Generation of achromatic, uniform-phase, radially polarized beams Toshitaka Wakayama,1,* Oscar G. Rodríguez-Herrera,2 J. Scott Tyo,2 Yukitoshi Otani,3 Motoki Yonemura,1 and Toru Yoshizawa4 1
School of Biomedical Engineering, Saitama Medical University, Hidaka, Saitama, 350-1241, Japan 2 College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA 3 Utsunomiya University, Center of Optical Research and Education, Utsunomiya, Tochigi, 321-8585, Japan 4 NPO 3D Associates, Yokohama, Kanagawa, 230-0078, Japan * [email protected]
Abstract: Axially symmetric half-wave plates have been used to generate radially polarized beams that have constant phase in the plane transverse to propagation. However, since the retardance introduced by these waveplates depends on the wavelength, it is difficult to generate radially polarized beams achromatically. This paper describes a technique suitable for the generation of achromatic, radially polarized beams with uniform phase. The generation system contains, among other optical components, an achromatic, axially symmetric quarter-wave plate based on total internal reflection. For an incident beam with a constant phase distribution, the system generates a beam with an extra geometrical phase term. To generate a beam with the correct phase distribution, it is therefore necessary to have an incident optical vortex with an azimuthally varying phase distribution of the form exp( + iθ). We show theoretically that the phase component of radially polarized beam is canceled out by the phase component of the incident optical vortex, resulting in a radially polarized beam with uniform phase. Additionally, we present an experimental setup able to generate the achromatic, uniform-phase, radially polarized beam and experimental results that confirm that the generated beam has the correct phase distribution. ©2014 Optical Society of America OCIS codes: (260.5430) Polarization; (260.1440) Birefringence.
References and links 1.
M. R. Dennis, K. O'Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics (Elsevier, 2009). 2. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1– 57 (2009). 3. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). 4. A. V. Nesterov and V. G. Niziev, “Laser beam with axially symmetric polarization,” J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000). 5. J. R. Fontana and R. H. Pantell, “A high-energy laser accelerator for electrons using the inverse Cherenkov effect,” J. Appl. Phys. 54(8), 4285–4288 (1983). 6. M. Kraus, M. A. Ahmed, A. Michalowski, A. Voss, R. Weber, and T. Graf, “Microdrilling in steel using ultrashort pulsed laser beams with radial and azimuthal polarization,” Opt. Express 18(21), 22305–22313 (2010). 7. Y. Kozawa, T. Hibi, A. Sato, H. Horanai, M. Kurihara, N. Hashimoto, H. Yokoyama, T. Nemoto, and S. Sato, “Lateral resolution enhancement of laser scanning microscopy by a higher-order radially polarized mode beam,” Opt. Express 19(17), 15947–15954 (2011). 8. Y. Kozawa and S. Sato, “Single higher-order transverse mode operation of a radially polarized Nd:YAG laser using an annularly reflectivity-modulated photonic crystal coupler,” Opt. Lett. 33(19), 2278–2280 (2008). 9. M. Endo, “Azimuthally polarized 1 kW CO2 laser with a triple-axicon retroreflector optical resonator,” Opt. Lett. 33(15), 1771–1773 (2008). 10. M. Endo, M. Sasaki, and R. Koseki, “Analysis of an optical resonator formed by a pair of specially shaped axicons,” J. Opt. Soc. Am. A 29(4), 507–512 (2012).
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3306
11. W. M. Gibbons, P. J. Shannon, S. T. Sun, and B. J. Swetlin, “Surface-mediated alignment of nematic liquid crystals with polarized laser light,” Nature 351(6321), 49–50 (1991). 12. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). 13. 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). 14. K. Yamane, Y. Toda, and R. Morita, “Ultrashort optical-vortex pulse generation in few-cycle regime,” Opt. Express 20(17), 18986–18993 (2012). 15. Y. Tokizane, K. Oka, and R. Morita, “Supercontinuum optical vortex pulse generation without spatial or topological-charge dispersion,” Opt. Express 17(17), 14517–14525 (2009). 16. Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using spacevariant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79(11), 1587–1589 (2001). 17. M. Beresna, M. Gecevicius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98(20), 201101 (2011). 18. K. J. Moh, X. C. Yuan, J. Bu, R. E. Burge, and B. Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46(30), 7544–7551 (2007). 19. Q. Zhan and J. R. Leger, “Microellipsometer with radial symmetry,” Appl. Opt. 41(22), 4630–4637 (2002). 20. T. Wakayama, K. Komaki, Y. Otani, and T. Yoshizawa, “Achromatic axially symmetric wave plate,” Opt. Express 20(28), 29260–29265 (2012). 21. T. Wakayama, Y. Otani, and T. Yoshizawa, “An interferometric observation of topological effect by novel axially symmetrical wave plate,” Proc. SPIE 8493, 849306 (2012). 22. G. A. Swartzlander, Jr., “Achromatic optical vortex lens,” Opt. Lett. 31(13), 2042–2044 (2006). 23. X. C. Yuan, J. Lin, J. Bu, and R. E. Burge, “Achromatic design for the generation of optical vortices based on radial spiral phase plates,” Opt. Express 16(18), 13599–13605 (2008).
1. Introduction Axially symmetric polarized beams have recently attracted great interest [1, 2]. Radially polarized beams are a class of axially symmetric polarized beams used in a number of applications. When these beams are focused by an objective lens with a large numerical aperture they generate a longitudinal electric field in the focal region [3, 4]. The longitudinal electric field can be used, for instance, to accelerate electrons or probe sub-wavelength scattering features of objects at focus [5]. Laser processing, super-resolution microscopy, and laser trapping have also been proposed as applications of the longitudinal electric field [6–11]. Most of the applications that have been proposed in the literature require a radially polarized beam with uniform phase in the plane perpendicular to the propagation direction. An angularly varying half-wave plate is a convenient optical element to generate the radially polarized beams [12–14]. This and other techniques that have been proposed to generate this kind of beams are shown in Table 1 [12–15]. Techniques that use optical elements with polarization controlling nanostructures can also be found in the literature [16–18]. However, most of techniques are limited to use with monochromatic beams. Therefore, a new method is needed to generate a uniform-phase, radially polarized beam achromatically. The optical configuration shown in row A of Table 1 transforms a linearly polarized beam into a radially polarized beam after passing through the angularly varying half-wave plate, which is known as an axially symmetric half-wave plate (ASH) [12–14]. The fast axis of the ASH is oriented at an angle θ/2 from the x-axis, where θ is the azimuth in the beam’s aperture. In this case, the Stokes vector of the transmitted beam is (1/2){1, cosθ, sinθ, 0}T. In the optical configuration in row B of Table 1, a linearly polarized beam becomes a circularly polarized beam after passing through a quarter-wave plate. The circularly polarized beam is then transformed into the axially symmetric polarized beam after passing through an angularly varying linear polarizer, which is known as a radial polarizer (RP) [19]. The transmitting axis of the RP is oriented at an angle θ with respect to the x-axis. The Stokes vector in this case is (1/4){1, cosθ, sinθ, 0}T. Finally, in the system shown in row C of Table 1, a linearly polarized beam is converted into a circularly polarized beam that is then transformed into an axially symmetric polarized beam after passing through an angularly varying quarter-wave plate, referred to as an axially symmetric quarter-wave plate (ASQ) herein. The ASQ is similar to the ASH, but with segments made out of quarter-wave plates rather than half-wave plates. The fast axis of the ASQ is oriented at an angle θ from the #202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3307
x- axis. After passing through the ASQ, the combination of two half-wave plates oriented at 0° and 22.5° performs as a polarization rotator (PR). The Stokes vector in this case is (1/2){1, cosθ, sinθ, 0}T. Neglecting the reduced intensity of the beam produced by the system in the second row of Table 1, the Stokes vectors are the same for these three optical configurations. However, the Jones vectors shown in Table 1 are different in all three cases. The Jones vectors shown in the second and third rows of the table have an additional phase component of the form exp(-iθ), which is not present in the Jones vector in the first row. Therefore, we refer to the beam shown in the first row of Table 1 as a uniform-phase, radially polarized beam. The other two beams are referred to as spiral-phase, radially polarized beams. From the results in Table 1, it is clear that the ASH is the only suitable element to generate the uniform-phase, radially polarized beam of the elements discussed herein. However, the ASH is strongly dependent on the wavelength, which limits its applicability in the generation of achromatic, radially polarized beams. Table 1. Conventional generation techniques of radially polarized beams [12–15] Optical configurations
Stokes vectors
Jones vectors
1 cos θ 1 cos θ A sin θ 2 sin θ 0 1 π i 4 cos θ 1 cos θ − iθ e B e 4 sin θ 2 sin θ 0 1 π i cos θ 1 cos θ − iθ e4 C e 2 sin θ sin θ 0 In this paper, we use the Jones polarization formalism to present a method to generate achromatic, uniform-phase, radially polarized beams using a novel, achromatic ASQ. We present an experimental setup able to generate radially polarized beams and verify the correct phase distribution of the generated beams using a Mach-Zehnder interferometer. In our experimental setup, an achromatic, axially symmetric quarter-wave plate (AASQ) based on Fresnel reflections [20, 21] is the key element to generate the achromatic, uniform-phase, radially polarized beams.
Fig. 1. Generation system of the achromatic, uniform-phase, radially polarized beam.
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3308
2. Generation of an achromatic, uniform-phase, radially polarized beam
Figure 1 presents a diagram of the proposed system to generate achromatic, uniform-phase, radially polarized beams. This system is composed of two subsystems: i) the generation optics (GO), used to generate an incident achromatic optical vortex, and ii) the conversion optics (CO), used to generate the achromatic, radially polarized beam. In this system, we use the generation technique for achromatic optical vortices discussed in Refs [15, 22, 23]. The GO is comprised of an axially symmetric retarder (ASR) placed between a pair of linear polarizers (LP) and achromatic quarter-wave plates (AQ). The first and second polarizers are set at 0° with respect to the x-axis. The first and second AQs are oriented at −45°. The retardance of the ASR has the form δ(λ), where λ is the design wavelength. The azimuthal angle of the ASR varies as θ/2 from the x-axis as a function of the azimuth direction θ. According to the Jones formalism, the resulting electric field is given by E1 = LP ⋅ AQ ⋅ ASR ⋅ AQ ⋅ LP ⋅ E0 ,
(1)
where E0 and E1 are the Jones vectors of the input and output, respectively. Therefore, the Jones vector of the output can be expressed as Ex δ ( λ ) 1 + iθ E1 = = i sin ⋅ ⋅e . E 2 0 y
(2)
It is clear from Eq. (2) that the output beam is linearly polarized, as shown in Fig. 2(a). At any point ‘1’ in the output beam, the beam has a spiral phase component of the form exp( + iθ). The interaction of the output beam E1 with the CO for achromatic, radially polarized beams can be expressed as E2 = APR ⋅ AASQ ⋅ AQ ⋅ E1.
(3)
In this case, the AQ is set at + 45°, and the AASQ and achromatic polarization rotator (APR) are parallel. The APR is composed of two achromatic half-wave plates set at 0° and −22.5°, respectively. In terms of the Jones formalism this can be written as
δ ( λ ) cos θ E x i 3π E2 = = e 4 ⋅ sin ⋅ , E 2 sin θ y
(4)
where exp(i3π/4)⋅sin(δ (λ)/2) represents an amplitude factor, and (cosθ, sinθ)T is the radially polarized beam. In this case, there are no phase factors of the form exp( + iθ). However, the amplitude factor depends on the wavelength. As a result, the output beam created by the system in Fig. 1 is a uniform-phase, radially polarized beam at any point ‘2’ in the beam, as shown in Fig. 2(b). Therefore, it is possible to transform a linearly polarized incident beam into an achromatic uniform-phase, radially polarized beam using the system in Fig. 1.
Fig. 2. Details of the output beam.
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3309
Fig. 3. Optical configuration of the systems to generate and measure uniform-phase and spiralphase, achromatic, radially polarized beams.
3. Theoretical considerations for radially polarized beams
We employed a Mach-Zehnder interferometer to verify the phase distribution of both the uniform-phase and the spiral-phase, radially polarized beams. Figure 3 is a schematic representation of the optical configuration used in each case. In the following calculations, the retardation of the ASR, δ(λ), is 90° and the reference beam is circularly polarized. Therefore, the Jones vector of the reference beam, Eref, is given as Eref . =
a 1+ i ⋅ , 2 −1 + i
(5)
where a is the amplitude of the electric field. When a spiral-phase, radially polarized beam is used as the input, as show in Fig. 3(a), its Jones vector can be expressed as π i cos θ − iθ Eobj1 = b ⋅ e 4 ⋅ ⋅e , sin θ
(6)
where b is the amplitude of the electric field. The intensity distribution at the output of the optical configuration in Fig. 3(a) that results from the interference between the reference and objective beams is given from Eqs. (5) and (6) as
(
)
I = I 0 ⋅ a 2 + b2 + 2ab ⋅ cos 2θ ,
(7)
where I0 is a constant of proportionality. In the case of a uniform-phase, radially polarized objective beam, like the one shown in Fig. 3(b), the Jones vector can be expressed as
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3310
i
Eobj 2
3π
c⋅e 4 = 2
cos θ ⋅ , sin θ
(8)
where c is the amplitude of the electric field. In this case, the intensity distribution at the output of the optical configuration in Fig. 3(b) is given by Eqs. (5) and (8) as π c2 I = I 0 ⋅ a 2 + − ac ⋅ cos θ + . 2 2
(9)
Equations (7) and (9) show that the spiral-phase and uniform-phase, achromatic, radially polarized beams produce different total intensity distributions when they interfere with the same reference beam. The azimuthal variation of the total intensity is 2θ for the spiral-phase and θ for the uniform-phase beam. Therefore, by introducing a slight tilt to the reference beam and counting the resulting fringes it is possible to distinguish between uniform-phase and spiral-phase, radially polarized beams. In the following section we present experimental results that demonstrate that the GO system in Fig. 1 compensates the geometrical phase introduced by the AASQ and can be used to create an achromatic, uniform-phase, radially polarized beam. 4. Experimental results Figure 4(a) shows the achromatic, axially symmetric quarter-wave plate developed and introduced in a previous work [20]. The AASQ is made out of SiO2 (ShinEtsu QUARTZ Co., Ltd.). This element has a diameter of 30 mm, a width of 20 mm, and it is designed to have 90° of retardance in the visible. The principle of an achromatic, axially symmetric waveplate is based on Fresnel reflections. The waveplate has a concave conical surface created by the cross-section of a Fresnel rhomb rotated, with one of its vertices fixed, about a line that becomes the optical axis of the AASQ. The incident beam propagates parallel to the optical axis and impinges on the vertex of the concave conical surface. It then forms a cone-shaped beam that reflects on the sloping edge of the waveplate and creates a ring-shaped beam that propagates parallel to the optical axis [20]. Figure 4(b) shows a polarization image of the AASQ placed between two crossed linear polarizers. From the intensity distribution shown in Fig. 4(b), it is clear that the angle of linear polarization at the output of the AASQ varies along the azimuthal direction. The dark regions correspond to light linearly polarized at 0°.
Fig. 4. Pictures of the achromatic, axially symmetric quarter-wave plate.
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3311
Fig. 5. Mach-Zehnder interferometer for the observation of radially polarized beams.
To determine whether a radially polarized beam is uniform- or spiral-phase, we built the Mach-Zehnder interferometer shown in Fig. 5, which is based on the diagrams in Fig. 3. In this experiment we used a He-Ne laser as a light source. In addition, we perform experiments to observe the intensity distribution based on polarization states using three other lasers with different wavelengths. Here, the main element in our experimental setup is the AASQ, which has been demonstrated to be achromatic in our previous work [20]. The intensity of the linearly polarized incident laser beam was controlled with the combination of a polarizer and an attenuator. The laser beam was expanded, collimated, and sent to a non-polarizing beam splitter (NPBS) to create the reference and objective beams. In this experiment we neglect the depolarizing effects of the NPBS and folding mirrors as being small compared with other error sources in the experiment. The reference beam was transformed into a circularly polarized beam by passing it through a polarizer and a quarter-wave plate. The objective beam was transformed from a linearly polarized beam with no topological charge into a linearly polarized beam with a phase factor exp( + iθ) using the GO system. To create the uniform-phase, radially polarized beam, the linearly polarized vortex beam is sent to the CO system. Alternatively, to create the spiral-phase, radially polarized beam the GO is removed from the experimental setup and the incident beam is sent directly to the CO.
Fig. 6. Axially symmetric quarter-wave plate (ASQ).
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3312
Fig. 7. Output beam after the CO.
Figure 6(a) is a picture of an ASQ plate fabricated with 9 segments of a retarder film with their fast axis varying azimuthally. Figure 6(b) is the intensity distribution obtained when the axially symmetric quarter-wave plate is located between crossed polarizers. With a slightly tilted reference beam and the ASQ in Fig. 6(a) taking the place of the ASR in the GO system, we observe the interference fringes shown in Fig. 6(c) when the CO is removed. It is clear that the central bright fringe is split in two. This tuning fork shape is a consequence of the exp( + iθ) phase distribution in the beam created by the GO. Inserting the CO back into the system, the beam becomes doughnut-like after passing through it, as shown in Fig. 7. In this demonstration, we employed laser beams with three different wavelengths (405nm, 550nm and 632nm). The intensity distributions obtained by passing this beam through a linear polarizer at 0°, 45° and 90° are shown in Figs. 7(a)–7(i). These arrows in Fig. 7 indicate the orientation angles of the linear polarizer. These figures show that the CO converts the state of polarization of the beam from linear polarization into radial polarization. However, the intensity distributions in these figures are the same for both the uniform- and spiral-phase radially polarized beams. To prove that the beam generated by the CO has the correct phase distribution, we investigated the fringes that result from the interference of the beam generated by the CO with a slightly tilted reference beam.
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3313
Fig. 8. Experimental interference fringes for the radially polarized beams.
Fig. 9. Numerical interference fringes for the radially polarized beams.
Figures 8(a) and 8(b) show the intensity distribution of the radially polarized beams for both cases. The nearly circular rings have dark spots like doughnut shape beams. In contrast, the interference patterns are shown in Figs. 8(c) and 8(d) for the spiral-phase and the uniformphase beams, respectively. Although the effects of significant aberrations are apparent on the fringe shapes in the center area, the important point is not the specific fringes shape at the center, but rather the fringe behavior on the circular rings as shown in Figs. 8(c) and 8(d). The aberrations on the dark spots of the center area are caused by the scattering for the small amount of light that passes through the center of the AAS-QWP. Naturally, additional aberrations also arise from the optical system. Let us consider interference fringes on the circular rings to understand the generation of the radially polarized beam. The total number of interference fringes is different for the
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3314
clockwise (red numbers) and the counterclockwise (green numbers) direction in both figures. The absolute value of the difference between the number of fringes in the clockwise and counterclockwise directions in Figs. 8(c) and 8(d) is |14-12| = 2 and |14-13| = 1, respectively. These differences are consistent with what is expected from the azimuthal angle dependence of the irradiance distributions in Eqs. (7) and (9). That is to say, we can understand whether the output beam has uniform-phase or spiral-phase from the interference fringes. To better understand the experimental results, we performed a numerical calculation of the interference fringes obtained for both types of radially polarized beams when the reference beam is slightly tilted. These results are shown in Fig. 9. Although the numerical results have no aberrations, the difference in the number of fringes corresponds with the difference observed in our experimental results. Therefore, the results in Figs. 8(d) and 9(b) show that the optical configuration in Fig. 2 can be used to generate a radially polarized beam with the correct phase distribution (i.e., uniform-phase). 7. Conclusions A method to generate an achromatic, uniform-phase, radially polarized beam has been presented. In our method, a linearly polarized beam is transformed into an optical vortex with phase distribution exp( + iθ) using a vortex generation optical system. The optical vortex is converted into an axially symmetric polarized beam using a conversion optical system that employs an achromatic, axially symmetric quarter-wave plate based on Fresnel’s reflections. Calculations performed using the Jones formalism have shown that our method is able to generate an achromatic, uniform-phase, radially polarized beam. We have shown that if the vortex generation optics is omitted, the beam generated by the conversion optics has the incorrect phase distribution, i.e., is a spiral-phase, radially polarized beam. Finally, we have verified our theoretical results experimentally using an AASQ designed and fabricated at our group. Acknowledgments This work was partially supported by the Japan Science and Technology Agency, Adaptable and Seamless Technology Transfer Program through target-driven R&D with number AS242Z01381K. O. G. Rodríguez-Herrera and J. S. Tyo were supported by the Air Force Office of Scientific Research (AFOSR) under grant FA9550-10-0114.
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3315
School of Biomedical Engineering, Saitama Medical University, Hidaka, Saitama, 350-1241, Japan 2 College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA 3 Utsunomiya University, Center of Optical Research and Education, Utsunomiya, Tochigi, 321-8585, Japan 4 NPO 3D Associates, Yokohama, Kanagawa, 230-0078, Japan * [email protected]
Abstract: Axially symmetric half-wave plates have been used to generate radially polarized beams that have constant phase in the plane transverse to propagation. However, since the retardance introduced by these waveplates depends on the wavelength, it is difficult to generate radially polarized beams achromatically. This paper describes a technique suitable for the generation of achromatic, radially polarized beams with uniform phase. The generation system contains, among other optical components, an achromatic, axially symmetric quarter-wave plate based on total internal reflection. For an incident beam with a constant phase distribution, the system generates a beam with an extra geometrical phase term. To generate a beam with the correct phase distribution, it is therefore necessary to have an incident optical vortex with an azimuthally varying phase distribution of the form exp( + iθ). We show theoretically that the phase component of radially polarized beam is canceled out by the phase component of the incident optical vortex, resulting in a radially polarized beam with uniform phase. Additionally, we present an experimental setup able to generate the achromatic, uniform-phase, radially polarized beam and experimental results that confirm that the generated beam has the correct phase distribution. ©2014 Optical Society of America OCIS codes: (260.5430) Polarization; (260.1440) Birefringence.
References and links 1.
M. R. Dennis, K. O'Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics (Elsevier, 2009). 2. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1– 57 (2009). 3. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). 4. A. V. Nesterov and V. G. Niziev, “Laser beam with axially symmetric polarization,” J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000). 5. J. R. Fontana and R. H. Pantell, “A high-energy laser accelerator for electrons using the inverse Cherenkov effect,” J. Appl. Phys. 54(8), 4285–4288 (1983). 6. M. Kraus, M. A. Ahmed, A. Michalowski, A. Voss, R. Weber, and T. Graf, “Microdrilling in steel using ultrashort pulsed laser beams with radial and azimuthal polarization,” Opt. Express 18(21), 22305–22313 (2010). 7. Y. Kozawa, T. Hibi, A. Sato, H. Horanai, M. Kurihara, N. Hashimoto, H. Yokoyama, T. Nemoto, and S. Sato, “Lateral resolution enhancement of laser scanning microscopy by a higher-order radially polarized mode beam,” Opt. Express 19(17), 15947–15954 (2011). 8. Y. Kozawa and S. Sato, “Single higher-order transverse mode operation of a radially polarized Nd:YAG laser using an annularly reflectivity-modulated photonic crystal coupler,” Opt. Lett. 33(19), 2278–2280 (2008). 9. M. Endo, “Azimuthally polarized 1 kW CO2 laser with a triple-axicon retroreflector optical resonator,” Opt. Lett. 33(15), 1771–1773 (2008). 10. M. Endo, M. Sasaki, and R. Koseki, “Analysis of an optical resonator formed by a pair of specially shaped axicons,” J. Opt. Soc. Am. A 29(4), 507–512 (2012).
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3306
11. W. M. Gibbons, P. J. Shannon, S. T. Sun, and B. J. Swetlin, “Surface-mediated alignment of nematic liquid crystals with polarized laser light,” Nature 351(6321), 49–50 (1991). 12. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). 13. 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). 14. K. Yamane, Y. Toda, and R. Morita, “Ultrashort optical-vortex pulse generation in few-cycle regime,” Opt. Express 20(17), 18986–18993 (2012). 15. Y. Tokizane, K. Oka, and R. Morita, “Supercontinuum optical vortex pulse generation without spatial or topological-charge dispersion,” Opt. Express 17(17), 14517–14525 (2009). 16. Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using spacevariant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79(11), 1587–1589 (2001). 17. M. Beresna, M. Gecevicius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98(20), 201101 (2011). 18. K. J. Moh, X. C. Yuan, J. Bu, R. E. Burge, and B. Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46(30), 7544–7551 (2007). 19. Q. Zhan and J. R. Leger, “Microellipsometer with radial symmetry,” Appl. Opt. 41(22), 4630–4637 (2002). 20. T. Wakayama, K. Komaki, Y. Otani, and T. Yoshizawa, “Achromatic axially symmetric wave plate,” Opt. Express 20(28), 29260–29265 (2012). 21. T. Wakayama, Y. Otani, and T. Yoshizawa, “An interferometric observation of topological effect by novel axially symmetrical wave plate,” Proc. SPIE 8493, 849306 (2012). 22. G. A. Swartzlander, Jr., “Achromatic optical vortex lens,” Opt. Lett. 31(13), 2042–2044 (2006). 23. X. C. Yuan, J. Lin, J. Bu, and R. E. Burge, “Achromatic design for the generation of optical vortices based on radial spiral phase plates,” Opt. Express 16(18), 13599–13605 (2008).
1. Introduction Axially symmetric polarized beams have recently attracted great interest [1, 2]. Radially polarized beams are a class of axially symmetric polarized beams used in a number of applications. When these beams are focused by an objective lens with a large numerical aperture they generate a longitudinal electric field in the focal region [3, 4]. The longitudinal electric field can be used, for instance, to accelerate electrons or probe sub-wavelength scattering features of objects at focus [5]. Laser processing, super-resolution microscopy, and laser trapping have also been proposed as applications of the longitudinal electric field [6–11]. Most of the applications that have been proposed in the literature require a radially polarized beam with uniform phase in the plane perpendicular to the propagation direction. An angularly varying half-wave plate is a convenient optical element to generate the radially polarized beams [12–14]. This and other techniques that have been proposed to generate this kind of beams are shown in Table 1 [12–15]. Techniques that use optical elements with polarization controlling nanostructures can also be found in the literature [16–18]. However, most of techniques are limited to use with monochromatic beams. Therefore, a new method is needed to generate a uniform-phase, radially polarized beam achromatically. The optical configuration shown in row A of Table 1 transforms a linearly polarized beam into a radially polarized beam after passing through the angularly varying half-wave plate, which is known as an axially symmetric half-wave plate (ASH) [12–14]. The fast axis of the ASH is oriented at an angle θ/2 from the x-axis, where θ is the azimuth in the beam’s aperture. In this case, the Stokes vector of the transmitted beam is (1/2){1, cosθ, sinθ, 0}T. In the optical configuration in row B of Table 1, a linearly polarized beam becomes a circularly polarized beam after passing through a quarter-wave plate. The circularly polarized beam is then transformed into the axially symmetric polarized beam after passing through an angularly varying linear polarizer, which is known as a radial polarizer (RP) [19]. The transmitting axis of the RP is oriented at an angle θ with respect to the x-axis. The Stokes vector in this case is (1/4){1, cosθ, sinθ, 0}T. Finally, in the system shown in row C of Table 1, a linearly polarized beam is converted into a circularly polarized beam that is then transformed into an axially symmetric polarized beam after passing through an angularly varying quarter-wave plate, referred to as an axially symmetric quarter-wave plate (ASQ) herein. The ASQ is similar to the ASH, but with segments made out of quarter-wave plates rather than half-wave plates. The fast axis of the ASQ is oriented at an angle θ from the #202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3307
x- axis. After passing through the ASQ, the combination of two half-wave plates oriented at 0° and 22.5° performs as a polarization rotator (PR). The Stokes vector in this case is (1/2){1, cosθ, sinθ, 0}T. Neglecting the reduced intensity of the beam produced by the system in the second row of Table 1, the Stokes vectors are the same for these three optical configurations. However, the Jones vectors shown in Table 1 are different in all three cases. The Jones vectors shown in the second and third rows of the table have an additional phase component of the form exp(-iθ), which is not present in the Jones vector in the first row. Therefore, we refer to the beam shown in the first row of Table 1 as a uniform-phase, radially polarized beam. The other two beams are referred to as spiral-phase, radially polarized beams. From the results in Table 1, it is clear that the ASH is the only suitable element to generate the uniform-phase, radially polarized beam of the elements discussed herein. However, the ASH is strongly dependent on the wavelength, which limits its applicability in the generation of achromatic, radially polarized beams. Table 1. Conventional generation techniques of radially polarized beams [12–15] Optical configurations
Stokes vectors
Jones vectors
1 cos θ 1 cos θ A sin θ 2 sin θ 0 1 π i 4 cos θ 1 cos θ − iθ e B e 4 sin θ 2 sin θ 0 1 π i cos θ 1 cos θ − iθ e4 C e 2 sin θ sin θ 0 In this paper, we use the Jones polarization formalism to present a method to generate achromatic, uniform-phase, radially polarized beams using a novel, achromatic ASQ. We present an experimental setup able to generate radially polarized beams and verify the correct phase distribution of the generated beams using a Mach-Zehnder interferometer. In our experimental setup, an achromatic, axially symmetric quarter-wave plate (AASQ) based on Fresnel reflections [20, 21] is the key element to generate the achromatic, uniform-phase, radially polarized beams.
Fig. 1. Generation system of the achromatic, uniform-phase, radially polarized beam.
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3308
2. Generation of an achromatic, uniform-phase, radially polarized beam
Figure 1 presents a diagram of the proposed system to generate achromatic, uniform-phase, radially polarized beams. This system is composed of two subsystems: i) the generation optics (GO), used to generate an incident achromatic optical vortex, and ii) the conversion optics (CO), used to generate the achromatic, radially polarized beam. In this system, we use the generation technique for achromatic optical vortices discussed in Refs [15, 22, 23]. The GO is comprised of an axially symmetric retarder (ASR) placed between a pair of linear polarizers (LP) and achromatic quarter-wave plates (AQ). The first and second polarizers are set at 0° with respect to the x-axis. The first and second AQs are oriented at −45°. The retardance of the ASR has the form δ(λ), where λ is the design wavelength. The azimuthal angle of the ASR varies as θ/2 from the x-axis as a function of the azimuth direction θ. According to the Jones formalism, the resulting electric field is given by E1 = LP ⋅ AQ ⋅ ASR ⋅ AQ ⋅ LP ⋅ E0 ,
(1)
where E0 and E1 are the Jones vectors of the input and output, respectively. Therefore, the Jones vector of the output can be expressed as Ex δ ( λ ) 1 + iθ E1 = = i sin ⋅ ⋅e . E 2 0 y
(2)
It is clear from Eq. (2) that the output beam is linearly polarized, as shown in Fig. 2(a). At any point ‘1’ in the output beam, the beam has a spiral phase component of the form exp( + iθ). The interaction of the output beam E1 with the CO for achromatic, radially polarized beams can be expressed as E2 = APR ⋅ AASQ ⋅ AQ ⋅ E1.
(3)
In this case, the AQ is set at + 45°, and the AASQ and achromatic polarization rotator (APR) are parallel. The APR is composed of two achromatic half-wave plates set at 0° and −22.5°, respectively. In terms of the Jones formalism this can be written as
δ ( λ ) cos θ E x i 3π E2 = = e 4 ⋅ sin ⋅ , E 2 sin θ y
(4)
where exp(i3π/4)⋅sin(δ (λ)/2) represents an amplitude factor, and (cosθ, sinθ)T is the radially polarized beam. In this case, there are no phase factors of the form exp( + iθ). However, the amplitude factor depends on the wavelength. As a result, the output beam created by the system in Fig. 1 is a uniform-phase, radially polarized beam at any point ‘2’ in the beam, as shown in Fig. 2(b). Therefore, it is possible to transform a linearly polarized incident beam into an achromatic uniform-phase, radially polarized beam using the system in Fig. 1.
Fig. 2. Details of the output beam.
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3309
Fig. 3. Optical configuration of the systems to generate and measure uniform-phase and spiralphase, achromatic, radially polarized beams.
3. Theoretical considerations for radially polarized beams
We employed a Mach-Zehnder interferometer to verify the phase distribution of both the uniform-phase and the spiral-phase, radially polarized beams. Figure 3 is a schematic representation of the optical configuration used in each case. In the following calculations, the retardation of the ASR, δ(λ), is 90° and the reference beam is circularly polarized. Therefore, the Jones vector of the reference beam, Eref, is given as Eref . =
a 1+ i ⋅ , 2 −1 + i
(5)
where a is the amplitude of the electric field. When a spiral-phase, radially polarized beam is used as the input, as show in Fig. 3(a), its Jones vector can be expressed as π i cos θ − iθ Eobj1 = b ⋅ e 4 ⋅ ⋅e , sin θ
(6)
where b is the amplitude of the electric field. The intensity distribution at the output of the optical configuration in Fig. 3(a) that results from the interference between the reference and objective beams is given from Eqs. (5) and (6) as
(
)
I = I 0 ⋅ a 2 + b2 + 2ab ⋅ cos 2θ ,
(7)
where I0 is a constant of proportionality. In the case of a uniform-phase, radially polarized objective beam, like the one shown in Fig. 3(b), the Jones vector can be expressed as
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3310
i
Eobj 2
3π
c⋅e 4 = 2
cos θ ⋅ , sin θ
(8)
where c is the amplitude of the electric field. In this case, the intensity distribution at the output of the optical configuration in Fig. 3(b) is given by Eqs. (5) and (8) as π c2 I = I 0 ⋅ a 2 + − ac ⋅ cos θ + . 2 2
(9)
Equations (7) and (9) show that the spiral-phase and uniform-phase, achromatic, radially polarized beams produce different total intensity distributions when they interfere with the same reference beam. The azimuthal variation of the total intensity is 2θ for the spiral-phase and θ for the uniform-phase beam. Therefore, by introducing a slight tilt to the reference beam and counting the resulting fringes it is possible to distinguish between uniform-phase and spiral-phase, radially polarized beams. In the following section we present experimental results that demonstrate that the GO system in Fig. 1 compensates the geometrical phase introduced by the AASQ and can be used to create an achromatic, uniform-phase, radially polarized beam. 4. Experimental results Figure 4(a) shows the achromatic, axially symmetric quarter-wave plate developed and introduced in a previous work [20]. The AASQ is made out of SiO2 (ShinEtsu QUARTZ Co., Ltd.). This element has a diameter of 30 mm, a width of 20 mm, and it is designed to have 90° of retardance in the visible. The principle of an achromatic, axially symmetric waveplate is based on Fresnel reflections. The waveplate has a concave conical surface created by the cross-section of a Fresnel rhomb rotated, with one of its vertices fixed, about a line that becomes the optical axis of the AASQ. The incident beam propagates parallel to the optical axis and impinges on the vertex of the concave conical surface. It then forms a cone-shaped beam that reflects on the sloping edge of the waveplate and creates a ring-shaped beam that propagates parallel to the optical axis [20]. Figure 4(b) shows a polarization image of the AASQ placed between two crossed linear polarizers. From the intensity distribution shown in Fig. 4(b), it is clear that the angle of linear polarization at the output of the AASQ varies along the azimuthal direction. The dark regions correspond to light linearly polarized at 0°.
Fig. 4. Pictures of the achromatic, axially symmetric quarter-wave plate.
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3311
Fig. 5. Mach-Zehnder interferometer for the observation of radially polarized beams.
To determine whether a radially polarized beam is uniform- or spiral-phase, we built the Mach-Zehnder interferometer shown in Fig. 5, which is based on the diagrams in Fig. 3. In this experiment we used a He-Ne laser as a light source. In addition, we perform experiments to observe the intensity distribution based on polarization states using three other lasers with different wavelengths. Here, the main element in our experimental setup is the AASQ, which has been demonstrated to be achromatic in our previous work [20]. The intensity of the linearly polarized incident laser beam was controlled with the combination of a polarizer and an attenuator. The laser beam was expanded, collimated, and sent to a non-polarizing beam splitter (NPBS) to create the reference and objective beams. In this experiment we neglect the depolarizing effects of the NPBS and folding mirrors as being small compared with other error sources in the experiment. The reference beam was transformed into a circularly polarized beam by passing it through a polarizer and a quarter-wave plate. The objective beam was transformed from a linearly polarized beam with no topological charge into a linearly polarized beam with a phase factor exp( + iθ) using the GO system. To create the uniform-phase, radially polarized beam, the linearly polarized vortex beam is sent to the CO system. Alternatively, to create the spiral-phase, radially polarized beam the GO is removed from the experimental setup and the incident beam is sent directly to the CO.
Fig. 6. Axially symmetric quarter-wave plate (ASQ).
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3312
Fig. 7. Output beam after the CO.
Figure 6(a) is a picture of an ASQ plate fabricated with 9 segments of a retarder film with their fast axis varying azimuthally. Figure 6(b) is the intensity distribution obtained when the axially symmetric quarter-wave plate is located between crossed polarizers. With a slightly tilted reference beam and the ASQ in Fig. 6(a) taking the place of the ASR in the GO system, we observe the interference fringes shown in Fig. 6(c) when the CO is removed. It is clear that the central bright fringe is split in two. This tuning fork shape is a consequence of the exp( + iθ) phase distribution in the beam created by the GO. Inserting the CO back into the system, the beam becomes doughnut-like after passing through it, as shown in Fig. 7. In this demonstration, we employed laser beams with three different wavelengths (405nm, 550nm and 632nm). The intensity distributions obtained by passing this beam through a linear polarizer at 0°, 45° and 90° are shown in Figs. 7(a)–7(i). These arrows in Fig. 7 indicate the orientation angles of the linear polarizer. These figures show that the CO converts the state of polarization of the beam from linear polarization into radial polarization. However, the intensity distributions in these figures are the same for both the uniform- and spiral-phase radially polarized beams. To prove that the beam generated by the CO has the correct phase distribution, we investigated the fringes that result from the interference of the beam generated by the CO with a slightly tilted reference beam.
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3313
Fig. 8. Experimental interference fringes for the radially polarized beams.
Fig. 9. Numerical interference fringes for the radially polarized beams.
Figures 8(a) and 8(b) show the intensity distribution of the radially polarized beams for both cases. The nearly circular rings have dark spots like doughnut shape beams. In contrast, the interference patterns are shown in Figs. 8(c) and 8(d) for the spiral-phase and the uniformphase beams, respectively. Although the effects of significant aberrations are apparent on the fringe shapes in the center area, the important point is not the specific fringes shape at the center, but rather the fringe behavior on the circular rings as shown in Figs. 8(c) and 8(d). The aberrations on the dark spots of the center area are caused by the scattering for the small amount of light that passes through the center of the AAS-QWP. Naturally, additional aberrations also arise from the optical system. Let us consider interference fringes on the circular rings to understand the generation of the radially polarized beam. The total number of interference fringes is different for the
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3314
clockwise (red numbers) and the counterclockwise (green numbers) direction in both figures. The absolute value of the difference between the number of fringes in the clockwise and counterclockwise directions in Figs. 8(c) and 8(d) is |14-12| = 2 and |14-13| = 1, respectively. These differences are consistent with what is expected from the azimuthal angle dependence of the irradiance distributions in Eqs. (7) and (9). That is to say, we can understand whether the output beam has uniform-phase or spiral-phase from the interference fringes. To better understand the experimental results, we performed a numerical calculation of the interference fringes obtained for both types of radially polarized beams when the reference beam is slightly tilted. These results are shown in Fig. 9. Although the numerical results have no aberrations, the difference in the number of fringes corresponds with the difference observed in our experimental results. Therefore, the results in Figs. 8(d) and 9(b) show that the optical configuration in Fig. 2 can be used to generate a radially polarized beam with the correct phase distribution (i.e., uniform-phase). 7. Conclusions A method to generate an achromatic, uniform-phase, radially polarized beam has been presented. In our method, a linearly polarized beam is transformed into an optical vortex with phase distribution exp( + iθ) using a vortex generation optical system. The optical vortex is converted into an axially symmetric polarized beam using a conversion optical system that employs an achromatic, axially symmetric quarter-wave plate based on Fresnel’s reflections. Calculations performed using the Jones formalism have shown that our method is able to generate an achromatic, uniform-phase, radially polarized beam. We have shown that if the vortex generation optics is omitted, the beam generated by the conversion optics has the incorrect phase distribution, i.e., is a spiral-phase, radially polarized beam. Finally, we have verified our theoretical results experimentally using an AASQ designed and fabricated at our group. Acknowledgments This work was partially supported by the Japan Science and Technology Agency, Adaptable and Seamless Technology Transfer Program through target-driven R&D with number AS242Z01381K. O. G. Rodríguez-Herrera and J. S. Tyo were supported by the Air Force Office of Scientific Research (AFOSR) under grant FA9550-10-0114.
#202763 - $15.00 USD Received 9 Dec 2013; revised 17 Jan 2014; accepted 17 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003306 | OPTICS EXPRESS 3315
