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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 22 — Oct. 25, 2010
  • pp: 22772–22780
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Modulation of optical focusing by using optimized zone plate structures

Jia-Han Li, Chih-Hong Lin, Yao-Jen Tsai, Yi-Wei Cheng, and Tony Wen-Hann Sheu  »View Author Affiliations


Optics Express, Vol. 18, Issue 22, pp. 22772-22780 (2010)
http://dx.doi.org/10.1364/OE.18.022772


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Abstract

The focusing properties of the optimized zone plate structures which have upper and lower zones with different thicknesses are studied by the three-dimensional finite-difference time-domain method. Two kinds of materials are chosen, including silver representing metal and BK7 glass representing dielectric. An optimization algorithm is applied to tune the parameters of zone plate structures. Several optimized zone plate structures with smaller circular-shape focus are presented. By using the angular spectrum representation method, we found that the cases with smaller focal sizes have larger high-k components; however, the intensities of side lobes also become larger in comparison with the main beam. It is also found that the phase differences between different spatial field components can have the influences on focusing properties. A special case with two focuses is shown by changing the cost function of the same optimization algorithm. Our findings suggest that the optimized zone plate structures can reconstruct the light intensity distribution and have a great potential for the applications in imaging, lithography, and data storage.

© 2010 OSA

1. Introduction

The Fresnel zone plate structures are widely used as the focusing lenses in optical applications, such as microscopy, imaging, lithography, etc. Recent studies in plasmonic zone plate structures show that better focusing properties can be achieved. For example, the zone plate structures with metal and dielectric material have a highly-directional focal beam with the focal size smaller than the diffraction limit in the visible wavelength region [1

1. Y. Fu, W. Zhou, L. E. N. Lim, C. L. Du, and X. G. Luo, “Plasmonic microzone plate: Superfocusing at visible regime,” Appl. Phys. Lett. 91(6), 061124 (2007). [CrossRef]

3

3. Y. Fu, Y. Liu, X. Zhou, Z. Xu, and F. Fang, “Experimental investigation of superfocusing of plasmonic lens with chirped circular nanoslits,” Opt. Express 18(4), 3438–3443 (2010). [CrossRef] [PubMed]

] or infrared light region [4

4. A. G. Curto, A. Manjavacas, and F. J. García de Abajo, “Near-field focusing with optical phase antennas,” Opt. Express 17(20), 17801–17811 (2009). [CrossRef] [PubMed]

]. The zone structure covered with the other dielectric layer [5

5. H. C. Kim, H. Ko, and M. Cheng, “Optical focusing of plasmonic Fresnel zone plate-based metallic structure covered with a dielectric layer,” J. Vac. Sci. Technol. B 26(6), 2197–2203 (2008). [CrossRef]

] or metal-dielectric multilayer [6

6. H. C. Kim, H. Ko, and M. Cheng, “High efficient optical focusing of a zone plate composed of metal/dielectric multilayer,” Opt. Express 17(5), 3078–3083 (2009). [CrossRef] [PubMed]

] is found to have a better focus. The phase of surface plasmon polaritons can be modified by varying the dielectric structure on a metal surface, and an in-phase phase modulation Fresnel zone plate is proposed to either enhance or suppress the field intensity [7

7. Q. Wang, X. Yuan, P. Tan, and D. Zhang, “Phase modulation of surface plasmon polaritons by surface relief dielectric structures,” Opt. Express 16(23), 19271–19276 (2008). [CrossRef]

]. By analyzing the fields in the focal plane, larger high-k components are seen in the k-domain for the silver zone plates than for the glass zone plates, and the smaller focus can be obtained for the silver zone plates [8

8. J. Li, Y. Cheng, Y. Chue, C. Lin, and T. W. Sheu, “The influence of propagating and evanescent waves on the focusing properties of zone plate structures,” Opt. Express 17(21), 18462–18468 (2009). [CrossRef]

]. The focuses of different metal coated Fresnel zone plate structures in the visible range are smaller than the diffraction limit because the evanescent field affects the near field focusing [9

9. R. G. Mote, S. F. Yu, B. K. Ng, W. Zhou, and S. P. Lau, “Near-field focusing properties of zone plates in visible regime - New insights,” Opt. Express 16(13), 9554–9564 (2008). [CrossRef] [PubMed]

]. A higher resolution can be obtained by the radiationless electromagnetic interference [10

10. R. Merlin, “Radiationless electromagnetic interference: Evanescent-field interference lenses and perfect focusing,” Science 317(5840), 927–929 (2007). [CrossRef] [PubMed]

13

13. R. Gordon, “Proposal for superfocusing at visible wavelengths using radiationless interference of a plasmonic array,” Phys. Rev. Lett. 102(20), 207402 (2009). [CrossRef] [PubMed]

]. The contribution of the evanescent waves to the total electromagnetic field behind the subwavelength apertures has been studied [14

14. L. E. Helseth, “The almost perfect lens and focusing of evanescent waves,” Opt. Commun. 281(8), 1981–1985 (2008). [CrossRef]

,15

15. L. E. Helseth, “Radiationless electromagnetic interference shaping of evanescent cylindrical vector waves,” Phys. Rev. A 78(1), 013819 (2008). [CrossRef]

]. Also, the Poynting flux in these structures was found to have optical singularities in the near field region [16

16. M. Perez-Molina, L. Carretero, P. Acebal, and S. Blaya, “Optical singularities and power flux in the near-field region of planar evanescent-field superlenses,” J. Opt. Soc. Am. A 25(11), 2865–2874 (2008). [CrossRef]

]. Without the help of evanescent waves, a superoscillating focusing design with only propagating waves was proposed in [17

17. F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9(3), 1249–1254 (2009). [CrossRef] [PubMed]

]. These findings show that the plasmonic zone plate structures with specific geometries and materials can reconstruct the light intensity distribution and have a great potential in nanophotonic applications.

In this paper, we assume the zone plate structures with the upper zones and lower zones which may have the same or different materials. We use an optimization algorithm to tune the zone thicknesses and study the field in the focal plane by the finite-different time-domain method. Several optimized structures are obtained and analyzed by the angular spectrum representation. The focusing properties of these optimized zone plate structures are studied and their physical mechanisms are described. One special case with two focuses is also demonstrated by modifying the cost function used in the same optimization procedures.

2. Zone plate structures and simulation setup

A standard Fresnel zone plate having the same thickness of each zone can be used to focus the plane wave. We use an optimization procedure by tuning the thickness of the zones to find better focusing properties. Figure 1
Fig. 1 A cartoon of the investigated zone plate structure with different thickness in each zone, where (a) is the top view and (b) is the cross view.
shows the cartoon of the zone plate structure under current study. The upper zones and lower zones with the same radius have different thickness. The transparent zones are air and the dark zones are silver or BK7 glass. The radius of the zones can be calculated by the equation Rn = ((d + nλ0/2)2 -d 2)1/2 [18

18. F. L. Pedrotti, S. J. and L. S. Pedrotti, Introduction to Optics (Prentice-Hall International, 2nd ed., 1993), Chap. 18.

], where Rn is the radius of nth zone, n is the number of the zone, d is the focal distance, and λ0 is the wavelength of the incident light. We assume that the radius of the outermost zone, which looks like the substrate, is infinite.

We apply the three-dimensional finite-difference time-domain method to study the focal properties of zone plate structures. The mesh is a cubic with the length 10 nm. The perfectly matched layers are applied at the outer boundaries to simulate the infinite space. We consider the incident light with x polarization and wavelength 632.8 nm. The focal plane is at z = 1 μm. The dielectric constants of the silver and BK7 glass can be found in [19

19. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, San Diego, 1985).

,20

20. X. He, J. Wu, X. Li, X. Gao, L. Zhao, and L. Wu, “Synthesis and properties of silicon dioxide films prepared by pulsed laser deposition using ceramic SiO2 target,” Appl. Surf. Sci. 256(1), 231–234 (2009). [CrossRef]

].

3. Optimization algorithm and optimized structures

By tuning the parameters and the materials of the zone plate structures, the size, shape and distance of focus change accordingly. The focusing properties can be evaluated by the full width half maximum (FWHM) of the focal size in the focal plane. The focus with a smaller focal size and circular shape are preferred. If there are N parameters and M steps in total for each parameter, there are MN possibilities of different structures. For example, the 4-air-zone structure with different thickness of upper and lower zones shown in Fig. 1 has 2710 possibilities of the structures if the zone thickness varies for every 20 nm from 20 nm to 540 nm. This demands a huge amount of computational time if the simulations for all structures need to be done. Thus, it is required to have an algorithm or the procedure to find the optimized zone plate structures. Here, we propose an optimization algorithm and describe the procedures in below to find the optimized zone plate structures with smaller focus.

4. Results and discussion

Figure 3
Fig. 3 (a), (c), (e), (g), and (i) are the simulated Pz(x, y) for the cases standard, 1, 2, 3, and 4 in Table 1, and (b), (d), (f), (h), (j) are the zoom-in plots of (a), (c), (e), (g), and (i) where the white dashed lines are the contours of half intensity of the focus.
shows P z(x, y), the average power density normalized by the incident plane wave in the focal plane at z = 1 μm, in space domain for all cases in Table 1, and their focal sizes can be estimated by the contours of half intensities of the focuses being shown as the white dashed lines in Fig. 3(b), 3(d), 3(f), 3(h), and 3(j). The focal size of each case is also listed in Table 1. For example, the focal size of case 4 is about 4.6*104 nm2 which is similar to the standard zone plate with 25 air zones.

To understand the focusing properties of the standard and optimized cases, Fig. 4
Fig. 4 The simulated normalized Pz(x, y) divided by its maximum along the lines of (a) y = 0 and (b) x = 0 in the focal planes (z = 1 μm) for the cases tabulated in Table 1.
shows the normalized z-component of the average power density divided by its maximum along x and y directions in space domain in the focal plane of all cases in Fig. 3 or listed in Table 1. The results show that the optimized structures have smaller sizes and smaller intensities in focus than the standard zone plate structure. Although the focal sizes of the optimized cases are smaller than the standard case, the side lobes get larger and the shapes of some focuses for optimized cases are not circular as the standard case.

By using the angular spectrum representation [21

21. L. Novotny, and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), Chap. 2.

], the field in the k-domain in a specific plane can be calculated by taking the Fourier transform of the field in the space domain for the same plane. When observing the field in the k-domain for the focal plane, we know the intensities of the fields with different kx and ky in the focal plane. The field in the range of kx 2 + ky 2 < k0 2 is the propagating wave with the propagating constant kz = (k0 2 - kx 2 - ky 2)1/2. The field in the range of kx 2 + ky 2 > k0 2 is the evanescent wave with an attenuation constant αz = (kx 2 + ky 2 - k0 2)1/2. Figure 5
Fig. 5 The simulated normalized Pz(kx, ky) divided by its maximum along the lines of (a) ky = 0 and (b) kx = 0 in the focal planes (z = 1 μm) for the cases tabulated in Table 1.
shows the normalized z-component of the average power density divided by its maximum along kx and ky directions in the k-domain in the focal plane of all cases in Table 1. Comparing to Fig. 4 and Fig. 5, the cases with smaller focal sizes have larger high-k components and side lobes. The components of the field in the k-domain can be varied by tuning the thickness and material of zone plate structures, and the field in space domain changes accordingly. In the investigated cases, we found that the side lobes get larger as the focal size becomes smaller during the optimization procedures.

Combining the discussion of influences of phase and the intensities of high-k and evanescent waves, the focusing properties of the optimized zone plate structures have a better physical explanation. Thus, the focusing properties of a structure can be modified by varying the parameters of the structure and controlling the phase effect and the field distribution in the k-domain.

Modifying the cost function Eq. (1) to c(tu, tl) = fx and removing the limitations of Eqs. (2) and (3) in the optimization procedures and keeping the other conditions unchanged, a structure with silver upper zones tu = {40, 40, 40, 380, 540} and BK7 glass lower zones tl = {420, 60, 100, 20, 400} is found to have two focuses in its centric of the focal plane. The z-component of the average power density normalized by the incident plane wave for this structure is shown in Fig. 7(a)
Fig. 7 Two-focus case: (a) The simulated Pz(x, y) which is normalized to the incident plane wave, and (b) is the zoom-in plot of (a) where the white dashed lines are the contours of half intensity of the focuses.
, and Fig. 7(b) is the zoom-in plot of Fig. 7(a). The focal size of each focus can be estimated by the contours of half intensity of the focus, which is shown as the white dashed lines in Fig. 7(b). The focal size is about 1.2*105 nm2. The distance of two focuses shown as the cross marks in Fig. 7(b) is about 800 nm. It should be noticed that the intensities of the side lobes in Fig. 7(a) are larger than the intensities of the two focuses in the centric of the focal plane because of the removal of the two imposed limitations in Eqs. (2) and (3).

5. Conclusions

Based on the finite-difference time-domain method and the proposed optimization algorithm, several optimized plasmonic zone plate structures are found to have better focusing properties with smaller focal sizes and acceptable side lobe intensities. From the angular spectrum representation method, the smaller focus can be generated and has larger high-k components in the k-domain; however, the intensities of the side lobes also get larger. The focusing properties are influenced by the phase differences between different spatial field components. We also present another two-focus design by changing the cost function and removing the limitation conditions in the optimization procedures. These studies show that the functional optical structures can be designed solely by varying the thicknesses of the zone plate structures, and it is expected that a better performance can be achieved if more parameters are considered in the optimization procedures, such as the material, zone radius, etc.

Acknowledgments

This work was supported by the National Science Council, Taiwan (NSC-96-2221-E-002-133-MY3, NSC-98-2120-M-002-004, NSC-98-2120-M-009-007, NSC-98-2221-E-002-164-MY3), and the National Taiwan University Innovative Research Funds (NTU-98R0333). We are grateful to the National Center for High-Performance Computing, Taiwan, for providing us the requested computer time and facilities.

References and links

1.

Y. Fu, W. Zhou, L. E. N. Lim, C. L. Du, and X. G. Luo, “Plasmonic microzone plate: Superfocusing at visible regime,” Appl. Phys. Lett. 91(6), 061124 (2007). [CrossRef]

2.

Y. Fu and W. Zhou, “Modulation of main lobe for superfocusing using subwavelength metallic heterostructures,” Plasmonics 4(2), 141–146 (2009). [CrossRef]

3.

Y. Fu, Y. Liu, X. Zhou, Z. Xu, and F. Fang, “Experimental investigation of superfocusing of plasmonic lens with chirped circular nanoslits,” Opt. Express 18(4), 3438–3443 (2010). [CrossRef] [PubMed]

4.

A. G. Curto, A. Manjavacas, and F. J. García de Abajo, “Near-field focusing with optical phase antennas,” Opt. Express 17(20), 17801–17811 (2009). [CrossRef] [PubMed]

5.

H. C. Kim, H. Ko, and M. Cheng, “Optical focusing of plasmonic Fresnel zone plate-based metallic structure covered with a dielectric layer,” J. Vac. Sci. Technol. B 26(6), 2197–2203 (2008). [CrossRef]

6.

H. C. Kim, H. Ko, and M. Cheng, “High efficient optical focusing of a zone plate composed of metal/dielectric multilayer,” Opt. Express 17(5), 3078–3083 (2009). [CrossRef] [PubMed]

7.

Q. Wang, X. Yuan, P. Tan, and D. Zhang, “Phase modulation of surface plasmon polaritons by surface relief dielectric structures,” Opt. Express 16(23), 19271–19276 (2008). [CrossRef]

8.

J. Li, Y. Cheng, Y. Chue, C. Lin, and T. W. Sheu, “The influence of propagating and evanescent waves on the focusing properties of zone plate structures,” Opt. Express 17(21), 18462–18468 (2009). [CrossRef]

9.

R. G. Mote, S. F. Yu, B. K. Ng, W. Zhou, and S. P. Lau, “Near-field focusing properties of zone plates in visible regime - New insights,” Opt. Express 16(13), 9554–9564 (2008). [CrossRef] [PubMed]

10.

R. Merlin, “Radiationless electromagnetic interference: Evanescent-field interference lenses and perfect focusing,” Science 317(5840), 927–929 (2007). [CrossRef] [PubMed]

11.

A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: Subdiffraction focusing with patterned surfaces,” Science 320(5875), 511–513 (2008). [CrossRef] [PubMed]

12.

A. Grbic and R. Merlin, “Near-field focusing plates and their design,” IEEE Trans. Antenn. Propag. 56(10), 3159–3165 (2008). [CrossRef]

13.

R. Gordon, “Proposal for superfocusing at visible wavelengths using radiationless interference of a plasmonic array,” Phys. Rev. Lett. 102(20), 207402 (2009). [CrossRef] [PubMed]

14.

L. E. Helseth, “The almost perfect lens and focusing of evanescent waves,” Opt. Commun. 281(8), 1981–1985 (2008). [CrossRef]

15.

L. E. Helseth, “Radiationless electromagnetic interference shaping of evanescent cylindrical vector waves,” Phys. Rev. A 78(1), 013819 (2008). [CrossRef]

16.

M. Perez-Molina, L. Carretero, P. Acebal, and S. Blaya, “Optical singularities and power flux in the near-field region of planar evanescent-field superlenses,” J. Opt. Soc. Am. A 25(11), 2865–2874 (2008). [CrossRef]

17.

F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9(3), 1249–1254 (2009). [CrossRef] [PubMed]

18.

F. L. Pedrotti, S. J. and L. S. Pedrotti, Introduction to Optics (Prentice-Hall International, 2nd ed., 1993), Chap. 18.

19.

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, San Diego, 1985).

20.

X. He, J. Wu, X. Li, X. Gao, L. Zhao, and L. Wu, “Synthesis and properties of silicon dioxide films prepared by pulsed laser deposition using ceramic SiO2 target,” Appl. Surf. Sci. 256(1), 231–234 (2009). [CrossRef]

21.

L. Novotny, and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), Chap. 2.

OCIS Codes
(050.1970) Diffraction and gratings : Diffractive optics
(240.6680) Optics at surfaces : Surface plasmons
(050.1965) Diffraction and gratings : Diffractive lenses

ToC Category:
Diffraction and Gratings

History
Original Manuscript: July 23, 2010
Revised Manuscript: September 17, 2010
Manuscript Accepted: October 6, 2010
Published: October 13, 2010

Citation
Jia-Han Li, Chih-Hong Lin, Yao-Jen Tsai, Yi-Wei Cheng, and Tony Wen-Hann Sheu, "Modulation of optical focusing by using optimized zone plate structures," Opt. Express 18, 22772-22780 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-22772


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References

  1. Y. Fu, W. Zhou, L. E. N. Lim, C. L. Du, and X. G. Luo, “Plasmonic microzone plate: Superfocusing at visible regime,” Appl. Phys. Lett. 91(6), 061124 (2007). [CrossRef]
  2. Y. Fu and W. Zhou, “Modulation of main lobe for superfocusing using subwavelength metallic heterostructures,” Plasmonics 4(2), 141–146 (2009). [CrossRef]
  3. Y. Fu, Y. Liu, X. Zhou, Z. Xu, and F. Fang, “Experimental investigation of superfocusing of plasmonic lens with chirped circular nanoslits,” Opt. Express 18(4), 3438–3443 (2010). [CrossRef] [PubMed]
  4. A. G. Curto, A. Manjavacas, and F. J. García de Abajo, “Near-field focusing with optical phase antennas,” Opt. Express 17(20), 17801–17811 (2009). [CrossRef] [PubMed]
  5. H. C. Kim, H. Ko, and M. Cheng, “Optical focusing of plasmonic Fresnel zone plate-based metallic structure covered with a dielectric layer,” J. Vac. Sci. Technol. B 26(6), 2197–2203 (2008). [CrossRef]
  6. H. C. Kim, H. Ko, and M. Cheng, “High efficient optical focusing of a zone plate composed of metal/dielectric multilayer,” Opt. Express 17(5), 3078–3083 (2009). [CrossRef] [PubMed]
  7. Q. Wang, X. Yuan, P. Tan, and D. Zhang, “Phase modulation of surface plasmon polaritons by surface relief dielectric structures,” Opt. Express 16(23), 19271–19276 (2008). [CrossRef]
  8. J. Li, Y. Cheng, Y. Chue, C. Lin, and T. W. Sheu, “The influence of propagating and evanescent waves on the focusing properties of zone plate structures,” Opt. Express 17(21), 18462–18468 (2009). [CrossRef]
  9. R. G. Mote, S. F. Yu, B. K. Ng, W. Zhou, and S. P. Lau, “Near-field focusing properties of zone plates in visible regime - New insights,” Opt. Express 16(13), 9554–9564 (2008). [CrossRef] [PubMed]
  10. R. Merlin, “Radiationless electromagnetic interference: Evanescent-field interference lenses and perfect focusing,” Science 317(5840), 927–929 (2007). [CrossRef] [PubMed]
  11. A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: Subdiffraction focusing with patterned surfaces,” Science 320(5875), 511–513 (2008). [CrossRef] [PubMed]
  12. A. Grbic and R. Merlin, “Near-field focusing plates and their design,” IEEE Trans. Antenn. Propag. 56(10), 3159–3165 (2008). [CrossRef]
  13. R. Gordon, “Proposal for superfocusing at visible wavelengths using radiationless interference of a plasmonic array,” Phys. Rev. Lett. 102(20), 207402 (2009). [CrossRef] [PubMed]
  14. L. E. Helseth, “The almost perfect lens and focusing of evanescent waves,” Opt. Commun. 281(8), 1981–1985 (2008). [CrossRef]
  15. L. E. Helseth, “Radiationless electromagnetic interference shaping of evanescent cylindrical vector waves,” Phys. Rev. A 78(1), 013819 (2008). [CrossRef]
  16. M. Perez-Molina, L. Carretero, P. Acebal, and S. Blaya, “Optical singularities and power flux in the near-field region of planar evanescent-field superlenses,” J. Opt. Soc. Am. A 25(11), 2865–2874 (2008). [CrossRef]
  17. F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9(3), 1249–1254 (2009). [CrossRef] [PubMed]
  18. F. L. Pedrotti, S. J. and L. S. Pedrotti, Introduction to Optics (Prentice-Hall International, 2nd ed., 1993), Chap. 18.
  19. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, San Diego, 1985).
  20. X. He, J. Wu, X. Li, X. Gao, L. Zhao, and L. Wu, “Synthesis and properties of silicon dioxide films prepared by pulsed laser deposition using ceramic SiO2 target,” Appl. Surf. Sci. 256(1), 231–234 (2009). [CrossRef]
  21. L. Novotny, and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), Chap. 2.

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