Enhancement and modulation of photonic spin Hall effect by defect modes in photonic crystals with graphene

Enhancement and modulation of photonic spin Hall effect by defect modes in photonic crystals with graphene

Accepted Manuscript Enhancement and modulation of photonic spin Hall effect by defect modes in photonic crystal with graphene Jie Li, Tingting Tang, L...

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Accepted Manuscript Enhancement and modulation of photonic spin Hall effect by defect modes in photonic crystal with graphene Jie Li, Tingting Tang, Li Luo, Jianquan Yao PII:

S0008-6223(18)30345-2

DOI:

10.1016/j.carbon.2018.03.094

Reference:

CARBON 13037

To appear in:

Carbon

Please cite this article as: Jie Li, Tingting Tang, Li Luo, Jianquan Yao, Enhancement and modulation of photonic spin Hall effect by defect modes in photonic crystal with graphene, Carbon (2018), doi: 10.1016/j.carbon.2018.03.094 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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ACCEPTED MANUSCRIPT Enhancement and modulation of photonic spin Hall effect by defect modes in photonic crystal with graphene Jie Li, Tingting Tang*, Li Luo, and Jianquan Yao

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Dr. Tingting Tang, Jie Li, Li Luo Information Materials and Device Applications Key Laboratory of Sichuan Provincial Universities, Chengdu University of Information Technology Chengdu 610225, China E-mail: [email protected]

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Prof. Jianquan Yao Information Materials and Device Applications Key Laboratory of Sichuan Provincial Universities, Chengdu University of Information Technology Chengdu 610225, China Key Laboratory of Optoelectronics Information Technology (Tianjin University), Ministry of Education, Institute of Laser & Opto-electronics, School of Precision Instruments and Optoelectronics Engineering, Tianjin University Tianjin 300072, China Keywords: photonic spin Hall effect; magneto-optical modulation; photonic crystals Abstract: Photonic spin Hall effect (PSHE) holds great potential applications for manipulating photon spins. However, the efficient control of PSHE in a particular optical structure is still

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difficult to realize. In this paper, we report enhanced and magnetically tunable PSHE of reflected light in one-dimensional photonic crystals (1D-PC) with a defect layer. By inserting monolayer graphene into the defect layer of 1D-PC, large polarization rotation and strong

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photon spin-orbit interaction of reflected light are obtained. Due to the optical field confinement by defect mode, the Kerr rotation angle of reflected light is near 90 degrees and

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the spin-dependent splitting is about 15 times the wavelength at a specific incident wavelength. In addition, a static perpendicular magnetic field increasing from 1T to 11T shows greatly modulation effect on Kerr rotation angle and spin-dependent splitting. These results pave the way toward the design of spin-based photonic devices in the future. 1. Introduction Magneto-optical Kerr effect (MOKE) arises from the time reversal symmetry breaking when linearly polarized light reflects from surface of magnetic materials. According to the direction of the applied magnetic field, there are three types of MOKE namely polar magneto-optical 1

ACCEPTED MANUSCRIPT Kerr effect(PMOKE), longitudinal magneto-optical Kerr effect (LMOKE), and transverse magneto-optical Kerr effect (TMOKE). The first two types can cause a polarization rotation called Kerr rotation, the latter produces a non-reciprocal phase shift(NRPS) in reflected light[1]. Recently, MOKE has been widely used in many fields, such as high-density data

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storage technology [2], magnetic interlayer coupling [3], magnetoplasmonics [4] and so on.

As a zero band gap semiconductor material, graphene has a unique Dirac cone band and linear dispersion relationship, which brings it many unique optical properties.

[5-7]

In the

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quantum system, the quantum Hall effect in graphene will be revealed, but this requires highquality graphene samples in a specific environment, [8]and several literatures have reported [9, 20]

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quantized PSHE and GH (Goos-Hänchen) shift in graphene.

But in more general cases,

such as ordinary graphene samples at room temperature, studies have shown that conductivity of graphene satisfies the Drude model in terahertz and far-infrared bands.[10,11] More interestingly, the optical conductivity of graphene can be adjusted by applying a static electric

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field or magnetic field. The above characteristics of graphene resulting in the adjustment of many other optical phenomena, such as the magneto-optical (MO) Faraday effect, electromagnetic induction transparent (EIT)

[16-18]

[12-15]

the

and beam shifts.[19-24] In particular,

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graphene has a stronger magneto-optical response and can generate a very large Kerr rotation angle in terahertz region.[25] Compared with ordinary magneto-optical materials, monolayer

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graphene is thinner and can be more conducive to the miniaturization of optical devices. At the same time, using graphene can modulate the magneto-optical effect more flexibly, because we can not only change the applied magnetic field strength, but also change the Fermi level and other parameters of graphene. The photonic spin Hall effect means that when a linearly polarized light beam is propagated in a non-homogeneous medium, the photons with opposite spin state drift in the opposite direction perpendicular to the refractive index gradient, resulting in the beam splitting into two beams of circularly polarized light.[26-28] In recent years, the PSHE has attracted great 2

ACCEPTED MANUSCRIPT attentions and has been studied extensively.[29-34] In order to achieve the applications of PSHE, researchers try to use a variety of methods to manipulate the spin-dependent splitting, such as using different material interfaces

[35-37]

and metasurfaces.[38-41] However, almost all those

methods control the PSHE by changing the type of materials or the structural parameters of

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the metasurfaces, it is not very conducive to applications. It will be exciting if we can achieve the manipulation of PSHE by external electromagnetic fields rather than changing the structure itself. The MO effect of graphene in terahertz region helps us to realize magnetic-

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field modulated PSHE.

One-dimensional photonic crystals (1D-PC) are formed by periodically overlapping two

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kinds of materials, and photonic bandgap can be observed in a specific incident wavelength range. The defect layers break the photonic bandgap effectively and a defect mode can be generated.[42,43] The highly selectivity on frequency and great field locality of the defect mode make it been widely used in integrated optics, such as filters [44,45] and microcavity lasers.[46,47]

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Based on the above considerations, using graphene as a defect layer may significantly increase the concentration of in-band transition carriers in graphene. On the other hand, magneto-optical effect of graphene is caused by the circular motion of intra-band transition

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carriers in a magnetic field [12,13], then the Kerr rotation can be greatly enhanced when the defect mode is excited near graphene.

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In this paper, we insert monolayer graphene into the defect layer of one- dimensional photonic crystal to enhance and modulate the PSHE. In the appropriate external magnetic field, greatly enhanced and tunable spin-dependent splitting of reflected light beam can be observed due to giant Kerr rotation caused by nonreciprocity of graphene and field localization of defect mode. The influences of several parameters on Kerr rotation and magnetically tunable PSHE are discussed in detail. 2. Model and theory

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Figure 1. (a) Schematic of PSHE in 1D-PC with a defect layer; (b) Two kinds of defect materials: graphene and SiC.

In this paper, we consider a model of 1D-PC with a defect layer as shown in Figure 1a. To illustrate the great influence of graphene on PSHE, we will study three types of defect layers, namely monolayer graphene, monolayer SiC, and graphene-SiC bilayer structure. A p-

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polarized Gaussian beam is incident on the surface of the defective photonic crystal with an incident angle of θi, and the spin-dependent splitting of reflected light occurs due to the spinorbit interaction of photons. A static magnetic field (B) is applied perpendicularly to the

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photonic crystal.

In terahertz and far-infrared bands, energy of photons satisfies ℏω < 2EF , where ω is

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angular frequency and EF is Fermi level of graphene, thus the main contribution to the conductivity of graphene arises for intraband transitions.[6] In an external magnetic field, due to the cyclotron motion of charge carriers, the conductivity tensor of graphene can be written as

 σ xx

σG = 

 σ yx

σ xy  σ yy 

The diagonal elements and off-diagonal elements of σ G are given by [13]

4

(1)

ACCEPTED MANUSCRIPT σ xx (ω ,B)=

2D

π



1 / τ − iω ω − (ω + i/ τ ) 2

(2)

2 c

σ xy (ω ,B)=-σ yx (ω ,B) = −

2D

π



ωc ω − (ω + i/ τ )2 2 c

(3)

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In which D = 2σ0 E F / ℏ is the Drude weight and ωc = eBvF2 EF is cyclotron frequency. Here vF is the Fermi velocity (vF =9.5×105 m/s) and e (e=1.6×10-19 C) is the electronic charge. Then the elements in dielectric tensor of graphene can be obtained by [16]

(4)

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ε g = 1 + iσ / ωε 0tg

where tg is the effective thickness of a single layer graphene (tg=0.5 nm).

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Before calculating the Kerr rotation angle and PSHE shift of reflected light, we first need to obtain the Fresnel reflection coefficients (including rpp , rss , rps , rsp ) of the multilayer structure by using magneto-optical transfer matrix method.[48] The Kerr rotation angle θ K (when incident light is p polarized) can be calculated by [49]

2 Re(rsp / rpp )

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tan(2θ K ) =

1 − rsp / rpp

2

=

2 Re( χ ) 1− χ

2

(5)

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where χ = rsp / rpp .The PSHE is usually described by a general propagation model using angular spectrum theory. [31] The angular spectrum of a light beam reflected from anisotropic

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materials can be written as [50]

kry  r − ( rps − rsp ) cot θi pp  k0 Eɶ rH    ɶV  =  Er   r + kry (r − r ) cot θ  sp pp ss i k0 

 (rpp − rss ) cot θi  H ɶ k0   Ei  ɶV    E kry rss − (rps − rsp ) cot θi  i  k0 

rps −

kry

(6)

ɶ H and E ɶ V denote the horizontal (H) and vertical (V) components of angular spectrum, here E r r and k ry is y component of reflected wave vector. For a linearly polarized reflected beam we have [51]

5

ACCEPTED MANUSCRIPT  rpp  irps Eɶ r (H) = exp(ik yδ H′ )  Eɶ r + exp(ik yδ H ) − rpp 2   +

(7)

 rpp  irps exp(ik yδ H′ )  Eɶ r −  exp(−ik yδ H ) + rpp 2  

  irsp exp(ik yδV′ )  Eɶ r +  exp(ik yδV ) + rpp   ir  ir  + ss exp(−ik yδV ) − sp exp(−ik yδV′ )  Eɶ r − rss 2 

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−ir Eɶ r (V) = ss 2

(8)

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where δ H′ = cot θ i (1 − rsp / rps ) / k0 , δV′ = cot θi (1 − rps / rsp ) / k0 , Eɶ r + and Eɶ r − denote the angular

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spectrum of the left and right circularly polarized (LHCP and RHCP) components, respectively. The terms exp( ±ik yδ H′ ,V ) in Eq(7) and Eq(8) predicts that the Kerr rotation angle

θ K causes additional photonic spin-orbit coupling [51], resulting in asymmetric spin dependent splitting [48]. Once the Kerr rotation disappears, the asymmetry of PSHE will also disappear.

quantified by factor χ .

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From Eq(12) in Ref[48] we can see that spin dependent splitting and Kerr rotation are

Then we can obtain the angular spectrum of LHCP and RHCP components of reflected

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beam, as well as their electric field

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Er (x r , y r , z r ) = ∫∫ Eɶ r (k rx , k ry ) exp[i(k rx x r + k ry y r + k rz zr )]dk rx dk ry

(9)

Finally, the spin-dependent shift of reflected light can be calculated as [31]

δ

±

∫∫ E ⋅ E y dx dy = ∫∫ E ⋅ E dx dy r±

∗ r±

∗ r±



3. Results and discussion

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r

r

r

r

r

(10)

ACCEPTED MANUSCRIPT (a) 1.0

(b) 150 Magnetic field(A/m)

0.8 SiC GF GF

0.2

GF,SiC

0.0

GF,SiC 9.0

9.5

10.0

10.5

(d) GF,sub

6 PSHE shift(λ )

GF,sub

0 0.8 0.6 0.4 0.2 0.0

10.2

Frequency(THz)

GF,sub

SiC

2

10.0

10.4

0.07

GF

4

0 10.0

40000

x(nm)

10.2

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Kerr rotation(deg)

20000

GF,SiC

GF

30

-90

0

8

GF,SiC

60

-60

-50

-100

11.0

(c)

-30

0

SiC

Frequency(THz) 90

GF,SiC

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0.4

50

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Rpp

0.6

11.54THz 10.13THz 8.82THz

100

10.4

Frequency(THz)

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Figure 2. (a) Reflectivity of Si/SiO2 photonic crystal with three types of defect layer in the incident frequency change; (b) Magnetic field distribution (Hy, section) of the graphene(GF)/SiC defect structure at three different incident frequencies; (c) Kerr rotation angle and (d) spin-dependent splitting of reflected light (LHCP) in photonic crystal with graphene defects, graphene / SiC defects and single layer graphene with SiC substrate. All the values of spin-dependent splitting are converted to multiples of the wavelength.

Before analyzing PSHE in photonic crystals systematically, we first calculate the reflectivity of photonic crystals with three types of defect layers using magneto-optical transfer matrix

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method, as shown in Figure 2a. According to the Bragg condition n1d1 = n2 d 2 = λ 0 / 4

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(assume that λ0=30µm), which is widely used in photonic crystals, we can obtain the value of d1 and d2, where n1, n2, d1, d2 are the refractive indices and thicknesses of two repeating media. Using a layer of λ 0 / 2 / n SiC thick SiC as defect layer, a significant absorption peak can be observed on the reflection line, as shown by the blue curve in Figure 2a. It means that there is a defect mode with high concentration of incident energy in the SiC layer, the resonant frequency is just around 10THz. In order to utilize the energy concentration effect of the defect mode, we add a single layer graphene to the upper side of SiC layer, and the reflectivity is shown by the red curve in Figure 2a. It can be found that the absorption peak undergoes a 7

ACCEPTED MANUSCRIPT slight movement and the reflectivity is slightly increased, since the presence of graphene changes the resonant condition of the defect mode. As comparison, we calculated the structure which has a defect layer of monolayer graphene, as shown by black curve in Figure 2a. It is easy to see the reflection absorption peak is broadly widened, since the equivalent permittivity

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of graphene has a metallic characteristic near 10THz, which brings greater energy loss.

To analyze the energy distribution of the defect mode more intuitively, we use finite element method (FEM) to calculate the magnetic field distribution (y component, section) of

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the graphene/SiC defect structure at three different incident frequencies, as shown in Figure 2b. Without loss of generality, we choose one frequency that can excite the defect mode and

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two frequencies that excite no defect mode. For example, we can choose 8.82THz, 10.13THz and 11.54THz, where 10.13THz is the resonant frequency. It can be found that the energy of the incident light is concentrated near the defect layer at 10.13THz, which can enhance the interaction between the light wave and graphene, and less energy can penetrate the defect

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layer at the other two incident frequencies(8.82THz, 11.54THz).

We are about to prove that defect mode does enhance the interaction between graphene and photons, and then enhance the magneto-optical Kerr effect (MOKE) and PSHE. As shown in

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Figure 2c, we calculated the Kerr rotation angle of reflected light in the traditional structure of single layer graphene with a SiC substrate and photonic crystals with graphene/SiC defects,

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single-layer graphene defects. The inset at left side of Figure 2c shows the reflectance spectrum of graphene/SiC substrate structure, which presents a significant cyclical oscillation due to Fabry-Perot (F-P) resonance effect (a Fabry-Perot interferometer or etalon is typically made of a transparent plate with two reflecting surfaces, or two parallel highly reflecting mirrors) with appropriate substrate thickness. In Ref [14], the enhancement of F-P resonance on Faraday rotation is studied. Similarly, we find that F-P resonance also enhances the magneto-optical Kerr effect as shown in Figure 2c, with a maximum Kerr rotation angle of about 45 degrees. As shown in Figure 2c by the black and red curves, Kerr rotation of 8

ACCEPTED MANUSCRIPT approximately 90 degrees can be observed in the two defective 1-D PCs, indicating that defect mode in 1D-PC with graphene can greatly enhance the magneto-optical Kerr effect. Recent studies have shown that in order to satisfy the transversality of photon polarization, the plane wave component of the reflected light on the graphene surface will undergo a

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polarization rotation, which leads to strong photon spin-orbit interaction and significant PSHE.[52] We may wonder if it is possible to further enhance the spin-orbit interaction of photons when the polarization rotation of reflected light is enhanced. To confirm this

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prediction, we calculate the spin-dependent splitting of the three structures in Figure 2c, as shown in Figure 2d. In the absence of graphene (only SiC in the defect layer), there is a small

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peak in the spin dependent splitting of reflected terahertz beam, because the presence of defect modes increases the value of rs/rp and then enhance the PSHE. This situation is similar to defective photonic crystals for visible light.[53] It can be found that the maximum transverse shift of LHCP enhanced by F-P resonance is close to twice of the incident wavelength.

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the wavelength.

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However, in 1D-PC with graphene, the maximum transverse shift achieves about 7 times of

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ACCEPTED MANUSCRIPT (b) 90 B=1T B=3T B=5T B=7T B=9T

0 -30 -60 -90 10.0

10.1

10.2

0 -30 -60 -90 10.0

-60 10.0

10.1

10.2

10.3

Frequency(THz)

90 Kerr rotation(deg)

Kerr rotation(deg)

30

-30

(d)

Ef=0.1eV Ef=0.2eV Ef=0.3eV

60

0

-90

10.3

Frequency(THz)

(c) 90

30

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30

N=1 N=2 N=3

60 Kerr rotation(deg)

60

τ=0.1ps τ=0.2ps τ=0.3ps

60 30 0 -30 -60

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90

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Kerr rotation(deg)

(a)

-90

10.1

10.2

Frequency(THz)

10.3

10.0

10.1

10.2

10.3

Frequency(THz)

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Figure 3. Kerr rotation angle in 1D-PC with graphene/SiC defects for different value of (a)magnetic field B, (b) repetition number N, (c) Fermi level EF and (d) relaxation time τ; Where EF=0.2 eV, B=5T in (b) and (d), τ=0.1ps in (c). Since the spin dependent splitting is directly related to the Kerr rotation, we first study the influence of several factors on Kerr rotation in the graphene/SiC defect structure to analysis

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the PSHE systematically. As shown in Figure 3a, the applied magnetic field shows a significant modulation effect on Kerr effect of reflected light. When B is 5T or 7T, the rotation

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angle is close to 90 degrees. Such a large value is due to the nonreciprocity of graphene in magnetic field and the light field localization caused by optical multiple interference in the defective photonic crystal. When B increases from 1T to 11T, the reversal point of Kerr rotation is also moving toward higher frequencies, which can be traced back to the characteristics of Dirac's quasi-particles in graphene. [54] The variation of the curve in Figure 3a is very similar to that of the Faraday effect in the graphene ribbon arrays.[13] It is well known that the field concentration of defect mode is not only related to the incident wavelength but also to repetition number (N) in photonic crystals. 10

ACCEPTED MANUSCRIPT Figure 3b shows the Kerr rotation angle of reflected light when applied magnetic field is 5T and N has different values. It can be found that while N is equal to 1 or 2, the rotation angle is about 90 degrees, but the curve is narrower when N equals to 2. When N = 3, the rotation angle becomes smaller suddenly, it may be due to fewer incident light energy entering the

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defect layer and resulting in a weaker interaction between light and graphene.

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Figure 4. Spin-dependent splitting in photonic crystal with graphene/SiC defects for different values of N, (a) LHCP, (b) RHCP. All the values are converted to multiples of the wavelength. Where EF=0.2 eV and τ=0.1ps. It has been shown that the spin-dependent splitting of reflected light on the structure with or

longitudinal

magneto-optical

Kerr

effects

will

show

a

distinct

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polar

asymmetry.[48,51]According to this conclusion, asymmetric spin splitting should also occur in

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photonic crystal with a graphene/SiC defect layer. Our calculations prove this, as shown in

Figure 4, where a-c and d-f denote the transverse shift of LHCP and RHCP components for different values of N. We can find that the PSHE shows obvious asymmetry, both in the distance of spin splitting and the trend of change. On the other hand, the areas of LHCP and RHCP light where PSHE can be observed become narrower gradually with the increase of N, which is similar with that of the Kerr rotation angle in Figure 3b. It is worth mentioning that in Figure 4f the transverse shift of RHCP changes from negative to positive when N is equal

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4d and Figure 4e.

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Figure 5. Spin-dependent splitting in photonic crystal with graphene/SiC defects for different values ofτ, (a) LHCP, (b) RHCP. All the values are converted to multiples of the wavelength. Where EF=0.2 eV. Figure 5 shows the influence of relaxation time on PSHE. From the figure we can find that when the relaxation time changes from 0.1ps to 0.3ps, both the PSHE shift of LHCP and

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RHCP light change obviously, but it is slower compared with the change caused by N, especially RHCP component. This is consistent with the modulation effect of N and τ on

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Kerr rotation in Figure 3 From Figure 5a-c we can find that the PSHE shift of LHCP tends to decrease when the relaxation time changes from 0.1 ps to 0.3 ps, and the frequency range with

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apparent spin splitting becomes smaller. It can be found from Figure 5a that there is a small area in which the PSHE shift is negative, and it moves to the left side gradually when τ increases. It can be seen from Figure 5d-f that the PSHE shift curve of LHCP component is gradually narrowed when the relaxation time is increased, but much slower than that of RHCP. Meanwhile the PSHE shift of RHCP is always negative in our calculated magnetic field range, but its maximum value corresponds to a larger magnetic field when τ increases.

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Figure 6. Spin-dependent splitting in photonic crystal with graphene/SiC defects for different value of EF, (a) LHCP, (b) RHCP. All the values are converted to multiples of the wavelength. Where τ=0.1ps.

Graphene has an optical conductivity with Drude model in terahertz band and the Fermi level of graphene can be changed by electrical doping. Figure 6 shows the PSHE shift of reflected light in graphene/SiC defect structure when Fermi level is 0.1eV, 0.15eV and 0.2eV,

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respectively. It can be found that the spin-dependent splitting does not attenuate as fast as that in Figure 4 and Figure 5, but shows a large splitting in the area with larger external magnetic field and higher incident frequencies. It means that when the Fermi level of graphene

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increases, the modulation effect of magnetic field on the PSHE becomes weaker. Particularly, when EF = 0.1eV, another region of large splitting appears in the higher frequency and larger

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magnetic field range i.e., the lower right corner of Figure 6a and Figure 6d.

Figure 7. (a) Kerr rotation angle and spin-dependent splitting of (b) LHCP and (c) RHCP light in photonic crystal with double graphene/SiC defects (ie. (Si/SiO2)N (graphene/SiC)Si(graphene/SiC)(SiO2/Si)N).

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optical field can be concentrated in more than one frequency ranges. Therefore, large spindependent splitting can be obtained in different frequency ranges and the modulation of magnetic field on PSHE can be further enhanced. Figure 7 shows the Kerr rotation and PSHE

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shift of 1D-PC with double defects. It can be found that the reflected light has a large Kerr rotation and PSHE shift when incident frequency is near 9THz or 11THz, but the Kerr effect

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and PSHE near the two frequencies show different properties. First, when the incident frequency is near 9THz, the Kerr rotation angle decreases gradually after the magnetic field increases to 10T. At the same time, the PSHE shift of LHCP in the corresponding region changes from positive to negative, while spin dependent splitting of RHCP appears obvious

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broadening and the shift distances decreases. However, there is no similar phenomenon when the incident frequency is near 11THz.

In experiments, weak measurement in quantum mechanics is widely used to detect the spin-

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dependent splitting in PSHE.[33,34] Weak measurement can amplify the weak effects and

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amplified optical signals can be captured by CCD camera. Based on the recent advances in terahertz technology, like terahertz lasers, detectors, and terahertz elements, the experimental observation of the photonic spin Hall effect in terahertz region becomes possible. In addition, the photonic spin Hall effect has also been studied on the layered structure[57], and the PSHE and Goos-Hanchen effect on the surface of graphene have also been observed via weak measurements[58,59]. Here we propose an experimental scheme for measuring spin-dependent splitting of reflected terahertz beam, as shown in Figure8. A terahertz laser is used as a light source. Similar to the condition of visible light band, a terahertz half-wave plate is employed 14

ACCEPTED MANUSCRIPT to adjust the light intensity in the measurement process. Two different terahertz lenses are used for beam focusing and collimation, two terahertz polarizers are used for polarization selection. At last, a high-performance terahertz camera is employed to measure the shape and the centroid position of terahertz beam. A tunable magnetic field is applyed along the

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direction perpendicular to the sample through an tunable electromagnet.

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Figure 8. Experiment scheme for characterizing the PSHE of terahertz beam reflected from 1D-PC with a defect layer. A terahertz laser is used as a light source; A terahertz wave plate is used to adjust the intensity captured by terahertz camera; L1 and L2, terahertz lens with different effective focal lengths; P1 and P2, terahertz polarizers; A special adjustable electromagnet can be used to provide the magnetic field.

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Based on the above experimental scheme, we calculated the normalized PSHE shift of the reflected terahertz beam from 1-D PC with a graphene/SiC defect layer after the weak

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measurement amplification, assuming the incident angle is 10 degrees and the incident frequency is 10.13THz. According to Figure 9a we can find that the displacement increases significantly when the magnetic field increases. When the sign of the amplifying angle changes from negative to positive, the spin dependent splitting changes from positive to negative. Figure 9b shows the light intensity received by terahertz camera when B=3T and the amplifying angle is -3 degrees, 0 degrees and 3 degrees respectively, the movement of the beam centriod is consistent with Figure 9a.

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Δ=-3deg

Δ=0deg

Δ=3deg

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(b)

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(a)

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y

x

x

x

Figure 9. Theoretical results of PSHE after weak measurement amplification when N=2, EF =0.15eV, τ=0.1ps. (a) Normalized PSHE shift of reflected terahertz beam with different amplify angle ( ∆ ). (b) Light intensity received by CCD in case of ∆ =-3deg, ∆ =0deg and ∆ =3 deg. 4. Conclusion

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In summary, we systematically study magnetically tunable PSHE enhanced by defect mode in 1D-PC with graphene. We first analyze photonic crystals with three kinds of defect layers by

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using magneto-optical transfer matrix method and find that graphene can greatly enhance the magneto-optic Kerr effect and PSHE. Then the influences of several parameters on Kerr

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rotation angle and spin-dependent splitting of reflected light are analyzed comprehensively, especially the modulation effect of magnetic field on PSHE. The Kerr rotation angle of near 90 degrees and the spin-dependent splitting of about 15 times the wavelength are obtained at a specific incident wavelength. After that we calculate the Kerr rotation angle and spindependent splitting in the photonic crystal with double defect layers and find that enhanced PSHE occurs near both defect modes but show different properties around the two resonant frequencies. Finally, we proposed a possible experimental scheme for measuring spindependent splitting of reflected terahertz beam and calculated the PSHE shift and field 16

ACCEPTED MANUSCRIPT intensity after the weak measurement amplification, further confirming the modulation effect of the external magnetic field. These results show that the proposed method for PSHE manipulation have potential applications in design of spin-based photonics devices.

Acknowledgements

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This work is supported by Natural National Science Foundation of China (NSFC) (61505016, 61735010, 61675147, 61605141, 61475031, 51522204 and 11674234) and Project of Sichuan Provincial Department of Education (15ZA0183).

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