Polarization controllable generation of flat superimposed OAM states based on metasurface

Ming Chen; Wenwen Gao; Houquan Liu; Chuanxin Teng; Shijie Deng; Hongchang Deng; Libo Yuan

doi:10.1364/OE.27.020133

1. Introduction

Optical angular momenta are classified as spin angular momentum (SAM) and orbital angular momentum (OAM). The SAM is arisen from the circular polarization state of the light. In contrast, the OAM is associated with azimuthal phase structure of the light beam. It is not until 1992 that researchers recognized that light beam can carry well-defined OAM [1]. Light possessing OAM exhibits a helical phase front in the mathematical form of $\exp (i l φ)$ , where $φ$ is the azimuthal angle and $l$ is the topological charge. The significance of the OAM is that a high dimensional Hilbert space can be constructed by using the OAM state of a single photon to achieve high-dimensional quantum information encoding [2]. In recent years, the OAM superposition states have received extensive attention from researchers [3]. It has been shown that the OAM superposition states have many novel application values in both classical physics and quantum sciences [4]. For instance, an equal-weight linear combination of OAM modes of $| l | = 1$ with opposite signs can produce vector beams [5]. This kind of beams have unique advantages in laser processing and optical imaging. Arbitrary superpositions of atomic rotational states can be generated in a Bose-Einstein condensate via multi-OAM state [6]. And the superposition state of high-order OAM modes can be used for ultrasensitive angular measurement and spin object detection [7]. Therefore, the efficient generation of the superimposed-OAM-state beams is suggested to be important.

Liquid crystal spatial light modulator (SLM) is excellent for generating OAM superposition states [8,9]. However, the SLM is large in size and do not meet the current demand of high integration devices. In our previous work [10], we found that the quadratic-power-exponent-phase (QPEP) vortex beam can be expanded as the superposition of a series of canonical spiral phase vortices with approximately equal power (the power variation less than 3 dB). It is an interesting phenomenon, based on which, a phase plate to realize QPEP modulation is proposed to generate OAM superposition states with flat power spectrum. The advantages of the phase plate are that it is simple in structure and can be integrated. However, the phase plate lacks adjustability, i.e., once it is prepared, the resultant OAM superposition state is fixed.

Metasurface [11], a new type of 2D artificial material with subwavelength discrete structure [12–14] or continuous structure [15,16], can provide superior characteristics in local control of the amplitude, phase and polarization of light beams [17–20]. Its ultra-thin and easy-to-manufacture features make the metasurface an ideal platform for device miniaturization and system integration [21–24]. Therefore, a metasurface device can be designed to provide QPEP modulation to generate OAM superposition states. Compared to our previous wave plate [10], there will be significant differences and advantages by using metasurface devices. Firstly, a metasurface is based on abrupt geometric phase and can operate effectively at subwavelength thickness, while our previous wave plate is based on the propagation phase, resulting its thickness much larger than the working wavelength. Secondly, the light phase control of a metasurface is generally light polarization dependent. Thus, polarization controllable OAM superposition state generators can be designed by using metasurface. In addition, in generally, metasurface devices always have good broadband response characteristics.

Thus, in this paper, two kind of polarization controllable OAM-superposition-state generators are proposed by using all-dielectric metasurface. One of them can generate OAM superposition states with power variation between the OAM modes less than 3 dB, and the other can generate OAM superposition states with power variation between the OAM modes less than 0.3 dB. In addition, the two generators can effectively operate in wide wavelength range of 635nm-730nm. In the following, we will introduce the two generators in detail.

2. Theory and design of the metasurface

Firstly, the QPEP vortex beam can be expressed as

E = E_{0} \exp [i 2 π M {(\frac{φ - α}{2 π})}^{2}],

where $E_{0}$ is the amplitude, $φ$ is the azimuth angle ranging from 0 to 2π, $α$ is a variable angle and $M$ is a dimensionless constant. The value of M determines the phase variation of the field with $φ$ . A larger M value leads to a faster phase variation. When $α = 0$ , the phase factor $2 π M {(φ / 2 π)}^{2}$ ranges from 0 to $2 π M$ . In this case, the physical meaning of M is clear, which is: the absolute value of M is the number of 2π-phase shifts that occurs across one revolution of $φ$ from 0 to $2 π$ , and its sign determines the handedness of the phase helix. Itcan be seen from the phase map of $E = E_{0} \exp [i 2 π M {(φ / 2 π)}^{2}]$ under M = 10 given in Fig. 1, in which there are 10 2π-phase shifts. While in the case of $α \neq 0$ , there is no definite physical meaning of M. From Fig. 1 one can also see that the phase variation of the QPEP vortex with $φ$ is gradually intensified due to the power-exponent phase term.

Fig. 1 The phase map of $E = E_{0} \exp [i 2 π M {(φ / 2 π)}^{2}]$ for M = 10.

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The Eq. (1) can be expanded as the superposition of a series of canonical spiral phase vortices, in which the relative power of the Nth-order vortex (the OAM mode with topological l = N) is given by [10]

P (N) = {| F (N) |}^{2},

where

F (N) = \frac{1}{2 π} \int_{0}^{2 π} \exp [i 2 π M {(\frac{φ - α}{2 π})}^{2}] e^{- i N φ} d φ,

is the Nth-order Fourier expansion coefficient. As long as M≥2, the obtained OAM spectrum exhibits a flat region (more detail about this will be given in the following). The results of [10] show that the power variation of the OAM modes within the flat region is less than 3 dB and, the topological value of the center OAM mode of the flat region moves with the change of

α

. When

α = π

, the topological value of the center OAM mode is 0, while when

α \neq π

, the topological value of the center OAM mode will deviate from 0. Since the OAM spectrum of Eq. (1) must be symmetric with that of its complex conjugate light with respect to the topological charge value of 0, it is not difficult to imagine that as long as

α \neq π

, the OAM spectra of Eq. (1) and its complex conjugate light must be partially separated from each other. This is realizingly shown by the calculation results of Fig. 2, in which the red line is the OAM spectrum of Eq. (1) and the blue line is the OAM spectrum of its complex conjugate light. Inthe calculation M = 10 is used, and the results of Figs. 2(a), 2(b), 2(c) and 2(d) are corresponding to

α = 0

,

α = π / 2

,

α = π

and

α = 3 π / 2

, respectively. In the Fig. 2(a), the blue arrows are used to mark out the bandwidth of the flat region with power variation less than 3 dB of the OAM spectrum of Eq. (1). Introducing

η

to present the number of the OAM modes within the flat region of OAM spectrum of Eq. (1), the relationship between it and M can be calculated numerically. The calculation is performed for a series of discrete M (integers ranging from 1 to 20). In the calculation, we first calculate the OAM spectrum under each discrete M, then extract the corresponding

η

one by one. The results are shown in Fig. 3. One can see that

η = 1

when M = 1. It means that for this case although the QPEP modulation can raise the power of multi-OAM modes, there is no a flat region. While when M≥2, the flat region appears. One can also see that the calculated data points are not in a straight line. It is because the number of the OAM modes must be an integer.

Fig. 2 The OAM spectrum of Eq. (1) (the red line) and its complex conjugate light (the blue line). In the calculation M = 10 is used. The results of (a), (b), (c) and (d) are corresponding to $α = 0$ , $α = π / 2$ , $α = π$ and $α = 3 π / 2$ , respectively. The blue arrows in (a) are used to mark out the bandwidth of the flat region with power variation less than 3 dB of the OAM spectrum of Eq. (1).

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Fig. 3 Relationship between the number $η$ of the OAM modes within the flat region of OAM spectrum of Eq. (1) and M.

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The results of Fig. 2 clearly shown that when $α = π$ , the OAM spectra of Eq. (1) and its complex conjugate light are coincident. While when $α \neq π$ , the two OAM spectra are gradually separated from each other and, when $α = 0$ they are completely separated. Therefore, as long as we design such a polarization-dependent metasurface device, when light beams with different polarizations passing through the device can obtain phase factor $\exp [i 2 π M {(φ / 2 π)}^{2}]$ and its complex conjugate phase $\exp [- i 2 π M {(φ / 2 π)}^{2}]$ respectively, the OAM spectra obtained under the two-polarization cases are completely separated, thereby a polarization controllable OAM-superposition-state generator can be obtained.

The metasurface sub-wavelength dipole unit can be seen as a waveplate, and its regulation of the beam phase can be represented by a transmission matrix [25]:

T = \cos \frac{δ}{2} (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) - i \sin \frac{δ}{2} (\begin{matrix} \cos 2 θ & \sin 2 θ \\ \sin 2 θ & - \cos 2 θ \end{matrix}),

where

δ

is the relative phase delay between the two light components polarized along the two axial directions of the dipole unit,

θ

is the rotation angle of the dipole unit with respect to x axial of the global coordinate system.

If considering the incident light is circularly polarized, i.e., $E_{i n} (r, φ) = E_{0} {[1, σ i]}^{T}$ , where $σ = \pm 1$ represent left-handed and right-handed circularly polarization, respectively. The output beam can be expressed as

E_{o u t} = T E_{i n} = E_{0} \cos \frac{δ}{2} (\begin{array}{l} 1 \\ σ i \end{array}) - i E_{0} \sin \frac{δ}{2} \exp (2 σ i θ) (\begin{array}{l} 1 \\ - σ i \end{array}) .

It can be seen from Eq. (5) that the output light contains two items. The polarization of the first item remains unchanged, while the polarization of the second item is changed. And only the second item gains a phase factor

\exp (2 σ i θ)

, which means only the phase of this part is modulated by the metasurface. Therefore, when designing the metasurface devices, the first item should be minimized. In theory, when

δ = π

, the first item can be eliminated completely. In order to design a metasurface device to realize the QPEP modulation, the rotation angle

θ

needs to be a function of the azimuth angle

φ

. By considering that the OAM spectra generated under different-polarization conditions are separated completely, i.e.,

α =0

, the relationship between the rotation angle

θ

and the azimuth angle

φ

should satisfy:

θ = π M {(\frac{φ}{2 π})}^{2} .

It can be known from the Eqs. (5) and (6) that when left-handed circularly polarized light (

σ = 1

) passes through the metasurface, the right-handed circularly polarized component of the output beam will obtain phase factor

\exp [i 2 π M {(φ / 2 π)}^{2}]

and, when right-handed circularly polarized light (

σ = - 1

) passes through the metasurface, the left-handed circularly polarized component of the output beam will gain the complex conjugate phase

\exp [- i 2 π M {(φ / 2 π)}^{2}]

. This is exactly what we want.

We design the metasurface device using amorphous silicon dipole unit. The dipole unit of the amorphous silicon material cannot only reduce the absorption of light, but also suppress the reflection effectively, thereby increasing the transmittance of the device. We consider the elliptical cylinder structure as the dipole unit, as shown in Fig. 4(a). By using the FDTD method to simulate at a center wavelength of 670nm, it can be obtained that when the geometric parameters of the elliptical cylinder structure are: length (long axis) l = 280 nm, width (short axis) w = 110 nm and height h = 370 nm, and the interval between two adjacent dipole units is 300 nm, the relative power (relative to the incident power) of the polarization changed part in the output beam can reach $T_{c r} = α {sin}^{2} (δ /2)=97 .5%$ , where $α$ is the attenuation coefficient caused by material absorption. $T_{c r}$ is generally called the cross-transmission coefficient. Since the material absorption is taken into account in the FDTD simulation, the cross-transmission coefficient cannot reach 100% even though $δ = π$ .

Fig. 4 (a) The schematic diagram of the dipole unit, in which the upper layer is an amorphous silicon elliptical cylinder structure with length (long axis), width (short axis) and height being l, w and h, respectively, and the lower layer is silicon dioxide substrate. (b) The relationship between the cross-transmission coefficient and the wavelength (red line), and the relationship between the relative phase delay $δ$ and the wavelength (black line). (c) The schematic diagram of the working principle of the metasurface device, in which a left-handed circularly polarized light beam passes through the metasurface, the emitted light contains part of right-handed circularly polarized light carrying phase $\exp [i 2 π M {(φ / 2 π)}^{2}]$ and part of left-handed circularly polarized light.

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It can be seen from Eq. (5) that the phase modulation of the metasurface dipole unit on the light beam is determined by the rotation angle $θ$ and the polarization state of the incident light, which is wavelength independent. So, in generally, metasurface devices always have good broadband response characteristics [26]. Therefore, we can further analyze the broadband response characteristics of the designed metasurface structure. As the phase delay $δ$ is a function of the wavelength (see Fig. 4(b)), when $δ \neq π$ , the output beam should contain two parts, i.e., polarization unchanged and changed parts, as shown in Fig. 4(c). Simulating by FDTD, the cross-transmission coefficient of the metasurface device in the wavelength range of 600nm-750nm is obtained, the result is shown in Fig. 4(b). It can be found that the cross-transmission coefficient can reach 90% in the range of 635 nm-730 nm.

Until now, a polarization controllable metasurface-based flat OAM superposition-state generator is proposed. The power variation of the OAM modes of the generated OAM superposition states is less than 3 dB. In the following, we will show that the flatness of the OAM spectrum could be further optimized to less than 0.3 dB, and then put forward the other OAM superposition-state generator. To achieve this, we find that the specific form of the light should be:

E_{1} = E_{0} [1 + e^{i π / 2} e^{\frac{i π \cos φ}{2}}] \exp [i 2 π M {(\frac{φ}{2 π})}^{2}] .

The phase map of this field for M = 10 is shown in Fig. 5. And its OAM spectrum could be obtained via its Fourier transformation. The relative power of the Nth-order OAM mode is

P_{1} (N) = {| F_{1} (N) |}^{2},

with

F_{1} (N) = \frac{1}{2 π} \int_{0}^{2 π} [1 + e^{i π / 2} e^{\frac{i π \cos φ}{2}}] \exp [i 2 π M {(\frac{φ}{2 π})}^{2}] e^{- i N φ} d φ .

According to Eq. (8), Fig. 6 gives the numerical results of the OAM spectra of Eq. (7) (the red line) and its complex conjugate light (the blue line) for a series of

M

. Figures 6(a), 6(b), 6(c), and 6(d) are the results of M = 10, M = 14, M = 16 and M = 19, respectively. The blue arrows in Fig. 6(a) are used to mark out the bandwidth of the flat region with power variation less than 0.3 dB of the OAM spectrum of Eq. (7).

Fig. 5 The phase map of light field of Eq. (7) for M = 10.

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Fig. 6 The OAM spectrum of Eq. (7) (the red line) and its complex conjugate light (the blue line), in which (a), (b), (c), and (d) are the results of M = 10, M = 14, M = 16 and M = 19, respectively. The blue arrows in (a) are used to mark out the bandwidth of the flat region with power variation less than 0.3 dB of the OAM spectrum of Eq. (7).

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It can be seen that the OAM spectrum shows a flat region. The power variation of the OAM modes within the flat region is less than 0.3 dB, and the number of the OAM modes within the flat region can reach dozens and is determined by $M$ . In Fig. 6, except the number of OAM modes in the flat region, there is no significant difference between Figs. 6(a)-6(d). The power variations between the OAM modes within the four flat regions of Figs. 6(a)-6(d) are all about 0.3 dB. Actually, it could be verified that for other M bigger than 4 (see Fig. 7), the obtained OAM spectrum exhibits a flat region, in which the power variation between the OAM modes is about 0.3 dB. This property of the OAM spectra for cases of M = 4,5,6…,30 was confirmed one by one (the OAM spectra are not shown in this paper). The Fig. 7 is the relationship between the number $η_{1}$ of OAM modes in the flat region with power variation less than 0.3 dB of the OAM spectrum of Eq. (7). Here, $η_{1}$ it is extracted same as $η$ in Fig. 3. From Fig. 7 we can found that as long as M≥4, the flat region of the OAM spectrum of filed Eq. (7) appears.

Fig. 7 Relationship between the number $η_{1}$ of the OAM modes within the flat region with power variation less than 0.3 dB of OAM spectrum of Eq. (7) and M.

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Another important thing we can learn from Fig. 6 is that the OAM spectra of Eq. (7) and its complex conjugate light are separated from each other. So, to put forward a polarization-dependent OAM superposition-state generator, our task is to design such a metasurface device, when light beams with different polarizations passing through the device can obtain phase factor $(1 + e^{i π / 2} e^{i π \cos φ / 2}) \exp [i 2 π M {(φ / 2 π)}^{2}]$ and its complex conjugate one, respectively. To achieve this, two set of dipole unit arrays are designed in single matesurface. The dipole units of the two sets have same length, width, height and material. Thus, they work for same frequency. The only difference of the two sets is the orientation of the dipole units. The orientation of the first set is designed to assign phase factor $\exp [i 2 π M {(φ / 2 π)}^{2}]$ ( $\exp [- i 2 π M {(φ / 2 π)}^{2}]$ ) to the light beam when the incident light polarization is left-handed (right-handed) circular polarization, i.e., if there only exists the first set of dipole units, the generated field is $E_{0} \exp [i 2 π M {(φ / 2 π)}^{2}]$ ( $E_{0} \exp [- i 2 π M {(φ / 2 π)}^{2}]$ ) when the incident light polarization is left-handed (right-handed) circular polarization. While the second set is designed to assign phase factor $e^{i π / 2} e^{i π \cos φ / 2} \exp [i 2 π M {(φ / 2 π)}^{2}]$ ( $e^{- i π / 2} e^{- i π \cos φ / 2} \exp [- i 2 π M {(φ / 2 π)}^{2}]$ ) to the light beam when the incident light polarization is left-handed (right-handed) circular polarization. Therefore, by nesting the two sets evenly to each other (see Fig. 8(a)), we can obtain a novel metasurface structure to offer light modulation of $(1 + e^{i π / 2} e^{i π \cos φ / 2}) \exp [i 2 π M {(φ / 2 π)}^{2}]$ and its complex conjugated form when the incident polarization is left-handed and right-handed circular polarization, respectively. It is exactly the polarization-dependent OAM superposition-state generator that we want. This method of achieving combined functions by combining multi-metasurface structures has been proved to be feasible in [14,27].

Fig. 8 (a) The OAM-superposition-state generator obtained by combining two set of dipole units. (b) The schematic diagram of the working principle of the generator, in which a left-handed circularly polarized light beam passes through the metasurface, the emitted light contains part of left-handed circularly polarized light and part of right-handed circularly polarized light carrying phase factor $\exp (i π / 2 + i π \cos φ / 2) \exp [i 2 π M {(φ / 2 π)}^{2}]$ .

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To fabricate the combined metasurface structure, the relationship between the rotation angle $θ$ of the dipole unit of the first set and the azimuth angle $φ$ is same as Eq. (6), and the relationship between the rotation angle $θ$ of the dipole unit of the second set and the azimuth angle is

θ = \frac{π}{4} + \frac{π \cos φ}{4} + π M {(\frac{φ}{2 π})}^{2} .

In the combined metasurface, the dipole unit and the interval between two adjacent dipole units can be totally designed same as that in Fig. 4. In this case, its broadband response is also given by Fig. 4(b). In addition, similar as Fig. 4(c), when

δ \neq π

, the output beam contains two parts, i.e., the modulated and unmodulated parts, as shown in the schematic diagram in Fig. 8(b).

3. Conclusion

In conclusion, two OAM-superposition-state generators based on all-dielectric metasurface are proposed, which are light polarization dependent. The generated OAM superposition states can be controlled by controlling the left-handed and right-handed circular polarization states of the incident light. The power spectra of the OAM superposition states show a flat region. The power variation of the OAM modes within the flat region of the OAM spectra from one generator is less than 3 dB, and the power variation of the OAM modes within the flat region of the OAM spectra from the other generator is less than 0.3 dB. In addition, the number of the OAM modes within the flat region can reach dozens and, the two generators can operate efficiently on a wide wavelength range of 635nm to 730nm. This working wavelength range depends on the design parameters, i.e., the length, width and height of the metasurface dipole unit. By changing these design parameters, the generators can be easy extended to other wavelength ranges, such as radio frequency region, within which metasurface based artificial modulation of the electromagnetic wave may suggest promising applications in wireless communication systems [28]. The generated OAM superposition states from our generators are collinear superposition of OAM modes. This kind of OAM superposition states is not suitable for using in OAM multiplexing communication, since in OAM multiplexing communication, the OAM modes of collinear superposition states should firstly be separated from each other to code message separately, which is not better than directly producing multiple independent OAM beams. Our devices may have potential applications in OAM multicasting [29–31], where data on single OAM channel needs to be duplicated onto multiple OAM channels and then be sent to multiple potential users. Our generators enable us to control the potential users simply by changing the light polarization and distribute the power approximately equally to each OAM channels. Moreover, if embedding our devices within wavelength division multiplexing system to realize OAM multicasting at multi-wavelength, the number of the potential users can remain the same for different wavelengths. It is because the bandwidth of the flat region of the OAM spectrum is independent of the wavelength. Due to the ultra-thin feature of metasurface, our work may be applied for chip integrated OAM multicasting devices. In the multicasting, the potential users can decode the message in their respective OAM channels within the collinear superposed OAM state by separating the OAM modes via traditional OAM sorting method, such as that using in [32–34]. Besides multicasting, if carving our metasurface devices on the tip of optical fiber, it may also provide some novel optical manipulations since recent studies have shown that superimposed OAM beams are more potential in particle manipulation [35,36]. In addition, our devices may also act as broadband OAM modulator to realize OAM shift keying.

Funding

National Natural Science Foundations of China (NSFC) (61705050, 61805050, 61675052, 61535004, 61735009, 61765004, 61640409, 11604050); Guangxi Project for ability enhancement of young and middle-aged university teacher (2018KY0200); National Key R&D Program of China (2017YFB0405501); Guangxi project (AD17195074); National Defense Foundation of China (6140414030102); Guangxi Provincial Natural Science Foundation (2017GXNSFAA198048)

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Polarization controllable generation of flat superimposed OAM states based on metasurface

Abstract

1. Introduction

2. Theory and design of the metasurface

Firstly, the QPEP vortex beam can be expressed as

3. Conclusion

Funding

References

Cited By

Figures (8)

Equations (10)

Optics Express