## Optimal design of SPP-based metallic nanoaperture optical elements by using Yang-Gu algorithm |

Optics Express, Vol. 19, Issue 10, pp. 9512-9522 (2011)

http://dx.doi.org/10.1364/OE.19.009512

Acrobat PDF (1885 KB)

### Abstract

An optimization method for design of SPP-based metallic nanoaperture optical elements is presented. The design process is separated into two steps: Firstly, derive the amplitude and phase modulation of isolating single slit with different width; Secondly, realize the optimal design of element by using an iteration procedure. The Yang-Gu algorithm is expanded to perform this design. Three kinds of lenses which can achieve various functions have been designed by using this method. The rigorous electromagnetical theory is employed to justify and appraise the performances of the designed elements. It has been found that the designed elements can achieve the preset functions well. This method may provide a convenient avenue to optimally design metallic diffractive optical elements with subwavelength scale.

© 2011 OSA

## 1. Introduction

1. R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. **106**, 874–881 (1957). [CrossRef]

20. B. Y. Gu, G. Z. Yang, and B. Z. Dong, “General theory for performing an optical transform,” Appl. Opt. **25**, 3197–3206 (1986). [CrossRef] [PubMed]

*i.e.*the FDTD method, is used to justify and appraise the performances of the design element and show the deviation degree.

## 2. Beaming structure and design theory

*h*. The air-slit width is

*w*and the concrete width of each slit will be designed according to the function of the element. The interspace between neighboring slits is

*d*. When a TM-polarized plane wave illuminates on the grating from the left side, the SPPs can be excited at slit entrances. Then SPPs propagate inside the slits in specific waveguide modes till they arrive at the exit plane where they can radiate into free space. The exit plane of the structure, denoted as P

_{0}, is laid on the

*x*

_{0}–

*y*

_{0}plane with

*z*= 0. Because of different slit widths, a specific phase and amplitude wavefront is generated on the exit plane. So when the diffracted light with a specific wavefront radiates into the free space, particular shape of the field distribution can be formed on the observation plane, denoted as P, on the

*x*–

*y*plane with

*z*=

*f*. Here,

*f*denotes the spacing between the exit plane and the observation plane along the

*z-*axis. The coordinates on the P

_{0}plane (P plane) is denoted as

*r⃗*

_{0}= (

*x*

_{0}

*, y*

_{0}

*, z*= 0) (

*r⃗*= (

*x,y,z*=

*f*)). The wave function of the magnetic on the P

_{0}plane is denoted as

*H*

_{y0}and the corresponding magnetic field on the P plane is denoted as

*H*

_{y}, respectively.

*y*direction is infinite. The magnetic field

*H⃗*is parallel to the

*y-*axis and the electric field in the plane of incidence. Therefore, it is more convenient to calculate the magnetic field. The electric field is also easily deduced from Maxwell’s equations. It is free space propagation from P

_{0}to P plane. So if the phase and amplitude distributions on the P

_{0}plane can be derived, the field distribution on the P plane can be calculated by using the Green’s function.

### 2.1. Extraction of complex amplitude distribution on the structure’s exit plane

_{0}plane, we have employed the following approximation: (

*i*) Neglecting the coupling of SPPs during the propagation in the slits provided that the metallic walls between any two adjacent slits are much thicker than the skin depth in metal. (

*ii*) Omitting the coupling effect occurred on the exit surface from neighboring slits, compared with the intensity of directly radiation light through slits. So we can express the complex amplitude distributions on the P

_{0}plane of the structure as the sum of phase and amplitude distributions of each slit.

*β*in the slit is determined by the following equation: where where

*k*

_{0}is the wave vector of illuminating light in free space.

*ε*

_{m}and

*ε*

_{d}are the relative dielectric constants of the metal and dielectric material, respectively.

*β*is the propagation constant for SPPs in a slit. So the phase retardation of light transmitted through the slit can be approximately expressed as [21]

22. P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett. **95**, 263902 (2005). [CrossRef]

9. H. W. Kihm, K. G. Lee, D. S. Kim, J. H. Kang, and Q. Park, “Control of surface plamson generation efficiency by slit-width tuning,” Appl. Phys. Lett. **92**, 051115 (2008). [CrossRef]

_{0}plane can be derived by summing the phase and amplitude of each slit.

### 2.2. Two dimensional Green function diffraction formula

_{0}plane to P plane via free space propagation. After knowing the field distribution on the P

_{0}plane, the near/far-field distribution at any point on the observation plane P is given by [23

23. B. Hu, B. Y. Gu, B. Z. Dong, Y. Zhang, and M. Liu, “Various evaluations of a diffractive transmitted field of light through a one-dimensional metallic grating with subwavelength slits,” Cent. Eur. J. Phys. **8**, 448–454 (2009). [CrossRef]

_{0}, and it can be expressed as So, Eq. (5) is rewritten as where Equation (7) can be simply expressed as where and

*H*

_{1}(

*k*

_{0}|

*r⃗*–

*r⃗*

_{0}|) is the first-order Hankel function of the second kind.

### 2.3. Design procedure of the Yang-Gu algorithm

*w*(

*x*

_{1}) is their first derivatives are Then we have the variational argument

*δw*as combined with the variation argument

*δϕ*we have with where

*Â*

_{D}(

*Â*

_{ND}) is the diagonal (off-diagonal) element of

*Â*.

*w*distribution, thus known

*H*

_{y0}(i.e.,

*p*(

*w*)

*,q*(

*w*)),

*p*

^{′}(

*w*)

*,q*

^{′}(

*w*); (ii) Using Eq. (13), calculate

*H*

_{y}; (iii) Using Eq. (22) and desired distribution

*ϕ*

_{0}in the inner loop, thus from

*w*=

*q*

^{−1}(

*ϕ*

_{0}), calculating

*ρ*

_{0}=

*p*(

*w*), obtaining new

*H*

_{y0}for the next generation

*H*

_{y0}; (vi) Repeat step (iii) and fixed

*H*

_{y}until two adjacent generation

*H*

_{y0}results are close or the derivation is less than a given value; then (v) go to the outer loop, using Eq. (13), estimate the new

*H*

_{y}; (vi) Repeat the step (ii)–(v), until the

*D*

^{2}approaches zero. The calculation is terminated.

## 3. Design and numerical simulations

*ε*

_{m}= −17.36 + 0.715

*i*at the wavelength of

*λ*= 650nm, and the surrounding medium is air with

*ε*

_{d}= 1.0. The thickness of the structure is

*h*= 450nm and the slit interspacing is

*d*= 200nm (center to center) while each slit width is uncertain and will be determined by the Yang-Gu algorithm. The number of nano-slits is chosen as 32 to ensure that the nanolens is in the scale needed.

### 3.1. SPPs lens with one focal spot

16. H. F. Shi, C. T. Wang, C. L. Du, X. G. Luo, X. C. Dong, and H. T. Gao, “Beam manipulating by metallic nano-slits with variant widths,” Opt. Express **13**, 6815–6820 (2005). [CrossRef] [PubMed]

*f*

_{0}= 600nm. The objective intensity distribution on the observation plane P is given in Fig. 2(a) with the black line. The Yang-Gu algorithm is utilized to decide the width of each slit. The iteration will not be terminated until the given objective is achieved. The normalized intensity distribution on the observation plane which is obtained using Eq. (9) is shown in Fig. 2(a) with the blue line. The design result corresponds to the required one well except some small oscillations. The distribution of the slit widths is shown in Fig. 2(b) by the star symbols. For comparison, the distribution of slit widths designed by the EOPP method is also presented in Fig. 2(b) with circle symbols. Because the Yang-Gu algorithm considers not only the phase modulation but also the amplitude modulation of the silt during the design procedure, the result obtained by this method is different with that obtained by the EOPP method. Furthermore, the propagation from the structure’s exit plane to the observation plane is calculated by the 2D Green function method instead of the Fresnel transformation in the EOPP.

*H*

_{y}| is presented in Fig. 2(c), where the exit plane of the structure is laid on

*z*= 0nm. The cross section of the focal spot along the

*z*direction is given in Fig. 2(d). The full width at half maximum (FWHM) along the

*z*direction is 544nm and the focal spot is at

*z*= 608nm which has been denoted by the dashed line in Fig. 2(c). The real focal length is

*f*= 608nm, which deviates from our objective

*f*

_{0}= 600nm very little. The cross section of focal spot along the

*x*direction is given in Fig. 2(a) with the red line. The FWHM along the

*x*direction is 230nm. It is less than the diffraction limit. These results demonstrate that the performance of the optimally designed nanolens agrees well with the preset objective.

*z*= 608nm which is real focal plane for working wavelength

*λ*= 650nm. It shows that the designed nanolens can achieve good focusing at wavelength with a range of ±100nm with the reference wavelength

*λ*= 650nm, the focal length becomes short with the wavelength increasing.

### 3.2. SPPs lens with two focal spots

*z*= 600nm from the structure. The objective intensity distribution on the observation plane P is given in Fig. 4(a) with the black line. The intensity ratio for two spots is 1 : 2 and the positions of them are at

*x*= ±885nm. After optimization, the structure of nanolens is derived. The normalized intensity distribution on the observation plane which is obtained using Eq. (9) is shown in Fig. 4(a) with the blue line. The intensity ratio of the two peaks is 1:1.92. The corresponding distribution of slit widths is shown in Fig. 4(b) by the star line.

*|H*

_{y}

*|*is presented in Fig. 4(c). The exit plane of the structure is laid on

*z*= 0nm and the real focal plane is at

*z*= 584nm. The cross section of focal spots along the

*x*direction is given in Fig. 4(a) with the red line. The positions deviate from the preset value

*x*= ±885nm to

*x*= ±910nm. The intensity ratio of two peaks is 1:1.96. The FWHM along the

*x*direction is 320nm. These values demonstrates that the designed lens can achieve the preset objective well. The cross sections of focal spots along the

*z*direction are shown in Fig. 4(d). The red line corresponds to the first spot along

*x*= −910nm and the blue line to the second spot along

*x*= 910nm. In order to achieve the preset intensity ratio, the maximum of two spots are not on a same

*x – y*plane.

### 3.3. SPPs lens with three focal spots

*z*= 600nm. The objective intensity distribution on the observation plane P is given in Fig. 6(a) with black line. The coordinates are (−1680nm, 600nm), (0nm, 600nm), and (1680nm, 600nm) for three focal spots, respectively. The intensity ratio of three focal spots is 2:3:4. The normalized intensity distribution of the designed structure on the observation plane obtained using Eq. (9) is shown in Fig. 6(a) with the blue line. The intensity ratio of three focal spots is 2:3.08:4.17. The optimal distribution of the slit width is shown in Fig. 6(b).

*H*

_{y}| is demonstrated in Fig. 6(c). The real focal length is

*f*= 610nm, which deviates from the objective with 1.7%. The cross section of focal spots along the

*x*direction are given in Fig. 6(a) with the red line. It demonstrates that the real coordinates of the three focal spots are (−1700nm, 610nm), (66nm, 610nm), and (1590nm, 610nm), respectively. The good agreement between the calculation result and the required one can be found. The cross sections of focal spots along the

*z*direction is given in Fig. 6(d). The red line corresponds to the first spot along

*x*= −1700nm, the blue line to the second spot along

*x*= 66nm, and the black line to the third spot along

*x*= 1590nm. In order to achieve the preset intensity ratio, the maximum of three spots are not on a same

*x – y*plane.

## 4. Summary

## Acknowledgments

## References and links

1. | R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. |

2. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature |

3. | H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Merono, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science |

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

5. | P. B. Catrysse, “Beaming light into the nanoworld,” Nat. Phys. |

6. | D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyonds the diffractin limit,” Nat. Photonics |

7. | A. G. Curto, A. Manjavacas, and F. J. García de Abajo, “Near-field focusing with optical phase antennas,” Opt. Express |

8. | I. P. Kaminow, W. L. Mammel, and H. P. Weber, “Metal-clab optical waveguides: analytical and experimental study,” Appl. Opt. |

9. | H. W. Kihm, K. G. Lee, D. S. Kim, J. H. Kang, and Q. Park, “Control of surface plamson generation efficiency by slit-width tuning,” Appl. Phys. Lett. |

10. | J. Lindberg, K. Lindfors, T. Setälä, M. Kaivola, and A. T. Friberg, “Spectral analysis of resonant transmission of light through a single sub-wavelength slit,” Opt. Express |

11. | P. Lalanne, J. P. Hugonin, S. Astillean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. |

12. | L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. |

13. | Z. J. Sun and H. K. Kim, “Refractive transmission of light and beam shaping with metallic nano-optic lenses,” Appl. Phys. Lett. |

14. | T. Xu, C. T. Wang, C. L. Du, and X. G. Luo, “Plasmonic beam deflector,” Opt. Express |

15. | Z. J. Sun, “Beam splitting with a modified metallic nano-optic lens,” Appl. Phys. Lett. |

16. | H. F. Shi, C. T. Wang, C. L. Du, X. G. Luo, X. C. Dong, and H. T. Gao, “Beam manipulating by metallic nano-slits with variant widths,” Opt. Express |

17. | L. Verslegers, P. B. Catrysse, Z. F. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. H. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. |

18. | W. M. Saj, “Light focusing on a stack of metal-insulator-metal waveguides sharp edges,” Opt. Express |

19. | G. Z. Yang and B. Y. Gu, “On the amplitude-phase retrieval problem in optical system,” Acta Phys. Sin. |

20. | B. Y. Gu, G. Z. Yang, and B. Z. Dong, “General theory for performing an optical transform,” Appl. Opt. |

21. | T. Xu, C. L. Du, C. T. Wang, and X. G. Luo, “Subwavelength imaging by metallic slab lens with nanoslits,” Appl. Phys. Lett. |

22. | P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett. |

23. | B. Hu, B. Y. Gu, B. Z. Dong, Y. Zhang, and M. Liu, “Various evaluations of a diffractive transmitted field of light through a one-dimensional metallic grating with subwavelength slits,” Cent. Eur. J. Phys. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(240.3990) Optics at surfaces : Micro-optical devices

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: January 31, 2011

Revised Manuscript: March 23, 2011

Manuscript Accepted: April 8, 2011

Published: May 2, 2011

**Citation**

Qiaofen Zhu, Jiasheng Ye, Dayong Wang, Benyuan Gu, and Yan Zhang, "Optimal design of SPP-based metallic nanoaperture optical elements by using Yang-Gu algorithm," Opt. Express **19**, 9512-9522 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9512

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### References

- R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. 106, 874–881 (1957). [CrossRef]
- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998). [CrossRef]
- H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Merono, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002). [CrossRef] [PubMed]
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). [CrossRef] [PubMed]
- P. B. Catrysse, “Beaming light into the nanoworld,” Nat. Phys. 3, 839–840 (2007). [CrossRef]
- D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyonds the diffractin limit,” Nat. Photonics 4, 83–91 (2010). [CrossRef]
- A. G. Curto, A. Manjavacas, and F. J. García de Abajo, “Near-field focusing with optical phase antennas,” Opt. Express 17, 17801–17811 (2009). [CrossRef] [PubMed]
- I. P. Kaminow, W. L. Mammel, and H. P. Weber, “Metal-clab optical waveguides: analytical and experimental study,” Appl. Opt. 13, 396–405 (1974). [CrossRef] [PubMed]
- H. W. Kihm, K. G. Lee, D. S. Kim, J. H. Kang, and Q. Park, “Control of surface plamson generation efficiency by slit-width tuning,” Appl. Phys. Lett. 92, 051115 (2008). [CrossRef]
- J. Lindberg, K. Lindfors, T. Setälä, M. Kaivola, and A. T. Friberg, “Spectral analysis of resonant transmission of light through a single sub-wavelength slit,” Opt. Express 12, 623–632 (2004). [CrossRef] [PubMed]
- P. Lalanne, J. P. Hugonin, S. Astillean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2, 48–51 (2000). [CrossRef]
- L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90, 167401 (2003). [CrossRef] [PubMed]
- Z. J. Sun and H. K. Kim, “Refractive transmission of light and beam shaping with metallic nano-optic lenses,” Appl. Phys. Lett. 85, 642–644 (2004). [CrossRef]
- T. Xu, C. T. Wang, C. L. Du, and X. G. Luo, “Plasmonic beam deflector,” Opt. Express 16, 4753–4759 (2008). [CrossRef] [PubMed]
- Z. J. Sun, “Beam splitting with a modified metallic nano-optic lens,” Appl. Phys. Lett. 89, 261119 (2006). [CrossRef]
- H. F. Shi, C. T. Wang, C. L. Du, X. G. Luo, X. C. Dong, and H. T. Gao, “Beam manipulating by metallic nano-slits with variant widths,” Opt. Express 13, 6815–6820 (2005). [CrossRef] [PubMed]
- L. Verslegers, P. B. Catrysse, Z. F. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. H. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. 9, 235–238 (2009). [CrossRef]
- W. M. Saj, “Light focusing on a stack of metal-insulator-metal waveguides sharp edges,” Opt. Express 17, 13615–13623 (2009). [CrossRef] [PubMed]
- G. Z. Yang and B. Y. Gu, “On the amplitude-phase retrieval problem in optical system,” Acta Phys. Sin. 30, 410–413 (1981).
- B. Y. Gu, G. Z. Yang, and B. Z. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206 (1986). [CrossRef] [PubMed]
- T. Xu, C. L. Du, C. T. Wang, and X. G. Luo, “Subwavelength imaging by metallic slab lens with nanoslits,” Appl. Phys. Lett. 91, 201501 (2007).
- P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett. 95, 263902 (2005). [CrossRef]
- B. Hu, B. Y. Gu, B. Z. Dong, Y. Zhang, and M. Liu, “Various evaluations of a diffractive transmitted field of light through a one-dimensional metallic grating with subwavelength slits,” Cent. Eur. J. Phys. 8, 448–454 (2009). [CrossRef]

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