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Abstract
Manipulating the polarization states of electromagnetic waves, a fundamental issue in optics, has attracted intense attention. However, most of the reported devices are either so bulky or with specific functionalities. Here we propose a conceptually new approach to design an ultra-thin meta-waveplate (MWP) with anomalous functionalities. By elaborately designing the structural units of the metasurface, the incident right circular polarized (CP) light carrying spin angular momentum can be coupled into two surface plasmon modes with opposite orbital angular momenta which interaction with each other in the near-field, degenerating to a linear polarized (LP) light in the far-filed. The incoming spin angular momentum is annihilated and the designed MWP can function as a quarter-waveplate. However, compared with the conventional quarter-waveplates, our designed MWP owns the unidirectional function (only converting CP light to LP light) with a certain output polarization angle, which provides an extra degree of freedoms in controlling the polarization. Moreover, the designed MWP can function as a chiral material and exhibiting optical rotation properties within a broad bandwidth.
© 2017 Optical Society of America
1. Introduction
Polarization, which is associated with the spin angular momentum of photon, is one of the fundamental characteristics of electromagnetic waves. Manipulating the polarization state of light is of great importance in various optical applications since many optical phenomena are inherently polarization dependent [1]. Conventional methods to control of the polarization depends on birefringent materials, which accumulate different phase retardance along the two orthogonal axes. In such a strategy, the optical components often suffer from specific thickness limitation and bulky configurations for phase accumulation along the optical paths, hindering their integration and miniaturization in nanophotonics.
Metasurfaces [2], planer thin structures with exotic electromagnetic properties, have drawn much attention due to their excellent capacity for efficient light manipulation. Metasurfaces are typically composed of resonant subwavelength elements, where arbitrary changes in amplitude and abrupt phase shifts can be introduced. In virtue of its flexibility and effectiveness in shaping the wavefront, metasurface thus enables a variety of functionalities such as anomalous refraction or reflection [3–7], focusing [8–16], optical vortex generation [17–21] and polarization control [22–28]. For this reason, metasurfaces promise to be an excellent candidate to replace the conventional bulky components for wavefront and polarization manipulation, imaging or on-chip integration.
Optical quarter-waveplate (QWP), one of the most important optical components, have attracted constant attention for their switching between linear polarization and circular polarization. Recently, advances in the field of metasurfaces have offered a new way for optical QWP miniaturization [22–25,28]. However, these types of QWPs are designed by optimizing the parameters of the unit cells so that to have equal amplitudes and a π/2 phase difference when illuminated by two orthogonal polarized light. What’s more, the QWPs based on such a design strategy can realize mutual transformations between the linear polarization light (LPL) and circular polarization light (CPL). To the best of our knowledge, how to arbitrarily control the LPL’s polarization rotation angle or achieve a unidirectional function has barely been investigated.
Here we propose a conceptually new approach to realize a meta-waveplate (MWP) by utilizing an ultra-thin plasmonic metasurface. The unit cell of the MWP is composed of two apertures, which transform the incident CPL carrying spin angular momentum into two surface plasmon modes with opposite orbital angular momenta. The generated two surface plasmon modes interact in the near-field, annihilating the spin angular momentum of the incident CPL. In the far-field, the light become linear polarized (LP), indicating the QWP’s functionality of our designed structure. Besides, for incident LP light with polarization angle ± 45°, the transmitted light is still LP. Compared with the previous QWP, our proposed MWP could realize unidirectional polarization conversion with a certain output polarization angle, providing a new approach for designing the QWP. Moreover, for LP incident light with other polarization angles, the non-chiral metasurfaces exhibits large optical activity that depends on the wavelength and incident polarization angle.
2. Theoretical analysis
As we known, the Dirac bracket notation can be utilized to define the polarization state of light. For a LP light, the light’s polarization state can be expressed as
where θ is defined as the angle between the polarization plane and x axis. Hence, |Jx> = [1,0]T and |Jy> = [0,1]T, two orthogonal components of the LP, represent the x-directional LP light and y-directional LP light, respectively. Light with any polarization can be written as a combination of the two orthogonal components. Therefore, the Dirac bracket forms for CPL can be written aswhere |JL> and |JR> correspond to Dirac brackets of the left-handed circular polarization (LCP) light and right-handed circular polarization (RCP) light, respectively.Figures 1(a) and 1(b) show the simulated Ez-field distribution of the aperture structures under normal RCP incidence light. It indicates that the ‘ + ’-shaped aperture supports the surface plasmon carrying one orbital momentum while the other L-shaped aperture with opposite momentum. As shown in Fig. 1(c), when placing the two apertures together to a certain value, the generated two plasmonic waves will interact strongly with each other. The interaction between the incident light and the metasurface can be expressed as [29,30]
where the operator Û represents the momentum conversion function of the metasurface, namely the total influence of the matasurface on the transmission and phase shift of the incident light. The operators Û1 and Û2 represent the contribution of the L-shaped aperture and ‘ + ’-shaped aperture, respectively. With this consideration, Eq. (3) can be written aswhere am and φm (m = 1,2) represent the amplitude of the transmission and phase shift introduced by the two apertures. Assuming that the transmission amplitudes of the two SPP waves have the same value a1 = a2 and substituting Eq. (2) into Eq. (4), we can finally derived the transmitted light aswhere △φ = abs(φ1-φ2) represents the phase difference between the two SPP sources. Compared Eq .(1) with Eq. (5), we can conclude that the CP light degenerates to LP light after transmitting through the elaborately designed metasurface and the polarization direction is governed by △φ/2. Figure 1(d) shows the corresponding schematic of the polarization conversion process, from which we can see that the incident CPL with spin angular momentum is converted into LPL without any angular momentum. From that point of view, we can say that the spin angular momentum is annihilated during this process.
Fig. 1 Simulated Ez-field distribution of the nanoapertures under normal incident RCP light. (a) ‘ + ’-shaped aperture, (b) L-shaped aperture, (c) ‘ + ’-shaped and L-shaped apertures with a separated distance 0.31μm. The detail parameters of the two apertures are L1 = 200nm, L2 = 190nm and W = 40nm, which are optimized by using the three-dimensional finite difference time domain (FDTD) method in the results and discussion part. For these simulations, perfectly matched layers (PML) is applied along the three axis. (d) The corresponding schematic of the polarization conversion process.
3. Designs and structure
Figure 2 shows the schematic of the designed MWP with anomalous functionalities. The dashed line in the inset shows the unit cell of the MWP, which is composed of a L-shaped and a ‘ + ’-shaped apertures milled in a gold film. The thickness of the gold film is 200nm, the width of all the apertures is W = 40nm and the lengths of the two apertures are set as L1 and L2, respectively. Conceptually different from the previous QWP, the design principle of our MWP arises from annihilating the spin angular momentum of the incident CP light, which are demonstrated detailedly in the theoretical analysis part. The upper right inset represents the super cell of the MWP, in which the apertures are cross-arrayed to better annihilate the spin angular momentum. From the theoretical analysis, it is possible to realize the polarization conversion from CP to LP on condition that the two generated CPLs own the same amplitude and opposite helicity. This can be achieved by properly designing the structure parameters of the two apertures. Here, the parameters of the nanoapertures are optimized by using the three-dimensional finite difference time domain (FDTD) method from Lumerical Inc and the remaining results are also simulated by utilizing the FDTD method. In our simulations, the spatial mesh steps were set as Δx = Δy = 5nm and Δz = 20nm. The permittivity of the gold is approximated by the Drude model, which defines as ε(ω) = 1-ωp2/(ω2 + iωγp), with the plasma frequency ωp = 1.37 × 1016Hz and the absorption coefficient γp = 4.08 × 1013Hz. These parameters are obtained by fitting the experimental results [31].
Fig. 2 Schematic of the designed MWP for polarization manipulation. The dashed line in the inset is the unit cell of MWP with L1 and L2 to be the lengths of the L-shaped and ‘ + ’-shaped apertures. All the apertures have the same width W = 40nm and the thickness of the gold film is 200nm. The center to center distance between the apertures is set as 310nm, which is optimized in the results and discussion part.
4. Results and discussion
According to our approach, the first issue for the QWP design is to optimize the parameters of the apertures and enable the conversion of RCP incident light into two surface plasmon waves with opposite orbital angular momenta. Figures 3(a) and 3(b) show the transmission T of the L-shaped and ‘ + ’-shaped apertures, respectively. The ellipticity is utilized to describe the polarization state of the output light, which is defined by the Stokes parameters as X = S3/S0 [23]. Figure 3(c) demonstrates the ellipticity X as a function of the L-shaped aperture’s length L and the period P. The values of X equal to 1 and −1 represents the perfect RCP and LCP light, respectively. The second key issue is to make sure that the excited two surface plasmon waves have comparable amplitudes. Based on the above results, the apertures that satisfy the requirements can be selected. The detail parameters are L1 = 200nm, W = 40nm, P1 = 310nm, T1 = 0.421, X1 = −0.96 for the L-shaped aperture and L2 = 190nm, W = 40nm, P2 = 310nm, T2 = 0.415, X2 = 1 for the ‘ + ’-shaped aperture. As we know, the ‘ + ’-shaped aperture will not introduce additional phase difference on the incident RCP light for its structural symmetry [32]. Hence, the output light is RCP with X = = 1 and the corresponding figure that similar to Fig. 3 (c) is not shown here. Figure 3(d) shows the phase φx of the two selected apertures as a function of the wavelength λ, from which we can obtain the relative phase difference is △φ = 0.82 for the design wavelength (λ = 720nm). When placing the two apertures together to a certain value, the output light is LP in the far field as a result of the near-filed angular momentum annihilation between the generated plasmonic waves. Here, to make sure that the parameters of the two apertures are optimal, the center to center distance between the two apertures is set as 310nm.
Fig. 3 (a) Normalized transmission T for the L-shaped aperture as a function of L1 and P1. (b) Normalized transmission T for the ‘ + ’-shaped aperture as a function of L2 and P2. (c) Ellipticity for the L-shaped aperture as a function of L1 and P1. For (a)-(c), the incident light is RCP with a wavelength λ = 720nm. (d) The corresponding phase φx of the two selected apertures on the dependence of wavelength λ. The black dashed line corresponds to the wavelength of λ = 720nm. The thickness of the gold film is 200nm and the width of the apertures is W = 40nm. For these simulations, periodic boundary conditions are applied along the x and y axis and perfectly matched layer is applied along the z axis.
4.1 The designed MWP with CP to LP polarization conversion
The degree of linear polarization (DoLP), defined as , is verified to be a good figure of merit to characterize the performance of the LP light [22–24]. This quantitatively describes the polarization state of the output light, for which the value equal to 1 represents the perfect LP light. The DoLP of the proposed MWP in the far-field are depicted in Fig. 4(a), where the value above 0.9 indicated by shadowed region can be regarded as perfect LP light. It is obvious that for a bandwidth of about 50 nm (shadowed region), the incident CPL light are converted into LP light, indicating the QWP’s functionality of our designed structure. Figure 4(b) demonstrates the normalized transmission of the MWP in the far-filed. In the corresponding bandwidth the largest transmission can nearly reaches 0.2. To further demonstrate the LP property of the output light, the phase profile of the MWP along the x-direction (φx = angle(Ex)) and y-direction (φy = angle(Ey)) are illustrated in Figs. 4(c) and 4(d), respectively. The constant phase values in the x-y plane (φx = −2.05, φy = 0.86) suggest nearly π phase difference existing between Ex and Ey, confirming that the output light is LP. Based on the calculation of individual ‘ + ’ and ‘L’ periodic arrays, the relative phase difference between the two selected apertures at the design wavelength (λ = 720nm) is △φ = 1.64. According to Eq. (5), the polarization angle of the output LPL is calculated as △φ/2 = 0.82 (expressed in degree as 47°), showing well agreement with the simulated value (49°). Therefore, a MWP with the improved functionality of a QWP (convert CPL to LPL with a certain polarization angle) is demonstrated.
Fig. 4 (a) Degree of linear polarization (DoLP) as a function of the wavelength for RCP incident light. (b) Normalized transmission for the designed MWP. (c) The phase profile in the x-y plane along the x-direction (φx = angle(Ex)) at the wavelength of 720nm. (d) The phase profile in the x-y plane along the y-direction (φy = angle(Ey)) at the wavelength of 720nm. The phase profiles are simulated at 5μm from the gold surface.
4.2 The designed MWP with unidirectional polarization conversion
Next, we will demonstrate the unidirectional functionality of the designed MWP. It is well known that for LP incident light with certain polarization angles ± 45°, the conventional QWP will convert the LP light back into CPL light. However, for our proposed MWP, the transmitted light is LP rather than CPL, which is totally different from the conventional QWP. This novel performance can be ascribed to the conceptually different designing principle of our MWP. Figures 5(a) and 5(b) show the simulated DoLP and angle of linear polarization (AoLP) of the proposed MWP for LP light with incident polarization angles θin = ± 45°, respectively. As shown in Figs. 5(a) and 5(b), the output light is LP with the AoLP keep unchanged. Hence, unidirectional polarization conversion (only from CPL to LPL) is achieved for the first time in our proposed MWP, which provides an extra degree of freedoms in controlling the polarization state.
Fig. 5 Simulated DoLP and angle of linear polarization (AoLP) of the proposed MWP for LP light with different incident polarization angle θin; (a) θin = 45°, (b) θin = −45°, (c) θin = 0°, (d) θin = 90° . The black arrows represent the incident polarization directions.
4.3 The designed MWP with large optical activity
Optical activity is defined as the rotation of the plane of LP light along the propagation direction while it passes through the optically active materials [33,34]. The strength of optical activity is conventionally determined by the chirality of the material and then have been demonstrated by utilizing chiral or non-chiral structures [34–37]. Here, a large optical activity can also be realized in our proposed non-chiral metasurface structure since it lacks the two-fold rotational symmetry. Figures 5(c) and 5(d) show the simulated DoLP and AoLP of the proposed MWP for LP light with incident polarization angles θin = 0° and 90°, respectively. It is revealed that in the shadowed region (the value of DoLP is above 0.9) the output light is LP with various rotational polarization angles that depend on the incident wavelengths. To further investigate the optical activity of the proposed MWP, we plot the DoLP and polarization rotation angle at the designed wavelength λ = 720nm in Fig. 6. The incident polarization angle θin ranges from 0° to 150° with a step length of 30°. Here, the polarization rotation angle is adopted to clearly characterize the optical activity [37]. It can be observed that the output light is LP and the polarization rotation angles exhibit different values at different incident polarization angle. The largest value of polarization rotation angle reaches 18.9° at incident polarization angle θin = 120°. These results will provide helpful guidelines in controlling the optical activity.
Fig. 6 Simulated DoLP and polarization rotation angle of the proposed MWP at the designed wavelength λ = 720nm. The incident light is LP with the incident polarization angles θin range from 0° to 150° with a step angle of 30°.
5. Conclusions
In conclusion, we have proposed a conceptually new approach to design an untra-thin MWP, for which the basic design principle is based on annihilating the spin angular momentum. With such a strategy, the presented MWP exhibits anomalous functionalities such as controllable polarization conversion (converting CPL to LPL with a certain output polarization angle), unidirectional polarization conversion (only converting CPL to LPL) and large optical activity (depends on the wavelength and incident polarization angle). These results give an extra degree of freedom in manipulating the polarization state and optical activity, which may provide more possibilities for creating compact, integrated, and multi-functional plasmonic circuits and devices.
Funding
State Key Program for Basic Research of China (2013CB632705), National Natural Science Foundation of China (11334008, 61290301 and 61521005), Fund of Shanghai Science and Technology Foundation (16JC1400400, 16ZR1445300), Shanghai Sailing Program (16YF1413200), and Youth Innovation Promotion Association CAS (2017285).
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