Twisted vector field from an inhomogeneous and anisotropic metamaterial

doi:10.1364/JOSAB.29.000572

« Show journal navigation

Twisted vector field from an inhomogeneous and anisotropic metamaterial

Ming Kang, Jing Chen, Xi-Lin Wang, and Hui-Tian Wang »View Author Affiliations

JOSA B, Vol. 29, Issue 4, pp. 572-576 (2012)
http://dx.doi.org/10.1364/JOSAB.29.000572

View Full Text Article

Acrobat PDF (1828 KB)

Journal Search
Article Lookup

Article Tools

Citations

Alert me when this article is cited
Export Citation/Save

Article Navigation

▸ Article Outline
– Abstract
– References and links

Article Tools
Full Text
Article Info
References
Cited By
Figures
Metrics
Download PDF

Viewing Equations in the
HTML Article

Optimal Viewing Recommendations

Full Text
Article Info
References (29)
Cited By
Figures (5)
Metrics

Abstract

We propose a metamaterial design for realizing inhomogeneous and anisotropic effective media based on the localized waveguide resonance mechanism. Such a design can be easily achieved in experiment and enables us to simultaneously manipulate the wavefront and the state of polarization of the transmitted electromagnetic field by the polarization-sensitive extraordinary optical transmission. Numerical simulations, including the generation of the hybridized vector fields (especially twisted vector fields that are azimuthally polarized carrying a helical phase), prove the feasibility of our proposal. It could be expected as a good candidate of the specially designed subwavelength element for creating the exotic vector fields beyond the functionality of the existing vector fields in a wide spectral regime, especially the terahertz and radio regimes.

Metamaterials, a kind of artificially subwavelength structures with extraordinary electromagnetic (EM) responses not available from materials existing in nature, have become a flourishing subject of intensive investigation, due to the exciting potential applications, such as subdiffraction imaging [1

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005). [CrossRef]

X. Zhang and Z. W. Liu, “Superlenses to overcome the diffraction limit,” Nature Materials 7, 435–441 (2008). [CrossRef]

], EM cloaking [3

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef]

], negative refraction [4

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef]

], and planar optical chirality [5

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97, 167401 (2006). [CrossRef]

]. Among all the attractive characteristics of metamaterials, the most notable one is the phenomena of extraordinary optical transmission (EOT) [6

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391, 667–669 (1998). [CrossRef]

], which has intrigued the development of the scientific subject termed “plasmonics” [7

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010). [CrossRef]

]. Various mechanisms are proposed to explain the extraordinary high transmission through subwavelength aperture arrays in metallic slabs, such as the excitation of surface plasmon polaritons and the resonance of localized waveguide modes [8

F. J. García-Vidal, Esteban Moreno, J. A. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005). [CrossRef]

R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, “Strong polarization in the optical transmission through elliptical nanohole arrays,” Phys. Rev. Lett. 92, 037401 (2004). [CrossRef]

–10

Z. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96, 233901 (2006). [CrossRef]

]. Note that the most frequently investigated property of EOT is its high transmittance in intensity [7

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010). [CrossRef]

], while its polarization-sensitive aspect is rarely involved [11

L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104, 083903 (2010). [CrossRef]

]. Nevertheless, the polarization provides in fact an additional degree of freedom in extending functions and applications of metamaterials. By tailoring the states of polarization (SoPs), the vector fields with spatially inhomogeneous distribution of SoPs have been successfully created in the optical frequency regime by using the active and passive ways [12

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009). [CrossRef]

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007). [CrossRef]

X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007). [CrossRef]

Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30, 3063–3065 (2005). [CrossRef]

K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a $c$ -cut $Nd : {YVO}_{4}$ crystal,” Opt. Lett. 31, 2151–2153 (2006). [CrossRef]

H. Kawauchi, Y. Kozawa, and S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a $c$ -cut crystal,” Opt. Lett. 33, 1984–1986 (2008). [CrossRef]

M. A. Ahmed, A. Voss, M. M. Vogel, and T. Graf, “Multilayer polarizing grating mirror used for the generation of radial polarization in Yb:YAG thin-disk lasers,” Opt. Lett. 32, 3272–3274 (2007). [CrossRef]

M. Fridman, G. Machavariani, N. Davidson, and A. A. Friesem, “Fiber lasers generating radially and azimuthally polarized light,” Appl. Phys. Lett. 93, 191104 (2008). [CrossRef]

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002). [CrossRef]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008). [CrossRef]

M. A. A. Neil, F. Massoumian, R. Juskaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27, 1929–1931 (2002). [CrossRef]

–23

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010). [CrossRef]

]. The active way is from the output of novel lasers with specially designed laser resonators [15

Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30, 3063–3065 (2005). [CrossRef]

H. Kawauchi, Y. Kozawa, and S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a $c$ -cut crystal,” Opt. Lett. 33, 1984–1986 (2008). [CrossRef]

–19

M. Fridman, G. Machavariani, N. Davidson, and A. A. Friesem, “Fiber lasers generating radially and azimuthally polarized light,” Appl. Phys. Lett. 93, 191104 (2008). [CrossRef]

]. The passive one is based on the wavefront reconstruction of the output field from the traditional laser [13

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007). [CrossRef]

,14

], with the aid of the flexible spatial light modulator or specially designed optical elements such as the space-variant subwavelength structure [20

] and the

q

-plates [21

,24

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006). [CrossRef]

]. Generation of vector fields requires the interaction of the EM field with an inhomogeneous and anisotropic medium [24

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006). [CrossRef]

], such as

q

-plates, which is not always available from materials existing in nature. Because of the advantages of flexible structure design, easy fabrication, and many peculiar EM properties, the metamaterials could be expected as good candidates of specially designed subwavelength elements for creating the exotic vector fields beyond the functionality of the existing vector fields in a wide spectral regime, especially in the terahertz and radio regimes.

The frequently used inhomogeneous and anisotropic structures are the

q

-plates and the space-variant subwavelength diffraction grating structures, which can introduce the Pancharatnam–Berry (geometric) phase [25

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam–Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27, 1141–1143 (2002). [CrossRef]

,26

E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005) and references therein. [CrossRef]

] and then modify the phase profile of the optical field toward novel applications. In this article, we propose a metamaterial design to realize an inhomogeneous and anisotropic effective medium based on the mechanism of localized waveguide resonance associated with EOT. The waveguide resonance could possess a strong polarization dependence (especially for the rectangular holes concerned in this article), so that for two orthogonal linear polarizations, only one can be effectively transmitted. This property enables us to formulate each single unit cell containing a subwavelength metallic waveguide as a linear polarizer element. The proper spatial arrangement of the waveguide orientations could then lead to the achievement of the desired inhomogeneous and anisotropic EM behavior.

The mechanism of localized waveguide resonance utilized in this article has many advantages over those of the

q

-plates [21

,24

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006). [CrossRef]

] and the space-variant subwavelength diffraction grating structures [20

], and would benefit the realistic applications. For instance, due to the existence of the EOT effect, any subwavelength waveguide (rectangular holes here) can strongly couple with the incident EM field. When the localized waveguide mode in the rectangular hole is excited, a strong polarization-dependent transmission resonance can be achieved, which is tunable by changing the geometry of the rectangular holes. The proposed structure is fabricated in metallic slab and has a compact size in the propagation direction. The EOT associated with localized waveguide mode is only weakly dependent on the period [10

Z. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96, 233901 (2006). [CrossRef]

] and incident angle [9

In this article, we verify that the inhomogeneous and anisotropic EM response can be achieved by spatially arranging the individual subwavelength rectangular holes. The generation of the hybridized vector field simultaneously shaping the wavefront and SoP of the transmitted EM field, especially the twisted vector field, being an azimuthally polarized vector field carrying a helical phase, is shown to be realizable. Here we add evidence to the important potential of shaping the wavefront and SoP by the use of subwavelength EM resonance. The exotic EM property of the transmitted field consolidates the desired vector field. We believe that such a capability is potentially important for generating the polarization-dependent optical devices [27

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011). [CrossRef]

Consider an EM field with an amplitude

E_{0}

propagating in the

+ z

direction, which is normally incident on a linear polarizer with its polarization direction forming an angle

α

by the

x

axis, the Jones matrix

M

presenting the transformation of the polarizer, as follows [28

F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999). [CrossRef]

M = [\begin{array}{cc} \cos^{2} α & \sin α \cos α \\ \sin α \cos α & \sin^{2} α \end{array}] .

(1)

Under the incidence of the circularly polarized field, the transmitted field should be

E_{T}^{\pm} = M \cdot E_{I N}^{\pm} = \frac{1}{\sqrt{2}} E_{0} \exp (\pm j α) [\begin{array}{cc} \cos α \\ \sin α \end{array}],

(2)

where the superscripts

+

and

-

indicate the right-handed and left-handed polarization states, and the subscripts

I N

and

T

describe the incident and transmitted EM fields. Evidently, the transmitted field at the exit surface is always linearly polarized, with the same polarization direction as the linear polarizer, for circularly polarized incidence. Under the right- or left-circularly polarized incidence, the transmitted field has always an intensity of

ε_{0} E_{0}^{2} / 4

that is a half of the incidence and acquires an additional geometric phase

+ α

(

- α

) [25

,26

E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005) and references therein. [CrossRef]

] with respect to the incident field. If only the term of

\exp (\pm j α)

holds, the transmitted field shapes the wavefront only, such as the conventional vortex field [27

]. If only the term of

{(\cos α, \sin α)}^{T}

holds, the transmitted field shapes the SoP only, such as the radially and azimuthally polarized fields [12

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009). [CrossRef]

]. From Eq. (2), the transmitted field could shape the wavefront and the SoP simultaneously to enrich the functionality of the existing vector fields.

If the local subwavelength linear polarizer can be realized, we can properly arrange local linear polarizers with different orientations to construct the desired space-variant structure, and then the hybridized vector field with the helical phase can be realized. Hence, the crucial issue is how to realize the local linear polarizer with subwavelength scale. As is well known, when an EM field is incident on the subwavelength metallic structure, one of the most notable attractive characteristics is the EOT phenomenon. One of the most frequently investigated characteristics is the high transmittance in intensity [7

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010). [CrossRef]

], while its polarization aspect is rarely involved. Here, we adopt the polarization-sensitive metallic aperture to realize the local linear polarizer with subwavelength scale, based on the EOT phenomenon.

Now let us verify the performance of the effective local subwavelength linear polarizer composed of the subwavelength rectangular metallic hole as a unit cell in this article. According to the theory of EOT [7

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010). [CrossRef]

], the transmission through a metallic rectangular hole has a strongly polarization-dependent transmission when the localized waveguide mode inside the rectangular hole is excited [8

F. J. García-Vidal, Esteban Moreno, J. A. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005). [CrossRef]

–10

Z. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96, 233901 (2006). [CrossRef]

]. By using the same parameters as in [10

Z. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96, 233901 (2006). [CrossRef]

], the waveguide-mediated EOT effect occurs at the frequency around

0.56 (c / a)

(i.e., at wavelength of

1.79 a

) for the polarization with its electric field

E

normal to the long side of the rectangular hole (side

0.9 a

in Fig. 1), where

c

is the velocity of light in vacuum. The transmittance ratio between the two orthogonal linear polarizations exceeds

10^{6}

, thus each unit cell exhibits a strongly anisotropic response and then can be equivalent to an effective local perfect polarizer.

Fig. 1. Geometry of a one-dimensional inhomogeneous anisotropic metamaterial composed of 10 rectangular holes with the orientation changed stepwisely from 0 to

π / 2

. The inset is the geometry of the unit cell that is a square metal slab punched into a rectangular hole.

We now can properly arrange the effective local linear polarizers with different orientations to construct the desired space-variant structure, and then the hybridized vector field can be realized. For this purpose, we would like to explore the transmission behavior through a one-dimensional inhomogeneous and anisotropic metallic slab in the

x

direction, as shown in Fig. 1. Any unit cell punched into a rectangular hole has the same dimension of

a \times a

and any rectangular hole has the same dimension of

0.9 a \times 0.2 a

, as shown in the inset of Fig. 1. The orientation of the rectangular hole is defined by an angle

α

formed by the

x

axis and the short side of the hole. For the 10 rectangular holes in Fig. 1, their orientations are changed from

α = 0

π / 2

by a step of

π / 18

along the

+ x

direction. We investigate the transmission behavior of such a one-dimensional metamaterial, shown in Fig. 1, by using the finite difference time domain (FDTD) algorithm. For the FDTD simulation, the thickness of the metallic slab is chosen to be

h = 0.2 a

; like the geometric parameters adopted in the literature, the metal used can be considered as a perfect conductor within the low frequency regime (especially in the radio regime). We measure the transmitted field distribution in a plane near the metallic slab (at

1.2 a \approx 0.67 λ

) to avoid the diffraction effect caused by the finite transverse dimensions.

By using the FDTD algorithm, Fig. 2 shows the transmission properties of the right-circularly polarized field at the EOT wavelength of

λ = 1.8 a

, under the normal incidence. The spatial distribution of SoPs shown in Fig. 2(a) implies that the transmitted field is indeed local linearly polarized. The polarization exhibits a homogeneous distribution in the

y

dimension, while it exhibits a space-variant distribution in the

x

dimension (the linear polarization direction is gradually changed from the horizontal direction to the vertical one). As mentioned above, when a circularly polarized scalar field is incident on a linear polarizer, a geometric phase is introduced into its transmitted field. Such a geometric phase has in fact no significance when a circularly polarized scalar field passes through a linear polarizer. However, the situation is quite different when a circularly polarized field is incident on the inhomogeneous material. Like the simulated result shown in Fig. 2(b) and the analytic result shown in Fig. 2(c), the geometric phase of the transmitted field in the one-dimensional inhomogeneous metamaterial shown in Fig. 1 is space-invariant in the

y

dimension, while it exhibits a linear growth from 0 to

π / 2

along the

+ x

direction. We should emphasize that the total transmitted intensity is space-invariant in both the

x

and

y

dimensions and has a 50% transmittance, although the rectangular hole occupies 18% in a single unit cell.

Fig. 2. Properties of the transmitted field passing through the one-dimensional inhomogeneous anisotropic metamaterial shown in Fig. 1, under the circular polarization incidence. (a) Spatial distribution of SoPs, (b) spatial distribution of geometric phase given by the simulation, and (c) spatial distribution of geometric phase given by the analysis.

Based on the above results, we may draw two important conclusions. For the first, any unit cell with the subwavelength rectangular hole exhibits the strongly polarization-dependent EOT effect, in particular, can be considered as a local linear polarizer, with its polarization direction normal to the long side of the rectangular hole and with an extinction ratio exceeding

10^{6} : 1

. For the second, the array composed of the differently orientated rectangular holes can be equivalent to an effective inhomogeneous and anisotropic metamaterial in the radio regime, and then the Jones formulism can be safely applied on the field at each point of the effective inhomogeneous and anisotropic metamaterial, as the

q

-plate [21

,24

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006). [CrossRef]

] and the space-variant subwavelength structure [20

Clearly, we can design an inhomogeneous and anisotropic metamaterial with the rotation symmetry, which is a punched metallic slab with the differently orientated rectangular holes, as shown Fig. 3(a). Eight (16) differently orientated rectangular holes arrange a spokewise shape in the inner (outer) ring shown by the dotted-line (solid-line) ring with a radius of

r = 0.75 a

(

R = 2 a

). All the rectangular holes have the identical size of

0.9 \times 0.2 a^{2}

. Therefore, the extraordinary EM transmission should locate at a wavelength of

λ = 1.8 a

, like the aforementioned result. Under the normal incidence of the right-circularly polarized field, the transmitted field passing through the inhomogeneous anisotropic metamaterial shown in Fig. 3(a) can be obtained when replacing

α

ϕ

(

α = ϕ + π / 2

) from Eq. (2). With Eq. (2), the transmitted field is shown in Fig. 3(b), in which the arrows stand for the polarization directions of the fields in rectangular holes and the gray filled-in rectangular holes show the corresponding geometric phases. We should emphasize that the result shown in Fig. 3(b) is also demonstrated by the FDTD simulation. Evidently, the transmitted field through the structure shown in Fig. 3(a) carries the helical (geometric) phase, which is easy understood below. As shown in Fig. 2, the transmitted field in the one-dimensional inhomogeneous metamaterial carries the linearly variant geometric phase. Therefore the transmitted field in the two-dimensional inhomogeneous metamaterial shown in Fig. 3(a) could be considered as a twisted azimuthally polarized-like vector field (because compared with the well-defined azimuthally polarized vector field [2

X. Zhang and Z. W. Liu, “Superlenses to overcome the diffraction limit,” Nature Materials 7, 435–441 (2008). [CrossRef]

], this field has the similarity for the distributions of SoP and phase but has also the difference that it is structured).

Fig. 3. (a) Geometry of the metamaterial composed of different orientated rectangular holes with the rotation symmetry. (b) Transmitted field through the metamaterial; the arrows indicate the linear polarization directions, and the gray filled-in rectangular holes show the phase.

We would like to explore the propagation behavior of the transmitted field shown in Fig. 3(b). The propagation of this transmitted field in free space exhibits a focusing-like effect with its focal plane having a distance of

3.168 a

from the structure, as shown in Fig. 4. From the simulated intensity distribution of the transverse component in Fig. 4(a), it can be found that the azimuthally polarized-like vector field is focused into a spot, due to the existence of a helical phase, which is similar to the azimuthally polarized vector field carrying the helical phase [29

X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35, 3928–3930 (2010). [CrossRef]

], while being completely different from a focus ring of the well-defined azimuthally polarized vector field. The full width at half maximum (FWHM) of the focal spot is about

1.5 a

, which is slightly smaller than the wavelength of

1.8 a

. Based on the geometric representation of Poincaré sphere

Σ

for SoP [23

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010). [CrossRef]

], Fig. 4(b) shows the distributions of SoPs of the focal field along the

x

and

y

axes. Clearly, SoP at the center of the focal field is right-circularly polarized because it is located at the north pole on

Σ

. When the position is far away from the center along the

\pm x

and

\pm y

directions, SoP will experience the evolution from the right-circular polarization though the right-elliptical polarization to the linear polarization on the northern hemisphere of

Σ

. The SoPs of the focal field exhibit the central-inverse symmetry. In addition, the trajectories composed of the points on

Σ

describing the SoPs along the

x

and

y

axes have the twofold rotation symmetry about the

S_{3}

axis on

Σ

. Figure 4(c) shows the geometric representation on

Σ

for the SoPs of four different homocentric rings [

A

B

C

, and

D

in Fig. 4(a)] in the cross section of the focal field. The SoP of any point at a certain ring locates at a corresponding latitude circle on

Σ

, and the trajectory composed of the points describing the SoPs at any ring has the axial symmetry about the

S_{3}

axis on

Σ

. It should be pointed out that a pair of symmetric points about the center at a certain ring have the same SoP.

Fig. 4. Focal behaviors of the transverse component of the self-focusing field. (a) The intensity distribution. (b) The geometric representations of Poincaré sphere for SoPs along the

x

and

y

axes. (c) The geometric representations of Poincaré sphere for SoPs in four different homocentric rings

A

B

C

, and

D

(with the respective radii of

0.5 a

0.75 a

1.0 a

, and

1.25 a

) shown in (a).

The longitudinal component at the focal plane can be approximately described by

E_{z} = \sum r_{i}^{- 1} e^{- j k_{0} r_{i} + j σ ϕ}

, where

r_{i}

is the distance between the

i

th aperture element to the observation point,

σ = 1

is the incident right-circular polarized plane field, and the field at the observation point is the summation of the contribution from each aperture element. Figure 5 shows the analytical and simulated longitudinal component of the focal field; clearly, both are in good agreement. One can see that the longitudinal component of the focal field exhibits a focus ring and carries a helical phase. For the azimuthally polarized-like vector field, the longitudinal component of its focal field originates from the diffraction effect of the rectangular holes, which is completely different from no longitudinal component of the focal field for the well-defined azimuthally polarized vector field.

Fig. 5. Focal behaviors of the longitudinal component of the self-focusing field. (a) and (b) are the simulated and analytic intensity distributions. (c) and (d) are the simulated and analytic phase distributions.

In conclusion, by benefiting from the burgeoning of metamaterial science, we propose a way to engineer the artificial material toward inhomogeneous and anisotropic EM response not limited by natural material. Under the circularly polarized field, the rotation-symmetry metamaterial composed of the differently oriented rectangular holes can create the twisted azimuthally polarized vector field. Such a vector field exhibits a focusing effect with a slight subwavelength focal spot. Our result creates a link between two important issues on metamaterials and vector fields. This idea can extend the functionality of many systems, find novel effects and phenomena, and facilitate the development of additional surprising applications in the subwavelength region.

ACKNOWLEDGMENTS

This work is supported by the National Basic Research Program (973 Program) of China under grant 2012CB921900 and the National Natural Science Foundation of China under grants 10934003, 10974102, and 11174157.

REFERENCES

1.	N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005). [CrossRef]
2.	X. Zhang and Z. W. Liu, “Superlenses to overcome the diffraction limit,” Nature Materials 7, 435–441 (2008). [CrossRef]
3.	J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef]
4.	R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef]
5.	V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97, 167401 (2006). [CrossRef]
6.	T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391, 667–669 (1998). [CrossRef]
7.	F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010). [CrossRef]
8.	F. J. García-Vidal, Esteban Moreno, J. A. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005). [CrossRef]
9.	R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, “Strong polarization in the optical transmission through elliptical nanohole arrays,” Phys. Rev. Lett. 92, 037401 (2004). [CrossRef]
10.	Z. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96, 233901 (2006). [CrossRef]
11.	L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104, 083903 (2010). [CrossRef]
12.	Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009). [CrossRef]
13.	C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007). [CrossRef]
14.	X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007). [CrossRef]
15.	Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30, 3063–3065 (2005). [CrossRef]
16.	K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a $c$ -cut $Nd : {YVO}_{4}$ crystal,” Opt. Lett. 31, 2151–2153 (2006). [CrossRef]
17.	H. Kawauchi, Y. Kozawa, and S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a $c$ -cut crystal,” Opt. Lett. 33, 1984–1986 (2008). [CrossRef]
18.	M. A. Ahmed, A. Voss, M. M. Vogel, and T. Graf, “Multilayer polarizing grating mirror used for the generation of radial polarization in Yb:YAG thin-disk lasers,” Opt. Lett. 32, 3272–3274 (2007). [CrossRef]
19.	M. Fridman, G. Machavariani, N. Davidson, and A. A. Friesem, “Fiber lasers generating radially and azimuthally polarized light,” Appl. Phys. Lett. 93, 191104 (2008). [CrossRef]
20.	Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002). [CrossRef]
21.	G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008). [CrossRef]
22.	M. A. A. Neil, F. Massoumian, R. Juskaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27, 1929–1931 (2002). [CrossRef]
23.	X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010). [CrossRef]
24.	L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006). [CrossRef]
25.	Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam–Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27, 1141–1143 (2002). [CrossRef]
26.	E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005) and references therein. [CrossRef]
27.	N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011). [CrossRef]
28.	F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999). [CrossRef]
29.	X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35, 3928–3930 (2010). [CrossRef]

OCIS Codes
(260.0260) Physical optics : Physical optics
(260.2110) Physical optics : Electromagnetic optics
(260.5430) Physical optics : Polarization
(160.3918) Materials : Metamaterials

ToC Category:
Physical Optics

History
Original Manuscript: August 12, 2011
Revised Manuscript: November 19, 2011
Manuscript Accepted: December 1, 2011
Published: March 6, 2012

Citation
Ming Kang, Jing Chen, Xi-Lin Wang, and Hui-Tian Wang, "Twisted vector field from an inhomogeneous and anisotropic metamaterial," J. Opt. Soc. Am. B 29, 572-576 (2012)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-29-4-572

Sort: Author | Year | Journal | Reset

References

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005). [CrossRef]
X. Zhang and Z. W. Liu, “Superlenses to overcome the diffraction limit,” Nature Materials 7, 435–441 (2008). [CrossRef]
J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef]
R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef]
V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97, 167401 (2006). [CrossRef]
T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391, 667–669 (1998). [CrossRef]
F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010). [CrossRef]
F. J. García-Vidal, Esteban Moreno, J. A. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005). [CrossRef]
R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, “Strong polarization in the optical transmission through elliptical nanohole arrays,” Phys. Rev. Lett. 92, 037401 (2004). [CrossRef]
Z. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96, 233901 (2006). [CrossRef]
L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104, 083903 (2010). [CrossRef]
Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009). [CrossRef]
C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007). [CrossRef]
X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007). [CrossRef]
Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30, 3063–3065 (2005). [CrossRef]
K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31, 2151–2153 (2006). [CrossRef]
H. Kawauchi, Y. Kozawa, and S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a c-cut crystal,” Opt. Lett. 33, 1984–1986 (2008). [CrossRef]
M. A. Ahmed, A. Voss, M. M. Vogel, and T. Graf, “Multilayer polarizing grating mirror used for the generation of radial polarization in Yb:YAG thin-disk lasers,” Opt. Lett. 32, 3272–3274 (2007). [CrossRef]
M. Fridman, G. Machavariani, N. Davidson, and A. A. Friesem, “Fiber lasers generating radially and azimuthally polarized light,” Appl. Phys. Lett. 93, 191104 (2008). [CrossRef]
Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002). [CrossRef]
G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008). [CrossRef]
M. A. A. Neil, F. Massoumian, R. Juskaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27, 1929–1931 (2002). [CrossRef]
X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010). [CrossRef]
L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006). [CrossRef]
Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam–Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27, 1141–1143 (2002). [CrossRef]
E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005) and references therein. [CrossRef]
N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011). [CrossRef]
F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999). [CrossRef]
X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35, 3928–3930 (2010). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Click here to see a list of articles that cite this paper