IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 16, NO. 2, FEBRUARY 2004 515
Wavelength-Insensitive Nonadiabatic Mode
Evolution Couplers
George T. Paloczi, Student Member, IEEE, Avishay Eyal, Member, IEEE, and Amnon Yariv, Life Fellow, IEEE
Abstract—We investigate the wavelength insensitivity and
fabrication error tolerance of the relative output amplitude
and phase for a new type of optical coupler. This new coupler
overcomes the long-length requirements of adiabatic couplers, yet
achieves similar desirable properties. The three-space geometrical
representation of coupled-mode theory, used as the design and
analysis method, is presented and an example of a 3-dB coupler is
examined.
Index Terms—Coupled-mode theory (CMT), integrated optics,
optical couplers, optical waveguides, Poincaré sphere.
I. INTRODUCTION
T
WO-BY-TWO optical couplers that maintain output dif-
ferential phase and relative amplitude conditions despite
operating wavelength variation and fabrication errors are of
utmost importance in integrated optics, especially for cascaded
filters and modulators [1]. To this end, Mach–Zehnder struc-
tures can be constructed with a good degree of tolerance in
the output amplitudes [2]; however, in such constructions, the
output differential phase changes maximally with wavelength
variations, rendering these devices less useful in filter applica-
tions. The most promising technology for achieving the desired
tolerances has been the adiabatic coupler [3], for which the
local system states follow adiabatically the local eigenstates of
the waveguiding system and, thus, mode conversion between
eigenstate populations is minimized. However, this requires
structures that change over distances much longer than the
beat length at any cross section of the coupler, thus limiting
the usefulness of adiabatic couplers. Here, we describe a
novel design approach, which is based on varying the local
eigenstates, while keeping their beat length constant. It is
shown that, by choosing the beat length with discretion while
relaxing adiabatic criteria, adiabatic-like wavelength variation
immunity and fabrication tolerance can be realized for much
shorter device lengths.
Manuscript received August 26, 2003; revised October 14, 2003. This work
was supported in part by the Office of Naval Research, by the Defense Advanced
Research Project Agency, and by the Air Force Office of Scientific Research.
G. T. Paloczi and A. Yariv are with the Department of Applied Physics,
California Institute of Technology, Pasadena, CA 91125 USA.
A. Eyal was with the Department of Applied Physics, California Institute
of Technology, Pasadena, CA 91125 USA. He is now with the Department of
Interdisciplinary Studies, Tel Aviv University, Tel Aviv 69978, Israel.
Digital Object Identifier 10.1109/LPT.2003.821254
II. THEORETICAL
BACKGROUND
Coupled-mode theory (CMT) establishes that the vector of
complex field amplitudes
for two weakly cou-
pled lossless waveguides evolves according to
(1)
where
represents the coupling matrix with local propaga-
tion constants of the individual waveguides
and
as diagonal elements and the coupling coefficients
between the waveguides
on the off diagonals
[4]. Here, we assume the convention that
is real. In converting
to the geometrical representation of CMT [5]–[7], the normal-
ized amplitude vector at some position
, , be-
comes a unit state vector in a real three-dimensional space using
the relation
(2)
where
are the Pauli spin matrices in the order
(3)
The coupling matrix
maps to the same real three-dimen-
sional space as a rotation vector
(4)
Using (2) and (4) in conjunction with (1), the transformed cou-
pled-mode equations are [6], [8]
(5)
The coupled-mode equations are now regarded dynamically
as sequential infinitesimal rotations of the instantaneous state
vector
on the unit sphere around the instantaneous rota-
tion axis
. As seen from (5), a state vector is stationary
when collinear with the rotation vector, thus, the antipodal
points
represent the two local eigenstates of the waveguide
system. For nonconstant
, a solution of (5) is obtained
by direct numerical integration. This pictorial representation
of a two-state system is analogous to the Poincaré sphere of
polarization optics [8].
III. C
OUPLER DESIGN
For the coupling matrix above, the rotation vector becomes
, where . Accordingly,
we consider only rotation vectors
that lie on the equator of the
sphere, with
. Factorization of the ro-
tation vector into
gives as
1041-1135/04$20.00 © 2004 IEEE
516 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 16, NO. 2, FEBRUARY 2004
Fig. 1. Power output ratio as a function of
j
j
for the raised cosine shape
function. Points A–C represent couplers with the desired 3-dB power output
ratio.
the two design degrees of freedom the magnitude of the rotation
vector
and the shape function . It should be noted that
while the rotation vector is confined to the basal plane, the state
vector might take any position on the sphere.
We now describe via an example how the above formalism
can be used for the design of novel couplers with unique
features. The device we examine is a 3-dB nonadiabatic
mode evolution coupler that, unlike a conventional directional
coupler, yields no phase shift between the output waveguides.
In choosing the shape function
to obtain the desired 3-dB
power output ratio and phase relationship, we require the initial
and final eigenstates, assuming light is input in waveguide 1, to
be
and , corresponding
in the geometrical representation to
and
. We thus need a smoothly changing function
with beginning and ending points
and . Various
shape functions have been compared [9], however, here, we
choose for illustrative purposes, the raised cosine function
(6)
where
is the half-length of the coupler and is an additive
constant preventing
(infinite waveguide separation).
The magnitude of the rotation vector is taken to be constant
throughout the length of the 3-dB portion of the coupler for sim-
plicity. Beyond the point where the desired 3-dB output is ob-
tained, the magnitude of the rotation vector can be exponentially
decreased, linearly separating the waveguides. This latter por-
tion is of little consequence in the current study and is ignored in
what follows. As discussed previously [10], to obtain the desired
power output ratio, it is important to carefully choose specific
ranges of the product
for a given shape function. As op-
tical component miniaturization and integration progresses and
as planar lightwave circuit complexity grows, coupler lengths
are severely restricted so it is more realistic in this point of view
to consider the length as restricted to a constant, and to care-
fully select ranges of
to obtain the desired output. Fig. 1
shows a plot of the power output ratio for the raised cosine shape
function as a function of
. As can be seen, there exist several
Fig. 2. Geometrical representation of the eigenstate (
) and system state (
2
)
trajectories. The system states begin occupying one eigenmode, i.e., light is
injected into waveguide 1, and end in the eigenmode given by equal output
power and phase. (a)–(c) depicts trajectories as calculated for
j
j
given by points
A–C, respectively, for Fig. 1. In (d), we show the coincident system state and
eigenstate trajectories of a long-length adiabatic coupler for comparison.
values of
where the desired power output ratio (0.5) is ob-
tained. Shape functions exhibiting smaller ripple exist, but at
the cost of larger
or longer length. Larger results in cou-
plers that change rapidly in shape, increasing scattering loss, and
alternatively longer lengths are against the spirit of this study.
Referring to Fig. 1, the perfectly adiabatic limit results from
.
Once the magnitude
and shape function are
chosen to achieve the desired output ratio and phase, the
refractive index profile is computed as follows. For simplicity,
we model the system as a slab waveguide structure of length
300
m with core and cladding refractive indexes of 1.55 and
1.5. The first element of the rotation vector immediately yields
the individual propagation constants
and
, assuming a constant average propagation
constant
that ensures single-mode operation.
From these, the waveguide widths are obtained by inverting
the solutions for the propagation constants of slab waveguides,
as detailed in [4]. Finally, the second element of the rotation
vector is used in the equation relating the coupling coefficients
to the waveguide widths and center-to-center separation.
IV. W
AVELENGTH INSENSITIVITY AND
FABRICATION
TOLERANCE
Consider points A–C in Fig. 1, where the magnitude of the
rotation vector yields couplers exhibiting the desired 3-dB
power splitting and phase relationship. To give a geometrical
understanding to couplers designed with
at these points, we
calculate the state vector trajectories and plot them in the ge-
ometrical representation along with the eigenstate trajectories
in Fig. 2(a)–(c), corresponding with the letter designations in
Fig. 1. These trajectories represent the crossing of the final state
and the final desired state. With increasing
, the trajectories
tend toward adiabatic behavior, for which the state vector and
eigenstate remain arbitrarily close, as in Fig. 2(d).
To illustrate the wavelength insensitivity of the power output
ratios and relative output phases for such nonadiabatic 3-dB
coupling structures, we calculate the state vector trajectories as
PALOCZI et al.: WAVELENGTH-INSENSITIVE NONADIABATIC MODE EVOLUTION COUPLERS 517
Fig. 3. (a) and (b) Power output ratios and relative output phases as a function
of wavelength and (c) and (d) width scaling factor. The dashed–dotted (–
1
–) line
represents a symmetric directional coupler for comparison. The dashed (–––)
and solid (—) lines give the output ratios and relative phases corresponding to
couplers designed with
j
j
as given by points A and B in Fig. 1, respectively.
a function of wavelength using the model parameters described
above. The power output ratios are plotted in Fig. 3(a), over a
wavelength range of 1300–1600 nm. For comparison, the wave-
length dependence of a standard symmetric directional coupler
is also shown. Clearly the coupler designed using
chosen
at point A is a marginal improvement over a directional cou-
pler, but the choice of
at point B yields a coupler with a
good degree of wavelength invariance. Point C gives even fur-
ther improvement, but is not shown, as it essentially matches
the desired 3-dB output throughout the band. The relative output
phases are shown in Fig. 3(b) and, as for the case of the power
output ratio, a good degree of phase invariance is obtained for
the choice of
at point B in Fig. 1.
In photo- or electron-beam lithography used for the produc-
tion of optical integrated circuits, an exposure error may result in
a disparity of the intended waveguide widths and actual values.
To simulate this, we calculate the power output ratio and output
phases while varying the intended waveguide width via a scaling
factor ranging from 0.75 to 1.25. The power output ratio results
as calculated using matrix methods are shown in Fig. 3(c) and
the corresponding relative output phase calculations are shown
in Fig. 3(d). Again, as in the case of wavelength variations, we
see marked improvement over a directional coupler when
is chosen as point B of Fig. 1.
As an independent corroboration of the results as calculated
using the matrix methods, beam propagation method (BPM)
simulations [11] were employed. Results of the simulations
agree quite well with the wavelength and scaling factor re-
sponses, as predicted by the matrix method.
V. C
ONCLUSION
We have presented a scheme for the design of optical couplers
with a high degree of wavelength insensitivity and fabrication
tolerance. It has been demonstrated that it is possible to ob-
tain properties similar to those of purely adiabatic couplers, i.e.,
good wavelength insensitivity and fabrication tolerances, if the
design parameters are chosen with care. The physical reason for
the adiabatic-like features is due to the fact that, as the wave-
length is varied or the waveguide shapes are scaled,
is
changed by some amount. The key feature, therefore, to at-
taining the desirable properties of adiabatic couplers, is the slope
of the curve shown in Fig. 1 near the points of 3-dB output.
The smaller the slope, the smaller the deviation from the de-
sired output.
A
CKNOWLEDGMENT
The authors gratefully acknowledge the helpful discussions
with J. Choi, W. Green, Dr. B. Marshall, Prof. S. Mookherjea,
and Dr. J. Scheuer.
R
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