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Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Stephen A. Burns
  • Vol. 24, Iss. 6 — Jun. 1, 2007
  • pp: 1793–1798
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Sharper focal spot formed by higher-order radially polarized laser beams

Yuichi Kozawa and Shunichi Sato  »View Author Affiliations


JOSA A, Vol. 24, Issue 6, pp. 1793-1798 (2007)
http://dx.doi.org/10.1364/JOSAA.24.001793


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Abstract

The intensity distributions near the focal point for radially polarized laser beams including higher-order transverse modes are calculated based on vector diffraction theory. For higher-order radially polarized mode beams as well as a fundamental mode ( R - TEM 01 * ) beam, the strong longitudinal component forms a sharper spot at the focal point under a high-NA focusing condition. In particular, double-ring-shaped radially polarized mode ( R - TEM 11 * ) beams can effectively reduce the focal spot size because of destructive interference between the inner and the outer rings with π phase shift. Compared with an R - TEM 01 * beam focusing in a limit of NA = 1 , the full width at half-maximum values of the focal spot for an R - TEM 11 * beam are decreased by 13.6% for the longitudinal component and 25.8% for the total intensity.

© 2007 Optical Society of America

1. INTRODUCTION

Radially polarized laser beams are known to generate a strong longitudinal electric field at the focal point in the case of high-numerical-aperture (NA) focusing.[1

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

] This strong longitudinal component forms a tight focal spot that will advance high-resolution microscopy[2

2. S. Quabis, R. Dorn, M. Eberler, O. Glöcke, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

, 3

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

] and the generation of an evanescent Bessel beam by the excitation of a surface plasmon.[4

4. Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006). [CrossRef] [PubMed]

] In addition, advantages of radially polarized beams were theoretically and experimentally found in many applications, such as optical trapping,[5

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

] material processing,[6

6. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999). [CrossRef]

] and particle acceleration.[7

7. S. C. Tidwell, G. H. Kim, and W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt. 32, 5222–5229 (1993). [CrossRef] [PubMed]

] In calculations, however, the focusing beam was assumed to be a flat-top[4

4. Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006). [CrossRef] [PubMed]

, 5

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

] or single-ring-shaped beam, which is often referred to as a radially polarized TEM01* (R-TEM01*) mode beam.[1

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

, 2

2. S. Quabis, R. Dorn, M. Eberler, O. Glöcke, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

, 3

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

] On the other hand, a double-ring-shaped beam was experimentally observed as a higher-order radially polarized mode (R-TEM11*) directly from a laser cavity.[8

8. T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707–713 (2005). [CrossRef]

] In addition, these higher-order mode beams may be generated by a particular laser cavity designed to oscillate only with radial polarization.[9

9. Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30, 3063–3065 (2005). [CrossRef] [PubMed]

, 10

10. 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, 2151–2153 (2006). [CrossRef] [PubMed]

] Recently, it was theoretically reported that a double-ring-shaped radially polarized beam has the potential to form a dark space surrounded by a intense light field around the focal point under a particular focusing condition.[11

11. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31, 820–822 (2006). [CrossRef] [PubMed]

] This dark space is a so-called optical bottle beam[12

12. J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000). [CrossRef] [CrossRef]

] or optical cage.[11

11. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31, 820–822 (2006). [CrossRef] [PubMed]

] In the case of an R-TEM11* mode, a double-ring structure accompanies with π phase shift between inner and outer rings, which brings the destructive interference of the longitudinal component near the focal point in high-NA focusing. Therefore, the intensity distribution near the focal point varies drastically with the degree of the truncation of the incident beam by a pupil.

The focusing of a higher-order radially polarized mode beam[11

11. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31, 820–822 (2006). [CrossRef] [PubMed]

, 13

13. V. G. Niziev, R. S. Chang, and A. V. Nesterov, “Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer,” Appl. Opt. 45, 8393–8399 (2006). [CrossRef] [PubMed]

] such as an R-TEM11* is similar to the superresolution techniques that require a complex annular pupil or phase mask in imaging systems.[2

2. S. Quabis, R. Dorn, M. Eberler, O. Glöcke, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

, 14

14. L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001). [CrossRef]

, 15

15. C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004). [CrossRef] [PubMed]

] A higher-order radially polarized mode beam intrinsically possesses a ring-shaped intensity distribution and phase shift in the beam cross section. Consequently, higher-order radially polarized laser beams can be expected to form a narrower focal spot compared with R-TEM01* beam focusing. In fact, calculation has shown that a tightly focused R-TEM11* beam generates a sharp focal spot of the longitudinal component.[11

11. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31, 820–822 (2006). [CrossRef] [PubMed]

]

In this paper we investigate the focusing properties of radially polarized beams including higher-order modes based on vector diffraction theory. In particular, the focal spot size for radially polarized beams is numerically studied. For higher-order radially polarized modes, as well as for a single-ring-shaped R-TEM01* mode, the strong longitudinal component forms a sharper spot at the focal point under high-NA focusing conditions. A double-ring-shaped R-TEM11* mode beam can effectively reduce the focal spot size because of the destructive interference between inner and outer rings. Compared with R-TEM01* beam focusing in a limit of NA=1, the full width at half-maximum (FWHM) values of the focal spot for an R-TEM11* beam are decreased by 13.6% for the longitudinal component and 25.8% for the total intensity (the sum of the longitudinal and transverse components).

In Section 2 we describe the electric field distribution of high-NA focusing for radially polarized beams on the basis of vector diffraction theory. Then the mathematical expression for higher-order R-TEMp1* beams is presented. In Section 3 the numerical studies of the intensity distributions near the focal point for radially polarized laser beams with a higher-order transverse mode are presented. In Section 4 the focusing properties of R-TEM11* mode beams are shown in detail.

2. THEORY

A radially polarized beam is frequently referred to as a null centered doughnut mode beam, TEM01*, which has completely cylindrical symmetry in both intensity and polarization distributions. The notation of R-TEM01* corresponds to a single ring-shaped radially polarized beam as a fundamental mode. Similarly, the higher-order modes with multiring beam patterns that are expressed as R-TEMp1* also have completely cylindrical symmetry and p+1 rings. The intensity distributions of cylindrically symmetric R-TEMp1* modes are represented by using Laguerre–Gaussian distributions.[16

16. A. A. Tovar and G. H. Clark, “Concentric-circle-grating, surface-emitting laser beam propagation in complex optical systems,” J. Opt. Soc. Am. A 14, 3333–3340 (1997). [CrossRef]

, 17

17. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998). [CrossRef]

, 18

18. A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000). [CrossRef]

] Figure 1 shows the calculated intensity distributions and the instantaneous polarization states for several modes, R-TEM01*, R-TEM11*, and R-TEM21*. Every mode has a ring-shaped intensity distribution as shown in the lower part of the figure. In the upper part, the phase relationship between the rings is depicted. For example, the R-TEM11* mode has two rings, and the phase difference between the rings is π. This phase difference of π will cause destructive interference in the focal region.[11

11. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31, 820–822 (2006). [CrossRef] [PubMed]

]

3. NUMERICAL APERTURE DEPENDENCE FOR FOCAL SPOT SIZE

Figure 2 shows the intensity profiles at the focal point (z=0) for various radial mode numbers p. For the sake of comparison with the result in Ref. [2

2. S. Quabis, R. Dorn, M. Eberler, O. Glöcke, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

], the NA is assumed to be 1 in our calculations. In addition, the value of β0 for each mode beam are determined so that 90% of the incoming intensity in front of the objective is focused. Each intensity profile is normalized to the maximum value of the total intensity (solid curves) for the R-TEM01* beam in Fig. 2a. In any mode beam focusing, only the strong longitudinal component (dashed curves) emerges on the beam axis, whereas no radial component (dotted curves) is seen on the axis. The longitudinal peak intensity becomes weaker for higher-order mode beam focusing, in contrast to the enhancement and broadening of the sidelobes.

The longitudinal and total intensity distributions for each mode are plotted in Fig. 3 . The peak intensities are normalized to 1. It is clearly seen that the center peaks of the longitudinal component for the higher-order modes are sharper than that of the R-TEM01* mode as shown in Fig. 3a, while the sidelobes become stronger. It is thought that such size reduction of the focal spots is attributed to the π phase shifts of the multiring beam pattern for the higher-order modes, which causes destructive interference of the longitudinal component near the focal point. The values of the FWHM for the center focal spot are obtained to be 0.464λ, 0.401λ, 0.385λ, and 0.370λ for the radial mode number p=0, 1, 2, 5, respectively. For the total intensity distributions shown in Fig. 3b, the corresponding FWHM are 0.582λ, 0.432λ, 0.403λ, and 0.378λ. Note that the focal spot size of the R-TEM11* beam is reduced by 13.6% in the longitudinal component and by 25.8% in total intensity compared with the R-TEM01* beam focusing. The focal spot sizes calculated by Quabis et al.[2

2. S. Quabis, R. Dorn, M. Eberler, O. Glöcke, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

] for radially polarized doughnut mode focusing in a high-NA limit (NA=1) were 0.160λ2 for the longitudinal component and 0.260λ2 for the total intensity. These spot sizes given in units of λ2, which are determined by the area at half the maximum value, correspond to FWHM of 0.451λ and 0.575λ for the longitudinal component and the total intensity, respectively. These values are very close to those obtained for the R-TEM01* beam in our calculation. Thus the higher-order mode beam can create a smaller spot size than can the doughnut mode beam.

Figure 4 shows the intensity distributions of the longitudinal component along the optical axis for each mode. The peak intensities are normalized to 1. From Fig. 4, we can see the elongation of the longitudinal component with increasing radial mode number. This behavior is in strong contrast to the results obtained for the spot size, which decreases with increasing radial mode number, as shown in Fig. 3. In other words, the spot size can become smaller at the cost of elongation in the axial direction. Thus, one can manipulate the distribution of the longitudinal component along the optical axis by choosing the proper mode number of the radially polarized beam.

For comparison, the focal spot sizes for a linearly polarized fundamental Gaussian (L-TEM00) beam calculated along the x and y axes are added to Table 1. The incident L-TEM00 beam is assumed to be polarized along the x axis. The beam width of the beam is determined so that 90% of the incoming intensity in front of the objective is focused. The FWHM values for L-TEM00 in Table 1 indicate that the focal spot stretches along its polarization direction because the FWHM along the x axis is larger than that along the y axis. This elongation of the focal spot in the direction of the polarization is known to be the effect of the longitudinal component for linearly polarized beam focusing with a high-NA lens.[19

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

] It is also known that this elongation effect can be reduced by radially polarized illumination. In fact, the center focal spots in total intensity for the radially polarized beam are sharper than those for the linearly polarized beam, especially in high-NA focusing as shown in Table 1. However, since the center focal spot of the radially polarized beams consists mainly of the longitudinal component, the formation of a sharper spot size by using the radially polarized beams is limited to high-NA focusing.

As shown in Table 1, the reduction of the focal spot size with increasing NA is dominant for the R-TEM11* beam compared with the R-TEM01* beam. Although higher-order modes than R-TEM21* can also reduce the focal spot size in the high-NA condition, the size reduction ratio becomes small. In addition, a gradual decrease of the peak intensity of the focal spot is shown in Fig. 2 besides the growth and the broadening of the sidelobes in the focal plane in the higher-order radially polarized beam focusing. Accordingly, an R-TEM11* mode beam may provide both a sharper focal spot size and a comparatively strong peak intensity in high-NA focusing.

4. BEAM WIDTH DEPENDENCE OF R-TEM11* MODE BEAM FOCUSING

Figure 5a shows the peak intensity of the longitudinal component, Ez2, at the focal point for the R-TEM01* mode (dashed curve) and R-TEM11* mode (solid curve) focusing with NA=1 when β0 varies in the range of 0<β04. From Fig. 5a it can be seen that the intensity of the longitudinal component for the R-TEM11* beam changes drastically with the β0 values, in contrast to the R-TEM01* beam. The maximum and minimum-peak intensities in the R-TEM11* beam focusing in the range of β01 are obtained at β0=1.926 and β0=1.285, respectively. Note that the minimum intensity at β0=1.285 corresponds to the optical cage that forms a dark space surrounded by an intense light field at the focal point.[11

11. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31, 820–822 (2006). [CrossRef] [PubMed]

] The corresponding FWHM values of the longitudinal spot at the focal point are shown in Fig. 5b. For β01.285, the FWHM of the R-TEM11* is always smaller than that of the R-TEM01*. Thus, Fig. 5 suggests that both a strong longitudinal component and a sharper focal spot are satisfactorily formed at β0=1.926 for an R-TEM11* mode beam.

Figures 6a, 6b, 6c, 6d, 6e, 6f show the intensity profiles at the focal point of the R-TEM11* beam with β0=4.0, 3.0, 1.926, 1.5, 1.285, 1.2, respectively. The NA is assumed to be 1 in this calculation. These profiles in Figs. 6 are normalized to the peak intensity at the focal point in Fig. 6c. From Fig. 6 it is clearly seen that the intensity distribution in the focal plane drastically varies with β0 as well as the intensity peak at the focal point as shown in Fig. 5a. In the case of β0=1.926 in Fig. 6c, we can see that the center spot for the total intensity (solid curve) at the focal point is also sharper. In our calculation, the FWHM of the center spot for the longitudinal and total intensity in Fig. 6c are 0.399 and 0.428, respectively.

As can be seen in Fig. 6, a double-ring-shaped R-TEM11* mode beam has two different focusing properties due to destructive interference near the focal point. One is the dark spot formation at the focal point, the so-called optical cage property,[11

11. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31, 820–822 (2006). [CrossRef] [PubMed]

] and the other is the sharper focal spot of the longitudinal component being discussed here. The situation can be conveniently selected by choosing proper incident beam width, i.e., the value of β0, in front of an objective. In other words, one can generate the optical cage near the focal point by simply focusing a single R-TEM11* beam with appropriate truncation (β0=1.285). On the other hand, the sharper spot of the longitudinal component with proper peak intensity near the focal point can be obtained when the incident beam width is adjusted so as to generate the longitudinal component efficiently (β0=1.926).

5. CONCLUSION

The intensity distributions near the focal point for radially polarized laser beams including higher-order transverse modes are calculated based on vector diffraction theory. For higher-order radially polarized beams, the strong longitudinal component forms a sharper spot at the focal point under a high-NA focusing condition. In particular, double-ring-shaped radially polarized mode (R-TEM11*) beams, for which we have already reported the optical cage property,[11

11. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31, 820–822 (2006). [CrossRef] [PubMed]

] can effectively reduce the focal spot size because of destructive interference between the inner and the outer rings with a π phase shift. Compared to R-TEM01* beam focusing in a limit of NA=1, the full width at half-maximum values of the focal spot for an R-TEM11* beam are decreased by 13.6% for the longitudinal component and 25.8% for the total intensity.

Y. Kozawa’s e-mail address is yk-zw @mail.tagen.tohoku.oc.jp.

Table 1. FWHM (λ1) of the Focal Spot for High-NA Focusinga

table-icon
View This Table
Fig. 1 Bottom, theoretical intensity distributions of radially polarized TEM01*, TEM11*, TEM21* modes. Top, corresponding instantaneous polarization states.
Fig. 2 Calculated intensity profiles near the focal point (z=0) for the focusing of (a) R-TEM01*, (b) R-TEM11*, (c) R-TEM21*, (d) R-TEM51* modes, with a NA of 1. The total intensity (solid curves), the longitudinal component (dashed curves), and the radial component (dotted curves) are drawn in each figure. Every profile is normalized to the peak intensity in (a). The horizontal axis is in units of wavelength.
Fig. 3 Intensity profiles of (a) longitudinal components and (b) total intensity in the focal plane for different transverse modes. The peak intensity for each mode is normalized to 1.
Fig. 4 Calculated intensity profiles of the longitudinal component along the z axis for R-TEM01*, R-TEM11*, R-TEM21*, R-TEM51* mode focusing. The peak intensity for each mode is normalized to 1.
Fig. 5 Peak intensity of the longitudinal component (Ez2) at the focal point plotted against the beam width parameter β0 for R-TEM01* (dashed curve) and R-TEM11* (solid curve) mode focusing. (b) FWHM variation of the focal spot corresponding to (a). The FWHM value is in units of wavelength.
Fig. 6 Calculated intensity profiles near the focal point (z=0) of R-TEM11* mode focusing for β0= (a) 4.0, (b) 3.0, (c) 1.926, (d) 1.5, (e) 1.258, (f) 1.2 with a NA=1. The total intensity (solid curves), the longitudinal component (dashed curves), and the radial component (dotted curves) are drawn in each figure. Every profile is normalized to the peak intensity in (c). The horizontal axis is in units of wavelength.
1.

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

2.

S. Quabis, R. Dorn, M. Eberler, O. Glöcke, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

3.

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

4.

Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006). [CrossRef] [PubMed]

5.

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

6.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999). [CrossRef]

7.

S. C. Tidwell, G. H. Kim, and W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt. 32, 5222–5229 (1993). [CrossRef] [PubMed]

8.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707–713 (2005). [CrossRef]

9.

Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30, 3063–3065 (2005). [CrossRef] [PubMed]

10.

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, 2151–2153 (2006). [CrossRef] [PubMed]

11.

Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31, 820–822 (2006). [CrossRef] [PubMed]

12.

J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000). [CrossRef] [CrossRef]

13.

V. G. Niziev, R. S. Chang, and A. V. Nesterov, “Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer,” Appl. Opt. 45, 8393–8399 (2006). [CrossRef] [PubMed]

14.

L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001). [CrossRef]

15.

C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004). [CrossRef] [PubMed]

16.

A. A. Tovar and G. H. Clark, “Concentric-circle-grating, surface-emitting laser beam propagation in complex optical systems,” J. Opt. Soc. Am. A 14, 3333–3340 (1997). [CrossRef]

17.

A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998). [CrossRef]

18.

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000). [CrossRef]

19.

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

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(140.3300) Lasers and laser optics : Laser beam shaping
(260.3160) Physical optics : Interference
(260.5430) Physical optics : Polarization

ToC Category:
Physical Optics

History
Original Manuscript: October 17, 2006
Manuscript Accepted: December 19, 2006
Published: May 9, 2007

Citation
Yuichi Kozawa and Shunichi Sato, "Sharper focal spot formed by higher-order radially polarized laser beams," J. Opt. Soc. Am. A 24, 1793-1798 (2007)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-24-6-1793


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References

  1. K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000). [CrossRef] [PubMed]
  2. S. Quabis, R. Dorn, M. Eberler, O. Glöcke, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000). [CrossRef]
  3. R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]
  4. Q. Zhan, "Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam," Opt. Lett. 31, 1726-1728 (2006). [CrossRef] [PubMed]
  5. Q. Zhan, "Trapping metallic Rayleigh particles with radial polarization," Opt. Express 12, 3377-3382 (2004). [CrossRef] [PubMed]
  6. V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999). [CrossRef]
  7. S. C. Tidwell, G. H. Kim, and W. D. Kimura, "Efficient radially polarized laser beam generation with a double interferometer," Appl. Opt. 32, 5222-5229 (1993). [CrossRef] [PubMed]
  8. T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, "Polarization-selective grating mirrors used in the generation of radial polarization," Appl. Phys. B 80, 707-713 (2005). [CrossRef]
  9. Y. Kozawa and S. Sato, "Generation of a radially polarized laser beam by use of a conical Brewster prism," Opt. Lett. 30, 3063-3065 (2005). [CrossRef] [PubMed]
  10. 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, 2151-2153 (2006). [CrossRef] [PubMed]
  11. Y. Kozawa and S. Sato, "Focusing property of a double-ring-shaped radially polarized beam," Opt. Lett. 31, 820-822 (2006). [CrossRef] [PubMed]
  12. J. Arlt and M. J. Padgett, "Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam," Opt. Lett. 25, 191-193 (2000). [CrossRef]
  13. V. G. Niziev, R. S. Chang, and A. V. Nesterov, "Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer," Appl. Opt. 45, 8393-8399 (2006). [CrossRef] [PubMed]
  14. L. E. Helseth, "Roles of polarization, phase and amplitude in solid immersion lens systems," Opt. Commun. 191, 161-172 (2001). [CrossRef]
  15. C. J. R. Sheppard and A. Choudhury, "Annular pupils, radial polarization, and superresolution," Appl. Opt. 43, 4322-4327 (2004). [CrossRef] [PubMed]
  16. A. A. Tovar and G. H. Clark, "Concentric-circle-grating, surface-emitting laser beam propagation in complex optical systems," J. Opt. Soc. Am. A 14, 3333-3340 (1997). [CrossRef]
  17. A. A. Tovar, "Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams," J. Opt. Soc. Am. A 15, 2705-2711 (1998). [CrossRef]
  18. A. V. Nesterov and V. G. Niziev, "Laser beams with axially symmetric polarization," J. Phys. D 33, 1817-1822 (2000). [CrossRef]
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