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OPTICS LETTERS / Vol. 39, No. 19 / October 1, 2014

Entangled transverse optical vortex S. T. Chui1,* and Zhifang Lin1,2 1

Bartol Research Institute and Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA

2

State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China *Corresponding author: [email protected] Received May 9, 2014; revised August 4, 2014; accepted September 2, 2014; posted September 4, 2014 (Doc. ID 211450); published September 30, 2014 We discuss a new kind of optical vortex with the angular momentum perpendicular to the flow direction and entangled in that it is a coherent combination of different orbital angular momentum states of the same sign. This entangled state exhibits many unexpected physical properties. The transverse optical vortex can be generated from the reflection of an electromagnetic wave off an array of ferrite rods. Its vorticity can be reversed by switching the direction of the magnetization of the rods, which usually takes only a nanosecond. © 2014 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (260.2110) Electromagnetic optics; (260.5740) Resonance; (350.4010) Microwaves; (160.3918) Metamaterials; (350.4238) Nanophotonics and photonic crystals. http://dx.doi.org/10.1364/OL.39.005732

Over the last thirty years, there has been much interests in physical states that are topological in nature, as they are not destroyed by local perturbations. An example of this state is the vortex, which exists not only in fluids but also in optics [1]. The optical vortex with ∇ × P ≠ 0 where P is the Poynting vector has found many different interesting potential applications. In principle, this state can be used to encode information. Because of its topological nature, it would be less susceptible to the noise caused by the fluctuation in the intensity of the signal. This photonic vortex state, which exhibits an orbital angular momenta parallel to the beam direction, is generated with a spatial light modulator using liquid crystals with a low switching time (milliseconds) and thus not suitable for information encoding. The vortex state is not entangled in that it is not a coherent combination of different orbital angular momentum states of the same sign. In this Letter, stimulated by recent interest in the nonreciprocal properties [2–4] of magnetic photonic crystals (MPCs) [5], we discuss a new kind of photonic states with finite angular momenta perpendicular to the beam direction, the transverse optical vortex (TOV). The TOV photonic state generated is entangled. This entanglement leads to many unexpected physical effects. The TOV can be generated from the reflection of an electromagnetic wave off an array of ferrite rods. Its vorticity can be reversed by switching the direction of the magnetization of the rods, which is fast and usually takes only a nanosecond. Thus this state is a good candidate to encode microwave information that is not susceptible to the problem of the fluctuation of the intensity of the signal (fading). It is noted that the optical states in this study operates in a different regime of frequency from the conventional photonic vortex states generated by spatial light modulators. They, however, share the same principle for the applications in encoding information while exhibiting fundamentally different types of vortex. We now discuss our results in detail. The TOV states can be generated from the reflection of a wave packet from a MPC that consists of an array of ferrite rods with a thickness that can be as thin as 1∕15 to 1∕20 of a wavelength. This is illustrated in Fig. 1. 0146-9592/14/195732-04$15.00/0

We have considered an example of a MPC that consists of a single layer of ferrite cylinders of radius r  0.25a and lattice constant a  8 mm in air arranged uniformly along the x direction. The axis of the cylinder and the magnetization M s are along the z direction. For our calculation, we use an effective local magnetic field H 0  900 Oe, 4πM s  1750 Oe, and the permittivity of the ferrite rod ϵs  15. The corresponding frequencies are denoted by ω0  γH 0 , ωm  γ4πM s , where γ is the gyromagnetic ratio. To illustrate our result, we pick the most dramatic case with the direction of the electromagnetic (EM) wave coming in along the y direction perpendicular (kx  0) with the electric field along the cylinder. The EM wave is uniform along z. Our results are obtained using the multiple scattering method. In the multiple scattering theory, the scattered field b from a photonic crystal is related to the external field p by the full T matrix: b  Tp. This T matrix can be obtained from the single cylinder t matrix and a structure factor S; which depends on the geometry of the system as

reflected beam

reflected beam

y incident beam

x

O Fig. 1. Schematic diagram of the geometry of the problem. The beam is at normal incidence on an array of ferrite rods. At some frequency, the reflected beam exhibits an angular shift and carries a transverse vorticity perpendicular to the beam propagation direction. The sign of these effects can be switched (from blue to green or vice versa) within a timescale of a nanosecond when the external magnetic field is reversed. © 2014 Optical Society of America

October 1, 2014 / Vol. 39, No. 19 / OPTICS LETTERS

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Fig. 2. S x y as a function of y. Results at three different times for a normally incident Gaussian pulse reflected from the MPC are shown. This illustrates that the pulse carries a transverse vortex perpendicular to the direction of pulse propagation.

T −1  t−1 − S. The single cylinder t matrix is diagonal in the angular momentum representation. The mth matrix element can be expressed in terms of the scattering phase shift ηm by t−1 m  1 − i cot ηm . Our work is motivated by recent interests in the broken time-reversal symmetry in magnetic systems. This results in an asymmetric scattering of light from magnetic cylinders between angular momentum channels m and −m [6–8]. Our result is obtained by computing T and hence b. To illustrate the vorticity, we consider a pulse of finite width in time [9] and Rshow in Fig. 2 the average Poynting vector ∞ P x dx in the direction perpendicular to the S x y  −∞ propagation direction as a function of distance y along the direction of propagation. For states with no transverse vortex, S x vanishes independent of y. In the present case, S x is finite; it is positive at large y and becomes zero at a “center”; after that it changes sign, thus forming a vortex. The scattered EM field from a cylinder can be expressed as a sum of orthonormal basis functions with coefficients bm for different angular momentum m. Our results jbm kx  0j, normalized by the incoming wave, as a function of the dimensionless frequency f c∕a, are shown in Fig. 3. They are obtained by full-wave numerical calculation based on multiple scattering theory. For the ferrite, there are two frequencies that are of interest, the p spin wave frequency ωSW  ω0 ω0  ωm   0.727c∕a (f SW a∕c  0.1157) and the magnetic surface plasmon (MSP) frequency ωMSP  ω0  ωm ∕2  0.8359c∕a. At frequencies close to the ferromagnetic-resonance (spin wave) frequency, the reflected field is a linear combination of states of finite transverse vorticities of the same sign. In Fig. 3, only b0 , b1 are significant. b−1 is small. Because of the nature of the magnetic material bm ≠ b−m ; the average vorticity is finite. Because more than one angular momentum component is important, the state is entangled. The different angular momentum components are coherent because they are due to the same incoming wave. The phase lags between the different outgoing angular momentum components and the incoming wave is fixed and does not exhibit any statistical fluctuation. The physical properties of the system can be tuned by an external field along the direction of the rods. The sign of the vorticity can be changed by a reversal of the direction of the external field. This usually takes a nanosecond, much faster than the response time of liquid crystals. As the

Fig. 3. Magnitude jbm j of the scattering coefficients of different angular momentum channels m’s (upper panel) and the lateral shift hxi of beam center at y  2λ (lower panel) as a function of frequency.

external field is reversed, the average vorticity is changed; b−1 and b0 becomes much bigger than b1 . A surprising implication of the entanglement comes from a finite average Poynting vector in the direction perpendicular to the mean beam direction. The average transverse displacement as a function of the distance away from the interface is shown in Fig. 4 for different frequencies. The position of the beam in the top panel is extracted from the mean position of the beam at different distances from the ferrite rods, the shapes of which are shown in the two lower panels in the same figure. This demonstrates that the reflected wave packet has a finite component of the group velocity parallel to the interface and hence a finite P x ! The two curves for the top panel of Fig. 4 shows that the sign of the velocity changes when the frequency changes by only 5%. We have considered different widths of the Gaussian beam w. The results are qualitatively the same. We next describe analytic calculations that provides a direct connection between the entanglement and this transverse average velocity. At a given frequency, the total reflected wave f 0k from an incoming plane wave can be obtained by summing up the contributions from each partial waves of different angular momenta scattered from each of the cylinders. For a single array of cylinders, this can be carried out analytically by using Floquet’s theorem, see e.g., Ref. [10]. For the reflected wave, the x component of the wave vector can differ from that of the incoming wave by a reciprocal lattice vector Gj  2πj∕a:kjx  kx  Gj . The corresponding y wavevector for the reflected wave is given by kjy  k2 − k2jx 1∕2 . The reflected field is expressed in terms of planar modes of different orders j [10], f 0k 

X j

expikj · rhkj ;

(1)

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OPTICS LETTERS / Vol. 39, No. 19 / October 1, 2014

hLz i ∝

Fig. 4. Upper panel: the beam center of reflected a Gaussian beam of waist radius w0  2λ, hxi, as a function of the distance y from an array of ferrite rods of radius r s  0.25a at two different frequencies, f a∕c  0.101 and 0.105. The linear dependence of hxi on y demonstrates angular shift of different signs for the reflected beam at a small change of frequency of less than 5%. Middle panel: profile of jEj at different distance y from the array for f a∕c  0.101. Lower panel: the same as middle panel except for f a∕c  0.105.

where hk  −2 m bm ∕im expimϕk ∕ky a, with ϕk  tan−1 ky ∕kx . hk being a sum of terms of different vorticities m. There has been recent interest in optical vortices [1] where the electric field has a Fourier component proportional to expimϕk⊥  where k⊥ is the component of wavevector k in the x–z plane perpendicular to the direction of propagation. In our case, the angle ϕk is in the longitudinal plane, as will be shown in detail below, and the vortex is transverse. We are interested in an incoming Gaussian beam with the electric field polarized along the cylinder axis with a spatial distribution given byR E i x  2∕π1∕4 w−1∕2 ∞ exp−x2 ∕w2 ; at y  0 so that −∞ jEj2 dx  1. Typical widths w we have considered are of the order of 2–4 times the wavelength. We denote the Fourier transform of the incoming wave by E~ i k. For an incoming Gaussian wave packet, the reflected wave is given by the sum over the different Fourier components: E r 

m

mjbm j2 ∕

 X jbm j2 :

(3)

m

Because bm ≠ b−m , there is a net angular momentum, as is advertised. Now the Poynting vector P is given by P  1∕2 ReE × H . For a wave packet, we get P x R hkx i∕ k0 . From Eq. (2), we get P x  1∕2 Re dkx kx j momentum channel m f k E~ i kj2 ∕k0 . If a single angular R dominates, we get P x  dkx kx jbm E~ i k∕2aky j2 . This is odd in kx and equal to zero at perpendicular incidence. On the other hand, now the state is entangled and two channels, the m  0, and the m  1 angular momentum channels, are R important, and we get a contribution P x  1∕2 Re dkx k2x ib0 b1 − b0 b1 ∕kjE 0 k∕2aky j2 , which is even in kx and is not equal to zero. Because of the beam, it will experience a net transverse force. A finite P x of the wave packet is equivalent to a nonzero group velocity R along x. This implies that the average position hxi  dxxjErj2 will change as the wave packet moves away from the surface. We describe our numerical result for this effect next. To provide a better physical understanding and to directly relate these to the entanglement effect, we provide next an approximate analytic description of hxi. Using Eq. (2), the expression for hxi can be converted to an integral with respect to k. We evaluate the k integral for hxi approximately with the saddle point method. We found that Er ∝ hks exp−0.5x  x0 2 ∕w2 ∕2  iy∕k  iky; (4) where

P

s

X

Z dkf k expik · r∕2π;

(2)

with f k  hk E~ i k. The z component of the angular momentum, Lz  x∂y − y∂x ∕i of this state at a fixed can be R frequency s rL E s r  i  drE easily evaluated. We obtain hL z z R dkf k ∂ϕk ∕if k . Thus

x0  ky

X

h X i mbm ∕im expimϕks ∕ k2 bm ∕im expimϕks  ; (5)

the saddle point wave vector is determined from the equation ks;x  ix  x0 ∕w2 ∕2  iy∕k. Now the Gaussian beam appears to be shifted by x0 . Near the SW bands, usually only a few scattering phase shifts are close to resonance and the corresponding scattering amplitudes bm are significant. In the present case, only the m  0 and the m  1 components are significant. Let us examine the expression for x0 . If a single scattering channel at angular momentum m dominates, we obtain a real shift with x0  mky ∕k2 . For our example, both the m  0 and m  1 scattering channels are important, we obtain x0 ≈ −iky b1 expiϕks ∕k2 b0 − ib1 expiϕks : Now x0 contains an imaginary part! The expression for the electric field has an imaginary phase factor equal to iv · r  px  ky where the effective velocity v  p; k with p−

Imx0  : w2 ∕2  iy∕k

Thus the phase velocity has an x component, as is illustrated in Fig. 4. This transverse velocity comes from an

October 1, 2014 / Vol. 39, No. 19 / OPTICS LETTERS

“interference” effect between the two angular momentum channels. In the lower panel of Fig. 3 we show our numerical result for the average position of the reflected wave packet normalized by the wavelength, hxi∕λ, as a function of the frequency of the incoming wave. As can be seen from Fig. 3, the shift hxi becomes significant and shows a dramatic dependence on the frequency in the region around the spin wave frequency. The Goos Hanchen (GH) effect [11] predicts a shift in position of a reflected wave. In the GH effect, the longitudinal displacement is zero at perpendicular incidence. Furthermore, the GH effect predicts that the shift at an angle is proportional to the film thickness, up to the skin depth. The light is coming at perpendicular incidence. Our film is of thickness of 1∕20 of a wavelength. Yet hxi∕λ is significant. All these are different from the predictions of the GH effect. In conclusion, in this Letter, we discuss a different kind of photonic states with finite angular momenta, the TOV. For the “longitudinal” optical vortex previously discussed, the direction of the angular momentum is parallel to the beam direction. For the TOV, the angular momentum is perpendicular to the beam direction. This TOV state can be generated as the reflected wave packet from a single layer of an array of ferrite rods near the spin wave frequency. The reflected wave is a linear combination of states of finite transverse vorticites of the same sign. Because of the interference between components of different transverse vorticities, the Poynting vector has a finite average in the direction perpendicular to the mean beam direction; the angle of incidence is no longer equal to the angle of reflection. This deviation is of the order of a few degrees for a layer of rods with a thickness of the order of 1∕20 of the wavelength. As the

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external field and hence the spin wave frequency is changed by about 5%, the angle of reflection can change sign. With use of the fast switching time and the topological nature of the state, we think this state can provide for a way to encode information for microwave communication that is not susceptible to the usual problem of the fluctuation in magnitude of the signal (fading). ZL was supported by the China 973 Project (Nos. 2011CB922004 and 2013CB632701) and China National Natural Science Foundation (No. 11174059). References and Note 1. See, for example, A. M. Yao and M. J. Padgett, Adv. Opt. Photon. 3, 161 (2011). 2. J. X. Fu, R. J. Liu, and Z. Y. Li, Europhys. Lett. 89, 64003 (2010). 3. J. X. Fu, R. J. Liu, and Z. Y. Li, Appl. Phys. Lett. 97, 041112 (2010). 4. J. Lian, J. X. Fu, L. Gan, and Z. Y. Li, Phys. Rev. B 85, 125108 (2012). 5. See, for example, M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, Phys. Rev. B 56, 959 (1997). 6. S. Y. Liu, W. L. Lu, Z. F. Lin, and S. T. Chui, Appl. Phys. Lett. 97, 201113 (2010). 7. See also, S. T. Chui and Z. F. Lin, J. Phys. Condens. Matter 19, 406233 (2007). 8. S. T. Chui, S. Y. Liu, and Z. F. Lin, J. Phys. Condens. Matter 22, 182201 (2010). 9. We have considered a pulse exp−f − f 0 2 ∕Δf 2  centered at a frequency f 0 a∕c  0.165 with a width Δf a∕c  0.0083. 10. J. J. Du, Z. F. Lin, S. T. Chui, W. L. Lu, H. Li, A. M. Wu, Z. Sheng, J. Zi, X. Wang, S. C. Zou, and F. W. Gan, Phys. Rev. Lett. 106, 203903 (2011). 11. See, for example, J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), p. 308.

Entangled transverse optical vortex.

We discuss a new kind of optical vortex with the angular momentum perpendicular to the flow direction and entangled in that it is a coherent combinati...
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