Transmission, Reflection and Out-Coupling
Dependence on Duty Cycle for Second Order
Nai-Hsiang Sun1, Chia-Ming Hu1, Jiun-Jie Liau1, Jung-Sheng Chiang1, Jerome K. Butler2, Martin Achtenhagen3,
Linglin Jiang3, Vibhavie Amarasinghe3 and Gary A. Evans2
1
Department of Electrical Engineering, I-Shou University
1 Sec. 1, Syuecheng Rd., Dashu Township, Kaohsiung county, Taiwan
2
Department of Electrical Engineering, Southern Methodist University, P.O. Box 750338, Dallas, TX 75275-0338, U. S. A.
3
Photodigm, Inc., 1155 Collins Blvd., Suite 200, Richardson, TX 75081, U.S.A.
E-mail:snh@isu.edu.tw
Abstract: The dependence of reflectivity, transmission
and out-coupling of second order gratings on duty cycle are
investigated. Optimum duty cycles for maximum
out-coupling (~43%) or maximum reflection (>70%) are
determined.
2. RESULTS
We consider a second-order DBR structure a generic 780
nm laser structure consisting of a top dielectric layer with an
index of 2.01 and a thickness of 0.05 μm. The gratings are
etched into the second layer which has an index of 3.3179
and 0.215μm-thick. The grating length is 500 µm. The third
layer is p-cladding where the index is 3.3179 and a thickness
of 0.125 μm. Layers 4 and 6 are 0.1 µm thick and are graded
layers varying from an index of 3.3179 to 3.4531. Layer 5 is
a 0.008 µm thick quantum well with an index of 3.6301.
Layer 7 is the n-cladding with an index of 3.3179 and a
thickness of 1.9 µm. The GaAs substrate has an index of
3.677. For simplicity, we assume that there is no material
loss in the layers.
-6.E-4
αΛ -4.E-4
-2.E-4
0.E+0
6.290
6.295
1.889
1.888
Stop band
1.887
1.886
1. INTRODUCTION
1.885
1.884
6.270
6.275
6.280
βΛ
6.285
(a)
-5.E-3
-4.E-3
-3.E-3
6.275
6.280
αΛ
-2.E-3
-1.E-3
0.E+0
6.285
6.290
6.295
1.890
1.889
1.888
K0×Λ
The transmission, reflection, and outcoupled radiation are
important properties in the design of second-order gratings
for distributed Bragg reflector (DBR) lasers and
Grating-Outcoupled Surface-Emitting (GSE) lasers [1,2].
Second-order gratings for DBR lasers are often less
expensive or simpler to fabricate than first-order gratings,
especially for short wavelength (< 850 nm) laser diodes.
This paper studies the reflection, transmission and
out-coupling properties of second order gratings at and near
the Bragg condition and explores the relationship between
these characteristics as a function of duty cycle. A complete
second-order grating formulation is derived by using the
Floquet-Bloch Theory (FBT) [3] to numerically calculate
the dispersion relationship of corrugated waveguides.
-8.E-4
K0×Λ
Keywords: Bragg Reflectors, second order gratings
-1.E-3
1.890
1.887
1.886
1.885
1.884
1.883
1.882
6.270
βΛ
(b)
Fig. 1. The normalized wavenumber as a function of the real part and the
imaginary part of the normalized propagation constants with a duty cycle of
(a) 40% and (b) 70%.
Figures 1 (a) and (b) show the normalized wavenumber as
a function of the real and imaginary parts of the normalized
propagation constant near the second Bragg condition with
duty cycles of 40% and 70%, respectively. For a duty cycle
of 40%, the stop band region of the second Bragg in Fig. 1(a)
is not obvious and shows weak coupling. On the other hand,
Power %
100
90
Radiate to superstrate
80
Transmission
70
Radiate to substrate
60
50
40
30
20
Reflection
10
0
0.2335 0.2338 0.2341 0.2344 0.2347 0.2350
Λ(μm)
(a)
100
90
80
70
60
50
40
30
20
10
0
Power (%)
Reflection Transmission
0
20
40
60
Duty Cycle
80
100
(a)
100
90
80
70
60
50
40
30
20
10
0
Radiate to substrate
0
Power %
100
Reflection
90
Radiate to substrate
80
Transmission
70
60
50
Radiate to
40
superstrare
30
20
10
0
0.2335
0.2338
0.2341
0.2344
0.2347
Λ(μm)
power rapidly increases to 100% as the duty cycle
approaches 99%.
Radiation loss (%)
the real part of the dispersion curve for a 70% duty cycle
represents a typical stop band diagram. Moreover, the
maximum normalized attenuation constant (αΛ) for duty
cycles of 40% and 70% are -2.78×10-4 and -3.21×10-3,
respectively. The results show that the maximum attenuation
through a second-order grating with a 70% duty cycle is
over 10 times of that of a second-order grating with a 40%
duty cycle.
20
Radiate to superstrate
40
60
Duty Cycle
80
100
(b)
Fig. 3. (a) The transmission and reflection (b) the radiation power
efficiencies as a function of the duty cycle.
3. CONLCUSIONS
0.2350
(b)
Fig. 2. The reflection, transmission, and radiation efficiencies as a function
of the grating period with a duty cycle of (a)40% and (b)70%.
The percent of power reflected, transmitted and radiated
towards the superstrate (+x direction) and the substrate (-x
direction) as a function of the grating period are shown in
Fig. 2(a) and 2(b) for duty cycles of 40% and 70%,
respectively. A grating length of 500 µm is assumed in all
cases. For a grating with a duty cycle of 40%, at resonance,
the reflected power is only 14%, and the radiated power to
the superstrate and the substrate are 30% and 16%,
respectively with about 40% of the power transmitted. For a
grating with a duty cycle of 70% at resonance, the reflection
is greater than 90% with radiation losses in both directions
less than 10% and with almost 0% transmitted power.
Figure 3 shows the transmitted, reflected and radiated
power as a function of duty cycle, showing that if the duty is
43%, the reflected power has relative minimum of 12%
while the total outcoupled power is 50%. If the duty cycle is
70% or greater, the reflected power is over 90% and the
transmitted and radiated powers approach zero. The
reflected power rapidly drops to 0% and the transmitted
In this paper, we used the Floquet-Bloch theory to
calculate the spectrum of a periodic DBR waveguide with
second-order Bragg gratings. Our results show that the
transmitted, reflected and radiated power vary greatly with
duty cycle. If we want to use the gratings as a DBR reflector,
the duty cycle should be greater than 70%. If the second
order grating is designed for out-coupling, the duty cycle
should be ~ 40%.
This work was supported in part by the Texas Higher
Education Coordinating Board under Grants 003613-00412003, 003613-0019-2001 and 003613-0038-2001, and
National Science Council of the Republic of China under
Grant 96-2221-E-214-023-MY3.
REFERENCES
[1] Taha Masood et. al., “Single-frequency 1310-nm AlInGaAs-InP
grating- outcoupled surface-emitting lasers,” IEEE PTL, vol. 16,
pp.726-728, 2004.
[2] G. A. Evans, and J. M. Hammer, Surface emitting semiconductor lasers
and arrays, Academic Press, 1993.
[3] J. K. Butler, N. H. Sun, G. A. Evans, Lily Pang, and Phil Congdon,
“Grating-Assisted Coupling of Light Between Semiconductor and Glass
Waveguides,” IEEE JLT, vol. 16, pp. 1038-1048, 1998