A. Balanis, Antenna Theory: Analysis and Design (Wiley, 2005), p. 286.

23.

E. J. Rothwell and M. J. Cloud, Electromagnetics (CRC Press, 2008), p. 418.

24.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), p. 424.

25.

K. Preston, S. Manipatruni, A. Gondarenko, C. B. Poitras, and M. Lipson, “Deposited silicon high-speed integrated electro-optic modulator,” Opt. Express 17(7), 5118–5124 (2009). [CrossRef] [PubMed]

26.

Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s carrier-injection-based silicon micro-ring silicon modulators,” Opt. Express 15(2), 430–436 (2007). [CrossRef] [PubMed]

27.

O. Boyraz, X. Sang, E.-K. Tien, Q. Song, F. Qian, and M. Akdas, “Silicon based optical pulse shaping and characterization,” Proc. SPIE 7212, 72120U (2009). [CrossRef]

28.

W. P. Dumke, “Minority-carrier injection and storage into a heavily doped emitter—approximate solution for Auger recombination,” Solid-State Electron. 24(2), 155–157 (1981). [CrossRef]

29.

C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol. 24(3), 1433–1439 (2006). [CrossRef]

OCIS Codes
(230.7390) Optical devices : Waveguides, planar
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Optical Devices

History
Original Manuscript: January 14, 2011
Revised Manuscript: March 8, 2011
Manuscript Accepted: April 11, 2011
Published: April 20, 2011

Citation
Qi Song, Salvatore Campione, Ozdal Boyraz, and Filippo Capolino, "Silicon-based optical leaky wave antenna with narrow beam radiation," Opt. Express 19, 8735-8749 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8735


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References

  1. P. Ghenuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopic mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett. 101(11), 116805 (2008). [CrossRef] [PubMed]
  2. S. Wedge, J. A. E. Wasey, W. L. Barnes, and I. Sage, “Coupled surface plasmon-polariton mediated photoluminescence from a top-emitting organic light-emitting structure,” Appl. Phys. Lett. 85(2), 182–184 (2004). [CrossRef]
  3. R. L. Olmon, P. M. Krenz, A. C. Jones, G. D. Boreman, and M. B. Raschke, “Near-field imaging of optical antenna modes in the mid-infrared,” Opt. Express 16(25), 20295–20305 (2008). [CrossRef] [PubMed]
  4. T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13(3), 429–432 (2002). [CrossRef]
  5. Q. Song, F. Qian, E. K. Tien, I. Tomov, J. Meyer, X. Z. Sang, and O. Boyraz, “Imaging by silicon on insulator waveguides,” Appl. Phys. Lett. 94(23), 231101 (2009). [CrossRef]
  6. C. K. Toth, “R&D of mobile LIDAR mapping and future trends,” in Proceeding of ASPRS 2009 Annual Conference (Baltimore, Maryland, 2009).
  7. D. R. Jackson, J. Chen, R. Qiang, F. Capolino, and A. A. Oliner, “The role of leaky plasmon waves in the directive beaming of light through a subwavelength aperture,” Opt. Express 16(26), 21271–21281 (2008). [CrossRef] [PubMed]
  8. A. Micco, V. Galdi, F. Capolino, A. Della Villa, V. Pierro, S. Enoch, and G. Tayeb, “Directive emission from defect-free dodecagonal photonic quasicrystals: a leaky wave characterization,” Phys. Rev. B 79(7), 075110 (2009). [CrossRef]
  9. E. Colak, H. Caglayan, A. O. Cakmak, A. D. Villa, F. Capolino, and E. Ozbay, “Frequency dependent steering with backward leaky waves via photonic crystal interface layer,” Opt. Express 17(12), 9879–9890 (2009). [CrossRef] [PubMed]
  10. R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678–1687 (2006). [CrossRef]
  11. Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]
  12. L. Friedman, R. A. Soref, and J. P. Lorenzo, “Silicon double-injection electro-optic modulator with junction gate control,” J. Appl. Phys. 63(6), 1831–1839 (1988). [CrossRef]
  13. C. K. Tang and G. T. Reed, “Highly efficient optical-phase modulator in SOI waveguides,” Electron. Lett. 31(6), 451–452 (1995). [CrossRef]
  14. K. Van Acoleyen, W. Bogaerts, J. Jágerská, N. Le Thomas, R. Houdré, and R. Baets, “Off-chip beam steering with a one-dimensional optical phased array on silicon-on-insulator,” Opt. Lett. 34(9), 1477–1479 (2009). [CrossRef] [PubMed]
  15. S. M. Csutak, S. Dakshina-Murthy, and J. C. Campbell, “CMOS-compatible planar silicon waveguide-grating-coupler photodetectors fabricated on silicon-on-insulator (SOI) substrates,” IEEE J. Quantum Electron. 38(5), 477–480 (2002). [CrossRef]
  16. P. Cheben, S. Janz, D. X. Xu, B. Lamontagne, A. Delage, and S. Tanev, “A broad-band waveguide grating coupler with a subwavelength grating mirror,” IEEE Photon. Technol. Lett. 18(1), 13–15 (2006). [CrossRef]
  17. A. A. Oliner and D. R. Jackson, “Leaky-wave antennas,” in Antenna Engineering Handbook, J. Volakis, ed. (McGraw Hill, 2007), pp. 11.11–11.56.
  18. F. Capolino, D. R. Jackson, and D. R. Wilton, “Field representation in periodic artificial materials excited by a source,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 12.11.
  19. S. Campione and F. Capolino, “Linear and Planar Periodic Arrays of Metallic Nanospheres: Fabrication, Optical Properties and Applications,” in Selected Topics in Metamaterials and Photonic Crystals, A. Andreone, A. Cusano, A. Cutolo, and V. Galdi, eds. (World Scientific Publishing, 2011).
  20. D. R. Jackson and A. A. Oliner, “Leaky-wave antennas,” in Modern Antenna Handbook, C. A. Balanis, ed. (Wiley, 2008), pp. 325–367.
  21. R. E. Collin, Antennas and Radiowave Propagation (McGraw Hill, 1985), p. 164.
  22. C. A. Balanis, Antenna Theory: Analysis and Design (Wiley, 2005), p. 286.
  23. E. J. Rothwell and M. J. Cloud, Electromagnetics (CRC Press, 2008), p. 418.
  24. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), p. 424.
  25. K. Preston, S. Manipatruni, A. Gondarenko, C. B. Poitras, and M. Lipson, “Deposited silicon high-speed integrated electro-optic modulator,” Opt. Express 17(7), 5118–5124 (2009). [CrossRef] [PubMed]
  26. Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s carrier-injection-based silicon micro-ring silicon modulators,” Opt. Express 15(2), 430–436 (2007). [CrossRef] [PubMed]
  27. O. Boyraz, X. Sang, E.-K. Tien, Q. Song, F. Qian, and M. Akdas, “Silicon based optical pulse shaping and characterization,” Proc. SPIE 7212, 72120U (2009). [CrossRef]
  28. W. P. Dumke, “Minority-carrier injection and storage into a heavily doped emitter—approximate solution for Auger recombination,” Solid-State Electron. 24(2), 155–157 (1981). [CrossRef]
  29. C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol. 24(3), 1433–1439 (2006). [CrossRef]

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