A Numerical Investigation of Two-Dimensional Photonic Crystal Slab

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A Numerical Investigation of Two-Dimensional Photonic

Crystal Slab

Yan-Ju Chiang

1,*

Since the pioneering work on photonic bandgap structure (PBG) was proposed by Yablonovitch and John in 1987 [1, 2], the PBG concept has inspired an extensive area of research with the capacity for significant influences on the fundamental material sciences, applied optoelectronics, and engineering [3-8, 9-17]. In these nanostructures, a periodic variation of the dielectric constant over finite-size thin films (2D) [18] or self-organizing photonic crystals (3D) [19, 20] has been reported to cause the prohibition of light propagation in certain frequency bands (i.e., the photonic bandgaps). Defect engineering over such periodic dielectric structures allows an exact control over the localized radiation of the electromagnetic (EM) waves trapped into the defects embedded in the structures. Recently due to the technological progress of thin-film lithography, various PBG devices based on defect engineering has been successively realized, including PBG based optical waveguides [9, 10], optical communication devices [11], vertical cavity surface-emitting lasers (VCSELs) [12-15], solar cells [16], and light emitting diodes (LEDs) [17].

Meanwhile, the theoretical study of finite-size, two-dimensional photonic crystal thin films has focused on the importance of predicting on the variation of spectral characteritics of local density of state (LDOS)

over the disorder positions of defects in these structures. As has been reported in [21], a complete full vectorial method is essential for the analysis of out-of-plane EM waves radiation from the defects of the 2D photonic crystal film. However, the analysis of in-plane EM waves propagation along the surface of the 2D photonic crystal film can be simplified by decoupling the EM wave modes into transverse-electric (TE) and transverse-magnetic (TM) modes. Furthermore, over the past two decades the methods to analyze such in-plane EM wave propagation can be classified into two categories: numerical methods and semi-analytic methods. Specifically, various kinds of numerical methods has been proposed, such as plane wave expansion (PWE) method [22-24], finite element method (FEM) [25], finite difference time domain method [26], and finite difference frequency domain method [27]. Among these methods, it is worth to mention that although the PWE solver has an advantage to deal with the 2D photonic crystal film with circular rods in reciprocal space [22] (due to the reciprocal dielectric function can be expressed in an analytic, Bessel form), the calculation is inefficient due to requiring large number of plane waves to expand both EM waves and the dielectric function, provided that the resulting bandstructure or LDOS spectrum converges to

1 _{Oriental Institute of Technology}

* _{Correspondence author: Yan-Ju Chiang}

an adequate degree of precision. The calculation efficiency of the FEM solver also relies on the order of shape functions (e.g., linear [25] or curvilinear functions [21]) used in each element.

Wave Equation for Out-of-Plane Analysis

Proposition: The wave equation which characterizing the in-plane electric field E



Ex Ey 0



in a structure with relative permittivity r_,z_r__

i.e., the structure is uniform in the z-direction can be written as: 0 1 1 2 2 2 2 2 2 2                                       y r r x r x x x r r x E y x E c z E y E E x x x E       (1.1) 0 1 1 2 2 2 2 2 2 2                                       x r r y r y y y r r y E x y E c z E y E E y y x E       (1.2) Proof:

Assume the wave field oscillates at a single angular frequency



. The in-plane field E



E_x E_y 0



can be expressed as:

 

      _      Re , exp ) , , ( zt E z i t E_x  _x   _(1.3)

 

      _      Re , exp ) , , ( z t E z i t E_y  _y   _(1.4)

Maxwell’s equation with respect to the field phasor expressions E_ z, _ , H_ z, _ , D_ z, _ and B_ z, _

can be expressed as:

H i E   (1.5) E i H    (1.6) 0    H (1.7) 0    D (1.8)

In equation (1.8) we assume the structure with dielectric constant _,z___ is a source-free

medium. Applying a curl operation  to equation

(1.5), we derive: H i E       (1.9)

Using the identity of the vector analysis to the left-hand side of equation (1.9):

 

E E E 2     (1.10)

where the vector operator _2 _{is a Laplacian.}

Since equation (1.8) can be rewritten as:

 

0                      D E  E  E _(1.11)

where we can derive E in equation (1.10) as:

E E      (1.12)

Thus the left-hand side of equation (1.9) becomes:

E E2      (1.13)

On the other hand, applying equation (1.6) to the right-hand side of equation (1.9), we can derive the wave equation for the electric field phasor E_ z, _ as:

0 ) ( 2 2 _{ }            E c E E r    (1.14)

Consider the structure is uniform in z-direction. In this case, the derivative of relative permittivity with respect to z is zero: 0    z r  (1.15)

After substituting the electric field phasor representations (1.3), (1.4) and the structure uniformity condition (1.15) into equation (1.14), we can derive the wave equations (1.1) and (1.2).

Problem Definition

The computational grid is a supercell as a unit building block to construct a two-dimensional photonic crystal slab with a thickness 2H at horizontal (x,y) plane. As shown in Fig. 1, the position vector i,j denotes

every mesh-point in a supercell. In the vertical z-direction, the height of a photonic crystal slab between a cover (region I) and a substrate region (region III) is 2H. Here we assume the slab at region II is uniform in z-direction, i.e., the relative permittivity

 

z

r ,

 of the slab satisfies the equation (1.15). Together with the in-plane field assumption



E_x E_y 0



E in section 1, we have a well-defined

problem to be resolved: We are going to find out the in-plane electric field profile E



Ex Ey 0



where the

field propagates along the vertical z-direction (out-of-plane wave analysis) of the photonic crystal slab surface.

Fig. 1 The supercell definition

Band Structure and Dispersion Relationship

We would like to employ equations (1.1) and (1.2) to resolve the electric field profile E



E_x E_y 0



at each mesh points of region II in Fig.1. In the first place we decouple the equations (1.1) and (1.2) by assuming derivative of the relative permittivity r_ z, _ is zero

with respect to x and y. i.e., 1

 

x r

 _{and. This}

assumption is reasonable if we select mes 1

 

y r

 _h

points at the position far away from the boundary between air holes (dielectric pillars) and surrounding dielectric media (surrounding air media). In particular, we select the central points of the air holes as mesh points.

Applying uniform permittivity r_ z, _ to equations (1.1) and (1.2), the wave equations becomes:

0 ) ( 1 2 2 2 2 2 2 2                 x x x x r E c z E y E x E   (3.1) 0 ) ( 1 2 2 2 2 2 2 2                 y y y y r E c z E y E x E   (3.2)

where we have the same wave equation form for electric fieldsExandEy. Without loss of generality we

may use a series of plane waves _eKm,n _{to expand} electric fields ExorEyas:

iqz j i j i iqz j i j i j i II(, ,z)A(, )(, )e B(, )(, )e  (3.3)

where _II(_i,j,z) denotes the electric field x

E orE_y at mesh point  . _i,j A(i, j), B(i, j) and

H q

2 2

 denote the field amplitudes, and propagation

constant along the vertical z-direction in region II, respectively. . (i, j) and (i, j) denote the

basis wave functions in the forward and backward direction. In particular, (i, j) and (i, j) can be

written as:

 

     _NxN _ m L n K i n m j i b e mn 1 1 , , ) , (   (3.4)

 

     _NxN _ m L n K i n m j i b e mn 1 1 , , ) , (   _(3.5)

where the wave vector Km,nk0Gmgn. Gm is a reciprocal lattice vector. gn is a supercell reciprocal lattice vector in 1x1 first Brillouin zone, as shown in [1]. On the other hand, the periodic dielectric function

r



1 can also be expanded by a series of plane waves as in [2].:



       m m m r G i G K( ) exp( ) ) ( 1 ^    (3.6)

After substituting equations (3.3), (3.4), (3.5) and (3.6) into equation (3.1), we can derive a linear operator equation as: 0 , ^        _B c q L  _(3.7) F o r g i v e n ggn , _  _  _     b a b a G G K c q L ^ , ^ , 2 2 2 0                   c q G g k n b  is a NxN linear operator and





T n NxN n n b b b B _{1, 2,} .... _, . Equation (3.7) can be

employed to compute the dispersion relationship of a two-dimensional photonic crystal slab with an arbitrary slab thickness. Here we give an example to illustrate the correctness of our theoretical formulation. As shown in [3], S. M. Hsu and H. C. Chang proposed a full-vectorial, finite-element method to analyze both in-plane and out-of-plane wave propagation in a triangular arranged slab with air holes surrounding by silica (n=1.45). The radius of air holes r=0.351a, and

8 2 2   a H

qa  where a is lattice constant. Fig.2 shows

the corresponding band structure in [3].

Fig.3 shows the band structure derived by using the same parameters of the slab and equation (3.7). Compared with Fig.2, we find there are some

c k bands missing in Fig.3 and the bandgap is narrower. It might be due to the bandstructure in Fig.2 contains both in-plane and out-of-plane solutions (as shown in [3]), however Fig.3 contains only out-of-plane solutions as shown in equation (3.1) or (3.2).

Fig. 2 Band structure in [3].

Fig.3 Band structure derived by employing the equation (3.7)

Fig. 4(a) and (b) represent the computational result of normalized frequency c a   2 as a function of slab

thickness (q-value) in case of the wave-vector k0

results we demonstrate the correctness of the equations (3.7) while it is employed to compute the band structure of an arbitrary finite-thickness, two-dimensional photonic crystal slab.

Fig. 4(a) Normalized frequency as a function of slab thickness at

k

₀=M

Fig. 4(a) Normalized frequency as a function of slab thickness at

k

₀=K

Boundary Matching Scheme and

Future Work

As we follow the conclusion in section 3, once we derive the information of thickness of the slab, the radius of air holes and the refractive index of media surrounding air holes, we may correctly compute the bandstructure and normalized frequency

c a

 

2 . In

addition, the corresponding values of





T n NxN n n b b b B _{1, 2,} .... _, , (_{i, j}) _{and ( )} , j i   _in

equations (3.4) and (3.5) can be derived and then can be used to compute the electric field



II

(



i,j

,

z

)

at each

mesh point of the supercell in equation (3.3). However, we have not determined the field amplitude

) ( i, j

A  and B(i, j) at each mesh point. Here we

propose a boundary matching scheme to derive the field amplitude A(i, j) and B(i, j) by matching the

tangential components of electric field and magnetic field for each mesh point at the boundary between region I and region II, and between region II and region III in Fig. 1, respectively.

As shown in Fig. 1, we assume the tangential components of electric field E



Ex Ey 0



in region I and region III to be a plane wave form as:

) ( ) ( K z H I z Ce I   in region I (4.1) ) ( ) ( K z H III z De III   in region III (4.2)

where _I(z)and _III(z)denote the electric field x

E or E_y in region I and region III. The wave vectors

I

K and K_III can be derived by computing the

corresponding plane wave equation in bulk region I and region III. Note that we assume KIKIII here because

the growth of a two-dimensional photonic crystal slab is usually carried out on a substrate in region III (e.g., the epitaxial layer of a wafer or a sapphire) and the ambient environment of the slab is usually air (region I). The constants C and D represent the amplitude of the plane wave in region I and region III, respectively.

Equations (3.3), (4.1) and (4.2) characterize the electric field profiles in a tangential direction (in-plane) of the slab in all three regions of Fig. 1. However, even we have derived (i, j) and (i, j) for each

We still have four unknown variables to determine, i.e., )

( _{i, j}

A  , B(_{i, j}),

C

and

D

. Fortunately, we have four boundary conditions for each mesh points:

) ( ) , ( _i,j z H _I z H II      (4.3) ) ( ) , ( _i,j z H _III z H II      (4.4) z H z z H z _I j i II        (_, , ) _ ( ) (4.5) z H z z H z _III j i II          (, , )  ( ) (4.6)

Equations (4.3) and (4.4) represent the electric fields continuity in the tangential direction at the boundaries between region I and region II and between region I and region III, respectively. Equations (4.5) and (4.6) represent the magnetic fields continuity in the tangential direction at the same boundaries. After applying equations (4.3)-(4.6) to electric field profiles in equations (3.3), (4.1), and (4.2), we derive two linear equations involving A(_{i, j}) and B(_{i, j}) for each mesh point  : i,j 0 1 ) ( 1 ) ( ) ( , 3 , ,                                     iqH I j i iqH III j i iqH III I j i e iq K B e iq K B e iq K K A       (4.7)   0 1 ) ( 1 ) ( ) ( , 3 , ,                               iqH I III j i iqH III I j i iqH I III j i e iq K K B e iq K K B e K K A       (4.8)

This is a series of linear, homogeneous equations of A(_{i, j}) and B(_{i, j}). Before we proceed to find solutions of equations (4.7) and (4.8), there are some critical issues we need to care:

For mesh points at the boundary of the supercell, what is the boundary solution?

What is the relationship between KI and KIII

to make equations (4.7) and (4.8) have unique or at least nonzero solutions?

Reference

[1] Y. C. Chang,” Electronic properties of the reconstructed Si (111) 7×7 surface,” J. Vac. Sci. Technol. B, vol. 1, pp. 709-713, 1983.

[2] M. Philhal, and A. A. Maradudin,” Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B, vol. 44, pp. 8565-8571, 1999.

[3] S. M. Hsu, and H. C. Chang,” Characteristic investigation of 2D photonic crystals with full material anisotropy under out-of-plane propagation and liquid-crystal-filled photonic-band-gap-fiber applications using finite element methods,” Opt. Exp., vol. 16, pp. 21355-21368, 2008.