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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 15 — Jul. 18, 2011
  • pp: 13750–13756
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Finite-difference time-domain (FDTD) analysis on the interaction between a metal block and a radially polarized focused beam

Kyoko Kitamura, Kyosuke Sakai, and Susumu Noda  »View Author Affiliations


Optics Express, Vol. 19, Issue 15, pp. 13750-13756 (2011)
http://dx.doi.org/10.1364/OE.19.013750


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Abstract

Radially polarized focused beams have attracted a great deal of attention because of their unique properties characterized by the longitudinal field. Although this longitudinal field is strongly confined to the beam axis, the energy flow, i.e., the Poynting vector, has null intensity on the axis. Hence, the interaction of the focused beam and matter has thus far been unclear. We analyzed the interactions between the focused beam and a subwavelength metal block placed at the center of the focus using three-dimensional finite-difference time-domain (FDTD) calculation. We found that most of the Poynting energy propagates through to the far-field, and that a strong enhancement of the electric field appeared on the metal surface. This enhancement is attributed to the constructive interference of the symmetric electric field and the coupling to the surface plasmon mode.

© 2011 OSA

1. Introduction

Radially polarized beams have great promise for next-generation laser applications and devices because of their ability to generate tighter focal spots with longitudonal electric fields [1

1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

]. This singular characteristic makes them an attractive prospect for advances in three-dimensional super-resolution microscopy [2

2. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001). [CrossRef] [PubMed]

] [3

3. K. Yoshiki, K. Ryosuke, M. Hashimoto, and N. Hashimoto, and T. Araki, “Second-harmonic-generation microscope using eight-segment polarization-mode converter to observe three-dimensional molecular orientation,” Opt. Lett. 32(12), 1680–1682 (2007). [CrossRef] [PubMed]

], optical particle trapping [4

4. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

] [5

5. K. Sakai and S. Noda, “Optical trapping of metal particles in doughnut-shaped beam emitted by photonic-crystal laser,” Electron. Lett. 43(2), 107 (2007). [CrossRef]

] [6

6. J.-Q. Qin, X.-L. Wang, D. Jia, J. Chen, Y.-X. Fan, J. Ding, and H.-T. Wang, “FDTD approach to optical forces of tightly focused vector beams on metal particles,” Opt. Express 17(10), 8407–8416 (2009). [CrossRef] [PubMed]

], Raman spectroscopy [7

7. N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85(25), 6239 (2004). [CrossRef]

], and sub-wavelength imaging [8

8. A. Ono, J. Kato, and S. Kawata, “Subwavelength optical imaging through a metallic nanorod array,” Phys. Rev. Lett. 95(26), 267407 (2005). [CrossRef] [PubMed]

]. To develop these applications, several methods for producing radially polarized beams have been explored including interferometric techniques [9

9. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 (1990). [CrossRef] [PubMed]

] or modifying the design of the inner lasing cavity [10

10. Y. Mushiake, K. Matsumura, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60(9), 1107–1109 (1972). [CrossRef]

] [11

11. K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31(14), 2151–2153 (2006). [CrossRef] [PubMed]

]. Most recently, extremely compact, semiconductor photonic crystal surface-emitting lasers have attracted much attention for their potential to produce the beams necessary for these applications [12

12. E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Photonics: lasers producing tailored beams,” Nature 441(7096), 946 (2006). [CrossRef] [PubMed]

15

15. D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express 12(8), 1562–1568 (2004). [CrossRef] [PubMed]

]. We have previously found that a narrow-width annular beam enhances the longitudinal field and extends the focal depth [16

16. K. Kitamura, K. Sakai, and S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18(5), 4518–4525 (2010). [CrossRef] [PubMed]

] to allow stronger interaction with matter. Despite the fact that this unprecedented longitudinal electric field forms strong intensity on the beam axis, it has been found that the energy flow, i.e., the Poynting energy, has null intensity on the axis [4

4. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

]. This mismatch in the intensity distribution raises the question of how the radially polarized focused beam interacts with matter on the beam axis. In this study, we use a three-dimensional (3D) finite-difference time-domain (FDTD) calculation to investigate the interaction of the focused beam with matter in the specific case where a sub-wavelength metal block is placed at the center of the focus. Compared to the focused linearly polarized Gaussian beam, we clarify the unique properties of radially polarized focused beams, in terms of the transmitted Poynting energy to the far-field and also the electric field enhancement in the near-field of the metal block. This study demonstrates that the focused beam strongly couples to the surface plasmon mode on the metal block and also propagates through to the far-field. In section 2, we describe the calculation model based on the 3D FDTD method and show the focused beam propagation in the free space both for a radially polarized beam and a linearly polarized Gaussian beam. In section 3, the Poynting energy flow and field enhancement on the particle surface will be discussed. Concluding remarks appear in section 4.

2. Calculation model using FDTD method

In this study, we simulated the electromagnetic field around the focus of the linearly and radially polarized beams using the 3D FDTD method. Figure 1(a)
Fig. 1 Calculation model of three-dimensional FDTD analysis.(a) 3D model, (b) x-z plane
shows the 3D calculation model and Fig. 1(b) shows the cross section of the x-z plane, where we show the calculation results. We set the focal plane at the center (z/λ = 0) and the beam axis at x/λ = 0.0. We placed the excitation plane 2.5 λ below the focus (z/λ = −2.5), where we carefully reproduced the electromagnetic distributions that was obtained using vector diffraction theory in the case for a focusing lens with NA = 0.9 [17

17. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253(1274), 358–379 (1959). [CrossRef]

] [18

18. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]

]. When both of the electric and magnetic fields on this plane were excited with an adequate phase over a sufficiently large area, more than 10λ × 10λ, the electromagnetic fields only propagated upward and focused at (z/λ = 0), showing the same field-distributions as produced by vector diffraction theory. We excited the electromagnetic fields on this plane with a Gaussian modulated sinusoid field [19

19. M. Yokoyama and S. Noda, “Finite-Difference Time-Domain Simulation of Two-Dimensional Photonic Crystal Surface-Emitting Laser Having a Square-Lattice Slab Structure,” IEICE Trans. Electron . 87-C, 386 (2004).

]. We discretized the unit length λ to 20 mesh points (Δx = Δy = Δz = λ/20) and set a perfect matched layer (PML) at all surrounding boundaries. To simulate the interaction with the metal particle, we placed a gold cube (εmetal = −40.27 + 2.794i at λ = 980 nm) with a side length of λ/2 and a gold block with a size of one wavelength in the x and y direction at the center of the focus (x/λ = 0.0, y/λ = 0.0, z/λ = 0.0). To include the gold permittivity, we used the surface impedance method [20

20. J. R. Wait, “Exact surface impedance for a cylindrical conductor,” Electron. Lett. 15(20), 659 (1979). [CrossRef]

] [21

21. S. Kellali, B. Jecko, and A. Reineix, “Implementation of a surface impedance formalism at oblique incidence in FDTD method,” IEEE Trans. 35, 347 (1993).

].

Figure 2
Fig. 2 Time-averaged distributions of the electric field intensity (Ei: i = total, x, y, z) and the Poynting vector (Sz) calculated by the FDTD method: (a) for a linearly (x) polarized Gaussian beam, and (b) for a radially polarized beam.
shows time-averaged intensity distributions of the electric fields (E i:i = total,x,y,z) and the Poynting vector (Sz) on the x-z plane without the metal particle in free-space. The linearly (x-direction) polarized Gaussian beam (a) forms strong intensity on the beam axis, which mostly consists of x-polarized components (Ex), and shows almost identical intensity distributions in the electric field and the Poynting vector. Note that longitudinally (z) polarized components also exist in the vicinity of the beam axis, whereas its intensity is smaller than Ex by one order of magnitude. In contrast, the radially polarized beam (b) has strong intensity in the longitudinally polarized components (Ez) on the beam axis. The radially polarized components (Ex in this x-z plane) also form some intensity around the beam axis, which broadens the distribution on the focal plane. We should also note that there is a large discrepancy in the distribution of the electric field and the Poynting vector (|Sz|). On the beam axis, there is a strong electric field (Ez), whereas there is no intensity of the Poynting vector.

3. Calculation results with the sub-wavelength metal block in the focal spot

Figure 3
Fig. 3 Total electric field intensity profiles when interacting with a gold block placed at the center of the focus (at x/λ = 0.0, y/λ = 0.0, z/λ = 0.0) calculated by the FDTD method: (a) and (c) for a linearly (x) polarized Gaussian beam and (b) and (d) for a radially polarized beam. Gold cubes half a wavelength on a side produce a significant difference in the intensity distribution between the two beams in (a) and (b). When the size of the gold block increased in the x and y direction, the transmitted energy decreased for both focused beams. The size of the block in the x and y direction is one wavelength in (c) and (d).
shows cross-sectional time-averaged electric field intensity profiles with a gold block at the center of the focal spot for the linearly polarized beam ((a) and (c)) and the radially polarized beam ((b) and (d)). It is clearly evident that, in the case of the linearly polarized beam, the metal cube reflects the incident focused light, thus producing interference patterns in the area of z/λ<0.0, as shown in Fig. 3 (a). To estimate the amount of reflected light, we calculated the ratio of the reflected Poynting energy (|Sz|) over the total incoming Poynting energy in region A between the two lower dashed lines, and found that it was 38%. We also calculated the Poynting energy that has passed through region B between the upper dashed lines and found that the ratio over the total incident energy was 31%. The rest of the Poynting energy (31%) was assumed to be absorbed or scattered away from the enclosed regions of the white dashed lines. In contrast, in the case of the radially polarized beam, the ratio of the reflected Poynting energy was found to be significantly small (1%), whereas the propagated energy was as high as 91%, as shown in Fig. 3(b). This phenomenon is attributed to the distribution of the Poynting vector, which has two robed profiles with intensity peaks at around x/λ = ± 0.6λ, as shown in Fig. 2 (b). The metal cube mostly covers the center null region of the Poynting vector distribution. When the width of the cube was increased in the x and y directions, the transmitted energy for both beams drastically decreased, as shown in Fig. 3 (c) and (d). These results clearly indicate that the transmitted energy is determined by the amount of the Poynting vector that was covered by the gold block.

Another significant feature is the enhancement of the electric field at the upper surface (z/λ≥ + 0.25) of the metal cube in the case of the radially polarized beam. Compared to the incident side surface (z/λ≤−0.25), the electric field at the upper side surface is strongly enhanced, which seems to be a similar phenomenon as that pointed out by Sender for a spheroidal nanoparticle [22

22. K. Şendur and A. Sahinöz, “Interaction of radially polarized focused light with a prolate spheroidal nanoparticle,” Opt. Express 17(13), 10910–10925 (2009). [CrossRef] [PubMed]

]. This enhancement is attributed to the constructive interference at the upper surface and the efficient coupling into the surface plasmon mode. Figure 4 (a) and (b)
Fig. 4 Time dependence of total electrical intensity profiles and electrical vectors interacting with a gold block at the focal spot (the center at x/λ = 0.0, y/λ = 0.0, z/λ = 0.0) by the FDTD method. (a) is for a linearly (x) polarized Gaussian beam and (b) is for a radially polarized beam.
show the time-evolutional electric field distributions in the vicinity of the metal cube for the linearly polarized beam and the radially polarized beam, respectively. Black arrows indicate the electric field vectors. It is clear that the electric field vectors at the upper surface destructively interfere and only produce intensity at the edges for the case of the linearly polarized beam. In contrast, the electric field vectors of the radially polarized beam constructively interfere at the upper surface and hence produce strong intensity. It seems that the metal cube edges allow the radially polarized input beam to couple to the surface plasmon propagation mode that also constructively interfere at the upper surface. Figure 5
Fig. 5 Calculated total electrical intensity profiles interacted with half-wavelength-sized perfect conductivity at a focal spot (the center at x/λ = 0.0, y/λ = 0.0, z/λ = 0.0) by the FDTD method. (a) is the time-averaged intensity profile and (b) is the intensity profile and electrical vectors with time.
shows the electric field intensity profile for a perfect conductor cube. The maximum intensity was 35% smaller than the case with an Au cube shown in Fig. 3 (b), which is attributed to the lack of surface plasmon mode enhancement. These phenomena indicate the unique focusing properties of the radially polarized beam: the distribution of the Poynting vector (|Sz|) is unequal to that of electric fields as a result of the creation of z-polarized components near the focus.

4. Conclusion

We analyzed the interaction of a radially polarized focused beam with a sub-wavelength-sized gold cube placed on the beam axis. FDTD analysis revealed that most of the Poynting energy flows around the gold cube, because the distribution of the Poynting vectors exhibited null intensity on the beam axis. It was also found that strong electric field enhancement was obtained on the transmitted-side surface of the cube, which is attributed to the constructive interference on the beam axis and the coupling to the surface plasmon mode. The fact that the electric field enhancement occurred in spite of the small loss energy flow assists in our understanding and enables the design of new applications using radially polarized beams.

Acknowledgments

This work was partly supported by the Japan Society for the Promotion of Science, the Global COE program of Kyoto University, and the Special Coordination Funds for Promoting Science and Technology (SCF) commissioned by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. We thank Dr. K. Ishizaki for helpful discussion. The authors are grateful to Mr. K. Mochizuki for dedicated help with the FDTD coding.

References and links

1.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

2.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001). [CrossRef] [PubMed]

3.

K. Yoshiki, K. Ryosuke, M. Hashimoto, and N. Hashimoto, and T. Araki, “Second-harmonic-generation microscope using eight-segment polarization-mode converter to observe three-dimensional molecular orientation,” Opt. Lett. 32(12), 1680–1682 (2007). [CrossRef] [PubMed]

4.

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

5.

K. Sakai and S. Noda, “Optical trapping of metal particles in doughnut-shaped beam emitted by photonic-crystal laser,” Electron. Lett. 43(2), 107 (2007). [CrossRef]

6.

J.-Q. Qin, X.-L. Wang, D. Jia, J. Chen, Y.-X. Fan, J. Ding, and H.-T. Wang, “FDTD approach to optical forces of tightly focused vector beams on metal particles,” Opt. Express 17(10), 8407–8416 (2009). [CrossRef] [PubMed]

7.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85(25), 6239 (2004). [CrossRef]

8.

A. Ono, J. Kato, and S. Kawata, “Subwavelength optical imaging through a metallic nanorod array,” Phys. Rev. Lett. 95(26), 267407 (2005). [CrossRef] [PubMed]

9.

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 (1990). [CrossRef] [PubMed]

10.

Y. Mushiake, K. Matsumura, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60(9), 1107–1109 (1972). [CrossRef]

11.

K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31(14), 2151–2153 (2006). [CrossRef] [PubMed]

12.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Photonics: lasers producing tailored beams,” Nature 441(7096), 946 (2006). [CrossRef] [PubMed]

13.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316 (1999). [CrossRef]

14.

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293(5532), 1123–1125 (2001). [CrossRef] [PubMed]

15.

D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express 12(8), 1562–1568 (2004). [CrossRef] [PubMed]

16.

K. Kitamura, K. Sakai, and S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18(5), 4518–4525 (2010). [CrossRef] [PubMed]

17.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253(1274), 358–379 (1959). [CrossRef]

18.

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]

19.

M. Yokoyama and S. Noda, “Finite-Difference Time-Domain Simulation of Two-Dimensional Photonic Crystal Surface-Emitting Laser Having a Square-Lattice Slab Structure,” IEICE Trans. Electron . 87-C, 386 (2004).

20.

J. R. Wait, “Exact surface impedance for a cylindrical conductor,” Electron. Lett. 15(20), 659 (1979). [CrossRef]

21.

S. Kellali, B. Jecko, and A. Reineix, “Implementation of a surface impedance formalism at oblique incidence in FDTD method,” IEEE Trans. 35, 347 (1993).

22.

K. Şendur and A. Sahinöz, “Interaction of radially polarized focused light with a prolate spheroidal nanoparticle,” Opt. Express 17(13), 10910–10925 (2009). [CrossRef] [PubMed]

OCIS Codes
(260.1960) Physical optics : Diffraction theory
(260.5430) Physical optics : Polarization

ToC Category:
Physical Optics

History
Original Manuscript: March 28, 2011
Revised Manuscript: June 8, 2011
Manuscript Accepted: June 19, 2011
Published: July 1, 2011

Citation
Kyoko Kitamura, Kyosuke Sakai, and Susumu Noda, "Finite-difference time-domain (FDTD) analysis on the interaction between a metal block and a radially polarized focused beam," Opt. Express 19, 13750-13756 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-13750


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References

  1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]
  2. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001). [CrossRef] [PubMed]
  3. K. Yoshiki, K. Ryosuke, M. Hashimoto, and N. Hashimoto, and T. Araki, “Second-harmonic-generation microscope using eight-segment polarization-mode converter to observe three-dimensional molecular orientation,” Opt. Lett. 32(12), 1680–1682 (2007). [CrossRef] [PubMed]
  4. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]
  5. K. Sakai and S. Noda, “Optical trapping of metal particles in doughnut-shaped beam emitted by photonic-crystal laser,” Electron. Lett. 43(2), 107 (2007). [CrossRef]
  6. J.-Q. Qin, X.-L. Wang, D. Jia, J. Chen, Y.-X. Fan, J. Ding, and H.-T. Wang, “FDTD approach to optical forces of tightly focused vector beams on metal particles,” Opt. Express 17(10), 8407–8416 (2009). [CrossRef] [PubMed]
  7. N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85(25), 6239 (2004). [CrossRef]
  8. A. Ono, J. Kato, and S. Kawata, “Subwavelength optical imaging through a metallic nanorod array,” Phys. Rev. Lett. 95(26), 267407 (2005). [CrossRef] [PubMed]
  9. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 (1990). [CrossRef] [PubMed]
  10. Y. Mushiake, K. Matsumura, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60(9), 1107–1109 (1972). [CrossRef]
  11. K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31(14), 2151–2153 (2006). [CrossRef] [PubMed]
  12. E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Photonics: lasers producing tailored beams,” Nature 441(7096), 946 (2006). [CrossRef] [PubMed]
  13. M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316 (1999). [CrossRef]
  14. S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293(5532), 1123–1125 (2001). [CrossRef] [PubMed]
  15. D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express 12(8), 1562–1568 (2004). [CrossRef] [PubMed]
  16. K. Kitamura, K. Sakai, and S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18(5), 4518–4525 (2010). [CrossRef] [PubMed]
  17. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253(1274), 358–379 (1959). [CrossRef]
  18. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]
  19. M. Yokoyama and S. Noda, “Finite-Difference Time-Domain Simulation of Two-Dimensional Photonic Crystal Surface-Emitting Laser Having a Square-Lattice Slab Structure,” IEICE Trans. Electron . 87-C, 386 (2004).
  20. J. R. Wait, “Exact surface impedance for a cylindrical conductor,” Electron. Lett. 15(20), 659 (1979). [CrossRef]
  21. S. Kellali, B. Jecko, and A. Reineix, “Implementation of a surface impedance formalism at oblique incidence in FDTD method,” IEEE Trans. 35, 347 (1993).
  22. K. Şendur and A. Sahinöz, “Interaction of radially polarized focused light with a prolate spheroidal nanoparticle,” Opt. Express 17(13), 10910–10925 (2009). [CrossRef] [PubMed]

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