**Computer modeling offers accurate simulation to speed design of WDM components. Benefits include optimizing coupling efficiency, predicting performance degradation due to alignment errors or polarization-dependent loss, and determining critical tolerances in manufacturing and reliability.**

**Matthew P. Rimmer**

A common passive optical component in a fiberoptic system gets input from a single-mode fiber and sends the output to another fiber. These components perform optical functions such as isolation, multiplexing, demultiplexing, coupling, or filtering. A typical component has a pair of separated collimators with additional optics in the space between the collimators to perform the optical function. The beam between the collimators is on the order of 1 mm in diameter with an optical path ranging from a few tens of mm to several hundred mm. There are a number of design issues associated with such a component, a critical one being the prediction of performance degradation due to manufacturing and alignment errors. Accurate computer modeling allows different scenarios to be investigated quickly.

One type of modeling uses the Code V program. Issues such as free space beam propagation, gradient-index materials, insertion loss, polarization-dependent loss, optimization, and tolerancing for manufacturing can be addressed by this method (see "Addressing free space beam propagation," p. XX). This set of tools provides an efficient way of evaluating different designs before fabrication.

In a layout of a generic fiberoptic component having two gradient index (GRIN) collimators and a space between for additional optics (see Fig. 1), some of the ray paths through the gradients are shown. A small space is used for illustrative purposes; actual space will vary with application. In the design example, a space of 50 mm is used, with a 1.8-mm diameter aperture at the mid-point of the space. The collimators are 1.8 mm in diameter and are made of SLC1.8 SELFOC material from Nippon Sheet Glass (NSG).

The ends of the collimators near the fibers are wedged by eight degrees to keep back-reflected light from propagating back into the fiber. The wedge is large enough to keep the reflected light outside the numerical aperture of the fiber. The other end of the collimators can also reflect light back into the fibers but, in this case, the input fiber can be shifted about 10 µm from the center of the collimator to keep the back-reflected focused spot away from the fiber core. All surfaces in a system like this have anti-reflection coatings to further reduce the reflections. This system is designed for the C-band (1520 to 1560 nm) using SMF-28 fibers.

**DESIGN ISSUES**

The first step in setting up the system is to determine the length of the GRIN collimators to image the fiber mode at the end of the second collimator. Nominally, the collimators would be 1/4-pitch lenses, the length of which produces a collimated beam based on ray tracing. However, ray tracing does not adequately represent the propagation of the fiber mode through the system. The Gaussian nature of the fiber mode, diffraction spreading within the system, and clipping by the finite size of the optics will affect the focal position. An accurate optical propagation calculation was used to determine the optimum length of the lenses.^{1} For this system, the collimator length is 4.902 mm, somewhat larger than the 1/4-pitch length. The collimators are identical in order to produce a 1:1 magnification.

The variation of insertion loss and polarization-dependent loss (PDL) with wavelength from 1520 to 1560 nm is shown in Figure 2. The insertion loss is calculated by propagating the Gaussian mode from the input fiber through the system and integrating the resulting optical field with the fiber mode. These calculations take account of diffraction spreading within the system as well as clipping by the finite size of the elements and apertures. Insertion loss is due to a variety of factors including the mismatch between the optical field and the fiber mode, absorption and scattering within the system, and clipping by apertures. The insertion loss data show the effects of the mismatch between the optical field and the fiber mode.

The PDL is the maximum variation of insertion loss as a function of input polarization. It is obtained by calculating the complex vector fields for two orthogonal polarization states and doing the overlap integral between these and the fiber mode. Since any polarization state is a linear combination of the two orthogonal states, this information is enough to get the maximum and minimum values. In this system, the tilted faces on the collimators cause the PDL.

Misalignment of the output fiber with respect to the optical field exiting the second collimator will produce further insertion losses. The insertion loss varies with tilt and decenter of the fiber (see Fig. 3).

**TOLERANCES**

Tolerancing is a critical part of designing an optical system. The goal is to specify manufacturing, alignment, and environmental tolerances for the system and to predict the yield from the process (see Table 1). Compensators are parameters that can be adjusted during the manufacturing or assembly process to minimize the effect of the error in the parameter being toleranced. These parameters give looser tolerances than might be obtained without a compensator. Errors such as the wedge error on the faces of the collimators next to the air space will deviate the beam. In the assembly process, the position of the output fiber can be adjusted to compensate for this deviation. The compensator for an error in the SELFOC constant, which describes the radial gradient, is the length of the element, but that adjustment would be done during the manufacturing process.

Some parameters do not have compensators associated with them. A thermal soak is an environmental perturbation and, in a passive system like this, has no compensator. The thermal soak in this example was modeled by assuming a uniform growth of the elements due to the thermal expansion of the glass and a change in the air space resulting from the expansion of an aluminum spacer. In addition, a change in index of refraction due to the index sensitivity to temperature (dn/dT) was applied. The expansion coefficient of the glass was assumed to be 10 x 10-6°C and for the aluminum, 23.5 x 10-6°C. NSG does not provide a dn/dT value for its SELFOC materials so an estimate of 1.5 x 10-6°C was used.

In the case of the air space between the collimators, there is no effective compensator because the beam is close to collimated in this space. However, as the results will show, this is a very insensitive parameter and can have a large tolerance. Another example of a parameter without a compensator, although not shown here, is the mode size.

The basic sensitivities (including compensation) of the various parameters that are being toleranced can be determined (see Table 2). For each tolerance, the change in insertion loss (in dB) is given for both a positive and negative change in the parameter. Note that the changes are not necessarily symmetric. The insertion loss in the vicinity of the nominal design varies quadratically with the change in the parameter and the peak of the quadratic may not be at the nominal design. Thus, it is possible for the insertion loss to improve in one direction and degrade in the other over a small range of perturbations. The main drivers of insertion loss in this example are the GRIN lengths and the decenter of the input fiber.

Results of a Monte Carlo analysis of this system, using the tolerances in Table 1, are shown in Figure 4. For each sample, the parameters were varied randomly within the specified tolerance ranges, and a statistical summary was generated. The graph shows the probability of reaching a given insertion loss level. For example, there is an 80% probability of generating a system with an insertion loss better than -0.07 dB with these tolerances. This information gives an estimate of the yield of a manufacturing process.

**ACKNOWLEDGMENT**

The author would like to thank Tom Bruegge and Tom Kuper, both of Optical Research Associates, for helpful discussions.

**REFERENCE**

- T. J. Bruegge et al., Proc. SPIE, 3780, 14 (1999).

**Matthew P. Rimmer** is chief scientist at Optical Research Associates, 3280 E. Foothill Blvd., Pasadena, CA 91107. He can be reached at 626-795-9101 or mrimmer@opticalres.com.

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**ADDRESSING FREE SPACE BEAM PROPAGATION**

For many conventional optical systems such as a camera lens, optical ray tracing provides a very accurate algorithm for determining the performance of the system. For the typical fiberoptic component, ray tracing is usually not sufficient. For example, in a system with two gradient index collimators separated by 50 mm, where the beam in the nominally collimated space is about 0.5 mm or less in diameter, ray tracing is a poor approximation. As the beam propagates, instead of maintaining the same diameter, as ray tracing would predict, it actually spreads out because of diffraction effects. So it is necessary to converge the initial beam slightly to account for this effect. This adjustment results in the center of the beam being about 5% smaller in diameter than the ends of the beam.

A common algorithm used to address this problem is free space beam propagation. It is based on Fraunhofer or Fresnel propagation where the Fresnel number of the propagation indicates the formulation to be used. The Fresnel number is a function of the wavelength, the beam diameter, and the propagation distance, and indicates whether the far field (Fraunhofer) approximation is appropriate. When the Fresnel number is very small, Fraunhofer propagation is used; otherwise Fresnel propagation is used.

Also, the Fresnel propagator can be used for propagation between spherical reference surfaces, with some modification. When evaluating a system, the algorithm starts with the initial optical field (typically, the fiber mode) and propagates it through each space (air or glass) in the system until it gets to the receiver (typically, another fiber). At each optical interface, refraction or reflection takes place.