## Directional coupler and nonlinear Mach-Zehnder interferometer based on metal-insulator-metal plasmonic waveguide |

Optics Express, Vol. 18, Issue 20, pp. 21030-21037 (2010)

http://dx.doi.org/10.1364/OE.18.021030

Acrobat PDF (1017 KB)

### Abstract

Directional coupler (DC) and nonlinear Mach-Zehnder interferometer (MZI) based on metal-insulator-metal (MIM) plasmonic waveguide are investigated numerically. We show that the coupling length increases almost linearly with the wavelength and this property is utilized in the design of wavelength division multiplexer (WDM). A nonlinear MZI, with one branch filled with Kerr nonlinear medium, is built to ensure controlling light with light. Employing nonlinear processes including self-phase modulation (SPM) and cross-phase modulation (XPM), intensity-based router and all-optical switch are realized.

© 2010 OSA

## 1. Introduction

1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**(6950), 824–830 (2003). [CrossRef] [PubMed]

2. R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today **9**(7-8), 20–27 (2006). [CrossRef]

7. Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Krasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. **34**(3), 310–312 (2009). [CrossRef] [PubMed]

2. R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today **9**(7-8), 20–27 (2006). [CrossRef]

8. Z. Han, L. Liu, and E. Forsberg, “Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface Plasmon polaritons,” Opt. Commun. **259**(2), 690–695 (2006). [CrossRef]

11. G. Veronis and S. Fan, “Crosstalk between three-dimensional plasmonic slot waveguides,” Opt. Express **16**(3), 2129–2140 (2008). [CrossRef] [PubMed]

12. Z. Han, A. Y. Elezzabi, and V. Van, “Wideband Y-splitter and aperture-assisted coupler based on sub-diffraction confined plasmonic slot waveguides,” Appl. Phys. Lett. **96**(13), 131106 (2010). [CrossRef]

14. C. Min, P. Wang, C. Chen, Y. Deng, Y. Lu, H. Ming, T. Ning, Y. Zhou, and G. Yang, “All-optical switching in subwavelength metallic grating structure containing nonlinear optical materials,” Opt. Lett. **33**(8), 869–871 (2008). [CrossRef] [PubMed]

15. Z.-J. Zhong, Y. Xu, S. Lan, Q.-F. Dai, and L.-J. Wu, “Sharp and asymmetric transmission response in metal-dielectric-metal plasmonic waveguides containing Kerr nonlinear media,” Opt. Express **18**(1), 79–86 (2010). [CrossRef] [PubMed]

## 2. Principle and the wavelength-selective direction coupler

11. G. Veronis and S. Fan, “Crosstalk between three-dimensional plasmonic slot waveguides,” Opt. Express **16**(3), 2129–2140 (2008). [CrossRef] [PubMed]

_{0}is derived from the poles of transmission function in the evanescent region [3

3. D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. **87**(26), 261114 (2005). [CrossRef]

*k*stands for the vacuum wave vector,

_{0}*ε*and

_{m}*ε*are the permittivities for the silver and dielectric media, respectively, and

_{d}*w*is the slab width. Drude model

*ε*(ω) =

_{m}*ε*-

_{∞}*ω*

_{p}^{2}/(

*ω*

^{2}+ i

*ωγ*) is employed in numerical simulations with

*ε*= 5.0,

_{∞}*γ*= 0.0987eV, and

*ω*= 9.5eV for silver [16

_{p}16. C. Oubre and P. Nordlander, “Optical Properties of Metallodielectric Nanostructures Calculated Using the Finite Difference Time Domain Method,” J. Phys. Chem. B **108**(46), 17740–17747 (2004). [CrossRef]

*n*is defined as the ratio of

_{eff}*β*and

*k*, which can be calculated by solving Eq. (1) over the complex plane. The propagation length, defined as the distance when the intensity decrease to 1/e of its original value, can be calculated by 1/(2Im(β)) and tends to decrease as the slab width and mode confinement increase, which is shown in Fig. 1 . Since we are more concerned about the deep sub-wavelength property of MIM waveguide, the slab width is chosen as 100nm even the propagation length is only 5.8μm at wavelength 1.064μm.

_{0}*d*, forming four metal-dielectric interfaces (the inset of Fig. 2 ). The slabs are further connected to parallel waveguides with 400nm center-to-center separation by 1μm-long S bends. If the gap width

*d*is small enough, a significant fraction of power is transferred from one slab to the other slab, and for this reason this structure is called directional coupler (DC).

*H*(

_{s}*x*) and

*H*(

_{a}*x*) are the amplitudes of supermodes as illustrated in the inset of Fig. 2.

*z*= 0, the amplitudes of magnetic field for two waveguides at

*z*are where

*β*and

_{a}*β*are complex propagation constants of the anti-symmetrical and symmetrical supermodes,

_{s}*β*and

*κ =*(

*β*-

_{a}*β*)/2 are propagation constant of single plasmonic waveguide and coupling coefficient of two parallel waveguides, respectively.

_{s}19. J. Chilwell and I. Hodgkinson, “Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A **1**(7), 742–753 (1984). [CrossRef]

20. C. Chen, P. Berini, D. Feng, S. Tanev, and V. Tzolov, “Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media,” Opt. Express **7**(8), 260–272 (2000). [CrossRef] [PubMed]

*k*, are illustrated in Fig. 2.

_{0}*ε*with strong dispersion becomes more negative as the wavelength increases,

_{m}*β*+

_{s}*β*)) is of great importance for maximum power transfer [11

_{a}11. G. Veronis and S. Fan, “Crosstalk between three-dimensional plasmonic slot waveguides,” Opt. Express **16**(3), 2129–2140 (2008). [CrossRef] [PubMed]

^{2}is degraded by this so-called extreme power position offset effect. This can be well understood by considering loss effect. As shown in Fig. 2, the symmetrical supermode has longer propagation length than the antisymmetrical supermode (Im(

*β*)<(

_{s}*β*)). So

_{a}*κ*is a complex value and the phase shift between a(z) and b(z) is not 90° anymore and there is a position offset between the extreme values of |

*a*(z)| and |

*b*(z)|. That is to say, when |a(z)| gets to its maxima, |b(z)| does not reach its minima [8

8. Z. Han, L. Liu, and E. Forsberg, “Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface Plasmon polaritons,” Opt. Commun. **259**(2), 690–695 (2006). [CrossRef]

*d*decreases, the real and imaginary parts of the coupling coefficient

*κ*will increase, which results the decreased coupling length and enhanced extreme power position offset effect. The error of CMT will also become larger for such strong coupling. For these reasons, the refractive index of dielectric

*n*and gap width

*d*are fixed at 1.535 (corresponding to the refractive index of BCB (Benzocyclobutene) polymer at the light wavelength of 1.55 μm) and 20nm (a typical value chosen in literature [8

8. Z. Han, L. Liu, and E. Forsberg, “Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface Plasmon polaritons,” Opt. Commun. **259**(2), 690–695 (2006). [CrossRef]

9. H. Zhao, X. G. Guang, and J. Huang, “Novel optical directional coupler based on surface plasmon polaritons,” Physica E **40**(10), 3025–3029 (2008). [CrossRef]

*x*= 2 nm, Δ

*z*= 4 nm, which are small enough to capture the change of the field at the interfaces between the dielectric layers and the silver layers. The error due to staircase approximation of the S bends also vanishes in this case. Perfect Matched Layer (PML) boundary condition is employed for all the boundaries. The transverse magnetic mode was excited in the waveguide and observation points and lines are added at the four ports as detectors of magnetic field and power. The spectral response is calculated by Fast Fourier Transform (FFT) of time-domain results at the output ports (Fig. 4 ). Coupling ratio near 50% at a wavelength of 1064nm is observed. Here the coupling ratio is defined as the ratio of output power in the coupled branch to total output power in both branches. Since signal is injected into Port1 in our simulations, the coupling ratio is 100% if the energy is fully coupled to Port-4 (corresponding to the minimum power at Port-3) and 50% if the energy is spitted equally between Port-3 and Port-4. In the second case, the directional coupler is said to be a 3-dB DC. The coupling length evaluated from FDTD (e.g., 1.2μm at wavelength of 1064nm) is shorter than that calculated using CMT (e.g., 1.9μm at 1064nm in Fig. 3) due to the additional mode coupling in the S bends and error resulting from the approximation of CMT [17].

## 3. Nonlinear Mach-Zehnder interferometer

*D*= 300nm,

*d*= 20nm,

*l*= 1000nm. We choose MEH-PPV [poly(2-methoxy-5-(28-ethylhexyloxy)-PPV)] as the nonlinear material in our structure due to its high optical nonlinearity and good patterning behavior [21

_{1}21. M. A. Bader, G. Marowsky, A. Bahtiar, K. Koynov, C. Bubeck, H. Tillmann, H.-H. Hörhold, and S. Pereira, “Poly(*p*-phenylenevinylene) derivatives: new promising materials for nonlinear all-optical waveguide switching,” J. Opt. Soc. Am. B **19**(9), 2250–2262 (2002). [CrossRef]

### 3.1 Intensity-based router

### 3.2 All-optical switch

*λ*= 1064nm is injected into port-1 and is coupled to the lower branch Arm-2 totally by the first coupler (i.e., the coupler exhibits a coupling ratio 100% at

_{p}*λ*) and then coupled to Port-3 completely by the second identical coupler with

_{p}*l*= 1.6μm (less than the calculated coupling length 1.9μm as shown in Fig. 3, the difference attributes to the error of CMT and additional coupling in the S bends). At the same time, a signal CW at a wavelength of

_{2}*λ*= 860nm is injected into port-2 and is split equally between both arms (the calculated coupling length for

_{s}*λ*= 860nm is 1.39μm as shown in Fig. 3). The phase shift of signal light due to cross-phase modulation (XPM) is the same as Eq. (7) while the refractive index change is Δ

_{s}*n = n*(

_{2}*I*), where

_{1}+ 2I_{2}*I*and

_{1}*I*are the intensity of signal and pump light. Here,

_{2}*l*is chosen as 3μm corresponding to a signal phase shift of Δ

_{3}*φ*= π between Arm-1 and Arm-2 at low-intensity pump light input. Note that the refractive index change induced by XPM is much larger than that induced by SPM at the same input power intensity because the near 100% coupling ratio of the pump light.

## 4. Conclusion

## Acknowledgments

## References and Links

1. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

2. | R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today |

3. | D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. |

4. | J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chipscale propagation with subwavelength-scale localization,” Phys. Rev. B |

5. | R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons,” Opt. Express |

6. | B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. |

7. | Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Krasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. |

8. | Z. Han, L. Liu, and E. Forsberg, “Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface Plasmon polaritons,” Opt. Commun. |

9. | H. Zhao, X. G. Guang, and J. Huang, “Novel optical directional coupler based on surface plasmon polaritons,” Physica E |

10. | R. A. Wahsheh, Z. Lu, and M. A. G. Abushagur, “Nanoplasmonic Directional Couplers and Mach-Zehnder Inerferometers,” Opt. Commun. |

11. | G. Veronis and S. Fan, “Crosstalk between three-dimensional plasmonic slot waveguides,” Opt. Express |

12. | Z. Han, A. Y. Elezzabi, and V. Van, “Wideband Y-splitter and aperture-assisted coupler based on sub-diffraction confined plasmonic slot waveguides,” Appl. Phys. Lett. |

13. | B. E. A. Saleh, and M. C. Teich, |

14. | C. Min, P. Wang, C. Chen, Y. Deng, Y. Lu, H. Ming, T. Ning, Y. Zhou, and G. Yang, “All-optical switching in subwavelength metallic grating structure containing nonlinear optical materials,” Opt. Lett. |

15. | Z.-J. Zhong, Y. Xu, S. Lan, Q.-F. Dai, and L.-J. Wu, “Sharp and asymmetric transmission response in metal-dielectric-metal plasmonic waveguides containing Kerr nonlinear media,” Opt. Express |

16. | C. Oubre and P. Nordlander, “Optical Properties of Metallodielectric Nanostructures Calculated Using the Finite Difference Time Domain Method,” J. Phys. Chem. B |

17. | A. Yariv, and P. Yeh, |

18. | C. L. Chen, |

19. | J. Chilwell and I. Hodgkinson, “Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A |

20. | C. Chen, P. Berini, D. Feng, S. Tanev, and V. Tzolov, “Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media,” Opt. Express |

21. | M. A. Bader, G. Marowsky, A. Bahtiar, K. Koynov, C. Bubeck, H. Tillmann, H.-H. Hörhold, and S. Pereira, “Poly( |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(230.7370) Optical devices : Waveguides

(240.6680) Optics at surfaces : Surface plasmons

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: July 27, 2010

Revised Manuscript: September 5, 2010

Manuscript Accepted: September 16, 2010

Published: September 20, 2010

**Citation**

Mingbo Pu, Na Yao, Chenggang Hu, Xuecheng Xin, Zeyu Zhao, Changtao Wang, and Xiangang Luo, "Directional coupler and nonlinear Mach-Zehnder interferometer based on metal-insulator-metal plasmonic waveguide," Opt. Express **18**, 21030-21037 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-21030

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### References

- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]
- R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9(7-8), 20–27 (2006). [CrossRef]
- D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87(26), 261114 (2005). [CrossRef]
- J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chipscale propagation with subwavelength-scale localization,” Phys. Rev. B 73(3), 035407 (2006). [CrossRef]
- R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons,” Opt. Express 13(3), 977–984 (2005). [CrossRef] [PubMed]
- B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91(8), 081111 (2007). [CrossRef]
- Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Krasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. 34(3), 310–312 (2009). [CrossRef] [PubMed]
- Z. Han, L. Liu, and E. Forsberg, “Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface Plasmon polaritons,” Opt. Commun. 259(2), 690–695 (2006). [CrossRef]
- H. Zhao, X. G. Guang, and J. Huang, “Novel optical directional coupler based on surface plasmon polaritons,” Physica E 40(10), 3025–3029 (2008). [CrossRef]
- R. A. Wahsheh, Z. Lu, and M. A. G. Abushagur, “Nanoplasmonic Directional Couplers and Mach-Zehnder Inerferometers,” Opt. Commun. 282(23), 4622–4626 (2009). [CrossRef]
- G. Veronis and S. Fan, “Crosstalk between three-dimensional plasmonic slot waveguides,” Opt. Express 16(3), 2129–2140 (2008). [CrossRef] [PubMed]
- Z. Han, A. Y. Elezzabi, and V. Van, “Wideband Y-splitter and aperture-assisted coupler based on sub-diffraction confined plasmonic slot waveguides,” Appl. Phys. Lett. 96(13), 131106 (2010). [CrossRef]
- B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics (Wiley Intersceince, Hoboken, NJ, 2007), 2nd ed.
- C. Min, P. Wang, C. Chen, Y. Deng, Y. Lu, H. Ming, T. Ning, Y. Zhou, and G. Yang, “All-optical switching in subwavelength metallic grating structure containing nonlinear optical materials,” Opt. Lett. 33(8), 869–871 (2008). [CrossRef] [PubMed]
- Z.-J. Zhong, Y. Xu, S. Lan, Q.-F. Dai, and L.-J. Wu, “Sharp and asymmetric transmission response in metal-dielectric-metal plasmonic waveguides containing Kerr nonlinear media,” Opt. Express 18(1), 79–86 (2010). [CrossRef] [PubMed]
- C. Oubre and P. Nordlander, “Optical Properties of Metallodielectric Nanostructures Calculated Using the Finite Difference Time Domain Method,” J. Phys. Chem. B 108(46), 17740–17747 (2004). [CrossRef]
- A. Yariv, and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University Press, New York, NY, 2006), 6th ed.
- C. L. Chen, Foundations for Guided-Wave Optics (Wiley, 2006), chap.6.
- J. Chilwell and I. Hodgkinson, “Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1(7), 742–753 (1984). [CrossRef]
- C. Chen, P. Berini, D. Feng, S. Tanev, and V. Tzolov, “Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media,” Opt. Express 7(8), 260–272 (2000). [CrossRef] [PubMed]
- M. A. Bader, G. Marowsky, A. Bahtiar, K. Koynov, C. Bubeck, H. Tillmann, H.-H. Hörhold, and S. Pereira, “Poly(p-phenylenevinylene) derivatives: new promising materials for nonlinear all-optical waveguide switching,” J. Opt. Soc. Am. B 19(9), 2250–2262 (2002). [CrossRef]

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