Abstract

We demonstrate a technique for generating azimuthally and radially polarized beams using a nematic liquid crystal spatial light modulator and a π phase step. The technique is similar in concept to prior techniques that interfere TEM01 and TEM10 laser modes, but the presented technique removes the requirement of interferometric stability. We calculate an overlap integral of >0.96 with >70% efficiency from an input Gaussian mode. The technique can easily switch between beams with azimuthal and radial polarization.

©2010 Optical Society of America

1. Introduction

In recent years considerable attention has been devoted to light beams with radially and azimuthally polarized fields (RPFs and APFs) due to their unusual properties and potential applications. For example, focusing RPFs with a high numerical aperture lens produces a strong longitudinal field component in the focus that is smaller than the diffraction limit for the linearly polarized light [16]. Some of the applications of these beams include optical guiding and trapping [7,8], metal cutting [9,10], determination of the orientation of single molecules [11,12], optimizing photon collection of molecular-based single-photon sources [13,14], charged-particle acceleration [15], and scanning optical microscopy [16,17].

Numerous techniques have been employed to generate these beams externally to the laser cavity, having different degrees of stability, complexity, efficiency, and cost. Techniques with segmented waveplates [18], spatially-varying liquid crystal plates [19], and custom optical fibers [20,21] can efficiently generate these fields, but are not commercially available or are not versatile. Standard multimode optical fibers can produce vector beams, but lack intrinsic stability and are inefficient unless the input beam is already preshaped. Sub-wavelength gratings [2224] work well in the infrared, but have not been demonstrated for visible light. Ferroelectric liquid crystal spatial light modulators (FLC SLMs) can produce arbitrary polarization states only with low efficiency [25]. Efficient and arbitrary polarization modification can be done with a pair of nematic liquid crystal SLMs [26,27], but have the highest cost.

One of the earliest and simplest techniques combines orthogonal TEM01 and TEM10 laser modes to create a beam with azimuthal symmetry [28,29], commonly called the TEM01* doughnut mode. In those works, the TEM10 and TEM01 modes are generated approximately in separate arms of a Michelson interferometer by inserting π - phase steps into the paths of the Gaussian beams in each path. While this technique is efficient and uses relatively inexpensive, commercially available components, a significant drawback is the requirement of interferometric stability.

In this work, we present a stabilized version of the interference of modes technique using polarization multiplexing on a nematic SLM and a π-phase step. Our approach eliminates the requirement on the phase stability by effectively combining two interferometer arms into a single path. While this technique cannot produce truly arbitrary 2D polarization encoding, it allows easy reconfiguration between radially, azimuthally, or hybrid vector beams. Our analysis shows that the mode purity and mode purification through spatial filtering can achieve beams with excellent cylindrical symmetry and over 98% TEM01* mode purity.

2. Experimental setup

The schematic of our technique is depicted in the Fig. 1 . To assist with the visualization, we show in Fig. 2 the relative phases of the light for x- and y-polarizations in four quadrants corresponding to points A, B, and D of the Fig. 1, and the phase function of the SLM (Boulder Nonlinear Systems Model P512) at point C. Initially our technique requires that horizontally and vertically polarized light components are equal, so a half-wave plate is used to orient the linear polarization of the 780 nm, TEM00 laser light to 45 degrees as shown in Fig. 1. 
Figure 2(a) shows the identical phases, which we chose as 0 in all four quadrants for both polarizations. The black arrows in Fig. 2 depict the phase of each polarization component in each quadrant. Also shown is the resultant polarization (third column). Figure 2(b) shows the phases after passing through the π - phase step operating on the left (x < 0) portion of the beam (see Fig. 1). The phase step affects both polarizations equally. The π - phase step was made by depositing a 390nm-thick layer of SiN on half of a quartz wafer. By adjusting the angle of the phase step plate we could use it with different optical wavelengths. Next we change the phases for the y-polarization only using the SLM as shown in the Fig. 2(c). The combined operation results in the quadrant approximation to the radial polarization field shown in the third column of Fig. 2(d). Mathematically, the generated field is

ERPF=C0exp(r2ω02)(sign(x)x^+sign(y)y^),
where C 0 = 1/πω02 is a normalization constant, and the waist, ω 0, is the 1/e2 radius of the Gaussian (TEM00) input beam. This is an approximation to the desired RPF formed by the superposition of TEM01 and TEM10 modes:
ERPFTRUE=C1exp(r2ω12)(xx^+yy^)=C1exp(r2ω12)(rcos(θ)x^+rsin(θ)y^),
where C 1 = 2/(ω12π) is the normalization constant, θ is the azimuthal coordinate, and ω 1 is the waist parameter. The lenses L1 and L2 in Fig. 1 are used to relay and register the image of the phase step onto the SLM; in principle a π/2 phase step could be used directly in front of the reflective SLM, and these lenses could be removed. The lens L3 and the CCD camera can both be moved relative to L2 either to image the SLM or the focal plane of L2. With our choice of the phase step orientation and the SLM phases shown in Fig. 2(c) we create the condition needed for RPF generation similar to that of the interference of modes approach. The main advantage of our technique is realized by keeping both polarizations in the same optical beam, which eliminates interferometric stability requirements. Overall measured efficiency here is only 50% due to uncoated lenses and imperfect SLM reflectivity.

 figure: Fig. 1

Fig. 1 Experimental setup used to produce radially and azimuthally polarized beams.

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 figure: Fig. 2

Fig. 2 Parameters required for producing radially polarized beams. Row letters (A - D) correspond to the positions in Fig. 1. Columns depict x-, y-, and combined polarizations at those points.

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To form an APF, both the SLM phase pattern and the π - phase step are rotated by 90 degrees. Figure 3 shows the phases needed to produce an APF. In analogy with the RPF:

EAPF=C0exp(r2ω02)(sign(y)x^sign(x)y^),
where C 0 = 1/πω02. In addition to the APF beam described by Eq. (3), other modes can also be produced using this technique by the appropriate phase choices in the four quadrants of the SLM. For example, to form the HE21 cylindrical waveguide mode, which alternates between radial and azimuthal polarization, either the SLM phase pattern or the π - phase step is rotated by 90 degrees. We note that one can also convert between radial, azimuthal, and hybrid modes simply by adding two standard half waveplates.

 figure: Fig. 3

Fig. 3 Parameters required for producing azimuthally polarized beams. Row letters (A - D) correspond to the positions in Fig. 1. Columns depict x-, y-, and combined polarizations at those points.

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We note that a ferroelectric liquid crystal SLM (FLC SLM) may be used in place of the nematic SLM to generate these approximate polarization profiles with one caveat. In FLC SLMs, each pixel acts as an orientable half-wave-plate. For most materials used in FLC SLMs, the maximum polarization rotation is ~90 degrees, so that the π - phase step used here is still required in order to have 180 degree polarization rotations. However, to DC balance the FLC SLM, the waveplate orientation of each pixel must oscillate on frequency scales of ~1 Hz or faster, such that the desired polarization field is only present with 50% duty cycle. For some applications this duty cycle may still be acceptable.

3. Results and theory

The resulting images in the focus of L3 with and without a polarizer present are shown in 
Fig. 4(a) , 4(b), along with calculated images. The arrows in the images show polarizer transmission axis. The images lack cylindrical symmetry because, like some of the techniques presented in the introduction, we are not combining pure TEM10 and TEM01 modes. Mode purity is calculated through the overlap integral of these target modes with our approximate fields ERPF or EAPF. By symmetry, we consider only the overlap integral of x-polarization components of Eq. (2) and Eq. (1) (the overlap integral for the APF will be identical). Specifically

ERPF*ERPFTRUEdA=0C0C1exp(r2ω02)exp(r2ω12)r2drπ/2π/22cos(θ)dθ,
where we have used the symmetry of this integral in x (x sgn(x) = |x|) to integrate over θ from –π/2 to π/2. The purity of the TEM10 mode is determined by maximizing Eq. (4) with respect to ω 1. A straightforward calculation shows that this is achieved when ω1=ω0/2, at which value the overlap integral is 8/33π = 0.87, or a mode purity of 64/27π = 0.75.

 figure: Fig. 4

Fig. 4 Experimental (top row) and calculated (bottom row) RPF images at the focus of lens L3. When the polarizer is present, its direction is shown by the arrows.

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To produce purer TEM01 modes efficiently we have to remove higher order modes selectively. To achieve this, we spatially filter the beam using a pinhole [30]. Because the π-phase step strictly eliminates the TEM00 mode contribution, and the TEM01 mode is the first higher order mode, this traditional spatial filtering is very effective at isolating the TEM01 modes. Figure 5 shows the setup used to spatially filter the beam shown in the Fig. 4(a). After the pinhole we also used an aperture to reject the diffraction rings and re-imaged the pinhole with a triplet lens L4. Experimentally, we also adjusted the aperture size in Fig. 1 to control the size of the focused beam on the pinhole to improve the final beam quality. The resultant image is shown together with the calculated image in Fig. 5. Comparing Figs. 4 and 5 it is clear that the quality of the mode in the experiment is also significantly improved. In Fig. 5, we also show an image of the beam interfered with a horizontally polarized TEM00 beam. In this case, the generated beam was an APF, so interference contrast is maximized at the top and bottom of the beam, where the polarization is purely horizontal, and is zero on the left and right sides, where polarization is vertical. Furthermore, the interferogram explicitly shows the π phase shift between the top and bottom halves of the beam as a lateral shift in the interference pattern by half a fringe period.

 figure: Fig. 5

Fig. 5 Schematic of the setup used to spatially filter the beam shown in the Fig. 4(a). The experimental and calculated final beams are shown. Also shown is the interferogram of the APF beam with the horizontally polarized TEM00 beam, and horizontally polarized APF beam before and after the pinhole.

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The TEM01 mode purity is increased by spatial filtering at the expense of power throughput. In Fig. 6 we show overlap integral and the pinhole throughput as a function of the pinhole diameter. The curves in this figure were calculated with parameters approximating experimental conditions. The solid red curve shows the overlap integral, which achieves a maximum value near 300 microns. The measured mode purities before and after the pinhole, given by the square of the overlap integral of the measured APF beam with the perfect calculated mode, are 0.85 and 0.97, respectively.

 figure: Fig. 6

Fig. 6 Calculated overlap integral and the pinhole throughput as a function of the pinhole size for optimizing the TEM01 mode. The unmodified TEM00 waist diameter was 130 microns.

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Although making the pinhole smaller generates a beam with better cylindrical symmetry, the overlap integral is reduced due to the extra diffraction rings that are generated. The aperture in Fig. 5 eliminates these and, therefore, the overlap integral with the “useful” portion of the beam continues to increase with smaller pinhole size (dashed red line). For scale we point out that the beam waist diameter for an unmodified TEM00 beam propagating through the system (i.e. no phase step or SLM present) is ~130 microns, so that the pinhole size is approximately 2.3 times larger than the beam at the peak of the solid red curve. Also shown in Fig. 6 is the relative power transmitted through the pinhole (dashed blue line). In practice, overall power throughput is also reduced by the reflection efficiency of SLM, which in our case was 60% at 780 nm.

4. Conclusion

In summary we demonstrated an improved interference of modes technique that does not require interferometric stability, is reconfigurable and could be used at different wavelengths. We also demonstrated a technique that can be used to improve the mode quality by stripping away higher order modes. Experimental results agree well with the calculations.

Acknowledgments

This work was supported by the Office of Naval Research.

References and links

1. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959). [CrossRef]  

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

3. D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express 9(10), 490–497 (2001), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-10-490. [CrossRef]   [PubMed]  

4. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

5. Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-7-324. [PubMed]  

6. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-2-77. [CrossRef]   [PubMed]  

7. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997). [CrossRef]  

8. K. T. Gahagan and G. A. Swartzlander Jr., “Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap,” J. Opt. Soc. Am. B 16(4), 533–537 (1999). [CrossRef]  

9. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999). [CrossRef]  

10. A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33(15), 1817–1822 (2000). [CrossRef]  

11. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000). [CrossRef]   [PubMed]  

12. 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]  

13. J. Azoulay, A. Debarre, R. Jaffiol, and P. Tchenio, “Original tools for single-molecule spectroscopy,” Single Mol. 2(4), 241–249 (2001). [CrossRef]  

14. F. Treussart, R. Alléaume, V. Le Floc’h, L. T. Xiao, J. M. Courty, and J. F. Roch, “Direct measurement of the photon statistics of a triggered single photon source,” Phys. Rev. Lett. 89(9), 093601 (2002). [CrossRef]   [PubMed]  

15. C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams,” Appl. Phys. B 74, S83–S88 (2002). [CrossRef]  

16. K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75–85 (2000). [CrossRef]  

17. E. Descrovi, L. Vaccaro, L. Aeschimann, W. Nakagawa, U. Staufer, and H. P. Herzig, “Optical properties of microfabricated fully-metal coated near-field probes in collection mode,” J. Opt. Soc. Am. A 22(7), 1432–1441 (2005). [CrossRef]  

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

19. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21(23), 1948–1950 (1996). [CrossRef]   [PubMed]  

20. S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34(16), 2525–2527 (2009). [CrossRef]   [PubMed]  

21. T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203(1-2), 1–5 (2002). [CrossRef]  

22. U. D. Zeitner, B. Schnabel, E. B. Kley, and F. Wyrowski, “Polarization multiplexing of diffractive elements with metal-stripe grating pixels,” Appl. Opt. 38(11), 2177–2181 (1999). [CrossRef]   [PubMed]  

23. Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating,” Appl. Phys. Lett. 79(11), 1587–1589 (2001). [CrossRef]  

24. G. M. Lerman and U. Levy, “Space-variant subwavelength periodic element at a wavelength of 1064 nm,” Opt. Lett. 33, 22782–22784 (2008).

25. M. A. A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27(21), 1929–1931 (2002). [CrossRef]   [PubMed]  

26. M. R. Beversluis, L. Novotny, and S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14(7), 2650–2656 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2650. [CrossRef]   [PubMed]  

27. J. A. Davis, D. E. McNamara, D. M. Cottrell, and T. Sonehara, “Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator,” Appl. Opt. 39(10), 1549–1554 (2000). [CrossRef]   [PubMed]  

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

29. N. Passilly, R. de Saint Denis, K. Aït-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 22(5), 984–991 (2005). [CrossRef]  

30. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32(11), 1468–1470 (2007). [CrossRef]   [PubMed]  

References

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  • |

  1. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
    [Crossref]
  2. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
    [Crossref]
  3. D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express 9(10), 490–497 (2001), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-10-490 .
    [Crossref] [PubMed]
  4. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
  5. Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-7-324 .
    [PubMed]
  6. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-2-77 .
    [Crossref] [PubMed]
  7. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
    [Crossref]
  8. K. T. Gahagan and G. A. Swartzlander., “Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap,” J. Opt. Soc. Am. B 16(4), 533–537 (1999).
    [Crossref]
  9. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999).
    [Crossref]
  10. A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33(15), 1817–1822 (2000).
    [Crossref]
  11. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
    [Crossref] [PubMed]
  12. 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]
  13. J. Azoulay, A. Debarre, R. Jaffiol, and P. Tchenio, “Original tools for single-molecule spectroscopy,” Single Mol. 2(4), 241–249 (2001).
    [Crossref]
  14. F. Treussart, R. Alléaume, V. Le Floc’h, L. T. Xiao, J. M. Courty, and J. F. Roch, “Direct measurement of the photon statistics of a triggered single photon source,” Phys. Rev. Lett. 89(9), 093601 (2002).
    [Crossref] [PubMed]
  15. C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams,” Appl. Phys. B 74, S83–S88 (2002).
    [Crossref]
  16. K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75–85 (2000).
    [Crossref]
  17. E. Descrovi, L. Vaccaro, L. Aeschimann, W. Nakagawa, U. Staufer, and H. P. Herzig, “Optical properties of microfabricated fully-metal coated near-field probes in collection mode,” J. Opt. Soc. Am. A 22(7), 1432–1441 (2005).
    [Crossref]
  18. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [Crossref] [PubMed]
  19. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21(23), 1948–1950 (1996).
    [Crossref] [PubMed]
  20. S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34(16), 2525–2527 (2009).
    [Crossref] [PubMed]
  21. T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203(1-2), 1–5 (2002).
    [Crossref]
  22. U. D. Zeitner, B. Schnabel, E. B. Kley, and F. Wyrowski, “Polarization multiplexing of diffractive elements with metal-stripe grating pixels,” Appl. Opt. 38(11), 2177–2181 (1999).
    [Crossref] [PubMed]
  23. Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating,” Appl. Phys. Lett. 79(11), 1587–1589 (2001).
    [Crossref]
  24. G. M. Lerman and U. Levy, “Space-variant subwavelength periodic element at a wavelength of 1064 nm,” Opt. Lett. 33, 22782–22784 (2008).
  25. M. A. A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27(21), 1929–1931 (2002).
    [Crossref] [PubMed]
  26. M. R. Beversluis, L. Novotny, and S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14(7), 2650–2656 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2650 .
    [Crossref] [PubMed]
  27. J. A. Davis, D. E. McNamara, D. M. Cottrell, and T. Sonehara, “Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator,” Appl. Opt. 39(10), 1549–1554 (2000).
    [Crossref] [PubMed]
  28. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 (1990).
    [Crossref] [PubMed]
  29. N. Passilly, R. de Saint Denis, K. Aït-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 22(5), 984–991 (2005).
    [Crossref]
  30. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32(11), 1468–1470 (2007).
    [Crossref] [PubMed]

2009 (1)

2008 (1)

G. M. Lerman and U. Levy, “Space-variant subwavelength periodic element at a wavelength of 1064 nm,” Opt. Lett. 33, 22782–22784 (2008).

2007 (1)

2006 (1)

2005 (2)

2003 (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]

2002 (5)

F. Treussart, R. Alléaume, V. Le Floc’h, L. T. Xiao, J. M. Courty, and J. F. Roch, “Direct measurement of the photon statistics of a triggered single photon source,” Phys. Rev. Lett. 89(9), 093601 (2002).
[Crossref] [PubMed]

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams,” Appl. Phys. B 74, S83–S88 (2002).
[Crossref]

Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-7-324 .
[PubMed]

M. A. A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27(21), 1929–1931 (2002).
[Crossref] [PubMed]

T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203(1-2), 1–5 (2002).
[Crossref]

2001 (5)

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating,” Appl. Phys. Lett. 79(11), 1587–1589 (2001).
[Crossref]

D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express 9(10), 490–497 (2001), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-10-490 .
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

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]

J. Azoulay, A. Debarre, R. Jaffiol, and P. Tchenio, “Original tools for single-molecule spectroscopy,” Single Mol. 2(4), 241–249 (2001).
[Crossref]

2000 (5)

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33(15), 1817–1822 (2000).
[Crossref]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[Crossref] [PubMed]

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75–85 (2000).
[Crossref]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-2-77 .
[Crossref] [PubMed]

J. A. Davis, D. E. McNamara, D. M. Cottrell, and T. Sonehara, “Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator,” Appl. Opt. 39(10), 1549–1554 (2000).
[Crossref] [PubMed]

1999 (3)

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

1996 (1)

1990 (1)

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[Crossref]

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

Aeschimann, L.

Aït-Ameur, K.

Alléaume, R.

F. Treussart, R. Alléaume, V. Le Floc’h, L. T. Xiao, J. M. Courty, and J. F. Roch, “Direct measurement of the photon statistics of a triggered single photon source,” Phys. Rev. Lett. 89(9), 093601 (2002).
[Crossref] [PubMed]

Azoulay, J.

J. Azoulay, A. Debarre, R. Jaffiol, and P. Tchenio, “Original tools for single-molecule spectroscopy,” Single Mol. 2(4), 241–249 (2001).
[Crossref]

Beversluis, M. R.

M. R. Beversluis, L. Novotny, and S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14(7), 2650–2656 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2650 .
[Crossref] [PubMed]

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]

Biss, D. P.

Bomzon, Z.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating,” Appl. Phys. Lett. 79(11), 1587–1589 (2001).
[Crossref]

Brown, T. G.

Cottrell, D. M.

Courjon, D.

T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203(1-2), 1–5 (2002).
[Crossref]

Courty, J. M.

F. Treussart, R. Alléaume, V. Le Floc’h, L. T. Xiao, J. M. Courty, and J. F. Roch, “Direct measurement of the photon statistics of a triggered single photon source,” Phys. Rev. Lett. 89(9), 093601 (2002).
[Crossref] [PubMed]

Davis, J. A.

de Saint Denis, R.

Debarre, A.

J. Azoulay, A. Debarre, R. Jaffiol, and P. Tchenio, “Original tools for single-molecule spectroscopy,” Single Mol. 2(4), 241–249 (2001).
[Crossref]

Descrovi, E.

Dorn, R.

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Ford, D. H.

Gahagan, K. T.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Grosjean, T.

T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203(1-2), 1–5 (2002).
[Crossref]

Hasman, E.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating,” Appl. Phys. Lett. 79(11), 1587–1589 (2001).
[Crossref]

Hecht, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[Crossref] [PubMed]

Herzig, H. P.

Hierle, R.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Jackel, S.

Jaffiol, R.

J. Azoulay, A. Debarre, R. Jaffiol, and P. Tchenio, “Original tools for single-molecule spectroscopy,” Single Mol. 2(4), 241–249 (2001).
[Crossref]

Juškaitis, R.

Kimura, W. D.

Kleiner, V.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating,” Appl. Phys. Lett. 79(11), 1587–1589 (2001).
[Crossref]

Kley, E. B.

Kristensen, P.

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Le Floc’h, V.

F. Treussart, R. Alléaume, V. Le Floc’h, L. T. Xiao, J. M. Courty, and J. F. Roch, “Direct measurement of the photon statistics of a triggered single photon source,” Phys. Rev. Lett. 89(9), 093601 (2002).
[Crossref] [PubMed]

Leger, J. R.

Lerman, G. M.

G. M. Lerman and U. Levy, “Space-variant subwavelength periodic element at a wavelength of 1064 nm,” Opt. Lett. 33, 22782–22784 (2008).

Leuchs, G.

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Levy, U.

G. M. Lerman and U. Levy, “Space-variant subwavelength periodic element at a wavelength of 1064 nm,” Opt. Lett. 33, 22782–22784 (2008).

Lumer, Y.

Machavariani, G.

Massoumian, F.

McNamara, D. E.

Meir, A.

Moshe, I.

Nakagawa, W.

Neil, M. A. A.

Nesterov, A. V.

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33(15), 1817–1822 (2000).
[Crossref]

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999).
[Crossref]

Niziev, V. G.

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33(15), 1817–1822 (2000).
[Crossref]

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999).
[Crossref]

Novotny, L.

M. R. Beversluis, L. Novotny, and S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14(7), 2650–2656 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2650 .
[Crossref] [PubMed]

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]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[Crossref] [PubMed]

Passilly, N.

Piché, M.

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams,” Appl. Phys. B 74, S83–S88 (2002).
[Crossref]

Quabis, S.

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Ramachandran, S.

Richards, B.

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

Roch, J. F.

F. Treussart, R. Alléaume, V. Le Floc’h, L. T. Xiao, J. M. Courty, and J. F. Roch, “Direct measurement of the photon statistics of a triggered single photon source,” Phys. Rev. Lett. 89(9), 093601 (2002).
[Crossref] [PubMed]

Roch, J.-F.

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Schadt, M.

Schnabel, B.

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Sick, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[Crossref] [PubMed]

Sonehara, T.

Spajer, M.

T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203(1-2), 1–5 (2002).
[Crossref]

Stalder, M.

Staufer, U.

Stranick, S. J.

Swartzlander, G. A.

Tchenio, P.

J. Azoulay, A. Debarre, R. Jaffiol, and P. Tchenio, “Original tools for single-molecule spectroscopy,” Single Mol. 2(4), 241–249 (2001).
[Crossref]

Tidwell, S. C.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Treussart, F.

N. Passilly, R. de Saint Denis, K. Aït-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 22(5), 984–991 (2005).
[Crossref]

F. Treussart, R. Alléaume, V. Le Floc’h, L. T. Xiao, J. M. Courty, and J. F. Roch, “Direct measurement of the photon statistics of a triggered single photon source,” Phys. Rev. Lett. 89(9), 093601 (2002).
[Crossref] [PubMed]

Vaccaro, L.

Varin, C.

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams,” Appl. Phys. B 74, S83–S88 (2002).
[Crossref]

Wilson, T.

Wolf, E.

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[Crossref]

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

Wyrowski, F.

Xiao, L. T.

F. Treussart, R. Alléaume, V. Le Floc’h, L. T. Xiao, J. M. Courty, and J. F. Roch, “Direct measurement of the photon statistics of a triggered single photon source,” Phys. Rev. Lett. 89(9), 093601 (2002).
[Crossref] [PubMed]

Yan, M. F.

Youngworth, K. S.

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]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-2-77 .
[Crossref] [PubMed]

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75–85 (2000).
[Crossref]

Zeitner, U. D.

Zhan, Q.

Appl. Opt. (3)

Appl. Phys. B (2)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams,” Appl. Phys. B 74, S83–S88 (2002).
[Crossref]

Appl. Phys. Lett. (1)

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating,” Appl. Phys. Lett. 79(11), 1587–1589 (2001).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

J. Phys. D (2)

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999).
[Crossref]

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33(15), 1817–1822 (2000).
[Crossref]

Opt. Commun. (1)

T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203(1-2), 1–5 (2002).
[Crossref]

Opt. Express (4)

Opt. Lett. (5)

Phys. Rev. Lett. (5)

F. Treussart, R. Alléaume, V. Le Floc’h, L. T. Xiao, J. M. Courty, and J. F. Roch, “Direct measurement of the photon statistics of a triggered single photon source,” Phys. Rev. Lett. 89(9), 093601 (2002).
[Crossref] [PubMed]

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

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[Crossref] [PubMed]

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]

Proc. R. Soc. Lond. A Math. Phys. Sci. (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[Crossref]

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

Proc. SPIE (1)

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75–85 (2000).
[Crossref]

Single Mol. (1)

J. Azoulay, A. Debarre, R. Jaffiol, and P. Tchenio, “Original tools for single-molecule spectroscopy,” Single Mol. 2(4), 241–249 (2001).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Experimental setup used to produce radially and azimuthally polarized beams.
Fig. 2
Fig. 2 Parameters required for producing radially polarized beams. Row letters (A - D) correspond to the positions in Fig. 1. Columns depict x-, y-, and combined polarizations at those points.
Fig. 3
Fig. 3 Parameters required for producing azimuthally polarized beams. Row letters (A - D) correspond to the positions in Fig. 1. Columns depict x-, y-, and combined polarizations at those points.
Fig. 4
Fig. 4 Experimental (top row) and calculated (bottom row) RPF images at the focus of lens L3. When the polarizer is present, its direction is shown by the arrows.
Fig. 5
Fig. 5 Schematic of the setup used to spatially filter the beam shown in the Fig. 4(a). The experimental and calculated final beams are shown. Also shown is the interferogram of the APF beam with the horizontally polarized TEM00 beam, and horizontally polarized APF beam before and after the pinhole.
Fig. 6
Fig. 6 Calculated overlap integral and the pinhole throughput as a function of the pinhole size for optimizing the TEM01 mode. The unmodified TEM00 waist diameter was 130 microns.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

E R P F = C 0 exp ( r 2 ω 0 2 ) ( s i g n ( x ) x ^ + s i g n ( y ) y ^ ) ,
E R P F T R U E = C 1 exp ( r 2 ω 1 2 ) ( x x ^ + y y ^ ) = C 1 exp ( r 2 ω 1 2 ) ( r cos ( θ ) x ^ + r sin ( θ ) y ^ ) ,
E A P F = C 0 exp ( r 2 ω 0 2 ) ( s i g n ( y ) x ^ s i g n ( x ) y ^ ) ,
E R P F * E R P F T R U E d A = 0 C 0 C 1 exp ( r 2 ω 0 2 ) exp ( r 2 ω 1 2 ) r 2 d r π / 2 π / 2 2 cos ( θ ) d θ ,

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