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Description  |
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FIELD OF THE INVENTION
This invention relates to monolithic optical waveguide filters and, in
particular, to a new type of monolithic filter providing plural optical
paths wherein each optical path corresponds to a harmonic component of a
Fourier series comprising the filter transmission function.
BACKGROUND OF THE INVENTION
Optical filters are important devices in optical fiber communications
systems. Monolithic optical waveguide filters are particularly promising
because they can perform complex circuit functionalities and because they
can be made by mass production integrated circuit techniques.
The requirements of optical filters vary with applications. Many
applications require a rectangular wavelength response in order to
maintain a low-loss and wavelength-independent transmission in a passband
and a high-level rejection to all wavelengths in a stopband. For example,
anticipated telecommunications applications seek a 1.3/1.551 .mu.m WDM
filter a flat and low-loss passband at 1.280-1.335 .mu.m and a-50 dB
stopband at 1.525-1.575 .mu.m. Another desired application is a gain
equalization filter to flatten the gain of an Er-doped fiber amplifier
chain. This requires an equalization filter with an amplitude response
which is essentially the inverse of the amplifier gain.
Various devices have been proposed to fill these new, demanding
requirements but none are fully satisfactory. Multilayer thin-film filters
can be used to construct optical filters in bulk optics, but they are
undesirable because they cannot be readily integrated and because of
difficulties in coupling light to and from fibers. Mach-Zehnder (MZ)
interferometers have been widely employed, but they have a sinusoidal
response, giving rise to strongly wavelength-dependent transmission and a
narrow rejection band. Other designs have encountered a variety of
practical problems. Accordingly, there is a need for a new type of
monolithic optical waveguide filter.
SUMMARY OF THE INVENTION
In accordance with the invention, a new type of monolithic optical
waveguide filter comprises a chain of optical couplers of different
effective lengths linked by different differential delays. The transfer
function of the chain of couplers and delays is the sum of contributions
from all possible optical paths, each contribution forming a term in a
Fourier series whose sum forms the optical output. A desired frequency
response is obtained by optimizing the lengths of the couplers and the
delay paths so that the Fourier series best approximates the desired
response. The filter is advantageously optimized so that it is insensitive
to uncontrolled fabrication errors and is short in length. The wavelength
dependence of practical waveguide properties is advantageously
incorporated in the optimization. Consequently, the filter is highly
manufacturable by mass production. Such filters have been shown to meet
the requirements for separating the 1.3 and 1.55 .mu.m telecommunications
channels and for flattening the gain of Er amplifiers.
BRIEF DESCRIPTION OF THE DRAWINGS
The advantages, nature and various additional features of the invention
will appear more fully upon consideration of the illustrative embodiments
now to be described in detail in connection with the accompanying
drawings. In the drawings:
FIGS. 1a, 1b and 1c are schematic design layouts of 1.3/1.55 .mu.m
wavelength division multiplexing filters (WDM filters);
FIGS. 2a and 2b are diagrams useful in illustrating the design principle
that the filter output is the sum of all optical paths;
FIGS. 3a, 3b and 3c are schematic examples of four-coupler chain filters of
consecutive odd Fourier harmonics;
FIGS. 4a, 4b, and 4c illustrate the approximation of a rectangular filter
response by a Fourier series;
FIGS. 5a, 5b, 5c, 5d, 5e and 5f show the effect of various steps to
optimize a five coupler chain filter;
FIG. 6 illustrates the basic functions of a coupler-delay chain as a four
port optical filter;
FIGS. 7a, 7b and 7c are graphical illustrations of the transmission spectra
for embodiments of WDM filters of the designs shown in FIGS. 1a, 1b and
1c, respectively;
FIG. 8 shows the amplitude response of a gain equalization filter in
accordance with the invention designed to flatten the overall gain of
Er-doped amplifiers; and
FIG. 9 is a schematic top view of a simple filter according to the
invention.
It is to be understood that these drawings are for purposes of illustrating
the concepts of the invention and, except for graphical illustrations, are
not to scale.
DETAILED DESCRIPTION
This description is divided into five parts. In part I, we describe the
basic elements of a simple filter in accordance with our invention. In
part II we describe the physical fabrication of the filter. Part III is
directed to the design of the configuration of waveguides to obtain a
desired filter response. Part IV discusses practical considerations which
assist in the fabrication of practical filters; and Part V discusses
preferred uses of the filters in optical fiber communications systems.
I. The Basic Elements Of A Simple Filter
Referring to the drawings, FIG. 9 is a schematic top view of a simple form
of a monolithic optical waveguide filter 10 in accordance with the
invention comprising a pair of optical waveguides 11 and 12 on a substrate
13 configured to form a plurality N of at least three optical couplers 14,
15, and 16 alternately connected by a plurality of N-1 delay paths 17 and
18. Each coupler is comprised of a region of close adjacency of the two
waveguides where the exponential tail of light transmitted on each of
waveguides 11 and 12 interacts with the other, coupling light from one
waveguide to the other. The amount of power coupled from one waveguide to
the other is characterized by the effective length of the coupler. The
effective lengths of the couplers preferably differ from each other by
more than 5%. The effective length of any coupler is within 5% of at most
one other coupler.
Each delay path comprises a pair of waveguide segments between two
couplers, for example segments 17A and 17B between couplers 14 and 15. The
segments are configured to provide unequal optical path lengths between
the two couplers, thereby providing a differential delay. For example in
FIG. 9 upper segment 17A is longer than lower segment 17B, providing a
differential delay which can be denoted positive. Differential delays
associated with longer lower segments can be denoted negative. Delay path
18 provides a negative delay because lower segment 18B is longer than
upper segment 18A. In the preferred form of filter 10, at least one
differential delay differs from at least one other by 10% or more and at
least one differential delay is opposite in sign from at least one other.
In operation, an optical input signal is presented at an input coupler,
e.g. along waveguide 11 to coupler 14, and a filtered output is presented
at an output coupler, e.g. along waveguide 12 at coupler 16. The sequence
of couplers and delays provide light at the input with a plurality of
paths to the output. In general there will be 2.sup.N-1 paths where N is
the number of couplers. For example, the FIG. 9 device presents the
following four paths:
1) segment 17A--segment 18A
2) segment 17A--segment 18B
3) segment 17B--segment 18A
4) segment 17B--segment 18B
In accordance with an important aspect of the invention, each of the
optical paths of the filter provide light corresponding to a harmonic
component in a Fourier series whose summation constitutes the transmission
function of the filter. By proper choice of parameters one can closely
approximate a desired transmission function. By choice of N and the set of
differential delays, one can design a filter presenting a summation of odd
Fourier components particularly useful for fabricating a filter with a
rectangular response such as a 1.3/1.55 .mu.m WDM filter. With a different
N or a different set of differential delays, one can also design a filter
presenting a summation of both even and odd Fourier components. A
preferred odd harmonic filter can be made by providing differential delays
with a normalized ratio of.+-.1/.+-.2/.+-.2/ . . . /.+-.2 in any order,
and a preferred all harmonic filter can be made by providing differential
delays with a normalized ratio of .+-.1/.+-.1/.+-.2/.+-.2/ . . . /.+-.2 in
any order. Moreover, in the above ratios, any but not all of the
differential delays of .+-.2 can be replaced by .+-.4 and any but not all
of the differential delays of .+-.4 can in turn be replaced by .+-.8. In
addition, for broadband filters, the wavelength dependence of practical
waveguide properties is advantageously taken into account, which can alter
the above proportional differential delays by up to .+-.25%. Thus in one
preferred embodiment the delay paths provide, in any order, within
.+-.25%, one differential delay of proportion .+-.1 and one or more
differential delays of proportion .+-.2 or .+-.4 or .+-.8. In another
preferred embodiment the delay paths provide, in any order, within
.+-.25%, one differential delay of proportion .+-.1, one more differential
delay of proportion .+-.1, and one or more differential delays of
proportion .+-.2 or .+-.4 or .+-.8. Advantageously the filter can be
combined with other filters, as by connecting the other filters to the
waveguide outputs, thereby producing filter networks.
II. Physical Fabrication
The FIG. 9 structure is advantageously fabricated using planar optical
waveguide technologies. Doped silica optical waveguides are preferred
because they have low loss, low birefringence, are stable, and can couple
to standard fibers well. However, the invention is equally applicable to
other integrated optical waveguides including III-V semiconductor optical
waveguides and optical waveguides diffused in lithium niobate. A
description of the above waveguide technologies can be found in R. G.
Hunsperger, "Integrated Optics: Theory and Technology", 3rd ed.
(Springer-Verlag, Berlin, Heidelberg, New York 1991) which is incorporated
herein by reference.
With doped silica plannar waveguides, the FIG. 9 structure can be
fabricated much as described in C. H. Henry et al., "Glass Waveguides On
Silicon For Hybrid Optical Packaging," J. Lightwave Technol., vol. 7, pp.
1530-39 (1989). In essence a base layer of silica glass (SiO.sub.2) is
grown on a silicon or quartz substrate. A thin core layer of doped silica
glass is then deposited on the base layer. The core layer can be
configured to a desired waveguide structure, such as that shown in FIG. 9,
using standard photolithographic techniques. Subsequently another layer of
silica glass is deposited to act as a top cladding. The waveguide cores
have a higher refractive index than the base and top cladding layers,
thereby guiding the lightwave much as a fiber. In contrast to a fiber,
however, planar waveguide technologies are capable of more precise control
of the waveguide structures and of integrating many devices into complex
circuits.
In the specific examples discussed below, the following procedure was used
to fabricate the filter. First a.about.15 .mu.m thick base layer of
undoped SiO.sub.2 (HiPOX) is formed by oxidation of Si under high pressure
steam. A core layer of 5 .mu.m thick 7% P-doped SiO.sub.2 (p-glass) is
then deposited using low-pressure chemical vapor deposition (LPCVD). The
core layer is annealed in steam at 1000.degree. C. is patterned for single
mode waveguides of 5 .mu.m width with reactive ion etching (RIE). The
center-to-center separation of waveguides in the coupler regions is
9.25-9.5 .mu.m, and the minimum waveguide bend radius is 15 cm. A flow
layer of about 1.5 .mu.m thick B- and P-doped SiO.sub.2 (BPTEOS) is
subsequently deposited with LPCVD and annealed at 900.degree. C. to fill
the narrow gaps in between waveguide cores in the coupler regions.
Finally, two layers of BPTEOS of 7.5 .mu.m thickness each are deposited
and annealed as the top cladding. The top cladding has almost the same
refractive index (1.45 at .lambda.=1.4 .mu.m) as the base, and the core
has a refractive index of about 0.63% higher than that of the base and top
cladding.
III. Configuration Design And Optimization
The basic structure of our filters consists of a chain of N arbitrary
couplers and N-1 differential delays, where N>2. The transmission spectrum
of such a chain (referred to as an N-coupler chain) is the sum of
contributions from 2.sup.N-1 optical paths, each of which forms a term in
a Fourier series. The length of the couplers and delay paths can be
optimized so that this Fourier series best approximates a desired
frequency response. As an initial example we describe design of the
aforementioned 1.3/1.55 .mu.m WDM filter with a rectangular response.
FIGS. 1a, 1b and 1c are layout examples of 1.3/1.5 .mu.m WDM filters. FIG.
1a shows a basic five coupler chain. Arrows indicate input and output
ports. FIG. 1b shows a doubly filtered five-coupler configuration. FIG. 1c
show a combination of double and triple filtering of four-coupler and
five-coupler chains. When fabricated with the aforementioned doped silica
waveguide technology, the total lengths of the 1a, 1b and 1c filters are
43, 75 and 75 .mu.m, respectively, and the heights are 0.3, 0.6 and 0.6
mm, respectively. The vertical scale has been expanded 20 times for
clarity.
The Principle of Sum Over All Possible Optical Paths
The electric fields at the two output ports of an ideal coupler of two
identical waveguides are related to those at the input ports by a transfer
matrix
##EQU1##
where i=.sqroot.-1,.phi.=.pi.1'/2L,1' is the geometric length of the
coupler, and L is the coupling length which is a measure of the strength
of coupling between the two waveguides (not to be confused with the
geometric length of the coupler). We refer to 1'/L as the effective length
of the coupler.
Similarly, the transfer matrix characterizing the differential delay
between two identical waveguides is
##EQU2##
where .theta.=.pi.sn/.lambda.=.pi.snv/c, s is the difference in the
lengths of the two waveguides, n is the effective refractive index of the
waveguides, and .lambda., .nu., and c are the optical wavelength,
frequency, and velocity in free space. Note that a common phase factor has
been ignored in Eq. 2 because it is non-essential to the filter response.
The transfer matrix of our waveguide filter, consisting of a chain of N
couplers and N-1 differential delays characterized by
.phi..sub.1,.phi..sub.2, . . . .phi..sub.N, and .theta..sub.1,
.theta..sub.2, . . . .theta..sub.N -1, respectively, is given by
T.sub..phi..spsb..theta. =T.sub..phi..sbsb.N T.sub..theta..sbsb.N-1. . .
T.sub..theta..sbsb.2 T.sub..phi.2 T.sub..theta.1 T.sub..phi.1(3)
From the above matrix product we see that the transfer function from any
input port to any output port consists of a sum of the form
t.sub..phi..spsb..theta. =.SIGMA.f(.phi..sub.1, .phi..sub.2, . . .
.phi..sub.N)e.sup.i(.+-..theta..sbsp.1.sup..+-..theta..spsb.2 . . .
.sup..+-..theta..sbsb.N-1.sup.) (4)
Note that .theta. is proportional to .nu. which is proportional to
1/.lambda. if we ignore the wavelength dependence of n.
Physically, equation 4 can be interpreted as the following principle that
the response is the sum over all possible optical paths. The transfer
function from any input port to any output port of a chain of N couplers
and N-1 differential delays consists of the unweighted sum of
contributions of 2.sup.N-1 distinct optical paths. Each of such
contributions is a product of 2N-1 factors: traversing a coupler gives
cos.phi. without crossing and isin.phi. with crossing; traversing the
longer arm of a differential delay gives e.sup.i.theta. and the shorter
arm gives e.sup.-i.theta..
This principle is illustrated in FIGS. 2(a) and 2(b). FIG. 2a diagrams the
contributions from the paths in a coupler and a differential delay. FIG.
2b shows the cross state of a three coupler chain. Where we have used the
abbreviations c.tbd.cos.phi. and s.tbd.sin.phi.. The letter t designates
the transfer function. For N=3 there are four distinct optical paths from
any input port to any output port, and the transfer function shown in FIG.
2(b) is a sum of four terms. For non-ideal couplers, cos.phi. and
isin.phi. should be corrected accordingly, but the above principle is
still valid.
Construction of a Fourier Series
With arbitrary choices of the .theta.'s, the sum in Eq. (4) is normally not
a Fourier series because the terms in the sum do not normally represent
harmonics. However, Eq. (4) becomes a truncated Fourier series if the
ratio among the lengths of the differential delays satisfies certain
conditions. In the following, we define
.theta..sub.j (.nu.).vertline..gamma..sub.j.theta.0 (.nu.), j=1,2, . . . ,
N-1, (5)
where .theta..sub.0 contains the common wavelength-dependent part of
.theta..sub.1, .theta..sub.2, . . . .theta..sub.N-1 and has the same
period as the fundamental harmonic in the Fourier series. The unnormalized
ratio .gamma..sub.1 /.gamma..sub.2 / . . . /.gamma..sub.N-1 will be
referred to as the .theta.-ratio.
For WDMs with a rectangular response, because of the odd symmetry of the
required transfer function, we need a Fourier series with only odd
harmonics (see FIG. 4). We first consider the following two extreme
conditions which give consecutive odd harmonics:
Condition A: If N is even and the .theta.-ratio is.+-.1/.+-.1/ . . . /.+-.1
with any sign combinations, t.sub.100 .spsb..theta. is a Fourier series
with harmonics .+-..theta..sub.0, .+-.3.theta..sub.0, . . . ,
.+-.(N-1).theta..sub.0.
Condition B: If the .theta.-ratio is .+-.1/.+-.2/.+-.4/ . . .
/.+-.2.sup.N-2, in any order and with any sign combinations, t.sub.100
.spsb..theta. is a Fourier series with harmonics .+-..theta..sub.0,
.+-.3.theta..sub.0, . . . , .+-.(2.sup.N-1 -2).theta..sub.0.
For a given even N, condition B gives the maximum number of consecutive odd
harmonics while condition A gives the minimum. There are many other
conditions in between, such as:
Condition C: If the 74-ratio is .+-.1/.+-.2/.+-.2/ . . . /.+-.2, in any
order and with any sign combinations, t.sub..phi..theta. is a Fourier
series with harmonics .+-..theta..sub.0, .+-.3.theta..sub.0, . . . ,
.+-.(2N-3).theta..sub.0.
FIGS. 3a, 3b and 3c illustrate the construction of the Fourier harmonics
using N=4 with examples of .theta. ratios of 1/1/1, 1/2/4, and 2/1/-2,
each corresponding to a special case of conditions A, B, and C,
respectively. Note that a negative sign in the .theta. ratio corresponds
to an interchange of the longer and shorter delay arm. For clarity, we
have used .theta. in the drawing for .theta..sub.0. A negative .phi.
cannot be physically realized. However, .pi.-.phi. is equivalent to -.phi.
in that the transfer functions of the two cases differ only by a constant
phase.
While consecutive odd harmonics are needed for WDMs, other filter functions
may need even harmonics as well. A Fourier series with consecutive even
harmonics is constructed when one of the sections corresponding to the
.+-.1's in conditions A, B, and C is taken out of the N-coupler chain,
resulting in an (N-1)-coupler chain. A Fourier series with all consecutive
harmonics can also be constructed by halving all the delays in conditions
A, B, and C and adding another section with the minimum unit delay. The
.theta.-ratios for conditions A, B, and C become .+-.1/2/.+-.1/2/ . . .
/.+-.1/2(and N is odd),.+-.1/2/.+-.1/2/.+-.1/ .+-.2/ . . . /.+-.2.sup.N-4,
and 35 1/2/.+-.1/2/.+-.1/ . . . /.+-.1, respectively. The corresponding
normalized ratios of the differential delays are .+-.1/.+-.1/ . . .
/.+-.1,.+-.1/.+-.1/.+-.2/.+-.4/ . . . /.+-.2.sup.N-3, and
.+-.1/.+-.1/.+-.2/ . . . /.+-.2, respectively.
Optimization for a Given Filter Response
We have shown that the transfer function of a chain couplers and delays can
form a truncated Fourier series. FIG. 4 further demonstrates the basis of
approximating the frequency response of a desired filter by such a
truncated Fourier series (for the purpose of illustration, we have ignored
the phase). The MZ interferometer (N=2) only has the fundamental harmonic
and its frequency response sinusoidal as shown by the solid curve in FIG.
4(a). In contrast, the desired rectangular response of a filter is shown
by the solid curve in FIG. 4(b). For N>2 under conditions A, B, or C, the
transfer function also contains higher order harmonics, shown by the
broken curves in FIG. 4(a). The sum of the fundamental and higher order
harmonics, as shown by the broken curve in FIG. 4(b), can approximate the
rectangular response if their amplitudes and phases are chosen correctly.
In the following, in order to give a clear physical picture, we first
ignore the wavelength dependence of L (and hence the .phi.'s are
constants) and n and discuss the more general case later.
First, the common factor .theta..sub.0 of the differential delays can be
determined solely by the positions of the passband and the stopband,
similar to the case of a simple MZ. We want the fundamental harmonic to
have a phase of 0 or .pi. at the center of the stopband and .pi./2 at the
center of the passband. Therefore
.theta..sub.0s /.pi..tbd.sn/.lambda..sub.s =m, and .theta..sub.0.sbsb.p
/.pi..tbd.sn/.pi..sub.p =m.+-.1/2, (6)
where m is an integer, which we refer to as the order of the filter, and
the subscripts p and s refer to the center of passband and stopband. Since
the output ports can be interchanged, there is another configuration
corresponding to the interchange of s and p in Eq. 6. For our 1.3/1.55
.mu.m WDM filter, the best solutions are (m=3, .lambda..sub.p =1.322
.mu.m, .lambda..sub.s =1.542 .mu.m) and (m=3, .lambda..sub.p =1.566 .mu.m,
.lambda..sub.s =1.305 .mu.m). Here we have explicitly used the fact that
the transfer function expressed by the Fourier series is periodic in
frequency.
We will now find the values of the .phi.'s under a given condition that
renders the Fourier series with consecutive odd harmonics. Note that,
except under condition A, the number of harmonics exceeds the number of
couplers N. Therefore, the number of Fourier coefficients to be determined
is generally larger than the number of free variables available. Moreover,
for broadband filters, the phase response is not important, and we only
require .vertline.t.sub..phi..spsb..theta. .vertline..sup.2 to have the
desired response which is rectangular in our 1.3/1.55 WDM. We therefore
minimize an error function to solve for the .phi.'s under a given
.theta.-ratio as in the following:
E.sub..phi..spsb..theta. .tbd..intg.d
.nu..multidot.w(.nu.)(.vertline.t.sub..phi..spsb..theta.
.vertline.-t.sub.desired (.nu.).vertline.).sup.2 =min (7)
where w(.nu.) is a positive weighting function, and the integration is done
in the passband and stopband of interest. Since for our 1.3/1.55 .mu.m WDM
we do not constrain the transition between the passband and stopband, we
set w(.nu.)=0 in those regions (as in FIG. 4(c)). Eq.(7) represents a
nonlinear minimization problem which can be solved numerically by
iterative methods such as the simplex or conjugate gradient methods.
Although in our design of the 1.3/1.55 .mu.m WDM we have ignored the phase
response, if a particular phase response is desired, it can also be put
into the above equation and optimized together with the amplitude.
Note that E.sub..phi..spsb..theta. has many local and equivalent minima
which correspond to different configurations having similar or the same
amplitude response. For example, if one of the couplers in the middle of
an N-coupler chain is a full coupler (.phi.=(n+1/2) .pi. where n is an
integer) or a null coupler (.phi.=n.pi.), this coupler and the two
differential delays around it degenerate into the equivalence of a single
differential delay, and we effectively have an (N-1)-coupler chain. Thus
we can reproduce a minimum in E.sub..phi..spsb..theta. of an
(N-1)-coupler chain with an N-coupler chain, which is a local minimum.
Moreover, if the length of a coupler is changed in such a way that .phi.
is replaced by .phi.+2n.pi. or (2n-1).pi.-.phi., then .vertline.t.sub.100
.spsb..theta. .vertline. is unchanged, and we have an equivalent minimum.
Furthermore, a different sequence of .theta.'s or a different set of .+-.
signs in front of the .theta.'s gives different equivalent and local
minima. The different combinations of these variations give rise to large
number of configurations. Interchanging the two output ports brings in
another set of distinct configurations (which correspond to .phi..sub.1
=.pi./4 and .phi..sub.2 =3.pi./4 for the conventional MZ). Therefore,
physical insight is often needed to get good starting parameters for the
iterations to arrive to the best solution. We usually start with the
original MZ (N=2) and successively increase N, taking the results of N-1
as the starting parameter of N.
We still have to find the best choice of the .theta.-ratio that gives the
closest approximation of the required filter response for a given N. In
the extreme of condition A, all the Fourier coefficients can be satisfied
independently, but the number of harmonics in the series is small. In the
other extreme, under condition B, although the number of harmonics is
large, they are greatly constrained by the number of free variables.
Therefore, there is an optimum condition in between A and B. Our numerical
calculations show that condition C is in fact the optimum for filters with
a rectangular response.
FIGS. 5a through 5f illustrate the effect of various steps to optimize the
configuration design. FIG. 5a shows the bar and cross state power
transmission of a conventional Mach-Zehnder interferometer in a semi-log
plot.
The solid curves in FIG. 5(b) are the bar- and cross-state transmission
spectrum of a five-coupler chain satisfying condition C with a
.theta.-ratio of 1/2/-2/-2 (see FIG. 1 (a) for the layout), optimized when
assuming L and n are independent of .lambda.. Compared to the spectrum of
the corresponding MZ shown in FIG. 5(a), the width and flatness of the
passband, as well as the width and rejection of the stopband, are greatly
improved. The dashed curves in FIG. 5(b) show the corresponding
transmission of a six-coupler chain with a .theta.-ratio of 1/1/1/1/1
(i.e., condition A). Compared to the solid curves of the five-coupler
1/2/-2/-2 chain, the passband becomes narrower and the rejection lower. In
fact, the response of the this six-coupler chain is the same as a
four-coupler chain with a .theta.-ratio of 1/2/2. Similarly, when we
replace one or more of the 2's in the 1/2/-2/-2 chain by 4 or 8, we find
that the filter response becomes worse as well as the device length
becomes longer. We therefore conclude that condition C gives the most
efficient WDM configuration.
Wavelength Dependence of the Coupling Length and the Effective Refractive
Index
So far we have assumed that the coupling length L and the effective
refractive index n do not depend on wavelength, which is only valid for
narrow-band WDMs. In reality, L inevitably decreases as the the wavelength
is increased, because the optical field is more confined at shorter
wavelength. With our planar waveguide fabrication process, L increases by
about a factor of two as .lambda. changes from 1.55 .mu.m to 1.3 .mu.m,
implying that a 3 dB coupler at 1.3 .mu.m becomes a full coupler at about
1.55 .mu.m. Moreover, n also changes with .pi., because of the change in
confinement and the dispersion of the waveguide material, but the relative
change is much smaller (about 0.5% for our waveguide) than L. FIG. 5(c)
shows the response of the same five-coupler chain as in FIG. 5(b) (which
was designed for L=L.vertline..sub..lambda.=1.42 .mu.m =constant and
n=n.vertline..sub..lambda.=1.42 m =constant), except now the wavelength
dependent L and n are used in calculating the spectrum. It is apparent
that for wide-band WDMs, such as our 1.3/1.55 .mu.m WDM, the .lambda.
dependence of L and n is advantageously considered in the design.
With the .lambda. dependent L and n, each term in Eq. (4) is no longer
completely periodic, and it becomes necessary to optimize the .theta.'s in
Eq. 6 as well as the .phi.'s to get the best filter response. This is
illustrated by the solid curves in FIGS. 5(d) and (e). FIG. 5(d) shows the
response of the five-coupler 1/2/-2/-2 chain when only the coupler lengths
have been optimized with the .lambda.-dependent L and n, while FIG. 5(e)
shows the same when both the coupler lengths and the delay lengths have
been optimized. Only in the latter case have we recovered the rectangular
response comparable to that for constant L and n. With our planar
waveguide for the 1.3/1.55 .mu.m WDM example, the .theta.-ratio change is
from 1/2/-2/-2 to 1.187/1.978/-1.849/-2.031. In the re-optimization, we
usually use the results for constant L and n as starting parameters.
In summary, in our optical filter with a chain of arbitrary couplers and
differential delays, the basic building block has a quasi-periodic
transfer function. The transfer function of the chain is the sum of
contributions from all possible optical paths, each of which can form a
term in a Fourier series. The task of designing a filter is to optimize
the lengths of the couplers and the differential delays so that this
Fourier series best approximates the desired filter response. Fourier
expansion not only gives a direct and intuitive description of the
physical principle of the device, but also provides powerful and flexible
design procedures. Our Fourier expansion approach enables us to find the
most efficient .theta.-ratio (which is not 1/1/1/1/ . . . used in lattice
filters), to include the .lambda.-dependence of L and n which renders the
.theta.-ratio non-integral, and, as will be shown next, to arrive at fully
optimized configurations most immune to dominant fabrication errors which
involve negative signs in the .theta.-ratio.
IV. Practical Considerations
Because of the .lambda.-dependence of L and n, many of the equivalent
configurations discussed earlier become inequivalent, i.e., the
degeneracies are removed. We thus have many filter configurations which
give somewhat different response. The number of such configurations is
large.
The transmission of two of these configurations for our 1.3/1.551 .mu.m WDM
is shown in FIGS. 5(e) and (f). FIG. 5(f) shows a five-coupler chain the
same as that in FIG. 5(e) except the .theta.-ratio is approximately
1/2/2/2. The filter response is also similar to that in FIG. 5(e) except
for some fine details. However, a dramatic difference develops between the
two cases when the coupling length deviates from the nominal values, as
shown by the dashed curves in FIG. 5(e) and (f) for which a 10% overall
increase in L has been used in calculating the spectra. The design in FIG.
5(e) is less sensitive to the change in L than the design in FIG. 5(f).
For practical fabrication, the coupling length is the parameter most
susceptible to uncontrolled fabrication errors (e.g., errors in the
geometry and refractive index of the core), and the design in FIG. 5(f)
would have a low yield. Therefore, we always choose out of many
configurations the few which are most stable against overall changes in
the coupling length as well as with the best filter response and short
couplers.
We have designed filters with various response shapes and bandwidths, and
for all cases the configurations most stable against overall changes in L
have negative signs in the .theta.-ratio. While the problem of stability
is complicated in nature partly due to the wavelength-dependence of L, a
simple physical interpretation is as follows. For a filter with
rectangular response to be stable against fabrication errors, the partial
sum of the fundamental and, successively, those of the low-order harmonics
in the Fourier series of the transfer function should be stable near their
zeros. If half of the .theta.'s have negative signs, the fundamental and
low-order harmonics correspond to the optical paths crossing the smallest
number of couplers, which can be regarded as the dominant optical paths.
The zeros of the partial sums produced by these dominant paths are stable
against errors in L because they tend to depend only on the ratio of the
lengths of the small number of crossed couplers. Filters with negative
.theta.'s usually also have short couplers. For the stable 1.3/1.55 .mu.m
WDM example shown earlier, the .theta.-ratio is 1.187/1.978/-1.849/-2.031.
This stable design is important for manufacturability.
Our design approach based on Fourier expansion also gives us clear
guidelines to determine the number of stages of the chain. In principle,
as the number of coupler stages is increased, the flatness of the passband
and the rejection in the stopband will be constantly improved. In
practice, however, several factors limit N from being too large. The first
limitation is chip size. With our current fabrication process on a
five-inch wafer, the largest N is about 14 for short delay arms (such as
those used in our 1.3/1.55 .mu.m WDMs) without using waveguide U-turns
(which would consume a large space on the wafer). This limit of N can be
increased, however, if high-delta waveguides or U-turns are used. The
second limitation is excess insertion loss, due to the bends at the ends
of each coupler and intrinsic loss in the waveguides. A more subtle yet
important consideration is fabrication accuracy and non-ideal effects of
couplers and delays. As N is increased, higher order Fourier components,
with smaller Fourier coefficients, come into play. However, if the
fabrication error exceeds the accuracy required by the smallest Fourier
coefficient, increasing N no longer improves the performance of a real
device. Similarly, because of non-ideal effects such as excitations to
higher order modes at the ends of couplers and asymmetric loss in the
delay paths, Eq. (4) is only an approximate description of the transfer
function of a real device, and N should be small enough that every term in
Eq. 4 is meaningful. For our 1.3/1.55 .mu.m WDMs we find that N=3 to 7 is
adequate and practical.
To enhance the stopband rejection to >30 dB, We have adopted double- and
triple-filtering schemes using short chains of N=3 to 7, as illustrated in
FIG. 1 (b) and (c). FIG. 1 (b) is a double filtered version of (a) and (c)
is a combination of double and triple filtering of four-coupler and
five-coupler chains. For such a multistage filter, the transfer function
is simply the product of that of each stages:
##EQU3##
The unwanted light power is thrown away in the extra ports instead going to
the other output port, and the crosstalk is reduced at the expense of
rounding the flatness of the passband. This can be regarded as a further
optimization of the filter under the constraints of chip length and
fabrication errors.
V. System Applications
In this section, we discuss the potential applications of our filters in
optical fiber communication systems.
Our new filter (first, without double or triple filtering) has the
following properties. First, the two output ports are complimentary in
that the power in the two ports always sums up to the input power. Thus
the passband of one output is the stopband of the other (and for this
mason we have often used the terms "passband" and "stopband" without
specifying exactly where they are). Second, the device is symmetric and
reciprocal such that if the two input ports and simultaneously the two
output ports are interchanged, or if the input ports are interchanged with
the output ports, the transmission remains the same up to a constant phase
factor. Thus the device only has two distinct transmission states: the bar
state and the cross state. Third, the device is highly directional, i.e.,
the light propagating in one direction is independent of the light
propagating in the reverse direction.
The basic functions of the device are illustrated in FIG. 6, where
.lambda..sub.1 and .lambda..sub.2 are the passband of the bar and cross
state respectively, or vice versa. (Also, not shown here for simplicity,
both .lambda..sub.1 and .lambda..sub.2 can be groups of wavelengths.) As
an add-drop filter, it transmits .lambda..sub.1 from port 1 to port 3,
while dropping .lambda..sub.2.sup.- into port 4 and adding the
.lambda..sub.2.sup.+ from port 2. Note that the direction of any of the
signals can be reversed. If .lambda..sub.2.sup.- is absent in port 1, the
device works as a wavelength division multiplexer which combines
.lambda..sub.1 and .lambda..sub.2.sup.+ in port 3. Similarly, if
.lambda..sub.2.sup.+ is absent in port 2, the device works as a
wavelength division de-multiplexer which separates .lambda..sub.1 to port
3 and .lambda..sub.2.sup.- to port 4. In the de-multiplexer, if the light
in one of the output ports is simply discarded, the device works as a
simple filter.
With double or triple filtering, the situation is more complicated. The
filter now has more than four ports, not all of which are used. However,
all the above functions can still be realized with different
configurations using the appropriate ports.
Multiplexing and De-multiplexing of the 1.3 and 1.55 .mu.m Communication
Bands
Most existing fiber communication systems use the 1.3 .mu.m band mainly
because fibers have zero dispersion and relatively low loss around 1.3
.mu.m and other components were also first developed for 1.3 .mu.m. Fibers
have even lower loss away from the dispersion zero around 1.55 .mu.m, and
Er-doped fiber amplifiers (EDFA) are also readily available in the same
wavelength range. Therefore, the 1.55 .mu.m band is expected to coexist
with the 1.3 .mu.m band in many future communication systems, and 1.3/1.55
82 m WDM filters will become a key component in such systems.
To accommodate analog-signal transmission and future upgrade, 1.3/1.55
.mu.m branching devices used for some controlled environments should have
passbands wider than 1.280-1.335 .mu.m and 1.525-1.575 .mu.m for the two
output channels respectively, and the crosstalk should be lower than -50
dB. For other applications, the required passbands are even wider (100
nm). Our new filters fabricated with the planar waveguide technology are
ideal candidates for these WDMs, because they have the rectangular
response required, they are monolithic and reliable, and they can be
integrated with other components.
Using the double and triple filtering approach, we have designed 1.3/1.55
.mu.m WDMs meeting the above specifications, which correspond to the
layouts in FIGS. 1(b) and (c). FIG. 1(b) is a double filtered version of
(a) and FIG. 1(c) is a combination of double and triple filtering of
four-coupler and five-coupler chains. The triple filtering in FIG. 1(c) is
only in the 1.3 .mu.m path to enhance the rejection at 1.55 .mu.m. They
have 10 or 12 couplers in series in total and when fabricated with the
aforementioned doped silica waveguide technology they are 75 mm long and
0.6 mm wide. For the five-coupler chains in FIG. 1(a), (b), and (c), the
geometric lengths of the couplers are 757,795, 73, 1209, and 452 .mu.m,
and the geometric path differences are 3.754, 6.256, -5.849, and -6.42
.mu.m. For the four-coupler chain in FIG. 1(c), the geometric lengths of
the couplers are 677, 979, 199, and 1241 .mu.m, and the geometric path
differences are 2.483, 5.733, and -6.055 .mu.m.
The solid curves in FIG. 7 show our preliminary results for the designs
shown by the corresponding dashed curves. The layout of the three designs
is shown in FIGS. 1(a), 1(b) and 1(c). For all the designs the measured
data closely resembles the designed response. Wide and flat passbands
around both 1.31 .mu.m and 1.55 .mu.m are apparent. Also as designed, the
stopbands are wide (.about.100 nm) and the transitions are sharp. These
aspects of our WDMs are better than those of any broadband monolithic WDMs
previously reported.
Gain Equalization Filters for EDFA Systems
Er-doped fiber amplifiers (EDFA) have great advantages over other optical
amplifiers used in fiber communication systems, but have a highly
wavelength-dependent gain. For long-range (>100 km) transmission of
lightwave signals through optical fibers, EDFAs are inserted at spans of
every .about.50 km to compensate the attenuation of signals in the fiber.
In such a system where many EDFAs are cascaded, in order to use the full
bandwidth of the EDFAs (1.53-1.56 .mu.m), a gain equalization filter must
be used along with each EDFA to flatten the overall system gain. The
response of these filters is roughly the inverse of the gain of the EDFA,
and has a peak at 1.538 .mu.m with asymmetric wings. Despite the irregular
shape required, these filters can be readily designed with our Fourier
expansion approach and fabricated using the planar waveguide technology.
The solid curve in FIG. 8 shows a designed EDFA gain equalization filter
using a seven-coupler chain. The circles represent the required filter
response, optimized for maximum end-to-end flatness over a 30 | | |
