Diffraction-limited near-spherical focal spot with controllable arbitrary polarization using single objective lens

Chenhao Wan; Yanzhong Yu; Qiwen Zhan

doi:10.1364/OE.26.027109

1. Introduction

The engineering of optical focal fields has been of great interest to scientists because the spatial intensity, phase and polarization distributions in the focal region play an indispensable role when utilizing light beams in various disciplines including high-resolution imaging [1,2], particle trapping and manipulation [3], laser direct writing, lithography, data storage, spin-orbit interaction [4–6], and spin-directional coupling [7–10].

For spatial intensity engineering, focusing light beams into a subwavelength spherical volume is a challenging task. Several studies on the development of novel lenses [11–13], beam shaping [14] and polarization engineering [15,16] have proven to be successful in enhancing the transverse resolution. The axial extension, however, is always significantly larger (at least three times larger) than the transverse ones because the light beams are focused from only one side of the focal point [17]. The 4Pi microscopy technique in conjunction with amplitude, phase or polarization engineering solves this problem by focusing the light beams with two objective lenses facing each other at the expense of finicky adjustment of the double objective lenses and the double optical paths [18–21]. Similar results can be obtained by replacing one of the objective lenses with a mirror. The light beams are focused into two spots and they interfere constructively by reflecting one spot onto the other spot with a mirror [22].

The spatial polarization engineering in the focal region has also attracted significant attention and has been utilized in magneto-optic Kerr imaging [23], polarimetric imaging microscopy [24], and molecular orientation imaging [25–30]. A vector point spread function technique has been proposed to generate a linear, radial or azimuthal state of polarization in the focal region [31,32]. A time-reversal method has been demonstrated to generate arbitrary three-dimensional polarization near the focus of a high numerical aperture objective lens [33,34].

In the current paper, we propose a scheme to generate a subwavelength spherical focal spot with arbitrary three-dimensional polarization using single objective lens. The dipole antenna radiation theory and the Richards-Wolf vectorial diffraction method are combined to calculate the required input optical field in the pupil plane of an aplanatic objective lens in a time-reversal manner. The control of both subwavelength near-spherical focal volume and arbitrary three-dimensional state of polarization using single objective lens enables the method suitable for high-density data storage, laser micromachining, single-molecular imaging, tip enhanced Raman spectroscopy, spin-directional coupling, and anisotropic particle trapping and manipulation.

2. Time-reversal method

The time-reversal theory states that by sending in time reversed order, the field radiated by an infinitesimal dipole antenna in an arbitrary environment, one forms an optimal light spot at the source location [16,33,35,36]. Figure 1 shows the schematic of the proposed scheme. To create arbitrary three-dimensional polarization, three dipole antennae (labeled Dx₁, Dy₁ and Dz₁, oscillating along x-, y- and z-axis respectively) are situated at Point(0,0,-z₀). A flat reflective mirror is placed in the z = 0 plane. The corresponding image dipole antennae (labeled Dx₂, Dy₂ and Dz₂, oscillating along x-, y- and z-axis respectively) are situated at Point(0,0,z₀). The reflective fields from the mirror are equivalent to the radiation fields generated from the image dipole antennae if the Fresnel reflection coefficients are accurately accounted for. The radiation fields generated from the two sets of dipole antennae are coherently combined in the curved plane Ω. The radiation fields in plane Ω are expressed in terms of two unit vectors ${\vec{e}}_{θ}$ and ${\vec{e}}_{ϕ}$ that are along the elevation and azimuthal directions respectively. The radiation field of a dipole oscillating along the z direction is given by [37]:

{\vec{E}}_{3} = E_{3 θ} {\vec{e}}_{θ} = j η \frac{k I_{0} l \exp (- j k r)}{4 π r} \sin θ {\vec{e}}_{θ},

where η is the impedance, k is the wave number, I₀ is the constant electric current, θ is the elevation angle, and r is radius of curvature of the wavefront Ω. Accordingly, the radiation fields of a dipole oscillating along the x direction and y direction can be derived and expressed by Eq. (2) and Eq. (3) respectively.

{\vec{E}}_{1} = E_{1 θ} {\vec{e}}_{θ} + E_{1 ϕ} {\vec{e}}_{ϕ} = j η \frac{k I_{0} l \exp (- j k r)}{4 π r} [- \cos θ \cos ϕ {\vec{e}}_{θ} + \sin ϕ {\vec{e}}_{ϕ}],

{\vec{E}}_{2} = E_{2 θ} {\vec{e}}_{θ} + E_{2 ϕ} {\vec{e}}_{ϕ} = j η \frac{k I_{0} l \exp (- j k r)}{4 π r} [- \cos θ \sin ϕ {\vec{e}}_{θ} - \cos ϕ {\vec{e}}_{ϕ}] .

It is worthy of noting that the radiation field generated from the x-dipole or y-dipole has both elevation and azimuthal components whereas the z-dipole generates only the elevation component.

Fig. 1 Schematic of the proposed method. Three dipole antennae (labeled Dx₁, Dy₁ and Dz₁, oscillating along x, y and z axis respectively) are situated at Point(0,0,-z₀) and their corresponding image dipole antennae (labeled Dx₂, Dy₂ and Dz₂, oscillating along x, y and z axis respectively) are situated at Point(0,0,z₀). The radiation fields from both sets of dipole antennae are collected by an aplanatic objective lens.

Download Full Size | PDF

The radiation fields generated from the image dipole antennae are equivalent to the reflected waves generated from the real dipole antennae with a mirror with the Fresnel reflection coefficients accurately accounted for. The reflection coefficients at the dielectric/metal interface for the s wave and p wave are given by [17]:

r_{s} = \frac{n_{1} \cos θ_{1} - (u_{2} + i v_{2})}{n_{1} \cos θ_{1} + (u_{2} + i v_{2})},

r_{p} = \frac{[n_{2}^{2} (1 - κ_{2}^{2}) + 2 i n_{2}^{2} κ_{2}] \cos θ_{1} - n_{1} (u_{2} + i v_{2})}{[n_{2}^{2} (1 - κ_{2}^{2}) + 2 i n_{2}^{2} κ_{2}] \cos θ_{1} + n_{1} (u_{2} + i v_{2})},

where n₁ is the refractive index of medium, (n₂,κ₂) is the refractive index of metal, θ₁ is the incident angle. Since the distance between the two sets of dipole antennae 2z₀ is much smaller than the focal length of the objective lens, θ₁ = θ. The functions u₂ and v₂ are expressed as:

u_{2} = \frac{1}{\sqrt{2}} \sqrt{n_{2}^{2} (1 - κ_{2}^{2}) - n_{1}^{2} \sin^{2} θ_{1} + \sqrt{{[n_{2}^{2} (1 - κ_{2}^{2}) - n_{1}^{2} \sin^{2} θ_{1}]}^{2} + 4 n_{2}^{4} κ_{2}^{2}}},

v_{2} = \frac{1}{\sqrt{2}} \sqrt{- n_{2}^{2} (1 - κ_{2}^{2}) + n_{1}^{2} \sin^{2} θ_{1} + \sqrt{{[n_{2}^{2} (1 - κ_{2}^{2}) - n_{1}^{2} \sin^{2} θ_{1}]}^{2} + 4 n_{2}^{4} κ_{2}^{2}}} .

The relative amplitude and phase of the three real dipole antennae are denoted by three complex coefficients: p₁, p₂ and p₃. According to the time-reversal theory, the required optical field in the plane Ω for the generation of the two sets of dipoles is given by:

\begin{array}{l} E_{Ω} (θ, ϕ) = p_{1} ((\exp (j k z_{0} \cos θ) E_{1 θ} - \exp (- j k z_{0} \cos θ) \frac{E_{1 θ}}{r_{s}}) {\vec{e}}_{θ} + (\exp (j k z_{0} \cos θ) E_{1 ϕ} - \exp (- j k z_{0} \cos θ) \frac{E_{1 ϕ}}{r_{p}}) {\vec{e}}_{ϕ}) \\ + p_{2} ((\exp (j k z_{0} \cos θ) E_{2 θ} - \exp (- j k z_{0} \cos θ) \frac{E_{2 θ}}{r_{s}}) {\vec{e}}_{θ} + (\exp (j k z_{0} \cos θ) E_{2 ϕ} - \exp (- j k z_{0} \cos θ) \frac{E_{2 ϕ}}{r_{p}}) {\vec{e}}_{ϕ}) \\ + p_{3} (\exp (j k z_{0} \cos θ) E_{3 θ} + \exp (- j k z_{0} \cos θ) \frac{E_{3 θ}}{r_{s}}) {\vec{e}}_{θ} \end{array}

The dipole antennae Dx₁ and Dy₁ oscillate out of phase with respect to their corresponding image dipole antennae Dx₂ and Dy₂ according to the induced dipole theory [22]. The phase terms $\exp (j k z_{0} \cos θ)$ and $\exp (- j k z_{0} \cos θ)$ indicate the extra phase caused by the additional optical path length travelled by the reflective waves.

The optical fields in plane Ω can be expressed either in ( ${\vec{e}}_{θ}$ , ${\vec{e}}_{ϕ}$ ) components or ( ${\vec{e}}_{x}$ , ${\vec{e}}_{y}$ , ${\vec{e}}_{z}$ ) components based on the following relationships:

{\vec{e}}_{θ} = \cos θ \cos ϕ {\vec{e}}_{x} + \cos θ \sin ϕ {\vec{e}}_{y} + \sin θ {\vec{e}}_{z},

{\vec{e}}_{ϕ} = - \sin ϕ {\vec{e}}_{x} + \cos ϕ {\vec{e}}_{y} .

To express the input optical field in the pupil plane (r,ϕ), the projection function of the objective lens needs to be incorporated [38]. For an objective lens that obeys the sine condition r = fsinθ, where f is the focal length of the objective lens, the projection function from (r,ϕ) coordinates to (θ,ϕ) coordinates is $\sqrt{\cos θ}$ . Consequently, the optical field in the pupil plane is ${\vec{E}}_{i} (r, ϕ) = {\vec{E}}_{Ω} (θ, ϕ) / \sqrt{\cos θ}$ , where $θ = \sin^{- 1} (r / f)$ .

The Richards-Wolf vectorial diffraction method is utilized to verify whether the input optical field will generate two focal spots located at (0,0,-z₀) and (0,0,z₀) respectively. When the mirror is placed in the z = 0 plane, the spot located at (0,0,z₀) will be reflected and overlapped with the spot located at (0,0,-z₀). The coherent sum of the two spots will form one near-spherical spot with prescribed state of polarization regulated by the three dipole coefficients p₁, p₂ and p₃. The electric field at any point P(r_p,Ψ,z_p) near the focal point can be calculated by [39,40]:

\vec{E} (r_{p}, Ψ, z_{p}) = \frac{i}{λ} \int_{0}^{θ_{\max}} \int_{0}^{2 π} E_{Ω} (θ, ϕ) \exp (j k r_{p} \sin θ \cos (ϕ - Ψ) + j k z_{p} \cos θ) \sin θ d θ d ϕ,

where λ is the wavelength,

θ_{\max} = \sin^{- 1} (N A)

, NA is the numerical aperture of the objective lens.

3. Numerical simulation

Three examples are considered to demonstrate that the method is capable of generating a subwavelength near-spherical focal spot with arbitrary three-dimensional polarization using a single objective lens and a mirror. The parameters used in the numerical simulation include: the wavelength λ = 589.3nm; the refractive index of the medium n₁ = 1; the refractive index of the silver mirror n₂ = 0.2, n₂κ₂ = 3.44 [17]; the numerical aperture of the objective lens NA = 1; the distance between the dipole antennae and the mirror z₀ = 2λ.

For the first example, the dipole coefficients are set to be p₁ = 0, p₂ = 0, p₃ = 1, i.e., a single dipole oscillating along the z direction. Figure 2(a) shows the optical field in the pupil plane that is a radially polarized beam with spatially varying intensity distribution. The phase distribution of E_x and E_y components are displayed in Fig. 2(b) and 2(c), respectively. Neighboring rings are found to be π out of phase for both E_x and E_y components. The optical field in the pupil plane is focused by a high-NA objective lens. To analyze the polarization state in the focal region, the focal field is decomposed into three components: the desired polarization state component (p₁ = 0, p₂ = 0, p₃ = 1) and two orthogonal polarization state components (p₁ = 1, p₂ = 0, p₃ = 0 and p₁ = 0, p₂ = 1, p₃ = 0). The intensity distribution of the desired polarization state component in the z = -z₀ plane (see Visualization 1 for other z loci) is shown in Fig. 2(d) and the intensity distribution of the two orthogonal polarization state components are shown in Fig. 2(e) and 2(f), respectively. It is obvious that the two orthogonal polarization state components are negligible compared to the desired polarization state component that is linearly polarized along the z direction. The intensity linescan of the focal spot along the three axes are plotted in Fig. 2(g). The full width at half maximum (FWHM) of the x-linescan, y-linescan and z-linescan are 0.402λ, 0.402λ and 0.574λ, respectively. A subwavelength nearly spherical focal spot linearly polarized along the z-axis is therefore produced. The difference between the transverse and longitudinal dimension has decreased significantly due to the constructive interference of the two counter-propagating waves. The remaining mismatch originates from the fact that the spot is formed at a distance of z₀ away from the focal spot (0,0,0) of the objective lens. The resulting aberrations lead to the size mismatch that is dependent on the three-dimensional polarization state and z₀. To minimize the size mismatch for a specific three-dimensional polarization state, one could use Herschel-type objective lens [41], reduce z₀, or employ the finite-length dipole model [42]. In Ref [43], closed formulas are presented for obtaining a maximum longitudinal component in the focal point. The optical field in the pupil plane is also radially polarized, the amplitude is an increasing function of the radial coordinate, and the phase is uniform. Our method applies modulation in amplitude, phase and polarization in the pupil plane and achieves control in both polarization and intensity distribution in the focal region.

Fig. 2 Dipole coefficients are set to be p₁ = 0, p₂ = 0, p₃ = 1. (a) Intensity and polarization distributions in the pupil plane; (b) E_x phase distribution in the pupil plane; (c) E_y phase distribution in the pupil plane; (d) Intensity distribution of the desired polarization state in the focal region (z = -z₀ plane, see Visualization 1 for other z loci); (e) Intensity distribution of the first orthogonal polarization state in the focal region (z = -z₀ plane); (f) Intensity distribution of the second orthogonal polarization state in the focal region (z = -z₀ plane); (g) FWHM of the focal spot along the x-axis, y-axis and z-axis.

Download Full Size | PDF

For the second example, the dipole coefficients are set to be p₁ = $\frac{1}{\sqrt{2}}$ , p₂ = 0, p₃ = $\frac{1}{\sqrt{2}}$ e^jπ/2, i.e., one dipole oscillating along the x-axis and the other dipole oscillating along the z-axis with a π/2 phase difference. Figure 3(a) shows the optical field in the pupil plane that is spatially varying in both intensity and polarization. The phase distribution of E_x and E_y components are shown in Fig. 3(b) and 3(c), respectively. The phase structure is more complex and π phase jumps occur between neighboring ring structures. To analyze the polarization state in the focal region, the focal field is decomposed into the desired polarization state component (p₁ = $\frac{1}{\sqrt{2}}$ , p₂ = 0, p₃ = $\frac{1}{\sqrt{2}}$ e^jπ/2) and two orthogonal polarization state components (p₁ = $\frac{1}{\sqrt{2}}$ , p₂ = 0, p₃ = $\frac{1}{\sqrt{2}}$ e^-jπ/2 and p₁ = 0, p₂ = 1, p₃ = 0). The intensity distribution of the desired polarization state component in the z = -z₀ plane (see Visualization 2 for other z loci) is shown in Fig. 3(d) and the intensity distribution of the two orthogonal polarization state components are shown in Fig. 3(e) and 3(f), respectively. The desired polarization state component is circularly polarized in the x-z plane and is dominant compared to the two orthogonal polarization state components. The intensity linescan of the focal spot along the three axes are plotted in Fig. 3(g). The FWHM of the x-linescan, y-linescan and z-linescan are 0.474λ, 0.402λ and 0.472λ, respectively. A subwavelength near-spherical focal spot with transverse spin angular momentum (SAM) is generated. Such an optical field has attracted an increasing amount of attention and may spur numerous applications in spin-directional coupling and spin-orbit interaction [4–10].

Fig. 3 Dipole coefficients are set to be p₁ = $\frac{1}{\sqrt{2}}$ , p₂ = 0, p₃ = $\frac{1}{\sqrt{2}}$ e^jπ/2. (a) Intensity and polarization distributions in the pupil plane; (b) E_x phase distribution in the pupil plane; (c) E_y phase distribution in the pupil plane; (d) Intensity distribution of the desired polarization state in the focal region (z = -z₀ plane, see Visualization 2 for other z loci); (e) Intensity distribution of the first orthogonal polarization state in the focal region (z = -z₀ plane); (f) Intensity distribution of the second orthogonal polarization state in the focal region (z = -z₀ plane); (g) FWHM of the focal spot along the x-axis, y-axis and z-axis.

Download Full Size | PDF

For the third example, the dipole coefficients are set to be p₁ = $\frac{1}{\sqrt{14}}$ , p₂ = $\frac{2}{\sqrt{14}}$ e^jπ/6, p₃ = $\frac{3}{\sqrt{14}}$ e^jπ/3, i.e., the first dipole oscillating along the x-axis, the second dipole oscillating along the y-axis with a π/6 phase difference, and the third dipole oscillating along the z-axis with a π/3 phase difference. Figure 4(a) shows the intensity and polarization distributions in the pupil plane. Figure 4(b) and 4(c) show the phase distributions in the pupil plane. To analyze the polarization state in the focal region, the focal field is decomposed into the desired polarization state component (p₁ = $\frac{1}{\sqrt{14}}$ , p₂ = $\frac{2}{\sqrt{14}}$ e^jπ/6, p₃ = $\frac{3}{\sqrt{14}}$ e^jπ/3) and two orthogonal polarization state components (p₁ = 0.9636i, p₂ = 0.0741 - 0.1284i, p₃ = 0.1926 - 0.1112i and p₁ = 0, p₂ = −0.7206 - 0.4160i, p₃ = 0.2774 + 0.4804i). The intensity distribution of the desired polarization state component in the z = -z₀ plane (see Visualization 3 for other z loci) is displayed in Fig. 4(d) and the intensity distribution of the two orthogonal polarization state components are displayed in Fig. 4(e) and 4(f), respectively. The desired polarization state component is elliptically polarized and is overwhelmed compared to the two orthogonal polarization state components. The intensity linescan of the focal spot along the three axes are plotted in Fig. 4(g). The FWHM of the x-linescan, y-linescan and z-linescan are 0.412λ, 0.440λ and 0.499λ, respectively. A subwavelength near-spherical focal spot with prescribed three-dimensional polarization is therefore generated.

Fig. 4 Dipole coefficients are set to be p₁ = $\frac{1}{\sqrt{14}}$ , p₂ = $\frac{2}{\sqrt{14}}$ e^jπ/6, p₃ = $\frac{3}{\sqrt{14}}$ e^jπ/3. (a) Intensity and polarization distributions in the pupil plane; (b) E_x phase distribution in the pupil plane; (c) E_y phase distribution in the pupil plane; (d) Intensity distribution of the desired polarization state in the focal region (z = -z₀ plane, see Visualization 3 for other z loci); (e) Intensity distribution of the first orthogonal polarization state in the focal region (z = -z₀ plane); (f) Intensity distribution of the second orthogonal polarization state in the focal region (z = -z₀ plane); (g) FWHM of the focal spot along the x-axis, y-axis and z-axis.

Download Full Size | PDF

4. Discussions and conclusions

In this work, we demonstrate a time-reversal method based on the Richards-Wolf vectorial diffraction theory to generate a subwavelength near-spherical focal spot with arbitrary three-dimensional polarization. The implementation requires only single objective lens and achieves both spatial intensity and polarization control within a subwavelength volume in the focal region. In principle, the flat mirror can be replaced by substrates of other shapes and materials. Consequently, the method is general and can be adapted for more complex focusing environment, inhomogeneous background, or even microfluidic devices [22,44]. The method is useful for focal field engineering and is promising for applications in high-resolution imaging, laser micromachining and direct writing, lithography, spin-directional coupling, and anisotropic particle trapping and manipulation.

Funding

National Natural Science Foundation of China (NSFC) (61505062, 61571271); Wuhan Youth Science and Technology Plan (2016070204010152).

References

1. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19(11), 780–782 (1994). [CrossRef] [PubMed]

2. S. W. Hell, “Far-field optical nanoscopy,” Science 316(5828), 1153–1158 (2007). [CrossRef] [PubMed]

3. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987). [CrossRef] [PubMed]

4. K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015). [CrossRef]

5. O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104(25), 253601 (2010). [CrossRef] [PubMed]

6. Y. Zhao, J. S. Edgar, G. D. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007). [CrossRef] [PubMed]

7. K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015). [CrossRef]

8. M. Neugebauer, T. Bauer, P. Banzer, and G. Leuchs, “Polarization tailored light driven directional optical nanobeacon,” Nano Lett. 14(5), 2546–2551 (2014). [CrossRef] [PubMed]

9. F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013). [CrossRef] [PubMed]

10. J. Petersen, J. Volz, and A. Rauschenbeutel, “Chiral nanophotonic waveguide interface based on spin-orbit interaction of light,” Science 346(6205), 67–71 (2014). [CrossRef] [PubMed]

11. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

12. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [CrossRef] [PubMed]

13. F. Lemoult, G. Lerosey, J. de Rosny, and M. Fink, “Resonant metalenses for breaking the diffraction barrier,” Phys. Rev. Lett. 104(20), 203901 (2010). [CrossRef] [PubMed]

14. A. Sentenac and P. C. Chaumet, “Subdiffraction light focusing on a grating substrate,” Phys. Rev. Lett. 101(1), 013901 (2008). [CrossRef] [PubMed]

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

16. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1–6), 1–7 (2000). [CrossRef]

17. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999), 7th (expanded) ed.

18. Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012). [CrossRef] [PubMed]

19. S. N. Khonina and I. Golub, “Engineering the smallest 3D symmetrical bright and dark focal spots,” J. Opt. Soc. Am. A 30(10), 2029–2033 (2013). [CrossRef] [PubMed]

20. T. Jabbour and S. Kuebler, “Vector diffraction analysis of high numerical aperture focused beams modified by two- and three-zone annular multi-phase plates,” Opt. Express 14(3), 1033–1043 (2006). [CrossRef] [PubMed]

21. S. N. Khonina, A. V. Ustinov, and S. G. Volotovsky, “Shaping of spherical light intensity based on the interference of tightly focused beams with different polarizations,” Opt. Laser Technol. 60, 99–106 (2014). [CrossRef]

22. E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, and A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105(20), 203903 (2010). [CrossRef] [PubMed]

23. W. K. Hiebert, A. Stankiewicz, and M. R. Freeman, “Direct observation of magnetic relaxation in a small permalloy disk by time-resolved scanning Kerr microscopy,” Phys. Rev. Lett. 79(6), 1134–1137 (1997). [CrossRef]

24. R. Oldenbourg and G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180(2), 140–147 (1995). [CrossRef] [PubMed]

25. 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]

26. J. T. Fourkas, “Rapid determination of the three-dimensional orientation of single molecules,” Opt. Lett. 26(4), 211–213 (2001). [CrossRef] [PubMed]

27. 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]

28. A. Débarre, R. Jaffiol, C. Julien, D. Nutarelli, A. Richard, P. Tchénio, F. Chaput, and J. P. Boilot, “Quantitative determination of the 3D dipole orientation of single molecules,” Eur. Phys. J. D 28(1), 67–77 (2004). [CrossRef]

29. S. Boichenko, “Theoretical investigation of confocal microscopy using an elliptically polarized cylindrical vector laser beam: Visualization of quantum emitters near interfaces,” Phys. Rev. A (Coll. Park) 97(4), 043825 (2018). [CrossRef]

30. X. Wang, S. Chang, L. Lin, L. Wang, and S. Hao, “Elimination of fluorescence intensity difference in orientation determination of single molecules by highly focused generalized cylindrical vector beams,” Optik (Stuttg.) 122(9), 773–776 (2011). [CrossRef]

31. M. R. Beversluis, L. Novotny, and S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14(7), 2650–2656 (2006). [CrossRef] [PubMed]

32. E. Y. S. Yew and C. J. R. Sheppard, “Second harmonic generation polarization microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. 275(2), 453–457 (2007). [CrossRef]

33. W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. 12(4), 045707 (2010). [CrossRef]

34. J. Chen, C. Wan, L. Kong, and Q. Zhan, “Experimental generation of complex optical fields for diffraction limited optical focus with purely transverse spin angular momentum,” Opt. Express 25(8), 8966–8974 (2017). [CrossRef] [PubMed]

35. R. Carminati, R. Pierrat, J. de Rosny, and M. Fink, “Theory of the time reversal cavity for electromagnetic fields,” Opt. Lett. 32(21), 3107–3109 (2007). [CrossRef] [PubMed]

36. G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315(5815), 1120–1122 (2007). [CrossRef] [PubMed]

37. A. Balanis, Antenna Theory: Analysis and Design (Wiley-Interscience, 2005).

38. M. Gu, Advanced Optical Imaging Theory (Springer, 1999).

39. 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]

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

41. N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29(17), 1968–1970 (2004). [CrossRef] [PubMed]

42. W. Chen and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. 34(16), 2444–2446 (2009). [CrossRef] [PubMed]

43. H. P. Urbach and S. F. Pereira, “Field in focus with a maximum longitudinal electric component,” Phys. Rev. Lett. 100(12), 123904 (2008). [CrossRef] [PubMed]

44. Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2(2), 110–115 (2008). [CrossRef] [PubMed]

Name	Description
Visualization 1	Polarization evolution through focal region.
Visualization 2	Polarization evolution through focal region.
Visualization 3	Polarization evolution through focal region.

Diffraction-limited near-spherical focal spot with controllable arbitrary polarization using single objective lens

Abstract

1. Introduction

2. Time-reversal method

3. Numerical simulation

4. Discussions and conclusions

Funding

References

Supplementary Material (3)

Cited By

Figures (4)

Equations (11)

Optics Express