## Focusing properties of surface plasmon polariton floating dielectric lenses

Optics Express, Vol. 16, Issue 5, pp. 3049-3057 (2008)

http://dx.doi.org/10.1364/OE.16.003049

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### Abstract

We have investigated the focusing properties of surface plasmon polariton floating dielectric lenses. An analysis of the scattering characteristics of surface plasmon polaritons using a floating dielectric block shows that the air-gap thickness between a floating dielectric block and a metal substrate can be an effective dynamic variable for modulating the amplitude and phase of the transmission coefficient of the surface plasmon polaritons. This property can be used to realize a variable-focusing surface plasmon dielectric lens with the air-gap thickness as the dynamic variable. The focusing properties of a Fresnel lens and a parabolic lens with respect to the air-gap thickness are compared and analyzed.

© 2008 Optical Society of America

## 1. Introduction

5. S. Kim, H. Kim, Y. Lim, and B. Lee, “Off-axis directional beaming of optical field diffracted by a single subwavelength metal slit with asymmetric dielectric structure surface gratings,” Appl. Phys. Lett. **90**, 051113 (2007). [CrossRef]

6. S. Kim, Y. Lim, H. Kim, J. Park, and B. Lee, “Optical beam focusing by a single subwavelength metal slit surrounded by chirped dielectric surface gratings,” Appl. Phys. Lett. **92**, 013103 (2008). [CrossRef]

7. I. P. Radko, S. I. Bozhevolnyi, A. B. Evlyukhin, and A. Boltasseva, “Surface plasmon polariton beam focusing with parabolic nanoparticle chains,” Opt. Express **15**, 6576–6582 (2007). [CrossRef] [PubMed]

8. W. Nomura, M. Ohtsu, and T. Yatsui, “Nanodot coupler with a surface plasmon polariton condenser for optical far/bear-field conversion,” Appl. Phys. Lett. **86**, 181108 (2005). [CrossRef]

9. R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young’s double-slit experiment,” Nature Nanotech. **2**, 426–429 (2007). [CrossRef]

10. F. Lopez-Tejeria, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Padko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nature Phys. **3**, 324–328 (2007). [CrossRef]

11. P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. **25**, 1092–1094 (2000). [CrossRef]

12. H. Kim, I.-M. Lee, and B. Lee, “Extended scattering-matrix method for efficient full parallel implementation of rigorous coupled-wave analysis,” J. Opt. Soc. Am A **24**, 2313–2327 (2007). [CrossRef]

9. R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young’s double-slit experiment,” Nature Nanotech. **2**, 426–429 (2007). [CrossRef]

## 2. Modulation of surface plasmon polariton eigenmode using a finite-size dielectric block floating over a metal substrate

*t*and

*h*, respectively. The length,

*t*, is a fixed structural parameter that cannot be controlled dynamically, while the air-gap,

*h*, is an effective dynamic structural variable that can be controlled using highly developed nano-scale actuators.

*ε*(=-9.5487+

_{m}*j*1.1327),

*ε*

_{a}(=1), and

*ε*

_{b}(=2.25), respectively.

*z*-direction be denoted by,

*n*is the mode index of the SPP eigenmode in the RCWA scheme, and the superscript + denotes the positive

_{spp}*z*-directional propagation of the SPP eigenmode. In the modal analysis framework of the RCWA [11

11. P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. **25**, 1092–1094 (2000). [CrossRef]

12. H. Kim, I.-M. Lee, and B. Lee, “Extended scattering-matrix method for efficient full parallel implementation of rigorous coupled-wave analysis,” J. Opt. Soc. Am A **24**, 2313–2327 (2007). [CrossRef]

*E*

^{±}

_{n}(

*x*,

*y*,

*z*) with coupling coefficients,

*C*

^{±}

_{n},

*E*

^{±}

_{n}(

*x*,

*y*,

*z*) are the eigenvectors of the linear eigenvalue equation. The coupling coefficients,

*C*

^{±}

_{n}, are determined by the mode matching (boundary) conditions at

*z*=0 and

*z*=

*t*. When solving the mode matching conditions, we can excite the SPP eigenmodes selectively in Region I, as indicated in Fig. 1. In the analysis of the low-dimensional diffraction of SPPs, we only need to consider the SPP mode in Region II,

*z*-direction. The transmission coefficient,

*x*-direction.

5. S. Kim, H. Kim, Y. Lim, and B. Lee, “Off-axis directional beaming of optical field diffracted by a single subwavelength metal slit with asymmetric dielectric structure surface gratings,” Appl. Phys. Lett. **90**, 051113 (2007). [CrossRef]

6. S. Kim, Y. Lim, H. Kim, J. Park, and B. Lee, “Optical beam focusing by a single subwavelength metal slit surrounded by chirped dielectric surface gratings,” Appl. Phys. Lett. **92**, 013103 (2008). [CrossRef]

*x*-direction Fourier spatial harmonics is set to 61, which is the number of spatial harmonics showing a reasonable convergence, and the

*x*-direction supercell periods,

*L*, is chosen to be 7µm. Figures 2 and 3 show the distribution of the transmission coefficient,

_{x}*x*-direction of 1µm and 2µm, respectively.

*t*, from 0µm to 10µm and the air-gap thickness,

*h*, from 0µm to 0.5µm. Within the air-gap region below 50nm, we can observe a trend that as the length increases, the amplitude of the transmission coefficient decreases monotonically because of ohmic loss and radiation loss. However, in the air-gap region above 50nm, a complex amplitude fluctuation along the length axis is observed. This amplitude fluctuation originates from the finite thickness of the dielectric block. This point can be visually understood by comparing Figs. 2(c) and 2(d). These figures show the

*x*-polarization and

*z*-polarization electric field distributions of the cases where the air-gap thickness of 0 and 150nm, respectively. In the latter case, due to the air-gap, the radiation into the dielectric block is significant, but the radiation field immediately becomes a guided multimode of the dielectric block because of the total internal reflection (TIR), as seen in Fig. 2(d). The wave bundle reflected by the ceiling of the dielectric block transfers some optical energy to the SPP eigenmode. Thus, we can observe a periodic amplitude fluctuation along the length axis in the air-gap region above a distance of 50nm. In addition, as shown in Fig. 2(b), we can observe a more sensitive change in the phase-varying rate along the length axis compared to the variation in air-gap thickness, and the phase-varying rate is lowered in the air-gap range above 50nm.

*h*, and length,

*t*. The real coefficient polynomial fitting functions of the amplitude and phase distribution are given by

*A*(

*h*,

*t*) and Φ(

*h*,

*t*), respectively. In particular, the phase fitting function, Φ(

*h*,

*t*), is a polynomial that approximates the unwrapped phase profile.

## 3. Focusing properties of surface plasmon polariton floating dielectric lenses

*y*,

*ϕ*(

*h*,

*y*) and Γ(

*h*,

*y*), respectively, are given by,

*k*is the wavenumber of the SPP eigenmode,

_{spp}*A*(

_{spp}*s*) is the modal amplitude of the SPP eigenmode that is propagated a distance,

*s*,

*t*(

*y*) is the lens surface profile function, and

*l*is the maximum longitudinal thickness of the lens, as shown in Fig. 4.

*π*modulo are considered. The lens profile function,

*t*(

*y*), for changing the incident SPP eigenmode to a spherically converging SPP wave was designed using the following procedure. The profile function,

*t*(

*y*), of a parabolic lens and a Fresnel lens that produce the same spherical wavefront, needs to satisfy the polynomial Eqs. (4a) and (4b), respectively,

*y*is limited to the range -

*y*

_{max}≤

*y*≤

*y*

_{max}, [·] is Gauss’ symbol, Re(

*s*) is the real part of a complex number

*s*, and

*f*is the focal length of the lens. We can easily find the value of

_{c}*t*(

*y*) by using standard numerical root-finding libraries. The incident SPP eigenmode is transformed to the SPP mode with a spherically converged wavefront, Γ(

*h*,

*y*)exp (

*jϕ*(

*h*,

*y*)), for -

*y*

_{max}≤

*y*≤

*y*

_{max}, and at the flat facet of the lens.

9. R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young’s double-slit experiment,” Nature Nanotech. **2**, 426–429 (2007). [CrossRef]

*α*;

*h*). Then, Π(

*α*;

*h*) is obtained as,

*α*is the

*y*-direction spatial-frequency component. Using the angular spectrum propagation formula [13], the SPP field distribution at the air/metal interface (

*x*=0) can be approximated as,

*T*, is the ratio of transmission power to the input power, defined by,

_{e}*π*modulo Fresnel lens, respectively, with the same focal length of 10µm. Figures 5(c) and 5(d) show the focusing properties of a parabolic lens and a Fresnel lens, respectively, with the same focal length of 5µm. The transmission efficiency,

*T*, of each lens is also shown in Fig. 5.

_{e}## 4. Conclusion

## Acknowledgment

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

2. | P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. |

3. | I.-M. Lee, J. Jung, J. Park, H. Kim, and B. Lee, “Dispersion characteristics of channel plasmon polariton waveguides with step-trench-type grooves,” Opt. Express |

4. | J. Takahara and T. Kobayashi, “Low-dimensional optical waves and nano-optical circuits,” Opt. Photon. News |

5. | S. Kim, H. Kim, Y. Lim, and B. Lee, “Off-axis directional beaming of optical field diffracted by a single subwavelength metal slit with asymmetric dielectric structure surface gratings,” Appl. Phys. Lett. |

6. | S. Kim, Y. Lim, H. Kim, J. Park, and B. Lee, “Optical beam focusing by a single subwavelength metal slit surrounded by chirped dielectric surface gratings,” Appl. Phys. Lett. |

7. | I. P. Radko, S. I. Bozhevolnyi, A. B. Evlyukhin, and A. Boltasseva, “Surface plasmon polariton beam focusing with parabolic nanoparticle chains,” Opt. Express |

8. | W. Nomura, M. Ohtsu, and T. Yatsui, “Nanodot coupler with a surface plasmon polariton condenser for optical far/bear-field conversion,” Appl. Phys. Lett. |

9. | R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young’s double-slit experiment,” Nature Nanotech. |

10. | F. Lopez-Tejeria, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Padko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nature Phys. |

11. | P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. |

12. | H. Kim, I.-M. Lee, and B. Lee, “Extended scattering-matrix method for efficient full parallel implementation of rigorous coupled-wave analysis,” J. Opt. Soc. Am A |

13. | J. W. Goodman, Introduction to Fourier Optics, 3rd ed., (Roberts & Company Publishers, Englewood, 2005). |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: January 10, 2008

Revised Manuscript: February 15, 2008

Manuscript Accepted: February 15, 2008

Published: February 20, 2008

**Citation**

Hwi Kim, Joonku Hahn, and Byoungho Lee, "Focusing properties of surface plasmon polariton floating dielectric lenses," Opt. Express **16**, 3049-3057 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-3049

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### References

- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003). [CrossRef] [PubMed]
- P. Berini, R. Charbonneau, and N. Lahoud, "Long-range surface plasmons on ultrathin membranes," Nano Lett. 7, 1376-1380 (2007). [CrossRef] [PubMed]
- I.-M. Lee, J. Jung, J. Park, H. Kim, and B. Lee, "Dispersion characteristics of channel plasmon polariton waveguides with step-trench-type grooves," Opt. Express 15, 16596-16603 (2007). [CrossRef] [PubMed]
- J. Takahara and T. Kobayashi, "Low-dimensional optical waves and nano-optical circuits," Opt. Photon. News 15, 54-59 (2004). [CrossRef]
- S. Kim, H. Kim, Y. Lim, and B. Lee, "Off-axis directional beaming of optical field diffracted by a single subwavelength metal slit with asymmetric dielectric structure surface gratings," Appl. Phys. Lett. 90, 051113 (2007). [CrossRef]
- S. Kim, Y. Lim, H. Kim, J. Park, and B. Lee, "Optical beam focusing by a single subwavelength metal slit surrounded by chirped dielectric surface gratings," Appl. Phys. Lett. 92, 013103 (2008). [CrossRef]
- I. P. Radko, S. I. Bozhevolnyi, A. B. Evlyukhin, and A. Boltasseva, "Surface plasmon polariton beam focusing with parabolic nanoparticle chains," Opt. Express 15, 6576-6582 (2007). [CrossRef] [PubMed]
- W. Nomura, M. Ohtsu, and T. Yatsui, "Nanodot coupler with a surface plasmon polariton condenser for optical far/bear-field conversion," Appl. Phys. Lett. 86, 181108 (2005). [CrossRef]
- R. Zia and M. L. Brongersma, "Surface plasmon polariton analogue to Young’s double-slit experiment," Nature Nanotech. 2, 426-429 (2007). [CrossRef]
- F. Lopez-Tejeria, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Padko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, "Efficient unidirectional nanoslit couplers for surface plasmons," Nat. Phys. 3, 324-328 (2007). [CrossRef]
- P. Lalanne and E. Silberstein, "Fourier-modal methods applied to waveguide computational problems," Opt. Lett. 25, 1092-1094 (2000). [CrossRef]
- H. Kim, I.-M. Lee, and B. Lee, "Extended scattering-matrix method for efficient full parallel implementation of rigorous coupled-wave analysis," J. Opt. Soc. Am A 24, 2313-2327 (2007). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics, 3rd ed., (Roberts & Company Publishers, Englewood, 2005).

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