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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 16 — Aug. 3, 2009
  • pp: 13646–13653
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Continuous Numerical Aperture Properties of a Cylindrically Polarized Light Illuminated Sub-wavelength Annular Aperture

Chun-Chieh Fang, Tsung-Dar Cheng, Jyi-Tyan Yeh, Kuang-Chong Wu, and Chih-Kung Lee  »View Author Affiliations


Optics Express, Vol. 17, Issue 16, pp. 13646-13653 (2009)
http://dx.doi.org/10.1364/OE.17.013646


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Abstract

We investigated the process of focusing a radially polarized (RP) light beam through a sub-wavelength annular aperture (SAA). We found that the result was a non-diffraction doughnut-shaped light beam which propagates in free space. After analyzing the electric field component of the focus generated by the SAA structure, we identified the relationship between the focal field generated by the SAA. We then compared it to a case with a traditional objective lens. From our findings, we propose that a SAA structure can be viewed as a continuous numerical aperture optical element.

© 2009 Optical Society of America

1. Introduction

The optical phenomenon of metallic nanostructures has been previously broadly discussed. One special phenomenon of metallic nanostructures is the existence of extraordinary transmission.[1

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

] In addition, the directionality of emitted light from metallic nanostructures known as directional beaming has also been studied.[2

2. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002). [CrossRef] [PubMed]

] Another noteworthy research during the last few years relates to the study of the optical characteristics of metallic sub-wavelength annular apertures (SAA). In a near field, the SAA was found to excite surface plasmon (SP) at the metal-dielectric interface.[3

3. Z. W. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Letters 5, 1726–1729 (2005). [CrossRef] [PubMed]

] At a far field region, a non-diffraction beam similar to the above-mentioned directional beaming can also be generated when a linearly polarized (LP) beam impinges onto the SAA structure.[4

4. D. Z. Lin, C. H. Chen, C. K. Chang, T. D. Cheng, C. S. Yeh, and C. K. Lee, “Subwavelength nondiffraction beam generated by a plasmonic lens ,” Appl. Phys. Lett. 92, 3 (2008).

] Non-diffraction beams, also called Bessel beams, were first proposed by Durnin et al. [5

5. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef] [PubMed]

]

In addition to linearly polarized beams, there are many other types of polarized beams which exist in free space. One type is called a cylindrically polarized beam. A generalized cylindrically polarized beam is composed of a linear superposition of radially and azimuthally polarized beams. Radially polarized beams are one special state of generalized cylindrical polarized beams. In 1972, radially polarized (RP) beams were found to be initially generated when the resonant cavity of the laser was modified.[6

6. Y. Mushiake, K. Matsumura, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proceedings of the IEEE 60, 1107–1109 (1972). [CrossRef]

] Since that time, various other ways have been found to generate RP beams. For example, an interferometric technique[7

7. N. Passilly, R. de Saint Denis, K. Ait-Ameur, F. Treussart, R. Hierle, and J.-F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” Journal of the Optical Society of America A: Optics and Image Science, and Vision 22, 984–991 (2005). [CrossRef]

] and liquid crystal device[8

8. M. Stalder, “Active and passive optical components using liquid crystals,” (SPIE -International Society for Optical Engineering, Bellingham WA, WA 98227–0010, USA, San Jose, CA, USA, 1996), pp. 30–39.

, 9

9. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Optics Letters 21, 1948 (1996). [CrossRef] [PubMed]

] were used to convert LP beams directly into RP beams. Herein, we utilized a liquid crystal device to generate the RP beams needed for our experiments.

2. Generating non-diffraction doughnut shape beam by a sub-wavelength annular aperture

Figure 4 shows the images measured at different distances away from the SAA surface when the RP beam was incident onto the SAA structure with a 12µm diameter. The FWHM of the smallest central spot size located at 14µm above the exiting surface was found to 0.729µm (Fig. 5.). We found that the central spot size can be maintained below 2µm for a distance up to 25µm above the SAA exiting surface. In addition to observing the small focal spot size, we also used other SAA structure diameters and analyzed their depth of focus (DOF). We normalized the intensity profile of the focused beam with respect to the maximum intensity. Figure 6 shows the normalized light intensity at different distances away from the focal point. We found that the length of the DOF depends on the diameter of the SAA structure itself. The non-diffraction characteristics were demonstrated by the slow energy decay rate of the focal spot along the z axis when compared to that of the Gaussian beams. This long depth of focus demonstrates the non-diffraction characteristics. As such, we can conclusively say that the SAA can be considered to be a new method of generating a nondiffraction doughnut shaped beam.

Fig. 1. SAA structure: (a) top view and (b) side view.
Fig. 2. (a) Experimental set-up (b) polarization after passage through LC polarizer (c) image of the RP beam focused with a 5X objective lens.
Fig. 3. The structure of liquid crystal polariz
Fig. 4. Images obtained at different distances above the exiting surface. (Note: the focal spot is a doughnut shape)
Fig. 5. Z cross-section intensity profile at the smallest central spot size
Fig. 6. Maximum intensity measured at different distances above the exit surface for SAAs with different diameters.

3. Continuous numerical aperture proposition to explain the non-diffraction doughnut beam of a sub-wavelength annular aperture

To provide insight into our experimental results, a finite-difference time-domain (FDTD) simulation was applied to compare the focus characteristics. The dielectric constant of silver at 442nm was set to be -5.735+j0.7536. Our total simulation dimension was 20µm×20µm×25µm and the maximum mesh size in the three-dimensional simulation region was 10nm. By using this material, we found that the surface plasmon interference occurred on the surface of SAA structure, which corresponds to the characteristics of silver. Figure 7 shows the polarization of generalized cylindrically polarized beam. The unit electric field of the cylindrically polarized beam can be expressed as follows:

E(r,φ)=P(cosϕ0er+sinϕ0eφ)
(1)

Where er is the unit vector in a radial direction, eφ the unit vector in an azimuthal direction, and P the relative amplitude of the field. The polarization of each position in the beam was rotated ϕ 0 from its radial position.

Fig. 7. Generalized cylindrically polarized beam with ϕ 0 rotation from radial direction

The long depth of focus (DOF) feature was found when the generalized cylindrically polarized beams were incident onto the SAA structure. Figure 8 shows the focusing feature when generalized cylindrically polarized beams were incident onto the SAA with different ϕ 0 values. The variable ϕ 0 was dependent on the electric field intensity ratio of the radial direction and the azimuthal direction. If the radial component is stronger than the azimuthal component, a point-shaped focal pattern can be observed near the SAA structure. In contrast, a doughnut-shape can be seen near a SAA structure.

Fig. 8. Light beam intensity when generalized cylindrically polarized beam incidence: (a) ϕ 0=25.56° (b) ϕ 0=45° (c) ϕ 0=63.43°

To verify our experiment, a radially polarized beam was generated when ϕ 0 had a value of zero. An interesting phenomenon at the focal point (see Fig. 9) shows that the focal pattern will be point-shaped near the SAA surface, but in far field, the focal pattern will become more of a doughnut shape. Results show that the electric component is split into an Er and an Ez component when the RP beam is incident to the SAA structure. In Fig. 10, the focal intensity contributed by the Er component was doughnut-shaped in size but was point-shaped in size when contributed by the Ez component. Due to the non-propagating nature of the light beam z component[11

11. B. Hao and J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Optics Express 15, 3550–3556 (2007). [CrossRef] [PubMed]

], it cannot be observed and measured accurately by using microscopy. We determined that the shape of the focal plane depends on the intensity ratio of the Ez and Er components. Thus, a long DOF can be assumed to be focused by adopting several lens equipped with different NA. Based on this perspective, we can then derive the corresponding NA at different focused positions. Results show that the focal plane away from the SAA structure can be focused by a low NA lens (z>10µm). In addition, the focal plane near the SAA structure can be viewed by focusing with a high NA lens (z<4µm).

Fig. 9. Shapes of the focal spots imaged at different focal planes: (a) z=4µm, (b) z=9µm, and (c) z=16µm.
Fig. 10. SAA structure light beam intensity: (a) r component and (b) z component.

Fig. 11 shows the maximum intensity ratio of the Ez and Er components at the focal plane for different derived NA values of the SAA structure by analyzing the simulation results. A continuous relationship was found between the maximum intensity ratio and the derived NA. To obtain more optical properties on the metallic SAA structure, two SAA structures of 6µm and 9µm diameters were fabricated and their characteristics were measured. We found the relationship was identical as that shown in Fig. 11. This result shows that the relationship seen in Fig. 11 will always hold no matter the diameter size of the SAA structure. Therefore, based on this observation, we propose that we can consider a SAA structure as a continuous NA lens.

Fig. 11. Comparison of the intensity distribution of the axial and radial components at the focus position. (Note: using objective lens and SAA structure with diameters of 6µm, 9µm, and 12µm)

To prove the feasibility of a continuous NA proposition, we calculated the electric component of the focal plane numerically by focusing the objective lens and comparing the focal field of the SAA structure to a traditional objective lens. The unit electric field of the RP beam can be expressed as follows:

E(r,ϕ)=er.
(2)

To compare the focal properties of the SAA structure to the objective lens, we put in an annular diaphragm before an objective lens (see Fig. 12). In this way, the electric component on the focal point was contributed by a certain angle. After the RP beam was focused by the objective lens, the focal electrical field components can be expressed as follows [12

12. Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Optics Express 10, 324–331 (2002). [PubMed]

, 13

13. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Optics Express 7, 77–87 (2000). [CrossRef] [PubMed]

]:

Er(r,z)=Aθminθmaxcos12(θ)P(θ)sinθcosθJ1(krsinθ)dθ
(3)
Ez(r,z)=Aθminθmaxcos12(θ)P(θ)sin2θJ0(krsinθ)dθ
(4)
P(θ)=exp[β02(sinθsinθmax)2]J1[2β0sinθsinθmax]
(5)

where A is the constant which can be set to unity, P(θ) is the pupil apodization function, θmax and θmin are derived by the slit width, r the position in the focal plane, k the wave vector, θmax=sin-1 (NA), θ min=θ max-(R 2-R 1)/R 1cscθ max, β 0 is the radius ratio of the lens and incidence light, and J n(x) represents the Bessel function of the first kind of order n. All measurements are in units of wavelength, therefore β 0=1.5, A=1, k=2π/λ, R1=6, R2=5.85.

The intensity of the focal point is a combination of the Ez and Er components. The Er component can be described in the first order of the Bessel function, so that the focal intensity contributed by the r component shows up to be shaped like a doughnut. The Ez component at zero order of the Bessel function is such that the focal intensity contributed by the z component shows up as a point shape. We calculated the focal pattern focused by different NA lenses (see Fig. 13). The focal pattern, as determined by the intensity ratio of the r and z component, was a doughnut-shaped for the case where there was a low NA lens and was point-shaped for the case with a high NA lens. These focal patterns correspond to the focused beam pattern of the SAA structure for different derived NA values.

Fig. 12. RP beam incident into objective lens from a certain angle.
Fig. 13. Using Matlab to calculate the focal pattern when the RP beam is incident into SAA structure with an annular aperture: (a) NA=0.342, (b) NA=0.5, and (c) NA=0.766.

We found that as the NA value increases, the Ez component becomes stronger than the Er component. The relationship can be seen in Fig. 14 which shows the ratio of the maximum intensities of the Ez and Er components at different NA values. We chose the NA value to be in the range of 0.18 to 0.94. When comparing the relative curves to the one obtained from the SAA structure, we see the same tendency or characteristics in all the curves. Thus, it appears that both the focal pattern and the electric component distribution can support a continuous NA proposition of the SAA structure.

Fig. 14. Comparison of the intensity distribution of the r and z components at the focus position. (Note: using objective lens and SAA structure with diameter of 12µm)

4. Conclusions

We propose a new method to generate non-diffraction beams with a cylindrically polarized beam incidence, using a SAA structure. Both simulation and experimental results show that a sub-wavelength focal spot and long depth of focus can be achieved by using a SAA structure. A comparison of the focal field characteristics between a SAA structure and an objective lens by a RP beam incidence supports the idea of a continuous NA in a SAA structure. Moreover, these results provide new insight into SAA structures which will be useful for future studies.

Acknowledgements

This work was financially supported by the Materials & Chemical Research Laboratory of the Industrial Technology Research Institute (ITRI) and Taiwan’s National Science Council under Grant No. 97-2221-E-002-159-MY3. We appreciate C.Y. Wang and other group members for their assistance with this work.

References and links

1.

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

2.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002). [CrossRef] [PubMed]

3.

Z. W. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Letters 5, 1726–1729 (2005). [CrossRef] [PubMed]

4.

D. Z. Lin, C. H. Chen, C. K. Chang, T. D. Cheng, C. S. Yeh, and C. K. Lee, “Subwavelength nondiffraction beam generated by a plasmonic lens ,” Appl. Phys. Lett. 92, 3 (2008).

5.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef] [PubMed]

6.

Y. Mushiake, K. Matsumura, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proceedings of the IEEE 60, 1107–1109 (1972). [CrossRef]

7.

N. Passilly, R. de Saint Denis, K. Ait-Ameur, F. Treussart, R. Hierle, and J.-F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” Journal of the Optical Society of America A: Optics and Image Science, and Vision 22, 984–991 (2005). [CrossRef]

8.

M. Stalder, “Active and passive optical components using liquid crystals,” (SPIE -International Society for Optical Engineering, Bellingham WA, WA 98227–0010, USA, San Jose, CA, USA, 1996), pp. 30–39.

9.

M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Optics Letters 21, 1948 (1996). [CrossRef] [PubMed]

10.

C. Lopez-Mariscal, J. C. Gutierrez-Vega, and S. Chavez-Cerda, “Production of high-order Bessel beams with a Mach-Zehnder interferometer,” Appl. Optics 43, 5060–5063 (2004). [CrossRef]

11.

B. Hao and J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Optics Express 15, 3550–3556 (2007). [CrossRef] [PubMed]

12.

Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Optics Express 10, 324–331 (2002). [PubMed]

13.

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Optics Express 7, 77–87 (2000). [CrossRef] [PubMed]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(260.1960) Physical optics : Diffraction theory
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Diffraction and Gratings

History
Original Manuscript: June 16, 2009
Revised Manuscript: July 9, 2009
Manuscript Accepted: July 10, 2009
Published: July 23, 2009

Citation
Chun-Chieh Fang, Tsung-Dar Cheng, Jyi-Tyan Yeh, Kuang-Chong Wu, and Chih-Kung Lee, "Continuous numerical aperture properties of a cylindrically polarized light illuminated sub-wavelength annular aperture," Opt. Express 17, 13646-13653 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13646


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References

  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003). [CrossRef] [PubMed]
  2. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002). [CrossRef] [PubMed]
  3. Z. W. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, "Focusing surface plasmons with a plasmonic lens," Nano Lett. 5, 1726-1729 (2005). [CrossRef] [PubMed]
  4. D. Z. Lin, C. H. Chen, C. K. Chang, T. D. Cheng, C. S. Yeh, and C. K. Lee, "Subwavelength nondiffraction beam generated by a plasmonic lens," Appl. Phys. Lett. 92, 3 (2008).
  5. J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-Free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
  6. Y. Mushiake, K. Matsumura, and N. Nakajima, "Generation of radially polarized optical beam mode by laser oscillation," Proc. IEEE 60, 1107-1109 (1972). [CrossRef]
  7. N. Passilly, R. de Saint Denis, K. Ait-Ameur, F. Treussart, R. Hierle, and J.-F. Roch, "Simple interferometric technique for generation of a radially polarized light beam," J. Opt. Soc. Am. A: Opt. Image Sci. Vision 22, 984-991 (2005). [CrossRef]
  8. M. Stalder, "Active and passive optical components using liquid crystals," (SPIE -International Society for Optical Engineering, Bellingham WA, WA 98227-0010, USA, San Jose, CA, USA, 1996), pp. 30-39.
  9. M. Stalder and M. Schadt, "Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters," Opt. Lett. 21, 1948 (1996). [CrossRef] [PubMed]
  10. C. Lopez-Mariscal, J. C. Gutierrez-Vega, and S. Chavez-Cerda, "Production of high-order Bessel beams with a Mach-Zehnder interferometer," Appl. Opt. 43, 5060-5063 (2004). [CrossRef]
  11. B. Hao, and J. Leger, "Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam," Opt. Express 15, 3550-3556 (2007). [CrossRef] [PubMed]
  12. Q. Zhan and J. R. Leger, "Focus shaping using cylindrical vector beams," Opt. Express 10, 324-331 (2002). [PubMed]
  13. K. S. Youngworth, and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000). [CrossRef] [PubMed]

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