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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 27 — Dec. 17, 2012
  • pp: 28479–28484
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Graphene-assisted control of coupling between optical waveguides

Andrea Locatelli, Antonio-Daniele Capobianco, Michele Midrio, Stefano Boscolo, and Costantino De Angelis  »View Author Affiliations


Optics Express, Vol. 20, Issue 27, pp. 28479-28484 (2012)
http://dx.doi.org/10.1364/OE.20.028479


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Abstract

The unique properties of optical waveguides electrically controlled by means of graphene layers are investigated. We demonstrate that, thanks to tunable losses induced by graphene layers, a careful design of silicon on silica ridge waveguides can be used to explore passive PT-symmetry breaking in directional couplers. We prove that the exceptional point of the system can be probed by varying the applied voltage and we thus propose very compact photonic structures which can be exploited to control coupling between waveguides and to tailor discrete diffraction in arrays.

© 2012 OSA

1. Introduction

Graphene is a single layer of carbon atoms packed in a honeycomb lattice which has been attracting increasing interest from the scientific community since 2010, when the Nobel Prize in Physics was assigned to Novoselov and Geim for their groundbreaking experimental results [1

1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004). [CrossRef] [PubMed]

].

Graphene is a special material which is characterized by exceptional properties. In particular mobility of charge carriers is extremely high, thus it is expected that graphene can have an important role for future evolution of high-speed electronics, likely by improving performance of conventional silicon devices rather than by replacing them [2

2. K. Kim, J. Y. Choi, T. Kim, S. H. Cho, and H. J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature (London) 479, 338–344 (2011). [CrossRef]

].

The optical properties of graphene display intriguing features over a very wide bandwidth that covers the mid-infrared and the visible spectrum. Indeed, infrared plasmonic devices based on graphene have already been proposed [3

3. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011). [CrossRef] [PubMed]

] whereas a plethora of applications at optical frequencies, ranging from photovoltaics and light-emitting devices [4

4. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photon. 4, 611–622 (2010). [CrossRef]

] to the realization of novel components for communication systems [2

2. K. Kim, J. Y. Choi, T. Kim, S. H. Cho, and H. J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature (London) 479, 338–344 (2011). [CrossRef]

, 5

5. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London) 474, 64–67 (2011). [CrossRef]

9

9. J. T. Kim and C. G. Choi, “Graphene-based polymer waveguide polarizer,” Opt. Express 20, 3556–3562 (2012). [CrossRef] [PubMed]

] have been reported, in spite of the small absorption which is typical of graphene layers under normal optical excitation.

In this context, deposition of graphene layers onto/into silicon waveguides which permit interaction length to be increased appears to be a promising method to realize compact broadband optical modulators [2

2. K. Kim, J. Y. Choi, T. Kim, S. H. Cho, and H. J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature (London) 479, 338–344 (2011). [CrossRef]

,5

5. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London) 474, 64–67 (2011). [CrossRef]

7

7. M. Midrio, S. Boscolo, M. Moresco, M. Romagnoli, C. De Angelis, A. Locatelli, and A.-D. Capobianco, “Graphene-assisted critically-coupled optical ring modulator,” Opt. Express 20, 23144–23155 (2012). [CrossRef]

]. These structures exploit a simple operation principle: graphene behaves as a loss element controlled by an electric voltage, which properly tunes the electric Fermi level near the Dirac point of the conical dispersion relation [10

10. Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. 4, 532–535 (2008). [CrossRef]

]. In this way, graphene conductivity can be switched from 0 to an universal frequency-independent value.

In this work we present numerical results which demonstrate the huge potential of graphene as a mean to control coupling between optical waveguides. The possibility of tuning losses in each waveguide by acting on a thin loss element permits symmetry of the system of coupled waveguides to be broken without introducing a strong perturbation of the single waveguide. Remarkably we demonstrate that tunable losses induced by graphene and a careful design of ridge waveguides allows to probe passive PT-symmetry breaking in directional couplers [11

11. S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008). [CrossRef] [PubMed]

13

13. S. Yu, G. X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A 86, 031802 (2012). [CrossRef]

]. Moreover we prove that the exceptional point of the proposed structure can be dynamically reached by varying the applied voltage. We will thus explore this property to mould energy exchange between waveguides and to finely tune discrete diffraction in waveguide arrays.

2. Optimal control of losses in an optical waveguide

We considered the behavior of silicon waveguides on a silica substrate in a wavelength range between 1350 and 1600 nm. The structure has been inspired by the modulator proposed in [2

2. K. Kim, J. Y. Choi, T. Kim, S. H. Cho, and H. J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature (London) 479, 338–344 (2011). [CrossRef]

]. In particular, a layer of silicon with thickness equal to 50 nm is deposited onto the substrate. The 400-nm-wide ridge waveguide is composed of a lower and a higher layer made of silicon (both with thickness 200 nm) which sandwich a central region including three alternating layers of alumina (thickness 7 nm) and two absorption layers composed of three graphene monolayers with thickness 0.34 nm. Graphene can be electrically controlled in order to tune doping (and then conductivity), as suggested in [2

2. K. Kim, J. Y. Choi, T. Kim, S. H. Cho, and H. J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature (London) 479, 338–344 (2011). [CrossRef]

] and [6

6. M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12, 1482–1485 (2012). [CrossRef] [PubMed]

]. The dielectric constants of silicon, silica and alumina were taken equal to 12.1, 2.1 and 3. Figure 1(a) displays a schematic view of the structure.

Fig. 1 (a) Schematic view of the waveguide structure, with a detail of the central region with graphene layers. (b) Losses of the single waveguide (in dB/μm) when graphene is in OFF state (null voltage), and x-component of the electric field of the TE-like mode (inset).

The behavior of graphene in the optical regime has been numerically modeled by following the approach suggested in [3

3. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011). [CrossRef] [PubMed]

]. Indeed, we assigned to each graphene monolayer with thickness Δ a volume conductivity equal to σg,v = σ2D/Δ, where σ2D is the conductivity of the 2D sheet. It was demonstrated that, as a first approximation, few-layer graphene is characterized by the same band structure (and then by the same excellent electronic properties) of the monolayer, and plus conductivity of N-layer graphene (N = 3 in our design) can be evaluated as N times conductivity of the single layer if N is small enough [5

5. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London) 474, 64–67 (2011). [CrossRef]

].

Analytical studies and experimental results (see [10

10. Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. 4, 532–535 (2008). [CrossRef]

,14

14. G. W. Hanson, “Dyadic Green’s functions for an anisotropic, non-local model of biased graphene,” IEEE Trans. Antennas Propagat. 56, 747–757 (2008). [CrossRef]

]) which provide the 2D conductivity from microwaves to the optical regime as a function of the driving voltage have been reported in the literature. All these works demonstrate that the optical conductivity of graphene is dominated by the interband component, which can be tuned by acting on the control voltage. In particular, the real part of graphene conductivity is close to zero below a well defined threshold frequency corresponding to the onset of interband transitions, whereas over this abrupt threshold conductivity recovers the universal value σ2D = πe2/2h [10

10. Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. 4, 532–535 (2008). [CrossRef]

]. The threshold frequency fth is located around hfth = 2EF, where h is Planck’s constant and EF is the Fermi level, which can be moved by acting on the applied voltage. This threshold shifts to larger frequencies with increasing voltage, and few volts are sufficient to move it in the near-infrared.

3. Coupled-mode theory

4. Control of coupling between optical waveguides

Fig. 2 Schematic view of the 300-nm-gap coupler (left), with the electric field of the low-(center) and high-loss mode (right) at 1530 nm. Graphene layers are in ON-OFF states.

The dispersive properties of the two modes have been characterized through full-wave and CMT simulations, and these quantitative results confirm the intuitive analysis we have reported above. Indeed, in Fig. 3(a) losses of the two supermodes are depicted when graphene layers are in the ON-OFF state. In this case symmetry is broken, as a consequence one mode is characterized by absorption which is close to zero, whereas losses of the other one are large, very close to those of a single lossy waveguide. It is worth noting that this effect tends to blur with increasing wavelength due to the dependence of coupling coefficient on frequency (C gets larger with increasing wavelength). A thorough treatment on phenomena arising from the wavelength dependence of the PT-symmetry condition is reported in [13

13. S. Yu, G. X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A 86, 031802 (2012). [CrossRef]

]. The noticeable agreement between simulations performed by using a full-wave mode solver and results evaluated by using CMT (in the latter case the imaginary part of λ1,2 is reported) allows to confirm the accuracy of CMT.

Fig. 3 (a) Losses of low- (red line) and high-loss mode (blue line) from mode solver (solid line) and CMT (dashed-dotted line). (b) Normalized attenuation constant of low- (red line with squares) and high-loss mode (blue line with circles) vs. normalized attenuation constant of the single waveguide at λ = 1530 nm. The vertical thin line indicates α = αmax.

These phenomena stem from breaking of passive PT-symmetry in complex potentials. Indeed, in Fig. 3(b) we plot the attenuation constants of the two supermodes, evaluated by using CMT, as a function of the attenuation constant of the single waveguide α. Data are normalized with respect to twice the coupling coefficient, so that we have the exceptional point when the abscissa is equal to 1. The vertical dotted line indicates αmax, i.e. the value of α when our structure is in OFF state, and it is straightforward to see that we can work beyond the exceptional point, in agreement with the results in Fig. 3(a). It is worth to emphasize that graphene-based waveguides exhibit superior properties with respect to waveguides wherein losses are introduced by depositing metal layers [12

12. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009). [CrossRef] [PubMed]

]. Losses induced by sandwiching graphene layers inside silicon waveguides can be orders of magnitude larger (thousands of cm−1 with respect to tens of cm−1), therefore it is possible to probe the exceptional point even in structures characterized by strong coupling. Last, but not least, it is important to note that graphene is electrically tunable, therefore losses in each single waveguide can be varied between 0 (ON state) and a maximum value αmax determined only by geometry (OFF state).

We envisage that switching of the state of one waveguide can be exploited to finely tune coupling between waveguides. In order to verify the effectiveness of this approach we applied CMT to our reference structure at the wavelength of 1530 nm, and we show the results in Fig. 4. When the coupler is in the ON-ON state losses are zero, and the predicted beat length LB = π/(βevenβodd) is around 80 μm. Viceversa, when graphene layers are ON and OFF in the input and output channel the two waveguides tend to decouple and field intensity in the first waveguide is larger than in the second one. It is possible to justify this behavior by recalling that when we inject light into the waveguide in ON state the low-loss supermode is mainly excited.

Fig. 4 Field amplitude in (a) first and (b) second waveguide of the coupler. Graphene layers are in ON-ON (black line), and ON-OFF (red line) states.

These results have been validated by comparison with simulations of the 80-μm long coupler performed by using the commercial software CST Microwave Studio, which allows to solve Maxwell’s equations in the time domain through the finite-integration technique. Indeed, the ratio between output and injected power evaluated by using CMT is −3 and −12 dB if coupler is in ON-OFF state and we consider as output port waveguides 1 and 2. CST simulations exhibit a good agreement, in fact the corresponding calculated values are about −5 and −13 dB.

5. Control of discrete diffraction in optical waveguide arrays

Fig. 5 Discrete diffraction along the array. (a) All the graphene layers are in ON state. (b) Only graphene layers inside the central waveguide are in ON state.

6. Conclusion

We have presented properties of graphene-based coupled waveguides. In particular, we have shown that tuning the state of graphene layers in each single waveguide allows energy exchange between the two channels of a coupler and discrete diffraction in arrays to be controlled.

Acknowledgments

AL and CDA acknowledge financial support from Cariplo Foundation under grant no. 2010-0595, and from US Army under grant no. W911NF-12-1-0202.

References and links

1.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004). [CrossRef] [PubMed]

2.

K. Kim, J. Y. Choi, T. Kim, S. H. Cho, and H. J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature (London) 479, 338–344 (2011). [CrossRef]

3.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011). [CrossRef] [PubMed]

4.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photon. 4, 611–622 (2010). [CrossRef]

5.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London) 474, 64–67 (2011). [CrossRef]

6.

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12, 1482–1485 (2012). [CrossRef] [PubMed]

7.

M. Midrio, S. Boscolo, M. Moresco, M. Romagnoli, C. De Angelis, A. Locatelli, and A.-D. Capobianco, “Graphene-assisted critically-coupled optical ring modulator,” Opt. Express 20, 23144–23155 (2012). [CrossRef]

8.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photon. 5, 411–415 (2011). [CrossRef]

9.

J. T. Kim and C. G. Choi, “Graphene-based polymer waveguide polarizer,” Opt. Express 20, 3556–3562 (2012). [CrossRef] [PubMed]

10.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. 4, 532–535 (2008). [CrossRef]

11.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008). [CrossRef] [PubMed]

12.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009). [CrossRef] [PubMed]

13.

S. Yu, G. X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A 86, 031802 (2012). [CrossRef]

14.

G. W. Hanson, “Dyadic Green’s functions for an anisotropic, non-local model of biased graphene,” IEEE Trans. Antennas Propagat. 56, 747–757 (2008). [CrossRef]

15.

D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988). [CrossRef] [PubMed]

16.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. 88, 093901 (2002). [CrossRef] [PubMed]

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(130.2790) Integrated optics : Guided waves
(230.2090) Optical devices : Electro-optical devices
(250.7360) Optoelectronics : Waveguide modulators
(130.4815) Integrated optics : Optical switching devices

ToC Category:
Integrated Optics

History
Original Manuscript: October 12, 2012
Revised Manuscript: November 14, 2012
Manuscript Accepted: November 14, 2012
Published: December 7, 2012

Citation
Andrea Locatelli, Antonio-Daniele Capobianco, Michele Midrio, Stefano Boscolo, and Costantino De Angelis, "Graphene-assisted control of coupling between optical waveguides," Opt. Express 20, 28479-28484 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-28479


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References

  1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306, 666–669 (2004). [CrossRef] [PubMed]
  2. K. Kim, J. Y. Choi, T. Kim, S. H. Cho, and H. J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature (London)479, 338–344 (2011). [CrossRef]
  3. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science332, 1291–1294 (2011). [CrossRef] [PubMed]
  4. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photon.4, 611–622 (2010). [CrossRef]
  5. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London)474, 64–67 (2011). [CrossRef]
  6. M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett.12, 1482–1485 (2012). [CrossRef] [PubMed]
  7. M. Midrio, S. Boscolo, M. Moresco, M. Romagnoli, C. De Angelis, A. Locatelli, and A.-D. Capobianco, “Graphene-assisted critically-coupled optical ring modulator,” Opt. Express20, 23144–23155 (2012). [CrossRef]
  8. Q. Bao, H. Zhang, B. Wang, Z. Ni, C. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photon.5, 411–415 (2011). [CrossRef]
  9. J. T. Kim and C. G. Choi, “Graphene-based polymer waveguide polarizer,” Opt. Express20, 3556–3562 (2012). [CrossRef] [PubMed]
  10. Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys.4, 532–535 (2008). [CrossRef]
  11. S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett.101, 080402 (2008). [CrossRef] [PubMed]
  12. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett.103, 093902 (2009). [CrossRef] [PubMed]
  13. S. Yu, G. X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A86, 031802 (2012). [CrossRef]
  14. G. W. Hanson, “Dyadic Green’s functions for an anisotropic, non-local model of biased graphene,” IEEE Trans. Antennas Propagat.56, 747–757 (2008). [CrossRef]
  15. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett.13, 794–796 (1988). [CrossRef] [PubMed]
  16. T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901 (2002). [CrossRef] [PubMed]

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