1.Introduction

Since the concept of photonic crystal(PC)was first proposed by Yablonovitch and John in 1987[1,2],PC has attracted considerable attention for potential applications[3,4].PC is an artificial material and periodically distributed in different media with the order of magnitude of the incoming electromagnetic wavelength.It can be divided into 1D,2D and 3D PC according to the spatial arrangement.PC will form photonic band gap

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(PBG)for the electromagnetic interference of multiple Bragg scattering.PBG is one of the most remarkable features of PC because electromagnetic waves cannot propagate in it.For conventional PC,its PBG cannot be changed once it has been fabricated.Therefore,it is very important to obtain the tunable PBG.In 2004,Hojo and Mase deduced the dispersion relation of the propagation of electromagnetic waves in 1D plasma photonic crystal(PPC)by solving the Maxwell equations,and they obtained the tunable PBG of 1D PPC by adjusting the plasma density[5].By introducing plasma into the PC,the tunable PBG can be achieved because the refractive index of plasma can be tuned by the external magnetic field[6]or the plasma thickness[7].And by introducing a superconductor into the PC,the tunable PBG can also be obtained because the permittivity of the superconductor can be controlled by the ambient temperature[8].It is well known that the 1DPC is more easily processed and prepared than the two other dimensional PCs.Furthermore,tunable potential applications based on 1DPCs have been extensively studied,such as the tunable omnidirectional absorber[9]and multichannel absorber[10],tunable alloptical diode[11],tunable sensor[12],tunable omnidirectional reflectors[13,14],tunable terahertz(THz) filters[15,16]and multichannel gigahertz(GHz) filters[17],etc.

In our previous works[18,19],tunable multichannel filters were reported by single-negative materials and grapheme based on 1DPCs,respectively.In this paper,the THz filter has been designed and studied,which is composed of 1DPC by introducing a defect layer of semiconductor GaAs.Based on the transfer matrix method(TMM),we have derived the dispersion relation of 1D defective PC(1DDPC)and its transmission expressions.Because the permittivity of semiconductor GaAs can be controlled by the external pressure in the THz range,the electromagnetic filtering properties of the 1DDPC can be tuned by this external pressure.Then,the electromagnetic plane wave simulations have been performed on the finite element software,and the results demonstrate that such a 1DDPC containing a defect layer of semiconductor GaAs can realize the external pressure-controlled THz filter.

The format of this paper is as follows.Section 1 is the introductory part.In section 2,we give the formulations that will be used in our theoretical model.Then,the results and discussion are given in section 3.Finally,a summary is presented in section 4.

2.Theoretical model

The schematic diagram of the 1DDPC in symmetric structure and asymmetric structure is displayed in figure 1.In figure 1,A is high refractive index material of Si,B is low refractive index material of SiO2,and C is a defect layer of GaAs.According to[20],the permittivity of semiconductor GaAsεs as a function of frequency and the external pressure in the THz region,is expressed as:

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Here,

These parameters have been reported in[20]:the external pressure　p　∈[0　,10] GPa,the damping factor γ = 0.072THz,ωTOis the angular frequency of the transverse optical phonons,ε∞ and ε0are respectively the high-frequency permittivity and static permittivity;in addition, and ε(0)0　　 =13.18 when the external pressure p = 0GPa.

In this work,the dielectric constants of the Si and SiO2 materials are unchanged by the external pressure.According to equations(1)and(2)in the THz region,the variation relation of the permittivity of defect layer GaAs as a function of frequency and external pressure is:(a)the real part;(b)the imaginary part,which is depicted in figure 3.It is obvious that the permittivity of GaAs can be controlled by the external pressure.In figure 3(a),the real part of the permittivity of GaAs rises with the increasing of frequency at the fixed pressure,and the real part of the permittivity of GaAs decreases with the increasing of pressure at the fixed frequency.From figure 3(b),we can find that the defect layer GaAs has certain losses,the imaginary part of the permittivity of GaAs rises remarkably with the increasing of frequency at the fixed pressure,and the imaginary part of the permittivity of GaAs decreases slightly with the increasing of pressure at the fixed frequency.

For the oblique electromagnetic wave incidence in the structures of the 1DPC,the electromagnetic tangential components of the nth sub-layer based on the TMM,can be given by[21]:

Figure 1.Schematic diagram of the 1DDPC in symmetric structure and asymmetric structure.

where ω　is angular frequency,c is the speed of light,θ is the electromagnetic wave incident angle.

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In terms of the TMM,the electromagnetic field vector at the interface can be obtained as:

where M = M1M2…MN−1MNis the multilayer eigen matrix.The transmission coefficient t and reflection coefficient r of the 1DPC can be written as:

If the 1DPC is considered to be embedded in air,thuscosθ,the transmittance T,the reflectance R and the absorbance A can be calculated as:

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Figure 2.PBG of the 1DPC of(AB)Nat a = b = 5 μm:(a)dispersion relation;(b)variation relation of transmittance with frequency;(c) first PBG.

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For a multilayer medium,the total transmission matrix is described as:

In addition,for different modes of the electromagnetic plane wave,they are namely:

Figure 3.Variation relation of the permittivity of GaAs as a function of frequency and pressure:(a)real part;(b)imaginary part.

According to the definite condition　the dispersion relations of the defect-free 1DPC of(AB)Ncan be derived as:

where

We assume the defect-free 1DPC of(AB)Nembedded in air;a is the layer A thickness and b is the layer B thickness,the structural unit d = a + b.According to the Bloch theorem,the electromagnetic field vector at the interface is given by[21]as:

3.Result and discussion

The defect-free 1DPC of(AB)Nis supposed to be embedded in air,in which A is the high refractive index material of Si and B the low refractive index material of SiO2.The refractive indices are respectively nSi = 3.3 and nSiO2 = 2.25 at the THz frequency in[15].At normal incidence(i.e.θ = θa = θb = 0),the transmission properties of the transverse electric(TE)mode plane wave propagates in the 1DPC of(AB)N,which is the same as the transverse magnetic mode plane wave.According to equations(7)and(10),the PBG of the 1DPC of(AB)Nat layer thickness a = b = 5 μm(i.e.a different optical thickness):(a)the dispersion relation;(b)variation relation of transmittance with frequency;(c) first PBG,which is plotted in figure 2.In[21],we can calculate the center frequency of the first PBG f0 = 5.408THz.Figure 2(c)illustrates that the frequency range of the first PBG　fP　BG　　∈[4　.77, 6.02]THz.Thus,the size of the first PBG Δf = 1.25THz.Although figures 2(a)and(b)are calculated by two formulas,the PBG in figure 2(a)is in very good in agreement with figure 2(b).

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We first study the electromagnetic propagation properties of the 1DDPC containing a defect layer GaAs in symmetric structure(i.e.air/(Si/SiO2)N/GaAs/(SiO2/Si)N/air)at normal incidence.These parameters are assumed as follows:the layer A and B thickness a = b = 5 μm,and the defect layer thickness is 11 μm.Figure 4 displays the variation relation of transmittance of the 1DDPC in symmetric structure as a function of periods and frequency at external pressure P = 0GPa.From figure 4,we can see that the transmission peak at the frequency of 4.82 THz in the first PBG fP　BG　　∈[4　.77, 6.02]THz.Irrespective of the number of cycles,the filtering peak is at the same frequency of 4.82 THz as long as the defect layer thickness is constant.Due to the loss of the defect layer GaAs,the transmittance cannot reach 1 at 4.82THz.We choose the cycles N = 9 for the maximum transmittance T = 0.79.

It is well known that the defect layer thickness can change the electromagnetic wave propagation distance in the 1DDPC based on the Bragg scattering theory.These parameters are assumed as follows:the layer A and B thickness a = b = 5 μm,and the cycles N = 9.Figure 5 shows that the variation relation of the transmittance of the 1DDPC in symmetric structure as a function of defect layer thickness and frequency at P = 0GPa.From figure 5,the filtering effect strongly depends on the defect layer thickness,and the filtering effect is best with the defect layer thickness dc = 11 μm.Thus,we select the defect layer thickness dc = 11 μm in our design.

Figure 4.Variation relation of the transmittance of the 1DDPC in symmetric structure as a function of periods and frequency at P = 0 GPa.

Figure 5.Variation relation of the transmittance of the 1DDPC in symmetric structure as a function of defect layer thickness and frequency at P = 0 GPa.

Figure 6.Variation relation of the transmittance of the 1DDPC in symmetric structure as a function of pressure and frequency.

In the above two paragraphs,we have discussed the cycles and the defect layer thickness influence on the electromagnetic wave propagation properties in the 1DDPC,and obtained the optimization results:the cycles N = 9 and the defect layer thickness dc = 11 μm.To investigate the effect of the external pressure on the electromagnetic wave propagation properties in this 1DDPC,we plot the variation relation of the transmittance of the 1DDPC in symmetric structure as a function of pressure and frequency,as shown in figure 6.According to equation(1)and figure 3,the electromagnetic wave filtering property depends on the external pressure because the permittivity of GaAs can be controlled by the external pressure.From figure 6,we can achieve the filtering frequency f and the transmittance T at three different pressures:(1)f = 4.82 THz and T = 0.79 at P = 0GPa;(2)f = 4.90THz and T = 0.78 at P = 5GPa;(3)f = 4.99 THz and T = 0.72 at P = 10 GPa.The filtering transmittances are all more than 70%,and the filtering frequency gradually shifts to the direction of high frequency and the filtering　transmittance　decreases slightly　with　the increasing of the external pressures.

Figure 7.Comparison of electric field modulus with different pressures at two frequencies.

Figure 8.Variation relation of the transmittance of the 1DDPC in asymmetric structure as a function of frequency at P = 5 GPa.

To further demonstrate the external pressure tunable filtering properties of the 1DDPC in symmetric structure,the TE mode plane wave simulation is carried out by using the finite element method.In our simulation,this 1DDPC is excited by a uniform TE mode plane wave incoming from the left to right in figure 1 with electric field amplitude equal to 1Vm−1.We select the external pressure P = 0GPa and P = 10 GPa as examples at their own best transmission frequencies,and perform a comparative analysis of the electromagnetic full-wave simulation.Figure 7 illustrates the comparison diagram of electric field modulus with different pressures at two frequencies.For the fixed frequency f = 4.82 THz,the electric field modulus of the output in this 1DDPC is minimum and almost near zero at the external pressure P = 10 GPa,which shows that the electromagnetic wave cannot propagate through this 1DDPC.For the same fixed frequency f = 4.82 THz with the different external pressure P = 0GPa,the electric field modulus of the defect layer in this 1DDPC is maximum,which shows that the electromagnetic wave can propagate to this defect layer,and the electric field modulus of the output exhibits a certain intensity,although the defect layer GaAs has certain loss and absorption,which shows that the electromagnetic wave can still transfer to this output.From figures 7(c)and(d),we can also get a similar result at another fixed frequency f =4.99 THz with the two different external pressures.The tunable filtering effect is very obvious when the external pressure is modulated in this 1DDPC in symmetric structure.It is further demonstrated that our filtering model is effective and feasible.

In the above discussion,we studied the electromagnetic propagation properties of the 1DDPC containing a defect layer GaAs in symmetric structure(i.e.air/(Si/SiO2)N/GaAs/(SiO2/Si)N/air)at normal incidence.Next,we investigated the electromagnetic propagation properties of this 1DDPC　in　asymmetric　structure　(i.e.air/(Si/SiO2)N/GaAs/(Si/SiO2)N/air)at normal incidence.To compare the transmission properties of the 1DDPC with the two structures in figure 1,we assumed the same parameters in the two structures:the layer A and B thickness a = b = 5 μm,the defect layer thickness dc = 11 μm,the cycles N = 9 and the external pressure P = 5GPa.Figure 8 shows the variation relation of the transmittance of the 1DDPC in asymmetric structure as a function of frequency at P = 5GPa.We obtained the filtering frequency f = 5.43 THz and the transmittance T = 0.04 in asymmetric structure.By comparing the filtering properties of the 1DDPC in the two structures,we found that the loss of defect layer GaAs in asymmetric structure is more sensitive to symmetric structure.As a result,the filtering transmittance in symmetric structure is far higher than in asymmetric structure.The result is consistent with the literature[15]because the filtering frequency is near the center frequency of the first PBG in asymmetric structure.

4.Conclusion

In summary,we have designed and investigated the electromagnetic wave transmission properties of the 1DDPC in the symmetric structure of air/(Si/SiO2)N/GaAs/(SiO2/Si)N/air and asymmetric structure of air/(Si/SiO2)N/GaAs/(Si/SiO2)N/air by introducing a defect layer of semiconductor GaAs.Based on the TMM,we have derived the dispersion relation of this 1DDPC and its transmission expressions.Because the permittivity of semiconductor GaAs can be modulated by the external pressure in the THz range,the tunable THz electromagnetic filtering properties of the 1DDPC can be obtained.First,we studied the filtering properties of the 1DDPC in symmetric structure at normal incidence,and further obtained the optimized parameters:the defect layer of GaAs thickness dc = 11 μm and the periods N = 9 when the layer A of Si and B of SiO2thickness a = b = 5 μm.The filtering transmittances are all more than 70%,and the filtering frequency gradually shifts to the direction of high frequency and the filtering transmittance decreases slightly with the increasing of the external pressures.Second,we simulated the propagation of TE mode plane wave in this 1DDPC by the finite element method,and the results demonstrated that such a 1DDPC can realize the external pressure controlled filtering in the THz region.Finally,we compared the electromagnetic wave propagation properties of this 1DDPC in the two structures.By comparing the filtering properties of the 1DDPC in the two structures,we found that the transmittance is more sensitive in asymmetric structure than in symmetric structure for the loss of defect layer GaAs.The numerical simulation results con firmed that the tunable THz filter is effective and feasible.

This work was partially supported by National Natural Science Foundation of China(Grant Nos.11174147,11175152 and 11704326),the Funding of Jiangsu Innovation Program for Graduate Education(Grant No.KYLX15_0316)and the Fundamental Research Funds for the Central Universities.

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