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Submitted on 1 Jan 1993
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An electrooptic switching structure on III-V
semiconductors using the coherent coupling of radiation modes
Diaa Khalil, Smail Tedjini
To cite this version:
Diaa Khalil, Smail Tedjini. An electrooptic switching structure on III-V semiconductors using the coherent coupling of radiation modes. Journal de Physique III, EDP Sciences, 1993, 3 (9), pp.1769- 1776. �10.1051/jp3:1993236�. �jpa-00249038�
Classification Physic-s Abstiacts
42.82 42.BOK 42.BOL
An electrooptic switching structure on III-V semiconductors
using the coherent coupling of radiation modes
Diaa Khalil and Smail Tedjini
LEMO, CNRS-URA 833, 23 avenue des martyrs, B.P. 257, 38016 Grenoble Cedex, France (Receii'ed lo Noi,ember 1992, revised 20 April 1993, accepted 5 May 1993)
Rdsumk. Dans cet article nous proposons une nouvelle structure pour la commutation
dlectrooptique utilisantle couplage cohdrent des modes rayonnds. La structure proposde est dtudide
en utilisant h la fois l'analyse non modale BPM (Beam Propagation Method) et une technique
modale basde sur l'analyse du spectre des modes rayonnds gdndrds h chaque discontinuitd optique.
Cette analyse montre que la structure propos6e a un taux de distinction de 12 dB et des pertes de l'ordre de 3,5 dB dans l'dtat ON.
Abstract. In this paper we propose a new structure for electrooptic switching using the coherent
coupling of radiation modes. The proposed structure is studied using both the non-modal Beam Propagation Method (BPM) and a modal technique based on the analysis of the radiation mode spectrum generated at each optical discontinuity in the structure. The analysis shows that the
proposed structure may have
a distinction ratio of 12 dB with radiation losses in the order of 3.5 dB in the ON state.
1. Introduction.
Integrated electrooptic switches are important elements that have a wide range of applications especially in optical communication systems. These devices contain intrinsic optical discon- tinuities which are usually considered as sources of radiation losses. One of the design objectives of these elements is thus to minimize the power coupled to the radiation field to
reduce the losses. However, recent works have shown that the radiation energy may be
coherently coupled to the guided mode at another optical discontinuity ]. In previous works,
we have investigated theoretically the effects of this coherent coupling on the losses of passive
structures like waveguide bends [2], Y-junctions [3] and Mach-Zehnder interferometer, as
well as the active electrooptic modulator [4]. In this work we propose a new point of view for the design of the switching structures based on this coherent coupling. Instead of considering
the radiation modes as a source of losses, we may make use of their energy to perform the required switching. The proposed electrooptic switching structure, based on the interference
1770 JOURNAL DE PHYSIQUE III N° 9
between the guided and radiation modes, is analysed using two techniques the standard BPM
as a non-modal technique to study its overall performances and a modal technique based on the spectrum of the radiation modes generated in the structure.
2. Analysis.
The schematic diagram of the proposed structure is shown in figure where 4 main parts could be observed :
I) the input optical waveguide ;
2) a bend section to excite the radiation modes with a relatively confined spectrum ;
3) the interaction section which is simply an optical waveguide loaded by a metallic electrode ; 4) the output branching Y-junction.
~i L2 L3 L4
m ~l~ »'« ~- ~
p~
z~
,W I - " +
' ' III,, '
~
-l ~
W
dl d ' p2
- - V
+ II
- I Elec~ode
j '
Input x Output
a)
d=S pm Metalization
-- /
p+~~p~~
As 2 prn
GaAS
~ ~ Gai-xAl As
Substrate N+GaAS b)
Fig, I. Proposed structure for the switch d
= 5 ~m, dj = 5.5 ~m, W 2dj = ~m, L, 400 ~m
and L,
= 4 300 ~m. a) Optical path. b) Cross-section of the guiding structure.
The guiding structure is a GaAs/GaAl,Asj optical ridge waveguide with ~r
= 0.03 whose cross-section is shown in figure 16. Using the effective index technique, the two-dimensional
waveguide could be replaced by a one-dimensional waveguide with an equivalent refractive index n~,, = 3.392, an equivalent step index 8n
= 2.4 x 10~ ~ and a width d
=
5 ~Lm where
the Gais refractive index is assumed
to be no =3.4 at the operating wavelength
A
= 1.3 ~Lm. The optical guides are designed to be single mode at = 1.3 ~Lm. The input guide is assumed to be excited by its fundamental mode.
2. NON-MODAL ANALYSIS. The performances of the structure can be analysed using the
non-modal Beam Propagation Method (BPM). In this technique, the wave equation is solved
for a propagation distance A= so small that the guiding effect of the waveguide and the
propagation action could be treated separately. Thus, the waveguide is replaced by a model
that consists of a series of lenses AZ apart from each other. Through the distance
AZ the propagation is assumed to be in a homogeneous medium [5] and the guiding effect is then introduced as a phase correction in the spatial domain. Neglecting the reflected field, the
wave equation is reduced to the following recurrence relation :
w(a.,Az)=P.Q.P.w(x,0)+O(Az~) (1)
where : P
= exp (- j AZ(VII[(V) + k(n()'~~ + no ko])) is the operator representing the propagation in
a homogeneous medium of refractive index no, V, =
d~/dx~ is the transverse del operator, Q
= exp (- ilAz ko 8n(.r, z dz is the operator representing the phase correction of the
equivalent lens, O(Az~ is an error term arises from the non-commutation of the two operators, ko is the propagation constant of plane waves in free space, no is the refractive index of the
substrate, 8n(a.)
=
the non-homogeneous part of the refractive index, and ~(.r, z) is the slowly varying amplitude of the electric field E(x, z) such that
E(x, z) = (x, z) e~~'""° (2)
The repeated application of equation (I) could be used to advance the solution from E(x, 0) to E(x, z). The application of the operator Q is a simple multiplication in the spatial
domain. However, to apply the propagation operator P, it is required to represent the field in its spectral domain (where the operator is applied in the exponential form). For the propagation
in a homogeneous medium, this spectral domain consists of the plane wave vectors. This
simplifies greatly the problem as it enables to use the FFT and the IFFT between the two domains. The application of the FFT technique requires a finite calculation window that should be large enough with respect to the guide itself. Besides, a suitable absorbent is required to
simulate the infinite physical structure. The absorbent could be simply a negative imaginary part in the refractive index. The form of the absorbent should be chosen such that it minimizes the reflections from the boundaries of the calculation window. In our work we use a Gaussian
form which we found to be the most suitable one.
Using the BPM, the field distribution in the structure is calculated for two applied voltages
V
=
0.0 and V
=
26 V. In the two cases, we observe the existence of the field oscillations in the interaction guide of length L~. These oscillations are due to the interference between the
even guided mode and a group of odd radiation modes. This group of modes is generated by the
optical discontinuity represented by the bend of length L~ and angle @. Depending on the bend geometry, the generated group of radiation modes will have a certain spectrum. When this spectrum is relatively confined, the group of radiation modes may behave as a quasi-guided
mode (or a leaky wave). The basic idea of the structure depends on the control of the field oscillations in the interaction section of length L~. The spatial periodicity of these oscillations
depends on the difference between the propagation constant of the guided mode and the effective propagation constant of the group of radiation modes. Applying an electric field leads to a differential phase shift between the guided and radiation modes due to the different overlap
of this field with them. This changes the spatial periodicity and hence deviates the field maximum at the input of the Y-junction. Such a deviation favours one output with respect to the other and then the switching could be achieved.
2.2 MODAL ANALYSIS. To analyse the behaviour of the group of radiation modes, a modal
IOURNAL DE PHYSIQUE>l> -T 1, N'9 SEPTEMBER lo91 61
1772 JOURNAL DE PHYSIQUE III N° 9
I ~°°° ~_
~ - ~
n4 n4
@ @
u u
f ~
j ~
W W '
( 4ggg j
C
~
f 2000 fC
~ ~ ~
© ©
O ~ C
~ ~
O O
ij @ ij @
0
-15.0 s-G 15.0 .0 s-G ls.0
~) X( um) b) X( um)
Fig. 2. Field distribution in the switch for two applied voltages. a) V 0.0 V. b) V 26 V.
analysis is required. For this purpose the Radiation Spectrum Method [RSM], previously reported by the authors [2], is used. In this technique the input field is decomposed in the form of the guided and radiation modes of the waveguide. Assuming that we have a TE field, where
the extension for the TM field is quite similar, the input electric field at any point
E(,i, ±) could be written in the form [6]
E(~i, ?)
= jja~ ~,(~K, p~)expj- jp, zj + ~ A(p) ~(.K, p)expj- jp=j dp. (3) Where ~ (.r, p represents the profile in the,v-direction of a mode with a propagation constant
p and p is the spatial frequency given by
P~ " k(o P~ (4)
where k~~ = n~~ x ko is the equivalent substrate propagation constant.
The first summation in equation (I extends over the guided modes, while the integration is
performed over the continuum of the radiation modes. The second summation is used to
consider the even and odd radiation modes. The complex amplitudes a, and A(p) are
determined from the exciting input field using the orthogonality of modes and the normalization conditions. The power carried by the group of radiation modes is, thus, given by
Pm
= jS(p)dp =
jA(p)j2 j~~ j~(>., p)j2da( ~ dp (5)
where
w is the optical frequency, p is the magnetic permeability and S(p) is the power
spectral density of the radiation modes at the spatial frequency p.
As the set of guided and radiation modes is a complete one, any field distribution could be described using equation (3) however this requires an infinite number of radiation modes to
represent the continuous spectrum. In practical cases the optical field of interest is usually
located in the space such that we can sample the radiation spectrum. Moreover, as the spatial variations of the optical field do not contain, usually, strong oscillations with high spatial frequencies, the radiation modes far from cutoff do not contain significant energy and can be
easily neglected. This enables us to discretize both the optical field and its radiation spectrum.
Thus using equation (7), the projection of the input field over the selected mode could be
calculated numerically. However, the normalization of the radiation modes should be
calculated analytically. For this purpose, the simplified method introduced in reference [7] is used.
To use this modal technique for the study of the bend structure, the output of the first waveguide « I » is considered as the exciting field of the second guide « II » which has a direction of propagation z' making a small angle with z as shown in figure I. Assuming that the guide I is excited by its guided mode, the input field of the guide II could be written as
E,~~~(,i', z'
= o)
= ~~(~i) e~~~'° (at z'
=
o)
= ~j(x' cos ) e
~~'' sin (6)
where z' and ~r' are the new set of coordinates related to [z,.<] by the transformation
z =
z'cos -.r'sin 0
,
a. = z'sin + x'cos
Using equation (3) with equation (6), the amplitudes of the radiation and guided modes in the second waveguide II can be calculated. The propagation of each mode results in a phase shift
pL2 where p is the propagation constant of the corresponding mode. At the second
discontinuity the field is transformed again to the spatial domain, with the help of equation ( ), and another rotation of the axis by an angle « o » is used to obtain the input field of the third
waveguide III. To follow the propagation of the field, the projection on the guided and radiation modes of guide III could be used.
With this technique, the radiation spectrum S(p) at the inputs of the second and third
waveguides is calculated. The obtained spectrum is shown in figure 3. We observe that the
amplitudes of the odd radiation modes are more important than those of the even modes which is expected from the asymmetric excitation. At the input of the second guide, the normalized
~ ~
~'~ ~~~~°~~ i
°.6 ----~$~~$~$~nt.
~ ii
~ <-~
~ ,<
g , G j1
0A ', § 0 4
~
i > i
u i , t
( ', (
~ ,
, ~ i '
1 0.2 <
~ 0.2
w ~ ,
m ,N
m ~ ,
i f
( ( 0.0
0.0 0.5 10 0.0 0.5 1.0
spatial ~equency (pm-I) spatial Frequency (pm- I)
a) b)
Fig. 3. Spectrum of the radiated field for H 0.7° and L~ 600 ~m. a) Even and odd radiation modes at the input of guide II. hi Odd radiation modes at the input of guides II and III.
1774 JOURNAL DE PHYSIQUE III N° 9
power of the odd radiation modes, defined by equation (5), is in the order of 0.137 while that of the even modes is about 0.018 ; an order of magnitude less. We remark also that the spectral
width of the group of odd radiation modes is narrower than that of the even modes and then the effects of the odd modes are expected to persist in the guide for a longer distance.
The obtained spectrum explains well the oscillatory behaviour of the field. While the central
spatial frequency component gives approximately the periodicity of the field oscillations, the mixture of the different spectral components results in the damping of these oscillations. We may thus define an effective propagation constant p~~~ as well as an effective spectral width
«p to be associated with the radiation spectrum. For p~~~ the value of the propagation constant of each mode could be weighted by its power spectral density to give
Petr
" S(p p dp/P~
The spectral width could be defined with respect to this effective propagation constant to be
«j
= s(p )(P Pe~r)~ dp/Prrr.
Using these two parameters we may define two lengths, a beating length L~ at which the
effective phase shift between the guided mode and the group of radiation modes is
2 w
Lb
~
2 WI(pgn Pefr)
and a coherence length at which the phase shift between the radiation modes themselves
becomes effectively w :
~C ~~~#
These parameters are useful to characterize the radiation field behaviour in the waveguide. The
beating length gives approximately the periodicity of the field oscillations while the coherence
length gives an indication to the distance at which the radiation mode interference with the
guided mode is significant. For the spectrum of the odd radiation modes generated at the input
of guide II (shown in Fig. 3) we get
L~ = 658.7 ~Lm and L~ =
024.516 ~Lm.
These parameters are functions of both the guiding properties and the geometry of the bend.
2.3 PERFORMANCES AND DIscussIoN. The overall performances of the proposed structure
could be obtained using the standard BPM. To evaluate this performance, the output field is projected on the guided mode of each of the two output guides and 2. For the electrooptic
effect calculation the classical formula given in reference [8] are used and the applied electric field is assumed to be constant within the guide width
« d
» and zero outside. The normalized output intensity as a function of the applied voltage is shown in figure 4, where we can notice that the intensities at the two outputs vary in a complementary way with the applied voltage. A
distinction ratio in the order of 12 dB is predicted between the two outputs. The radiation losses of the structure in the ON state (between the input and output at zero voltage) is in the order of 3.5 dB which is quite reasonable for power dividing devices.
Actually, the device performances depend mainly on the radiation mode spectrum generated
at the input of the interaction section. To improve the distinction ratio, the power coupled to the
~ IO 2c
c Q6
+
- output I
( O.6
- output 2
(o O 4
2
o
IO 20 30
Applied voltage (V) Fig. 4. Predicted performances of the proposed structure.
group of odd radiation modes should be increased. However this may lead to an increase of the power coupled to the even radiation modes. As the coherence of these modes is relatively weak, their power are practically lost which results in an increase of the structure losses at the
ON state. This means that better performances of the structures may be obtained by further
optimization of the bend section geometry. For such an optimization, the two parameters L~ and 0 can be used. This enables also to reduce the dimensions of the device with respect to
conventional devices based on only the guided modes. When using only the guided modes, the
optical discontinuity should be relatively long to achieve a smooth geometrical changes and then to avoid any power coupling to the radiation modes. This leads to devices with small
angles and long transition lengths (L~ or L~) while in our case the reduction of the transition
lengths is required to maintain the coherence of the group of radiation modes. In addition, the
use of the radiation modes makes it possible to control geometrically the operating point of the device. This latter is a function of the length L~ of the interaction section where the change of L~ changes the position of the field maximum with respect to the axis of symmetry of the output
Y-junction. As the periodicity of the field oscillations, in this section, is in the order of a fraction of a millimetre, the control of the operating point is not difficult to be achieved
technologically. This shows also the importance of the spectral analysis for the optimization of
the structure.
Finally, the devices based on the radiation modes may show also an additional advantage in the application of high speed modulation. As we can modulate either the guided or the radiation field, the electrode can be situated on the substrate adjacent to the guide. As we have
reported in a previous communication [9]. This enables to separate the optical and electrical guides and then to optimize them separately. Such separation is of particular interest at high speed modulation.
3. Conclusion.
The performances of a switching device based on the radiation mode coherent coupling is studied in this work. Our analysis shows that such devices may be comparable to classical devices based on the propagation of only the guided modes. However, the use of the radiation modes enables us to optimize the device geometry by reducing the dimensions of the optical discontinuity~ In addition, it allows to control the operating point of the active device by a simple control of its geometry. On the other hand the coherent coupling of radiation modes
