Twisting light with hyperbolic metamaterials - OSA Publishing

Twisting light with hyperbolic metamaterials Jingbo Sun,1,* Jinwei Zeng,1 and Natalia M. Litchinitser1 1

Department of Electrical Engineering, The State University of New York at Buffalo, Buffalo, NY 14260, USA * [email protected]

Abstract: We propose a novel, miniaturized astigmatic optical element based on a single biaxial hyperbolic metamaterial that enables the conversion of Hermite-Gaussian beams into vortex beams carrying an orbital angular momentum and vice versa. As an example, we design a biaxial anisotropic metamaterial that introduces a π/2 phase shift between two orthogonal components of a Hermite–Gaussian beam due to the optical path difference and at the same time astigmatically focuses these orthogonal components such that they recombine in a symmetric Laguerre-Gaussian beam. We design the proposed device using an array of silver nanowires in an MgF2 matrix. The advantages of the proposed approach over the existing bulk optics based techniques include compactness and therefore, compatibility with ultra-compact opto-electronic circuits, potential reconfigurability and an increased tolerance to misalignment. ©2013 Optical Society of America OCIS codes: (160.3918) Metamaterials; (260.6042) Singular optics; (160.4670) Optical materials.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

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Received 3 Apr 2013; revised 9 May 2013; accepted 10 May 2013; published 17 Jun 2013 17 June 2013 | Vol. 21, No. 12 | DOI:10.1364/OE.21.014975 | OPTICS EXPRESS 14975

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1. Introduction Hyperbolic metamaterials (HMMs) are a class of strongly anisotropic materials having their principle elements of dielectric permittivity or magnetic permeability tensors of opposite signs. These materials have enabled such novel properties and potential applications as omnidirectional negative refraction, high density of states and, imaging beyond the diffraction limit using a so-called hyperlens [1–3]. Despite significant theoretical and experimental progress in recent years, no studies of propagation of complex light beams such as those carrying the orbital angular momentum and/or vector beams in HMMs have been reported. It is noteworthy that hyperbolic metamaterial can be considered as an astigmatic anisotropic element such that rays propagating in two perpendicular planes have different foci. The degree of astigmatism can be designed and even potentially tuned. In particular, we propose a new approach that enables a transformation of uncharged Hermite-Gaussian beams into vortex beams carrying an orbital angular momentum (OAM) of a particular order in a biaxial HMM. Beams with different OAM states have a strong potential to enable a new degree of freedom for increasing capacity through space division multiplexing or for building higher dimensional quantum encryption systems [4–11]. Very first wirelesses as well as specialty fiber based links, utilizing OAM states, have been recently reported [4, 12]. Previously, several approaches have been developed for generating and manipulating OAM, including fork holograms and spatial light modulators, spiral phase plates, q-plates, and cylindrical lens mode converters [13–23]. All these methods developed to date rely on bulky free space optics elements and have drawbacks with respect to alignment, bandwidth, and incompatibility with future ultra-compact opto-electronic circuits. Only recently have first attempts at the realization of miniaturized vortex elements, including those based on metasurfaces [24, 25] and on silicon optical waveguides [28], been reported. In this paper, we propose an alternative, simple, compact, hyperbolic metamaterial based beam transformer that converts uncharged Hermite-Gaussian modes into a Laguerre-Gaussian

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vortex beam of a particular order. The basic principle of a proposed device is as follows. A 45° polarized Hermite-Gaussian mode beam can be transformed into a vortex beam of Laguerre-Gaussian mode by the optical path difference along the two perpendicular crystal orientations resulting from the biaxial anisotropy of the HMM. As compared to existing approaches, the proposed device is compact, does not require a precise alignment and is potentially reconfigurable. We discuss basic theory, design, and potential experimental realization of the proposed device. 2. Theory We consider a Hermite-Gaussian beam (HG) polarized at 45° to the principal axes x and y of the biaxial HMM propagating along the z axis E (r , z ) = E 0

  −r 2   2 w0 r2 − ikz − ik + i ς (z )  H 1  exp  2 (x + y )  , (1) w (z ) 2R ( z )  w (z )  w 0  2

 z  where w( z ) = w0 1 +   is the radius of the beam at z, w0 is the waist of the beam, ζ is the  zR   2  π w02 Gouy phase, H1  is the Raleigh ( x + y )  is the first order Hermite polynomial, zR = λ  w0  range of the beam, and λ is the wavelength. Such a kind of HG beam can be decomposed into an in-phase HG1, 0 and HG0, 1 modes. We design an HMM such that when the beam propagates in a biaxial hyperbolic medium, negative refraction resulting from hyperbolic dispersion relation leads to astigmatic focusing inside and outside of the HMM. Simultaneously, due to the anisotropy, a phase difference between the two modes arises after the beam transmitting through the HMM. By optimizing a certain thickness of the HMM so that the phase difference between the two modes is π/2, the output beam turns out to be a vortex beam of LG1,0 mode [15]. As shown in Fig. 1, the incident HG beam from its source at z = 0 is incident into the biaxial HMM which is located at z. Thus the incident angle θi can be obtained by:  w (z )  ,  z 

(2)

0   0 , ε zz 

(3)

θi = arc tan  For an HMM with dielectric tensor:  ε xx ε =  0  0  

0

ε yy 0

where ε xx > 0, ε yy > 0 (ε xx ≠ ε yy ) and ε zz < 0, the dispersion relations illustrated by the equal

frequency contours (EFCs) in yz-plane and xz-plane shown in Fig. 1(b) and 1(c) can be written as: k x2

ε zz

k y2

ε zz

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+ +

k z2

ε xx

k z2

ε yy

= =

ω2 c2

ω2 c2

,

(4) .


Fig. 1. Schematic of negative refraction in HMM (a) and the equi-frequency contours of the HMM in xz-plane (b) and yz-plane (c). The black circles are the EFCs of the air and the blue hyperbola are the EFCs of the hyperbolic material. ki and Si are the incident wave vector and Poynting vector, respectively. kr and Sr are the refractive wave vector and Poynting vector, respectively. Due to the indefinite property between xy-plane and z orientation, both the HG1,0 mode and the HG0,1 mode experience a negative refraction. The astigmatic beam is focused inside the indefinite beam.

ω2

If the two EFCs are normalized by the EFC of air k x2 + k z2 = 2 , we define the c   normalized incident wave vector k i and the wave vector of refracted wave k r along different directions as:   k i , x = k i , y = sin(θ i ) =

tan 2 (θ i

)

1 + tan (θ i 2

)

 , k i , z = cos(θ i ) =

1 , 1 + tan 2 (θ i )

    k i ,x = k i , y = k r,x = k r, y ,   k r2, x  k r , xz = ε xx  1 −  ε zz 

  k r2, y    , k r , yz = ε yy  1 −   ε zz  

(6)  .  

Thus, the refractive indices in x and y orientations are:     n kr ,xz = k r2, xz + k r2, x , n kr ,yz = k r2, yz + k r2, y ; The wave vector refraction angles in xz-plane and yz-plane are:    k r, y  k r, x  θ kr , zx = arc tan    , θ kr , yz = arc tan   k  k  r, xz   r , yz

(5)

(7)

(8)

 ,  

(9)

The Poynting vector refraction angles in xz-plane and yz-plane are:

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   k r, y ε yy   k r, x ε xx  θSr , xz = arc tan   (10) .  , θSr , yz = arc tan   k ε  k ε  r, z zz  r, z zz    Because the input surface and the output surface of the HMM are parallel to each other, the incident angle θi and the output angle θo are also equal to each other in Fig. 1(a). If we consider the mode of the beam in xy-plane, the optical paths of in yz-plane and xz-plane in the air will be same. As shown in Fig. 1(a), the optical path difference in the HMM between the xz-plane and yz-plane is: Δl o =

d cos(θSr , zx + θ kr , zx ) cos(θSr , zx )

n kr , zx −

d cos(θSr , yz + θ kr , yz ) cos(θSr , yz )

n kr , yz ,

(11)

After some algebra deduction:   k r2, x Δl o = d  K −  ε zz K 

  Δn  

(12)

 k r2, x

, a constant for a fixed incident angle, and Δn = ε xx − ε yy is the ε zz difference of the refractive indices between x and y orientations. If Δl0 = λ/4, the phase difference between the two beams in the yz-plane and xz-plane is π/2, which means that the composed beam is a vortex of LG1,0 mode. in which K = 1 −

3. Design

One of the most common structures used to realize the HMM is the array of metal nanowires embedded in a dielectric matrix [29]. Here, we design a tetragonal silver nanowire array in a MgF2 matrix to realize the biaxial HMM, as shown in Fig. 2. The permittivity of the silver can be described by the Drude model:

ε m (ω ) = ε ∞ −

ω p2 . ω (ω + i γ c )

(13)

where ε∞ = 6 is the bulk permittivity at infinite frequency, ωp = 1.5 × 1016rad/s and γc = 7.73 × 1013 rad/s are the plasma frequency and the collision frequency, respectively [29, 30]. According to the Maxwell-Garnett theory, we can obtain the permittivity parallel to the nanowire (ε∥) and vertical to the nanowires (ε⊥) [31, 32]:

ε || = p ε m + (1 − p ) ε d ,

ε ⊥ = εd +

p εd (ε m − εd

(14)

)

. (15) ε d + (1 − p )( ε m − ε d ) qeff here, qeff is the effective depolarization factor perpendicular to the nanowires and is equal to 1/2 if the wavelength is much larger than the radius of the nanowires [29]. The permittivity of the MgF2 is 1.9 at 532nm. For an HMM with unit size of a = 124nm, b = 117nm, and radius r = 30 nm at 532 nm, filling ratio of the silver along x-, y- and z-axis are px = 0.185, py = 0.205, pz = 0.195, respectively. According to Eq. (14) and (15), the permittivities of the metamaterial are εxx = 2.93 + 0.006i, εyy = 3.08 + 0.007i, and εzz = −0.8 + 0.07i. Considering a 45° polarized HG1,0 Gaussian beam with wavelength of 532nm and waist also 532nm propagating along z axis, the thickness of the HMM with above parameters should be 3.48μm according to Eq. (12).

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Fig. 2. Structure of the biaxial HMM.

4. Numerical results

To demonstrate the performance of the vortex beam produced by biaxial HMM, we performed finite-element method based simulations (using Comsol Multiphysics 4.2) for the HG1, 0 beam propagating through a biaxial HMM consisting of silver nanowires array in the MgF2 matrix with the design parameters mentioned above. Results in Fig. 3 show the time averaged energy density distribution and the phase distribution of the output beam at different positions along the z-direction. In the range of 8.5~10.5μm, the shape of the beam exhibits a doughnut shape with a helical phase change in the cross-section, confirming that the 45° polarized HG1,0 beam was transformed into a LG1,0 mode of a good quality in this range. Due to the angular dispersion of the HMM, the thickness d can be varied in the range of 3.23.7μm.

Fig. 3. The schematic of the device and the results including energy density and the phase distributions of the output beam around the focal length.

Figure 4 shows the evolution of the beam shape and the phase along the direction of propagation. The first pair on the left is the incident HG1,0 beam with a pattern of two lobes. The second to fourth pairs are inside the HMM. The third pair around the focusing point has a doughnut shape due to the focusing effect of the HMM; however, its phase is not helical. The last six pairs on the right are in the air. Although, the doughnut shape only occurs around the focal range in the air (outside the HMM), the phase of the beam remains helical anywhere after its transmission through the HMM, indicating that the output beam carries an OAM.

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Fig. 4. The time averaged energy density and phase distributions in the cross section of the beam along the propagation direction.

7. Conclusion

In conclusion, we proposed and designed a new approach to generating a vortex beam by transforming the HG mode into the LG mode by using a biaxial HMM. We demonstrated that the indefinite dielectric tensor combined with the biaxial anisotropy enables conversion of a 45° polarized HG beam into a vortex beam owing to different focusing effects in two orthogonal planes inside the HMM and therefore, different optical paths. We design the HMMs such that the optical path difference results in a π/2 phase shift between the two orthogonal components, resulting in an output beam carrying an OAM. The proposed approach offers several advantages over the existing techniques, including compactness and therefore, compatibility with ultra-compact opto-electronic circuits, robustness and potentially re-configurability. Acknowledgments

The authors appreciate discussions with A. N. Cartwright, Zh. A. Kudyshev, M. I. Shalaev, and X. Wang, and acknowledge support of the U.S. Army Research Office under the award W911NF-11-1-0333.

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