## Graphene-assisted control of coupling between optical waveguides |

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

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]

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

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. 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]

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]

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]

## 2. Optimal control of losses in an optical waveguide

**479**, 338–344 (2011). [CrossRef]

**479**, 338–344 (2011). [CrossRef]

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

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

*σ*=

_{g,v}*σ*

_{2D}/Δ, where

*σ*

_{2D}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]

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. 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]

*σ*

_{2D}=

*πe*

^{2}/2

*h*[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]

*f*is located around

_{th}*hf*= 2

_{th}*E*, where

_{F}*h*is Planck’s constant and

*E*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.

_{F}*μ*m, which corresponds to 1100 cm

^{−1}) and almost constant over the entire bandwidth: indeed, a 6-dB modulation contrast between ON and OFF states can be achieved with a 12.5

*μ*m-long waveguide. In the next paragraphs we will study properties of coupled waveguides wherein the described structure is the basic building block.

## 3. Coupled-mode theory

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]

*β*of each isolated waveguide are affected by the status of graphene layers, whereas switching between ON and OFF states has the effect of turning off and on losses in the single waveguide, which are modeled by the attenuation constant

*α*. Notice that modal evolution reads as exp(

*iβz*)exp(−

*αz*).

*A*

_{1,2}, which are the modal field amplitudes in the first and second waveguide of a directional coupler composed of two identical graphene-based waveguides, can be approximated as where

*α*

_{1,2}can be tuned between 0 (ON state) and

*α*(OFF state) by controlling the voltage applied to the graphene layers, and

_{max}*C*is a complex coupling coefficient [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]

*α*

_{1}= 0 and

*α*

_{2}=

*α*the eigenvalues read as

*α*= 2

_{c}*C*. When

*α*<

*α*the two supermodes have different propagation constants and the same attenuation constant

_{c}*α*/2. Beyond the critical loss supermodes coalesce, indeed they are characterized by equal propagation constant

*β*and different losses. In particular, one supermode experiences increasing losses with increasing

*α*, whereas the other one is characterized by the opposite trend [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]

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]

*α*is much larger than

*C*, one supermode is characterized by losses which are close to those of a single lossy waveguide, whereas the other one tends to be loss-free (see next section).

## 4. Control of coupling between optical waveguides

*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]

*λ*

_{1,2}is reported) allows to confirm the accuracy of CMT.

*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

*α*, i.e. the value of

_{max}*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]

^{−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

*α*determined only by geometry (OFF state).

_{max}*L*=

_{B}*π*/(

*β*−

_{even}*β*) is around 80

_{odd}*μ*m. 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.

*μ*m 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

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]

*μ*m). Moreover, we assume that state of graphene layers in each waveguide can be controlled independently from each other.

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]

## 6. Conclusion

## Acknowledgments

## 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 |

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) |

3. | A. Vakil and N. Engheta, “Transformation optics using graphene,” Science |

4. | F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photon. |

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) |

6. | M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. |

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 |

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. |

9. | J. T. Kim and C. G. Choi, “Graphene-based polymer waveguide polarizer,” Opt. Express |

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. |

11. | S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. |

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. |

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 |

14. | G. W. Hanson, “Dyadic Green’s functions for an anisotropic, non-local model of biased graphene,” IEEE Trans. Antennas Propagat. |

15. | D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. |

16. | T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. |

**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

- 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]
- 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]
- A. Vakil and N. Engheta, “Transformation optics using graphene,” Science332, 1291–1294 (2011). [CrossRef] [PubMed]
- F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photon.4, 611–622 (2010). [CrossRef]
- 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]
- M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett.12, 1482–1485 (2012). [CrossRef] [PubMed]
- 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]
- 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]
- J. T. Kim and C. G. Choi, “Graphene-based polymer waveguide polarizer,” Opt. Express20, 3556–3562 (2012). [CrossRef] [PubMed]
- 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]
- S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett.101, 080402 (2008). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett.13, 794–796 (1988). [CrossRef] [PubMed]
- 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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