The high-transmission photonic crystal heterostructure Y-branch waveguide operating at photonic band region

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The high-transmission photonic crystal heterostructure Y-branch waveguide operating

at photonic band region

Ting-Hang Pei and Yang-Tung Huang

Citation: Journal of Applied Physics 109, 034504 (2011); doi: 10.1063/1.3532048 View online: http://dx.doi.org/10.1063/1.3532048

View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/109/3?ver=pdfcov Published by the AIP Publishing

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Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu, 30010 Taiwan, Republic of China

共Received 11 May 2010; accepted 1 December 2010; published online 3 February 2011兲

We design a high-efficiency two-dimensional photonic crystal共PhC兲 Y-branch waveguide formed as a heterostructure with two different triangular PhCs composed of air holes. At photonic-band frequency regions corresponding to circular equifrequency surfaces, the two PhCs could be taken as effective homogeneous media with effective refractive indices. A triangular coupler composed of the PhC is designed at the input port in order to divide the incident beam into two parts. The two parts finally propagate into different channels. A case demonstrated here shows that the total transmission of light passing through the Y-branch waveguide is about 91.2%. The propagation phenomena can be explained by the mechanism of total internal reflection very well. © 2011 American Institute of

Physics.关doi:10.1063/1.3532048兴

I. INTRODUCTION

Photonic crystals共PhCs兲 are formed with dielectric pe-riodic structures and exhibit new electromagnetic phenomena.1,2 They show some properties analog to semi-conductors, such as the photonic band structures共PBSs兲 in-cluding photonic passing bands and photonic band gaps 共PBGs兲, and complicate dispersion relations. In analogous to the electron transport in semiconductors, the Bloch theorem is applied to describe electromagnetic waves propagating in the PhC very well.

Semiconductor heterostructures, such as the quantum wells3can confine electrons in the nanoscale region, combine at least two different materials. Recently, PhC heterostructures4like their semiconductor counterparts have been introduced and shown attractively optical features. Such as omnidirectional resonance modes can be generated in one-dimensional 共1D兲 periodic PhC heterostructures stacked with alternate ␮-negative and ␧-negative materials;5–7 the photonic multiple heterostructures consist-ing of different PhCs can be used to enlarge the nontransmis-sion frequency range of PhCs.8Moreover, the confinement of the photonic envelope wave function in a two-dimensional photonic heterostructure quantum well implemented with quasiperiodic array of vertical-cavity surface emitting lasers as a model system has been discussed.9It is possible to con-trol the characteristics of the photonic envelope wave func-tions, and control photonic states in quasiperiodic media.

The notion of localized states formed at PhC impurities or defects1 has been utilized to guide light in waveguides.10–13 Light could be confined and propagate in the waveguide efficiently even through sharp bends. The fre-quency of the propagation mode is chosen at the PBG region. Recently, a PhC heterostructure waveguide with PhC core surrounded by a uniform low-index material has been analyzed.14 In that paper, the structure is similar to the con-ventional dielectric-slab waveguide, in which the mechanism

of guiding light is the total internal reflection at the crystal-material interface. Each mode in this waveguide needs to satisfy the transverse resonance condition because of the ex-tra ex-transverse phase shift.15The frequency of light in the PhC defects waveguide needs to be operated within the PBG but the PhC heterostructure waveguide has no such limitation. Besides, the advantage of the PhC heterostructure waveguide is the core wide enough for efficiently coupling to conven-tional waveguides.

The PBS shows different optical responses when light propagates inside the PhC at different frequencies. The PhC can be taken as an effective anisotropic material that presents birefringence effect. It can also be taken as an effective ho-mogeneous material with an effective dielectric constant un-der certain condition.16In this paper, we propose a PhC het-erostructure waveguide combined two different PhCs possessing different effective dielectric constants. The PhC heterostructure waveguide is formed with a PhC core and PhC claddings. We not only show that the PhC heterostruc-ture Y-branch waveguide is an optical high-transmission de-vice but also show that the total internal reflection takes place at the interface of two PhCs. Especially, the operating frequency belongs to the photonic passing band. All light propagations in this paper are demonstrated by using the finite-difference time-domain共FDTD兲 method.17

II. CONCEPT AND DESIGN OF PHC

HETEROSTRUCTURE Y-BRANCH WAVEGUIDE A. Structure of the PhC heterostructure waveguide

We consider a two-dimensional PhC heterostructure Y-branch waveguide lying in the x-y plane as shown in Fig.

1. It is composed of air holes embedded into the low-index dielectric material in a triangular array with a lattice constant

a. A triangular coupler made of the PhC, whose composition

is the same as the core, is designed at the input port. It is utilized to couple light from the outside region into the PhC heterostructure waveguide. The triangular coupler used here is equivalent to a traditional prism composed of a dielectric

a兲_{Electronic mail: [email protected].}

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material and each angle of it is 60°. The triangular coupler is a flexible design, whose effective refractive index can be varied with the radius of air holes. The triangular coupler can divide the incident Gaussian wave into two parts. The two divided waves go toward interface 1 and interface 2共dotted lines兲 as shown in Fig. 1. Except for the triangular coupler, the whole structure is divided into three regions A, B, and C. Light meets different interfaces when propagates within A, B, and C regions. Light meets the interface along the ⌫M direction in the A and C regions and meets the interface along the ⌫⌲ direction in the B region, respectively. The effective refractive index is chosen to be almost fixed for all propagation direction at the operating frequency. Light propagating inside the PhC heterostructure Y-branch wave-guide is designed to obey the mechanism of the total internal reflection. So it needs to calculate the relation between nd and ncsatisfying the total internal reflection in all regions, where nd and nc are the effective refractive indices for the claddings and core, respectively. In Fig. 1, the possible propagation processes satisfying the total internal reflections in all regions are simply shown by the ray traces with dotted-line arrows. In Fig. 2, the curve represents the limit of the

relation between ndand nc. Values under the curve satisfy the condition of total internal reflection in all regions.

B. Effective refraction index

The refractive angle of a light beam from one material into the PhC is determined by the equifrequency surfaces 共EFSs兲 of light in the material and in the PhC. Each EFS corresponds to certain frequency. The group velocity is nor-mal to the EFS at a certain wave vector and defined as v៝g =ⵜk៝␻ in which k៝ and ␻ are wave vector and frequency, respectively. Because the propagation direction is parallel to the group velocity, we can determine it in the PhC once k៝is given. 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 with frequency and incident angle will determine the refractive wave vector and the refractive angle. By analyzing the EFS, we can find out the operating frequency at which optical response of the PhC is like that of a homogeneous material.

In our design as shown in Fig.1, the background refrac-tive indices of dielectric materials in the core and claddings are both 1.50. The radius r₁of air holes in the core is 0.12a and the radius r₂ of air holes in the claddings is 0.42a. The width of the core in the A region of the PhC heterostructure waveguide is d1 and its length along the⌫M direction is L.

The distance between two channels is d3in the C region. The

width of each channel d2is equal to half d1. From the

calcu-lation of EFSs in the first band, the shapes of EFSs are round the same as shapes of EFSs in homogeneous materials as shown in Fig. 3共a兲. When light is incident from air to the interface along the⌫⌲ direction or the ⌫M direction, we can define an effective refractive index Nefffrom each EFS with

the corresponding frequency as shown in Fig. 3共b兲. The up-per and lower curves are the Neff-frequency relations for r1

and r2, respectively. The radius of each EFS increases as the

operating frequency increase, so each Neffis positive defined FIG. 1. The schematic of the PhC heterostructure Y-branch waveguide with a triangular coupler. The two dotted lines represent two interfaces and the extended rectangular region at the input represents the triangular coupler region. The arrows with dotted line denote the propagation direction when light propagates in the Y-branch waveguide.

FIG. 2. The curve represents the limit of the relation between ndand nc. The values under the curve satisfy the total internal reflection in all A, B, and C regions.

034504-2 T.-H. Pei and Y.-T. Huang J. Appl. Phys. 109, 034504共2011兲

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here. The operating frequency is chosen as 0.30共c/a兲. At this frequency, the Neff共nc兲 for the core is 1.478 and the Neff共nd兲 for claddings is 1.205. Both N_effvalues are below the curve in Fig.2and simultaneously satisfy the condition of the total internal reflection. In Sec. III, we calculate transmissions of the PhC heterostructure Y-branch waveguide and show it is a high-transmission device.

C. The approximate wave in the PhC

The propagation wave in the PhC is the Bloch wave which satisfies the periodic condition. However, the round EFS also implies that the Bloch wave should be very close to the propagation wave in the effective medium with an effec-tive refraceffec-tive index Neff. In the following, we calculate the

Bloch wave in the PhC by using the plane-wave expansion method in order to verify the justification of the effective refractive index. The Bloch wave can be expressed as sum of all the Fourier series18

Ez,k៝n共r៝兲 = eik៝·r៝

兺

G៝

Ez,k៝n共G៝兲eiG

៝_·r_៝

, 共1兲

where k៝ is the wave vector in the first Brillouin zone, n is a band index, and G៝= G៝pq= pG៝1+ qG៝2 with integers p and q.

The elementary lattice vectors are a៝1=共a,0兲 and a៝2

=共a/2,

冑

3a/2兲, and the elementary reciprocal lattice vectors in the k-space are G៝₁= 2␲/a共1,−1/

冑

3兲 and G៝₂ = 2␲/a共0,2/

冑

3兲, respectively. According to PBSs in Figs.

4共a兲and4共b兲, where the first Brillouin zone is shown in Fig.

4共c兲, wave numbers of k៝ in Eq. 共1兲 are chosen as k₁ = 0.4435共2␲/a兲 and k2= 0.3627共2␲/a兲, respectively. These

choices correspond to the frequency 0.30共c/a兲. Furthermore, wave number k can be expressed as 2␲N_eff/a where N_effare 1.478 and 1.205 for r₁and r₂, respectively. When TM waves with frequency 0.30共c/a兲 propagate along the ⌫M direction in PhCs with radii of r₁ and r₂, the real part of Ezfields are, respectively, shown in Figs. 5共a兲and5共b兲. These fields can be approximately described by Eq.共1兲 in which several

co--5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 Γ Μ Κ Vg 0.24 (c/a)

(a)

(b)

FIG. 3. 共a兲 The EFSs for the radius r1of air holes. The frequencies of the inner and the outer circles are 0.24共c/a兲 and 0.34共c/a兲, respectively. 共b兲 The

Neff-frequency relations in共a兲 for r1and r2, respectively.

Μ Γ Κ Μ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Wavevector Fre qu en cy (c/a)

(a)

(b)

(c)

Μ Γ Κ Μ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Wavevector Fr eque nc y (c/a)

FIG. 4. 共a兲 The PBS when the radius r is 0.12a. 共b兲 The PBS when the radius r is 0.42a. 共c兲 The first Brillouin zone of the triangular PhC.

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efficients more than 0.01 are included. The maximum one is the G៝00term, which are 0.999 and 0.9934 for r1and r2cases,

respectively. All higher ones are very small and less than 0.01 in the r1 case. In the case of r2, the other coefficients

more than 0.01 are G៝−1−1, G៝0−1, and G៝10 terms which are

⫺0.0298, ⫺0.0160, and ⫺0.160, respectively. So the waves ␾_PhC共1兲 _and_␾

PhC

共2兲 _{in these two PhCs can be approximately}

ex-pressed as ␾PhC共1兲共x,y兲 ⬇ Ae ik៝₁·r៝_{⫻ 0.999e}iG៝₀₀·r៝_⬇_␾ eff 共1兲 ⫻共x,y兲, 共r1 case兲, 共2兲 ␾PhC共2兲共x,y兲 ⬇ Ae

ik៝₂·r៝_{⫻ 0.9934e}iG៝₀₀·r៝_{− 0.0298e}−iG៝0−1·r៝

+ 0.0160共e−iG៝−1−1·r៝_{+ e}−iG៝10·r៝兲 ⬇␾

eff 共2兲_共x,y兲

⫻ 关0.9934 − 0.0298e−i共4␲/冑3a兲y

+ 0.0160共e−i共2␲/a兲共x+y/冑3兲

+ ei共2␲/a兲共x−y/冑3兲_{兲兴, 共r}

2 case兲, 共3兲

where A is the constant amplitude. Because coefficients of

G៝₀₀ term in these two cases are at least 33 times larger than those of all higher ones, Eqs. 共2兲 and 共3兲 can be further simplified by dropping all higher terms. Finally, waves in these two PhCs are very close to those in two effective ho-mogeneous media, that is, ␾_eff共1兲共x,y兲=Aeik៝1·r៝ _and ␾

eff 共2兲_共x,y兲

= Aeik៝2·r៝. It means that these two PhCs at frequency 0.30共c/a兲

can be much appropriately replaced with two effective ho-mogeneous media of Neff= 1.478 and Neff= 1.205 as well as

optical responses.

When TM waves with frequency 0.30共c/a兲 propagate along the ⌫⌲ direction in PhCs with radii of r1 and r2, Ez fields can be, respectively, approximated to

␾PhC共1兲共x,y兲 ⬇ Ae

ik៝₁·r៝_{⫻ 0.999e}iG៝₀₀·r៝_{− 0.0118共e}−iG៝−1−1·r៝

+ e−iG៝−10·r៝兲 ⬇␾ eff 共1兲_{共x,y兲 ⫻ 关0.999 − 0.0118} ⫻ 共e−i共2␲/a兲共x+y/冑3兲 + e−i共2␲/a兲共x−y/冑3兲兲兴, 共r1 case兲, 共4兲 ␾PhC共2兲共x,y兲 ⬇ Ae

ik៝₂·r៝_{⫻ 0.9934e}iG៝₀₀·r៝_{− 0.0241共e}−iG៝−1−1·r៝

+ e−iG៝−10·r៝兲 ⬇␾

eff

共2兲_{共x,y兲 ⫻ 关0.999 − 0.0241}

⫻ 共e−i共2␲/a兲共x+y/冑3兲

+ e−i共2␲/a兲共x−y/冑3兲兲兴, 共r2 case兲. 共5兲

Equations 共4兲 and 共5兲 can be also further simplified by the same reason mentioned above. Waves propagating along⌫⌲ direction in these two PhCs can also be treated as plane waves propagating in two effective homogeneous media of

Neff= 1.478 and Neff= 1.205. So the effective refraction index

can perform the optical performance of the PhC very well.

III. TRANSMISSIONS OF THE PHC

HETEROSTRUCTURE Y-BRANCH WAVEGUIDE

In order to investigate the transport phenomenon, the FDTD method is used to simulate light incident from the triangular coupler into the PhC heterostructure Y-branch waveguide. A TM-mode Gaussian beam with the electric field perpendicular to the x-y plane is lunched into the de-vice. L and d3are chosen as 80

冑

3a and 54a, respectively. In

our design, d2is equal to half d1. Using these design

param-eters, we calculate transmissions of different d1values from

14a to 30a as shown in Fig.6共a兲. The best transmission from the A region to the C region is about 45.6% at each output channel when d1 is 28a and d2 is 14a. In advanced, we

calculate the transmission of each output channel at different lengths of the A region from 50

冑

3a to 320

冑

3a when d1 is

28a and d2is 14a. The results are recorded in Fig. 6共b兲. We

find the minimal transmission case takes place at 190

冑

3a. The second maximum transmission case takes place at 300

冑

3a. The period of the transmission between two succes-sive maximum peaks is 220

冑

3a in our calculations.

Ezfield of the best transmission case is demonstrated in Fig. 7. It can be found that the triangular coupler divides incident beam into two parts at the input and then propagate toward two interfaces where total internal reflections take place. It is noticeable that the steady interface mode does not exist at interfaces 1 and 2, so no energy transfers from core region to the interface and energy is almost kept inside the waveguide. In the B region, total internal reflections take place explicitly that light reflects twice from the interfaces. When light enters into the C region, light propagates for-wardly and is confined in the waveguide very well.

(a) (b)

FIG. 5.共Color online兲共a兲 Ezfield in the PhC the radius r is 0.12a.共b兲 Ezfield in the PhC the radius r is 0.42a.

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A demonstration of the FDTD simulation at first minimal transmission is shown in Fig. 8. A lot of light leaking out from the B region results in energy loss and simultaneously decreases the transmission from the A region to the C region. Besides, total internal reflections are not satisfied in the C region in which some energy is also lost. The transmission from the A region to the C region is about 27.3% at each output channel. The transmission decreases 18.3% at each channel when the length increases from 80

冑

3a to 190

冑

3a.

The second maximum transmission case at 300

冑

3a is demonstrated in Fig.9. Even light in the A region propagates through a distance almost four times longer than that of the first maximal transmission case, it is still confined in the waveguide very well. The transmission is close to 45%. Only 1.2% energy is lost when the length of A region increases from 80

冑

3a to 300

冑

3a.

Finally, we investigate another high-efficiently transmis-sion case by broadening the distance d₃. The length in the A region is still held at 80

冑

3a and the widths d₁ and d₂ are fixed at 28a and 14a, respectively. d3 is chosen twice larger

than that in Fig. 7, that is, 108a. The field distribution is shown in Fig.10. In the B region, the multiple total internal reflections take place explicitly. From FDTD calculation, the transmission is still close to 45%. Only 1.2% energy is lost when the distance between two channels is broadened from 54a to 108a.

IV. COMPARISON OF THE GUIDING MECHANISMS BETWEEN THE PHC DEFECT WAVEGUIDE

AND THE PHC HETEROSTRUCTURE WAVEGUIDE A. PBG guiding

A lot of researches about PhC defect waveguides pay much attention on defect modes. It is well-known that fre-quencies of defect modes are within PBGs. Often, the PhC defect waveguide is created by removing or adding some specific pattern of air holes or rods which are used to guide light from one place to another. Generally speaking, most of electromagnetic field is localized in the core region and at-tenuates exponentially when enters into the cladding region formed by the PhC because the defect mode has a complex wave vector in it.

The PhC heterostructure waveguide use the mechanism of total internal reflection to transmit energy in the wave-guide. If the incident angle at the interface between the core and claddings is smaller than the critical angle, energy will leak out from the core into claddings. On the contrary, en-ergy transmitted in the PhC defect waveguide has nothing to do with the critical angle. It only depends on frequency of light whether or not locates within the PBG.

Jamois et al.19 has given a review of the properties of silicon-based two-dimensional PhCs. The dielectric constant of silicon they used is 11.6. They have shown the guiding of TE mode in the PhC defect waveguide. Here we use the same structure to demonstrate an example of the guiding of TM mode in it. The structure of it formed by a triangular lattice of air holes where a row of air holes are removed along the ⌫M direction is shown in Fig. 11共a兲. The lattice constant is a and the radius of all air holes is 0.43a. The supercell method is used to calculate the PBS and defect

FIG. 7. 共Color online兲 Ezfield in the PhC heterostructure Y-branch wave-guide when L, d1, d2, and d3are 80冑3a, 28a, 14a, and 54a, respectively.

FIG. 8. 共Color online兲 Ezfield in the PhC heterostructure Y-branch wave-guide where L, d1, d2, and d3are 190冑3a, 28a, 14a, and 54a, respectively.

FIG. 6. The transmission of light incident from air into the PhC heterostructure Y-branch waveguide共a兲 when L is fixed at 80冑3a and d1is changed from 14a

to 30a,共b兲 when d1is fixed at 28a and L is changed from 50冑3a to 320冑3a.

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modes, where the supercell denoted as a rectangular region with area a⫻

冑

3a is shown in Fig.11共a兲. The first Brillouin zone in the k-space is shown in Fig. 11共b兲. Calculations of the PBS and defect modes along M-⌫-X-M are shown in Fig.

12共a兲. Ezfield of a defect mode is shown in Fig.12共b兲. Fre-quency of this defect mode corresponding to the star symbol in the PBG is 0.40共c/a兲. It can be investigated that most of

Ezfield is confined in the core region. The distance of trans-verse distribution is less than one air-hole width in the clad-ding region. It is not the case that total internal reflection takes place at the interface between a high-index medium and a low-index one. The defect mode propagates in the waveguide without considering the critical angle because it is meaningless here.

B. Index guiding

We have mentioned that the guiding mechanism in the PhC heterostructure waveguide is index guiding. In Sec II B, effective refraction indices are calculated from EFSs and used to replace with optical responses of the PhCs. It has been proved in Sec. II C that Bloch waves in the PhCs can approximate to waves in effective homogeneous media with effective refraction indices. From the viewpoint of effective refraction index, it is the case that the core of the PhC het-erostructure waveguide is made of a high-index medium and the cladding of it is made of a low-index medium. Unlike the frequency of the defect mode within the PBG, the operating frequency of the wave in the PhC heterostructure waveguide is within the photonic conducting band where the wave vec-tor of light is real in the PhC and light can propagate in it. So the mechanism of PBG guiding cannot use to explain the wave propagation in the PhC heterostructure waveguide.

In the following, we calculate the propagation directions of energy flow in one arm region enclosed by dashed-line rectangle in Fig. 10. It has been showed that the group ve-FIG. 9. 共Color online兲 Ezfield in the PhC heterostructure Y-branch

wave-guide where L, d1, d2, and d3are 300冑3a, 28a, 14a, and 54a, respectively.

FIG. 10. 共Color online兲 Ezfield in the PhC heterostructure Y-branch wave-guide where L, d1, d2, and d3are 80冑3a, 28a, 14a, and 108a, respectively.

a

(a) (b)

x y

FIG. 11. 共a兲 The supercell with size of a⫻3冑3a in the PhC removed a row of air hole.共b兲 The first Brillouin zone in the k-space. The band structures are calculated along M-⌫-X-M.

-Μ Γ Χ Μ 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 Wavevector Fr eq ue nc y (c/a)

(a)

(b)

x

y

FIG. 12.共Color online兲共a兲 Defect modes exist between 0.387 and 0.410共c/a兲 in the structure of Fig.11共a兲, where a triangular array of air holes are embedded in the material with dielectric constant 11.6, and the radius r of all air holes is 0.43a. The two horizontal lines denote the PBG edges.共a兲 Ezfield is drawn along the PhC waveguide when the frequency of the defect mode is 0.40共c/a兲.

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locity v៝g is equal to the averaged energy velocity 具v៝e典 in a unit cell, that is,具v៝e典=v៝g=ⵜk៝␻.20具v៝e典 is defined as

具v៝e典 = 1 S兰P៝k៝dr៝储 1 S兰Uk៝dr៝储 ⬅ 具P៝k៝典 具Uk៝典 , 共6兲

where S is the area of a unit cell and the integral range, P៝k៝is the time-averaged Poynting vector, and Uk៝is time-averaged energy density. The energy direction of each unit cell in this rectangle region is shown in Fig.13. The green arrow orien-tation means the direction of averaged energy velocity in each unit cell. The length of each arrow also represents the magnitude of energy flow in each unit cell. The large black arrows show the trend of energy flow from region A to re-gion C in the waveguide. Most of transmitted energy is con-fined in the core and total internal reflection of light takes place four times at the interface between the core and clad-ding. If we extend or reduce the length of region A, total internal reflection may not satisfy that some energy will leak out from the core into the cladding. It is obvious that some energy leaks out when length of region A increases from 80

冑

3a to 190

冑

3a. In summary, the guiding mechanism of the wave propagation in the PhC heterostructure waveguide can be explained very well by Snell’s law.

C. Transverse phase shift

From the ray-optic point of view,21 one needs only Snell’s law of total internal reflection to explain the propa-gating of light in a waveguide like our effective structure. Each mode in the waveguide can be derived by using geo-metric optics due to wave propagation in it can be repre-sented by two plane waves. One of the plane waves may be considered as the incident wave, and the other may be treated as reflected one. They move in zigzag paths like waves propagating in the B region as shown in Fig. 10. However, the total internal reflection is only a necessary condition which is not sure whether all waves trapped in the core re-gion constitute a mode. In order to establish a guided mode,

tion can thus be represented by plane waves traveling at an angle␪. The wave number in the core region is k0ncwhere k0

is the wave number in vacuum. Then we obtain relations ␤ = k0ncsin␪ and h = k0nccos␪. According to the condition known as transverse resonance,22 the net phase shift mea-sured during a round trip, where a zigzag path forms from one interface to the other one and back again to original interface, must be a multiple of 2␲. This leads to the condi-tion,

2ht − 2␾= 2m␲, 共7兲

where t is the thickness of the core region, m is the mode number, and␾is the phase shift upon total internal reflection at one of the interfaces. In our case, the phase shift␾can be expressed as ␾= 2 tan−1共nc 2 sin2␪− nd 2_兲1/2 nccos␪ , 共8兲

where nc and the nd are the effective refraction indices of core and claddings, respectively. We use the guided wave inside the B region in Fig.10to verify the condition of Eq.

共7兲. The averaged propagation direction␪ between two suc-cessive reflections shown in Fig.10is 60.4°. The thickness t of the core region is 7

冑

3a so the ht is 5.311␲. Substituting␪ into Eq.共8兲,␾= 0.350␲can be obtained. Next, ht and␾are both substituted into Eq. 共7兲, we obtain 2ht − 2␾= 9.92␲ which is very close to 10␲. It figures out that the mode number m is 5. So the condition transverse resonance is sat-isfied in the heterostructure PhC waveguide.

V. CONCLUSION

In conclusion, in the case of our design, the operating frequency in the PhC heterostructure Y-branch waveguide is not at the PBG region but belongs to the photonic band re-gion. The mechanisms of PBG guiding and index guiding are different and discussed in detail. The guided mechanism of the PhC heterostructure waveguide can be explained by total internal reflection in the waveguide. Even the two PhCs are grating structures, light almost obeys total internal reflection at each interface. Generally speaking for dielectric PhCs, the frequency region of photonic bands belonging to our case is often much larger than that of PBGs. So the usable frequency regions for our device are more than that for the PhC defect waveguide guiding light at the PBG regions. We also have more choices for designing a PhC heterostructure waveguide due to flexible values of the nd and nc. Finally, the results also show the fine way to use PhCs as effective media cor-responding to circular EFSs, which will enrich the optical device design in the future.

ACKNOWLEDGMENTS

We are grateful to Dr. C. M. Kwei for offering valuable comments on this work.

FIG. 13. 共Color online兲 Directions of energy flow in one arm region en-closed by dashed-line rectangle in Fig.10.

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