Download Non-Magneto-Optic Waveguide Isolator Namkhun Srisanit , Non-member

148
ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.3, NO.2 AUGUST 2005
Non-Magneto-Optic Waveguide Isolator
Namkhun Srisanit, Non-member
ABSTRACT
We report on a novel technique to realize a
waveguide optical isolator without using conventional magneto-optical materials and external magnets. The simulations are repeatedly performed using
Beam Propagation Method (BPM) to confirm the accuracy. The simulation results show that the forward
and backward propagations reach 2.54 and 15.68 dB,
respectively.
Keywords: Waveguide devices, beam propagation
methods, and waveguide isolator.
1. INTRODUCTION
Optical isolators are passive components that prevent laser beams from being reflected back to the laser
source after optical alignments, which would result in
laser power instability. To the best of our knowledge,
all available published work on optical isolators [1-13]
uses magneto-optics materials to create a Faraday effect on the waveguide layers to achieve satisfactory
isolation ratios as shown in Figure 1.
Polarizer
(450 azimuth)
B
Magneto-optic
450 rotator
Polarizer
(00 azimuth)
Fig.1: A demonstration of the conventional optical
isolator using magneto-optical materials.
The longitudinal magneto-optic or Faraday effect
is exploited in the design of optical isolators and circulators. An optical beam is sent through a magnetooptic crystal parallel to an applied magnetic field.
04PRN01: Manuscript received on January 31, 2005 ; revised
on March 23, 2005.
The author are with Department of Electrical Engineering, Faculty of Engineering Srinakharinwirot University 107
Rangsit-Nakhonnayok Rd., Nakhonnayok, 26120, Thailand.
Phone: 0-2664-1000 Ext.2041, Fax: 0-3732-2605, Email:
[email protected]
Linearly polarized light traveling with the field will
find its polarization direction rotated as a right-hand
screw, while light traveling against the field is rotated
in the opposite direction. This rotation is similar to
that experienced in optically active crystals with one
important difference: the magnetic medium experiences right- and left-hand rotation, while the sense of
rotation in an optically active medium is not changed
by reversing the propagation direction. If the path
length in the magneto-optic crystal achieved a 45 rotation, we are able to devise one-way transmission
characteristics that constitute an optical isolator by
the simple addition of two polarizers at suitable azimuth orientations.
Waveguide modes in planar dielectric waveguides
have their electric vectors vibrating in orthogonal
planes. The simple optical isolator described earlier
will therefore not carry over to waveguide applications, because the input and output polarizations are
at 45◦ , not 0◦ or 90◦ . A second active element must
be included so that the output polarization is brought
back to 0◦ azimuth. A 45◦ optically active rotator
with a left-hand sense of rotation is inserted between
the magnetic rotator and the second polarizer. This
brings light vibrating at 45◦ azimuth to the vertical for both directions of propagation for the optical
waveguide isolator. Thus, we can see intuitively that
we need two active media to make an optical waveguide isolator.
In this paper, we proposed a novel technique to
realize a waveguide optical isolator without using
conventional magneto-optical materials and external
magnets using Beam Propagation Method (BPM).
The simulation results would benefit to be a guideline
for the fabrication of the waveguide isolators without
using conventional magneto-optical materials and external magnets in the future.
2. SUPPORTED THEORIES
The advancement of optical communication, and
especially the progress of integrated optical circuit research, leads to the requirement of waveguide optical isolators. Recent development of multiwavelength laser arrays [14], which support wavelength division multiplexing (WDM) applications, demands that waveguide optical isolators be placed between the monolithically integrated laser array and
the other integrated photonic components (including
array waveguide grating (AWG) devices). Using a
commercially available bulk fiber coupled optical isolator for these applications has the following prob-
Non-Magneto-Optic Waveguide Isolator
149
lems:
1. A bulk isolator with fiber pigtail is suitable for
single fiber-in and single fiber-out arrangements. Array fiber-in and array fiber-out devices using a single
Faraday rotator have not yet been demonstrated.
2. Using a fiber pigtailed bulk optical isolator cannot
support monolithic integration. These isolators cannot be placed on the common laser array chip with
other photonic devices.
3. Using the bulk optical isolator is bulky for the array laser structure.
To satisfy the monolithic integration requirement on a laser array chip, it is preferred that
the waveguide optical isolator is fabricated on any
non-magneto-optical waveguide and that the use of
the conventional non-reciprocal Faraday effect be
avoided. Since almost all optical propagations are
reciprocal, such waveguide isolator design represents
a great challenge to scientific research.
In a conventional straight channel waveguide, the
mode intensity profiles are symmetrical with respect
to the waveguide center. Lower order modes are those
with a fewer number of intensity peaks in the lateral
direction. The lowest order mode is the fundamental
mode with its lateral intensity distribution being near
a Gaussian shape. Higher order modes are those with
many intensity peaks in the lateral direction.
In a bent optical waveguide, however, the situation is more complicated. When the bending radius is large, the waveguide mode behavior is similar to the straight channel waveguide, with lower order modes having a fewer number of intensity peaks
and higher order modes having more intensity peaks
in the lateral direction. As the bending radius decreases, the bending effect becomes more and more
significant, which affects the total internal reflection
behavior of the modes in the bent waveguide. Some
light rays will remain trapped in the waveguide to
form guided modes while some light rays will escape
from the waveguide to form leaky waves. Figure 2
shows the concept of light propagation in the bent
optical waveguide.
In a sharply bent waveguide, only those light rays
that are totally internally reflected by the outer bending boundary of the waveguide form stable guided
modes (Eigen-modes), while those modes bounced
between the inner and outer surface boundary of the
waveguide form unstable modes. The unstable modes
are partially converted to the stable modes through
an overlap integral with these stable modes. The remaining non-converted portion forms leaky waves as
shown in Figure 2 and escapes from the bent waveguide structure. The ray drawing could help visualize
the mode propagation behavior although mode calculation would be more accurate to represent the mode
propagation. Therefore, in the bent optical waveguide, the lower order modes are again those with minimal intensity peaks near the waveguide cross-section
center. The higher order modes are those with intensity peaks near the outer bending surface due to total
internal reflection from the outer boundary only. In
this case, the higher order modes do not necessarily
have more intensity peaks because their total internal reflections do not involve the inner bent waveguide boundary. In fact, the highest order mode is
the one with only a single intensity peak located very
close to the outer waveguide boundary. This may
be called the “shifted center-of-gravity” mode. This
phenomenon was predicted by Marcatili in 1969 [15].
For a bent multimode waveguide with a few modes,
we may not have many higher order stable modes.
Possibly, the highest order mode is the only one that
can be considered stable. This issue will be considered during the design of the device.
Theoretical studies performed by Marcatili [15]
and others [16-20] have substantiated the relationship
between bending loss, mode conversion efficiency,
channel width, channel index differential between
core and surroundings, bending curvature, guided
mode, and light wavelength. For example, as formulated by Marcatili [15], the loss per radian of a bent
channel waveguide is
αc R =
k A 21/2
1
n2 −1/2 n3 kx0 a 2 A 3
x0
1−
1 − 32
2
n1
n1
πa
π
2 2 3/2
kx0 A
R
2c
1 + akx0
exp − 3 1 −
π
1− 1−
n23
n21
kx0 A
π
2
A 2 −1/2
n2 A
+ 2 n32 a 1 − kx0
π
1
(1)
Bent Channel
Waveguide
Leaky Wave
Stable Bent
Waveguide Mode
Straight Channel
Waveguide
Fig.2: Light propagation in a bent channel waveguide.
The mode conversion coefficient “c” from fundamental mode (lowest order mode) to higher order modes
is given by
1 πa 3 1
,
(2)
c=
2kx0 a A where
=
2π 2 R
n2 3/2
k3 = 2 21 1 − 32
R,
2
3
kz0 A
kz0
n1
(3)
150
ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.3, NO.2 AUGUST 2005
and kx0 and kz0 are channel waveguide propagating constants at corresponding coordinate axes when
there is no bending. The geometry and parameters
of the channel waveguide is shown in Figure 3. k1 is
the bulk propagation constant in the waveguide core.
A is defined by
λ
A= 1/2 .
2
2 n1 − n23
incident beam direction. The presence of additional
loss to the return beam makes the return beam depart
further from the original incident beam. This is very
important since it shows that the mode conversion
does not violate any physical principle, especially the
second law of thermodynamics.
(4)
R is the radius of the curvature. c2 measures the
power that the fundamental mode would couple to
higher order modes.
Fig.3: Parameters of bent channel waveguide used
in the formulation above [15].
The loss per radian increases when the confined
3
decreases. Thus, to
index differential ∆ = n1n−n
3
reduce the bending loss, a larger waveguide to surrounding index difference (n1 − n3 ) would be helpful.
In addition, the loss per radian can be reduced by increasing the waveguide width “a” for a fixed parameter “A”. Figure 4 shows the calculation result of loss
per radian versus a/A ratio performed by Marcatili
[15]. As the a/A ratio increases, the loss reduces dramatically while the mode conversion coefficient “c”
approaches 1 (see dashed curves). A statement had
thus been made by Marcatili that the “conversion to
higher order modes are found more significant than
radiation loss resulting from curvature” in a multimode waveguide [15]. Therefore, using a multimode
bent channel waveguide, we expect the incident of
unstable modes will be converted to the higher order
stable modes with good conversion efficiency while
the associated loss will be low.
Figure 5 shows the calculated highest order mode
that has only a single intensity peak near the curved
waveguide outer boundary. The bending loss can be
considered as the trade-off for the mode conversion.
The bending loss pays for the mode conversion and,
thus, for the non-reciprocal beam propagation as well.
Because of the bending loss, the non-reciprocal beam
propagation concept is valid. Without making up the
proper forward lost beams (with proper phase factors), the return beam may not return to the original
Fig.4: Calculated losses as a function of curvature
and index difference for fundamental TE dominating
mode [15]. For TM dominating mode, the curves are
similar since ∆ << 1 .
Fig.5: Calculated highest order mode with a single
intensity peak near the outer curvature boundary [15].
The above results show clearly the tendency of
mode conversion in a bent channel waveguide. It provides guidelines for a bent channel waveguide design
that reduces curvature loss and increases mode conversion. The unidirectional mode conversion can be
used for the waveguide isolator design since it does
not involve any magneto-optical material or external
magnet. However, it cannot be exactly applied to
our optical isolator design. The primary reason is
that the conversion coefficient “c” is for the fundamental mode to higher order modes (not only highest
order mode). It is not the conversion coefficient from
a specific incident unstable mode to the highest order
Non-Magneto-Optic Waveguide Isolator
151
stable mode. Therefore, to exactly predict the behavior of the new waveguide isolator structure, we must
perform a device simulation using a beam propagation method.
Output Single
Mode Beam
3. SIMULATION RESULTS
Based on the bent waveguide mode conversion discussion above, we can imagine a waveguide optical
isolator device with a single mode input, a multimode bent waveguide mode conversion section, and
a single mode output as shown in Figure 6. The input single mode laser beam is launched to the single
mode isolator input channel. Through a taper, the
single mode beam is coupled to the multimode bent
waveguide section and excites the bouncing modes.
Such bouncing modes gradually convert to the highest order mode (or “shifted center-of-gravity” mode)
along with some conversion loss. Using a waveguide taper, we merge the converted beams into the
single mode output channel as the isolator output.
During isolator return beam propagation as shown
in Figure 7, the single-mode return beam excites
the multimode beams in the bent waveguide section through the waveguide taper. Again, the mode
conversion occurs in the bent waveguide. The converted beams or modes are leaning on the outer bent
waveguide boundary, resulting in minimal interaction
with the original single-mode input channel waveguide and thus minimal beam return coupling to the
input channel. High optical return isolation can thus
be achieved.
The optical isolator structure was simulated using a Computer-Aided Design (CAD) tool simulation program called BeamPROPT M from Rsoft Inc.
The software provides a general simulation package
for computing the propagation of light waves in arbitrary waveguide geometries. This is a complex problem. The computational core of the program is based
on a finite difference beam propagation method (FDBPM) [21,22].
The parameters for the simulations are set as follows:
1. Wavelengths: Varied from visible to near infrared
2. Light wave polarizations: TE and TM
3. Single mode waveguide width: Varied
4. Waveguide depth: 2.5 µm
5. Gap widths between two waveguides: Varied
6. Waveguide structure: Diffused
7. Index of refraction of the substrate: 1.517
8. Cover Index: 1 (air)
9. Index difference: Varied
From the simulations, the higher order modes in
the waveguide structure are not totally converted to
the highest order mode (with single intensity peak) as
the light wave exits the bend waveguide. This is due
to the fact that, as pointed out by Marcatili in 1969
[15], the radius of the curvature of the bent waveguide
should be small enough for an efficient mode conver-
Taper
Waveguide
Bent
Waveguide
Loss
Multimode
Waveguide
Coupling Portion
from Single Mode
to Multimode
(gap width)
Single Mode
Waveguide
The separation
between single
mode and
mutimode
waveguides
Input Single
Mode Beam
Fig.6: The designed structure of the proposed optical
isolator for the forward propagation.
sion. Unfortunately, the BeamPROPT M simulation
results showed repeatedly that if the radius of curvature of the bend waveguide is too small, all the energy
in the waveguides would be lost because the index difference is too small. (We use a relatively small index
difference because a small difference can be easily realized in many fabrication processes). Therefore, the
radius of the curvature of the bent waveguide of our
designs is selected to be relatively large to minimize
waveguide loss. If the radius of curvature of the bend
waveguide were large, the desired mode conversion
would not be complete.
To solve this problem, a waveguide taper [23]−a
waveguide that gradually decreases or increases in
width along the propagation axis is introduced to
the design. As the results, the waveguide taper completely converts all the higher order modes back to the
fundamental mode. This raises a question: why do
we have to keep the bent waveguide in the structure
since it does not convert the bouncing modes to the
single-peak highest order mode? The answer to the
question is that the bent waveguide mode conversion
is not complete in the present design. It needs help
from the waveguide taper in order to merge the beams
152
ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.3, NO.2 AUGUST 2005
The separation
between single
mode and
mutimode
waveguides
propagation optical isolators, Iid is the input power
from the forward propagation optical isolators, Is is
the output power from the straight waveguide, and
Iis is the input power from the straight waveguide.
To calculate the backward isolation ratios, we compare the outputs between single mode and multimode
waveguides as follows:
1. Measure the output power (I2 ) of the multimode
waveguide for backward propagation, and then measure the input power (Iin2 ) of the multimode waveguide for backward propagation;
2. Measure the output power (I1 ) of the single mode
waveguide for backward propagation, and then measure the input power (Iin1 ) of the single mode waveguide for backward propagation.
Therefore the isolation ratio (ηBackward ) for the
backward propagation is
I1 IIin2
in2
, (7)
ηBackward = 10log
I1 IIin2
+ I2 + ILoss
in2
Output
Beam
Single Mode
Waveguide
Multimode
Waveguide
Coupling Portion
from Single Mode
to Multimode
(gap width)
Loss
Bent
Waveguide
Taper
Waveguide
Input Single
Mode Beam
Fig.7: The designed structure of the proposed optical
isolator for the backward propagation.
into the single-mode output channel for the forward
propagation. Although the mode conversion is incomplete, it is significant enough to differentiate the
forward and backward mode propagation behaviors
in the bent waveguide and to support the waveguide
optical isolation function.
The performance of the waveguide isolators is measured in terms of forward and backward propagation loss. The forward and backward propagations
losses are measured by the simulations. To calculate the forward propagation loss, we simulated a
straight waveguides along with the optical isolators
to compare the output powers between the optical
isolators and straight waveguides fabricated on the
same samples. Therefore, the forward propagation
loss (ηF orward ) is
ηF orward
I
Isolator
=
10log
=
10log IIids ,
IStraight
I d
,
(5)
(6)
Iis
where Id is the output power from the forward
where ILoss is the propagation loss of the waveguides plus the coupling loss between the single mode
and multimode waveguides. Here we assume that
ILoss is so small that we can neglect it.
Figures 8, 9, 10, and 11 are the simulation results
for different parameter sets. In Figure 8, the gap
widths of the waveguide isolators are varied from 0
to 4 µm. In Figure 9, the refractive index differences
are varied from 0.004 to 0.010. In Figure 10, the single mode waveguide widths are varied from 3 to 6.5
µm. And in Figure 11, the input wavelengths are
varied from 400 to 850 nm. It can be summarized
in the table 1 that the forward and backward propagations reach 2.54 and 15.68 dB, respectively, at the
gap width of 1.85 µm, the refractive index difference
of 0.006, single mode width of 5 µm, and the wavelength of 632.8 nm. It should be noted that some of
the results in Figure 11 is not considered the single
mode.
Recently, we have searched for some publications
on the waveguide isolators as shown in the Table 2.
It is found that for the research point of view, the
isolation ratio of more than 15 dB is acceptable for
novel techniques. However, for the commercial optical isolators, the isolation ratio of more than 30 dB is
required. As mentioned earlier, this paper is intended
to be an audacious guideline for the real fabrications
of the designed waveguide isolators in the future.
4. CONCLUSION
We successfully demonstrated the first waveguide
optical isolator without using a magneto-optical material or external magnet by means of the BPM CAD
tool. There is a large difference between the forward and return propagation losses, demonstrating
that the waveguide mode propagation in the isolator
structure is non-reciprocal in nature. The best results
Non-Magneto-Optic Waveguide Isolator
153
Table 1: Summary of the simulation results.
Fig.
Gap Width
(µm)
∆n
Single mode
Width (µm)
Wavelength
(nm)
8
9
10
11
Varied
1.85
1.85
1.85
0.006
Varied
0.006
0.006
5
5
Varied
5
632.8
632.8
632.8
Varied
Best performance (dB)
Forward/Backward
propagation
2.54/15.68
4.32/34.55
2.69/16.81
2.42/16.94
Table 2: Summary of the recently published work on the waveguide isolators.
Waveguide isolator inventors
Columbia University
Electrotech Laboratory
AT&T Bell Laboratory
Pondicherry Engineering
College, India
Our Best Simulations
Optics For Research (OFR)
Inc.
Oz Optics, Ltd., Canada
Photop Technologies, Inc.
Oplink Corp.
Techniques Used
Mach-Zehnder
Interferometer
Nonreciprocal
Ferromagnetic
Nonreciprocal
Phase Shift
Thin-Film
Gyrotropic
Bent waveguides
Faraday Rotation
Performance (dB)
20
Refence)
1
18
2
17
3
15
4
15
40
This paper
5
Faraday Rotation
Faraday Rotation
Faraday Rotation
35
30
35
6
7
8
40
30
30
Propagation Loss (-dB)
Propagation Loss (-dB)
25
20
15
10
20
10
5
0
0.003
0.004
0.005
0.006
0
0
1
TE Forward Propagation
TM Forward Propagation
TE Backward Propagation
TM Backward Propagation
2
3
4
5
Gap Width ( um)
Fig.8: The simulation results of the optical isolator
at different channel waveguide gaps.
show that the forward and backward propagations are
2.54 and 15.68 dB, respectively.
From the simulation results, it is found that the
waveguide gap widths are a trade-off between the
forward and backward propagations, that is the narrower the gap is, the better the forward propagations,
but the worse the backward propagations. Moreover,
we run the simulations on both TE (Transverse Electric) and TM (Transverse Magnetic) modes to confirm the fact that our designed waveguide isolator is
TE Forward Propagation
TM Forward Propagation
TE Backward Propagation
TM Backward Propagation
0.007
0.008
0.009
0.010
0.011
n
Fig.9: The simulation results of the optical isolator
as a function of waveguide refractive index difference.
polarization-free. It is obvious that our simulations
show that with different polarizations, the waveguide
isolator still performs well compared to conventional
waveguide isolators that require polarizers with fixed
polarization as explained in Figure 1. In addition,
Figure 11 shows that our waveguide isolators have
wide operating bandwidths, from 400 nm to 850 nm.
The infrared wavelengths require further modifications.
154
ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.3, NO.2 AUGUST 2005
20
18
Propagation Loss (-dB)
16
14
12
10
8
6
4
2
0
2
3
4
5
6
7
Single Mode Waveguide Width ( um)
TE Forward Propagation
TM Forward Propagation
TE Backward Propagation
TM Backward Propagation
Fig.10: The simulation results of the optical isolator at different single mode waveguide widths. When
the width is larger than 6 µm, the waveguide becomes
multimode.
30
Propagation Loss (-dB)
25
20
15
10
5
0
300
400
500
600
700
800
900
Wavelength (nm)
TE Forward Propagation
TM Forward Propagation
TE Backward Propagation
TM Backward Propagation
Fig.11: The simulation results of the optical isolator
at different input wavelengths.
References
[1] J. Fujita, M. Levy, R. M. Osgood Jr., L. Wilkens,
and H. Dotsch, “Waveguide Optical Isolator
Based on Mach-Zehnder Interferometer,” Applied
Physics Letters, vol. 76, pp. 2158-2160, 2000.
[2] W. Zaets and K. Ando, “Optical Waveguide Isolator Based on Nonreciprocal Loss/Gain of Amplifier Covered by Ferromagnetic Layer,” IEEE Photonics Technology Letters, vol. 11, pp. 1012-1014,
1999.
[3] R. Wolfe, W.-K. Wang, D. J. DiGiovanni, and
A. M. Vengsarkar, “All-Fiber Magneto-Optic Isolator Based on the Nonreciprocal Phase Shift in
Asymmetric Fiber,” Optics Letters, vol. 20, pp.
1740-1742, 1995.
[4] A. Alphones and L. L. Joseph, “Optical Isola-
tor on a Thin-Film Gyrotropic Waveguide,” Microwave and Optical Technology Letters, vol. 14,
pp. 151-153, 1997.
[5] Optics for Research,
Internet:
Isolators,
January 2005. [Online] available:
http://www.ofr.com/index.htm.
[6] Oz Optics Ltd., Internet:
Fiber Optics
Isolator, January 2003. [Online] available:
http://www.ozoptics.com/ALLNEW PDF/
DTS0016.pdf
[7] Photop Technologies,
Inc.,
Optical Isolator,
January 2005. [Online] available:
http://www.photoptech.com/new05/product/
koncent pow isolators.php
[8] Oplink
Corp.,
Fiber
Isolator,
January
2005.
[Online]
available:
http://www.oplink.com/isolatorsHAP.html
[9] T. Sikora, “Waveguide Analysis of Multisectional
Magneto-Optical Isolator,” IEEE Transactions
on Magnetics, vol. 26, pp. 2783-2785, 1990.
[10] T. Shintaku and T. Uno, “Optical Waveguide Isolator Based on Nonreciprocal Radiation,”
Journal of Applied Physics, vol. 76, pp. 8155-8159,
1994.
[11] Karl Lambrecht Corp., Internet: Polarizing
Cube Isolators/Displacers, June 2003. [Online]
available: http://www.klccgo.com/
[12] J. Warner, “Nonreciprocal Magnetooptic Waveguides,” IEEE Transactions on Microwave Theory
and Techniques, vol. MTT-23, pp. 70-78, 1975.
[13] M. Levy, R. Scarmozzino, R. M. Osgood Jr.,
and R. Wolfe, “Permanent Magnet Film MagnetoOptic Waveguide Isolator,” Journal of Applied
Physics, vol. 75, pp. 6286-6288, 1994.
[14] D. V. Thourhout, A. V. Hove, T. V. Caenegem,
I. Moerman, P. V. Daele, R. Baets, X. J. M.
Leijtens, and M. K. Smit, “Packaged Hybrid Integrated Phased-Array Multi-Wavelength Laser,”
Electronics Letters, vol. 36, pp. 434-436, 2000.
[15] E. A. J. Marcatili, “Bends in Optical Dielectric
Guides,” Bell System Technical Journal, vol. 48,
pp. 2103-2132, 1969.
[16] H. F. Taylor, “Power Loss at Directional
Changes in Dielectric Waveguides,” Applied Optics, vol. 13, pp. 642-647, 1974.
[17] D. Marcuse, “Light Transmission Optics,” Van
Nostrand Reinhold, New York, NY, pp. 32-49,
1973.
[18] D. Marcuse, “Length Optimization of an SShaped Transition between Offset Optical Waveguides,” Applied Optics, vol. 17, pp. 763-768, 1978.
[19] L. D. Hutcheson, I. A. White, and J. J. Burke,
“Comparison of Bending Losses in Integrated Optical Circuits,” Optics Letters, vol. 5, pp. 276-278,
1980.
[20] V. Ramaswamy and M. D. Divino, “Low-Loss
Bends for Integrated Optics,” Conference on
Non-Magneto-Optic Waveguide Isolator
Lasers and Electro-Optics, paper THP1, Washington, DC, May 1981.
[21] R. Scarmozzino, A. Gopinath, R. Pregla, and
S. Helfert, “Numerical Techniques for Modeling
Guided-Wave Photonic Devices,” Journal of Selected Topics in Quantum Electronics, vol. 6, pp.
150-162, 2000.
[22] R. Scarmozzino and R. M. Osgood, Jr.,
“Comparison of Finite-Difference and FourierTransform Solutions of the Parabolic Wave Equation with Emphasis on Integrated-Opticss Applications,” Journal of the Optical Society of America A, vol. 8, pp. 724-731, 1991.
[23] W. K. Burns, A. F. Milton, and A. B. Lee, “Optical Waveguide Parabolic Coupling Horns,” Applied Physics Letters, vol. 30, pp. 28-30, 1977.
Namkhun Srisanit received Bachelor of Engineering from Srinakharinwirot University with 2nd class honors in 1996. He also received Master
of Science in Electrical and Computer
Engineering, and Doctor of Philosophy
from the University of Miami in 1999
and 2003, respectively. His research
focuses on Electro-Optics applications,
optoelectronics, and optical communications.
155