Generation of perfect vectorial vortex beams - OSA Publishing

Letter

Vol. 41, No. 10 / May 15 2016 / Optics Letters

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Generation of perfect vectorial vortex beams PENG LI,1,2 YI ZHANG,1 SHENG LIU,1 CHAOJIE MA,1 LEI HAN,1 HUACHAO CHENG,1

AND

JIANLIN ZHAO1,*

1

Key Laboratory of Space Applied Physics and Chemistry, Ministry of Education and Shaanxi Key Laboratory of Optical Information Technology, School of Science, Northwestern Polytechnical University, Xi’an 710072, China 2 e-mail: [email protected] *Corresponding author: [email protected] Received 16 March 2016; accepted 12 April 2016; posted 13 April 2016 (Doc. ID 261341); published 5 May 2016

We propose the concept of perfect vectorial vortex beams (VVBs), which not merely have intensity profile independent of the polarization order and the topological charge of spiral phase, but also have stable intensity profile and state of polarization (SoP) upon propagation. Utilizing a Sagnac interferometer, we approximately generate perfect VVBs with locally linear and elliptical polarizations, and demonstrate that such beams can keep their intensity profile and SoP at a certain propagation distance. These proposed VVBs can be expanded to encode information and quantum cryptography, as well as to enrich the conversion of spin and orbital angular momenta. © 2016 Optical Society of America OCIS codes: (260.5430) Polarization; (050.4865) Optical vortices; (260.6042) Singular optics; (050.1940) Diffraction. http://dx.doi.org/10.1364/OL.41.002205

The complex manipulation of cross-sectional amplitude, phase, and polarization of light beams has been intensively studied in the past two decades. Some specially tailored optical beams have been proposed, such as vortex beams possessing orbital angular momentum (OAM) associated with helical wavefronts [1] and vector beams with spatially varying states of polarization (SoP) [2]. Such beams have found increasing utilization in a variety of applications, including micromanipulation [1], optical microscopy [3], quantum information [1], and highcapacity optical communication [4,5]. Recently, vectorial vortex beams (VVBs), which possess the properties of vector beams and vortex beams, have been demonstrated to enable interesting optical phenomena and applications [6,7]. A VVB can be considered as the coherent superposition of two orthogonal circularly polarized vortices with different phase topological charges [8]. In this principle, some previous works for generating VVBs have been explored, for instance, via liquid crystal-based inhomogeneous birefringent elements [9], space-variant subwavelength gratings [8,10], and modified interferometers [11,12]. However, in these types of construction, the intensity profile and beam divergence of the constituent vortices are strongly dependent on their topological charges [13]. A major drawback of this property is that, to composite VVBs, the intensity profiles of the two constituent vortices cannot closely coincide with each other. Moreover, once the topological charges of the constituent 0146-9592/16/102205-04 Journal © 2016 Optical Society of America

vortices are different, the composited VVBs will develop unstable SoPs [14], and the intensity profile will collapse [8] upon propagation. More recently, the concept of perfect vortex whose annular intensity profile is independent of topological charge has been proposed [15–18]. However, the perfect vortices are obtained at a fixed propagation distance, and have never yet involved the vector polarization. In this Letter, we attempt to generate perfect VVBs, which not only have intensity profile independent of the polarization order and the topological charge of spiral phase, but also possess stable SoPs and intensity profiles upon propagation. Based on this concept, we develop a scheme of generating perfect VVBs. By measuring the distributions of intensity, SoP and Pancharatnam phase [8] of output and diffractive fields, we demonstrate the successful generation of perfect VVBs. We start considering a generalized VVB, which can be represented as a linear combination of two orthogonal circularly n polarized vortices with coefficients ψ m R and ψ L . Thus, the state of the VVB can be described as [19] n jψi ψ m R jR m i ψ L jLn i;

with

pffiffiffi jR m i expimθ − iθ0 ∕2ex iey ∕ 2 pffiffiffi jLn i expinθ iθ0 ∕2ex − iey ∕ 2;

(1)

(2)

where jR m i and jLn i represent the right- (RH) and left-handed (LH) circularly polarized vortices with topological charges of m and n, respectively, θ is the angle in polar coordinate system r; θ, ex and ey are the unit vectors along x and y axes in Cartesian coordinate system, and θ0 is a constant phase. To visually present the inhomogeneous SoP and phase states, the VVBs can also be represented with higher-order Poincaré spheres (PS) [19,20]. Here, we utilize a more general PS, named hybrid-order PS [21], to map the state described in Eq. (1), of which the Stokes parameters can be rewritten as 8 n 2 2 S jψ m > R j jψ L j > < 0 n S 1 2jψ m jjψ j cos ϕ R L ; (3) m n 2jψ jjψ j sin ϕ S > R L > : 2 n 2 2 S 3 jψ m R j − jψ L j n 2 2 where jψ m R j and jψ L j are the intensities of the jR m i and jLn i n eigenstates, respectively, and ϕ argψ m R − argψ L .

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Figure 1 schematically illustrates the hybrid-order PS, where the north and south poles correspond to the eigenstates jR m i and jLn i, respectively. It is clear that an arbitrary VVB can be represented on this sphere. For example, the state corresponding to a point on the equator has an azimuthally variant SoP, and its state is characterized by the polarization order of n − m∕2 and topological Pancharatnam charge of l p m n∕2, respectively [8]. Here, the topological Pancharatnam charge is analogous to the one of scalar vortices, corresponding to the OAM carried by the VVB. The points C and D, denoted by jH m;n i and jV m;n i [19], are the horizontal and vertical polarization bases. Figures 1(b)–1(e) schematically display the intensity distributions of states A–D on a hybridorder PS composed by m −1 and n 3 Laguerre–Gaussian (LG) modes. Noteworthily, the eigenstates jR −1 i and jL3 i have different intensity profiles. Hence, the SoPs of the jH −1;3 i and jV −1;3 i states actually are not completely linear polarized. Once the polarization order and the topological Pancharatnam phase change, the intensity profile and SoP of the VVB also change. According to the above discussions, generating constituent vortices with constant intensity profiles is crucial for the realization of perfect VVBs. Hence, we combine the generation methods of perfect vortices [15–18] with the Sagnac interferometer proposed in Refs. [22–24]. The experimental setup is schematically depicted in Fig. 2. A linearly polarized Gaussian beam (from Ar laser with λ 514.5 nm and ω0 2 mm) is input to the interferometer. The computer-generated hologram (CGH) displayed on the reflective phase spatial light modulator (PSLM) (Holoeye LETO) is shown in the inset (a), which has an annular photic region with transmission profile of t 0.25f2 cos2ax δ1 x; y cos2ay δ2 x; yg in region R − Δ∕2 < r < R Δ∕2. Here, R and Δ are the radius and width of the photic ring, respectively; a is the spatial frequency of the orthogonal oblique two-dimensional grating; δ1 x; y mθ − θ0 ∕2 and δ2 x; y nθ θ0 ∕2 are additional phases. Note that the PSLM is placed at the front focal plane of a 4f imaging system (consisting of lenses L1 and L2 with f 50 cm). Therefore, in the Fourier plane of L1 , after the filtering process of a circular aperture, the first-order spatial spectra corresponding to the jR m i and jLn i states separately possess OAMs of mℏ and nℏ per photon, and approximately present mth and nth order Bessel–Gauss patterns as [25–29] ψ v ρ; ϕ ∝ J v k r ρ exp−ρ2 ∕ω2 expivϕ expikz z;

(4)

where v m or n for the jR m i and jLn i states, respectively; J v is the vth order Bessel function of the first kind; kr k0 R∕f

Fig. 1. Schematic illustration of the hybrid-order PS. Insets: the intensity distributions at points A–D for the case of m −1 and n 3.

Fig. 2. Experimental setup for generating perfect VVBs. RT: reversed telescope; λ∕2: half-wave plate; PBS: polarizing beam splitter; M: mirror; PSLM: phase spatial light modulator; λ∕4: quarter-wave plate; L: lens; and F: filter. Insets: (a) CGH displayed on the PSLM; (b) fundamental and first-order Fourier spectra; (c) intensity profiles of Bessel–Gauss beams; and (d) intensity profile of perfect VVB.

and kz are the radial and longitudinal wave numbers and meet pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the relationship k k 2r k2z 2π∕λ, where λ is the wavelength; and ω λf ∕πω0 is the waist of the Gaussian background. Experimentally, we set m n; thus, the filtered firstorder spectra [marked by dotted circles in inset (b)] at the Fourier plane have a mth order Bessel–Gauss profile with linear polarization. Figure 2(c) shows the intensity patterns for four cases of m 0, 1, 2, and 4. The lens L2 can act as another optical Fourier transformer. According to the principle of Fourier transform and the orthogonality of Bessel functions, the output constituent vortices can be written as [18,30] 2 r R 2 2Rr ψ v r; θ ∝ exp − I expivθ; (5) v ω20 ω20 where I v is the vth order modified Bessel function of the first kind. Equation (5) can also be approximately represented as [18] r − R2 v : (6) ψ r; θ ∝ expivθ exp − ω20 From Eq. (6), one can see that just the term of spiral phase is related to the topological charges. Namely, at the immediate outlet of L2 , the constituent vortices exhibit an intensity profile independent of the topological charges [18]. Consequently, the linear combination of those two coaxial constituent vortices results in a VVB with an annular profile [as shown in inset (d)] independent of its polarization order and topological Pancharatnam charge. In addition, note that the apodizing effect in filtering would cut off the high order rings of Bessel– Gauss beams. After the inverse Fourier transformation, the VVBs have a wider annular profile with respect to the photic ring in the mask. We first verify the independence on the polarization order and topological Pancharatnam charge, by measuring the intensity, S 3 , polarization orientation, and Pancharatnam phases [8] of VVBs with different jH m;n i states after the immediate outlet, as shown in Fig. 3. In experiment, the parameters of photic region are R 0.9 mm and Δ 170 μm. The Stokes


Letter

Fig. 3. Distributions of intensity (top row), S 3 (second row), polarization orientation (third row) and Pancharatnam phases (bottom row) of different jH m;n i states. (a) m −2, n 2; (b) m −5, n 5; (c) m 1, n 3; (d) m 2, n 4; (e) m −1, n 3. The dimension for all images is 2.7 mm × 2.7 mm.

parameters are characterized by insetting a λ∕4 plate and a polarizer before the CCD. According to Eqs. (4)–(7), we can obtain the polarization orientation in cross section. The Pancharatnam phases are measured by use of a Mach–Zehnder interferometer [31]. In practice, we select the x-polarization of VVBs to interfere with the reference beam, and then extract phase profiles from the interference patterns via Fourier and inverse Fourier transforms. After removing the polarizationrelated phase, we present the Pancharatnam phases. Figures 3(a) and 3(b) depict the cases of m −n −2 and m −n −5, respectively. We can see that S 3 is mostly zero in two such cases, and the polarization orientations separately rotate 4π and 10π rad in a full circle. Remarkably, their Pancharatnam phases are mostly zero (i.e., l p 0). For comparison, Figs. 3(c) and 3(d) depict the cases of m 1, n 3, and m 2, n 4, respectively. It is found that the polarization orientation rotates 2π rad in a full circle while the Pancharatnam phases spirally vary 4π and 6π. This means that their polarizations are both first order, but the topological Pancharatnam charges are l p 2 and 3, respectively. Figure 3(e) corresponds to the case of m −1 and n 3. As expected, the jH −1;3 i state has a second-order polarization and topological Pancharatnam charge of l p 1. These measured results confirm that we successfully generated the VVBs with correct polarization orders and topological Pancharatnam charges. More importantly, this clearly indicates that these VVBs with different polarization orders and topological Pancharatnam charges have a constant intensity profile. We then observe the propagation of such generated VVBs. According to Fresnel diffraction theory, the diffractive field of Eq. (5) at any plane can be expressed as [30] ψ vz r; θ ∝ exp−F z r 2 R 2 I v 2RrF z expivθ; 1∕ω2z 2 1∕2

2

(7)

− ik∕2Rz, Rz z L ∕z, ωz where F z ω0 1 z∕L , and L πω20 ∕λ is the Rayleigh range of the Gaussian beam. For the case of a short propagation distance, z ≪ L, then F z ≈ 1∕ω20 . Hence, Eq. (7) has the same form as Eq. (5). This means that the constituent vortices barely

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change. Namely, the annular VVBs have a stability intensity profile in a certain propagation distance. As the propagation distance continues to increase, the distributions of modified Bessel and Gaussian functions undergo deformation, resulting in the obvious diffraction of VVBs. Note that, for large z, the argument of modified Bessel function in Eq. (7) tends to −ikRr∕z [30]. According to the transformation between Bessel functions, since I v −ikRr∕z i v J v −kRr∕z, the constituent vortices evolve into Bessel–Gauss vortices. Correspondingly, the VVBs evolve into fields with hybrid polarization. To confirm how the profile of annular VVBs affects its propagation dynamic, we have performed a series of numerical simulations for VVBs with different parameters. It is found that, for smaller width and radius, the distribution at the Fourier plane is much closer to the pure Bessel profile; inversely, the more obvious diffraction VVBs occur. Therefore, by selecting appropriate photic ring and filtering apertures, one can improve the stable propagation distance of such VVBs. Figure 4(a) shows the side view of a beam with the jH −1;3 i state propagating in a distance of 70 cm. As expected, the VVB keeps the annular profile well within 60 cm of propagation distance, and then a slight deformation occurs. Figures 4(b)–4(i) display the distributions of intensity, S 3 , polarization orientation and Pancharatnam phases of VVBs with jH −1;3 i and jH −2;2 i states at z 60 cm, respectively. One can see that the jH −1;3 i state carrying nonzero OAM has an identical intensity profile with the jH −2;2 i state carrying zeroth OAM. More importantly, comparing with the results shown in Figs. 3(a) and 3(e), it is discernible that the SoP and Pancharatnam phase profiles are invariant. Namely, the generated VVBs have stable SoP, intensity, and Pancharatnam phase profiles at such a distance. Figures 4(j) and 4(k) depict the intensity profiles for the cases of m −n 1 and 4 at far field, respectively. It can be seen that the VVBs approximately evolve into first- and fourth-order vector Bessel–Gauss beams. We also generate VVBs with elliptically polarized states corresponding to the points on upper and lower semispheres and observe the stability of such VVBs upon propagation. The sphere angles 2θ; 2φ can sufficiently characterize an arbitrary state on the hybrid-order PS, and can be given as

Fig. 4. (a) Side view of beam with jH −1;3 i state propagating in x–z plane. Distributions of intensity [(b) and (f)], S 3 [(c) and (g)], polarization orientation [(d) and (h)], and Pancharatnam phases [(e) and (i)] of VVBs with jH −1;3 i (b)–(e) and jH −2;2 i (f )–(i) states at z 60 cm. (j) and (k) Intensity distributions for the cases of m −n 1 and 4 at far field.

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Letter REFERENCES

Fig. 5. Schematic illustration of the hybrid-order PS composed by jR −1 i and jL3 i eigenstates. (b)–(e) Distributions of intensity (left column), S 3 (center column) and polarization orientation (right column) at points A (0, π∕6) and B (0, −π∕6) after immediate outlet (z 0) [(b) and (d)] and propagating 60 cm [(c) and (e)]. The dimension of images is 2.7 mm × 2.7 mm.

2θ tan−1 S 2 ∕S 1 and 2φ sin−1 S 3 ∕S 0 . Here 2φ represents the ellipticity angle. Figure 5 schematically depicts the hybrid-order PS constructed by the jR −1 i and jL3 i eigenstates, where the points A and B correspond to the points of (0, π∕6) and (0, −π∕6), respectively. Figures 5(b) and 5(d) separately show the intensity and SoP distributions of the VVBs with states at points A and B, recorded after the immediate outlet. As shown, the VVBs have homogeneous polarization ellipticity close to 0.5, with azimuthally varying polarization orientation. The measured intensity and polarization properties, after propagating 60 cm, are shown in Figs. 5(c) and 5(e). As can be seen, the SoP and intensity profiles of VVBs with locally elliptical polarization are also stable at such a propagation distance. These above measured output and diffractive fields indicate the successful generation of perfect VVBs. It is noteworthy that these VVBs can support spin angular momentum and OAM simultaneously [32,33]. To some extent, they are desired candidates for realizing optical trapping and enriching the conversion of OAM [34]. In addition, the VVBs have been proposed to encode information in free space communication [35,36]. It is conceivable that the characteristic of invariant intensity profiles enables perfect VVBs to be suitable for encoding information and quantum cryptography. In summary, we have proposed the concept of perfect VVBs with respect to traditional VVBs, and experimentally generated such beams. The generated perfect VVBs have profound manifestations that their intensity profiles are independent of the polarization order and topological charge. We have also demonstrated the stabilities of SoP, intensity, and Pancharatnam phase of perfect VVBs at certain propagation distances. Funding. National Natural Science Foundation of China (NSFC) (11404262, 61205001, 61377035); 973 Program (2012CB921900); Natural Science Basic Research Plan in Shaanxi Province of China (2015JQ1026); Fundamental Research Funds for the Central Universities (3102014JCQ01084, 3102015ZY057).

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