The equivalent structure and some optical properties of the periodic-defect photonic
crystal
Ting-Hang Pei and Yang-Tung Huang
Citation: Journal of Applied Physics 109, 073104 (2011); doi: 10.1063/1.3567296 View online: http://dx.doi.org/10.1063/1.3567296
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/109/7?ver=pdfcov Published by the AIP Publishing
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The equivalent structure and some optical properties of the periodic-defect
photonic crystal
Ting-Hang Pei and Yang-Tung Huanga)
Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu, Taiwan
(Received 10 October 2010; accepted 17 February 2011; published online 4 April 2011)
The concept of the effective medium is used to deal with problems of defects in photonic crystals (PhCs) throughout this paper. First, scattering phenomena in PhCs with defects are investigated and can be very well replaced by those in an effective medium with effective defects in interesting frequency region. It is based on the fact that the Bloch wave in the defect-free PhC very approximates to the wave in the effective medium. Then a periodic-defect PhC created by adding periodic defects into the defect-free PhC is proposed. It can be replaced with an effective PhC by utilizing the concept of effective periodic defects embedded into an effective medium. We verify the equivalence between the periodic-defect PhC and the effective PhC by comparing their photonic band structure (PBS) and photonic band gaps (PBGs) as well as transmissions. Next, the effective PhCs instead of the periodic-defect PhCs are used to investigate the variance of the PBGs by applying different periodic defects. The substitution allows us successfully predicting the negative refraction in the periodic-defect PhC through the PBS of the effective PhC. Finally, the field distributions and Fourier coefficients of waveguide modes in the effective PhC and the periodic-defect PhC are equivalent by the supercell method confirmations. VC 2011 American
Institute of Physics. [doi:10.1063/1.3567296]
I. INTRODUCTION
Photonic crystals (PhCs) are formed with dielectric peri-odic structures analog to solids and exhibit some properties, such as the photonic band structures (PBSs) which include photonic passing bands and photonic band gaps (PBGs), and complicate dispersion relations.1,2Generally speaking, PBSs have to be obtained before further discussing other applications of PhCs. They can be solved by the widely used plane-wave expansion method. Then the finite-difference time-domain (FDTD) method3is often used to calculate its real-time field and energy flow. If distributions of cylinders in the two-dimensional PhC are quasiperiodic or random, it needs a large supercell to calculate the PBS and a large PhC area for FDTD simulation. It is not only a time-consuming calcula-tion, but also a challenge on hardware equipment such as computer memory. If optical properties of PhCs can be re-placed with effective refractive indices, it is possible to design some special structures in PhCs and understand their optical performances based on the effective models. The problems on calculation time and equipments could be improved explicitly.
As we know, PBSs reveal different optical responses when light propagates inside the PhC at different frequen-cies. On the one hand, PhCs can be treated as an anisotropic material that presents negative refraction4–6and the super-prism effect.4,7On the other hand, they can be treated as a homogeneous material with an effective dielectric constant under certain condition.4Several effective-medium methods (EMTs) about disordered PhCs or metamaterials8–11 reveal the possibility that treats some PhCs as simple media with
effective refraction indices for all incident angles in certain frequency regions. In our previous work, a high-transmission PhC heterostructure Y branch waveguide is proposed.12 Unlike other defect-mode PhC waveguide operating at the photonic bandgap (PBG), the designed waveguide operates at photonic band region. It is composed of two different PhCs, each of them has different air-hole radius resulting in different effective refraction index within the first photonic band. Then these two effective media, just like the core and cladding of the conventional waveguide, confine light in the core region very efficiently. It can be explained by the mech-anism of total internal reflection between two PhCs and the transverse resonance condition is also satisfied. Especially, the boundary between two PhCs does not support any inter-face mode. This result tells us that even in complicated struc-tures such as PhCs and their compounds, the concept of effective medium still holds in certain frequency region without losing correctness of optical performances.
In other fields, Andrianovet al. have successfully used asymptotic homogenization method to calculate acoustic wave propagation in periodic composite materials in long-wavelength region.13,14 This method uses multiple-scale as-ymptotic expansions in powers of a natural small parameter e, which characterizes the rate of heterogeneity of the struc-ture, to represent fields in a composite. It is successfully applied in micromechanics of composites whose coefficients are rapidly varying periodic functions in spatial coordinates, for example, the hexagonal array of cylindrical fibers. Besides, these results will remain correct for other kinds of transport coefficients such as electrical conductivity, diffu-sion, magnetic permeability, etc. From the concept of ho-mogenization, the optical response of the PhC can be simplified that may help to design new composite materials a)Electronic mail: [email protected].
0021-8979/2011/109(7)/073104/9/$30.00 109, 073104-1 VC2011 American Institute of Physics
for a large variety of engineering applications. It also points out that the homogenization method is useful and important for both fundamental and applied points of view.
In this paper, we use the concept of homogenization to verify the equivalence between the periodic-defect PhC and the effective PhC in certain frequency region. The starting point of our goal is the scattering phenomena. First, we dis-cuss them in PhCs with some defects and compare with the results of the effective model composed of the effective me-dium and effective defects. Then the Fourier series expan-sion of the Bloch wave in the defect-free PhC is used to confirm the concept of the effective medium. Next, periodic defects are added into the defect-free PhC to form the periodic-defect PhC, and the effective PhC is proposed to replace with the former. We show that PBGs of the periodic-defect PhC are equal to those of the effective PhC. Due to the equivalence exists between the periodic-defect PhC and the effective PhC, it is easy to investigate tunability of the PBG through the PBS of the effective PhC by changing the distri-bution of periodic defects. We also predict the negative refraction from the PBS of the effective PhC and verify it by using the FDTD simulation. Finally, defect modes are investigated by removing a line of periodic defects in the periodic-defect PhC. The calculations show that field distri-butions and Fourier coefficients of the periodic-defect PhC is equivalent to those of an effective PhC.
II. SCATTERING PHENOMENA OF ONE AND TWO DEFECTS IN THE PHOTONIC CRYSTAL
A. One-defect scattering
Recently, the multiple-scattering theory has been used to study PBSs,15,16optical flows,17transmissions,18negative refractive phenomena,19 and other topics20,21 of PhCs. Because the PhC contain lots of periodic scatters, the wave in the PhC suffers multiple scattering processes by all scat-ters resulting in a spatially periodic distribution eventually. Due to the periodic distribution of the dielectric constant, the Bloch wave can exist in the PhC. In certain frequency regions, the PhC show a characteristic of a fixed effective re-fractive index for all angles. As a result, the Bloch wave propagating through the PhC is similar to the plane wave propagating through a homogeneous medium. If a cylindrical defect exists on the path of the propagation plane in the ho-mogeneous medium, the wave is scattered by the defect and form concentric circles. The similar phenomenon should also take place in the PhC. The investigation of the scattering phenomenon by defects in an originally defect-free PhC is an intuitive way to initially verify the correctness of the PhC replaced by an effective medium.
Before discussing the concept of an effective medium, effective refraction indices varied with incident angles of the PhC have to be obtained first. The normal direction of the equifrequency surface (EFS) is the direction of the group ve-locity as well as the propagation direction of light in the PhC, so EFS can give the relation between the incident and the refractive angles.4 According to the conservation rule, the incident and refractive wave vectors are continuous for the tangential components parallel to the interface. Given the
incident wave vector ki with a corresponding frequency x
and an incident angle hi, the refractive wave vector kr and
thus the refractive angle hr can be determined
simultane-ously. By using Snell’s law, effective refraction indices var-ied with incident angles of the PhC are acquired.
We consider the PhC composed of a two-dimensional triangular array of air cylinders on thex-y plane. The air cyl-inders with dielectric constant eb¼ 1.0 are embedded into a
background medium with the dielectric constant ea¼ 2.25.
UK and UM are parallel to x and y axes, respectively. The lat-tice constant of the triangular array isa and the radius r of all air cylinders is 0.20a. A defect is created by removing an air cylinder which is denoted as the dotted circle in Fig.1(a), so the dielectric constant edof the defect is 2.25. To remove
an air cylinder is also equivalent to add an additional me-dium with a dielectric constant eaebat the defect place. The
defect can perturb and scatter the propagating Bloch wave in the PhC. The EFS considered here is circular so that effec-tive refraction indices neffare constant for all angles. In the
following, an effective medium of which effective dielectric constant eeff¼ neff2 and the relative permeability l¼ 1 is
substituted for the defect-free PhC. At the same time, the defect with the radius 0.20a is treated as an effective defect, whose effective dielectric constant edis equal to eeffþ ea
eb. Then the structure of the PhC with a defect is replaced
with an effective one which includes the effective medium and an effective defect. The distribution of the effective dielectric constant in the effective structure is shown in Fig.1(b).
The scattering phenomenon in the one-defect PhC is shown by using the FDTD method. The frequency of the
FIG. 1. (a) A defect within a two-dimensional triangular PhC. (b) Equiva-lent structure of an effective medium with an effective defect.
incident Gaussian beam with TM-polarization (E-field is par-allel to the cylinder) is 0.30(c/a), where c is the speed of light in vacuum. The Gaussian beam is normally incident from air into the PhC and propagates along the UM direction. The dis-tance between the defect and the PhC boundary has to be large enough for the reason that the incident beam needs passing through an enough distance to become the Bloch wave. This distance is 25a in our simulation. The effective index neffcalculated from the EFS is 1.44, which indicates
that eeffand edare 2.0736 and 3.3236, respectively. The total
wave wtotal in the PhC is the sum of the incident wave
wincident and the scattered wave wscattered. So the scattered
wave by the defect can be obtained by subtracting the inci-dent wave from the total wave in the PhC as shown in Fig.
2(a). It is found out that the wave fronts of the scattered wave in the PhC form a series of near concentric circles away from the defect. Then we compare it with the corre-sponding scattering phenomenon in the effective structure. Figure 2(b) shows that the scattering wave is produced by the effective defect in the effective medium. Compare two scattered waves in Figs.2(a)and2(b), the scattered wave in an effective medium is very close to that in the PhC. The consistency reveals that the effective structure is a good approximation to deal with the scattering phenomenon by the defect in the PhC.
B. Two-defect scattering
In this section, the scattering phenomenon produced by two defects in the PhC is discussed. When the cylinder is shifted by a distance of d and does not overlap with other cylinders, the shifted cylinder can produce two defects within the PhC as shown in Fig. 3(a). One (defect 1) is located at the original position and the other (defect 2) is a distance ofd away from the original position. Using the con-cept of effective defects, the effective dielectric constant of the defect 1 is eaebmore than eeff, and that of the defect 2 is
eaebless than eeff. The distribution of the effective
dielec-tric constant in the effective structure is shown in Fig.3(b). Then we also use the FDTD method to compare the scattering phenomena in the PhC and the effective structure, respectively. The shifted distanced is chosen as 0.5a along the x (UK) direction. A Gaussian beam with the frequency 0.30 (c/a) is normally incident from air into the PhC and
propagates along y (UM direction). In Fig.4(a), the calcula-tion shows that the scattered wave is indeed from two posi-tions of these two defects. Next, we also simulate the same Gaussian beam incident on the effective structure, and com-pare with the calculation of the PhC case. The effective me-dium has an effective dielectric constant of 2.0736, and effective dielectric constants of the defect 1 and the defect 2 are 3.3236 and 0.8236, respectively. Both radii of two defects are still 0.20a. The scattered wave by these two defects in the effective structure is shown in Fig.4(b), and it is also very close to that in the PhC.
FIG. 2. (Color online) (a) The wave scattered by a defect in the PhC where ea, eb, ed, andr are 2.25, 1.0, 2.25, and 0.20a, respectively. (b) The wave
scattered by the effective defect in the effective medium where eeff, ed, and
r are 2.0736, 3.3236, and 0.20a, respectively. A Gaussian beam with fre-quency 0.30(c/a) is incident in both (a) and (b) cases.
FIG. 3. (a) Two defects within a two-dimensional triangular PhC by shifting a cylinder a distance of d away from the original position. The shifting cylin-der does not overlap with other cylincylin-ders. (b) Equivalent structure of an effective medium with two effective defects.
FIG. 4. (Color online) (a) The wave scattered by two defects whose dielec-tric constants edare individually 2.25 and 1.0 in the PhC. ea, eb, andr are
2.25, 1.0, and 0.20a, respectively. (b) The wave scattered by two effective defects whose effective dielectric constants edare individually 3.3236 and
0.8236 in the effective medium. eeffandr are 2.0736 and 0.20a, respectively.
A Gaussian beam with frequency 0.30(c/a) is incident and the shifted dis-tance d is 0.5a along the x (UK) direction in both (a) and (b) cases.
In order to verify that the effective structure can apply to defects with large dielectric constants, another demonstra-tion is shown in Fig.5(a), where two defects of Fig.4(a)are changed to 2.25 and 16.0801. The scattering phenomenon produced by two effective defects in the effective medium is shown in Fig.5(b)where effective dielectric constants of the defect 1 and the defect 2 are 3.3236 and 15.9037, respec-tively. In this case, the difference of the dielectric constant between the defect 1 and background medium is 1.25 and that for another defect is 13.83. According to the scattering formula of the 2D cylinder,22 the scattered wave is propor-tional to the difference of the dielectric constant between the cylinder and the background medium. So the scattered wave from the large-dielectric-constant defect is eleven times larger than that from the small-dielectric-constant one. In Figs. 5(a) and 5(b), both scattered waves seem from one defect only. But in fact, they are produced by two defects. The scattered wave from the small-dielectric-constant defect is very weak so that it is covered by the scattered wave from the large-dielectric-constant one. The scattered wave in Fig.
5(b) is consistent with that in Fig. 5(a). As a result, both Figs.4and5show that effective structures can exhibit very similar scattering phenomena to those in PhCs.
C. Comparison of the Bloch wave in the PhC and the propagation wave in the effective medium
Above results show the effective structure is a good approximation to describe the scattering phenomena in the PhC, and they also imply that the Bloch wave should be very close to the propagation wave in the effective medium. Then a demonstration of the Bloch wave in the defect-free triangu-lar PhC is calculated by the plane-wave expansion method.23 The elementary reciprocal lattice vectors in thek-space are
G1¼ 2p/a(1, 1/ ffiffiffi 3 p ) and G2¼ 2p/a(0, 2/ ffiffiffi 3 p ), respectively, and the reciprocal lattice vector is G¼ Gpq¼ p G1þ q G2,
wherep and q are integers. All parameters of the PhC are the same as above discussions, which are ea¼ 2.25, eb¼ 1.00,
andr¼ 0.20a. The electric field Ez of the TM-polarization
Bloch wave is expressed as sum of all Fourier series,23
E z; k*nðr * Þ ¼ ei k* r*X G * E z; k*nðG * ÞeiG* r*; (1)
wherek*is the wave vector in the first Brillouin zone andn is a band index. The real part ofEzat 0.30 (c/a) and the Fourier
coefficients Ez; k*nG*in Eq.(1) are shown in Fig.6(a)and
6(b), respectively. TheEzdistribution in Fig.6(a)shows the
wave propagates along the UM direction, in which circles represent air cylinders. By calculations, several coefficients more than 0.01 are denoted in Fig. 6(b)except for theG*00
term, and all the other undenoted terms are very small which can be ignored. TheG*00term is the maximum one, which is
0.9963. The Bloch wave can be approximately expressed as the summation of all these Fourier series whose coeffi-cients Ez; k*nG* are more than 0.01. On the other hand, according to the effective medium approximation, the wave in the effective medium can be described by /eff¼ exp
ik*eff r
*
¼ expik* r* with the wave vector k¼ keff
¼ 2pneff/a. If APhC represents the Bloch wave in the PhC,
we have the relation between APhCand Aeff: FIG. 5. (Color online) (a) The wave scattered by two defects whose
dielec-tric constants edare 2.25 and 16.0801 in the PhC. ea, eb, andr are 2.25, 1.0,
and 0.20a, respectively. (b) The wave scattered by two effective defects whose effective dielectric constants edare 3.3236 and 15.9037 in the
effec-tive medium. eeffandr are 2.07 and 0.20a, respectively. A Gaussian beam
with frequency 0.30(c/a) is incident and the shifted distance d is 0.5a along thex (UK) direction in both (a) and (b) cases.
FIG. 6. (Color online) (a) TheEzfield in the PhC where ea, eb, andr are
2.25, 1.0, and 0.20a, respectively. The frequency is 0.30(c/a) and the wave vector is 0.25(p/a) along the UM direction in the first Brillouin zone. (b) The coefficientsE
z; k*n
G*of theEzfield in (a).
/PhCðx; yÞ ei kr 0:9963ei G00r0:0357 ei G01r 0:0141 ðei G1 1rþ ei G1 0rÞ /eff x; yh0:99630:0357 eip4pffiffiffi3ay 0:0141 ei2pa xþyffiffi 3 p þ ei2pa xyffiffi 3 pi : (2)
From above equation, APhCis mainly composed of theG
*
00
-term harmonic wave. Other high-order harmonic ones com-paring to the G*00 term are very small so that they can be
ignored and only the G*00 term is left. Finally, we obtain
/PhC(x, y) /eff(x, y). It means that Aeffis approximate to
APhCwith very high accuracy in this case.
III. PERIODIC-DEFECT PHOTONIC CRYSTAL A. Effective photonic crystal
In Sec. II, the concept of the effective medium treats all defects in the PhC as effective defects in the effective me-dium. Following this concept, we can remove off some cyl-inders or replace some cylcyl-inders with different media to form periodic defects and then treat them as effective peri-odic defects or an array of effective dielectric cylinders. The periodic defects denoted by the empty circles are shown in Fig.7(a). The PhC with periodic defects is called the peri-odic-defect PhC. By the concept of the effective medium, the periodic-defect PhC can be approximate to another repre-sentation, the effective PhC, which consists of an effective medium replacing the defect-free PhC, and an array of effec-tive dielectric cylinders replacing periodic defects.
A periodic-defect PhC and its effective PhC are demon-strated here. The periodic defects are created by replacing air cylinders with a high refractive-index medium ofnc¼ 4.01
on a defect-free PhC, which is composed of a triangular array of air cylinders on a medium with the dielectric con-stant ea¼ 2.25. The radius r of each cylinder is 0.42a. When
the frequency is below 0.36 (c/a), neff¼ 1.20 is a good value
to represent the effective refraction index of the defect-free PhC for all angles. So the eeffof the effective medium is 1.44
and ed¼ eeffþ ec ebof each effective dielectric cylinder is
16.5201 in the effective PhC. The distance between two ad-jacent defects, which is called the defect lattice, is chosen as 2a. Hence the new lattice constant of the effective PhC is also 2a, and the ratio of the radius to the new lattice constant
becomes 0.21. The effective dielectric cylinders form a trian-gular array, too. The distribution of the effective dielectric constant in the effective PhC is shown in Fig.7(b).
B. Photonic band structure and transmissions
The PBS of the effective PhC with eeff¼ 1.44 and
ed¼ 16.5201 is shown in Fig. 8(a) where the first Brillouin
FIG. 7. (a) The periodic defects distribute in the PhC. (b) The distribution of the effective dielectric constant in the effective PhC.
FIG. 8. (a) The PBS of the effective PhC where eeff, ed,r and the defect
lat-tice are 1.44, 16.5201, 0.42a, and 2a, respectively. (b) The PBS of the peri-odic-defect PhC is calculated by the supercell method where ea, eb, ec, and
r are 2.25, 1.0, 16.0801, and 0.42a, respectively. The area of the supercell is shown in Fig. 7(a). In (a) and (b), the right figures are the first Brillouin zones.
FIG. 9. Transmissions of the periodic-defect PhC (dashed line) and the effective PhC (solid line), respectively.
zone is at right side. From the calculation, two full PBGs denoted as gray regions exist below frequency of 0.36 (c/a). These two frequency regions are from 0.116 to 0.187 (c/a) and from 0.236 to 0.251 (c/a), respectively. Next, the super-cell method24 is used to calculate the PBS of the periodic-defect PhC. The PBS with the first Brillouin zone is shown in Fig.8(b). The area of a supercell in the periodic-defect PhC is a rectangular region which is enclosed by solid line in Fig.7(a). By comparing Fig. 8(a) with Fig. 8(b), the posi-tions of the full PBGs almost coincide with each other.
In addition, the FDTD method is further used to calcu-late transmissions of the periodic-defect PhC and the effec-tive PhC, respeceffec-tively. Gaussian beams are normally incident from air into both PhCs and then propagate along the UM direction. In Fig.9, the dashed and solid lines repre-sent transmissions of the periodic-defect PhC and the
effec-tive PhC, respeceffec-tively. Both transmissions at the frequencies from 0.11 to 0.18 (c/a) and from 0.23 to 0.25 (c/a) are very close to zero. These zero-transmission regions just corre-spond to full PBGs in Figs.8(a)and8(b). Besides, frequen-cies from 0.315 to 0.33 (c/a) are another zero transmission region because it belongs to the local PBG, which is denoted as the left-upper gray block in Fig.8(a). From above discus-sions, PBGs of the effective PhC are coincident with those of the periodic-defect PhC. Except for these zero-transmission regions, both transmissions are also close to each other.
C. Tunability of the photonic bandgap due to periodic defects
Next, the effects of the periodic-defect distribution on the PBGs are discussed. Since the effective PhC is equivalent to the periodic-defect PhC, it is convenient to discuss the effects through the effective PhCs. The PBSs of three effec-tive PhCs with defect lattices of 3a, 4a, and 5a are calculated and shown in Figs.10(a),10(b), and10(c), respectively. The
FIG. 10. The PBSs of three effective PhCs with eeff¼ 1.44, ed¼ 16.5201,
and r¼ 0.42a where the defect lattices are (a) 3a, (b) 4a, and (c) 5a, respectively.
FIG. 11. (Color online) The refractive phenomena when the Gaussian beam with frequency of 0.15(c/a) incident from air into (a) the defect-free PhC where ea, eb, andr are 2.25, 1.00, and 0.42a, respectively, and (b) the
peri-odic-defect PhC where ea, eb, ed, r, and the defect lattice are 2.25, 1.0,
16.0801, 0.42a, and 5a, respectively.
effective dielectric constants of eeff¼ 1.44 and ed¼ 16.5201
are used, and the effective periodic defects are formed in a triangular array as that in Sec.III.B. Three PBGs exist below 0.36 (c/a) in the 3a case. Those are from 0.098 to 0.152 (c/a), from 0.220 to 0.221 (c/a), and from 0.317 to 0.321 (c/a), respectively. When compare with the 2a case, the position of the lowest PBG shifts 0.018 (c/a) downward and the width of it is narrowed down to 0.016 (c/a). In the 4a case, two PBGs are from 0.087 to 0.119 (c/a) and from 0.210 to 0.219 (c/a), respectively. In the 5a case, those are from 0.079 to 0.095 (c/a) and from 0.183 to 0.187 (c/a), respectively. The positions of the lowest PBG for the 3a, 4a, and 5a cases show the trend of shifting downward. It can be seen that the widths of the second PBG are decreased from 0.011 to 0.004 (c/a) as the defect lattice is increased from 3a to 5a. When the defect lattice is more than 7a, no any full PBG exists below 0.36 (c/a) by the calculation conclusion.
IV. NEGATIVE REFRACTION IN THE PERIODIC-DEFECT PHOTONIC CRYSTAL
From the PBSs of effective PhCs in Figs. 10(a),10(b), and 10(c), the photonic bands become denser when the defect lattice becomes larger. Due to the periodic defects, the optical characteristic of the periodic-defect PhC becomes more complex and is very different from that of the defect-free PhC. For example, when the defect lattice is 10a, more
than 100 photonic bands are below 0.36 (c/a). In the follow-ing, the FDTD method with a Gaussian beam is used to investigate the difference of the refractions between the defect-free and periodic-defect PhCs. The negative refrac-tions can be predicted through the PBS of the effective PhC.
The Gaussian beam with frequency of 0.15 (c/a) is inci-dent from air into the defect-free and periodic-defect PhCs at the incident angle 6. The defect-free PhC is the same as the one used in Sec. III. In Fig.11(a), the refractive phenomenon in the defect-free PhC shows a normal case as the Gaussian beam incident into a right-handed medium. When periodic defects are added into the defect-free PhC, the refractive phenomenon displays dramatic change. The defect lattice considered here is 5a so the incident frequency locates at the third band as shown in Fig.10(c). At this photonic band, the EFS shrinks as the frequency increase. It belongs to the neg-ative refraction region4and predicts that negative refraction takes place at 0.15 (c/a). In Fig.11(b), the Gaussian beam is incident with angle 6 into the periodic-defect PhC. The re-fractive phenomenon shows that the incident beam is deflected to the same side of the normal and the negative refraction takes place. It indicates that the PBS of the effec-tive PhC can predict the refraceffec-tive phenomenon very well, which is hard to figure out from the PBS of the periodic-defect PhC calculated by the supercell method. In this case, although the periodic defects only occupy 2pr2/25 pffiffiffi3a2of
FIG. 12. (a)The supercell with size of 14a 2pffiffiffi3a in the periodic-defect PhC. (a)The supercell with size of 14a 2pffiffiffi3a in the effective PhC. (c) The first Brillouin zone for both (a) and (b). The band structures are calculated alongM-U-X-M.
FIG. 13. (a) Defect modes exist between 0.1599 and 0.1754 (c/a) in the structure of Fig.11(a), where ea, eb, ec, andr are 2.25, 1.0, 16.0801, and
0.42a, respectively. (b) Defect modes exist between 0.1595 and 0.1749 (c/a) in the structure of Fig.11(b), where eeff, ed,r, and the defect lattice are 1.44,
16.5201, 0.42a, and 2a, respectively.
the PhC, which is about 2.56%, the refraction is very differ-ent from that of the defect-free PhC.
V. DEFECT MODES IN THE PERIODIC-DEFECT PHOTONIC CRYSTAL AND IN THE EFFECTIVE PhC
When a disorder is introduced to the PhC, defect modes would be created in the PBG region. In the PhC, if a row or a line of air-hole cylinders are removed, a photonic waveguide can be formed. Since previous results show that the periodic-defect and the effective PhCs have much similar performan-ces in the interesting frequency region, we further discuss the defect modes in both PhCs by using the supercell method, and check whether the effective PhC still works here. The PhC waveguide is created by removing a line of periodic defects. The size of the supercell is chosen as 14a 2pffiffiffi3a in the periodic-defect PhC shown in Fig.12(a).
In the periodic-defect PhC, dashed-line circles represent periodic defects with dielectric constant ec¼ 16.0801 and
solid-line circles represent air-hole cylinders, where the background dielectric constant ea¼ 2.25. Both radii of
peri-odic-defect and air-hole cylinders are 0.42a. One of those periodic defects at the center of the supercell is removed so that defect modes can be produced within the PBG region. The effective structure of Fig. 12(a)is shown in Fig. 12(b)
where periodic effective cylinders with dielectric constant ed¼ 16.5201 and radius r ¼ 0.42a are embedded in the
effec-tive medium with dielectric constant eeff¼ 1.44. The new
lat-tice constant is 2a and the ratio of radius to the latlat-tice constant is 0.21. An effective cylinder at the center of the supercell is also removed in order to match the structure in Fig. 12(a). The size of the supercell remains 14a 2pffiffiffi3a. The first Brillouin zone for both supercells is shown in Fig.
12(c). The PBSs are calculated alongM-U-X-M.
FIG. 14. (Color online) When the fre-quency is 0.17(c/a), field distributions of the defect mode (a) along the waveguide in the periodic-defect PhC and (b) along the waveguide in the effective PhC, and coefficient distributions of the defect mode (c) in the periodic-defect PhC and (d) in the effective PhC.
Because the supercells in Figs.12(a)and12(b)are very large, total 6241 plane waves are used to calculate the PBSs and defect modes. The results are shown in Figs.13(a)and
13(b), respectively. Between two dashed lines are the PBG regions. Defect modes exist between 0.1599 and 0.1754 (c/a) in the periodic-defect PhC, and those exist between 0.1595 and 0.1749 (c/a) in the effective PhC. Both defect modes are located at the PBG region. The error between two calcula-tions in Figs.13(a)and13(b)is less than 0.3%. It can be said that defect modes of the periodic-defect PhC are coincident with those of the effective PhC. Next, demonstrations of field distributions are shown to verify the point of view. When the frequency is 0.17(c/a), the field distributions along the wave-guide (UM or y direction) by using Eq.(1)are shown in Figs.
14(a)and14(b), respectively. In both cases, the defect modes are mostly confined in the lateral range of 10a. Coefficient distributions of Fourier series in these two cases are shown in Figs.14(c)and14(d). The field distribution in Fig. 14(a)
is much similar to that in Fig.14(b)because coefficient dis-tributions between two cases are also much close to each other. As a result, it exhibits again that the effective PhC is equivalent to the periodic-defect PhC in the discussions of PhC waveguides in the interesting frequency region.
VI. CONCLUSION
The concept of the effective medium is used to deal with problems of defects in PhCs throughout this paper. Scattering phenomena by defects in PhCs are an intuitive way to initially confirm this concept. It is found out that they are very similar to scattering phenomena in the effective structure composed of the effective medium and the effective defects in interesting frequency regions. Then we further dis-cuss the reason why the effective medium can substitute for a defect-free PhC. It is the factor that the Bloch wave in the defect-free PhC is mainly composed of the G*00-term
har-monic wave and all other higher-order ones can be ignored. TheG*00-term harmonic wave is the same as the wave in the
effective medium. If the Bloch wave is replaced with the G*00-term wave, the approximation is as high as 99%. So it is
a very good approximation to substitute the effective me-dium for the defect-free PhC.
Since scattering phenomena in PhCs are similar to those in effective media, it is reasonable to presume that there is an equivalence existing between the periodic-defect PhC and the effective PhC. Two fully PBGs and transmissions calcu-lated by the FDTD method verify this equivalence in the interesting frequency region. So the concept of the effective medium is successfully extended from few defects in the PhC to periodic defects in the PhC. It allows us changing the defect lattice to investigate the variance of the PBGs of the effective PhC instead of the period-defect PhC. Furthermore,
the PBS of the effective PhC with defect lattice 5a success-fully predicts the negative refraction when Gaussian beam propagates through the period-defect PhC. The difference of the FDTD calculations between the defect-free PhC and the periodic-defect PhC points out the optical performance can be dramatically changed by the periodic defects. Finally, waveguide modes in the periodic-defect PhC and the effec-tive PhC exhibit again that the effeceffec-tive PhC is equivalent to the periodic-defect PhC in the interesting frequency region.
In summary, the concept of the effective medium is im-portant for both fundamental and applied points of view. The PhC can display different effective refraction indices by varying radii of air-hole cylinders. It is similar to semicon-ductor materials which change their refraction indices by doping different concentrations of impurities. The PhC can be treated as an artificially material whose optical perform-ance different from the background medium and cylinders. So the effective representation of a PhC is much helpful to design optical devices composed of some materials not in nature.
ACKNOWLEDGMENTS
We are grateful to Dr. C. M. Kwei for offering valuable comments on this work.
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