Search of mode wavelengths in planar waveguides by using Fourier transform of wave equation
The article describes a numerical method based on Fourier transform for studying propagating optical waves in dielectric planar waveguides. The inverse problem to the known direct one in waveguide investigation is proposed, namely a search of light wavelengths according to taken values of propagatio...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2016
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nasplib_isofts_kiev_ua-123456789-1215202025-02-09T20:38:40Z Search of mode wavelengths in planar waveguides by using Fourier transform of wave equation Fitio, V.M. Romakh, V.V. Bobitski, Ya.V. The article describes a numerical method based on Fourier transform for studying propagating optical waves in dielectric planar waveguides. The inverse problem to the known direct one in waveguide investigation is proposed, namely a search of light wavelengths according to taken values of propagation constants. For each constant a set of wavelengths is obtained, among which an input value of wavelength from direct problem exists necessarily. A high accuracy of the method proposed is confirmed by exact values obtained by solution of transcendental dispersion equation. This method is tested on many examples, in particular, for waveguides of different permittivity profiles or for TE and TM modes propagate there. 2016 Article Search of mode wavelengths in planar waveguides by using Fourier transform of wave equation / V.M. Fitio, V.V. Romakh, Ya.V. Bobitski // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2016. — Т. 19, № 1. — С. 28-33. — Бібліогр.: 17 назв. — англ. 1560-8034 DOI: 10.15407/spqeo19.01.028 PACS 42.65.Wi, 77.22.Ch https://nasplib.isofts.kiev.ua/handle/123456789/121520 en Semiconductor Physics Quantum Electronics & Optoelectronics application/pdf Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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The article describes a numerical method based on Fourier transform for studying propagating optical waves in dielectric planar waveguides. The inverse problem to the known direct one in waveguide investigation is proposed, namely a search of light wavelengths according to taken values of propagation constants. For each constant a set of wavelengths is obtained, among which an input value of wavelength from direct problem exists necessarily. A high accuracy of the method proposed is confirmed by exact values obtained by solution of transcendental dispersion equation. This method is tested on many examples, in particular, for waveguides of different permittivity profiles or for TE and TM modes propagate there. |
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Fitio, V.M. Romakh, V.V. Bobitski, Ya.V. |
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Fitio, V.M. Romakh, V.V. Bobitski, Ya.V. Search of mode wavelengths in planar waveguides by using Fourier transform of wave equation Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Fitio, V.M. Romakh, V.V. Bobitski, Ya.V. |
| author_sort |
Fitio, V.M. |
| title |
Search of mode wavelengths in planar waveguides by using Fourier transform of wave equation |
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Search of mode wavelengths in planar waveguides by using Fourier transform of wave equation |
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Search of mode wavelengths in planar waveguides by using Fourier transform of wave equation |
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Search of mode wavelengths in planar waveguides by using Fourier transform of wave equation |
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Search of mode wavelengths in planar waveguides by using Fourier transform of wave equation |
| title_sort |
search of mode wavelengths in planar waveguides by using fourier transform of wave equation |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2016 |
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https://nasplib.isofts.kiev.ua/handle/123456789/121520 |
| citation_txt |
Search of mode wavelengths in planar waveguides by using Fourier transform of wave equation / V.M. Fitio, V.V. Romakh, Ya.V. Bobitski // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2016. — Т. 19, № 1. — С. 28-33. — Бібліогр.: 17 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT fitiovm searchofmodewavelengthsinplanarwaveguidesbyusingfouriertransformofwaveequation AT romakhvv searchofmodewavelengthsinplanarwaveguidesbyusingfouriertransformofwaveequation AT bobitskiyav searchofmodewavelengthsinplanarwaveguidesbyusingfouriertransformofwaveequation |
| first_indexed |
2025-11-30T14:08:53Z |
| last_indexed |
2025-11-30T14:08:53Z |
| _version_ |
1850224657357078528 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 1. P. 28-33.
doi: 10.15407/spqeo19.01.028
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
28
PACS 42.65.Wi, 77.22.Ch
Search of mode wavelengths in planar waveguides
by using Fourier transform of wave equation
V.M. Fitio1, V.V. Romakh1, Ya.V. Bobitski1,2
1Lviv Polytechnic National University, Department of Photonics,
12, Bandery str. 79013 Lviv, Ukraine
2University of Rzeszów, Faculty of Mathematics and Natural Sciences,
1, Pigonia str., 35959 Rzeszów, Poland
Phone/fax: +38(032)-258-25-81; e-mail: polyana@polynet.lviv.ua, vladykv@gmail.com, bobitski@polynet.lviv.ua
Abstract. The article describes a numerical method based on Fourier transform for
studying propagating optical waves in dielectric planar waveguides. The inverse problem
to the known direct one in waveguide investigation is proposed, namely a search of light
wavelengths according to taken values of propagation constants. For each constant a set
of wavelengths is obtained, among which an input value of wavelength from direct
problem exists necessarily. A high accuracy of the method proposed is confirmed by
exact values obtained by solution of transcendental dispersion equation. This method is
tested on many examples, in particular, for waveguides of different permittivity profiles
or for TE and TM modes propagate there.
Keywords: Fourier transform, eigenvalues, permittivity, propagation constant, spatial
frequency, wave equation.
Manuscript received 14.10.15; revised version received 12.01.16; accepted for
publication 16.03.16; published online 08.04.16.
1. Introduction
Dielectric planar waveguides are the structures used to
limit and control light in waveguide devices and
integrated optics circuits [1]. One of the most common
applications of such waveguides today is semiconductor
distributed feedback micro-lasers [2-4] using them as an
active layer. These lasers provide high speed data
transfer, and single-mode generation is achieved by
feedback filter in a form of corrugated layer. The filter is
formed inside waveguide semiconductor structure
parallel to the active layer.
For designing the devices based on planar
waveguides, it is necessary to know propagation
constants of waveguide modes that correspond to a taken
wavelength. A number of approximate methods are used
to determine propagation constants of localized modes in
gradient planar waveguides [5, 6], which for the first
time have been developed for analyzing the problems of
quantum mechanics. As the structure of wave equation
for modes of TE polarization is similar to that of the
stationary Schrödinger equation in quantum mechanics,
it is possible to use its analytical methods, particularly
WKB (Wentzel–Kramers–Brillouin) approximation [6],
for studying the planar waveguides.
A typical permittivity distribution of symmetric
gradient waveguide is shown in Fig. 1, where ε0 is the
substrate permittivity, ε1 is the maximum value of
permittivity in the active layer. For some profiles of
waveguide permittivity ε(x), the accurate analytical
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 1. P. 28-33.
doi: 10.15407/spqeo19.01.028
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
29
solutions are found [7-9]. As in a waveguide ε1 > ε0,
propagation of a localized guided mode with the
propagation constant β is possible, and the electric field
distribution is described by the following function:
E(x,z) = E(x) exp(–iβz), where x and z are the transverse
and longitudinal coordinates, respectively, E(x) is the
electric field amplitude, i – imaginary unit. But, even in
the simplest case (a step profile of permittivity as
particular case of the gradient one), a search of
propagation constants leads to solution of transcendental
algebraic equation [10]. Problem becomes more
difficult, if the permittivity varies according to a
complex function along the axis x. Well-known methods
to find propagation constants and waveguide mode fields
are mostly analytical, too cumbersome, and their
accuracy is rather low. For example, WKB
approximation allows to calculate with a high accuracy
these propagation constants that correspond to field
distributions with a large number of nodes (points where
electromagnetic field is zero). Mostly, in planar
waveguides situation is somewhat different: it is
necessary to find the values of propagation constants for
the lowest modes with a small number of nodes (1st, 2nd
order modes) or without them (basic mode) in
appropriate electromagnetic field.
The current state of computer hardware and
software sophistication allows using the numerical
methods effectively to search propagation constants and
fields of gradient planar waveguides. A numerical
method of wave equation solution for planar waveguides
in a coordinate domain is known. In this method, the
second derivative by coordinate is replaced by the
difference operator, and ultimately the problem is
reduced to a solution of the eigenvalue/eigenvector
problem [11, 12]. But this method is characterized by
low accuracy, because electromagnetic field may occupy
a large spatial range, especially for waveguide modes
with large indices (higher order modes).
In recent years, the numerical method for finding
the propagation constants based on the wave equation
Fourier transform was developed [13], and it is
characterized by high accuracy of analysis. By this
method, it is possible to find all the propagation
constants of localized modes and appropriate discrete
Fourier transforms of field distribution in a waveguide in
one calculation cycle [14]. The method was tested on
many gradient waveguides. For example, let waveguide
permittivity is described by a function ε(x) = ε0 + (ε1 –
ε0)/cosh2(2x/d) (see the figure), where d is the thickness
of active layer. Then for this waveguide, an exact
analytical solution exists, and exact values of
propagation constants are found [1, 7], which are listed
in the left column of Table 1 for a waveguide with the
following parameters: ε0 = 2.25, ε1 = 2.89, d = 5 μm.
Electro-magnetic wave of the length λ = 1 μm propa-
gates in the structure. At this wavelength, the waveguide
has 13 guided modes. Values shown in the right column
of Table 1 are the propagation constants calculated using
the numerical method described in [14].
Image of permittivity distribution for a symmetric gradient
waveguide.
The propagation constants of waveguide modes
with indices from 0 to 11 calculated by both methods are
the same, except the last one, appropriate fields of which
have a maximum length in the coordinate space; so for
them a small error is available. This numerical method
provides high calculation accuracy and, as the research
shows, it is characterized by high numerical stability. A
search of propagation constants by using the numerical
method [13, 14] is reduced to the problem on
eigenvalues (square of the propagation constants) and
eigenvectors (field discrete Fourier transforms in a
waveguide) that look like MV = β2V, where M is some
square matrix depending on the parameters mentioned
above, V is the vector-column, the elements of which are
actually eigenvectors.
It often happens that one needs to solve the inverse
problem, i.e., for a planar waveguide with certain
parameters the propagation constant of some guided
mode is known, and it is necessary to find the wave-
length that corresponds to the taken propagation
constant. This problem arises in the analysis of
Table 1. Values of propagation constants (μm–1) in a
gradient waveguide obtained by two methods.
Index of
propagation
constant
Exact method [1] Numerical method
[14]
0 10.59058151 10.59058151
1 10.41422086 10.41422086
2 10.25044271 10.25044271
3 10.09985917 10.09985917
4 9.96306855 9.96306855
5 9.84064604 9.84064604
6 9.73313382 9.73313382
7 9.64103073 9.64103073
8 9.56478192 9.56478192
9 9.50476895 9.50476895
10 9.46130077 9.46130077
11 9.43460608 9.43460608
12 9.42482808 9.42482739
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 1. P. 28-33.
doi: 10.15407/spqeo19.01.028
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
30
waveguide distributed feedback lasers by using the
coupled wave method [2]. In the course of this analysis,
all localized modes at which generation is possible are
determined [15]. So, the aim of the work is to show how
this problem can be solved successfully by using the
numerical method based on Fourier transform of wave
equation. The task is again reduced to another
eigenvalue/eigenvector problem where square of
wavelengths are eigenvalues: M1V = λ2M2V, where M1
and M2 are the square matrix with the dimension defined
by necessary accuracy of calculations; λ is the sought
wavelength, V – vector-column corresponding to the
Fourier transform of field distribution.
2. One-dimensional wave equations
and their Fourier transforms
If electric field is perpendicular to the plane of incident
wave on the interface of waveguide film and substrate,
i.e., E = {0, Ey, 0}, in this structure the transverse
electric modes (TE modes) are formed, for which the
wave equation is written as [15]
( ) ( ) ( ) ( )xExEx
dx
xEd 2
2
2
2 2
β=ε⎟
⎠
⎞
⎜
⎝
⎛
λ
π
+ . (1)
If electric field is parallel to the plane of incident
wave, i.e., H = {0, Hy, 0}, this case corresponds to the
transverse magnetic modes (TM modes), and wave
equation will look like:
( ) ( ) ( ) ( ) ( )xHxHx
dx
dH
dx
xd
dx
xHd 2
2
2
2 2ln
β=ε⎟
⎠
⎞
⎜
⎝
⎛
λ
π
+
ε
− . (2)
Functions E(x), H(x) describing fields in waveguide
localized modes and their first derivatives tend towards
zero at x → ±∞. That is why, for these functions, their
first and second derivatives the Fourier transform exists.
Function integrity in (1) and (2) is an important aspect of
this approach. As the expressions for field components
of appropriate modes are identical, it is sufficient to
introduce the Fourier transforms only for one of them,
e.g., for E(x) [16]:
( ) ( ) ( )uEdxuxixE =π−∫
∞
∞−
2exp , (3)
( ) ( ) ( )uuEidxuxi
dx
xdE
π=π−∫
∞
∞−
22exp , (4)
( ) ( ) ( ) ( )uEudxuxi
xd
xEd 2
2
2
22exp π−=π−∫
∞
∞−
, (5)
where u is the spatial frequency, E(u) – Fourier
transform of electric field.
Besides, for functions for which the Fourier
transforms exist, i.e., F{g(x)} = G(u), F{h(x)} = H(u),
the following equation is yet right [16]:
( ) ( ){ } ( ) ( )dvvHvuGxhxgF ∫
∞
∞−
−= , (6)
where F{…} is the Fourier transform. Equation (6)
describes the convolution theorem.
One takes the Fourier transforms of left and right
parts of (1) and (2) taking into account the expressions
(3) to (6). As a result, the transition from differential
equations to integral ones occurs, and we obtain next
wave equations in a frequency domain:
( ) ( ) ( ) ( )uEdvvEvuuEu 2
2
22 24 β=−ε⎟
⎠
⎞
⎜
⎝
⎛
λ
π
+π− ∫
∞
∞−
, (7)
( ) ( ) ( ) ( )
( ) ( ) ( ) .2
ln24
2
2
22
uHdvvHvu
dvvvHvu
dx
xdFuiuHu
β=−ε⎟
⎠
⎞
⎜
⎝
⎛
λ
π
+
+−
⎭
⎬
⎫
⎩
⎨
⎧ ε
π−π−
∫
∫
∞
∞−
∞
∞− (8)
The Fourier transform of permittivity is
( ) ( ) ( ){ }
( ) ( ) ( ){ },010
010
xfFu
xfFu
ε−ε+δε=
=ε−ε+ε=ε
where δ(u) is the Dirac delta function in a spatial
frequency domain, f(x) – function describing the
permittivity distribution. Besides, the following
condition is imposed on the function f(x):
( ) Mdxxf <∫
∞
∞−
, (9)
where M is a finite number.
3. A search of wavelengths according
to the taken propagation constant
To demonstrate a way of finding the wavelengths corres-
ponding to the taken propagation constant β, let’s
consider the equation (7) in another form:
( ) ( ) [ ] ( )uEudvvEvu 22222 44 π+βλ=−επ ∫
∞
∞−
. (10)
In (10), one can replace the integral by sum and go
to the equation in a discrete form. By changing
continuous values of u and v by the discrete ones, we
obtain:
( ) ( )
( )
( )
( )[ ] ( ) ,4
4
2222
21
21
2
ΔΔπ+βλ=
=ΔΔΔ−Δεπ ∑
−
−−=
sEs
kEks
N
Nk (11)
where N is the number of points in which electric field is
sought; s and k are the indices on which summation is
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 1. P. 28-33.
doi: 10.15407/spqeo19.01.028
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
31
Table 2. Propagation constants (μm–1) for several waveguides calculated for the wavelength 1 μm.
Index of propagation
constant Parabolic profile Gauss profile Exponential profile
0 10.58687909 10.52227751 10.47134576
1 10.39522986 10.21638483 10.21797941
2 10.20000124 9.94381586 10.06583391
3 10.00114042 9.71247251 9.93737112
4 9.79935523 9.53409763 9.83825131
5 9.59914463 9.43166891 9.75205958
6 9.43189030 9.68270923
… …
13 9.43890608
14 9.42980648
15 9.42530846
For this waveguide,
7 guided modes exist
For this waveguide,
6 guided modes exist
For this waveguide,
16 guided modes exist
done: ( ) 21, −≤ Nks ; Δ is the partitioning step of
maximum spatial frequency umax: Numax=Δ . Value of
N should be taken large enough and unpaired, while its
ratio with the frequency umax should give an optimum
value of partitioning step Δ. It is also assumed that the
function E(sΔ) is sought in the interval from 2maxu− to
2maxu , and beyond the field attenuates very quickly,
i.e., ( ) 02max →> uuE .
One can write the last equation for all discrete
spatial frequencies us = sΔ. Then, a set of these equations
will be written in a matrix form where the value of
square wavelength λ2 is common to all values of index s:
VMVM 21
2λ= , (12)
where M1 is the square symmetric matrix of elements
4π2ε(sΔ – kΔ), M2 – diagonal matrix of elements
β2 + 4π2(sΔ)2 in the main diagonal, V – vector-column of
elements E(sΔ).
So, the problem is reduced to the problem on
eigenvalues (square wavelength) and eigenvectors
(discrete Fourier transform of field E(x)) which
correspond to the found value of λ2. By carrying out the
inverse discrete Fourier transform of eigenvector, we
obtain field distribution along coordinate x for every
value of wavelength corresponding to the appropriate
localized mode.
Propagation constants βv of symmetric planar
waveguide for a taken wavelength λ satisfy the
following inequality:
λ
π
<β<
λ
π 10 22 nn
v , where
00 ε=n and 11 ε=n are the refractive indices of
substrate and active layer, respectively. For the inverse
problem, the wavelengths λv must satisfy the following
inequality accordingly to the known propagation
constant β:
β
π
<λ<
β
π 10 22 nn
v . (13)
For all the propagation constants from Table 1,
matching sets of wavelengths are found using the matrix
equation (12) and inequality (13). In every set, the
wavelength λ = 1 μm is available, which confirms
correctness of calculations. If we take an arbitrary
wavelength from all sets and use the equation
MV = β2V, we find appropriate propagation constant
among the set obtained again. A solution of both direct
and inverse problems was carried out using the
following numerical process parameters: number of
points N = 2001, maximum spatial frequency umax =
10 μm–1. They are selected from the subject to the
Whittaker-Shannon sampling theorem [16].
The permittivity of investigated waveguides is
described by functions of the following type:
ε(x) = 2.25 + 0.64 f (2x/d), where d = 5 μm. One can
check easily that all the functions f (2x/d) presented in
Tables 2 and 3 satisfy the condition (9). In these tables,
several examples for the waveguide permittivity profiles
are shown, so there is a possibility to check a direct
problem by using the inverse one. For the direct
problem, calculations are carried out at the light
wavelength λ = 1 μm, for the inverse one they are done
using the propagation constants with the indices 0, 1, 2,
m – 2, m – 1 and m from Table 2, where m is the last
number of β (it corresponds to the smallest propagation
constant according to its value).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 1. P. 28-33.
doi: 10.15407/spqeo19.01.028
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
32
Table 3. Light wavelength (μm) obtained via equations (12) and (13) for the propagation constants from Table 2.
Index of propagation
constant Parabolic profile Gauss profile Exponential profile
0 0.99999999946 1.00000000000 0.99999999978
1 0.99999999944 0.99999999998 1.00000000018
2 0.99999999965 0.99999999959 0.99999999999
m – 2 0.99999999930 0.99999999948 0.99999999935
m – 1 0.99999999909 1.00000000016 1.00000000007
m 0.99999999953 0.99999999983 1.00000000024
It is seen that for the taken parameters of
waveguide and numerical process, the largest number of
guided modes (n = 16) will be generated in the structure
with an exponential profile of permittivity, and the
smallest one (n = 6) will be in the waveguide with a
Gauss distribution.
Obviously, for any propagation constant from
Table 2, among the set of wavelengths found via
equations (12) and (13), it should be the wavelengths
very close to λ = 1 μm. The wavelength values for 6
different modes (3 lowest and 3 highest indices) are
adduced in Table 3.
Our analysis of results from Table 3 shows that
m101 9μ<−λ − , i.e., the relative error of computations
is less than 10–9. Therefore, for this problem accuracy of
calculations by using the proposed numerical method is
extremely high.
4. Conclusion
The results showed that the inverse problem of a search
of waveguide mode wavelengths corresponding to the
taken propagation constant is solved with sufficiently
high accuracy. This method is use comfortably for
planar gradient waveguides and for complex ones
consisting of several layers with certain thicknesses and
refractive indices. Mainly, these waveguides are used in
semiconductor lasers [15].
Accuracy of computations is defined by numerical
process parameters N and umax, i.e., these parameters
should have such values that the sampling theorem [16]
is practically performed. In this case, high accuracy of
solution can be provided. One can see from Table 1 that
the propagation constant of a waveguide mode with the
highest index is defined with the least accuracy. For this
mode, the electric field decreases slowly with increasing
the coordinate x. Therefore, N value should be taken
large enough, and Δ value should be done small, which
can be provided by a small spatial frequency umax. On
the other hand, umax cannot be taken quite small, as in
this case values of ( )2maxuE ± and ( )2maxu±ε will be
different from zero. To select the maximum spatial
frequency, one can use the criterion proposed in [17].
According to it, some function I on umax is sought using
the rule:
( ) ( ){ } duxfFuI
u
u
2
5.0
5.0
max
max
max
∫
−
, (14)
where F{f (x)} is the Fourier transform of function f (x).
This integral goes to saturation at increase of umax,
so the bottom boundary of acceptable values can be
determined from the condition that I (umax) is virtually
unchanged at change of spatial frequency. If these
conditions are provided, by the method proposed one
can find the solutions of both direct and inverse
problems that concern planar gradient waveguides with
high accuracy.
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