Letter
Vol. 40, No. 13 / July 1 2015 / Optics Letters
2961
Enhanced spin Hall effect of transmitted light through a thin epsilon-near-zero slab WENGUO ZHU
AND
WEILONG SHE*
State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510275, China *Corresponding author: [email protected] Received 24 April 2015; accepted 20 May 2015; posted 26 May 2015 (Doc. ID 239709); published 17 June 2015
Spin Hall effect (SHE) of transmitted light is small even near the Brewster angle in general case, e.g., at the air–glass interface. However, we find that the SHE can be enhanced when a linearly polarized Gaussian beam transmits through a thin epsilon-near-zero slab due to the difference between the Fresnel transmission coefficients t p and t s . For a vertically polarized incident beam, the spin-dependent transverse displacement of transmitted beam is enhanced near the Brewster angle, while it takes large value over a large angle scale for a horizontally polarized incident beam. © 2015 Optical Society of America OCIS codes: (160.3918) Metamaterials; (240.5440) Polarizationselective devices; (260.5430) Polarization. http://dx.doi.org/10.1364/OL.40.002961
As is well known, a bounded beam will not comply with the Snell’s law and Fresnel equations when reflected from or transmitted through a planar interface [1,2]. Due to the diffractive corrections, the beam undergoes displacements in directions parallel and perpendicular to the plane of incidence [1]. The later one is spin dependent and known as spin Hall effect (SHE) of light [2]. The SHE of light has recently drawn significant attention owing to both its physical interest and potential applications in nanophotonic devices [3–9]. A number of studies have investigated the SHE of light at various interfaces, such as air–glass [10], air–multilayer–film [11], and air– chiral medium interfaces [12]. In general, the spin-dependent displacements at these interfaces are limited by a fraction of the wavelength [1]. However, people found that the displacements of reflected light can be enhanced by launching the incident beam near the Brewster angle [13,14]. A displacement up to 10 wavelengths has been reported by a very recent experimental work [15]. But, the displacement of transmitted light cannot be enhanced simply by using this method [3]. So, we ask is there a way to enhance the SHE of a transmitted light. Epsilon-near-zero (ENZ) metamaterial is a special kind of artificial medium [16,17]. It has the ability to squeeze and tunnel electromagnetic waves in narrow waveguides [18,19], obtain a total reflection or transmission by introducing proper defect [20,21], and enhance the directive emission for an 0146-9592/15/132961-04$15/0$15.00 © 2015 Optical Society of America
embedded source [22]. Taking advantage of the directive emission of ENZ material, a high-resolution wide-field imaging was achieved by simply inserting a thin ENZ material between the sample and substrate, since the material convert diffracted near fields to quasi-zeroth-order far fields [23]. In 2012, Feng showed an anti-Snell’s law refraction phenomenon, when an electromagnetic wave passes from an arbitrary high refractive index media to ENZ metamaterial, the energy flow bends toward the interface normal for all incident angles [24]. Such all angle collimation of the incident wave results from the interplay between ENZ and material loss. Recently, the ENZ metamaterial in the visible spectral range has been achieved by using a metal–insulator–metal waveguide at cutoff [25], or by carefully sculpting parallel array of subwavelength silver and silicon nitride nanolamellae [26]. When a horizontally (H ) polarized plane wave impinges onto an ideal ENZ slab, there is no transmission through the slab, unless the impinging angle is exactly zero, which means the transmission coefficient t p is discontinuous [16]. However, the transmission coefficient for vertically (V ) polarization t s behaves in a very different way. It decreases gradually with the increase of incident angle. For a nonideal ENZ slab, the ratio Ret p ∕t s (Rex denotes the real part of x) is large near Brewster angle, and its reciprocal Ret s ∕t p can also be large at some other angles. The large ratio of Ret p ∕t s and Ret s ∕t p are helpful for the enhancements of SHE of light with V and H polarizations input, respectively. When a linearly H ∕V polarized Gaussian beam passes through a thin ENZ slab, the two opposite spin components of the transmitted beam split. The largest splittings is achieved near the Brewster angle for the case of V polarization input. The splitting for H polarization keeps large over a wide incident angle, and its peak displacement is larger than that for V polarization, when the thickness of slab d < 2λ. The spin-dependent splittings for both H and V polarizations input increase with d due to optical resonance effect. To investigate the SHE of transmitted light, let us consider a monochromatic Gaussian beam passing through a thin ENZ slab, as shown in Fig. 1, where the global coordinate system is x g ; yg ; z g , and the local coordinate systems attached to the incident and transmitted beams are identical and denoted as x; y; z. The angular spectrum of incident Gaussian profile can be written as G 2π−1∕2 w0 exp−k 20 w20 κ2x κ 2y ∕4,
2962
Vol. 40, No. 13 / July 1 2015 / Optics Letters
Letter
where k 0 is the wavevector in vacuum, w0 is the radius of beam waist. For the central wave of the incident beam, the wavevector lies in x g − z g plane, and forms an angle of θi with respect to the z g axis. However, a small in-plane deflection of wavevector κx will induce change in the incident angle of a noncentral wave: θi κ x . According to Ref. [1], the angular spectra of transmitted and incident light beams have following relationship: ˜H κ y t p κ x − t s κx cot θi t p κx Et κy t p κx − t s κ x cot θi t s κx E˜ Vt ˜H Ei ; (1) × E˜ Vi ˜V ˜H ˜V where E˜ H t , E t , E i , and E i are the H and V components of angular spectra of transmitted and incident beams, respectively. t p κ x and t p κy denote the transmission coefficients, which can be expanded into Taylor series. By making the first order approximation, we have t p;s κ y t p;s ∂t p;s ∕∂θi , where [16] tp
1 cos δ −
ts
2 θ −sin2 θ ∕ε i sin δ 1ε cosp iffiffiffiffiffiffiffiffiffiffiffi iffi 2 cos θi ε−sin2 θi
1 δ cos δ − i sin 2
cos 2θi ε
cos θi
pffiffiffiffiffiffiffiffiffiffiffiffi 2
;
:
(2a)
(2b)
ε−sin θi
The parameter δ k 0 d ε − sin2 θi 1∕2 with d and ε being the thickness and relative permittivity of the ENZ slab. One should notice that in the limit of ε 0, t p vanishes excepting normal incidence θi 0. However, at θi 0, t s takes nonzero value. Until now, the angular spectrum of transmitted field is ready to be deduced. And the corresponding complex amplitude in real space can be then calculated via angular spectrum theory. For an incident Gaussian beam with H polarization, the two circular components of the transmitted field (in circular basis eˆ eˆx iˆe y ) are calculated as 1 z0 k 0 x 2 y 2 p ffiffiffi exp − E H π w0 z 0 iz 2z 0 iz ix ∂t p yt s −t p cotθ ixy cotθ ∂t s ∂t p ∓ ∓ − ; × tp z 0 iz z 0 iz ∂θ z 0 iz2 ∂θ ∂θ (3)
displacements contain z dependent terms, which are known as angular Imbert–Fedorov shifts [1]. These shifts are small when z is not very large. However, they cause the wavevectors of the transmitted beams change directions, as shown in Fig. 1. In expression (5), if jt p j2 C 2 ≫ jt s − t p j2 , δ H is proportional to f1 − Ret s ∕t p g cot θ0 [27], which means a large ratio between two transmission coefficients will result in large transverse displacements. The transverse displacements of transmitted beam through a thin ENZ slab can be associated with the Berry phases across the beam: − cot θ0 [1]. And the difference between t p and t s is important in the generation of displacements. A linearly polarized beam can be considered as a superposition of right- and left-handed circularly polarized beams. After transmitted through a slab, a right (left)-handed polarized incident beam will convert partly to left (right) one, unless t p t s . As a result of conservation law of angular momentum for a single photon and the circular asymmetry of the angular spectrum about z axis, the converted beam component undergoes a shift along y axis [4,8]. Therefore, the linear polarized incident beam splits into two opposite circular components after passing through a thin slab. And the initial Gaussian profile gets heavily distorted, especially near Brewster angle. It should be noted that, the analytical displacements are valid only when the incident beam is away from Brewster angle. So, all the numerical results in this Letter are calculated without neglecting the terms with derivatives in Eq. (3). One can see from Eq. (4) or more obviously in Eq. (5) that the spin-dependent displacements depend on the parameters ε and d . We will investigate these dependences in detail in the next part. Figure 2(a) shows the dependences of the absolute values of transmission coefficients t p (solid line) and t s (dashed line) on the incident angle with the parameter d λ, ε 1.225 (pink), 0.01 (blue), and 0.001 (green), respectively. When ε 1.225, t p and t s have similar behaviors. However, their tendencies versus incident angle θi are totally different when ε 0.01 and 0.001. According to Eqs. (3) and (4), we calculate the
where z 0 k0 w20 ∕2 is the Rayleigh length. The transmitted field with V polarization input is obtained by making replacements t p → t s , t s → t p in Eq. (3). At an given plane z const, the transverse displacements of each field components of the transmitted beam compared to the geometrical optics prediction are defined as ZZ ZZ 2 2 ˜ δH ;V yjE H ;V j dxdy∕ jE˜ (4) H ;V j dxdy: The displacements must be calculated numerically. However, if we neglect the terms with derivatives in eq. (3), analytical expressions can be given: δ H Cw 0
jt p j2 − Ret p t s z∕z 0 Imt p t s ; jt p j2 C 2 jt s − t p j2
(5)
where C k0 w0 tan θi . The displacements for V polarization are still obtained by making replacements t p → t s , t s → t p . The
Fig. 1. Illustration of the SHE of transmitted light through an ENZ slab. When a linearly polarized beam is launched into an ENZ slab, the transmitted light beam splits into right- and left-handed circularly polarized components, which undergo transverse displacements δ and δ− , respectively. The reflected beam is not shown.
Letter
Fig. 2. (a) The dependences of transmission coefficients t p (solid lines) and t s (dashed lines) on incident angle. The displacements of right (solid lines) and left (dashed lines) circular components of transmitted light beam for V (b) and H (c) polarizations input. In calculation, the parameters w0 15λ, d λ, and ε 1.225 (pink), 0.01 (blue), and 0.001 (green), respectively. The inset in (b) zooms in the first 10°.
displacements of transmitted beam at z 0 and plot δ V in Fig. 2(b) and δ in 2(c), where the solid and dashed lines H represent the displacements for right and left circular components. From Figs. 2(b) and 2(c), one sees that two spin-opposite components shift toward opposite directions. And the displacements of ENZ slab (ε 0.01 or 0.001) are much larger than that of glass slab (ε 1.225), which is limited by fraction of wavelength. In the case of V polarization input, the displacements δ V for ε 0.01 and 0.001 change very fast near their Brewster angles, and the inset in Fig. 2(b) gives the zoom-in figure. The peak displacements of transmitted beam are obtained near Brewster angles, since the ratios Ret p ∕t s are large near these angles. When ε is near zero, the Brewster angle the critical angle of total internal reflection almost overlap. And a nearly π phase change occurs around the critical angle in t p , which reverses the directions of the displacements. The situation for H incident polarization is quite different. As shown in Fig. 2(c), the Brewster and critical angles give little influence to the spin-dependent displacements, since the ratio Ret s ∕t p is small near the angles. The displacements reach their peak values at θi 21° and 69° for ε 0.01, and at θi 5.8° and 84.5° for ε 0.001. The maximum displacement is up to 7.3λ for ε 0.001. One easily finds that, at θi 45°, the displacements are zero and change its sign. Moreover, they are almost symmetric about point (45°,0°). This is true for a nearly zero ε,
Vol. 40, No. 13 / July 1 2015 / Optics Letters
2963
Fig. 3. Same as Fig. (2), but for ε 0.04, and d λ (pink), λ∕2 (blue), and λ∕10 (green), respectively.
since under this condition, the displacements are proportional to cos 2θi f θi ; d with f θi ; d > 0. The dependences of the absolute values of transmission coefficients t p (solid line) and t s (dashed line) on the incident angle for different thickness of slab d are shown in Fig. 3(a), respectively. When d decreases, both t p and t s increase, but the steep peak near the Brewster angle becomes smooth gradually and the ratio Ret p ∕t s decreases. Therefore, the spindependent displacements of transmitted beam with V polarization input decreases, which is demonstrated in Fig. 3(b). The displacement peaks will shift outward the Brewster angle as d decreases. When d is small, the absolute values of the displacement peaks before Brewster angle are smaller than those after Brewster angle because of the different slopes of t p . Figure 3(c) shows the displacements of transmitted beam with H polarization input, from which one sees that the displacements decrease with the slab thickness d . And the symmetry property about point (45°,0°) is lost. The absolute values of the displacement peaks located in first half (θi < 45°) are smaller than those in the latter half (θi > 45°). In the following, we will investigate the dependences of transverse displacements δ V and δH on the thickness of slab d for three different incident angles θi · θi 4.92° corresponds to the peak displacements for V polarization input at d λ, while θi 21° to peak displacements for H polarization input, as shown by the blue lines in Figs. 2(b) and 2(c). Owing to the optical resonance effect in the ENZ slab [4], both displace ments δ V and δH increase with d , which are shown clearly in Figs. 4(a) and 4(b). And they will tend gradually to asymptotic
2964
Vol. 40, No. 13 / July 1 2015 / Optics Letters
Letter REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Fig. 4. Dependences of displacements of δ V (a) and δH (b) on slab thickness d . In calculation, ε 0.01, θi 4.92° (pink), 13° (blue), and 21° (green).
values. For example, the displacements for H polarization input reach their asymptotic values at d λ. For the case of V polarization input, however, the asymptotic values are approached in different speed. When θi 4.92°, it is after d 3λ that δ V is very close to its asymptotic value 4.4λ. In conclusion, we have demonstrated that large spindependent splittings of transmitted beam can be obtained for both H - and V -incident polarizations by passing a Gaussian beam through a thin ENZ slab. To get these large splittings, a small relative permittivity ε and a proper thickness of the slab d should be chosen, since the splittings increase with d , and approach asymptotic values. We believe that the large SHE of transmitted light could provide an alternative way for the design of nanophotonic devices. Ministry of Education of China (V200801); National Basic Research Program of China (2010 CB923200); National Natural Science Foundation of China (NSFC) (U0934002, 11274401).
14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
K. Y. Bliokh and A. Aiello, J. Opt. 15, 014001 (2013). K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. Lett. 96, 073903 (2006). O. Hosten and P. Kwiat, Science 319, 787 (2008). H. Luo, S. Wen, W. Shu, and D. Fan, Phys. Rev. A 82, 043825 (2010). D. Haefner, S. Sukhov, and A. Dogariu, Phys. Rev. Lett. 102, 123903 (2009). N. Hermosa, A. M. Nugrowati, A. Aiello, and J. P. Woerdman, Opt. Lett. 36, 3200 (2011). M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, Phys. Rev. A 82, 023817 (2010). W. Zhu and W. She, Opt. Lett. 39, 1337 (2014). X. B. Yin, Z. L. Ye, J. Rho, Y. Wang, and X. Zhang, Science 339, 1405 (2013). Y. Qin, Y. Li, H. Y. He, and Q. H. Gong, Opt. Lett. 34, 2551 (2009). B. Wang, Y. Li, M. Pan, J. Ren, Y.-F. Xiao, H. Yang, and Q. Gong, Phys. Rev. A 88, 043842 (2013). G. Xu, T. Zang, H. Mao, and T. Pan, Phys. Rev. A 83, 053828 (2011). H. Luo, X. Zhou, W. Shu, S. Wen, and D. Fan, Phys. Rev. A 84, 043806 (2011). L. Kong, X. Wang, S. Li, Y. Li, J. Chen, B. Gu, and H. Wang, Appl. Phys. Lett. 100, 071109 (2012). J. B. Götte, W. Löffler, and M. R. Dennis, Phys. Rev. Lett. 112, 233901 (2014). A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, Phys. Rev. B 75, 155410 (2007). J. Luo, W. X. Lu, Z. H. Hang, H. Y. Chen, B. Hou, Y. Lai, and C. T. Chan, Phys. Rev. Lett. 112, 073903 (2014). M. Silveirinha and N. Engheta, Phys. Rev. Lett. 97, 157403 (2006). R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, Phys. Rev. Lett. 100, 023903 (2008). J. Hao, W. Yan, and M. Qiu, Appl. Phys. Lett. 96, 101109 (2010). Y. Xu and H. Chen, Appl. Phys. Lett. 98, 113501 (2011). S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, Phys. Rev. Lett. 89, 213902 (2002). J. Kyoung, D. J. Park, S. J. Byun, J. Lee, S. B. Choi, S. Park, and S. W. Hwang, Opt. Express 22, 31875 (2014). S. Feng, Phys. Rev. Lett. 108, 193904 (2012). E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, Phys. Rev. Lett. 110, 013902 (2013). R. Maas, J. Parsons, N. Engheta, and A. Polman, Nat. Photonics 7, 907 (2013). X. Zhou, X. Ling, Z. Zhang, H. Luo, and S. Wen, Sci. Rep. 4, 7388 (2014).
Vol. 40, No. 13 / July 1 2015 / Optics Letters
2961
Enhanced spin Hall effect of transmitted light through a thin epsilon-near-zero slab WENGUO ZHU
AND
WEILONG SHE*
State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510275, China *Corresponding author: [email protected] Received 24 April 2015; accepted 20 May 2015; posted 26 May 2015 (Doc. ID 239709); published 17 June 2015
Spin Hall effect (SHE) of transmitted light is small even near the Brewster angle in general case, e.g., at the air–glass interface. However, we find that the SHE can be enhanced when a linearly polarized Gaussian beam transmits through a thin epsilon-near-zero slab due to the difference between the Fresnel transmission coefficients t p and t s . For a vertically polarized incident beam, the spin-dependent transverse displacement of transmitted beam is enhanced near the Brewster angle, while it takes large value over a large angle scale for a horizontally polarized incident beam. © 2015 Optical Society of America OCIS codes: (160.3918) Metamaterials; (240.5440) Polarizationselective devices; (260.5430) Polarization. http://dx.doi.org/10.1364/OL.40.002961
As is well known, a bounded beam will not comply with the Snell’s law and Fresnel equations when reflected from or transmitted through a planar interface [1,2]. Due to the diffractive corrections, the beam undergoes displacements in directions parallel and perpendicular to the plane of incidence [1]. The later one is spin dependent and known as spin Hall effect (SHE) of light [2]. The SHE of light has recently drawn significant attention owing to both its physical interest and potential applications in nanophotonic devices [3–9]. A number of studies have investigated the SHE of light at various interfaces, such as air–glass [10], air–multilayer–film [11], and air– chiral medium interfaces [12]. In general, the spin-dependent displacements at these interfaces are limited by a fraction of the wavelength [1]. However, people found that the displacements of reflected light can be enhanced by launching the incident beam near the Brewster angle [13,14]. A displacement up to 10 wavelengths has been reported by a very recent experimental work [15]. But, the displacement of transmitted light cannot be enhanced simply by using this method [3]. So, we ask is there a way to enhance the SHE of a transmitted light. Epsilon-near-zero (ENZ) metamaterial is a special kind of artificial medium [16,17]. It has the ability to squeeze and tunnel electromagnetic waves in narrow waveguides [18,19], obtain a total reflection or transmission by introducing proper defect [20,21], and enhance the directive emission for an 0146-9592/15/132961-04$15/0$15.00 © 2015 Optical Society of America
embedded source [22]. Taking advantage of the directive emission of ENZ material, a high-resolution wide-field imaging was achieved by simply inserting a thin ENZ material between the sample and substrate, since the material convert diffracted near fields to quasi-zeroth-order far fields [23]. In 2012, Feng showed an anti-Snell’s law refraction phenomenon, when an electromagnetic wave passes from an arbitrary high refractive index media to ENZ metamaterial, the energy flow bends toward the interface normal for all incident angles [24]. Such all angle collimation of the incident wave results from the interplay between ENZ and material loss. Recently, the ENZ metamaterial in the visible spectral range has been achieved by using a metal–insulator–metal waveguide at cutoff [25], or by carefully sculpting parallel array of subwavelength silver and silicon nitride nanolamellae [26]. When a horizontally (H ) polarized plane wave impinges onto an ideal ENZ slab, there is no transmission through the slab, unless the impinging angle is exactly zero, which means the transmission coefficient t p is discontinuous [16]. However, the transmission coefficient for vertically (V ) polarization t s behaves in a very different way. It decreases gradually with the increase of incident angle. For a nonideal ENZ slab, the ratio Ret p ∕t s (Rex denotes the real part of x) is large near Brewster angle, and its reciprocal Ret s ∕t p can also be large at some other angles. The large ratio of Ret p ∕t s and Ret s ∕t p are helpful for the enhancements of SHE of light with V and H polarizations input, respectively. When a linearly H ∕V polarized Gaussian beam passes through a thin ENZ slab, the two opposite spin components of the transmitted beam split. The largest splittings is achieved near the Brewster angle for the case of V polarization input. The splitting for H polarization keeps large over a wide incident angle, and its peak displacement is larger than that for V polarization, when the thickness of slab d < 2λ. The spin-dependent splittings for both H and V polarizations input increase with d due to optical resonance effect. To investigate the SHE of transmitted light, let us consider a monochromatic Gaussian beam passing through a thin ENZ slab, as shown in Fig. 1, where the global coordinate system is x g ; yg ; z g , and the local coordinate systems attached to the incident and transmitted beams are identical and denoted as x; y; z. The angular spectrum of incident Gaussian profile can be written as G 2π−1∕2 w0 exp−k 20 w20 κ2x κ 2y ∕4,
2962
Vol. 40, No. 13 / July 1 2015 / Optics Letters
Letter
where k 0 is the wavevector in vacuum, w0 is the radius of beam waist. For the central wave of the incident beam, the wavevector lies in x g − z g plane, and forms an angle of θi with respect to the z g axis. However, a small in-plane deflection of wavevector κx will induce change in the incident angle of a noncentral wave: θi κ x . According to Ref. [1], the angular spectra of transmitted and incident light beams have following relationship: ˜H κ y t p κ x − t s κx cot θi t p κx Et κy t p κx − t s κ x cot θi t s κx E˜ Vt ˜H Ei ; (1) × E˜ Vi ˜V ˜H ˜V where E˜ H t , E t , E i , and E i are the H and V components of angular spectra of transmitted and incident beams, respectively. t p κ x and t p κy denote the transmission coefficients, which can be expanded into Taylor series. By making the first order approximation, we have t p;s κ y t p;s ∂t p;s ∕∂θi , where [16] tp
1 cos δ −
ts
2 θ −sin2 θ ∕ε i sin δ 1ε cosp iffiffiffiffiffiffiffiffiffiffiffi iffi 2 cos θi ε−sin2 θi
1 δ cos δ − i sin 2
cos 2θi ε
cos θi
pffiffiffiffiffiffiffiffiffiffiffiffi 2
;
:
(2a)
(2b)
ε−sin θi
The parameter δ k 0 d ε − sin2 θi 1∕2 with d and ε being the thickness and relative permittivity of the ENZ slab. One should notice that in the limit of ε 0, t p vanishes excepting normal incidence θi 0. However, at θi 0, t s takes nonzero value. Until now, the angular spectrum of transmitted field is ready to be deduced. And the corresponding complex amplitude in real space can be then calculated via angular spectrum theory. For an incident Gaussian beam with H polarization, the two circular components of the transmitted field (in circular basis eˆ eˆx iˆe y ) are calculated as 1 z0 k 0 x 2 y 2 p ffiffiffi exp − E H π w0 z 0 iz 2z 0 iz ix ∂t p yt s −t p cotθ ixy cotθ ∂t s ∂t p ∓ ∓ − ; × tp z 0 iz z 0 iz ∂θ z 0 iz2 ∂θ ∂θ (3)
displacements contain z dependent terms, which are known as angular Imbert–Fedorov shifts [1]. These shifts are small when z is not very large. However, they cause the wavevectors of the transmitted beams change directions, as shown in Fig. 1. In expression (5), if jt p j2 C 2 ≫ jt s − t p j2 , δ H is proportional to f1 − Ret s ∕t p g cot θ0 [27], which means a large ratio between two transmission coefficients will result in large transverse displacements. The transverse displacements of transmitted beam through a thin ENZ slab can be associated with the Berry phases across the beam: − cot θ0 [1]. And the difference between t p and t s is important in the generation of displacements. A linearly polarized beam can be considered as a superposition of right- and left-handed circularly polarized beams. After transmitted through a slab, a right (left)-handed polarized incident beam will convert partly to left (right) one, unless t p t s . As a result of conservation law of angular momentum for a single photon and the circular asymmetry of the angular spectrum about z axis, the converted beam component undergoes a shift along y axis [4,8]. Therefore, the linear polarized incident beam splits into two opposite circular components after passing through a thin slab. And the initial Gaussian profile gets heavily distorted, especially near Brewster angle. It should be noted that, the analytical displacements are valid only when the incident beam is away from Brewster angle. So, all the numerical results in this Letter are calculated without neglecting the terms with derivatives in Eq. (3). One can see from Eq. (4) or more obviously in Eq. (5) that the spin-dependent displacements depend on the parameters ε and d . We will investigate these dependences in detail in the next part. Figure 2(a) shows the dependences of the absolute values of transmission coefficients t p (solid line) and t s (dashed line) on the incident angle with the parameter d λ, ε 1.225 (pink), 0.01 (blue), and 0.001 (green), respectively. When ε 1.225, t p and t s have similar behaviors. However, their tendencies versus incident angle θi are totally different when ε 0.01 and 0.001. According to Eqs. (3) and (4), we calculate the
where z 0 k0 w20 ∕2 is the Rayleigh length. The transmitted field with V polarization input is obtained by making replacements t p → t s , t s → t p in Eq. (3). At an given plane z const, the transverse displacements of each field components of the transmitted beam compared to the geometrical optics prediction are defined as ZZ ZZ 2 2 ˜ δH ;V yjE H ;V j dxdy∕ jE˜ (4) H ;V j dxdy: The displacements must be calculated numerically. However, if we neglect the terms with derivatives in eq. (3), analytical expressions can be given: δ H Cw 0
jt p j2 − Ret p t s z∕z 0 Imt p t s ; jt p j2 C 2 jt s − t p j2
(5)
where C k0 w0 tan θi . The displacements for V polarization are still obtained by making replacements t p → t s , t s → t p . The
Fig. 1. Illustration of the SHE of transmitted light through an ENZ slab. When a linearly polarized beam is launched into an ENZ slab, the transmitted light beam splits into right- and left-handed circularly polarized components, which undergo transverse displacements δ and δ− , respectively. The reflected beam is not shown.
Letter
Fig. 2. (a) The dependences of transmission coefficients t p (solid lines) and t s (dashed lines) on incident angle. The displacements of right (solid lines) and left (dashed lines) circular components of transmitted light beam for V (b) and H (c) polarizations input. In calculation, the parameters w0 15λ, d λ, and ε 1.225 (pink), 0.01 (blue), and 0.001 (green), respectively. The inset in (b) zooms in the first 10°.
displacements of transmitted beam at z 0 and plot δ V in Fig. 2(b) and δ in 2(c), where the solid and dashed lines H represent the displacements for right and left circular components. From Figs. 2(b) and 2(c), one sees that two spin-opposite components shift toward opposite directions. And the displacements of ENZ slab (ε 0.01 or 0.001) are much larger than that of glass slab (ε 1.225), which is limited by fraction of wavelength. In the case of V polarization input, the displacements δ V for ε 0.01 and 0.001 change very fast near their Brewster angles, and the inset in Fig. 2(b) gives the zoom-in figure. The peak displacements of transmitted beam are obtained near Brewster angles, since the ratios Ret p ∕t s are large near these angles. When ε is near zero, the Brewster angle the critical angle of total internal reflection almost overlap. And a nearly π phase change occurs around the critical angle in t p , which reverses the directions of the displacements. The situation for H incident polarization is quite different. As shown in Fig. 2(c), the Brewster and critical angles give little influence to the spin-dependent displacements, since the ratio Ret s ∕t p is small near the angles. The displacements reach their peak values at θi 21° and 69° for ε 0.01, and at θi 5.8° and 84.5° for ε 0.001. The maximum displacement is up to 7.3λ for ε 0.001. One easily finds that, at θi 45°, the displacements are zero and change its sign. Moreover, they are almost symmetric about point (45°,0°). This is true for a nearly zero ε,
Vol. 40, No. 13 / July 1 2015 / Optics Letters
2963
Fig. 3. Same as Fig. (2), but for ε 0.04, and d λ (pink), λ∕2 (blue), and λ∕10 (green), respectively.
since under this condition, the displacements are proportional to cos 2θi f θi ; d with f θi ; d > 0. The dependences of the absolute values of transmission coefficients t p (solid line) and t s (dashed line) on the incident angle for different thickness of slab d are shown in Fig. 3(a), respectively. When d decreases, both t p and t s increase, but the steep peak near the Brewster angle becomes smooth gradually and the ratio Ret p ∕t s decreases. Therefore, the spindependent displacements of transmitted beam with V polarization input decreases, which is demonstrated in Fig. 3(b). The displacement peaks will shift outward the Brewster angle as d decreases. When d is small, the absolute values of the displacement peaks before Brewster angle are smaller than those after Brewster angle because of the different slopes of t p . Figure 3(c) shows the displacements of transmitted beam with H polarization input, from which one sees that the displacements decrease with the slab thickness d . And the symmetry property about point (45°,0°) is lost. The absolute values of the displacement peaks located in first half (θi < 45°) are smaller than those in the latter half (θi > 45°). In the following, we will investigate the dependences of transverse displacements δ V and δH on the thickness of slab d for three different incident angles θi · θi 4.92° corresponds to the peak displacements for V polarization input at d λ, while θi 21° to peak displacements for H polarization input, as shown by the blue lines in Figs. 2(b) and 2(c). Owing to the optical resonance effect in the ENZ slab [4], both displace ments δ V and δH increase with d , which are shown clearly in Figs. 4(a) and 4(b). And they will tend gradually to asymptotic
2964
Vol. 40, No. 13 / July 1 2015 / Optics Letters
Letter REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Fig. 4. Dependences of displacements of δ V (a) and δH (b) on slab thickness d . In calculation, ε 0.01, θi 4.92° (pink), 13° (blue), and 21° (green).
values. For example, the displacements for H polarization input reach their asymptotic values at d λ. For the case of V polarization input, however, the asymptotic values are approached in different speed. When θi 4.92°, it is after d 3λ that δ V is very close to its asymptotic value 4.4λ. In conclusion, we have demonstrated that large spindependent splittings of transmitted beam can be obtained for both H - and V -incident polarizations by passing a Gaussian beam through a thin ENZ slab. To get these large splittings, a small relative permittivity ε and a proper thickness of the slab d should be chosen, since the splittings increase with d , and approach asymptotic values. We believe that the large SHE of transmitted light could provide an alternative way for the design of nanophotonic devices. Ministry of Education of China (V200801); National Basic Research Program of China (2010 CB923200); National Natural Science Foundation of China (NSFC) (U0934002, 11274401).
14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
K. Y. Bliokh and A. Aiello, J. Opt. 15, 014001 (2013). K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. Lett. 96, 073903 (2006). O. Hosten and P. Kwiat, Science 319, 787 (2008). H. Luo, S. Wen, W. Shu, and D. Fan, Phys. Rev. A 82, 043825 (2010). D. Haefner, S. Sukhov, and A. Dogariu, Phys. Rev. Lett. 102, 123903 (2009). N. Hermosa, A. M. Nugrowati, A. Aiello, and J. P. Woerdman, Opt. Lett. 36, 3200 (2011). M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, Phys. Rev. A 82, 023817 (2010). W. Zhu and W. She, Opt. Lett. 39, 1337 (2014). X. B. Yin, Z. L. Ye, J. Rho, Y. Wang, and X. Zhang, Science 339, 1405 (2013). Y. Qin, Y. Li, H. Y. He, and Q. H. Gong, Opt. Lett. 34, 2551 (2009). B. Wang, Y. Li, M. Pan, J. Ren, Y.-F. Xiao, H. Yang, and Q. Gong, Phys. Rev. A 88, 043842 (2013). G. Xu, T. Zang, H. Mao, and T. Pan, Phys. Rev. A 83, 053828 (2011). H. Luo, X. Zhou, W. Shu, S. Wen, and D. Fan, Phys. Rev. A 84, 043806 (2011). L. Kong, X. Wang, S. Li, Y. Li, J. Chen, B. Gu, and H. Wang, Appl. Phys. Lett. 100, 071109 (2012). J. B. Götte, W. Löffler, and M. R. Dennis, Phys. Rev. Lett. 112, 233901 (2014). A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, Phys. Rev. B 75, 155410 (2007). J. Luo, W. X. Lu, Z. H. Hang, H. Y. Chen, B. Hou, Y. Lai, and C. T. Chan, Phys. Rev. Lett. 112, 073903 (2014). M. Silveirinha and N. Engheta, Phys. Rev. Lett. 97, 157403 (2006). R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, Phys. Rev. Lett. 100, 023903 (2008). J. Hao, W. Yan, and M. Qiu, Appl. Phys. Lett. 96, 101109 (2010). Y. Xu and H. Chen, Appl. Phys. Lett. 98, 113501 (2011). S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, Phys. Rev. Lett. 89, 213902 (2002). J. Kyoung, D. J. Park, S. J. Byun, J. Lee, S. B. Choi, S. Park, and S. W. Hwang, Opt. Express 22, 31875 (2014). S. Feng, Phys. Rev. Lett. 108, 193904 (2012). E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, Phys. Rev. Lett. 110, 013902 (2013). R. Maas, J. Parsons, N. Engheta, and A. Polman, Nat. Photonics 7, 907 (2013). X. Zhou, X. Ling, Z. Zhang, H. Luo, and S. Wen, Sci. Rep. 4, 7388 (2014).
