Modeling Telecommunications Components
Limits of the Gaussian Approximation
Dr. Joe Shiefman
The term “crosstalk”
describes the phenomenon of power leaking from one channel of a wavelength division
multiplexing (WDM) system into a neighboring one. One of the most important design
issues for optical communications systems, crosstalk can seriously degrade system
performance because it represents noise power relative to the signal power of a
Determining the amount of crosstalk in a WDM
component under design is very dependent on the model that is used to represent
the mode field of the fibers carrying light into and out of the component. A single-mode,
step-index fiber is defined by a J0 Bessel function for the region inside the core
and by a decaying K0 Bessel function inside the cladding. This is referred to as
the fundamental fiber mode.
Often, modeling software for WDM components
represents the fundamental fiber mode with a Gaussian beam approximation. Historically,
this is because most photonics professionals are familiar with Gaussian beams and
because they enable determination of the beam parameters in any plane using simple,
Loss in accuracy
Despite the fact that a Gaussian beam appears
to have the same shape as the fundamental fiber mode, there are subtle and important
differences that can lead to performance problems. A Gaussian beam may offer reasonable
results for modeling component performance specifications such as insertion loss,
polarization-dependent loss, return loss and polarization mode dispersion, but it
is not good for modeling crosstalk.
To understand the loss in accuracy
caused by using the Gaussian approximation, consider a design for an optical add/drop
multiplexer, constructed using ASAP software from Breault Research Organization
Inc. of Tucson, Ariz. (Note that this is not intended to be a practical design.)
A component based on microelectromechanical
systems (MEMS) technology has two inputs (In and Add) and two outputs (Out and
Drop). Some of the channels, each of which is represented by a different center
wavelength, are removed from the input beam by sending them to the Drop fiber. Those
same channels are replaced by signals arriving via the Add fiber. In this way, it
is possible to control on a channel-by-channel basis whether the input signal from
each of the input channels goes to Drop or Out.
In an energy cross section of a fundamental
fiber mode superimposed on a Gaussian beam of a similar mode diameter for a Corning
SMF-28 fiber, the two modes appear to be very similar, but the Gaussian beam is
a little wider in a region around the 1/e2 energy value (Figure 1).
Figure 1. On a linear scale, the plots of relative energy as a function of position for an
SMF-28 single-mode fiber display little difference between the fundamental fiber
mode and the equivalent-size Gaussian beam mode approximation. In the graph, maximum
energy has been normalized to 1.
At distances beyond 7.25 μm from
the center of the mode, the two mode profiles again appear to be similar. What is
not apparent when the data is plotted on a linear scale such as this is the large
difference between the two modes in the low-energy tail region of the beam.
In the energy cross section of the
same functions plotted on a logarithmic decibel scale, it is evident that the two
energy plots cross in the region where the distance from the center of the beam
exceeds 7.25 μm and that the fundamental fiber mode begins to have significantly
more energy than the Gaussian representation (Figure 2). At the edge of the plot,
11 μm from the mode center, the energy in the fundamental fiber mode exceeds
that of the Gaussian beam by more than a factor of 10.
Figure 2. On a logarithmic scale, the plots of relative energy as a function
of position for the two mode representations display a significant
difference. This is most pronounced in the low-energy tails of the beams.
The model component
The add/drop multiplexer is composed of two identical
halves, an In/Drop half and an Add/Out half (Figure 3). A beam originating from
either an In or an Add input passes through a gradient-index lens that collimates
the beam and then is incident on a grating with a 2.3-μm period. The grating
has been constructed so that the incident light is diffracted into the —1 (Littrow)
order, and it is used to demultiplex the beam by diffracting the light of the different
channels at different angles, which will increase with wavelength in accordance
with the grating equation.
Figure 3. A ray trace
diagram illustrates two beams from neighboring channels through the In/Drop half
of an optical add/drop multiplexer based on microelectromechanical systems (MEMS).
Light entering through the In fiber is demultiplexed by the grating and imaged as
separate beams onto different micromirrors in the MEMS plane. After reflecting from
their respective micromirrors, the beams are multiplexed back into a single beam
that exits the component through the Drop fiber. By rotating individual micromirrors,
light entering through the In fiber can be directed out of plane to exit through
the Out fiber, which is in a plane parallel to the In/Drop plane with the Add fiber.
Similarly, light entering through the Add fiber can be directed into either the
Drop or the Out fiber.
For the sake of simplicity, Figure 3 shows only two beams from neighboring channels leaving the grating. The diffracted
beams pass through a lens that focuses them to separate spots in the MEMS plane.
The amount of the separation is approximately equal to the difference in their angles
multiplied by the focal length of the lens.
The spot is elliptical at the MEMS
plane, with the larger, Y-axis having a 1/e2 beam radius of 36 μm. This, combined
with the short (~7.25 mm) focal length of the focusing lens, drives the choice
of 20-nm wavelength spacing between channels, which is equivalent to an 84.5-μm
spacing between neighboring channels at the MEMS plane.
This 20-nm channel spacing corresponds
to a coarse WDM situation. Decreasing the channel spacing would require increasing
the focal length of the focusing lens and, hence, the package size. It also could
be facilitated by decreasing the grating period to increase the angular spacing
for a given wavelength separation and/or by increasing the beam size incident on
the grating so that it produces a smaller beam size at the MEMS plane.
The micromirrors in Figure 3 measure
80 x 80 μm and are the switching elements of the optical add/drop multiplexer.
When they are oriented parallel to the focusing lens, they direct light from the
In to the Drop fiber. By rotating the micromirrors so that they are no longer parallel
to the focusing lens, light is directed out of plane to the Out fiber. Similarly,
channels originating from the Add fiber can be directed either to the Out or the
Modeling insertion loss
Plots of insertion loss as a function of source
wavelength for both the fundamental fiber mode and the Gaussian beam mode approximation
can be compared to illustrate their differences (Figure 4). Insertion loss is the
fraction (in decibels) of the light that enters through an input fiber (In or Add),
reflects off a single micromirror and couples into the intended output fiber (Drop
and Out). The runs that generated the data points in this figure were for an ideal
optical add/drop multiplexer with no alignment or fabrication errors.
Figure 4. A plot of insertion loss as a function of wavelength for
the fundamental fiber mode and the equivalent-size Gaussian beam mode approximation
has a similar relative relationship to the energy plots of the two modes in Figure
2. The large difference in the low-energy tails of the beams results in significant
errors in the calculation of crosstalk between neighboring channels.
Each data point represents a run performed
with a different source wavelength, varying from the nominal channel wavelength
of 1.55 to 1.57 μm, which corresponds to the center wavelength of the neighboring
channel. This causes a shifting of the beam from its nominal location at the center
of the 1.55-μm channel micromirror to the center of the neighboring 1.57-μm
micromirror. By shifting the beam location in the MEMS plane, beam position errors
attributable to other sources may also be examined, including fabrication or misalignment
errors of the system elements, or environmental changes in the component.
Contributors to insertion loss include
Fresnel surface losses, the loss of that portion of the beam that falls off the
edge of the micromirror, edge diffraction and losses in coupling efficiency of the
return beam at the Drop fiber. The coupling efficiency is calculated via the overlap
integral of the return beam with the mode representing the fiber. When the optical
field of the return beam does not match the mode of the fiber in position, size,
shape or phase, it results in loss of coupling efficiency.
In Figure 4, it is obvious that the
relative relationship of the insertion loss between the two models mimics that in
Figure 2. The minimum insertion loss occurs at the nominal 1.55-μm wavelength
and is approximately —2.3 dB for both mode representation types.
As the wavelength increases above 1.55
μm, the fundamental fiber mode shows slightly more insertion loss than the
Gaussian until the wavelength reaches 1.563 μm. Above this point, the Gaussian
mode displays increasingly more insertion loss than the fundamental fiber mode.
At 1.57 μm, the difference is more than 11 dB. This difference and how it relates
to errors in the crosstalk calculation is crucial.
Figures 5a to 5d show the beam (using
the Gaussian beam mode approximation) incident on the 1.55-μm channel micromirror
and in a plane just in front of the Drop fiber. Figures 5a and 5b show the beam
at the micromirror and in a plane just in front of the Drop fiber for the 1.55-μm
source wavelength. In both cases, the beam is centered: on the micromirror in 5a
and on the fiber in 5b.
Figure 5. The beam from the channel’s 1.55-μm nominal
wavelength is imaged at the center of the micromirror (a), and it returns to a plane
just in front of and centered on the Drop fiber (b). A small portion of the tail
of the beam from the neighboring channel’s 1.57-μm nominal wavelength
spills over onto the micromirror of the 1.55-μm channel (c). The reflected
portion of the tail is significantly decentered relative to its output fiber (d).
Figures 5c and 5d examine the same
situations, but the source wavelength is 1.57 μm. In 5c, all but the tail portion
of the beam misses the 1.55-μm channel micromirror, so the amount of energy
reflecting from the micromirror is less than 1 percent of what it was for the 1.55-μm
source wavelength. The rest of the 1.57-μm beam (except the portion lost in
the 4.5-μm dead space between the micromirrors) is incident on the neighboring
1.57-μm channel micromirror. Because the fundamental fiber mode has more energy
in its tails, the amount of energy reflecting off the 1.55-μm channel micromirror
for the 1.57-μm source wavelength is greater than it is for the Gaussian mode.
In Figure 5d, the beam is in a plane
just in front of the Drop fiber. Although two-thirds of the reflected energy returns
to this plane, the return beam is shifted significantly from the center of the Drop
fiber (given by the intersection of the two white lines). The poor coupling efficiency
of this shifted beam drops the insertion loss almost five more orders of magnitude.
Here again, the insertion loss due
to poor coupling efficiency is less for the fundamental fiber mode than for the
Gaussian mode. This is because there is more overlap of the larger tail of the fundamental
fiber mode beam with the tail of the fundamental fiber mode representing the Drop
Effects on crosstalk
Plots such as Figure 4 offer information about
two system performance issues. The first is the amount of insertion loss as a function
of beam placement error. For acceptable systems, this issue appears in the region
of the plot near the nominal 1.55-μm wavelength. In this region, there is a
slightly favorable bias from a Gaussian mode approximation.
The second issue is crosstalk. The
effect of crosstalk appears in the region near the 1.57-μm wavelength, which
is the center wavelength of the neighboring channel. This wavelength value represents
the portion of power that leaks from the neighboring channel into the 1.55-μm
channel output because, even in a perfectly aligned system, a portion of the beam
from each channel spills onto the micromirrors of the neighboring channels. The
exact amount of crosstalk is a complicated function of this finite beam size, the
non-ideal system and the channel bandwidth.
Additionally, the amount of crosstalk
for a given system will vary in time because the system’s state varies with
its environment and because the channel bandwidth varies with the rate of signal
modulation. Nonetheless, it is clear that the Gaussian mode approximation produces
significant amounts of error in modeling crosstalk.
Moreover, the amount of crosstalk is
affected by the relative states of the neighboring micromirrors. Depending on
whether the positions of those micromirrors are the same or different, the main
and tail portions of the beam for a given channel can be directed either to the
same output fiber or to different fibers.
If the tail goes to a different output
fiber than the main portion (one to Drop and the other to Out), whichever portion
of the tail that couples into the fiber will produce crosstalk in the signals intended
for that output fiber. If another signal of the same wavelength (originating from
the other input fiber) is in that output fiber, it will add to it as homowavelength
or in-band crosstalk.3
Such a situation is illustrated in
Figure 6. The tail portion of the beam from the In fiber’s channel N ends
up at the Out fiber with the main signal beam (also channel N) from the Add fiber.
This type of crosstalk is a significant problem because it is from the same spectral
channel as the beam from the Add fiber and will therefore sum coherently with the
Figure 6. Depending on the relative
states of the neighboring micromirrors, the tail portion of the beam that entered
the optical add/drop multiplexer via the In fiber can exit in the same fiber as
the signal beam of the same channel that entered via the Add fiber. Although the
beam tail couples (albeit inefficiently) into the fiber, it leads to coherent, homowavelength crosstalk. This type of crosstalk
can be a significant problem. An analogous situation can occur for tails that originate
from either input fiber and that exit through either output fiber with the main
signal of the same channel that originated from the other input fiber.
This is true despite the fact that
the two signals originated from different laser sources, because the coherence time
of the pulses is relatively large compared with the signal detection time. Therefore,
the two sources will maintain a fixed phase relationship during signal detection.
Also, because coherent summing is based on amplitude rather than on amplitude squared,
as in the incoherent case, and because the phases from the two sources will vary
differently over time, there can be large time-varying signal fluctuations even
at very small levels of crosstalk.
If, however, the tail ends up at a different output
fiber than the main portion of the beam when no signal of the same wavelength is
present in that output fiber, then the problem is not as great. This is referred
to as heterowavelength or out-of-band crosstalk.3
Because the tail in that case does
not spectrally overlap with the other channels in the fiber, there is a good chance
that it will be removed from the other signals during demultiplexing and not show
up as crosstalk in the received signal. Even if it does show up in one of the other
channel’s final signal, it will sum incoherently because of its different
System crosstalk is one of the most
important design issues for optical communications. It is imperative that the software
model can accurately determine its impact in the design. Substituting a Gaussian
beam mode approximation for the fundamental fiber mode can significantly underestimate
the amount of crosstalk. This, in turn, could lead to WDM components that will not
meet system specifications.
1. G.P. Agrawal (1997). Fiber-optic communication
systems. John Wiley & Sons, p. 318.
2. J. Zhou et al (June 1996). Crosstalk
in multiwavelength optical cross-connect networks. JOURN. LIGHTWAVE TECH.,
3. G.P. Agrawal, op. cit., p. 319.
Meet the author
Joe Shiefman is an optical consultant at Shiefman
Consulting in Tucson, Ariz. He received his PhD from the Optical Sciences Center
at the University of Arizona in Tucson.
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