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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 10 — May. 9, 2011
  • pp: 9512–9522
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Optimal design of SPP-based metallic nanoaperture optical elements by using Yang-Gu algorithm

Qiaofen Zhu, Jiasheng Ye, Dayong Wang, Benyuan Gu, and Yan Zhang  »View Author Affiliations


Optics Express, Vol. 19, Issue 10, pp. 9512-9522 (2011)
http://dx.doi.org/10.1364/OE.19.009512


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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

Since the pioneering work of Ritchie in the 1950s [1

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

], the surface plasmon polaritons (SPPs) have been widely recognized in the field of surface science. SPPs are charge density waves resulting from the coupling of electromagnetic waves with oscillations of electrons in a metal and propagating along the interface between the dielectric and noble metal with amplitudes decaying into both layers. One of attractive properties of SPPs is that they can be concentrated and channeled in nanostructures which makes it possible to miniaturize the photonic devices in nano-scale.

This article is arranged as follows: Section 2 describes the structure of the elements to be designed and design theory. Section 3 presents the design results with discussion and Section 4 draws a conclusion.

2. Beaming structure and design theory

Fig. 1 Structural diagram of a SPPs lens formed on a thin metallic film.

For simplification, we consider that the structure along the 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 P0 to P plane. So if the phase and amplitude distributions on the P0 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

In order to derive the phase and amplitude distributions on the P0 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 P0 plane of the structure as the sum of phase and amplitude distributions of each slit.

Assuming the slit width is much smaller than the operating wavelength, it is reasonable to only consider the fundamental mode in the slit. Its complex propagation constant β in the slit is determined by the following equation:
pmεdkdεm=1ekw1+ekw,
(1)
where
kd=k0(βk0)2εd;
(2a)
pm=k0(βk0)2εm,
(2b)
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

21. 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).

]
ΔϕRe(βh).
(3)

The amplitude of light wave at the exit of each slit crucially depends on the slit width. It has been investigated theoretically [22

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]

] and experimently [9

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]

]. When other parameters are fixed, the slit width dependent SPPs intensity shows a sinusoidal behavior and the intensity ratio between the peak and dip values can reach 10. It demonstrates that amplitude information is quit important in the design procedure. The relation between the slit width and SPPs amplitude can be approximately expressed as
|Hy|sin(ksppw/2)kspp,
(4)
where kspp=k0εmεd/(εm+εd). Based on these approximations, the field distribution on P0 plane can be derived by summing the phase and amplitude of each slit.

2.2. Two dimensional Green function diffraction formula

The light transmits from P0 plane to P plane via free space propagation. After knowing the field distribution on the P0 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]

]
Hy(r)=P0Hy0(r0)G˜(r0,r)n^dx0,
(5)
where G˜(r0,r)n^ denotes differential of the two dimensional (2D) Green function along the inward normal to P0, and it can be expressed as
G˜(r0,r)n^=jk02zz0|rr0|H1(k0|rr0|),
(6)
So, Eq. (5) is rewritten as
Hy(r)=jk02P0Hy0(r0)H1(k0|rr0|)cosθdx0,
(7)
where
cosθ=zz0|rr0|.
(8)
Equation (7) can be simply expressed as
Hy(r)=G^Hy0(r0)=P0G(r0,r)Hy0(r0)dx0,
(9)
where
G(r0,r)=jk02H1(k0|rr0|)cosθ.
(10)
and H 1(k 0|r⃗r⃗ 0|) is the first-order Hankel function of the second kind.

2.3. Design procedure of the Yang-Gu algorithm

In this work, we consider a metal film perforated with a series of slits with different widths, which corresponds to amplitude-phase modulation on the exit surface. We denote the amplitude-phase modulation for a given slit-width configuration w(x 1) is
δw(ρ0)=ρ0(w(x0))=p(w),
(15)
δw(ϕ0)=ϕ0(w(x0))=q(w),
(16)
their first derivatives are
δρ0δw=p(w),
(17)
δϕ0δw=q(w).
(18)
Then we have the variational argument δw as
δw(D2)=δρ0(D2)δw(ρ0)+δϕ0(D2)δw(ϕ0)=0,
(19)
combined with the variation argument δϕ
δϕ(D2)=0,
(20)
we have
Hy0=(A^D+q)1(ieiarg(Hy0)p+G^+HyA^NDHy0),
(21)
ϕ0=arg[(A^D+q)1(ieiarg(Hy0)p+G^+HyA^NDHy0)],
(22)
with
A^=G^+G^,
(23)
A^=A^D+A^ND,
(24)
where  D ( ND) is the diagonal (off-diagonal) element of Â.

3. Design and numerical simulations

To illustrate the idea of the expanded Yang-Gu algorithm, three kinds of SPPs lenses to realize different functions have been designed. The parameters of the lenses are as follows: The used metal is silver whose dielectric constant is ε m = −17.36 + 0.715i 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.

The design process is as follows: At first, define the objective intensity on the observation plane according to the function of the element; Then, utilize the expanded Yang-Gu algorithm to optimize the width of each slit and derive the optimal structure; At last, simulate the performance of the achieved sturcutre using the FDTD method to verify and appraise the properties of the designed nanolens.

3.1. SPPs lens with one focal spot

As the first demonstration, the SPPs lens with one focal spot is designed to compare with the former method: Equal optical path principle (EOPP) [16

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]

]. The focal length of the desired lens is 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.

Fig. 2 SPPs lens with one focal spot designed by the Yang-Gu algorithm. (a) Objective and designed intensity distributions. (b) Distributions of the slit widths. (c) Normalized intensity distribution of |Hy| for the designed lens calculated using the FDTD method. (d) Cross section of the focal spot along the z direction.

In order to show the performance of the designed structure, the FDTD method is utilized to character the structure. The normalized intensity distribution of |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.

The chromatic dispersion of the designed nanolens has also been investigated. The intensity distributions around focal spot of the designed lens operating at different wavelengths are shown in Fig. 3. The up-side is close to the lens exit plane and the dashed line indicates the plane of 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.

Fig. 3 Chromatic dispersion of the SPPs lens with one focal spot, optimally designed for λ = 650nm, but operating at different wavelength, as indicated in each plot. The dashed line indicates the real focal plane for λ = 650nm.

3.2. SPPs lens with two focal spots

Then, the SPPs lens with two focal spots is designed to show the validity of new proposed method. Two spots are set on the same plane with distance 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.

Fig. 4 Lens with two focal spots designed by the Yang-Gu algorithm. (a) Objective and designed normalized intensity distributions. (b) Distribution of the slit widths. (c) Normalized intensity distribution of |Hy| for the designed lens calculated using the FDTD method. (d) Cross sections of the focal spots along the z direction.

The FDTD method is also used to check the performance of the designed lens. The normalized intensity distribution of calculated |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.

The chromatic dispersion of the designed two spots nanolens is shown in Fig. 5. Only in the range of 650 ± 20nm, the designed nanolens can keep its performance. Since the structure of this lens is more complex than that of one focal spot lens, it is more sensitive to the working wavelength.

Fig. 5 Chromatic dispersion of the SPPs lens with two focal spots, optimally designed for λ = 650nm, but operating at different wavelengths, as indicated in each plot. The dashed line indicates the focal plane for λ = 650nm.

3.3. SPPs lens with three focal spots

At last, we design the SPPs lens with three focal spots. All of spots are set on the plane of 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).

Fig. 6 Lens with three focal spots designed by the Yang-Gu algorithm. (a) Objective and designed intensity distributions. (b) Distribution of the slit widths. (c) Normalized intensity distribution of |Hy| for the designed lens calculated using the FDTD method. (d) Cross sections of focal spots along the z direction.

The designed nanolens has been checked by the FDTD method and the normalized intensity distribution of |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.

The chromatic dispersion of the designed three spots nanolens is shown in Fig. 7. Only in the range of 650 ± 10nm, the designed nanolens can keep its performance well. These results demonstrate that the device is sensitive to the operating wavelength and it should be specially designed for a fixed operating wavelength.

Fig. 7 Chromatic dispersion of the SPPs lens with three focal spots, optimally designed for λ = 650nm, but operating at different wavelength, as indicated in each plot. The dashed line indicates the real focal plane for λ = 650nm.

4. Summary

In summary, a method for optimal design of SPP-based metallic nano-optic elements is proposed. The Yang-Gu algorithm is expanded to carry out the iterative procedure. Three kinds of lens which can preform different functions have been designed and the designed structures have been validated by the FDTD method. It is found that the designed lenses can achieve the preset objectives well. This method may constitute a new basis for potential applications in photonic and plasmonic devices and for achieving optimal design of metallic diffractive optical elements with subwavelength scale.

Acknowledgments

This work was supported by the 973 Program of China (No. 2011CB301801) and the National Natural Science Foundation of China (No. 10904099).

References and links

1.

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

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 391, 667–669 (1998). [CrossRef]

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 297, 820–822 (2002). [CrossRef] [PubMed]

4.

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

5.

P. B. Catrysse, “Beaming light into the nanoworld,” Nat. Phys. 3, 839–840 (2007). [CrossRef]

6.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyonds the diffractin limit,” Nat. Photonics 4, 83–91 (2010). [CrossRef]

7.

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]

8.

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]

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]

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 12, 623–632 (2004). [CrossRef] [PubMed]

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. 2, 48–51 (2000). [CrossRef]

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. 90, 167401 (2003). [CrossRef] [PubMed]

13.

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]

14.

T. Xu, C. T. Wang, C. L. Du, and X. G. Luo, “Plasmonic beam deflector,” Opt. Express 16, 4753–4759 (2008). [CrossRef] [PubMed]

15.

Z. J. Sun, “Beam splitting with a modified metallic nano-optic lens,” Appl. Phys. Lett. 89, 261119 (2006). [CrossRef]

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]

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. 9, 235–238 (2009). [CrossRef]

18.

W. M. Saj, “Light focusing on a stack of metal-insulator-metal waveguides sharp edges,” Opt. Express 17, 13615–13623 (2009). [CrossRef] [PubMed]

19.

G. Z. Yang and B. Y. Gu, “On the amplitude-phase retrieval problem in optical system,” Acta Phys. Sin. 30, 410–413 (1981).

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]

21.

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).

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]

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]

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

  1. R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. 106, 874–881 (1957). [CrossRef]
  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 391, 667–669 (1998). [CrossRef]
  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 297, 820–822 (2002). [CrossRef] [PubMed]
  4. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). [CrossRef] [PubMed]
  5. P. B. Catrysse, “Beaming light into the nanoworld,” Nat. Phys. 3, 839–840 (2007). [CrossRef]
  6. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyonds the diffractin limit,” Nat. Photonics 4, 83–91 (2010). [CrossRef]
  7. 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]
  8. 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]
  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]
  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 12, 623–632 (2004). [CrossRef] [PubMed]
  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. 2, 48–51 (2000). [CrossRef]
  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. 90, 167401 (2003). [CrossRef] [PubMed]
  13. 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]
  14. T. Xu, C. T. Wang, C. L. Du, and X. G. Luo, “Plasmonic beam deflector,” Opt. Express 16, 4753–4759 (2008). [CrossRef] [PubMed]
  15. Z. J. Sun, “Beam splitting with a modified metallic nano-optic lens,” Appl. Phys. Lett. 89, 261119 (2006). [CrossRef]
  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]
  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. 9, 235–238 (2009). [CrossRef]
  18. W. M. Saj, “Light focusing on a stack of metal-insulator-metal waveguides sharp edges,” Opt. Express 17, 13615–13623 (2009). [CrossRef] [PubMed]
  19. G. Z. Yang and B. Y. Gu, “On the amplitude-phase retrieval problem in optical system,” Acta Phys. Sin. 30, 410–413 (1981).
  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]
  21. 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).
  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]
  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]

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