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  • Modeling Telecommunications Components

Photonics Spectra
May 2002
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 given channel.1,2

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, well-known formulas.

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 Drop fiber.

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 fiber.

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 intended signal.

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.

Out-of-band 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 wavelength.

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., pp. 1423-1435.

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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