## Extended long range plasmon waves in finite thickness metal film and layered dielectric materials

Optics Express, Vol. 14, Issue 25, pp. 12409-12418 (2006)

http://dx.doi.org/10.1364/OE.14.012409

Acrobat PDF (229 KB)

### Abstract

In this paper, we show that the propagation distance of the long range plasmon wave mode guided by a finite thickness gold metal film can be extended several orders of magnitude longer if we place intermediate dielectric layers on both sides of the metal film and choose the layer thickness properly. The propagation distance goes to infinite if the intermediate layer thickness approaches a critical thickness.

© 2006 Optical Society of America

## 1. Introduction

2. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. **182**, 539–554 (1969). [CrossRef]

3. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B **33**, 5186–5201 (1986). [CrossRef]

15. G. I Stegeman and J. J. Burke, “Long-range surface plasmons in electrode structures,” Appl. Phys. Lett. **43**, 221–223, (1983). [CrossRef]

16. F. Y. Kou and T. Tamir, “Range extension of surface plasmons by dielectric layers,” Opt. Lett.12, 367-(1987). [CrossRef] [PubMed]

16. F. Y. Kou and T. Tamir, “Range extension of surface plasmons by dielectric layers,” Opt. Lett.12, 367-(1987). [CrossRef] [PubMed]

17. L. Wendler and R. Haupt, “Long-range surface plasmon-polaritons in asymmetric layer structures,” J. Appl. Phys.59, pp. 3289–3291 (1986). [CrossRef]

## 2. The reflection pole method

_{0}) is in the bottom. A thin metal film (Au) layer (ε

_{2}=ε

_{m}) of thickness

*d*is in the middle. One intermediate dielectric layer (ε

_{2}_{1}) of thickness

*d*is place between the metal layer and the dielectric medium in the bottom cladding. Another intermediate dielectric layer (ε

_{1}_{2}) of thickness

*d*is place between the metal layer and the dielectric medium in the top cladding. Another semi-infinite dielectric medium (ε

_{2}_{4}) is on the top of the top intermediate dielectric layer. The two intermediate dielectric layers have the same dielectric constant (ε

_{1}=ε

_{3}) and the same thickness (

*d*=

_{1}*d*). The top dielectric cladding is the same material as the bottom dielectric cladding (ε

_{3}_{0}=ε

_{4}). Therefore, the plasmon waveguide structure is symmetrical with respect to the center of the metal layer.

18. J. Xia, A. K. Jordan, and J. A. Kong, “Electromagnetic inverse-scattering theory for inhomogeneous dielectrics: the local reflection model,” J. Opt. Soc. Am. A. **9**, 740–748 (1992). [CrossRef]

19. E. Anemogiannis, E. N Glytsis, and T. K Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. **17**, 929–941 (1999). [CrossRef]

*x*components of complex wave vectors in different regions,

*γ*is the propagation constant (complex) in the

*z*direction.

*A*and

_{i}*B*are coefficients corresponding to the propagating waves in

_{i}*x*and -

*x*directions in the

*i*-th region.

*k*is the wave number in the free space.

_{o}_{0}and ε

_{1}from the bottom of the waveguide structure. At each boundary, the tangential components of the magnetic field

*H*and the electrical field

_{yi}*γ*=(

*n*-j

*n*

_{i})

*k*

_{0}. The guided plasmon modes are the modes which propagate in the z direction and exponentially decay away from the metal layer in the x and -x directions. The guided modes correspond to the resonances of the layered waveguide structure, which happen at the poles of the reflection coefficient

*R*. The poles of the reflection coefficient are the solutions of

*t*

_{22}(

*γ*)=0.

*t*

_{22}(

*γ*)=0 is numerically challenging. Reflection pole method (RPM) monitors the phase of

*t*

_{22}(

*γ*) as a function of the real part index (

*n*) of the propagation constant. The plasmon wave guided modes correspond to the rapid changes of the phase of

*t*

_{22}. According to Bode plot theory [20], a peak of the derivative of the phase of

*t*

_{22}corresponds to the real part index of a guided mode, and the full width at the half maximum of the phase derivative curve is the imaginary part of mode index.

*ε*=

_{o}*ε*

_{4}=(1.45)

^{2}) at the 1.55 micron wavelength (d

_{1}=d

_{3}=0, d

_{2}=20 nm). The dielectric constant of gold film at 1.55 micron [21

21. P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

_{2}=-114.925 -11.0918j. Fig. 2 (a) is the plot of the phase of

*t*

_{22}versus the real part of the propagation constant index. Fig. 2 (b) is the derivative of the phase of

*t*

_{22}versus the real part of the propagation constant index. From the Fig. 2, we see two rapid changes of the phase of

*t*

_{22}. These two rapid phase changes correspond to two guided plasmon wave modes. Once we know the approximate values of the mode index from the plot, we can numerically solve the equation

*t*

_{22}=0 with the initial guess of its roots.

_{1}=1.45229459-0.00003345585j, n

_{2}=1.525501873-0.013489423j. Both modes are guided modes. The first mode is the symmetric mode (s

_{b}) which has a propagation distance (1/e of the intensity) of 3686.8 micron. The second mode is the anti-symmetric (a

_{b}) mode, which has a propagation distance of 9.144 micron. The mode size (full width at 1/e of the maximum intensity) of the symmetric mode is 3.0429 micron. The mode size of the anti-symmetric mode is 0.54047 micron.

## 3. Higher dielectric constant intermediate layer

_{1}=ε

_{3}=4.0) as the intermediate layer material and silicon dioxide (ε

_{0}=ε

_{4}=(1.45)

^{2}) as the top and bottom cladding material (

*ε*

_{1}=

*ε*

_{3}>

*ε*

_{0}=

*ε*

_{4}). The thickness of the gold metal layer is d

_{2}=20 nm. For the intermediate dielectric layer thickness (

*d*

_{1}and

*d*

_{3}) of 1000 nm, we plot the phase (ϕ) of

*t*

_{22}as the function of the real part mode index in Fig. 3 (a). We see the stair case phase π changes at certain values of the real part of the mode index.

*t*

_{22}. The imaginary part of the root of

*t*

_{22}corresponds to the width of the phase derivative curve (dϕ/dn) of

*t*

_{22}. Fig. 3 (b) shows the phase derivative versus the real part of the mode index. Once we know the approximate values of mode indices from the plot, we numerically solved the equation

*t*

_{22}=0 with the initial guesses of its roots. For the intermediate dielectric layer thickness of 1000 nm, we found four complex solutions: n1=1.732403005-0.0001470494j, n2=1.80394551-0.00522046879j, n3=1.989189193-0.000206846j, n4=2.194229366-0.0341893452j. To verify these solutions are indeed the poles of the reflection coefficient, we input these complex numbers to the

*t*

_{22}, and find |Re[

*t*

_{22}]|<10-21 and |Im[

*t*

_{22}]|<10-21. These four solutions are all guided plasmon wave modes.

*e*. The mode (c) is the second long range plasmon mode. The propagation distance of the second long range mode is 596.314 micron. The modes (b) and (d) are the two short range modes. The propagation distances of the short range modes are one order of magnitude less than the long range modes.

_{y}) of these four guided plasmon modes in Fig. 4. The modes (c) and (d) are the fundamental modes. The modes (a) and (b) are the higher order modes. We see that the long range modes (a) and (c) are symmetric modes. The short range modes (b) and (d) are anti-symmetric modes. The mode profiles are similar to the profiles of the dielectric waveguide modes, but have significant deviations from the dielectric waveguide modes near the metal layer due to the surface plasmon effect.

^{-7}. The propagation distance of this mode is 155.844 mm. This propagation distance is approximately 42 times of the propagation distance of the long range plasmon wave mode guided by the gold film of same thickness without intermediate dielectric layers.

*Cosine*function mode profile in the high index region or support additional higher order modes with the

*Sine*and

*Cosine*function mode profiles. With the presence of the metal film, the modes we have found here are due to the two effects: the total internal reflection effect and the surface plasmon effect. The mode profiles shown in Fig. 4 (b), (c), and (d) deviates significantly from the mode profiles of the dielectric waveguide due to the surface plasmon effect. The extended long travel ranges for the two symmetric modes are because the electromagnetic energies of the guided modes are located mostly in the dielectric media which are assumed as lossless in our calculations.

## 4. Lower dielectric constant intermediate layer

_{1}=ε

_{3}=(1.45)

^{2}at 1.55 micron) as the intermediate layer dielectric material and the high dielectric constant silicon nitride (ε

_{0}=ε4=4.0 at 1.55 micron) as the top and bottom cladding material (

*ε*

_{1}=

*ε*

_{3}<

*ε*

_{0}=

*ε*

_{4}). The thickness of the gold metal layer is kept the same at d

_{2}=20 nm. We searched the guided plasmon wave modes for different intermediate silicon dioxide dielectric layer thickness. Fig. 7 (a) shows the real part of the mode index for different intermediate dielectric layer thickness. The solid line curve is the real part of the mode index of the long range mode. The dashed line curve is the real part of the mode index of the short range mode. From Fig. 7 (a), we see that two guided modes are supported when the intermediate dielectric layer thickness is smaller than a critical thickness. Fig. 7 (b) shows the imaginary part of the mode index versus the low index intermediate layer thickness for the long range mode (solid line curve) and the short range mode (dashed line curve).

*nm*. At the intermediate layer thickness of 10.5 nm, the propagation distance of the symmetric mode is 43.037 mm. This propagation distance is approximately 38 times of the propagation distance of the long range mode without the intermediate layer. The size of the mode is 63.194 micron in the x direction. Beyond the critical thickness, the symmetric mode disappears. When we increase the intermediate layer thickness to about 59 nm, the real part of the effective mode index of the short range mode becomes smaller than the index of refraction of the cladding material. The short range mode becomes a leaky mode and no more guided plasmon wave mode exists.

_{y}) of two guided plasmon modes for the intermediate layer thickness of 10.5 nm. The long range symmetric mode magnetic field profile is shown in Fig. 9 (a). Fig. 9 (b) is the close look of the mode profile near the metal layer. We see that the symmetric mode has a small notch inside the metal layer. This small notch is due to the electromagnetic field absorption inside the metal material. The short range magnetic field profile is shown in Fig. 10 (a). Fig. 10 (b) is the close look of short range magnetic profile near the metal region. The anti-symmetric mode magnetic field reaches zero in the center of the metal layer.

## 5. Summary

## References and links

1. | H. Raether, |

2. | E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. |

3. | J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B |

4. | D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. |

5. | P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B |

6. | P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B |

7. | P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, “Characterization of long-range surface-plasmon-polariton waveguides,” J. Appl. Phys. |

8. | S. J. Al-Bader, “Optical transmission on metallic wires - fundamental modes,” IEEE J. Quantum Electron. |

9. | B. Lamprecht, J.R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, and F.R. Aussenegg, “Surface plasmon propagation in microscale metal stripes,” Appl. Phys. Lett. |

10. | A. Degiron and D. Smith, “Numerical simulations of long-range plasmons,” Opt. Express |

11. | R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, “Experimental observation of plasmon polariton waves supported by a thin metal film of finite width,” Opt. Lett. |

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

13. | K. Leosson, T. Nikolajsen, A. Boltasseva, and S. I. Bozhevolnyi, “Long-range surface plasmon polariton nanowire waveguides for device applications,” Opt. Express |

14. | L. Holland, |

15. | G. I Stegeman and J. J. Burke, “Long-range surface plasmons in electrode structures,” Appl. Phys. Lett. |

16. | F. Y. Kou and T. Tamir, “Range extension of surface plasmons by dielectric layers,” Opt. Lett.12, 367-(1987). [CrossRef] [PubMed] |

17. | L. Wendler and R. Haupt, “Long-range surface plasmon-polaritons in asymmetric layer structures,” J. Appl. Phys.59, pp. 3289–3291 (1986). [CrossRef] |

18. | J. Xia, A. K. Jordan, and J. A. Kong, “Electromagnetic inverse-scattering theory for inhomogeneous dielectrics: the local reflection model,” J. Opt. Soc. Am. A. |

19. | E. Anemogiannis, E. N Glytsis, and T. K Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. |

20. | A. Papoulis, |

21. | P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B |

22. | G. I. Stegeman, J. J. Burke, and D. G. Hall, “Surface-polariton like waves guided by thin, lossy metal films,” Opt. Lett. |

23. | J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B |

24. | J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: September 7, 2006

Revised Manuscript: November 3, 2006

Manuscript Accepted: November 17, 2006

Published: December 11, 2006

**Citation**

Junpeng Guo and Ronen Adato, "Extended long range plasmon waves in finite thickness metal film and layered dielectric materials," Opt. Express **14**, 12409-12418 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-12409

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### References

- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin Heidelberg, 1988).
- E. N. Economou, "Surface plasmons in thin films," Phys. Rev. 182, 539-554 (1969). [CrossRef]
- J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986). [CrossRef]
- D. Sarid, "Long-range surface-plasma waves on very thin metal films," Phys. Rev. Lett. 47, 1927-1930 (1981). [CrossRef]
- P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B 61, 10484-10503 (2000). [CrossRef]
- P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B 63, 125417 (2001). [CrossRef]
- P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, "Characterization of long-range surface-plasmon-polariton waveguides," J. Appl. Phys. 98, 043109 (2005). [CrossRef]
- S. J. Al-Bader, "Optical transmission on metallic wires - fundamental modes," IEEE J. Quantum Electron. 40, 325-329 (2004). [CrossRef]
- B. Lamprecht, J.R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, and F.R. Aussenegg, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett. 79, 51-53 (2001). [CrossRef]
- A. Degiron and D. Smith, "Numerical simulations of long-range plasmons," Opt. Express 14, 1611-1625 (2006). [CrossRef] [PubMed]
- R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, "Experimental observation of plasmon polariton waves supported by a thin metal film of finite width," Opt. Lett. 25, 844-846 (2000). [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, 977-984 (2005). [CrossRef] [PubMed]
- K. Leosson, T. Nikolajsen, A. Boltasseva, and S. I. Bozhevolnyi, "Long-range surface plasmon polariton nanowire waveguides for device applications," Opt. Express 14, 314-319 (2006). [CrossRef] [PubMed]
- 4. L. Holland, Vacuum Deposition of Thin Films (Chapman and Hall, London, 1966).
- G. I Stegeman and J. J. Burke, "Long-range surface plasmons in electrode structures," Appl. Phys. Lett. 43, 221-223, (1983). [CrossRef]
- F. Y. Kou and T. Tamir, "Range extension of surface plasmons by dielectric layers," Opt. Lett. 12, 367 (1987). [CrossRef] [PubMed]
- L. Wendler and R. Haupt, "Long-range surface plasmon-polaritons in asymmetric layer structures," J. Appl. Phys. 59, pp. 3289-3291 (1986). [CrossRef]
- J. Xia, A. K. Jordan, and J. A. Kong, "Electromagnetic inverse-scattering theory for inhomogeneous dielectrics: the local reflection model," J. Opt. Soc. Am. A. 9, 740-748 (1992). [CrossRef]
- E. Anemogiannis, E. N Glytsis, and T. K Gaylord, "Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method," J. Lightwave Technol. 17, 929-941 (1999). [CrossRef]
- A. Papoulis, Circuits and Systems (Holt, Rinehart and Winston, Inc., New York, 1980).
- P. B. Johnson and R. W. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]
- G. I. Stegeman, J. J. Burke, and D. G. Hall, "Surface-polariton like waves guided by thin, lossy metal films," Opt. Lett. 8, 383-386 (1983). [CrossRef] [PubMed]
- J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986). [CrossRef]
- J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," Phys. Rev. B 72, 075405 (2005). [CrossRef]

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