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Manifestation of the Gouy phase in vector-vortex beams Geo M. Philip,1 Vijay Kumar,1 Giovanni Milione,2 and Nirmal K. Viswanathan1,* 1 2
School of Physics, University of Hyderabad, Hyderabad 500046, India
The City College of New York of the City University of New York, New York 10031, USA *Corresponding author:
[email protected] Received March 27, 2012; revised May 9, 2012; accepted May 9, 2012; posted May 9, 2012 (Doc. ID 165562); published June 26, 2012
Experimental measurements of the twirl and changes in the anisotropy of the constant intensity ellipse, and the rotation of the polarization singular lemon pattern a generalized vector-vortex beam experiences around the two foci due to the converging and diverging conical waves and in between, are presented and interpreted as being due to the universal form of the Gouy phase, ϕG mπ∕2. © 2012 Optical Society of America OCIS codes: 260.0260, 260.6042, 260.2130, 060.2310.
Vector optical beams wherein the spatially inhomogeneous polarization characteristics vary dramatically around a phase singularity, resulting in the appearance of polarization singularities, are termed as vector-vortex beams (VVBs) [1–3]. The coherent superposition of a homogeneous orthogonal circularly polarized vortex beam and Gaussian beam results in the appearance of polarization singularities (PS) in the transverse plane: C-singularity where the polarization ellipse is circular, and L-singularity where the polarization ellipse degenerates into a straight-line segment [1,3]. The polarization ellipse pattern surrounding the C-point can be any of the six morphological types: elliptic or hyperbolic contours with lemon, star, or monstar lines [1,3]. In VVBs generated due to the off-axis interference of the constituent beams, the noncanonical constant intensity contour [4] and the elliptic lemon pattern around the C-point [5] experiences twirl (tw) (rotation of constant intensity and constant “b∕a” ellipses) and twist (rate of rotation of phase around the dislocation and C-points) upon propagation [6]. For the VVBs considered here in paraxial approximation, we track the twirl ϕtw and the anisotropy cosψ 2a∕b b∕a−1 of the constant intensity contour and the rotation ϕR of the lemon PS pattern around the C-point as the beam passes through the two foci of converging and diverging conical waves due to an axicon. Such measurements are realized by focusing the VVB, generated using a two-mode optical fiber [7,8], using the axicon. The twirl and the twist the VVB experiences during propagation, around the foci and in between, is a result of the Gouy phase experienced by the constituent beams and confirms that the beam carries orbital angular momentum (OAM) [9]. Further, these measurements verify the universal form of the Gouy phase: nonspherical converging/diverging waves passing through two successive foci experience π∕2 phase change per focus, adding up to π for two successive foci [10–12]. Up until now, the vector vortices [13] are experimentally realized using space-variant, subwavelength gratings [14], although the potential to realize such beams using liquid crystal devices via intracavity summation of laser modes and interferometric techniques exists. Missing in this repertoire of techniques is the inhomogeneous medium such as optical waveguides. Using a twomode optical fiber we have reported the generation of VVBs [7,8], including the generation of isolated optical 0146-9592/12/132667-03$15.00/0
singularities—a single C-point surrounded by a welldefined L-line contour [8]. It is expected that the phase evolution studies of these VVBs will enable several applications, including optical trapping, lithography, material processing, and free-space communication. The vortex beam at the output of an optical fiber can be treated as linear superposition of the Hermite-Gauss (HG) beams: Ex; y; z xF iyF HG10 iHG01 , where Fx; y; z is the host beam profile. In its generalized form, a single-charge optical vortex can be written as [4], Ex; y; z fpx expiπ∕2 φs qygFx; y; z, where p and q are constants. An additional phase φs is introduced between the HG10 and HG01 beams to render the vortex beam noncanonical. For φs 0, we get canonical vortex due to the fixed π∕2 phase difference between the HG beams. The intensity contour around the vortex core can be written as I E × E p2 x2 q2 y2 2 pq cosπ∕2 φs xy: (1) If we consider an ellipse with a, b, and ϕ as major and minor semi-axes and ellipse orientation angle, respectively, the ellipse equation, in terms of its orientation, can be written as 2 2 cos ϕ sin2 ϕ 2 sin ϕ cos2 ϕ 2 2 x y a2 b a2 b2 1 1 2 cos ϕ sin ϕ 2 2 xy: a b
(2)
Comparing Eqs. (1) and (2), we get cos ϕ sin ϕ π cos φs 2 pq 2
2
2
h
1 a2
b12
2
i ;
(3)
where p2 cosa2 ϕ sinb2 ϕ and q2 sina2 ϕ cosb2 ϕ. This noncanonical vortex beam, when focused using a lens, results in two Rayleigh ranges, Z Rx and Z Ry , due to astigmatism and/or beam misalignment [15]. Moving towards the focus, the constant intensity ellipse twirls around the axis with a simultaneous decrease in the core anisotropy, becoming isotropic at the focus due to the different Gouy © 2012 Optical Society of America
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phase acquired by the misaligned constituent HG10 and HG01 modes. The vector optical fields are also characterized by ellipse parameters [1,3]. In two dimensions, the superposition of two orthogonal, elliptically polarized fields results in the formation of C-point and L-line [1,3]. Rotation of the polarization ellipse orientation around the C-point by π corresponds to 1∕2 index [1,3]. The positive index PS occurs in two forms, lemon and monstar, whereas star is the only form with negative index singularity [1,3,5]. The PSs are of special interest, as they form the skeleton of the vector field and give its surroundings significant physical properties [3,6]. In the generation of PSs using optical fiber, a circularly polarized fundamental mode (Gaussian output beam) and orthogonal circularly polarized vortex mode are simultaneously excited and the x, y components of the resultant field are given by [1] E x εeiφF x ilyjlj eiφV ; E y iσ F εeiφF iσ V x ilyjlj eiφV ; where the first and second terms in the square brackets represent the fundamental (F) and vortex (V ) modes respectively, and ε is real amplitude corresponding to perturbed fundamental mode. If the polarization ellipse map is simulated using the above equations for a vortex of charge l 1 with circular polarization σ V −1 (LCP) and σ F 1 (RCP), a lemon-type PS pattern surrounding the C-point is generated. The orientation of the polarization pattern is determined by the relative phase difference between the fundamental and vortex modes, i.e., Δϕ ϕV ∼ ϕF . The resulting PS pattern rotates as the beam evolves around the foci due to the Gouy phase difference between the constituent modes [16]. Figure 1 shows the schematic of the experimental setup used for generating a noncanonical vortex beam with embedded isolated polarization singularity. The VVB generator consists of a He─Ne laser (λ 632.8 nm) and a horizontally held, 42.8 cm long two-mode fiber (TMF). The circularly-polarized Gaussian beam from the laser is coupled into the TMF as an offset-tilted beam [7,8]. The VVB with noncanonical vortex core and uniquely identifiable lemon PS pattern around the C-point (and within the L-contour) is achieved by adjusting the fiber launch and the input beam polarization [8]. The output beam from the TMF is collimated and its intensity is measured using a CCD camera and its polarization using spatiallyresolved Stokes parameter measurements [8]. The collimated VVB is then focused using an axicon (A) of open angle γ 0.5°. The first ring of the focused beam intensity (with constant intensity contour) and the PS pattern around the C-point (with constant “b∕a” contour), measured at a distance of z −25 cm, are
Fig. 1. Schematic of the experimental setup. BE, Beam expander; A, axicon; d1 , d2 two focal planes of the axicon.
shown in Figs. 2(a) and 2(b). It is seen from Fig. 2(a) that the vortex core is elliptic with anisotropy cos ψ 0.958, and is oriented at an angle ϕ −37° with respect to the horizontal (x) axis. Figure 2(b) shows that the PS pattern around the C-point (white dot) in the RCP field (region inside second white contour) has a lemon line pattern (white line) and an elliptic contour. The right circular polarization (RCP) field transforms into left circular polarization (LCP) (region outside second white contour) via linear polarization as one move radially away from the C-point. The simulated scalar and vector characteristics of the beam output from the TMF is carried out using the formalism discussed earlier [Eqs. (2) and (3)] and shown in Figs. 2(c) and 2(d), respectively, are found to match well with the experimental results. The intensity and polarization characteristics of the VVB are measured, after the axicon, at different transverse planes (in 10 mm intervals) along the propagation axis-from the first focus to the second through the propagation invariant (PI) region, covering the Rayleigh ranges of the converging and diverging regions of the conical waves. From the images captured using the CCD as a function of z, the average intensity of the first ring of the J 1 -BG (Bessel-Gauss) beam is plotted in Fig. 3 and is found to be similar to that of the linear axicon: linear increase in intensity within the region d1 < z < d2 [17]. More importantly, this result indicates that the conical waves from within the core region of the astigmatic vortex beam crosses the z-axis at z d1 ≠ 0 and that from the outer region at z d2 . The different Gouy phase acquired by the astigmatic beam (in the “xz” and “yz” planes) due to the misaligned constituent HG beams, results in two Rayleigh ranges, leading to the transformation of the noncanonical vortex to a canonical vortex. This is evidenced by the reduction in the core anisotropy, becoming more circular, cosψ ≈ 1 at z d1 . The canonical vortex remains so in the PI region and becomes noncanonical beyond z d2 . This behavior is clear from the plot of the anisotropy cosψ of the core region of the J 1 -BG beam, calculated from the measured intensity images, as a function of z (Fig. 3). As seen from the graph, the anisotropy is close to 1 in the PI region, indicating a canonical vortex core. The separation between the
Fig. 2. (Color online) (a), (b) Intensity and polarization ellipse of the output beam measured at z −25 cm; (c), (d) the corresponding theoretically calculated patterns. See text for details.
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2.0 1.0
0.8
0.6
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d2
d1 0.4
0.4
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0.0 -40
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-20
-10
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- Axicon - Lens
/ /
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z (cm)
Fig. 3. (a) Plot of normalized average intensity of the first ring of the J 1 -BG beam (□) and the anisotropy of the vortex core (•) as a function of z. Open box indicates the PI region.
beginning and the end of the canonical vortex region, d2 −d1 43 cm agrees well with the maximum PI region of 42.16 cm for the beam calculated using zmax ω0 k∕kr with kr ≈ n − 1γk [18]. Thus, even though the vortex core was aligned with the axicon apex, the generated J 1 -BG beam [18] due to the ellipticity of the input beam vortex core has two foci, characteristic of astigmatic wave fields [12]. We observed that the noncanonical, constant intensity vortex core twirls around by π∕2 (π∕4 within the Rayleigh range) in the neighborhood of the two foci of the conical waves but remain irrotational in the PI region, between d1 and d2 . The plot of the twirl ϕtw as a function of z shown in Fig. 4 confirms that the Gouy phase accumulated by the astigmatic beam passing through the two foci results in a total phase change of π. This behavior follows the universal form of the Gouy phase, ϕG z c arctanz∕zR , where c 1∕2 for a focal line (cylindrical wavefront) and c 1 for a focal point (spherical wavefront) [19,20]. In addition to the intensity, we also measured the polarization characteristics of the VVB at the different transverse planes along z. The measured rotation of the lemon pattern around the C-pointϕR quantifies the Gouy phase accumulated by the constituent beams during propagation. The VVB passing through the focus acquires mode-dependent phase change [16], and the resultant relative phase change between the propagating fundamental and vortex modes manifests as a rotation of the lemon PS pattern. We measured the axial phase change directly by tracking the rotation angle of the lemon PS pattern embedded in the J 1 -BG beam. Focusing the output beam from the fiber, which is a combination of the fundamental and vortex modes of orthogonal circular polarization, leads to a relative Gouy phase change between the modes, resulting in the rotation of the lemon PS pattern as a function of z. Measured along z, the PS pattern of the VVB focused by the axicon clearly shows that the orientation of the lemon pattern rotates fast around the two foci (d1 and d2 ) but remains irrotational in the propagation invariant region of the J 1 -BG beam, as shown in Fig. 4. A rotation angle of π∕2, following the relation ϕR z 1∕2 arctanz∕zR ϕG z, each is measured around the two foci of the conical waves, resulting in the overall Gouy phase change of π for the PI J 1 -BG beam as shown in Fig. 4. It is worth mentioning here that the constant “b∕a” elliptic contour around the C-point behaves the same way as cosψ shown
Fig. 4. Measured twirl of the constant intensity ellipse (open symbol) and rotation of the lemon PS pattern (filled symbol) as a function of z for axicon and lens focusing of the VVB and the corresponding fit.
in Fig. 3. The above measurements are further compared with that due to the focusing of the VVB by a spherical lens (f 40 cm) as shown in Fig. 4, which gives ϕtw ϕR ϕG π. Our results on the changes in the degree of polarization the VVB experiences will be reported elsewhere. In summary, we report here the twirl of the constant intensity ellipse and the rotation of the lemon PS pattern due to conical and spherical focusing of VVB. The associated changes in the intensity and the anisotropy of the PI J 1 -BG beam are also presented. The results presented are a manifestation of the universal form of the Gouy phase of astigmatic wave fields. Extending this to investigate the dynamics of dipoles, created in different VVBs, is expected to provide rich information on the total OAM of such beams. The authors acknowledge DST India for financial support for the project. GMP and VK acknowledge CSIR India for research fellowship. References 1. J. F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, 1999). 2. M. V. Berry, Proc. SPIE 4403, 1 (2001). 3. M. R. Dennis, Opt. Commun. 213, 201 (2002). 4. G. M. Terriza, E. M. Wright, and L. Torner, Opt. Lett. 26, 163 (2001). 5. M. R. Dennis, Opt. Lett. 33, 2572 (2008). 6. M. R. Dennis, J. Opt. A 6, S202 (2004). 7. N. K. Viswanathan and V. V. G. K. Inavalli, Opt. Lett. 34, 1189 (2009). 8. Y. V. Jayasurya, V. V. G. K. Inavalli, and N. K. Viswanathan, Appl. Opt. 50, E131 (2011). 9. J. Visser and G. Nienhuis, Phys. Rev. A 70, 013809 (2004). 10. L. G. Gouy, C. R. Acad. Sci. 110, 1251 (1890). 11. L. G. Gouy, Ann. Chem. Phys. 24, 145 6e series (1891). 12. T. D. Visser and E. Wolf, Opt. Commun. 283, 3371 (2010). 13. P. Paakkonen, J. Tervo, P. Vahimaa, J. Turunen, and F. Gori, Opt. Express 10, 949 (2002). 14. A. Niv, G. Biener, V. Kleiner, and E. Hasman, Opt. Express 14, 4208 (2006). 15. A. Y. Bekshaev, M. V. Vasnetsov, and M. Soskin, Opt. Spectrosc. 100, 910 (2006). 16. D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, Phys. Rev. Lett. 101, 050501 (2008). 17. A. T. Friberg, J. Opt. Soc. Am. A 13, 743 (1996). 18. J. Arlt and K. Dholakia, Opt. Commun. 177, 297 (2000). 19. R. Simon and N. Mukunda, Phys. Rev. Lett. 70, 880 (1993). 20. S. Feng and H. G. Winful, Opt. Lett. 26, 485 (2001).
