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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 5 — Mar. 1, 2010
  • pp: 4518–4525
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Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam

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


Optics Express, Vol. 18, Issue 5, pp. 4518-4525 (2010)
http://dx.doi.org/10.1364/OE.18.004518


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Abstract

We demonstrate the formation of a sub-wavelength focal spot with a long depth of focus using a radially polarized, narrow-width annular beam. Theoretical analysis predicts that a tighter focal spot (approximately 0.4λ) and longer depth of focus (more than 4λ) can be formed by a longitudinal electric field when the width of the annular part of the beam is decreased. Experimental measurements using a radially polarized beam from a photonic crystal laser agree well with these predictions. Tight focal spots with long depths of focus have great potential for use in high-tolerance, high-resolution applications in optical systems.

© 2010 OSA

1. Introduction

Recently, a number of applications have been proposed that require tailored optical beam modes. These include microscopy techniques that enable the observation of three-dimensional molecular orientation [1

1. 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]

,2

2. K. Yoshiki, K. Ryosuke, M. Hashimoto, T. Araki, and N. Hashimoto, “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 trapping of opaque particles [3

3. 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]

], and efficient material processing [4

4. M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. 86(3), 329–334 (2007). [CrossRef]

]. Many methods of producing tailored optical beams have been explored, including interferometric techniques [5

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

], modifying the design of the inner laser cavity [6

6. 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]

, 7

7. 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]

], and the use of a liquid crystal phase modulator [8

8. G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004). [CrossRef]

] or a polarization converter [9

9. S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81(5), 597–600 (2005). [CrossRef]

]. However, these methods all require additional optical elements that must also be aligned. If a range of optical beam modes could be made available in compact semiconductor lasers, it would become straightforward to realize the applications listed above. We have previously demonstrated that a range of tailored optical beam modes [10

10. 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]

] can be produced by appropriate design of the lattice points of semiconductor photonic crystal (PC) surface-emitting lasers or by the introduction of phase shifts within the PC structure [11

11. 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]

13

13. 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 (2004). [CrossRef] [PubMed]

]. When the PC is comprised of circular air holes, doughnut-shaped beams with radial or azimuthal polarization can be obtained.

Among the various optical beam modes reported, the radially polarized doughnut beam has attracted much attention because of the unique possibility that it provides to create smaller spot sizes [14

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

] and a longitudinal electric field [15

15. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 1, 1 (2008).

] at the focal point. This longitudinal electric field would allow novel applications to be realized, such as particle acceleration [16

16. D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368(5), 402–407 (2007). [CrossRef]

] and Raman spectroscopy [17

17. 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]

]. Furthermore, if the depth of focus could be extended while keeping the focal spot small, a longer interaction between the optical field and the object can be expected in the above applications, as well as simpler alignment procedures for many optical systems.

In this paper, we demonstrate that the depth of focus of a radially polarized beam can be extended and the spot size made smaller when the intensity distribution of the beam cross-section is changed. We have used vectorial diffraction theory [18

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

] to theoretically evaluate the change in electric field distribution at the focus that occurs when the annular intensity profile of the beam cross-section is made narrower. Moreover, we have used a radially polarized beam supplied by a PC laser to experimentally investigate the intensity profile near the focal plane when the annular width of the beam is decreased.

2. Theoretical calculation

The electric field components in the vicinity of the focus of a radially polarized doughnut beam are formulated using vectorial diffraction theory [18

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

, 19

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

] as follows:
e=[eρeϕez]=[A0αcos12θsin(2θ)l0(θ)J1(kρsinθ)eikzcosθdθ02iA0αcos12θsin2θl0(θ)J0(kρsinθ)eikzcosθdθ]..
(1)
Here, l 0(θ) is the apodization function given in the form:
l0(θ)=exp[β02(sinθsinα)2]J1(2β0sinθsinα).
(2)
We express the field components using cylindrical coordinates (ρ, ϕ, z) [see Fig. 1 (a)
Fig. 1 (a) Schematic sketch of the focusing of a radially polarized beam with δ=0.0. The red and blue arrows indicate the longitudinal and radial electric fields, respectively. (b) Cross-sectional intensity profiles of the electric field components at the focus. The sum of the radial (blue line) and longitudinal (red line) components gives the total intensity profile (black line).
], where the positive z-direction corresponds to the beam propagation direction and z=0 at the focal plane. In Eq. (1), A is the coefficient of power (A=1), β 0 is the ratio of the outer (pupil) radius to the beam waist, which we set at β 0=3/2, k is the wavenumber (k=2π/λ), and α is the maximum angle of the focus given by sin−1(NA/n), where NA is the numerical aperture. In this paper, we set NA to 0.9, the refractive index to n=1, and the wavelength to λ=980 nm. As described in Eq. (1), the radially polarized doughnut beam produces electric fields with radial (ρ) and longitudinal (z) components at the focus. Figure 1(a) shows a schematic picture of the beam focusing, where the electric field vectors are indicated by arrows; Fig. 1(b) shows the cross-sectional intensity profile at the focal plane (z/λ=0.0) along the ρ-axis. The length is normalized in units of wavelength. The longitudinal component produces strong intensity at the beam axis (ρ/λ=0.0), whereas the radial component produces a peak of intensity on each side of the axis. The total intensity profile at the focus takes the form of a single peak, which is much wider than that of the longitudinal component. Therefore, if the radial component can be reduced, making the longitudinal component dominant, we can expect a smaller focal spot.

In order to enhance the longitudinal component, which is mainly produced by the outer part of the annular input beam as shown in Fig. 1(a), it is important to make this outer part more intense. The characteristics of the modified beam obtained when the intensity is restricted to the outer part of the ring can be calculated by rewriting the apodization function l(θ) using a mask function T(θ) in the form:
l(θ)=T(θ)   l0(θ),
(3)
where
T(θ)={0(0θδα)1(δα<θα).
(4)
Here, δ is the ratio of the inner focusing angle θ min to α, as shown in Fig. 2(a)
Fig. 2 (a) Schematic sketch of the focusing of a radially polarized, narrow-width annular beam with δ=0.8. (b) Cross-sectional intensity profiles of the electric field components at the focus. The central peak of the total intensity (black line) is dominated by the longitudinal component (red line); the contribution of the radial component (blue line) has been decreased.
. We used Eqs. (3) and (4) instead of Eq. (2) to calculate the cross-sectional intensity profile at the focal plane for the modified input beam; the result is shown in Fig. 2(b) for δ=0.8. The contribution of the radial electric field component has been decreased and the total intensity profile is now dominated by the longitudinal component. Therefore, the width of the total intensity decreases when the parameter δ increases. Modification of the intensity distribution of the input beam not only changes the profile of the cross-section at the focal plane (z/λ=0.0), but also along the z-axis. Figure 3
Fig. 3 Intensity profiles of the electric fields through the focus (a) for δ=0.0 (the original doughnut beam) and (b) for δ=0.8 (a narrow-width annular beam). A long, sharp focal spot is formed for δ=0.8. The cross-sectional intensity profiles for δ=0.8 show that the longitudinal component (red line) dominates the total intensity (black line), with a much smaller contribution from the radial component (blue line). The focal plane lies at z/λ=0.0. (c) Focal spot size and depth of focus as a function of the ratio of inner and outer focusing angles δ.
shows the intensity distribution through the focus in the propagation direction (z), (a) for the original input beam with δ=0.0 and (b) for the modified beam with δ = 0.8. For the original input beam, the depth of focus (the full-width at half maximum (FWHM) along the z-direction) is approximately 2λ, whereas the depth of focus is longer than 4λ for the modified beam. In the latter case the radial component has been strongly suppressed, whereas the z-component dominates the total intensity profile through the focus, as shown in Fig. 3(b). Figure 3(c) shows the depth of focus and the focal spot size (the FWHM along the ρ-axis) of the central peak as a function of δ. When δ>0.9, the depth of focus is more than 10λ and the focal spot size is approximately 0.4λ, which is smaller than the width (0.55λ) of focus obtained for a linearly polarized Gaussian beam under the same focusing conditions (NA=0.9, n=1).

Such an extension of the focal depth can be explained if we consider the interference of the outermost and innermost rays of the beam axis in the region around the focal point, as shown Fig. 4
Fig. 4 Schematic sketch of the interference of optical rays near the focal point. The blue and red arrows indicate rays originating from the outermost and innermost parts of the input beam, respectively. The dotted lines indicate the wave fronts of these rays. For the original beam (a), the rays are only in phase at the focal plane (z=0), whereas for the narrow-width annular beam (b), the rays are in phase on the beam axis even away from the focal plane.
. When the original radially polarized beam is focused, the incident angles of the two rays are significantly different (α » θ min). Constructive interference takes place only at the focal plane (z/λ=0.0) on the beam axis (ρ/λ=0.0) because the two rays are in phase only at this point, as shown in Fig. 4(a). Therefore, strong intensity appears only in close vicinity to the focal plane. In contrast, the two corresponding rays for the modified, narrower annular-width input beam interfere constructively on the beam axis even away from the focal plane; because the incident angles are almost identical, the phases of the rays match well, as shown in Fig. 4(b). The modified, narrower annular-width input beam thus possesses a longer depth of focus. Such an extension of the optical field at the focal point for optical rays with limited focusing angles has previously been explained for non-diffracting beams by Durnin [20

20. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651 (1987). [CrossRef]

].

3. Experiment

Two methods of measuring the focal spot size of a radially polarized beam have been reported so far: knife-edge scanning [21

21. R. Dorn, S. Quabis, and G. Leuchs, “The focus of light−linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50, 1917 (2003).

] and a photo-resist exposure method [22

22. B. Hao and J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Opt. Express 15(6), 3550–3556 (2007). [CrossRef] [PubMed]

]. We employed the knife-edge scanning method in the current study because it gives spot sizes as a function of focal depth.

3.1.1. Detector

3.1.2. Measurement setup

Figure 5(c) shows our experimental set-up. We used the radially polarized doughnut beam emitted by a PC laser, which consisted of a semi-conductor laser structure with a PC laser cavity near the active layer [10

10. 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]

]. The beam was focused by an objective lens with 0.9 NA (W. D. =1.0 mm) after passing through a collimating lens and a non-polarizing beam splitter. The detector described above was placed at the focal plane and scanned using a high-resolution xyz stage with a positional precision of 10 nm. Part of the collimated beam was divided using a beam splitter and passed to a CCD camera to check the quality of the input beam. For precise alignment, the confocal image of the objective lens was monitored by another CCD camera, allowing the knife-edge position relative to the focal spot to be observed. The knife-edge was scanned in 50 nm steps by a distance of 20 μm through the focal spot using the high-resolution stage. The photo-current and the position of the knife-edge were measured at every step. The changes in photo-current and the corresponding knife-edge position give the intensity profile along the scanning direction. The intensity along the knife-edge was integrated in each measurement. In order to obtain a two-dimensional (2D) profile, the same procedure was carried out for several different scanning angles, as shown in Fig. 5(d). The data that we acquired allowed the 2D beam intensity distribution at the focus to be reconstructed using the Radon back-transformation [23

23. M. Born, and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, New York, 1999), chap. 4.

]. Although the intensity of the input beam in the current study is symmetrical over most of the cross-section perpendicular to the beam axis, a small region of the cross-section is asymmetric and we thus used data from a single scan performed along the most symmetrical direction to reconstruct the 2D profile.

In order to modify the cross-sectional intensity profile of the beam to obtain a narrower annular shape, we placed a shading mask with a circular opaque region on the glass such that the inner part of the input beam was blocked. The radii of the opaque area and the objective lens pupil then corresponded to the inner and outer radii of the modified beam, respectively. We changed the value of δ by varying the radius of the opaque area.

3.2. Results

Figure 6
Fig. 6 Normalized cross-sectional intensity profiles at the focal plane (z=0) and its vicinity. (a) Intensity profiles for the original beam (δ=0.0) show significant changes from a single peak to a double peak for a deviation of ±1 μm from the focal plane. The intensity profile at z=2 μm could not be measured due to the small signal intensity. (b) Intensity profiles for the narrower-width annular beam (δ=0.8), show a central peak for all observation planes from −2 μm to 2 μm.
shows the experimentally determined cross-sectional intensity profiles of the focal spot for the original radially polarized beam with δ=0.0 and for the modified beam with δ=0.8. The profiles were measured in the focal plane (z=0 μm) and in planes away from the focus (z=±1, ±2 μm). Plus signs indicate positions away from the objective lens with respect to the focal plane. For the original beam, shown in Fig. 6(a), the center of the intensity profile changes drastically from a single peak to a double peak when the observation plane is shifted by ± 1 μm from the focus; the peak intensity is less than half of that at z=0 μm. At z=2 μm the cross-sectional intensity profile could not be measured because it became widely dispersed with intensity changes smaller than the noise level. In contrast, the modified beam with δ=0.8 gave a central peak profile that was almost single-peaked for all observation planes from z=−2 μm to z=2 μm; the intensity of the central peak was maintained at more than half of the z=0 μm value between z=−1 μm and z=2 μm, as shown in Fig. 6(b). These results clearly show that the depth of focus is expanded when δ is enlarged. Furthermore, the modified beam with δ=0.8 produces a smaller spot at the focal plane (approximately 600 nm) than the original beam with δ=0.0 (approximately 1200 nm). The minor discrepancies between the calculated and experimental results, for example the dip in the central peak of the experimental profile at z = −1 μm in Fig. 6(b) and the larger side-lobe of Fig. 6(b) than Fig. 3(b), are probably due to scattering from the knife-edge or to the effect of coupling to a surface plasmon mode of the thin metal film, and will be reported in a future article.

4. Conclusion

Acknowledgments

This work was partly supported by the Japan Society of Promotion and Science, the Global COE program of Kyoto University, and 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 Y. Kurosaka and W. Kunishi for providing the PC lasers. We also thank K. Ishizaki for helpful discussions concerning the calculations.

References and links

1.

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]

2.

K. Yoshiki, K. Ryosuke, M. Hashimoto, T. Araki, and N. Hashimoto, “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]

3.

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]

4.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. 86(3), 329–334 (2007). [CrossRef]

5.

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

6.

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]

7.

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]

8.

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004). [CrossRef]

9.

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81(5), 597–600 (2005). [CrossRef]

10.

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]

11.

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]

12.

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]

13.

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 (2004). [CrossRef] [PubMed]

14.

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

15.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 1, 1 (2008).

16.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368(5), 402–407 (2007). [CrossRef]

17.

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]

18.

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

19.

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

20.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651 (1987). [CrossRef]

21.

R. Dorn, S. Quabis, and G. Leuchs, “The focus of light−linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50, 1917 (2003).

22.

B. Hao and J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Opt. Express 15(6), 3550–3556 (2007). [CrossRef] [PubMed]

23.

M. Born, and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, New York, 1999), chap. 4.

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

ToC Category:
Physical Optics

History
Original Manuscript: November 25, 2009
Revised Manuscript: January 20, 2010
Manuscript Accepted: February 9, 2010
Published: February 19, 2010

Citation
Kyoko Kitamura, Kyosuke Sakai, and Susumu Noda, "Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam," Opt. Express 18, 4518-4525 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-4518


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References

  1. 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]
  2. K. Yoshiki, K. Ryosuke, M. Hashimoto, T. Araki, and N. Hashimoto, “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]
  3. 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]
  4. M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. 86(3), 329–334 (2007). [CrossRef]
  5. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234 (1990). [CrossRef] [PubMed]
  6. 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]
  7. 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]
  8. G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004). [CrossRef]
  9. S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81(5), 597–600 (2005). [CrossRef]
  10. 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]
  11. 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]
  12. 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]
  13. 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 (2004). [CrossRef] [PubMed]
  14. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]
  15. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 1, 1 (2008).
  16. D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368(5), 402–407 (2007). [CrossRef]
  17. 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]
  18. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253(1274), 358–379 (1959). [CrossRef]
  19. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]
  20. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651 (1987). [CrossRef]
  21. R. Dorn, S. Quabis, and G. Leuchs, “The focus of light−linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50, 1917 (2003).
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