Jpn. J. Appl. Phys. Vol. 40 (2001) pp. 5920–5924
Part 1, No. 10, October 2001c
2001
The Japan Society of Applied Physics

Interfaces of Photonic Crystals for High Efficiency Light Transmission
Toshihiko BABA and Daisuke O HSAKI
Yokohama National University, Division of Electrical and Computer Engineering, 79-5 Tokiwadai, Hodogayaku, Yokohama 240-8501, Japan(Received May 22, 2001; accepted for publication July 12, 2001)

Input and output interfaces of photonic crystals are discussed for high efficiency transmission of light. The discussion focuseson the frequency range higher than the photonic bandgap, at which a small group velocity of light, the superprism effect, thestrong structural birefringence and the negative refractive index effect are expected. The two-dimensional photonic crystalof triangular lattice airholes in a high-refractive-index dielectric medium is assumed as the fundamental model. The finite
difference time domain simulation shows that two types of interfaces improve the transmission efficiency. One is composed ofsmall airholes and the other is composed of projected airholes. In particular, the latter generates a wide transmission band of∼ 0.1 times the center frequency with the maximum efficiency of −0.01 dB against the normal or slightly inclined incidence of
the TE-polarized wave.
KEYWORDS: photonic crystal, photonic band, superprism, FDTD

1. Introduction
Photonic crystals (PCs), i.e., multidimensional periodic
structures, have attracted much attention due to their novel
concept of optics combined with solid-state physics and theircrystallography, and also due to the potential of applications
in various devices in next-era optoelectronic systems. As applications, their use in defect lasers and waveguides is studied by many groups. Since they utilize the forbidden photonic bandgap (PBG), we can call them reflection-type devices. Also, there are some reports on transmission-type devices, which utilize peculiar dispersion characteristics of allowed bands above the PBG, e.g., a narrow band filter, a dispersion compensator and an optical deflector based on the
superprism effect,1–3) nonlinear devices based on the small
group velocity,3–9) a polarization filter based on the structural birefringence,10) and an image processor based on thenegative refractive index effect.11) However, discussions havemainly focused on the functions of PCs. Quantitative discussions on light transmission efficiency are limited, although
it is one of the most fundamental specifications of these devices. There are two major losses which degrade the transmission efficiency, i.e., the diffraction loss inside the PC andthe reflection loss at input and output ends of the PC. The
former is an essential problem related to the dispersion characteristics of photonic bands.1) It is negligible when the PC
is sufficiently large, or it may be reduced by using the collimating effect with a flat equi-frequency curve in a dispersion
surface.12) On the other hand, there are no discussions on the
reduction of reflection loss except for a brief comment.13)
In this paper, we discuss the reduction in reflection loss at
the ends of a two-dimensional (2-D) PC, through the finite
difference time domain (FDTD) calculation. As a fundamental model, we employ one of the most popular structures studied by many groups, i.e., PCs composed of triangular lattice
airholes in a high-refractive-index dielectric medium. First,
we show the relationship between photonic bands and transmission spectra, and discuss the mechanism of reflection loss.
Then, we discuss two types of input and output interfaces.
One is composed of small airholes and the other is composedof projected airholes. We show that the latter is effective for
reducing the reflection loss and improving the transmission
efficiency in a meaningful wide spectral range. We show thedependence of this effect on the polarization and angle of incident waves and discuss its possibility for practical use.
2. Condition of FDTD Calculation
For all calculations in this paper, we assume 2-D models
which are infinite in the third dimension. The fundamental model of the PC and the corresponding Brillouin zone
are shown in Fig. 1. As examples, the refractive index of
the background dielectric medium is assumed to be 3.065 forthe transverse-electric (TE) polarization with the electric field
E parallel to the 2-D plane, and 2.635 for the transversemagnetic (TM) polarization with E perpendicular to the 2-D
plane. These are modal indexes of an airbridge PC slab witha material index of 3.5 and thickness of 0.32 µm at a wavelength λ of 1.55 µm.14) Recently, such PC slabs have become
very popular in studies of PC waveguides and photonic circuits, because of their easy fabrication process. By assuming
the above indexes, we expect that the calculated results in thisstudy can be used in other experiments and discussions, notonly on deep 2-D PCs but also on PC slabs. Strictly speaking,PC slabs require a full 3-D calculation. Therefore, a quanti-

Fig. 1. Schematic of 2-D PCs of triangular lattice circular airholes in adielectric medium. (a) Fundamental structure A composed of 9 rows of
airholes, (b) structure B with small airhole interfaces, and (c) structure C
with projected airhole interfaces. (d) shows the Brillouin zone of a PC.

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Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 10

X
0

10

10

20
30

50

2r/a = 0.833
0

0.2
0.4
0.6
Normalized Frequency a/

0.8

(a)
Fig. 2.

Transmission [dB]

X
0

40

5921

k

PBG

k
Transmission [dB]

T. BABA and D. O HSAKI

20
30
40
50

2r/a = 0.520
0

0.2
0.4
0.6
Normalized Frequency a/

0.8

(b)

Photonic bands and transmission spectra for TE-polarized wave in structure A: (a) 2r/a = 0.833 and (b) 0.520.

tative discussion on PC slabs is difficult with the 2-D calculation. However, the qualitative discussion is possible, since
projected 3-D bands for PC slabs15) are roughly evaluated by
2-D bands with the modal index approximation.
In FDTD calculations of transmission spectra, we first assume that the PC is infinitely wide and its front and back ends
orient toward the –J direction in the Brillouin zone. Theseassumptions allows us to eliminate edge effects in a finitewidth PC and complicated scattering at uneven ends orienting
toward the –X direction, respectively, thus simplifying the
problem. Nine unit cells are assumed to be the length of the
PC in the –X direction. Except for some calculations shownin §5, the incident wave is an infinitely wide, TE-polarizedplane wave propagating along the –X direction. To simulate this condition, the calculation area is reduced to the unit
cell in the –J direction by applying the periodic boundary
condition. In the FDTD algorithm, 60 square Yee cells divide the calculation area in this direction. The front and backends of the analysis area are terminated by Berenger’s perfectmatched layer absorbing condition. For these ends, the reflection loss of more than 140 dB is confirmed against the incident
wave. The incident wave is excited for the field perpendicular
to the 2-D plane on a horizontal line close to the front end with
the Gaussian time function. Poynting powers of incident andreflected waves and that of the transmitted wave are measured
on other horizontal lines close to the front and back ends, respectively. The transmission spectrum is evaluated from theratio of the transmitted power to the incident power after the
fast Fourier transform.

and 2(b), the frequency range exhibiting very low transmission corresponds to the PBG. Almost perfect transmission
with some Fabry–Perot resonance is observed at lower frequencies than the PBG. On the other hand, spectra are much
more complicated at higher frequencies due to the strongly
modulated photonic bands exhibiting small group velocities,
ministopbands and uncoupled modes. For larger airholes,
the transmission efficiency at these frequencies is less than
−5 dB. Figure 3(a) shows the magnetic field distribution oflight. Here, we assume the finite-width PC and the calculation area absolutely terminated by the absorbing condition inorder to gain a better understanding of the field profile inside(a)

3. Transmission Spectra and Photonic Bands
The correspondence of transmission spectra using the
FDTD method and photonic bands using the plane-wave
expansion method16) is demonstrated in Fig. 2. Here,2r/a = 0.833 and 0.520 are assumed, where 2r is the airhole diameter and a is the pitch of the triangular lattice. To
minimize the digitizing error in the FDTD algorithm, the 2r
assumed in the FDTD is optimized so that the total area of
Yee cells in one airhole is closest to the target area πr 2 assumed in the plane wave expansion method. In Figs. 2(a)(b)
Fig. 3. Distribution of TE-polarized Gaussian-pulsed wave (magnetic fieldnormal to the 2-D plane) with center normalized frequency a/λ = 0.665
in structures A (a) and C (b) with 2r/a = 0.833. For structure C, b/2r = 8
is assumed.

Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 10

4. Small Airhole Interface
In general, a simple dielectric interface between two media
with refractive indexes n 1 and n 2 is made antireflective by in√serting another medium with an intermediate index of n 1 n 2
and an odd-multiple thickness of the quarter-wavelength in
the medium. The small airhole interface has a similar effect.
A line of small airholes has an effective intermediate index
between the effective index of the PC and the background index. Figure 4(a) shows transmission spectra for a PC with theinterfaces at both front and back ends, a PC with the interface at only the front end, and the PC without the interface.
Evidently, a pair of interfaces improves the transmission efficiency. In particular, at a/λ = 0.693, the maximum efficiency
is over −1 dB. Figure 4(b) shows the dependence of the efficiency on the normalized radius of the small airholes r  /r.
Maximum efficiency is achieved at r  /r = 0.59. This optimum radius is explained by the intermediate index, as mentioned above. The effective index of the PC, which is estimated from the slope of the lowest photonic band at the low
frequency limit, is 1.57, while that of a PC composed of thesmall airholes is 2.17. The latter is equal to the intermediateindex between the effective index of the PC and the background index of 3.065. However, as observed in Fig. 4(a), thehigh transmission range is too narrow to make it applicable tosome practical devices. In addition, the spectral behavior is socomplicated that further discussion on physical phenomena inthis structure may be difficult.
5. Projected Airhole Interface
Regarding antireflective structures, various projected interfaces have been studied in the fields of radio waves andoptics. In particular, a dielectric surface with many random
projections smaller than the optical diffraction limit acts as a
wideband antireflection coating. For PCs, on the other hand,
structure C with periodic projections is more reasonable to(a)Transmission [dB]and outside the PC and of the additional edge effects at the PCedge. The Bloch wave inside the PC at these frequencies spatially oscillates in both –X and –J directions. Therefore,
the coupling of the incident plane wave to the Bloch wave
is essentially inefficient. Uncoupled waves are converted toreflected waves and scattered in various directions that satisfy the Bragg diffraction condition. For smaller airholes, thetransmission efficiency is improved due to the shallow oscillation depth of the field in the –J direction and relatively slow
transition from the incident plane wave to the Bloch wave inthe crystal. A similar improvement is expected for a lower
index background medium. However, this weakens the interaction between the PC and the incident wave, and essentially
increases the required length of the target PC device. Therefore, such simple designs are not acceptable, and some additional modification of the structure is necessary, if one aims
to achieve a small device.
Regarding modifications, we investigate two types of input
and output interfaces. One is structure B with small airholes,
as shown in Fig. 1(b). It effectively realizes the intermediate refractive index layer between the PC and the backgroundmedium. The other is structure C with projected airholes, as
shown in Fig. 1(c). It realizes the adiabatic transition of lightwaves in the gradually transformed structure.

T. BABA and D. O HSAKI

0.5

0.6

0.7

(b)
Transmission [dB]

5922

0

0.2

0.4

0.6

0.8

1.0

Fig. 4. Transmission characteristics of TE-polarized wave in PCs with andwithout small airhole interface. 2r/a = 0.833. (a) Transmission spectra.
Solid and dotted lines indicate structures B and A, respectively. The dashedline indicates the intermediate structure in which only the input end has thesmall airhole interface. For small airholes, 2r  /a = 0.435 is assumed. (b)Dependence of the maximum transmission efficiency on normalized radiusof small airholes at a/λ = 0.693.

investigate, since the transmitted wave in the PC is not a freetraveling wave but a set of Bloch waves.
The transmission spectrum is shown in Fig. 5(a), where
the projection length b normalized by the airhole diameter 2r is assumed to be 3. In a wide frequency range ofa/λ = 0.59–0.67, the transmission efficiency is increased to> −0.5 dB. Maximum efficiency reaches −0.01 dB. The antireflection effect is clearly observed through the comparisonof three cases similar to that in Fig. 4(a). The frequency range
is shifted by changing the projection length. Similarly, the efficiency at a specified frequency depends on the projection
length, as shown in Fig. 5(b). The efficiency exhibits periodic characteristics with the change in the projection length.
This can be understood from the field profile, as shown in
Fig. 3(b). The phase difference of the incident wave occurs
between inside and outside the projected airholes. When thephase shift matches that of the Bloch wave inside the PC, thetransmission efficiency achieves a maximum. This is the reason of the periodic characteristic against the projection lengthand its frequency dependence. When the normalized projection length is 0.5–1, each projection has 45◦ –60◦ boundaries.
They act as corner mirrors which significantly suppress theJpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 10

T. BABA and D. O HSAKI

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b/2r = 3
2r/a = 0.833

Transmission [dB]

Transmission [dB](a)b/2r = 3
2r/a = 0.833

(a)
0.5

0.6

0.5

0.7

0.6

(b)

0

2

4

6

8

Transmission [dB]

Transmission [dB]

(b)

10

b/2r
Fig. 5. Transmission characteristics of TE-polarized wave in PCs with andwithout projected airhole interface. 2r/a = 0.833. (a) Transmission spectra. Solid and dotted lines indicate structures C and A, respectively. The
dashed line indicates the intermediate structure in which only the input
end has the projected airhole interface. For projected airholes, b/2r = 3
is assumed. (b) Dependence of transmission efficiency on the normalized
length of projected airholes. Solid line indicates a/λ = 0.548 and dash-dot
line 0.665.

transmission. Figure 5(b) indicates that the optimum b/2r
is nearly 4. However, this is not an essential length, but one
originating from the digitized model of the FDTD calculation.
At this b/2r or its odd multiple, the reflection at small stepsof the projected airholes expressed by Yee cells is negated byeach other. When b/2r ∼ 8, the maximum transmission efficiency in Fig. 5(b) is limited to around −1 dB due to the
opposite condition. Actually, however, this length will alsoachieve a high efficiency close to 0 dB.
Thus far, TE polarization has been assumed. Let us checktransmission characteristics for TM polarization. Figure 6(a)shows transmission spectra for PCs with and without projected airhole interfaces. The spectral shapes are very different from those for TE polarization, since there are two
PBGs. Even at frequencies outside the PBGs, the improvement of the transmission efficiency by the interfaces is notevident. The dependence of the efficiency on the normalized
projection length is shown in Fig. 6(b). Maximum efficiency
is limited to −3 dB. This low value is due not to the digitized model, but to the essential reflection of the structure.
This reflection cannot easily be explained quantitatively. We

0

2

4

6

8

10

b/2r
Fig. 6. Transmission characteristics of TM-polarized wave with and without projected airhole interfaces. 2r/a = 0.833. (a) Transmission spectra.
Solid and dotted lines indicate structures C and A, respectively. For projected airholes, b/2r = 3 is assumed. (b) Dependence of transmission
efficiency on the normalized length of projected airholes. Solid line indicates a/λ = 0.501 and dash-dot line 0.628.

consider that the TM-polarized wave undergoes large Fresnel
reflection at the inclined boundaries of the projected airholes,
while the existence of the Brewstar angle reduces this reflection for the TE-polarization.
Figure 7 shows the dependence of the maximum transmission efficiency for TE polarization on the angle of the incident
wave measured from the –X direction. Here, we used a PC
model with finite width. Even when the width is as large as 50
unit cells, the incident wave suffers strong diffraction in the
–J direction by the small group velocity effect and partly
by the superprism effect. This results in an excess loss ofmore than 3 dB. To investigate the angle dependence separately from the diffraction effect, results in Fig. 7 are plottedas the change of the efficiency against the normal incidence.
Although it is not explicitly displayed in Fig. 7, the absolutevalue of the efficiency is much higher in the PC with the projected airhole interface than without the interface. However,Fig. 7 indicates that the efficiency is degraded more sensitively when the PC has the interface. Efficiency over −1 dB
is only maintained for an inclination angle of less than 3◦ .

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Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 10

T. BABA and D. O HSAKI

Change of Transmission [dB]of the TE-polarized wave.
Acknowledgements
The authors would like to thank Professor Y. Kokubun,
Yokohama National University, and Professor K. Iga andProfessor F. Koyama, Tokyo Institute of Technology,
for helpful suggestions. This work was supported by
Grant-in-Aid #10210203 from the Ministry of Education,
Culture, Sports, Science and Technology and by CREST#530-13 of Japan Science and Technology Corporation.

0

2

4

6

8

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Fig. 7. Change of transmission efficiency with inclination angle θ ofTE-polarized incident wave. 2r/a = 0.833 and a/λ = 0.650 are assumed.

6. Conclusion
The FDTD simulation shows that the small airhole interface and the projected airhole interface improve the transmission efficiency at a frequency range higher than the PBG in atriangular lattice airhole PC. In particular, the latter allows
a high transmission efficiency of a maximum of −0.01 dBwithin the frequency range of 0.1 times the center frequency.
Such efficiency is sufficiently high and such a frequency range
is sufficiently wide, when phenomena such as the small groupvelocity and superprism effect of the PC are used in actual
applications. This interface is only effective for TE polarization with an inclination angle of less than 3◦ . Still, we
believe that this interface is practicable, since most of thefunctional effects in photonic crystals at the higher frequency
range are expected for normal or slightly inclined incidence

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