Download Modeling of Lithium Niobate Ridge Waveguides - Rays

Modeling of Lithium Niobate Ridge Waveguides
Anton Möller
[email protected]
under the direction of
Associate Professor Katia Gallo
Department of Applied Physics,
Quantum Electronics and Quantum Optics
Royal Institute of Technology
Research Academy for Young Scientists
July 10, 2014
Abstract
Waveguides are used in order to guide electromagnetic waves and an example of a waveguide is optical fibers. Problems arise with waveguides when propagation losses are induced, and the aim of this study is to model optical guidance, field profiles and effective
indices in ridge type waveguides and propagation losses in these. This is done by using a model of the waveguides with dimensions experimentally measured, in COMSOL.
Simulations were done for two waveguides with different dimensions at the wavelengths
λ = 1550 nm and its second harmonic λ = 775 nm. The results show that the waveguide
with smaller dimensions have properties which imply a larger confinement with less propagation modes. When studying these waveguides we could also see experimentally that a
second harmonic generation was generated. In the future, more exact modeling should be
done. The asymmetry of the waveguides as well as manufacturing imperfections should
be taken into account when building the simulation which should give even more reliable
data.
Contents
1 Introduction
1
2 Optical Waveguide Theory
1
2.1
General Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2.2
Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2.1
2D Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2.2
3D Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.3
Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.4
Second Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . .
5
2.5
Manufacturing of LiNbO3 Ridge Type Waveguides . . . . . . . . . . . . .
6
2.6
Propagation Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.6.1
Single-Mode Range . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.6.2
Full Width at Half Maximum . . . . . . . . . . . . . . . . . . . .
9
2.7
Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.8
Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.8.1
Wave Equation for TM Mode in a 2D-Slab Waveguide . . . . . .
11
2.8.2
Wave Equation for TE Mode in 2D-Slab Waveguide . . . . . . . .
11
Propagation Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.9.1
Fresnel losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.9.2
Coupling Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.9.3
Propagation Losses in the Waveguide . . . . . . . . . . . . . . . .
13
2.9.4
Determining the Total Power Loss . . . . . . . . . . . . . . . . . .
13
2.9.5
Determining the Experimental Sampling Rate . . . . . . . . . . .
14
2.9
3 Method
3.1
15
Technical Data of the Waveguides . . . . . . . . . . . . . . . . . . . . . .
4 Result
17
18
2
4.1
Mode Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
4.2
Comparison with Experimental data . . . . . . . . . . . . . . . . . . . .
21
5 Discussion
22
6 Acknowledgments
23
A Values of Coefficients for Sellmeier Equations
26
B Code: Full Width at Half Maximum
26
1
Introduction
Waveguides play an important role in the development of optical integrated circuits as it
can be viewed as one of the fundamental building blocks in this area. They are used in
order to guide electromagnetic waves and an example of a system where optical integrated
circuits are active is the fiber communications system. When dealing with waveguides, it is
important that the propagation losses are small to ensure high efficiency and transmission
of light. The ridge type waveguides used in this study are foremost utilized in order to
study the second harmonic generation and non-linear interactions and modeling these
waveguides is an important step in achieving this. The aim of this study was to model
ridge-type lithium niobate waveguides (LiNbO3 ) and propagation losses in these. This has
not been done previously, and this study may provide an approach for modeling other
types of waveguides as well.
2
Optical Waveguide Theory
In this Section, general optics which is necessary in order to fully understand the theory
of waveguides will be presented. After that, the underlying theory of this study will be
covered.
2.1
General Optics
Light travel through different media, and each medium has a specific refractive index,
commonly denoted nmedium and defined as
c
nmedium = ,
v
(2.1)
where c is the speed of light in vacuum and v is the speed of light in that medium. This
definition implies that the refractive index for vacuum is 1 and that the refractive index
describes how much faster light travels in vacuum than in the specific medium [6]. When
1
n2
θr
θi
(a)
θi
θc
θr
(b)
n1
(c)
Figure 1: (a). Light travels from a medium with refractive index n1 to a medium with
refractive index n2 with the incident angle θi and the refracted angle θr . This light is
refracted according to Snell’s law. (b). If the incident angle is greater than the critical
angle, total internal reflection occurs and light is confined in the medium. (c). When the
refracted angle is parallel to the interface (θr = π/2 rad), the incident angle is called the
critical angle, θc .
light travels from one medium to another with different refractive indices, it obeys Snell’s
law,
n1 sin θi = n2 sin θr ,
(2.2)
where n1 and n2 are the refractive indices for medium 1 and 2 respectively, θi is the
incident angle and θr is the refracted angle. The angles are measured from a line normal
to the interface between the two media [5]. When the incident angle causes a refraction
parallel to the interface (θr = π/2 rad), it is denoted the critical angle. If the incident
angle is greater than the critical angle; total internal reflection occurs and light cannot
penetrate the interface between the two media and is therefore confined in one of the
media [5]. These processes are illustrated in Figure 1.
2
Figure 2: A cross-section of a typical optical waveguide. The core has a higher refractive
index than the surrounding cladding, ncore > ncladding which means that the propagating
wave is propagating mostly in the core.
2.2
Waveguides
Waveguides guide electromagnetic waves in a specified direction. They are typically composed of several layers with different refractive indices. Innermost, there is the core in
which the propagation of light takes place. The core has a higher refractive index than
the surrounding layer in order to be able to guide the wave. The surrounding layer, also
known as the cladding or the substrate vary in size but is generally larger than the core [6].
A cross-section of a simple optical waveguide is seen in Figure 2.
2.2.1
2D Waveguides
Waveguides are classified either by how they guide waves or how their refractive indices
change. In two dimensional space; step-index waveguides and graded-index waveguides
are considered. A two dimensional waveguide confines light in one direction only. In a
step-index waveguide, the refractive index changes abruptly in the transition between core
and cladding. The graded-index waveguide on the other hand has a gradual change and
a hard line cannot be drawn between the two different media. This change in refractive
indices result in confinement of the wave as it is not able to escape the core. Because
of confinement in one direction only; the wave expands in the non-confinement direction
due to diffraction as it propagates. This means that the width of the beam increases [6].
3
(a) Buried type
(b) Ridge type
Figure 3: (a) The buried type is characterized by the core being submerged into the
cladding. This type of waveguide is suitable for constructing optical waveguide devices
due to relatively low propagation losses. (b) Ridge type waveguides are on the other hand
characterized by the core being on top of the cladding. Ridge waveguides typically induces
propagation losses due to tight confinement of the wave. [6].
2.2.2
3D Waveguides
When confinement takes place in two directions, x and y, the waveguide is a three dimensional waveguide. These waveguides are further divided into subtypes, depending on
the physical structure of the waveguide. Examples of this are the buried and ridge types.
In a buried type waveguide; the core is submerged into the cladding, and in a ridge type
waveguide the cladding is etched off around the core [6]. This is illustrated in Figure 3.
There are other types of three dimensional waveguides and descriptions of these can be
found elsewhere [6].
2.3
Refractive Index
A wave propagating inside the waveguide along the propagation axis sees the effective
index, neff , and not the refractive index, ncore of the core itself. The effective refractive
index is calculated by solving the wave equation described in Section 2.8 for the spatially
changing refractive index distribution.
n(λ, f, z) = nsubstrate (λ, f ) + ∆n(z, λ),
(2.3)
where ∆n(z, λ) is the refractive index increase induced by annealing process described
in Section 2.5, z is spatial coordinates and λ the wavelength. The refractive index of the
substrate, nsubstrate , is dependent on the wavelength of the light propagating and this
4
index is determined by using the Sellmeier equation for LiNbO3
s
nsubstrate (λ, f ) =
a1 + b 2 f +
λ2
a2 + b 2 f
a4 + b 4 f
− a6 λ 2 ,
+ 2
2
− (a3 + b3 f )
λ − a25
(2.4)
where ai and bi are coefficients determined experimentally [4]. These values are found in
Appendix A. f is a temperature parameter defined as
f = (T − 24.5)(T + 570.82),
(2.5)
where T is the current working temperature expressed in degrees Celsius [4].
In order to account for the calibrated dispersion factor of the annealing process, another Sellmeier equation is used [1],
r
∆n(λ) = c1 +
λ2
c2
,
− c23
(2.6)
where ci are coefficients which are found in Appendix A.
2.4
Second Harmonic Generation
In non-linear optics, generating a second harmonic (SHG) in optical devices is studied.
Suppose an incident wave which approaches an optical device with an angular frequency
of ω. When the wave is propagating through the device, depending on the structure,
two photons with individual energies of ~ω may merge into a single photon with the
combined energy of the two 2~ω. This conversion implies that the newly made photon
has a frequency twice that of the merging photons. This also means that the wavelength
of the second harmony is half than of the original wave. SHG is useful because the
conversion of a wave from a lower frequency to a higher enables more types of optical
integrated circuits to be constructed. One example of this is being able to convert a laser
of a specified wavelength into a laser with half that wavelength [3].
5
2.5
Manufacturing of LiNbO3 Ridge Type Waveguides
The manufacturing process of ridge type waveguides begins with a sample of a material,
in this case lithium niobate, LiNbO3 . Pure benzoic acid is then heated and one side of the
sample is then exposed to the melted substance. This is done in order to replace some of
the lithium ions with protons; hence the name proton exchange (PE) for the process. The
proton exchange process is used in order to locally change the refractive index (∆n(λ)
from Equation 2.6) of a region within the waveguide and this exchange is achieved due
to the melted benezoic acid increasing the optical refractive index of the material.
The depth of which the lithium is replaced by protons is determined by the temperature and duration of the process. When the PE has taken place for a fixed amount of
time; the sample is extracted. The sample is now a slab step-index waveguide with a
guiding layer composed of the volume in which the lithium has been replaced and the
manufacturing process could potentially stop here [2].
In order to refine the waveguide; the next step is annealing. During the annealing the
sample is heated for a fixed amount of time in order to allow the protons which have
replaced lithium to reposition themselves. This process allows the protons to sink further
down into the sample, eliminating the discontinuity of the refractive index creating a
graded-index slab waveguide [2].
In order to create the ridge-type waveguide; the now graded-index slab waveguide is
cut several times along the z-axis (the axis of propagation) and this yield the finished
waveguide. The waveguide itself is now finished, but in order to ensure a high quality
wave propagation its surfaces must be polished carefully. This includes both end faces and
the areas in beteween the individual ridges. When a waveguide has been manufactured
through PE and annealing it is generally called an Annealed Proton Exchange (APE)
waveguide. In Figure 4 a microscopic top-view image of a waveguide is shown.
6
Figure 4: Sample waveguide. If carefully looked upon, imperfections and residue from the
manufacturing process are seen. The waveguide is 7.48 µm wide.
2.6
Propagation Modes
Propagation of a wave in a waveguide only takes place according to different propagation
modes which depend on the wave characteristics and the waveguide. The simplest propagation mode is single-mode propagation. When in single-mode propagation all waves,
even with different wavelengths, travel with equal speed. Single-mode propagation is characterized by the output of the waveguide forming a single sharp dot, compared to the
multi-mode propagation which often forms different patterns due to interference between
waves with different propagation speeds [8].
Guided modes are further classified by the orientation of their electrical and magnetic
fields relative to the plane of propagation (polarization). The most common modes in this
context are called transverse magnetic (TM) and transverse electric (TE). Suppose that
there is an electromagnetic wave propagating in the z-direction. When the electrical field
of the wave is oscillating along the y-axis, the wave is polarized in transverse magnetic
mode. Conversely, when the magnetic field is oscillating in the y-direction, the wave is
polarized in transverse electric mode [8].
When describing a propagating mode the convention is TMmn or TEmn where the
7
(a) TM00
(b) TM10
Figure 5: Two different modes of propagation of transverse magnetic waves. The images
represent the electrical field distribution on a cross section of the waveguide 5 µm wide
and 4 µm high with a refractive index of 2.1401 in the substrate and are to scale. (a)
Single-mode propagation with the mode TM00 . (b) Multi-mode propagation with the
mode TM10 .
TM and TE part states if the mode is transverse magnetic or transverse electric while
m and n are two modal orders which are obtained by solving an eigenvalue equation
from the resulting process of solving the wave equations found in Section 2.8. For the
purpose of this study it is enough to say that m = n = 0 is the fundamental mode which
is equivalent with single-mode propagation and that all other combinations of m and n
imply multi-mode propagation [6].
In Figure 5 two different modes of propagation is seen for transverse magnetic polarized waves at the wavelength λ = 1550 nm.
Waveguides only guide a discrete set of modes, which is limited by a number of factors.
Firstly, the difference between the refractive index of the core and the substrate index,
∆n = ncore − nsubstrate , bounds the range of the solutions to the wave equation. This is
due to the constraints on the effective refractive index, nsubstrate < neff < ncore [6].
When solving the wave equation, the effective mode index is calculated and for a certain value of the index, the electrical field distribution is obtained. Although the number
of solutions already is limited, not all solutions to the wave equation guides a mode which
is seen by looking at their respective electric field distribution [3].
8
2.6.1
Single-Mode Range
The single-mode range of a waveguide is obtained by varying the width of the waveguide
in the COMSOL simulation and measuring the width at which the effective index goes
towards the refractive index of the substrate. This gives the lower boundary of the singlemode range (or cutoff for the TM00 mode). In order to find out the upper boundary value;
the width at which the TM10 mode starts appearing with an effective index meeting the
constraints from Section 2.6 is determined. The range in which single-mode is the only
propagating mode is then defined as wmin < w < wmax where w represents the width of
the waveguide. Observe that fundamental mode propagation is not limited to the singlemode range while the single-mode range specifies in which range the TM00 mode is the
only propagating mode [3].
2.6.2
Full Width at Half Maximum
Full width at half maximum (FWHM) is defined as the absolute value of the distance
between two x-coordinates where the y-coordinates is equal to half of the maximum yvalue. The coordinates were the normalized electrical field distribution versus location in
this study. FWHM is applied to a function representing a pulse or a wave where the data
can be approximated by either a Gaussian fit or a Lorentz fit. Matlab code to calculate
FWHM is found in Appendix B. This value is utilized when comparing and modeling
different modes since it describes in which span the propagation of the transverse side of
the mode takes place.
2.7
Coupling
Coupling is a way to connect a source of waves, typically a laser, to a waveguide. There
are several ways of doing this, and in this study end-fire coupling is described and used. In
order to excite modes in a waveguide; a wave with a similar profile to those modes is fired
directly onto the polished end face of the waveguide. If the wave does not have a similar
9
Figure 6: End-fire coupling. The waves from the light source are focused on the beginning
of the waveguide which lies in the focal point of the lens. The distance between the source,
lens and waveguide can be adjusted in order to obtain the most efficient incident angle.
profile as the waveguide, chances are that there will be no guidance. Often there is a lens
between the source and the waveguide which focuses the light in order to maximize the
efficiency. When utilizing this coupling method; it is important that the incident wave
profile is as similar as possible to that of the guided mode. This is due to the fact that
a waveguide which has an unlike wave profile with respect to the wave may not guide
well [6]. A diagram of an end-fire coupling arrangement is illustrated in Figure 6.
2.8
Wave Equations
From Maxwell’s equations for a lossless dielectric medium, it follows that for an electrical
field variation Ẽ and a magnetic field variation H̃
∇ × Ẽ = −µ0
∂ H̃
,
∂t
and
∇ × H̃ = 0 n2
∂ Ẽ
,
∂t
where µ0 is the magnetic permeability of free space, 0 the electric permittivity of free
space, n the refractive index for the medium, t the time and ∇ the curl operator.
Suppose that a is wave propagating in the orthogonal coordinate system (x,y,z) along
the z-axis with the propagation constant β = 2πneff /λ. The electric and magnetic fields
10
of the wave varies according to
Ẽ = E(x, y) · ei(ωt−βz) ,
(2.7)
H̃ = H(x, y) · ei(ωt−βz) ,
(2.8)
and
with the angular frequency ω = 2πc/λ, and the imaginary unit i, the wavelength λ and
the time t.
These fields are independent of y in a two-dimensional slab waveguide and taking the
partial derivatives yield
∂
∂
∂
= iω,
= iβ,
=0
∂t
∂z
∂y
.
Equations (2.7) and (2.8) result in two different modes with orthogonal polarized
states, the transverse electric and transverse magnetic modes. The TE mode has an electrical field orthogonal to the plane of propagation (the xz-plane), and therefore consists
only of the field components Ey , Hx and Hz . The TM mode would consequently only
have the field components Hy , Ex and Ez [8] [6]. With this information, one could express
the wave equations for both the TM and TE mode.
2.8.1
Wave Equation for TM Mode in a 2D-Slab Waveguide
∂ 2 Hy
+ (k02 n2 − β 2 )Hy = 0
∂x2



E = β H
x


Ez =
2.8.2
ω0 n2
(2.9)
y
(2.10)
∂Hy
1
iω0 n2 ∂x
Wave Equation for TE Mode in 2D-Slab Waveguide
∂ 2 Ey
+ (k02 n2 − β 2 )Ey = 0
∂x2
11
(2.11)



Hx = − β Ey
ωµ0
(2.12)


Hz = − 1 ∂Ey
iωµ0 ∂x
where k0 = 2π/λ.
When solving either of these wave equations; the result is an electrical or magnetic
field as a function of position in the x,y plane. From this the field distribution is utilized
in order to check whether a waveguide actually guides a mode or not. This is achieved by
determining if the field distribution stabilizes and forms a guiding shape such as a singleor multi-mode [6].
2.9
2.9.1
Propagation Losses
Fresnel losses
Between any two interfaces of two different refractive indices there is a Fresnel loss, R.
Fresnel losses occur due to light being reflected back towards the source by the interface.
This loss is estimated for weakly guiding structures by
R=
(neff − 1)2
,
(neff + 1)2
(2.13)
where neff = βλ/(2π) is the effective refractive index and β is the propagation constant.
This is a good approximation for waveguides with perfectly polished end-faces, but in
reality there are imperfections caused by the manufacturing process and polishing which
increases the losses [7][3].
2.9.2
Coupling Efficiency
During the coupling losses occur due to the waveguide and the source being misaligned.
Losses are also induced by misalignment of the focal point causing the light beam to
not being focused on the coupling point and by the shape of the incoming beam not
matching the mode in the waveguide. This loss is minimized by optimizing the coupling
12
experimentally [6].
The coupling loss is approximated by comparing a profile image of the waveguide to
a simulated one and compare the difference in area of the two. This yields an estimate of
how well the coupling is performed.
2.9.3
Propagation Losses in the Waveguide
When the light has entered the waveguide itself, a propagation loss may occur due to imperfections on the surface. This loss is interesting when trying to optimize the waveguides
because of the length dependent exponential power drop. The formula for calculating the
power at the end of the waveguide, Pend , with a starting power Pstart is
Pend = Pstart · e−αL ,
(2.14)
where α is the attenuation constant and L being the length of the waveguide [6].
2.9.4
Determining the Total Power Loss
Define transmission as T := 1 − R, i.e. how much power that is transmitted through an
interface due to some loss factor. Consider a waveguide for which the losses in power are
to be determined. If the incoming power is denoted Pin , the power which continues after
the interface, P1 , is calculated as follows where T is determined from R in Equation 2.13.
P1 = T Pin ,
(2.15)
During propagation through the waveguide the power is decaying according to Equation (2.14). This equation is rewritten as
P2 = P1 · e−αL ,
13
(2.16)
Figure 7: With an incoming power Pin at the first interface, the power P1 is the output
power from that interface due to a Fresnel loss. During propagation through the waveguide, P1 decreases to P2 according to Equation (2.16) and P2 hitting the end interface
which results in an output power of Pout which is calculated by Equation 2.18.
where P2 is the power at the end interface of the waveguide. At the end interface there
is once again a Fresnel loss which is calculated in a similar way of Equation (2.15).
Pout = T P2 ,
(2.17)
Combine equations (2.15), (2.16) and (2.17) as follows and add a term, ηcoupling , to
account for the coupling losses
Pout = (1 − R)2 Pin e(−αL) ηcoupling = T 2 Pin e(−αL) ηcoupling ,
(2.18)
in order to get an expression for the final output power. The different spots where the
power is considered are illustrated in Figure 7.
2.9.5
Determining the Experimental Sampling Rate
When sampling power as a function of the wavelength λ, it is important to use the correct
sampling step ∆λ in order to get useful data. The power as a function of wavelength is
sampled in order to be able to calculate the attenuation coefficient which describes how
the power attenuates as a wave propagates through a waveguide. Since the graph of this
sampling resembles the graph of a sinusoidal function, using a too big sampling step
would result in a data set where the crests and troughs are not visible whereas a too
14
small step would result in unnecessary data.
There are two ways of calculating this; and the differential method is used here.
Assume that
∆φ
∂φ
=
,
∂λ
∆λ
where φ = 2βL = 4πneff L/λ,. From the Taylor series of φ it follows that
∆φ =
∂φ
∆λ,
∂λ
where ∆φ = 2β(λ0 + ∆λ)L − 2β(λ0 )L = 2π and λ0 is an arbitrary point in space if the
higher order terms are neglected due to their minimal impact for this rough measurement.
Consequently,
∂φ
∆λ = 2π,
∂λ
∂
[4πneff Lλ−1 ]∆λ = 2π,
∂λ
−4πneff L
∆λ = 2π,
λ2
∆λ =
−1 λ2
,
2 neff L
(2.19)
The results given by this calculation is only an approximation of the distance between
two fringes of a curve. Therefore, the value given here cannot be used directly as the
sampling step while this would result in only getting measurements for every fringe and
nothing in between. The calculated value must therefore be reduced by a factor in order
to get good experimental results.
3
Method
In this study, two selected waveguides were studied at λ = 1550 nm and at the second
harmonic generation (λSHG = 775 nm). The two waveguides used are seen in Figure 8.
By utilizing Matlab and COMSOL, numerical solutions to Equation (2.9), the wave
15
equation, are obtained for a modeled waveguide. The COMSOL model consists of a ridge
waveguide defined by areas of different refractive indices. Matlab is used simultaneously
and provides COMSOL with nessecary data about how the refractive index of the core
changes depending on position, as the waveguide is a graded one which is seen in Section 2.5. The effective index is calculated in Matlab by using the Sellmeier equations
found in Section 2.3. COMSOL returns the mode electric field distribution and the mode
effective index. Therefore; multiple solutions are obtained for every simulation, one field
distribution per effective index.
By looking at the distribution by hand, the TM00 mode and its associated effective
index are selected. The electric field distribution is then compared with experimentally
found mode profiles of the waveguide to determine the coupling efficiency. The profile
image is taken by utilizing an infrared camera at the end of a coupled waveguide setup.
To get accurate results from experimental trials; the sampling step ∆λ is obtained
by using Equation (2.19) with approximate values for the length, effective index (which
is approximated as the refractive index of the core in this context) and the wavelength
around which the sampling takes place. This is done in order to model the power losses.
In order to properly model the waveguides, the dimensions of the waveguides must
be known. These values are measured using a surface imaging camera. Furthermore, the
refractive index of the substrate and the wavelength of the electromagnetic wave used
must be known or determined to simulate propagation.
For this study, the single-mode range, mode profiles and effective indices were calculated for the mode TM00 at the wavelength λ = 1550 nm and at the second harmonic
generation (λSHG = 775 nm) in order to model the waveguide. The calculations were based
upon measured dimensions of the waveguide. For the single-mode, FWHM is calculated
by using a Matlab function found in Appendix B.
16
(a) Waveguide 4
(b) Waveguide 6
Figure 8: Both images are taken with a microscope. (a) Microscopic image on the fourth
waveguide with a width of 8.47 µm. (b) Microscopic image of the sixth waveguide with
a width of 7.48 µm
3.1
Technical Data of the Waveguides
The technical specifications of the waveguides used in these experiments are found in
Table 1.
Table 1: Technical data of the waveguides used. Entries beginning with PE are specifications for the proton exchange process and entries beginning with APE are details about
the annealing process.
Property
Waveguide 4 Waveguide 6
Type
Ridge
Ridge
Length
10 mm
10 mm
Width
8.47 µm
7.48 µm
PE Environment
Benzoic acid
Benzoic acid
PE Duration
14 hours
14 hours
PE Temperature
160◦ C
160◦ C
PE Depth of exchange 700 nm
700 nm
APE Environment
Air
Air
APE Duration
26 hours
26 hours
◦
APE Temperature
330 C
330◦ C
17
Table 2: The effective refractive index for different propagation modes in the two
waveguides at the fundamental (λ = 1550 nm) and the second harmonic generation
(λSHG = 775 nm). These values were acquired from COMSOL simulations where the
waveguides had been modeled. A dash indicates that the mode is not guided and therefore no effective index exists for that specific waveguide at a certain wavelength.
Waveguide 4
Wavelength λ = 1550 nm λ = 775 nm
TM00
2.1554047
2.213887
TM10
2.148547
2.212509
TM20
−
2.210211
TM30
−
2.20699
TM40
−
2.202841
TM50
−
2.19776
TM60
−
2.19187
4
Waveguide 6
λ = 1550 nm λ = 775 nm
2.153548
2.21376
2.146548
2.212
−
2.209063
−
2.204945
−
2.19964
−
2.193139
−
−
Result
The single-mode range of the waveguides are 2.70 µm < w1550 < 5.70 µm and 0.895 µm <
w775 < 1.92 µm for the fundamental and second harmonic generation respectively. At the
fundamental wavelength, the refractive index of the substrate was determined according
to Equation (2.3) to be 2.1401 and at the second harmonic 2.1814. Table 2 shows the
effective refractive indices from the COMSOL simulations for the different propagation
modes and wavelength.
4.1
Mode Profiles
Waveguide 4 and 6 both theoretically support the TM00 and TM10 modes at a wavelength
of λ = 1550 nm. The mode profile images are seen in Figure 9. For the second harmonic
generation (λSHG = 775 nm), waveguide 4 supported modes ranging from TM00 to TM60 .
Waveguide 6 supported all modes that waveguide 4 did, except TM60 .
The cross section profiles of some of the modes are shown in Figure 10. For the TM00
modes, the full width at half maximum values are calculated and the results are shown
in Table 3.
18
(a) Waveguide 4 - TM00
(b) Waveguide 4 - TM00 (SHG)
(c) Waveguide 6 - TM00
(d) Waveguide 6 - TM00 (SHG)
Figure 9: Electric field distribution cross section of waveguide 4 and 6 at the fundamental
and second harmonic generation in the x,y-plane. (a) and (b) show the field distribution at
single-mode (TM00 ) for the fundamental and second harmonic generation respectively of
waveguide 4. (c) and (d) show the field distribution at the TM00 mode for the fundamental
and second harmonic generation of waveguide 6. All images are to scale.
Table 3: Calculated values for full width half maximum of the two different waveguides
at the fundamental and second harmonic generation. The data was retrieved from COMSOL simulations and a cross-section of the electrical field distribution used. FWHM is
expressed terms of micrometers.
Waveguide 6
Waveguide 4
Wavelength [nm]
1550
775
1550
775
Full Width Half Maximum [µm]
5.1826
5.1293
5.8226
5.7780
19
60
Electrical field norm [V/m]
Electrical field norm [V/m]
60
50
40
30
20
10
50
40
30
20
10
0
0
0
2
4
6
8
10
0
2
Position [µm]
(a) Waveguide 4 - TM00
6
8
10
(b) Waveguide 4 - TM00 (SHG)
60
Electrical field norm [V/m]
60
Electrical field norm [V/m]
4
Position [µm]
50
40
30
20
10
0
50
40
30
20
10
0
0
2
4
6
8
10
0
Position [µm]
2
4
6
8
10
Position [µm]
(c) Waveguide 6 - TM00
(d) Waveguide 6 - TM00 (SHG)
Figure 10: Cross section profiles of the electric field distribution in the x,y-plane of the two
different waveguides at the fundamental and second harmonic generation. These graphs
were then used in order to calculate the FWHM value.
20
Figure 11: Mode profile of waveguide 6 at the fundamental mode taken with an IR imaging
camera. The camera was an Alpha NIR InGaAs with a field of view of 30 × 30 µm with
an objective of 10x and with a focal length of 16.9 mm.
4.2
Comparison with Experimental data
The absolute value of the experimental sampling rate, |∆λ| was determined to be 0.056 nm
according to Equation (2.19) with the following values:




λ = 1550 nm




neff = 2.1401






L = 10 mm
and a sampling step of 0.005 nm was therefore used in the experiments.
With the effective refractive index the transmission factor for the Fresnel losses were
calculated according to (2.13) yielding T = 0.8682 at the fundamental mode. The coupling
efficiency, ηcoupling , was determined to be 0.41 by using Equation (2.18) with the following
21
values




Pout = 3.09585 mW








Pin = 10.0 mW




T = 0.8681








L = 10 mm





α = 0.8128
obtained by calculations based on experimentally found values.
5
Discussion
Both waveguides have a single-mode range of 2.70 µm < w < 5.70 µm at the fundamental
mode. This means that they should guide a combination of the TM00 and TM10 modes due
to the actual widths of the waveguides being above 7 µm. In Figure 11 we see the guided
mode profile and the single-mode propagation seems to be dominating which prohibits
us from seeing the multi-mode profile. Below the main concentration of the beam there
is a small dot, which probably is a diffracted part of the propagating waves or scattered
light.
Waveguide 4 is wider and therefore the refractive indices are greater than in waveguide 6 at the fundamental mode. Due to its width, waveguide 4 is theoretically able to
guide up to TM60 at the second harmonic generation but this is difficult to observe experimentally. Waveguide 6 has a smaller width which in this case only affects the second
harmonic generation limiting the guided modes to up until TM50 . Both waveguides supports the same modes at the fundamental wavelength. In Figure 9 the guided modes for
the fundamental and second harmonic generation are shown. Once again, observe that
the modes in waveguide 4 are wider and not as confined as in waveguide 6. If we look at
the FWHM, a narrower beam is observed at waveguide 6 due to the larger confinement
which is observed both at the fundamental and second harmonic generation.
22
In order to model a waveguide and its losses it is useful to simulate propagation
and determine characteristics such as the single-mode range and which modes that are
guided. These characteristics serves as a reference of what one should expect to see
in experiments but the modeling and experiments are dependent on one another. For
example, the simulations generate a single-mode range and an effective index while the
experiments determine dimensions and power measurements. One could determine the
effective index and single-mode range experimentally but the process is time-consuming
and inefficient. Improvements of this study would include an automated way of calculating
the single-mode range and some type of automation for the extraction of information from
COMSOL and Matlab when simulating the waveguides.
For future studies it would be interesting to model the second harmonic generation
more thoroughly and study the non-linear interactions in these waveguides. Another
study would be to model the imperfections and the end-face reflections in COMSOL and
to model the exact structure of the ridge and not assume that it is homogeneous. This
would yield a more accurate model and hence more accurate results.
6
Acknowledgments
First, I would like to thank my mentor Associate Professor Katia Gallo for all support
given and all time spent guiding me and for allowing me to work with her. I would then
like to thank all people at the department of Quantum Electronics and Optics for the
encouragement and helpfulness. Hampus Gummesson Svensson, another Rays student,
worked with a project similar to mine but with a different point of view. I would therefore
like to thank Hampus for this amazing journey we’ve been through together and for always
being there for me to bounce ideas off.
All the people making Rays possible have my sincere gratitude, especially my counselor
Dennis Alp, project director Philip Frick and of course counselors Mariam Andersson and
Agnes Nordquist.
23
I would also like to thank ABB, Teknikföretagen and The Royal Patriotic Society for
making Rays possible.
24
References
[1] Bortz, M., Arbore, M., and Fejer, M. Quasi-phase-matched optical parametric
amplification and oscillation in periodically poled LiNbO3 waveguides. Optics letters
20, 1 (1995), P. 59–60.
[2] Bremer, M. Periodically poled ridge waveguides in LiNbO3 . KTH physics, TritaFYS 2012:30 (2012), 17ff, 44ff.
[3] Hecht, E. Optics 4th edition. Optics, 4th Edition, Addison Wesley Longman Inc 1
(1998), P. 13f, 27f, 396ff, 641ff.
[4] Jundt, D. H. Temperature-dependent sellmeier equation for the index of refraction
in congruent lithium niobate. Optics Letters 22, 20 (1997), P. 1553–1555.
[5] Nilsson, K., Pålsgård, J., and Kvist, G. Ergo Fysik 2. Liber Förlag, 2010.
[6] Nishihara, H., Haruna, M., and Suhara, T. Optical integrated circuits. New
York (1989), P. 7–8, 10–15, 27ff, 225ff.
[7] Regener, R., and Sohler, W. Loss in low-finesse ti: LiNbO3 optical waveguide
resonators. Applied Physics B 36, 3 (1985), P. 143–147.
[8] Yariv, A., and Yeh, P. Photonics: Optical electronics in modern communications
(the oxford series in electrical and computer engineering). Oxford University Press,
Inc. (2006), P. 110–136.
25
A
Values of Coefficients for Sellmeier Equations
Table 4: Coefficients for the Sellmeier equation (equation 2.4) [4].
Coefficients
Value
a1
5.35583
a2
0.100473
a3
0.20692
a4
100
a5
11.34927
a6
1.5334 · 10−2
b1
4.629 · 10−7
b2
3.862 · 10−8
b3
−0.89 · 10−8
b4
2.657 · 10−5
Table 5: Coefficients for the Sellmeier equation (equation 2.6) [1].
Coefficients
B
Value
c1
3.43 · 10−5
c2
1.01 · 10−5
c3
0.326
Code: Full Width at Half Maximum
function [ fwhm ] = fw_hm( data )
%fw_hm Calculates the full width half maximum value for
% an input ’data’ with two column vectors x,y by fitting a gaussian
26
% curve to the vectors. By Anton Moeller and Philip Frick 2014-07-08.
load(data,’x’,’y’);
if isequal(size(x), size(y))
%Make them into orderd pairs
values = [x,y];
%Half max of y
half_max = max(y)/2;
%Curve fitting
[curve,gof]
= fit(x,y,’gauss8’);
%Output fitting data
rsquared = gof.rsquare
adjrsquared = gof.adjrsquare
%Prepare equation environment
syms f(x)
f(x) = formula(curve);
curve_coeffs = coeffvalues(curve);
%Load coeffficient values into the function
f(x) = subs(f(x),’a1’,num2str(curve_coeffs(1)));
f(x) = subs(f(x),’a2’,num2str(curve_coeffs(4)));
f(x) = subs(f(x),’a3’,num2str(curve_coeffs(7)));
f(x) = subs(f(x),’a4’,num2str(curve_coeffs(10)));
f(x) = subs(f(x),’a5’,num2str(curve_coeffs(13)));
27
f(x) = subs(f(x),’a6’,num2str(curve_coeffs(16)));
f(x) = subs(f(x),’a7’,num2str(curve_coeffs(19)));
f(x) = subs(f(x),’a8’,num2str(curve_coeffs(22)));
f(x) = subs(f(x),’b1’,num2str(curve_coeffs(2)));
f(x) = subs(f(x),’b2’,num2str(curve_coeffs(5)));
f(x) = subs(f(x),’b3’,num2str(curve_coeffs(8)));
f(x) = subs(f(x),’b4’,num2str(curve_coeffs(11)));
f(x) = subs(f(x),’b5’,num2str(curve_coeffs(14)));
f(x) = subs(f(x),’b6’,num2str(curve_coeffs(17)));
f(x) = subs(f(x),’b7’,num2str(curve_coeffs(20)));
f(x) = subs(f(x),’b8’,num2str(curve_coeffs(23)));
f(x) = subs(f(x),’c1’,num2str(curve_coeffs(3)));
f(x) = subs(f(x),’c2’,num2str(curve_coeffs(6)));
f(x) = subs(f(x),’c3’,num2str(curve_coeffs(9)));
f(x) = subs(f(x),’c4’,num2str(curve_coeffs(12)));
f(x) = subs(f(x),’c5’,num2str(curve_coeffs(15)));
f(x) = subs(f(x),’c6’,num2str(curve_coeffs(18)));
f(x) = subs(f(x),’c7’,num2str(curve_coeffs(21)));
f(x) = subs(f(x),’c8’,num2str(curve_coeffs(24)));
%Divide graph into two different sections and solve for the bounds
[trash index] = max(y);
center_x = values(index,1);
max_x = max(values(:,1));
%Solve for the x-coorinates of half max in both areas
solution_left = vpasolve(f(x) == half_max, [0 center_x]);
solution_right = vpasolve(f(x) == half_max, [center_x max_x]);
28
else
warning(’Input data does not have compatible dimensions.’);
end
fwhm = abs(solution_left-solution_right);
end
29