PIERS_2007 Beijing
Dispersion Relationships of AlGaInAs-InP DBR Gratings Using
Floquet-Bloch Theory
Jiun-Jie Liau 1, Jia-Huei Lin 1, Min-Xiu Jiang 1, Nai-Hsiang Sun 1, Jerome K. Butler 2, and Gary A. Evans 2
1
Department of Electrical Engineering, I-Shou University, Kaohsiung, Taiwan, R.O.C.
2
Department of Electrical Engineering, Southern Methodist University, Dallas, TX 75275-0338, U.S.A.
Abstract- Floquet-Bloch theory is used to calculate the dispersion relationship and the
spectrum of first order contra-directional coupling in the AlGaInAs system. A structure
of DBR gratings is calculated in this paper. Results from the Floquet-Bloch theory show
the region of stopband. The maximum reflection efficiency of the structure is obtained.
I. INTRODUCTION
To date, the AlxGayIn1-x-yAs system is widely applied to long-wavelength semiconductor lasers due
to its lager conduction band offset compared with that of the conventional GaxIn1-xAsyP1-y system.
This material system results in better high-temperature performance; for example, it reduces carrier
leakage from the quantum-well region under high temperature operation [1]. Recently, the
AlxGayIn1-x-yAs system is applied to grating-outcoupled surface emitting (GSE) lasers with single
wavelength of 1310 nm [2]. The geometry of a GSE laser consists of a ridge waveguide, a grating
outcoupler and a first order distributed Bragg reflector (DBR). The first-order DBR gratings provide
wavelength selective feedback to the laser cavity.
To design the first order DBR gratings, we need to calculate the dispersion relationship of
AlGaInAs-InP waveguides. The Floquet-Bloch theory provides rigorous solutions with high accuracy for
the structure of corrugated waveguides. In this paper, the Floquet-Bloch theory (FBT) is used to calculate the
dispersion curves of the first Bragg gratings. Moreover, by using the Mahmoud-Beal’s method and the FBT,
the reflection and transmission characteristics of first-order DBR gratings can be obtained [3]. This study is
helpful for us to analyze and design first-order gratings for the AlxGayIn1-x-yAs system.
II. PROBLEM FORMUATION AND RESULTS
Assume that a wave with time dependence of the form exp(jωt) propagates in the z direction. By
applying the Floquet-Bloch theory, the forward propagated and backward propagated electric fields for
TE mode are expressed as
E y ( x, z ) 
E yb ( x, z ) 

f
n  
n
( x)  exp(  jk kz z )  E y f  E y b
(1)
n
( x)  exp( jk kz z )  E yb f  E yb b ,
(2)

f
n  
respectively, where k zn   n  j  (   nK )  j is the complex propagation constant of the n-th spatial
harmonic, K=2/Λ is the grating wavenumber, Λ is the grating period, and n is the space harmonic order. Note
that the electric fields in (1) and (2) consist of forward harmonics ( E yf and E yfb ) and backward harmonics
b
( E yb and E yb
). The electric fields should satisfy the scalar Helmholtz equation in each layer.
Consider a temperature insensitive AlGaInAs/InP epitaxial laser structure. Figure 1 shows the total electric
field distributions for TE mode. The free space wavelength is 1310 nm. The refractive index profile is also
shown in Fig. 1. For the grating structure, the duty cycle is 50%, and the tooth height is 0.1 µm.
3.7
7.E+05
Ey
Index
6.E+05
3.6
5.E+05
4.E+05
3.4
3.E+05
Index n
Field Ey
3.5
3.3
Gratings
2.E+05
3.2
1.E+05
3.1
1.00
0.E+00
-2.50
-2.00
-1.50
-1.00
-0.50
Thickness x(mm)
Thickness
x (μm)
0.00
0.50
Fig. 1 The electric field distribution for TE mode. The index profile is also shown.
Figure 2 shows the normalized wavenumber as a function of the normalized propagation constant.
The grating period is 0.20175 µm. The first Bragg stopband occurs near the βΛ=π region, while the
normalized attenuation constant (αΛ) becomes negative. Since there is no radiation and the dielectric
material is lossless, the complex propagation constant indicates the occurrence of contra-directional
coupling. The range of the stopband is from the wavelength of 1309.82 nm to the wavelength of
1310.02 nm. The strongest coupling occurs at the middle of the stopband with the wavelength of
1309.92 µm. The maximum normalized attenuation constant (αΛ) is -2.665×10-4. Note that the
attenuation is zero if the wavelength is not in the region of stopband.
0.9680
0.9680
0.9679
Grating height=0.1µm
Duty cycle=0.5
0.9679
0.9678
k 0L
k 0L
0.9678
0.9677
0.9677
0.9676
0.9676
0.9675
0.9675
0.9674
3.1405
Grating height=0.1µm
Duty cycle=0.5
3.1408
3.1410
3.1413
3.1415
L
3.1418
3.1420
3.1423
3.1425
0.9674
-3.0E-04
-2.0E-04
-1.0E-04
0.0E+00
1.0E-04
L
Fig. 2 The dispersion curves around the first Bragg resonance. (a) The real part and (b) the imaginary
part of the normalized propagation constants.
PIERS_2007 Beijing
The reflection and transmission spectrum are shown in Fig. 3. The grating length is 500µm.
Figure 3 shows that the maximum resonance occurs at the wavelength of 1309.92 nm. When the
strongest coupling occurs, 31.82% of the power can be reflected, while 68.18% of the power is
transmitted through the grating. Note that the resonant region is from the wavelength of 1309.40 nm
to the wavelength of 1310.45 nm. The range is slightly larger than that from the dispersion
relationship (Fig. 2). As shown in Fig. 1, the power of the fundamental mode in the grating region is
small, which yields less coupling strength and reduces the reflection efficiencies.
100
Power
T(%) (%)
80
Grating height=0.1µm
Duty cycle=0.5
100
90
80
70
70
Transmission
60
60
50
50
40
40
30
30
Reflection
20
20
10
0
1.3080
R(%)
90
T
R
10
1.3085
1.3090
1.3095
1.3100
1.3105
Wavelength l(mm)
1.3110
1.3115
0
1.3120
Fig. 3 The reflection and transmission power efficiencies as a function of the wavelength.
III. CONCLUSION
In this paper we used the Floquet-Bloch theory and the Mahmoud-Beal’s method to calculate the
dispersion relationships and the spectrum of a periodic dielectric waveguide for the AlxGayIn1-x-yAs
systems. The dispersion curves of a DBR laser structure are analyzed. For the structure of Fig. 1,
the maximum reflection power of 31.82% is obtained at the wavelength of 1309.92 with the grating
period of 0.20175 µm.
REFERENCES
1. S. R. Selmic, T. M. Chou, J. P. Sih, J. B. Kirk, A. Mantie, J. K. Butler D. Bour, and G. A. Evans, “Design
and characterization of 1.3-um AlGaInAs-InP Multiple quantum well lasers,” IEEE Journal on Selected
Topics in Quantum Electronics, Vol. 7, 340-349, 2001.
2. Taha Masood, Steve Patterson, Nuditha V. Amarasinghe, Scott McWilliams, Duy Phan, Darren Lee, Zuhair
A. Hilali, Xiong Zhang, Gary A. Evans, and Jerome K. Butler, “Single-frequency 1310-nm AlInGaAs-InP
grating- outcoupled surface-emitting lasers,” IEEE Photon. Technol. Lett. Vol. 16, 726-728, 2004.
3. Chih-Cheng Chou, Nai-Hsiang Sun, Jerome K. Butler, and Gary A. Evans “Radiation loss of grating-assisted
directional couplers using the Floquet-Bloch theory,” Technical Digest CLEO/PR 2003, 350, Taipei, Taiwan,
2003.
1. Title of the paper: Dispersion Relationships of AlGaInAs-InP DBR Gratings Using
Floquet-Bloch Theory
2. Name, affiliation and email of each author:
Jiun-Jie Liau 1, Jia-Huei Lin 1, Min-Xiu Jiang 1, Nai-Hsiang Sun 1,
1
E-mail: snh@isu.edu.tw
Department of Electrical Engineering, I-Shou University
1 Sec. 1, Hsueh-Cheng Rd., Ta-Hsu Hsiang, Kaohsiung, Taiwan, R.O.C.
Tel. 886-7-6577711 ext. 6620, Fax. 886-7-6577205,
Jerome K. Butler 2, and Gary A. Evans 2
2
E-mail: jkb@smu.edu
Department of Electrical Engineering, Southern Methodist University
3145 Dyer Street, Dallas, TX 75275-0338, U. S. A.
Tel: 1-214-7684367
3. Corresponding author: Nai-Hsiang Sun
Presenting author: Jiun-Jie Liau
4. Topic or Session Organizer: 25 Optics and Photonics
Section Information: Large-Scale Passive Optical Waveguide Devices, Design and Simulation:
Hung-Wen Chang
5. Presentation is preferred