Abstract

We develop a method for generating focused vector beams with circular polarization at any transverse plane. Based on the Richards-Wolf vector model, we derive analytical expressions to describe the propagation of these set of beams near the focal area. Since the polarization and the amplitude of the input beam are not uniform, an interferometric system capable of generating spatially-variant polarized beams has to be used. In particular, this wavefront is manipulated by means of spatial light modulators displaying computer generated holograms and subsequently focused using a high numerical aperture objective lens. Experimental results using a NA = 0.85 system are provided: irradiance and Stokes images of the focused field at different planes near the focal plane are presented and compared with those obtained by numerical simulation.

© 2014 Optical Society of America

1. Introduction

The propagation of electromagnetic field distributions generated at the focal region has been extensively investigated in the last years [18]. Non-paraxial fields have demonstrated very useful in many fields for instance in high-resolution microscopy, particle trapping, high-density recording, tomography, electron acceleration, nonlinear optics, and optical tweezers [9]. Beam shaping in the focal area of a high numerical aperture objective lens requires a careful design of the input wavefront. In particular, full control of the complex amplitude and polarization distributions of the paraxial input field is required to generate focused fields adapted to the requirements of a specific problem [10]. Interestingly, several authors described inverse methods to find the pupil function from a predetermined field distribution in the focal area [11, 12]. Light shaping can be accomplished by using an optical setup able to generate beams with arbitrary polarization and shape distributions at a given plane. This is usually carried out by means of interferometric systems in combination with spatial light modulators and digital holography [1321]. The objective of this paper is to present a method for designing focused fields with transverse circular polarization at any plane. Among many others applications, circularly polarized tight focused beams are useful in resolution improvement [2224], third harmonic generation-based microscopy [2527], plasmonics and nano-optics applications [28, 29], optical activity and chemical related problems [30,31] or, conical refraction [32].

Using the Richards-Wolf vector diffraction theory, we derive analytical expressions to describe the propagation of these set of beams near the focal area. Complex amplitude and polarization of the input beam are manipulated by means of spatial light modulators (SLM) displaying computer generated holograms. Numerical calculations and experimental results are compared and analyzed. Accordingly, the paper is organized as follows: in section 2 we derive the equations for describing circularly-polarized focused fields at any transverse plane. The experimental setup and the holographic procedure required to synthesize the beam are reviewed in section 3. Experimental results including irradiance images and polarization analysis are presented in section 4. Finally, the main conclusions are summarized in section 5.

2. Circularly-polarized highly focused beams

The electromagnetic field in the focal area of a high numerical aperture objective lens that obeys the sine condition is described by the Richards-Wolf vector equation [33]

E(r,ϕ,z)=A0θ002πcosθ[f1(θ,φ)e1(φ)+f2(θ,φ)e2o(θ,φ)]×eikrsinθcos(ϕφ)eikzcosθsinθdθdφ,
where A is a constant, r, ϕ and z are the coordinates in the focal area, and angles φ and θ are the coordinates at the exit pupil; note that θ0 is the semi-aperture angle. Functions f1 (θ, φ) and f2 (θ, φ) are the azimuthal and radial components of incident field respectively,
f1(θ,φ)=ES(θ,φ)e1(φ)
f2(θ,φ)=ES(θ,φ)e2i(φ),
where ES(θ, φ) = (ESx, ESy, 0) is the input beam considered transverse and the dot stands for the inner product. Vectors e1 and e2i are unit vectors in the radial and azimuthal directions whereas e2o is the projection of e2i on the convergent wavefront surface, as shown in Fig. 1. This figure shows the geometrical variables used throughout this paper at different reference surfaces. Vectors e1, e2i and e2o are given by
e1(φ)=(sinφ,cosφ,0)
e2i(φ)=(cosφ,sinφ,0)
e20(θ,φ)=(cosθcosφ,cosθsinφ,sinθ).

To analyze the polarization structure of E(r, ϕ, z), an alternative base of mutually perpendicular unit vectors u+, u and uz is used

u±=12(1,±i,0)uz=(0,0,1),
thus E(r, ϕ, z) = E+ (r, ϕ, z)u+ + E (r, ϕ, z)u + Ez (r, ϕ, z)uz. According to Eq. (1), components (E+, E, Ez) read
E±(r,ϕ,z)=A20θ002πcosθ[if1(θ,φ)+cosθf2(θ,φ)]××eikrsinθcos(ϕφ)eikzcosθeiφsinθdθdφ
E±(r,ϕ,z)=A0θ002πcosθsinθf2(θ,φ)eikrsinθcos(ϕφ)eikzcosθsinθdθdφ.
Notice that E± represent the right (+) and left (−) circular content of the transverse field at the vicinity of the focus plane and Ez is the magnitude of the longitudinal component. Since our goal is to generate a focused field whose transverse component is circularly polarized at any plane z, either E+ or E has to be zero. This condition is fulfilled when f1 (θ, φ) = ±i cosθ f2 (θ, φ), which is equivalent to
f1(θ,φ)=±icosθg(θ,φ)
f2(θ,φ)=g(θ,φ)
where, g(θ, φ) is an arbitrary function. Additional characteristics of the global field can be obtained by choosing a suitable function g(θ, φ). For example, to obtain a non-zero longitudinal component at the axis implies that g(θ, φ) = g(θ).

 

Fig. 1 Notation and sketch of a highly focused optical system.

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In what follows and without loss of generality we choose the plus sign, i.e. we deal with right handed circularly polarized fields. For this kind of incident beams the transverse and longitudinal components of E(r, ϕ, z) become

E+(r,ϕ,z)=2A20θ002πcosθcosθg(θ)eiφeiksinθrcos(ϕφ)eikzcosθsinθdθdφ
E(r,ϕ,z)=0
Ez(r,ϕ,z)=A0θ002πcosθsinθg(θ)eiksinθrcos(ϕφ)eikzcosθsinθdθdφ.
Integrating over φ, field components E+ and Ez take the form
E+(r,ϕ,z)=i4πA2eiϕ0θ0cosθcosθg(θ)J1(krsinθ)eikzcosθsinθdθ
Ez(r,z)=2πA0θ0cosθsinθg(θ)J0(krsinθ)eikzcosθsinθdθ,
where J0(x) and J1(x) are the first kind Bessel functions of order 0 and 1 respectively. Interestingly, E+ presents topological charge e and |E+|2 and |Ez|2 show circular symmetry.

Figure 2 show irradiance maps for (a) the transverse component |E+|2, (b) the longitudinal component |Ez|2 and (c) the total field I = |E+|2 +|Ez|2 when a microscope objective NA = 0.85 (θ0 ≈ 1 rad) is used. The illumination is assumed to be Gaussian i.e. g(θ)=exp(1f0sinθsinθ0)2 and the filling factor f0 is set to 1. Notice that |Ez|2 presents high values at z = 0 and drops very fast out of the focal plane. On the other hand, |E+|2 = 0 at r = 0 at any plane z. Figure 2(d) shows the profiles of the total irradiance I at different distances from the focal plane (z = 0, z = −3λ, z = −5λ and z = −7λ).

 

Fig. 2 Irradiance maps for a circularly-polarized highly-focused beam (NA = 0.85): (a) |E+|2, (b) |Ez|2 and (c) I. (d) Profiles of I at z = 0 (red) z = −3λ (blue), z = −5λ (magenta) and z = −7λ (black).

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3. Synthesis of beam ES

Figure 3 depicts an experimental setup based on a Mach-Zehnder interferometer able to generate arbitrary spatially-variant polarized focused beams. An extended explanation on how this procedure can be used to generate beams with arbitrary polarization and shape can be found in [17]. A linearly polarized input beam Ein is split into two beams by means of polarizing beam splitter PBS1. Reflected by mirrors M1 or M2 the split beam (Ein1 or Ein2) passes through wave plates HWP and QWP which rotate the oscillating plane and set the modulator to the required desired modulation curve. Then, light passes through a translucent SLM (Holoeye HEO 0017) displaying cell-based double-pixel holograms to encode complex transmittances Cx(x, y) and Cy(x, y) [34].

 

Fig. 3 Sketch of the optical setup. Light source: HeNe laser λ= 633nm; P1, P2 and P3: linear polarizers; PBS1 and PBS2; polarizing beam splitters; M1 and M2: mirrors; HWP: half-wave plate: QWP: quarter-wave plate; SLM1 and SLM2: spatial light modulators; L1, L2 and L3: lenses; BS: beam splitter; MO: microscope objective.

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Precise alignment of the different optical components is required, especially a good match between the corresponding pixels of the two SLMs. This is carried out during the set up procedure by displaying the same distribution on SLM1 and SLM2 and imaging these scenes on camera 1. Note that both displays are controlled independently. Then, the scene displayed on one of the screens is shifted until a perfect match with the other one is accomplished. Shift values are used later to adapt the holograms displayed on both SLMs.

These beams are subsequently recombined by means of polarizing beam splitter PBS2 and fed into a 4f system. A spatial filter removes higher-order terms whereas allowing pass the synthesized field ES. The irradiance of this beam can be observed by means of camera 1. Afterward, ES is focused by means of a high numerical aperture microscope objective (MO) NA = 0.85. The beam in the focal area is reflected on a glass surface and imaged on camera 2. Polarization analysis is carried out by placing a polarizer (and a quarter-wave plate if required) next to camera 2.

According to Eqs. (2) and (6), the synthesized beam ES has to be

ES=(cosφicosθsinφ)g(θ)ex+(sinφ+icosθcosφ)g(θ)ey
where ex and ey are orthogonal Cartesian unit vectors as shown in Fig. 1. In order to synthesize ES the following complex valued distributions are coded on each SLM
Cx(ρ,φ)=cosφi1ρ2sinφ
Cy(ρ,φ)=sinφ+i1ρ2sinφ.
It is assumed that the radius of the entrance pupil is set to 1 and ρ = sinθ is the radial distance from the optical axis at the entrance pupil plane (see Fig. 1).

Figure 4 is a polar diagram displaying the values of the complex plane accessible by the codification method (gray small dots) and the set of physically accessible values by modulator SLM1 (red dots). A certain value C can be accessed as a combination of phasors ML and MR, that belong to the modulation curve, and EL and ER, that are diffracted off-axis and removed by the spatial filter at the focal plane of lens L1. As shown in this Figure, not all values of the complex plane are accessed by the encoding procedure. This drawback could be overcome using a light source with a shorter wavelength to improve the modulation response of the SLM [35]. However, if the subset of accessible values C within the circle of transmittance T = 0.3 is used (see inset in Fig. 4), almost any complex transmittance can be generated. Non accessible values are approximated to the closest one belonging to the subset.

 

Fig. 4 A certain complex value C is generated as a combination of phasors ML and MR (that belong to the modulation response curve), and EL and ER that are diffracted off-axis and removed. The inset shows the subset of C values used to generate the holograms.

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4. Experimental results

As explained in the previous section, the synthesized beam ES is focused by means of the objective lens and subsequently reflected on the cover slip, back-propagated through the objective and imaged on camera 2 aided by lens L2. The cover slip (observation plane) is mounted on a stage that enables to modify the observation distance z. Figure 5 (first row) shows the irradiance I at z = −3.5λ, −5λ and −7λ. Distance z is estimated by comparing the angular average of the experimental images with the numerical evaluation of I and |E+|2 (Eqs. (8a) and (8b)). These curves are presented in the second row of Fig. 5. Notice that the irradiance at the focal plane z = 0 is not analyzed due to lack of accuracy along the z-axis and insufficient resolution of the camera. Furthermore, the sudden increase in irradiance around the focal plane complicates the analysis, because camera is saturated.

 

Fig. 5 Experimental results at the observation plane: the first row corresponds to the image captured by camera 2. These images are normalized to its corresponding maximum. The second row shows the profile of the experimental images (black dots) and the numeric evaluation of I(red solid line) for z = −3.5λ, −5λ and −7λ.

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To analyze the polarization of the focused beam a measure of the Stokes parameters has been carried out. These parameters are obtained according to

S0=I(0°,0)+I(90°,0)
S1=I(0°,0)I(90°,0)
S2=I(45°,0)I(135°,0)
S3=I(45°,π/2)I(135°,π/2),
where I(α, β) stands for the recorded intensity when a polarizer is set at an angle α with respect to the x direction in front of camera 2; β is the retardation between the x and y directions [36]. Retardation β = π/2 is accomplished by using also a quarter wave plate. Once the Stokes parameters are found in each point of the beam, the polarization map can be generated. Figure 6 show the Stokes images S0, S1, S2, S3 for the focused field at z = −3.5λ. Notice that the values of images S1 and S2 are very close to zero, whereas the high values present in S3 demonstrates that the field is circularly polarized.

 

Fig. 6 Stokes images of the focused field at z = −3.5λ.

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In order to provide global parameters to describe the polarization of the whole beam, the following cumulative values Si are introduced:

Si2=Si2(k,l)S02(k,l)i=1,2,3
where (k, l) are the indexes of the pixels of the Stokes image. Table 1 shows the values of S1, S2, S3 for the three positions of the observation plane considered. A clear circular character of the beam along the z-axis is recognized since S32S12+S22 at any transverse plane z.

Tables Icon

Table 1. Si values for the three transverse planes z analyzed.

5. Conclusions

In this paper, a method for generating highly focused beams with circular polarization at any plane is presented. Using the Richards-Wolf diffraction formalism analytical expressions have been developed to design such fields. The analysis of the field in the focal area shows that the irradiance of the longitudinal component present very high values. The use of an interferometric setup for generating beams with arbitrary polarization combined with the use of digital holography techniques has enabled the experimental generation of such beams. Satisfactory practical results have been obtained showing a good agreement between theoretical predictions and the experimental behavior of the beam.

Acknowledgments

This work was funded in part by the Spanish Ministerio de Ciencia e Innovación, project FIS2010-17543.

References and links

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

2. N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29, 1318–1320 (2004). [CrossRef]   [PubMed]  

3. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14, 11277–11291 (2006). [CrossRef]   [PubMed]  

4. Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24, 1793–1798 (2007). [CrossRef]  

5. 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,” Nature Photon. 2, 501–505 (2008). [CrossRef]  

6. G. Lerman and U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (2008). [CrossRef]   [PubMed]  

7. X. Hao, C. Kuang, T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35, 3928–3930 (2010). [CrossRef]   [PubMed]  

8. S. N. Khonina and S. G. Volotovsky, “Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures,” J. Opt. Soc. Am. A 27, 2188–2197 (2010). [CrossRef]  

9. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009). [CrossRef]  

10. R. Martinez-Herrero, I. Juvells, and A. Carnicer, “On the physical realizability of highly focused electromagnetic field distributions,” Opt. Lett. 38, 2065–2067 (2013). [CrossRef]   [PubMed]  

11. M. R. Foreman, S. S. Sherif, P. R. T. Munro, and P. Török, “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express 16, 4901–4917 (2008). [CrossRef]   [PubMed]  

12. K. Jahn and N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields”, Opt. Commun. 288, 13–16 (2013). [CrossRef]  

13. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007). [CrossRef]  

14. H.-T. Wang, X.-L. Wang, Y. Li, J. Chen, C.-S. Guo, and J. Ding, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010). [CrossRef]   [PubMed]  

15. I. Moreno, C. Iemmi, J. Campos, and M. Yzuel, “Jones matrix treatment for optical Fourier processors with structured polarization,” Opt. Express 19, 4583–4594 (2011). [CrossRef]   [PubMed]  

16. F. Kenny, D. Lara, O. G. Rodríguez-Herrera, and C. Dainty, “Complete polarization and phase control for focus-shaping in high-na microscopy,” Opt. Express 20, 14015–14029 (2012). [CrossRef]   [PubMed]  

17. D. Maluenda, I. Juvells, R. Martínez-Herrero, and A. Carnicer, “Reconfigurable beams with arbitrary polarization and shape distributions at a given plane,” Opt. Express 21, 5432–5439 (2013). [CrossRef]   [PubMed]  

18. W. Han, Y. Yang, W. Cheng, and Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21, 20692–20706 (2013). [CrossRef]   [PubMed]  

19. E. H. Waller and G. von Freymann, “Independent spatial intensity, phase and polarization distributions,” Opt. Express 21, 28167–28174 (2013). [CrossRef]  

20. Z.-Y. Rong, Y.-J. Han, S.-Z. Wang, and C.-S Guo, “Generation of arbitrary vector beams with cascaded liquid crystal spatial light modulators,” Opt. Express 22, 1636–1644 (2014). [CrossRef]   [PubMed]  

21. C.-S. Guo, Z.-Y. Rong, and S.-Z. Wang, “Double-channel vector spatial light modulator for generation of arbitrary complex vector beams,” Opt. Lett. 39, 386–389 (2014). [CrossRef]   [PubMed]  

22. G. Brakenhoff, P. Blom, and P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979). [CrossRef]  

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

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

25. D. Biss and T. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28, 923–925 (2003). [CrossRef]   [PubMed]  

26. D. Oron, E. Tal, and Y. Silberberg, “Depth-resolved multiphoton polarization microscopy by third-harmonic generation,” Opt. Lett. 28, 2315–2317 (2003). [CrossRef]   [PubMed]  

27. O. Masihzadeh, P. Schlup, and R. A. Bartels, “Enhanced spatial resolution in third-harmonic microscopy through polarization switching,” Opt. Lett. 34, 1240–1242 (2009). [CrossRef]   [PubMed]  

28. Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008). [CrossRef]   [PubMed]  

29. L. Vuong, A. Adam, J. Brok, P. Planken, and H. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104, 083903 (2010). [CrossRef]   [PubMed]  

30. L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University, 2004). [CrossRef]  

31. Y. Inoue and V. Ramamurthy, Chiral Photochemistry (CRC, 2004).

32. A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Multiple rings formation in cascaded conical refraction,” Opt. Lett. 38, 1455–1457 (2013). [CrossRef]   [PubMed]  

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

34. V. Arrizón, L. González, R. Ponce, and A. Serrano-Heredia, “Computer-generated holograms with optimum bandwidths obtained with twisted-nematic liquid-crystal displays,” Appl. Opt. 44, 1625–1634 (2005). [CrossRef]   [PubMed]  

35. V. Arrizón, “Complex modulation with a twisted-nematic liquid-crystal spatial light modulator: double-pixel approach,” Opt. Lett. 28, 1359–1361 (2003). [CrossRef]   [PubMed]  

36. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999). [CrossRef]  

References

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  1. R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev, Lett. 91, 233901 (2003).
    [CrossRef]
  2. N. Davidson, N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29, 1318–1320 (2004).
    [CrossRef] [PubMed]
  3. M. Leutenegger, R. Rao, R. A. Leitgeb, T. Lasser, “Fast focus field calculations,” Opt. Express 14, 11277–11291 (2006).
    [CrossRef] [PubMed]
  4. Y. Kozawa, S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24, 1793–1798 (2007).
    [CrossRef]
  5. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
    [CrossRef]
  6. G. Lerman, U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (2008).
    [CrossRef] [PubMed]
  7. X. Hao, C. Kuang, T. Wang, X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35, 3928–3930 (2010).
    [CrossRef] [PubMed]
  8. S. N. Khonina, S. G. Volotovsky, “Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures,” J. Opt. Soc. Am. A 27, 2188–2197 (2010).
    [CrossRef]
  9. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
    [CrossRef]
  10. R. Martinez-Herrero, I. Juvells, A. Carnicer, “On the physical realizability of highly focused electromagnetic field distributions,” Opt. Lett. 38, 2065–2067 (2013).
    [CrossRef] [PubMed]
  11. M. R. Foreman, S. S. Sherif, P. R. T. Munro, P. Török, “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express 16, 4901–4917 (2008).
    [CrossRef] [PubMed]
  12. K. Jahn, N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields”, Opt. Commun. 288, 13–16 (2013).
    [CrossRef]
  13. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
    [CrossRef]
  14. H.-T. Wang, X.-L. Wang, Y. Li, J. Chen, C.-S. Guo, J. Ding, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
    [CrossRef] [PubMed]
  15. I. Moreno, C. Iemmi, J. Campos, M. Yzuel, “Jones matrix treatment for optical Fourier processors with structured polarization,” Opt. Express 19, 4583–4594 (2011).
    [CrossRef] [PubMed]
  16. F. Kenny, D. Lara, O. G. Rodríguez-Herrera, C. Dainty, “Complete polarization and phase control for focus-shaping in high-na microscopy,” Opt. Express 20, 14015–14029 (2012).
    [CrossRef] [PubMed]
  17. D. Maluenda, I. Juvells, R. Martínez-Herrero, A. Carnicer, “Reconfigurable beams with arbitrary polarization and shape distributions at a given plane,” Opt. Express 21, 5432–5439 (2013).
    [CrossRef] [PubMed]
  18. W. Han, Y. Yang, W. Cheng, Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21, 20692–20706 (2013).
    [CrossRef] [PubMed]
  19. E. H. Waller, G. von Freymann, “Independent spatial intensity, phase and polarization distributions,” Opt. Express 21, 28167–28174 (2013).
    [CrossRef]
  20. Z.-Y. Rong, Y.-J. Han, S.-Z. Wang, C.-S Guo, “Generation of arbitrary vector beams with cascaded liquid crystal spatial light modulators,” Opt. Express 22, 1636–1644 (2014).
    [CrossRef] [PubMed]
  21. C.-S. Guo, Z.-Y. Rong, S.-Z. Wang, “Double-channel vector spatial light modulator for generation of arbitrary complex vector beams,” Opt. Lett. 39, 386–389 (2014).
    [CrossRef] [PubMed]
  22. G. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
    [CrossRef]
  23. C. Sheppard, A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004).
    [CrossRef] [PubMed]
  24. K. Kitamura, K. Sakai, S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18, 4518–4525 (2010).
    [CrossRef] [PubMed]
  25. D. Biss, T. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28, 923–925 (2003).
    [CrossRef] [PubMed]
  26. D. Oron, E. Tal, Y. Silberberg, “Depth-resolved multiphoton polarization microscopy by third-harmonic generation,” Opt. Lett. 28, 2315–2317 (2003).
    [CrossRef] [PubMed]
  27. O. Masihzadeh, P. Schlup, R. A. Bartels, “Enhanced spatial resolution in third-harmonic microscopy through polarization switching,” Opt. Lett. 34, 1240–1242 (2009).
    [CrossRef] [PubMed]
  28. Y. Gorodetski, A. Niv, V. Kleiner, E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
    [CrossRef] [PubMed]
  29. L. Vuong, A. Adam, J. Brok, P. Planken, H. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104, 083903 (2010).
    [CrossRef] [PubMed]
  30. L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University, 2004).
    [CrossRef]
  31. Y. Inoue, V. Ramamurthy, Chiral Photochemistry (CRC, 2004).
  32. A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, J. Mompart, “Multiple rings formation in cascaded conical refraction,” Opt. Lett. 38, 1455–1457 (2013).
    [CrossRef] [PubMed]
  33. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” P. Roy. Soc. London A Mat. 253, 358–379 (1959).
    [CrossRef]
  34. V. Arrizón, L. González, R. Ponce, A. Serrano-Heredia, “Computer-generated holograms with optimum bandwidths obtained with twisted-nematic liquid-crystal displays,” Appl. Opt. 44, 1625–1634 (2005).
    [CrossRef] [PubMed]
  35. V. Arrizón, “Complex modulation with a twisted-nematic liquid-crystal spatial light modulator: double-pixel approach,” Opt. Lett. 28, 1359–1361 (2003).
    [CrossRef] [PubMed]
  36. M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
    [CrossRef]

2014 (2)

2013 (6)

2012 (1)

2011 (1)

2010 (5)

2009 (2)

2008 (4)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[CrossRef]

G. Lerman, U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (2008).
[CrossRef] [PubMed]

M. R. Foreman, S. S. Sherif, P. R. T. Munro, P. Török, “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express 16, 4901–4917 (2008).
[CrossRef] [PubMed]

Y. Gorodetski, A. Niv, V. Kleiner, E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef] [PubMed]

2007 (2)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Y. Kozawa, S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24, 1793–1798 (2007).
[CrossRef]

2006 (1)

2005 (1)

2004 (2)

2003 (4)

1979 (1)

G. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

1959 (1)

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

Adam, A.

L. Vuong, A. Adam, J. Brok, P. Planken, H. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104, 083903 (2010).
[CrossRef] [PubMed]

Arrizón, V.

Barends, P.

G. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

Barron, L. D.

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University, 2004).
[CrossRef]

Bartels, R. A.

Bernet, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Biss, D.

Blom, P.

G. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

Bokor, N.

K. Jahn, N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields”, Opt. Commun. 288, 13–16 (2013).
[CrossRef]

N. Davidson, N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29, 1318–1320 (2004).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[CrossRef]

Brakenhoff, G.

G. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

Brok, J.

L. Vuong, A. Adam, J. Brok, P. Planken, H. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104, 083903 (2010).
[CrossRef] [PubMed]

Brown, T.

Campos, J.

Carnicer, A.

Chen, J.

Cheng, W.

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[CrossRef]

Choudhury, A.

Dainty, C.

Davidson, N.

Ding, J.

Dorn, R.

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

Foreman, M. R.

Fürhapter, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

González, L.

Gorodetski, Y.

Y. Gorodetski, A. Niv, V. Kleiner, E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef] [PubMed]

Guo, C.-S

Guo, C.-S.

Han, W.

Han, Y.-J.

Hao, X.

Hasman, E.

Y. Gorodetski, A. Niv, V. Kleiner, E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef] [PubMed]

Iemmi, C.

Inoue, Y.

Y. Inoue, V. Ramamurthy, Chiral Photochemistry (CRC, 2004).

Jahn, K.

K. Jahn, N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields”, Opt. Commun. 288, 13–16 (2013).
[CrossRef]

Jesacher, A.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Juvells, I.

Kalkandjiev, T. K.

Kenny, F.

Khonina, S. N.

Kitamura, K.

Kleiner, V.

Y. Gorodetski, A. Niv, V. Kleiner, E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef] [PubMed]

Kozawa, Y.

Kuang, C.

Lara, D.

Lasser, T.

Leitgeb, R. A.

Lerman, G.

Leuchs, G.

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

Leutenegger, M.

Levy, U.

Li, Y.

Liu, X.

Loiko, Y. V.

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[CrossRef]

Maluenda, D.

Martinez-Herrero, R.

Martínez-Herrero, R.

Masihzadeh, O.

Maurer, C.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Mompart, J.

Moreno, I.

Munro, P. R. T.

Niv, A.

Y. Gorodetski, A. Niv, V. Kleiner, E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef] [PubMed]

Noda, S.

Oron, D.

Planken, P.

L. Vuong, A. Adam, J. Brok, P. Planken, H. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104, 083903 (2010).
[CrossRef] [PubMed]

Ponce, R.

Quabis, S.

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

Ramamurthy, V.

Y. Inoue, V. Ramamurthy, Chiral Photochemistry (CRC, 2004).

Rao, R.

Richards, B.

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

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Rodríguez-Herrera, O. G.

Rong, Z.-Y.

Sakai, K.

Sato, S.

Schlup, P.

Serrano-Heredia, A.

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[CrossRef]

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

Sherif, S. S.

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[CrossRef]

Silberberg, Y.

Tal, E.

Török, P.

Turpin, A.

Urbach, H.

L. Vuong, A. Adam, J. Brok, P. Planken, H. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104, 083903 (2010).
[CrossRef] [PubMed]

Volotovsky, S. G.

von Freymann, G.

Vuong, L.

L. Vuong, A. Adam, J. Brok, P. Planken, H. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104, 083903 (2010).
[CrossRef] [PubMed]

Waller, E. H.

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[CrossRef]

Wang, H.-T.

Wang, S.-Z.

Wang, T.

Wang, X.-L.

Wolf, E.

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

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[CrossRef]

Yang, Y.

Yzuel, M.

Zhan, Q.

Adv. Opt. Photon. (1)

Appl. Opt. (2)

J. Microsc. (1)

G. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

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

Nature Photon. (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[CrossRef]

New J. Phys. (1)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Opt. Commun. (1)

K. Jahn, N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields”, Opt. Commun. 288, 13–16 (2013).
[CrossRef]

Opt. Express (11)

M. R. Foreman, S. S. Sherif, P. R. T. Munro, P. Török, “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express 16, 4901–4917 (2008).
[CrossRef] [PubMed]

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

H.-T. Wang, X.-L. Wang, Y. Li, J. Chen, C.-S. Guo, J. Ding, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[CrossRef] [PubMed]

I. Moreno, C. Iemmi, J. Campos, M. Yzuel, “Jones matrix treatment for optical Fourier processors with structured polarization,” Opt. Express 19, 4583–4594 (2011).
[CrossRef] [PubMed]

F. Kenny, D. Lara, O. G. Rodríguez-Herrera, C. Dainty, “Complete polarization and phase control for focus-shaping in high-na microscopy,” Opt. Express 20, 14015–14029 (2012).
[CrossRef] [PubMed]

D. Maluenda, I. Juvells, R. Martínez-Herrero, A. Carnicer, “Reconfigurable beams with arbitrary polarization and shape distributions at a given plane,” Opt. Express 21, 5432–5439 (2013).
[CrossRef] [PubMed]

W. Han, Y. Yang, W. Cheng, Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21, 20692–20706 (2013).
[CrossRef] [PubMed]

E. H. Waller, G. von Freymann, “Independent spatial intensity, phase and polarization distributions,” Opt. Express 21, 28167–28174 (2013).
[CrossRef]

Z.-Y. Rong, Y.-J. Han, S.-Z. Wang, C.-S Guo, “Generation of arbitrary vector beams with cascaded liquid crystal spatial light modulators,” Opt. Express 22, 1636–1644 (2014).
[CrossRef] [PubMed]

G. Lerman, U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (2008).
[CrossRef] [PubMed]

M. Leutenegger, R. Rao, R. A. Leitgeb, T. Lasser, “Fast focus field calculations,” Opt. Express 14, 11277–11291 (2006).
[CrossRef] [PubMed]

Opt. Lett. (9)

N. Davidson, N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29, 1318–1320 (2004).
[CrossRef] [PubMed]

X. Hao, C. Kuang, T. Wang, X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35, 3928–3930 (2010).
[CrossRef] [PubMed]

C.-S. Guo, Z.-Y. Rong, S.-Z. Wang, “Double-channel vector spatial light modulator for generation of arbitrary complex vector beams,” Opt. Lett. 39, 386–389 (2014).
[CrossRef] [PubMed]

D. Biss, T. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28, 923–925 (2003).
[CrossRef] [PubMed]

D. Oron, E. Tal, Y. Silberberg, “Depth-resolved multiphoton polarization microscopy by third-harmonic generation,” Opt. Lett. 28, 2315–2317 (2003).
[CrossRef] [PubMed]

O. Masihzadeh, P. Schlup, R. A. Bartels, “Enhanced spatial resolution in third-harmonic microscopy through polarization switching,” Opt. Lett. 34, 1240–1242 (2009).
[CrossRef] [PubMed]

R. Martinez-Herrero, I. Juvells, A. Carnicer, “On the physical realizability of highly focused electromagnetic field distributions,” Opt. Lett. 38, 2065–2067 (2013).
[CrossRef] [PubMed]

V. Arrizón, “Complex modulation with a twisted-nematic liquid-crystal spatial light modulator: double-pixel approach,” Opt. Lett. 28, 1359–1361 (2003).
[CrossRef] [PubMed]

A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, J. Mompart, “Multiple rings formation in cascaded conical refraction,” Opt. Lett. 38, 1455–1457 (2013).
[CrossRef] [PubMed]

P. Roy. Soc. London A Mat. (1)

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

Phys. Rev, Lett. (1)

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

Phys. Rev. Lett. (2)

Y. Gorodetski, A. Niv, V. Kleiner, E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef] [PubMed]

L. Vuong, A. Adam, J. Brok, P. Planken, H. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104, 083903 (2010).
[CrossRef] [PubMed]

Other (3)

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University, 2004).
[CrossRef]

Y. Inoue, V. Ramamurthy, Chiral Photochemistry (CRC, 2004).

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Notation and sketch of a highly focused optical system.

Fig. 2
Fig. 2

Irradiance maps for a circularly-polarized highly-focused beam (NA = 0.85): (a) |E+|2, (b) |Ez|2 and (c) I. (d) Profiles of I at z = 0 (red) z = −3λ (blue), z = −5λ (magenta) and z = −7λ (black).

Fig. 3
Fig. 3

Sketch of the optical setup. Light source: HeNe laser λ= 633nm; P1, P2 and P3: linear polarizers; PBS1 and PBS2; polarizing beam splitters; M1 and M2: mirrors; HWP: half-wave plate: QWP: quarter-wave plate; SLM1 and SLM2: spatial light modulators; L1, L2 and L3: lenses; BS: beam splitter; MO: microscope objective.

Fig. 4
Fig. 4

A certain complex value C is generated as a combination of phasors ML and MR (that belong to the modulation response curve), and EL and ER that are diffracted off-axis and removed. The inset shows the subset of C values used to generate the holograms.

Fig. 5
Fig. 5

Experimental results at the observation plane: the first row corresponds to the image captured by camera 2. These images are normalized to its corresponding maximum. The second row shows the profile of the experimental images (black dots) and the numeric evaluation of I(red solid line) for z = −3.5λ, −5λ and −7λ.

Fig. 6
Fig. 6

Stokes images of the focused field at z = −3.5λ.

Tables (1)

Tables Icon

Table 1 Si values for the three transverse planes z analyzed.

Equations (24)

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

E ( r , ϕ , z ) = A 0 θ 0 0 2 π cos θ [ f 1 ( θ , φ ) e 1 ( φ ) + f 2 ( θ , φ ) e 2 o ( θ , φ ) ] × e i k r sin θ cos ( ϕ φ ) e i k z cos θ sin θ d θ d φ ,
f 1 ( θ , φ ) = E S ( θ , φ ) e 1 ( φ )
f 2 ( θ , φ ) = E S ( θ , φ ) e 2 i ( φ ) ,
e 1 ( φ ) = ( sin φ , cos φ , 0 )
e 2 i ( φ ) = ( cos φ , sin φ , 0 )
e 2 0 ( θ , φ ) = ( cos θ cos φ , cos θ sin φ , sin θ ) .
u ± = 1 2 ( 1 , ± i , 0 ) u z = ( 0 , 0 , 1 ) ,
E ± ( r , ϕ , z ) = A 2 0 θ 0 0 2 π cos θ [ i f 1 ( θ , φ ) + cos θ f 2 ( θ , φ ) ] × × e i k r sin θ cos ( ϕ φ ) e i k z cos θ e i φ sin θ d θ d φ
E ± ( r , ϕ , z ) = A 0 θ 0 0 2 π cos θ sin θ f 2 ( θ , φ ) e i k r sin θ cos ( ϕ φ ) e i k z cos θ sin θ d θ d φ .
f 1 ( θ , φ ) = ± i cos θ g ( θ , φ )
f 2 ( θ , φ ) = g ( θ , φ )
E + ( r , ϕ , z ) = 2 A 2 0 θ 0 0 2 π cos θ cos θ g ( θ ) e i φ e i k sin θ r cos ( ϕ φ ) e i k z cos θ sin θ d θ d φ
E ( r , ϕ , z ) = 0
E z ( r , ϕ , z ) = A 0 θ 0 0 2 π cos θ sin θ g ( θ ) e i k sin θ r cos ( ϕ φ ) e i k z cos θ sin θ d θ d φ .
E + ( r , ϕ , z ) = i 4 π A 2 e i ϕ 0 θ 0 cos θ cos θ g ( θ ) J 1 ( k r sin θ ) e i k z cos θ sin θ d θ
E z ( r , z ) = 2 π A 0 θ 0 cos θ sin θ g ( θ ) J 0 ( k r sin θ ) e i k z cos θ sin θ d θ ,
E S = ( cos φ i cos θ sin φ ) g ( θ ) e x + ( sin φ + i cos θ cos φ ) g ( θ ) e y
C x ( ρ , φ ) = cos φ i 1 ρ 2 sin φ
C y ( ρ , φ ) = sin φ + i 1 ρ 2 sin φ .
S 0 = I ( 0 ° , 0 ) + I ( 90 ° , 0 )
S 1 = I ( 0 ° , 0 ) I ( 90 ° , 0 )
S 2 = I ( 45 ° , 0 ) I ( 135 ° , 0 )
S 3 = I ( 45 ° , π / 2 ) I ( 135 ° , π / 2 ) ,
S i 2 = S i 2 ( k , l ) S 0 2 ( k , l ) i = 1 , 2 , 3

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