Nonlinear transmission and pseudospin in two-dimensional octagon and dodecagon photonic lattices

Jing Lyu; Zenrun Wen; Kun Han; Xinyuan Qi; Yuanmei Gao

doi:10.1364/OME.8.002713

1. Introduction

In the past decades, photonic lattices in photorefractive crystals induced by weak light nonlinear effect have attracted substantial research interests, which is directly associated with the wave dynamic equation. A lot of physical phenomena have been observed in photonic lattice, such as discrete diffraction, spatial solitons, Anderson localization, topological insulators and asymmetric propagation [1–4]. In variety of photonic lattices, compound periodic photonic lattices play an important role in physics. It is not only the change in the laws of electromagnetic wave propagation that gives rise to a novel phenomenon, but also the generality of this behavior appearing in solid state physics or optics that makes this field so highly fascinating for investigations of basic quantum and valleytronics. For instance, the honeycomb lattice, which is related to the graphene structure, has zero bandgaps [5]. And the kagome lattice, which can form localization transmission in linear condition [6]. Moreover, asymmetric propagation has been observed in spiral honeycomb lattice [4]. Hence, it is a matter of significance to design and study the new kind of lattices [7–12].

Nowadays, questions for studying the relationship between pseudospin and angular momentum in 2D compound periodic structures is still a challenge. Zhigang Chen and Daohong Song et al., unveil the pseudospin and angular momentum in photonic graphene. The result reveals the relationship between pseudospin and angular momentum in photonic graphene [5]. The pseudospin, as a new degree of freedom of electrons, plays an important role in understanding physical phenomena in graphene [13].

The purpose of this paper is to study two new kinds of compound photonic lattices, judging whether they have application value in mesoscopic artificial structure [14,15]. So, what novel physical phenomena arise in the new lattices, such as pseudospin? In this work, we fabricate the two new 2D photonic lattices, the octagon and dodecagon lattices with SBN. Based on the theory of nonlinear Schrödinger equation (NLSE) and Dirac points, we utilized Split-step Fourier method (SSFM) to simulate the nonlinear transmission and the vortex corresponding to the pseudospin. We discover that the transmission beam is localized in the unit cell eventually with the nonlinear effect getting stronger [2,16]. What’s more exciting, we observe the pseudospin in the octagon and dodecagon lattice both experimentally and numerically. The special designed probe beams are the square point lights and hexagon point lights in frequency domain that correspond to the sublattice of octagon and dodecagon lattices respectively.

2. Methods and experiments

To demonstrate our survey, we analysis the transmission and angular momentum in the two new compound lattice, the octagon and dodecagon lattice. The (NLSE) is applied to describe the light propagation in lattices:

i \frac{\partial ψ}{\partial z} + \frac{\partial ψ}{\partial x^{2}} + \frac{\partial ψ}{\partial y^{2}} + Δ n (x, y) ψ = 0

where

ψ (x, y, z)

is the electric field envelope of probe beam,

z

represents the horizontal propagation distance,

Δ n (x, y)

is an intensity-dependent nonlinear index charge of the form, and

Δ n (x, y) = 0.5 σ n_{0}^{3} γ_{e f f} U V (x, y) / L (1 + I)

, where

σ

is self-focusing nonlinearity

(σ = 1)

,

n_{0}

is the refractive index in the SBN

(n_{0} = 2.3)

,

γ_{eff}

is the electro-optic tensor

(γ_{eff} = 1340 p m / V)

,

U

is the voltage applied to the crystal

(U = 500 V)

,

L

is the length of SBN

(L = 0.5 c m)

,

I

is the intensity of optical beam

(I = {| V |}^{2} + {| ψ |}^{2})

and

V (x, y)

is the lattice light that induces SBN to form OL and DL [1,5,17,18]. The structure of lattice we design is a superposition of two equal sublattices (sublattice A and sublattice B). OL is a superposition of two square sublattices and DL is a superposition of two hexagon sublattices [5,19]. The

ψ_{A}

and

ψ_{B}

are special probe beam with Bloch momentum as follow:

\begin{array}{l} ψ_{A} = \sum_{0}^{n - 1} e^{i (K_{A} • r + ϕ_{A})} \\ ψ_{B} = \sum_{0}^{n - 1} e^{i (K_{B} • r + ϕ_{B})} \end{array}

Correspondingly, sublattice A or B plays the role of electron spins [5]. The

K_{A}

and

K_{B}

:

\begin{array}{l} K_{A} = [cos (\frac{2 j π}{n} + \frac{π}{n}) \sin θ, \sin (\frac{2 j π}{n} + \frac{π}{n}) \sin θ, \cos θ] \\ K_{B} = [cos (\frac{2 j π}{n} - \frac{π}{n}) \sin θ, \sin (\frac{2 j π}{n} - \frac{π}{n}) \sin θ, \cos θ] \end{array}

where θ is the angle between direction of beam propagation and z-axis [20].

ϕ_{A}

and

ϕ_{B}

are initial phases of OL and DL, respectively. When

n = 4

and

j = 0, 1, 2, 3

, Eqs. (2) and (3) represent the sublattice light of OL. When

n = 6

and

j = 0, 1, 2, 3, 4, 5

, it represents that of DL. Duo to that the two lattices, new compound lattice and photonic graphene have similar structure, and the structure distribution in frequency domain has two unequal points which are corresponding two K points in K-space, we deduce that Dirac points exist in OL and DL. So the lattice OL or DL is double degenerate in K-space. And we demonstrate that pseudospin arises in OL and DL in the form of vortex angular momentum both in simulation and experimental results [5,21].

The configuration in Fig. 1 displays our experimental setup. The laser beam (λ = 532nm, Power = 4.6mw) is modulated to form lattice light field by the spatial light modulator (SLM), which can induce compound lattice into SBN crystal to fabricate photonic lattice. we design from the first light path. The 4f system plays a role as spatial filter. The additional voltage applied along the c-axis of the SBN crystal was 1000 V/cm. We use the very thin Gauss beam who is focused by a short focal lens (f = 3cm) to measure the transmission in lattice from the second light path. The third path is reference light for interfering with the first path. Moreover, we use the Mask1 and Mask2 to acquire the probe beam in frequency domain which are square point lights and hexagon point lights corresponding to sublattices of OL and DL respectively. We use the SLM to generate the lattice light that induces periodic structures in the SBN. Moreover, the SLM size is 1920 × 1080 pixel. The focal length is 15 cm in the 4f imaging system. The Fourier transform of lattice light is a superposition pattern of two equal sublattices as the Fig. 2(d) and (h) show. Therefore, we acquire the probe beam that carries Bloch momentum and matches the lattice structure well on focal-plan of L3 in the 4f-system. And the probe beam in path one is used to measure the vortex in lattice. The image resolution of charge-coupled device (CCD) for capturing picture is 3.3 μm every pixel. To simulate the result of transmission and pseudospin, we use the SSFM. The process of SSFM is that to calculate every differential distance of light transmission (dz) along propagation direction and consider the nonlinear effect in the middle of dz, then consider the linear effect on the first half dz and the second half dz. Iterate every dz calculated and we’ll gain the simulation result eventually.

Fig. 1 The experiment to measure the transmission and pseudospin in the OL and DL. L1-L8: convex lens; PBS: polarizing beam splitter; λ/2: half -wave plate; SLM: Spatial Light Modulator; M: Mask 1 and Mask 2; A1and A2: apertures.

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Fig. 2 The simulation results of structure images of photonic lattice OL and DL. (a)-(d) are the lattice light field distribution, phase image, light intensity distribution in three dimension (3D) and the lattice light field in frequency domain corresponding to the Dirac points in K-space for OL respectively. (e)-(f) are the similar simulation results for DL.

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3. Results and discussion

In our survey, we study the OL and DL to examine whether some fancy phenomena arise. The lattices we design can be counted as superposition of two identical sublattices [20], which is the sublattice degree of freedom of OL and DL. First of all, we analyze the light field of the lattice. Next, we measure light nonlinear transmission in it. With the nonlinear effect getting stronger, the transmission beam is localized in the lattice unit eventually. We discover that the pseudospin is associated with the angular momentum in OL and DL [22]. We design the probe beam, which are the square point lights and hexgon point lights in frequency domain corresponding to Dirac points of octagon and dodecagon lattices respectively. The probe beam with Bloch momentum at Dirac point inspires the sublattice A or sublattice B. Then we detect the vortex at the lattice cell. Although the probe beam has no vortex at first, the beams break the sublattice degeneracy and the pseudospin arises as a form of vortex eventually in the lattice, which suggests the pseudospin has been converted to a vortex angular momentum in the experiment.

In the Fig. 2, we analyze the structure of OL and DL. The light field for inducing SBN to form OL has been given in Fig. 2(a). And the field is 2D periodic structure whose unit cell is a octagon lattice as shown in the bottom right corner of the image. As the Fig. 2(b) shows, there is a π- phase difference between two lattice points in the unit cell, which contributes to the regular period photonic structure and pseudospin. From the 3D light intensity image in Fig. 2(c), the pattern of intensity is like valleys that arrange periodically. Duo to the frequency domain image of lattice light field, we deduce that the Dirac points of OL in K-space are noted as K and K’ point in the Fig. 2(d) where the white points represent sublattice A and gray points represent sublattice B [23–25]. It’s a remarkable fact that OL is a superposition of two equal square sublattices. The following row in Fig. 2 is the analysis of DL, which is similar to OL. More over, the sublattice of DL is hexagon.

Using the experimental setup mentioned above, we fabricate the OL and DL. The unit length of phase pattern OL unit cell lattice that we input to the SLM is 135 pixel and unit length of DL is 180 pixel. Figure 3(a) and (c) are the photonic lattice image captured by CCD and the white arrows (a1 and a2) in the bottom left corner are basic vectors of OL and DL. The distance of two OL unit cells is 30 μm and that of DL is 36 μm. And we can see that the lattices are well formed. The Fig. 3(b) and (d) are the far-field diffraction images of OL and DL respectively, which are equivalent to the Fourier transform of lattice light field. The diffraction pattern in Fig. 3(b) is the superposition of two square lattices corresponding to the simulation results above. From Fig. 3(b) and (d), there are several weak light spots by the strong light spots. It was probably due to the waveguide coupling effect when the probe plane beam pass through the well formed lattices [26–29].

Fig. 3 The experiment result of OL and DL induced in SBN.(a) and (c) are the photonic lattices, OL and DL, captured by CCD. (b) and (d) are the far-field diffraction patterns of OL and DL.

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To explore the nonlinear phenomena in OL and DL, we use the SFM to simulate the transmission image. As shown in the Fig. 4(a), the Gauss beam passes through lattices and diffracts freely without the nonlinear effect. In Fig. 4(b)-(c), the beam is localized gradually with the nonlinear effect getting stronger. The similar results can been seen in Fig. 4(d)-(f). And the 3D light intensity distributions are show in the bottom right corner of every figures.

Fig. 4 The numerical simulation of nonlinear light transmission in OL and DL. (a)-(c) and (d)-(f) are transmission results in OL and DL respectively.

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The first row in the Fig. 5 is the transmission image in the OL and the second row is the image in DL. The SBN is applied voltage 500v/cm in Fig. 5(a) and (d) in order to keep the OL and DL’s shape. From Fig. 5(a), the thin Gauss beam passes through the OL and diffract without nonlinear effect. Then the SBN is applied voltage from 1000V/cm to 1400v/cm gradually. Thus, we can gain the nonlinear effect. The voltage is 1000v/cm in the Fig. 5(b) and Fig. 5(e) and the voltage is 1400v/cm in Fig. 5(c) and (f). Due to the anisotropy of SBN, the c axis (optic axis of SBN) is in the vertical direction. The carriers mobility is higher in the horizontal direction perpendicular to c axis than that in the direction along c axis. So we can see several lattice points are connected into line along the horizontal direction in Fig. 5(c) and (f). But we can observe the beam is localized in the unit lattice and converge at lattice points yet. As the effect gets stronger, the transmission beam is localized in the unit lattice and converge at lattice points eventually in Fig. 5(b)-(c). And we get a similar result in DL from Fig. 5(d)-(e). The experiment concerning the nonlinear transmission indicates that the new kind of lattice OL and DL can be applied in localized transmission of mesoscopic structure.

Fig. 5 The experimental nonlinear phenomena and the power spectrum corresponding to the transmission image. (a)-(c) are transmission results of OL. (d)-(f) are the similar results in DL.

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There comes a novel phenomenon when the sublattice was inspired selectively by the special probe beam. Based on the theory of NLSE and Dirac points, we simulate the transmission image and discover the vortex in OL and DL [30,31]. The Fig. 6 (a) is the probe beam we design and it corresponds with one of sublattice A or B. The Fig. 6(b) shows the probe beam transmission in OL and the light distribution is periodic. To measure whether a vortex arises in lattice, we simulate the interference. From the result in Fig. 6(c), we can observe the Y-shape bifurcation marked by the white circles. It indicates the vortex angular momentum in lattice. The similar results of DL are shown in the Fig. 6(d)-(f).

Fig. 6 The simulation results of speudospin in OL and DL. Fig (a)-(c) are probe beam, transmission image and interference pattern of lattice light and plane wave in OL. (d)-(f) are the simulations of DL.

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The experiment results show agreement with the numerical simulation from the Fig. 7 [19]. So, we deduce that the vortex-free probe beam with Bloch momentum obeys the Dirac equation:

\begin{array}{l} i \partial_{z} ψ_{A} + (\partial_{x} - i μ \partial_{y}) ψ_{B} = 0 \\ i \partial_{z} ψ_{B} - (\partial_{x} + i μ \partial_{y}) ψ_{A} = 0 \end{array}

where

μ = {(- 1)}^{m} = \pm 1

, and m = 0, ... ,7 for OL or 0,...11 for DL. The sublattice A and B in the lattice are equal to electron spins. The beam inspires one of the sublattices and break the degenerate in OL and DL. Then we observe the vortex in the rear surface of OL. And we can find it in the Fig. 7(c) and (f) where the vortex noted by gray arrows. Hence, we deduce that pseudospin is converted into the vortex angular momentum in the compound photonic lattices (OL and DL) when the degeneracy of lattice is broken by probe beam. To associate pseudospin with angular momentum, we give the total angular momentum along horizontal propagation as follow:

〈 J 〉 = 〈 L 〉 + 〈 S 〉 = \iint_{R^{2}} [- i ψ_{A}^{*} \frac{\partial}{\partial φ} ψ_{A} - i ψ_{B}^{*} \frac{\partial}{\partial φ} ψ_{B} + \frac{μ}{2} ({| ψ_{A} |}^{2} - {| ψ_{B} |}^{2})] d x d y

where the

μ = 1

, represents that the one of sublattice is inspired to affect the total angular momentum. In fact, the initial pseudospin is transferred to the final orbital angular momentum of the system, which explains the optical vortex angular momentum generation in OL and CL observed in our experiment.

Fig. 7 The experiment results of pseudospin in OL and DL. (a)-(c) are probe beam, transmission image and interference pattern of lattice light and plane wave in OL. (d)-(f) are the simulations of DL.

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The main purpose of this work is to explore fancy phenomena in the novel compound lattices, OL and DL. We have study the structural features and nonlinear transmission in the lattices. More over, we discover the transformation from the pseudospin to vortex angular momentum. However, the new lattices we fabricate are more complex in structure and we directly acquire the probe beam in the frequency domain, which means a initial beam that matches the structure of OL and DL well. We discover the pseudospin in the compound lattice, which indicates that the connection between pseudospin and vortex angular momentum commonly exists in photonic lattice who’s sublattice is degeneracy. Our study may provide two novel mesoscopic artificial structures for light control and light transmission field. Further, the discovery of the connection between pseudospin and angular momentum in lattice can supplies a new horizon for valleytronics in 2D materials [32–36].

4. Conclusion

We fabricated two novel compound lattices OL and DL and demonstrated the localization transmission and the pseudospin associate with vortex angular momentum both in simulations and experiments. This work is mainly based on the theory of nonlinear Schrödinger equation and Dirac equation and we calculate the light propagation and pseudospin in the lattices by Split-step Fourier method. The artificial structures OL and DL is made by the main instrument SLM and the material is SBN. Furthermore, the novel lattices supply a new model for mesoscopic artificial structure and the discovery of pseudospin in lattices will arouse more researches and applications in valleytronics and quantum field [37].

Funding

National Natural Science Foundation of China (Grant nos. 11304187, 11574185, and 11604183); China Postdoctoral Science Foundation (Grant no.2015M582126).

Acknowledgment

We thank professional Xinyuan Qi, from Department of Physics, Northwest University, Xi’an 710069, China, for supplying the experimental equipment and academic guidance.

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Nonlinear transmission and pseudospin in two-dimensional octagon and dodecagon photonic lattices

Abstract

1. Introduction

2. Methods and experiments

3. Results and discussion

4. Conclusion

Funding

Acknowledgment

References

Cited By

Figures (7)

Equations (5)

Optical Materials Express