Tight focusing of elliptically polarized vortex beams Baosuan Chen and Jixiong Pu* College of Information Science and Engineering, Huaqiao University, Quanzhou, Fujian 362021, China *Corresponding author:
[email protected] Received 30 October 2008; revised 17 December 2008; accepted 26 January 2009; posted 27 January 2009 (Doc. ID 103431); published 23 February 2009
We study the focusing properties of elliptically polarized vortex beams. Based on vectorial Debye theory, some numerical calculations are given to illustrate the intensity and phase distribution properties of tightly focused vortex beams. It is found that the spin angular momentum of the elliptically polarized vortex beam will convert to orbital angular momentum by the focusing. The influence of corresponding parameters on focusing properties is also investigated in great detail. It is shown that elliptical light spots can be obtained in the focal plane. Moreover the elliptical spot may rotate and the spot shape may change with the change of certain parameters. These properties are quite important for application of this kind of elliptically polarized vortex beam. © 2009 Optical Society of America OCIS codes: 050.1960, 110.1220, 260.5430.
1. Introduction
2.
Since the 1950s, tight focusing properties of laser beams have been investigated in great detail [1–3]. It has been found that beams focused by a high NA have unique properties, such as producing a strong longitudinal component and focusing to a tighter spot [4,5]. For its unique properties, tightly focused light beams have wide potential applications in optical data storage, microscopy, material processing, and optical trapping [6–10]. Therefore, behavior of tightly focused beams has attracted much attention from researchers [10–13]. Recently, a new kind of beam with an optical vortex became an issue of interest [14]. Different from common beams, this kind of beam has a screw wavefront that generates orbital angular momentum [15,16]. Laser beams with such orbital angular momentum have wide applications in micromanipulation, in an optical spanner, and in quantum information, and have been extensively studied in recent years [17–20]. Here, based on vectorial Debye theory [21], we extend the analysis to the tight focusing of elliptically polarized vortex beams, which can produce controllable elliptical beam spots.
Let us first consider the focusing of a linearly polarized beam by a high numerical aperture (NA). According to the classic paper by Wolf [1], the electric field Eðr; φ; zÞ in the focal plane can be expressed as
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Theoretical Model
2
Ex
3
ikf 6 7 Eðr; φ; zÞ ¼ 4 Ey 5 ¼ − 2π Ez
Z
α 0
Z
2π 0
Aðθ; ϕÞ
× exp½ikðz cos θ þ r sin θ pffiffiffiffiffiffiffiffiffiffi × cosðϕ − φÞÞ sin θ cos θ 3 2 cos2 ϕ cos θ þ sin2 ϕ 7 6 × 4 cos ϕ sin ϕðcos θ − 1Þ 5dϕdθ:
ð1Þ
sin θ cosðϕÞ where r, φ, and z are the cylindrical coordinates of an observation point, shown in Fig. 1. k ¼ 2π=λ is the wave vector, and f is the focal length of the high NA objective. α ¼ arcsin NA is the maximal angle determined by the NA of the objective. Aðθ; ϕÞ is the pupil apodization function at the objective aperture surface, which is related to the electric field of the incident beam. Elliptically polarized light is simply the superposition of two orthogonal linearly polarized beams
focal length of the objective. Therefore, the pupil apodization function of the linearly polarized LG beams can be expressed as
Fig. 1. Scheme of tight focusing.
pffiffiffi 2f sin θ jmj Amj ðθ; ϕÞ ¼ Amj ðθÞ expðimϕÞ ¼ E0j w0 2 2 f sin θ expðimϕÞ; ðj ¼ x; yÞ: ð5Þ × exp − w0 2
with a certain retardation β between them. Thus the electric field in the focal region when an elliptically polarized light is tightly focused can be calculated as
On substituting Eq. (5) into Eq. (2) and after some simplification, the x, y, and z components of the electric field in the focal region can be simplified as
2
Ex Ex 0
3
Z Z pffiffiffiffiffiffiffiffiffiffi ikf α 2π 7 6 sin θ cos θ exp½ikðz cos θ þ r sin θ cosðϕ − φÞÞ Eðr; φ; zÞ ¼ 4 Ey Ey 0 5 ¼ − 2π 0 0 Ez Ez 0 3 2 Ax ðθ; ϕÞðcos2 ϕ cos θ þ sin2 ϕÞ Ay ðθ; ϕÞeiβ cos ϕ sin ϕðcos θ − 1Þ 7 6 × 4 Ax ðθ; ϕÞ cos ϕ sin ϕðcos θ − 1Þ Ay ðθ; ϕÞeiβ ðcos2 ϕ þ sin2 ϕ cos θÞ 5dϕdθ:
ð2Þ
Ax ðθ; ϕÞ sin θ cosðϕÞ Ay ðθ; ϕÞeiβ sin θ sinðϕÞ
Laguerre–Gaussian (LG) beams have a helical phase structure of expðimϕÞ, therefore they can be treated as vortex beams. Here we consider the incident beam an elliptically polarized LG beam, the electric field of which can be expressed as E ðrÞ ¼ Ex ex Ey eiβ ey :
ð3Þ
Here, Eþ ðrÞ indicates the right-hand elliptical (RHE) polarized beam, while E ðrÞ indicates the left-hand elliptical (LHE) polarized beam. Ex and Ey are the electric fields of two orthogonal linear components, respectively. β is the retardation between the two beams. ex and ey are unit vectors along the x and the y directions, respectively. Considering p ¼ 0, the electric field of a linearly polarized LG beam in the source plane is given as pffiffiffi jmj r2 2r Emj ðr; ϕÞ ¼ E0j exp − 2 expðimϕÞ; w0 w0 ð4Þ ðj ¼ x; yÞ; where E0j and w0 are the constants representing amplitude and beam size. m is the topological charge. Since objectives are often designed to obey the sine condition [21], we get r ¼ f sin θ, where f is the
E;x ðr; φ; zÞ ¼ −
ikf 2
Z
α
0
pffiffiffiffiffiffiffiffiffiffi sin θ cos θ expðikz cos θÞ
× fAmx ðθÞð1 þ cos θÞim J m ðkr sin θÞ expðimφÞ 1 þ ðAmx ðθÞ ð−iÞAmy ðθÞeiβ Þðcos θ − 1Þimþ2 2 × J mþ2 ðkr sin θÞ exp½iðm þ 2Þφ 1 þ ðAmx ðθÞ iAmy ðθÞeiβ Þðcos θ − 1Þim−2 2 × J m−2 ðkr sin θÞ exp½iðm − 2Þφgdθ;
E;y ðr; φ; zÞ ¼ −
ikf 2
Z 0
α
ð6Þ
pffiffiffiffiffiffiffiffiffiffi sin θ cos θ expðikz cos θÞ
iβ
× fAmy ðθÞe ð1 þ cos θÞim J m ðkr sin θÞ expðimφÞ 1 þ ½ð−iÞAmx ðθÞ∓Amy ðθÞeiβ ðcos θ − 1Þimþ2 2 × J mþ2 ðkr sin θÞ exp½iðm þ 2Þφ 1 þ ðiAmx ðθÞ∓Amy ðθÞeiβ Þðcos θ − 1Þim−2 2 × J m−2 ðkr sin θÞ exp½iðm − 2Þφgdθ; 1 March 2009 / Vol. 48, No. 7 / APPLIED OPTICS
ð7Þ 1289
ikf E;z ðr; φ; zÞ ¼ − 2
Z 0
α
pffiffiffiffiffiffiffiffiffiffi sin2 θ cos θ expðikz cos θÞ
× f½Amx ðθÞ ð−iÞAmy ðθÞeiβ imþ1 J mþ1 ðkr sin θÞ × exp½iðm þ 1Þφ þ ðAmx ðθÞ iAmy ðθÞeiβ Þim−1 J m−1 ðkr sin θÞ × exp½iðm − 1Þφgdθ:
ð8Þ
From the above-derived equations we can see that, after being focused by a high NA objective, both E;x ðr; φ; zÞ and E;y ðr; φ; zÞ have three parts each with topological charge of l ¼ m and l ¼ m 2 and E;z ðr; φ; zÞ has two parts each with topological charge of l ¼ m þ 1 and l ¼ m − 1, indicating that, through focusing, the spin angular momentum (SAM) of the incident beam will covert to the orbital angular momentum (OAM) [17]. Based on Eqs. (6)– (8), the total intensity distribution and its x, y, and z components for the RHE- and LHE-polarized beam can be obtained, respectively. Moreover the experiment can be realized in the laboratory by the system shown in Fig. 1, which will be the subject of our further study. 3. Results and Discussion
We performed some numerical calculations on the focusing properties for the elliptically polarized beams. The fixed parameters for the calculations are λ ¼ 632:8 nm, f ¼ 1 cm, and w0 ¼ 1 cm. Figure 2 shows the total intensity distribution and its x, y, and z components in the focal plane for different topological charges and different polarization directions. All the components are normalized to their total intensity. It is shown that elliptical focal spots are obtained when the elliptically polarized beam is focused by a high NA objective. That may be attributed to the different amplitudes of the incident fields Emx ðr; ϕÞ and Emy ðr; ϕÞ, and the phase retardation between them. Elliptical spots are quite important when using laser beams to manipulate elliptical particles, for example, liquid crystalline molecules. When topological charge m ¼ 0 (RHE-polarized beams) [i.e., nonvortex; see Figs. 2(a), 2(d), 2(g), and 2(j)], the total intensity (I t ) and its x and y components (I x and I y ) in the focal plane are nonzero at the center, while the z component (I z ) has a zero central intensity. It is shown in Figs. 2(b), 2(e), 2(h), and 2(k) that, for topological charge m ¼ 1 (RHEpolarized beams), i.e., vortex beams, there is a dark core for I t, I x , and I y . However, it is interesting to discover that the z component I z has two dark cores, which indicates that there are two vortices in the intensity distribution of I z. The focusing properties for LHE-polarized beams with m ¼ 1 are shown in Figs. 2(c), 2(f), 2(i), and 2(l). Different from the RHE-polarized beams, the central intensity of I z is not hollow since the OAM is partly compensated by the OAM converted from the SAM of the elliptically polarized beam. 1290
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The phase contours of Ez ðr; ϕ; zÞ with different topological charges in the focal plane are shown in Fig. 3. It is shown that the phase contour of the m ¼ 0 beam exhibits a counterclockwise [Fig. 3(a), RHEpolarized beam] and a clockwise [Fig. 3(b), LHEpolarized beam] helical phase distribution, which provides effective evidence that the SAM of the incident beam will covert to the OAM when the beam is focused by a high NA objective. The phase contour in Fig. 3(c) exhibits a combined vortex with each helical phase changing from −π to π, which also explains why the z-component intensity distribution has two dark cores [shown in Fig. 2(k)]. From Fig. 3(d), it is seen that the phase of the center is fixed, which indicates that the wavefront of the LHE-polarized beam has no screw in the focal plane. That is because the OAM is partly compensated by the OAM converted from the SAM of the elliptically polarized beam. The number of vortices is directly correlated with the topological charge (l) of the beam, and the OAM of the beam can be calculated by lℏ [15], therefore, the change in the number of vortices indicates the change of OAM of the beam while focusing. It is shown that compared with the incident beam, the OAM of the focused RHE-polarized beam increases [see Fig. 3(a), 3(c), and 3(e)], however, the OAM of the focused LHEpolarized beam was found to be decreased [see Fig. 3(b), 3(d), and 3(f)]. That is because for the RHE-polarized beam, the OAM converted from SAM is in the same direction as the original OAM, enhancing the total OAM, while for the LHE-polarized beam, the OAM converted from SAM is in the opposition direction to the original OAM, reducing the total OAM. Phase contours of the x, y, and z components in the focal plane with m ¼ 1 are shown in Fig. 4. The phase contours present spiral structures in the focal plane, which is called screw wavefront dislocation [14]. It is shown that for both RHE- and LHE-polarized beams there is only one vortex in the phase contours of Ex ðr; ϕ; zÞ and Ey ðr; ϕ; zÞ, but the phase contours of Ez ðr; ϕ; zÞ exhibit a combined vortex and no vortex for RHE- and LHE-polarized beams, respectively, which indicates that the SAM to OAM conversion is less obvious in the phase distributions of Ex ðr; ϕ; zÞ and Ey ðr; ϕ; zÞ than that in Ez ðr; ϕ; zÞ. Then we investigate the influence of relative parameters on the focusing properties. Figures 5–8 are plotted to illustrate the influence of varying E0y on the total and three components of intensity distribution in the focal plane. Figure 5 shows the influence of E0y on the total intensity I t in the focal plane with m ¼ 1 (RHE-polarized beam). It is shown that when E0y is comparable to E0x , E0y and the phase retardation β that determine the incident beam polarization act together on the total intensity pattern in the focal plane and induce the intensity pattern to rotate at an angle that is correlated with the phase retardation, shown in Fig. 5(a). And with the increase of E0y, the intensity pattern rotates in a counterclockwise direction. Moreover, when E0y ¼ 40, i.e., E0y ≫ E0x ,
Fig. 2. Intensity distributions in the focal plane for different topological charges. (a), (d), (g), (j) m ¼ 0 (RHE-polarized beam); (b), (e), (h), (k) m ¼ 1 (RHE-polarized beam); (c), (f), (i), (l) m ¼ 1 (LHE-polarized beam). (a)–(c) total intensity It, (d)–(f) x component I x , (g)–(i) y component I y , (j)–(l) z component I z . The other parameters are chosen as NA ¼ 0:9, β ¼ π=4, E0x ¼ 1, E0y ¼ 1:5.
indicating that E0y becomes the main factor that influences the intensity distribution. In this case, the incident beam can be taken as a y-linearly polarized beam, the intensity pattern exhibits no rotation and becomes symmetric to both the x axis and the y axis. This phenomenon is also shown in the influence of E0y on I z distribution; see Fig. 8. The influence of varying E0y on I x distribution in the focal plane with m ¼ 1 (RHE-polarized beam) is illustrated in Fig. 6. It is found that, with the increase of E0y, the beam spot in the focal plane is elongated in the diagonal direction, and the intensity finally disperses on
two orthogonal ellipses. When E0y ¼ 40 (the incident beam can be taken as y-linearly polarized), the intensity distribution is shown in Fig. 6(d). The research finding indicates that E0y is an important parameter in controlling the focused field properties. This result is significant since it allows the possibility to control the intensity pattern to match specific applications. It can be seen from Fig. 7 that varying E0y has little influence on the y-component intensity distribution. Figure 9 shows the focused total intensity distribution with different phase retardation β. As β changes from 0 to π=2, the intensity pattern gradually 1 March 2009 / Vol. 48, No. 7 / APPLIED OPTICS
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Fig. 5. Influence of varying E0y on It in the focal plane with m ¼ 1 (RHE-polarized beam). (a) E0y ¼ 1, (b) E0y ¼ 2, (c) E0y ¼ 4, (d) E0y ¼ 40. The other parameters are the same as in Fig. 2. Fig. 3. Phase contours of Ez ðr; ϕ; zÞ in the focal plane. (a), (b) m ¼ 0; (c), (d) m ¼ 1; (e), (f) m ¼ 2; (a), (c), (e) RHE-polarized beam; (b), (d), (f) LHE-polarized beam. The other parameters are the same as in Fig. 2.
to the elliptical distribution, but with the longer axis perpendicular to that of the original ellipse. The influence of varying NA on the total intensity distribution in the focal plane is presented in Fig. 10. It is shown that with the increase of NA, the
changes from elliptical distribution into circular distribution. The central intensity gradually decreases and leads to a dark core distribution when β ¼ π=2, i.e., when the incident beam is a circularly polarized vortex beam. Then as the phase retardation β further increases, the intensity changes back
Fig. 4. Phase contours of the x, y, and z components in the focal plane with m ¼ 1. (a), (b), (c) RHE-polarized beam; (d), (e), (f) LHEpolarized beam; (a), (d) Ex ðr; ϕ; zÞ; (b), (e) Ey ðr; ϕ; zÞ; (c), (f) Ez ðr; ϕ; zÞ. The other parameters are the same as in Fig. 2. 1292
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Fig. 6. Influence of varying E0y on Ix in the focal plane with m ¼ 1 (RHE-polarized beam). (a) E0y ¼ 1, (b) E0y ¼ 4, (c) E0y ¼ 14, (d) E0y ¼ 40. The other parameters are the same as in Fig. 2.
Fig. 9. Influence of varying phase retardation β on I t in the focal plane with m ¼ 1 (RHE-polarized beam). (a) β ¼ 0, (b) β ¼ π=6, (c) β ¼ π=3, (d) β ¼ π=2, (e) β ¼ 2π=3, (f) β ¼ π. E0y ¼ 1. The other parameters are the same as in Fig. 2.
Fig. 7. Influence of varying E0y on I y in the focal plane with m ¼ 1 (RHE-polarized beam). (a) E0y ¼ 1, (b) E0y ¼ 4, (c) E0y ¼ 14, (d) E0y ¼ 40. The other parameters are the same as in Fig. 2.
characteristics of the intensity patterns stay unchanged except that the focal spot becomes smaller. That is because the increase of NA leads to a tighter focused transverse field. Last but not least, the influence of varying phase retardation β on the phase distributions of Ez ðr; ϕ; zÞ
Fig. 8. Influence of varying E0y on I z in the focal plane with m ¼ 1 (RHE-polarized beam). (a) E0y ¼ 1, (b) E0y ¼ 2, (c) E0y ¼ 4, (d) E0y ¼ 40. The other parameters are the same as in Fig. 2.
in the focal plane is indicated in Fig. 11. It is found that with β changing from 0 to π=2, the core of the two vortices will gradually get closer and will be combined when β ¼ π=2, i.e., when the incident beam is a circularly polarized vortex beam. Then by further increases in β, the two cores of the combined vortex separate again and return to the original phase distribution when β ¼ π, but with its direction perpendicular to that of the original distribution (when β ¼ 0).
Fig. 10. Influence of varying NA on the total intensity in the focal plane with m ¼ 1 (RHE-polarized beam). (a) NA ¼ 0:8, (b) NA ¼ 0:85, (c) NA ¼ 0:9, (d) NA ¼ 0:95. E0y ¼ 1. The other parameters are the same as in Fig. 2. 1 March 2009 / Vol. 48, No. 7 / APPLIED OPTICS
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Fig. 11. Influence of varying phase retardation β on phase distributions of Ez ðr; ϕ; zÞ with m ¼ 1 (RHE-polarized beam) in the focal plane. (a) β ¼ 0, (b) β ¼ π=6, (c) β ¼ π=3, (d) β ¼ π=2, (e) β ¼ 2π=3, (f) β ¼ π. E0y ¼ 1. The other parameters are the same as in Fig. 2.
4. Conclusions
Based on vectorial Debye theory, the tight focusing properties of elliptically polarized vortex beams have been analyzed. We have studied the intensity and phase properties in the focal plane and the influence of relative parameters on them. It is found that elliptical beam spots are obtained when the elliptically polarized vortex beams are focused by a high NA objective. And the SAM of the elliptically polarized beam will convert to OAM when the beam is tightly focused. Moreover, by adjusting relative parameters, the shape of the beam spot and the direction of the longer axis can be controlled. This research finding is important in the applications of tightly focused elliptical spots, especially the applications in elliptical particle trapping, manipulation, and so on. This research is supported by the Natural Science Foundation of Fujian Province under grant A0810012, and Key Project of Science and Technology of Fujian Province under grant 2007H0027. References 1. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959). 2. A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561– B1565 (1965).
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3. J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576–1578 (2002). 4. N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229–234 (2007). 5. T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314–319 (2007). 6. Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel–Gaussian beams through a high numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008). 7. G. M. Lerman and U. Levy, “Tight focusing of spatially variant vector optical fields with elliptical symmetry of liner polarization,” Opt. Lett. 32, 2194–2196 (2007). 8. E. P. Walker and T. D. Milster, “Beam shaping for optical data storage,” Proc. SPIE 4443, 73–92 (2001). 9. L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212, 343–352 (2002). 10. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77– 87 (2000). 11. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). 12. Z. Zhang, J. Pu, and X. Wang, “Tight focusing of radially and azimuthally polarized vortex beams through a uniaxial birefringent crystal,” Appl. Opt. 47, 1963–1967 (2008). 13. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). 14. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001). 15. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). 16. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). 17. Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phases in tightly focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006). 18. F. S. Roux, “Distribution of angular momentum and vortex morphology in optical beams,” Opt. Commun. 242, 45–55 (2004). 19. J. Courtial and M. J. Padgett, “Limit to the orbital angular momentum per unit energy in a light beam that can be focused onto a small particle,” Opt. Commun. 173, 269–274 (2000). 20. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997). 21. M. Gu, Advanced Optical Imaging Theory (Springer, 2000).
