Theoretical Computation of Optical Bandwidth of Photonic
Multiple Quantum Well using Semiconductor Heterostructure
Avisek Maity
Usmita Banerjee
RCC Institute of Information Technology
Canal South Road, Beliaghata
Kolkata, INDIA-700015
RCC Institute of Information Technology
Canal South Road, Beliaghata
Kolkata, INDIA-700015
avisek.m@gmail.com
usmitabanerjee@yahoo.in
Barnisa Chottopadhyay
Arpan Deyasi
RCC Institute of Information Technology
Canal South Road, Beliaghata
Kolkata, INDIA-700015
RCC Institute of Information Technology
Canal South Road, Beliaghata
Kolkata, INDIA-700015
barnisachottopadhyay@gmail.com
deyasi_arpan@yahoo.co.in
ABSTRACT
Transmittivity of AlxGa1-xN/GaN based photonic multiple
quantum well is numerically computed for normal and oblique
incidences of electromagnetic wave with p-polarization
condition. Transfer matrix technique is used to observe the
shift of optical filter bandwidth centered around 1.55 μm by
varying structural parameters. AlxGa1-xN and GaN layers are
considered as barriers and well respectively, and refractive
index of AlxGa1-xN is considered as a function of mole
fraction and operating wavelength following Adachi’s model.
Results are compared with conventional SiO2/air photonic
crystal with similar size and layers. Increasing no. of layers
makes it more effective filter by increasing reflectivity outside
the desired region of interest. Material composition has a
profound effect on the filter performance. Study reveals that
efficient bandwidth tuning can be made for semiconductor
heterostructure based device.
Keywords
Photonic Multiple Quantum Well, Optical Bandpass Filter,
Transfer Matrix Technique, Semiconductor Heterostructure,
P-polarized Wave
INTRODUCTION
Photonic multiple quantum well (PMQW) is the multilayer
periodic arrangement of dielectric/semiconducting materials
[1-2] with alternating regions of higher and lower potential
magnitudes where localization of electromagnetic wave can
be obtained by tuning the structural parameters of the device.
Alternatively, it can be treated as a photonic crystal with
periodic modulation of refractive indices helps to form
electromagnetic bandgap [3]. This property can be utilized to
design novel optical bandpass filter [4] by restricting e.m
wave of certain wavelengths and simultaneously allowing
other spectra; possible due to the formation of photonic
bandgap, may be exhibited in one, two or three dimensions.
The simplest 1D structure has already revolutionized optical
communication [5-8], integrated photonics [9], sensing [10],
quantum information science [11]. Electronic property of
PMQW structure is analyzed in recent past [12] though
optical properties and its possible integration in photonic
circuit is not well-studied as far the knowledge of authors.
Rudziński [13] analyzed electromagnetic wave propagation
inside 1D photonic crystal for both TE and TM mode using
transfer matrix technique. Jiang [12] calculated the
transmission property due to resonant tunneling between the
quantized states, experimentally confirmed by Xu [14] at
lower wavelength region. Lin [15] suggested high-Q resonant
cavity can be formed using photonic bandgap quantum
confined structures whose reflectivity is dependent on cavity
modal frequency. Gao [16] calculated the effect of refractive
index on transmission spectra for designing multi-narrow
channel band filter, can be used to design quantum-well
infrared photodetector [17]. The structure can also be used in
photonic integrated circuits [18].
In this paper, transmittivity of semiconductor heterostructure
based photonic multiple quantum well is numerically
calculated at 1.55 μm, and optical bandwidth is computed for
different structural parameters. Results are compared with
SiO2/air composition, which indicates that semiconductor is
better choice for filter performance.
MATHEMATICAL MODELING
Consider the smallest unit of 1D photonic crystal structure
comprising of GaN/AlxGa1-xN material composition where
forward and backward propagating waves are given by-
a2  t 21a1  r12b2
(1.1)
b1  t12b2  r21a1
(1.2)
where rij and tij are reflectivity and transmissivity in passing
from layer i to layer j. They are related to the refractive
indices of the materials following Fresnel’s equation as
rij   r ji 
ni  n j
(8)
Transmission coefficient is given by
T
1
2
(9)
M 11 (tot)
(2.1)
ni  n j
RESULTS
and
t ij  t ji  2
M tot  M N
ni n j
(2.2)
ni  n j
Eq. 2.1 and Eq. 2.2 are valid for normal incidence of input
wave. The reflectivities r and transmissivities t are coupled by
the relation
r2  t2 1
(3)
Using Eq. (9), transmittivity profile of photonic multiple
quantum well is computed as a function of wavelength for the
incident normal and p-polarized electromagnetic waves
respectively centered at 1.55 µm. In Fig 2a, transmittance of
11 layers GaN/Al0.35Ga0.65N PMQW is plotted for normal
incidence at 1.55 µm, and results are compared with
conventional photonic crystal using SiO2/Air composition
with similar structural parameters. It is observed from the plot
that both the structure may work as a bandpass filter in the
required optical domain with the choice of d1 and d2 as
mentioned in figure, where semiconductor heterostructure
provides higher bandwidth (40.8 THz) than the conventional
material composition (33.1 THz). It may also be seen that
outside of the desired wavelength region, the sharp fall of
transmittance magnitude is higher for the heterostructure than
the other which speaks for its efficient bandpass filtering.
Fig 1: Schematic picture of forward and backward waves
in smallest unit of 1D photonic crystal
For p-polarized incident wave at angle θ1, interface
reflectivities are given by
r12  r21 
n1 cos 2   n2 cos1 
n1 cos 2   n2 cos1 
(4)
From the wave equations, transfer matrix corresponding to the
interface can be obtained as
1 1
M T 1, 2  
t  r21,12
r21,12 

1 
(5)
Considering the phase factor of the field propagating through
uniform medium, propagation matrix is given as
0
 exp[ jk1, 2 d1, 2 ]


P1, 2  
0
 exp[ jk1, 2 d1, 2 ] 

(6)
Where di is the propagation length in ith layer, and ki is the
wavevector in that layer. Thus, transfer matrix for the
elementary cell is
M  M T 1 P1 M T 2 P2
(7)
For a perfectly periodic medium composed of N such
elementary cells, the total transfer matrix for such a structure
is
Figure 2a: Comparitive study of transmittivity profiles
between GaN/Al0.35Ga0.65N and SiO2/air at 1.55 µm
wavelength with normal incidence (d1= 1.94 µm, d2=0.8
µm, N=11)
Fig. 2b shows the comparitive analisis for 10° incident angle.
It may be observed from the plot that bandwith reduces for
both the material compositions compared to the case
calcualted for normal incidence, and the percentage of
decrement is higher for heterostructure (1.7%) than the
SiO2/Air (1.2%). This can be supported by the fact that
difference between the refractive indices of SiO2 and Air is
constant for centered frequency 1.55 μm; whereas the
difference between refractive indices for heterostructure
composition is higher for considered mole fraction (x = 0.35).
Considering the Fig 2b, where x equals to 0.35, heterostucture
has greater difference (8.2%) in refractive indices than the
conventional composition of SiO2/Air. Bandwidth becomes
higher with the high difference of refractive indices.
monotonically decreases, whereas it gives a fluctuating nature
for SiO2/air compositon for certain range of barrier width. For
a particular magnitude of barrier layer dimension, it increases
rapidly (for d1=1.64 μm, BW=81.5 THz), but then drastically
decreases when barrier width increases or decreases (for
d1=1.32 μm, BW =42.9 THz and for d1=1.94 μm, BW=32.7
THz). Hence bandwidth tuning is more efficient if
conventional photonic crystal strucutre can be replaced by
heterostrucutre with suitable materials.
Figure 2b: Comparitive study of transmittivity profiles
between GaN/Al0.35Ga0.65N and SiO2/air at 1.55 µm
wavelength with normal and 10° angle of incidences (d1=
1.94 µm, d2=0.8 µm, N=11)
For TE (p-polarized) mode of propagation, lower cut-off
wavelength increases and higher cut-off wavelength decreases
with the increase of refractive index difference. Therefore,
higher fall in bandwidth of heterostructure is acceptable. This
phenomenon may lead to the conclusion that with increase of
incident angle for p-polarized wave, optical bandwidth of the
filter decreases with higher rate for the heterostructure. Hence
tuning of bandwidth with incident angle is more suitable for
semiconductor based photonic crystal.
Figure 3b: Comparitive study of transmittivity profile
between GaN/Al0.35Ga0.65N and SiO2/air at 1.55 µm
wavelength with 10° angle of incidence for different well
width and constant barrier width
However, when well width is varied keeping barrier
dimension constant both convetional photonic crystal and
heterostructure share a fluctuating trend of bandwith variation,
as shown in Fig 3b. Bandwidth first decreases with increasing
well width, then increases for both material composition in a
selective dimensional range.
Figure 3a: Comparitive study of transmittivity profile
between GaN/Al0.35Ga0.65N and SiO2/air at 1.55 µm
wavelength with 10° angle of incidence for different
barrier width and constant well width
Fig 3a shows the transmittance profile as function of
wavelength for different barrier layer width of MQW
structure. From the plot, it may be observed that with increase
of barrier dimension keeping the well dimension constant,
optical bandwidth of the heterostrucutre-based PMQW
Figure 4: Transmittivity with wavelength for different
material composition of heterostructure
By varying the material composition of Al, it is observed from
Fig 4 that optical bandwidth is enhanced with increase of Al
content. This is due to the fact that Al xGa1-xN comprises of x
mole of AlN and (1-x) mole of GaN. As bandgap of former
material is higher, so with increase of AlN percentage, its
refractive index decreases which makes higher index
difference. This increases the separation between the higher
reflectivity zones, thus making increase of bandwidth. Also
with increase of Al mole fraction, sharp fall of magnitude of
transmittance increases, which makes it more efficient optical
bandpass filter. AlxGa1-xN also has an advantage of tunability
where dependence of refractive index on mole fraction is
accounted following Adachi’s model.
layers makes the performance of filter better since it increases
the magnitude of reflectivity outside the central wavelength.
Results are important for designing communication circuit at
THz frequency range.
Increasing number of layers makes higher reflectivity outside
the desired region of interest, which speaks that the filter is
more effective. The device becomes photonic multiple
quantum well as the number of layers is increased, where the
possible transmission is taken place once the incident
electromagnetic wave propagates through the entire structure
while being incident along the direction of quantum
confinement. This can be noted irrespective of the material
composition, as transmission probability at the edge of
passband decreases with increase of number of layers in the
photonic crystal structure. Hence reflectivity increases at the
lower and upper cut-off wavelengths, reflected by the
decrease in transmittivity, as depicted in Fig 5. Comparison
with SiO2/air composition reveals the fact that the
performance is better for semiconductor heterostructure in
terms of rejecting unwanted signal.
2. R. Loudon, “The Propagation of Electromagnetic Energy
through an Absorbing Dielectric”, Journal of Physics A, vol.
3, pp. 233-245, 1970.
REFERENCES
1. E. Yablonovitch, “Inhibited Spontaneous Emission in
Solid-State Physics and Electronics”, Physical Review
Letters, vol. 58, pp. 2059-2061, 1987.
3. I. S. Fogel, J. M. Bendickson, M. D. Tocci, M. J. Bloemer,
M. Scalora, C. M. Bowden, J. P. Dowling, “Spontaneous
Emission and Nonlinear Effects in Photonic Bandgap
Materials”, Pure and Applied Optics, 7, 393 (1998).
4. J. C. Chen, H. A. Haus, S. Fan, P. R. Villeneuve, J. D.
Joannopoulos, “Optical Filters from Photonic Band Gap Air
Bridges”, Journal of Lightwave Technology, 14, 2575 (1996).
5. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, P. O.
Hedekvist, “Fiber-based Optical Parametric Amplifiers and
their Applications,” IEEE Journal of Selected Topics on
Quantum Electronics, 8, 506 (2002).
6. P. Szczepański, “Semiclassical Theory of Multimode
Operation of a Distributed Feedback Laser”, IEEE Journal of
Quantum Electronics, vol. 24, pp. 1248-1257, 1988
7. J. Limpert, A. Liem, M. Reich, T. Schreiber, S. Nolte, H.
Zellmer, A. Tünnermann, J. Broeng, A. Petersson, C.
Jakobsen,
“Low-Nonlinearity
Single-Transverse-Mode
Ytterbium-Doped Photonic Crystal Fiber Amplifier,” Optic
Express, 12, 1313 (2004).
8. J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, T.
Tunnermann, R. Iliew, F. Lederer, J. Broeng, G.Vienne, A.
Petersson, C. Jakobsen, “High Power Air-Clad Large-ModeArea Photonic Crystal Fiber Laser,” Optic Express, 11, 818
(2003).
Figure 5: Transmittivity with wavelength for different
number of layers of PMQW strucutre
CONCLUSION
Photonic multiple quantum well using semiconductor
heterostructure can effectively be used as optical bandpass
filter at 1.55 μm under normal and oblique incidence of
electromagnetic wave with suitable choice of structural
parameters. Simulated results show that it works better than
conventional photonic crystal when computation is made in
terms of filter bandwidth, and efficient tuning can be made by
varying layer dimensions and material compositions. Increase
of incidence angle decreases the filter bandwidth due to
formation of incomplete photonic bandgap. Increasing no. of
9. K. Bayat, G. Z. Rafi, G. S. A. Shaker, N. Ranjkesh, S. K.
Chaudhuri and S. Safavi-Naeini, “Photonic-Crystal based
Polarization Converter for Terahertz Integrated Circuit”, IEEE
Transactions on Microwave Theory and Techniques, vol. 58,
pp. 1976-1984, 2010.
10. W. Belhadj, F. AbdelMalek, H. Bouchriha,
“Characterization and Study of Photonic Crystal Fibres with
Bends,” Material Science and Engineering: C, 26, 578 (2006).
11. H. Azuma, “Quantum Computation with Kerr-Nonlinear
Photonic Crystals”, Journal of Physics D: Applied Physics,
41, 025102 (2008).
12. Jiang, C. Niu and D. L. Lin, “Resonance Tunneling
through Photonic Quantum Wells”, Physical Review B, vol.
59, pp. 9981-9986, 1999.
13. A. Rudziński, “Analytic Expressions for Electromagnetic
Field Envelopes in a 1D Photonic Crystal”, ACTA Physics
Polonica A, vol. 111, pp. 323-333, 2007.
14. X. Xu, H. Chen, Z. Xiong, A. Jin, C. Gu, B. Cheng and D.
Zhang, “Fabrication of Photonic Crystals on Several Kinds of
Semiconductor Materials by using Focused-Ion Beam
Method”, Thin Solid Films, vol. 515, pp. 8297-8300, 2007.
15. S. Y. Lin, V. M. Hietala, S. K. Lyo and A. Zaslavsky,
“Photonic Bandgap Quantum Well and Quantum Box
Structures: A high-Q Resonant Cavity”, Applied Physics
Letters, vol. 68, pp. 3233-3235, 1996.
16. Y. Gao, H. Chen, H. Qiu, Q. Lu and C. Huang,
“Transmission Spectra Characteristics of 1D Photonic
Crystals with Complex Dielectric Constant”, Rare Metals, vol.
30, pp. 150-154, 2011.
17. P. Reininger, S. Kalchmair, R. Gansch, A. M. Andrews,
H. Detz, T. Zederbauer, S. I. Ahn, W. Schrenk and G.
Strasser, “Optimized Photonic Crystal Design for Quantum
Well Infrared Photodetectors”, Proc. of SPIE, vol. 8425, p.
84250A, 2012.
18. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve
and J. D. Joannopoulos, “High Transmission through Sharp
Bends in Photonic Crystal Waveguides”, Physical Review
Letters,
vol.
77,
pp.
3787-3790,
1996.