[2020] [paper notes] optically controlled spectral ratio adjustable y based on two-dimensional photonic crystal——

Preface

type
$Terahertz+Beam\; splitters$
Periodical
$Journal\; of\; infrared\; and\; millimeter\; wave$
author
$Jiang\; zongdan,Li\; Peili,Zhang\; Yuanfang$
Time
$2020$

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research objective

In Terahertz Technology , Terahertz energy beam splitter is the key device

Materials used to design terahertz beam splitters ： Metal grating 、 Fiber grating 、 Polyethylene plastic film 、 Photonic crystals

Photonic crystals are easier to control the propagation of light , And photonic crystals are small

The beam splitter based on photonic crystal has high efficiency 、 Low loss 、 High integration

Terahertz beam splitters based on photonic crystals are based on : Autocollimation effect 、 Multimode interference principle 、 Defect mode Beam splitter

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Research methods

The methods of studying and calculating two-dimensional photonic crystals are ：
Plane wave expansion method
Multiple scattering method
Transfer matrix method
Finite difference time domain method

This paper uses
Plane wave expansion method PWM
+
Finite difference time domain method FDTD

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Plane wave expansion method PWM

Based on Bloch theorem , In inverted space （ Positive lattice space undergoes reciprocal transformation ） Expand the electromagnetic wave into the form of plane wave superposition , Then through discrete Fourier space , Transformation of eigenproblem , take maxwell The equation is transformed into an eigenequation , And solve the eigensolution . The eigenfrequency of the propagating light wave mode of the photonic crystal and the dispersion relationship of the periodic structure are obtained

Bloch theorem ：
In the determined complete crystal structure , Bloch wave vector k Is a conserved quantity （ Take reciprocal lattice vector as module ）, That is, the group velocity of the electron wave is a conserved quantity
In a complete crystal , Electron motion can propagate without being scattered by lattice （ So this model is also called near free electron approximation ）, The resistance of crystalline conductors only comes from those crystal defects that destroy the periodicity of the potential field

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Finite difference time domain method FDTD

from maxwell Starting from curl equation , take E and H Press... In space and time Yee Cell discretization , Then use the method of alternating sampling , Sample electric and magnetic fields （ The interval is half a time step ）

Through this way of discretization in time and space maxwell The curl equation is transformed into a set of difference equations . According to the initial values and boundary conditions of electromagnetic problems in photonic crystal structures , Iterative solution can be carried out step by step on the time axis , Get the electromagnetic field distribution under the stable condition
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FDTD Solve the equation

Electromagnetic waves in two-dimensional photonic crystals $(x,y)$ In plane propagation , Yes TE（ Transverse wave ） and TM（ Transverse magnetic wave ） Two polarization modes

TE Modular maxwell Scalar form of equation ：
The electromagnetic component is only $(E_x,E_y,0),(0,0,H_z)$

$$\{\begin{array}{l}\frac{\partial H_z}{\partial x}=\epsilon (r)\frac{\partial E_y}{\partial t}\\ \\ \frac{\partial H_z}{\partial y}=\epsilon (r)\frac{\partial E_x}{\partial t}\\ \\ \frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}=-\mu \frac{\partial H_z}{\partial t}\end{array}$$

TM Modular maxwell Scalar form of equation ：
The electromagnetic component is only $(H_x,H_y,0),(0,0,E_z)$

$$\{\begin{array}{l}\frac{\partial E_z}{\partial x}=\mu \frac{\partial H_y}{\partial t}\\ \\ \frac{\partial E_z}{\partial y}=\mu \frac{\partial H_x}{\partial t}\\ \\ \frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}=\epsilon (r)\frac{\partial E_z}{\partial t}\end{array}$$

among $\epsilon ={\epsilon }_0{\epsilon }_r$, Refractive index $n_0=\sqrt{{\epsilon }_r}$,${\epsilon }_{0}and{\mu }_0$ They are vacuum permittivity and vacuum permeability

First, grid the space , Replace the above differential with the central difference , The finite difference time domain form is obtained ：

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General plan （ There may be a mistake ）
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The spatial relative position of each component of the electromagnetic field can properly describe the propagation characteristics of the electromagnetic field , Gradually obtain the distribution of space electromagnetic field at each time in the future

In order to ensure the iterative calculation The convergence , For example, meet the stability condition ：

$$\Delta t\le \frac{1}{C_{max}}[\frac{1}{(\Delta x{)}^2}+\frac{1}{(\Delta y{)}^2}{]}^{-1/2}$$

among $C_{max}$ Is to solve the maximum phase velocity of space light wave , commonly $\Delta t=\frac{min\{\Delta x,\Delta y\}}{2C_{max}}$
$\Delta x,\Delta y,\Delta t$ The smaller theta is , The denser the grid , The more accurate the calculation is

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Research models

Designed two-dimensional photonic crystal Y The terahertz wave beam splitter type a adopts Triangular lattice structure

In low refractive index medium air , array 65X71 cylindrical , High resistance silicon dielectric column

High resistance silicon in THz Low frequency band absorption , Refractive index $3.42$

Lattice constants $a=117\mu m$, radius $R=0.32a$

In a complete photonic crystal , Take out the front of the middle row 25 Media columns , Form the input waveguide ,V Type waveguide as beam splitting part , Its V The number of transverse columns occupied by type-A waveguide is 25 List to 48 Column , And the upper arm rotates counterclockwise with the middle input waveguide 30 degree , The lower arm rotates clockwise with the middle input waveguide 30 degree , stay V Type end 20 Xing He 52 Remove each row 17 Media columns , Respectively form the output waveguide 1,2
At the upper arm （ The first 34 That's ok 25 Column and the first 35 That's ok 26 Column ） Lead in radius $R_1=0.384a$ Point defects of dendrimer metal nanoparticles

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The dendritic metal nano polymer point defect is applied along the axial direction of the dielectric column The light intensity
—— Because dendritic metal nanopolymers have Third order nonlinear Kerr effect , The change of external light intensity changes the refractive index of dendritic metal nano polymer
$$n=n_0+\Delta n=n_0+n_{2}I$$ among $n_0$ Is the linear refractive index of dendrimers ,$\Delta n$ Is its nonlinear refractive index ,$I$ Is the optical pumping power intensity

$n_2=\frac{\pi \times 10^4\times Re{x}^{(3)}}{{\epsilon }_{0}c{n}_0^2}$ Is the nonlinear refractive index

$x^{(3)}$ Is the third-order nonlinear polarizability （ Here for $10^{-6}esu$）

$⋮$

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principle

THz Wave input from left , Due to the introduction of dendritic metal nano polymer point defects in the upper arm , So by right Dendritic metal Nanopolymer Point defects exert different light intensities , Its refractive index changes , So as to control the coupling of point defects and line defects , bring THz The transmission ratio of the wave in the two output waveguides is different , Realize adjustable spectral ratio

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Conclusion

In a complete two-dimensional triangular lattice silicon photonic crystal , Introduce line defects and point defects of nonlinear polymer materials , Using the coupling effect of line defect mode and point defect mode and nonlinear polymer material Third order nonlinear Kerr effect （ The refractive index changes due to the change of electric field ）, An optically controlled beam splitting ratio adjustable based on two-dimensional photonic crystal is proposed Y Type terahertz beam splitter （ Light control ）

$$Additional\; optical\; pump\; power\Rightarrow Change\; the\; refractive\; index\; of\; dendrimers\Rightarrow The\; spectral\; ratio\; is\; adjustable$$

—— Total transmittance 98% above , The insertion loss is less than 0.087dB, The adjustable range of spectral ratio is 0.08~1.29 Between , response time ps level

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problem

FDTD The relationship between solution introduction and actual calculation , How much help ？

Why not introduce the plane wave expansion method PWM The solution of ？

How to introduce point defects and line defects ？

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