# Single Mode Fiber Coupling: Sensitivities and Tolerancing

Purpose:  The purpose of this tutorial is to show the sensitivities for the coupling of two single mode fibers.  The sensitivities for the lateral and longitudinal displacement, angular misalignment, and the mode field radius mismatch of the two fibers will be discussed.  Also shown will be an example tolerancing of two single mode fibers to achieve a loss of only 10%.  This is an example where the linearity of the sensitivities can not be assumed and the interdependence of the perturbations has to be considered.

Figure 1.  Single Mode Fiber Coupling

## Introduction

Single mode fibers are frequently used in telecom and other applications.  The alignment tolerances to couple into a single mode fiber are fairly restrictive.  For example, a lateral offset of only 1um causes the coupling efficiency to drop by 4%.  This tutorial shows the sensitivity to lateral and longitudinal displacement, angular misalignment, as well as a mode field diameter mismatch.  The figure of merit is coupling efficiency. Unlike the tolerancing examples shown in class, this is an example where the linearity of the figure of merit to a small perturbation can not be assumed.  Additionally, unlike the problems shown in class the perturbations are not independent.  A tilt of the fiber not only causes a loss but also a lateral misalignment.  In spite of these issues an estimation of the tolerances required to achieve a loss of only 10% can be calculated.  A more detailed method like a Monte Carlo analysis would be needed to get more accurate results.  If needed a brief tutorial of the properties of a single mode fiber discussed are in Appendix A and the units of loss, decibels, are discussed in Appendix B.

## Single Mode Fiber Sensitivities

1. Lateral Misalignment

Figure 2.  Lateral Misalignment

Lateral misalignment is an offset of the two fibers as shown in Figure 1. It is probably best described as a radial offset ( r2 = x2 + y2 ) of the fiber core than as an offset in x and y separately. The percentage of loss due to a lateral misalignment is:

where ω0 is the mode field radius (1/e2).  Note that x/ω0 is the ratio of the offset to the mode field radius.  Figure 3 shows the loss for a wavelength of 1.3um with a mode field radius of 4.65um.  The fiber coupling is very sensitive to lateral misalignments.  When the two fibers are displaced by only 1um the loss is 4.2%.

2. Angular Misalignment

The loss due to an angular misalignment of the fiber has a very similar functional form to the lateral misalignment.

where ω0 is the mode field radius (1/e2), λ is the wavelength, n0 is the index of the material outside the fiber.  It is again of the functional form.  This can be seen by noting that there are a few constants out in front of a sin(θ) term.  For small angles sin(θ) is approximately θ.  The coupling is very sensitive to an angular misalignment.  The loss is about 4% for a 1 degree angular misalignment.

Figure 5.  Percent Loss due to an Angular Misalignment (λ = 1.3um ω0= 4.65um)

3. Longitudinal Displacement

Figure 6.  Longitudinal Displacement

The coupling is a little less sensitive to a longitudinal displacement of the fiber.  The loss due to a displacement is given by:

where ω0 is the mode field radius (1/e2), λ is the wavelength, n0 is the index of the material outside the fiber.  A 20 um displacement will cause a loss of about 3%.

Figure 7.  Percent Loss due to a Longitudinal Displacement (λ = 1.3um ω0= 4.65um)

4. Mode Field Diameter Mismatch

Figure 8.  Mode Field Mismatch

Another consideration for the coupling of two single mode fibers is the loss that occurs when the two fiber cores have different sizes.  There will then be a different mode field diameter in the two fibers leading to a coupling loss.  The loss is given by:

where ω1 is the mode field diameter of one of the fibers and ω2 is the mode field diameter of the other fiber.  Note the equation depends on the ratio of the two fibers.  This is shown in Figure 8.  If one of the fibers is 20% larger than the other fiber (ω1/ ω2 = 1.2) then the loss is about 3%.

Figure 9.  Percent Loss due to a Mode Field Diameter Mismatch (λ = 1.3um ω0= 4.65um)

## Back of the Envelope Tolerancing Analysis

Using the principles taught in class we can do a tolerancing analysis for coupling between two single mode fibers by calculating the sensitivities to each of these misalignments and then conducting a perturbation analysis to determine the tolerances for each misalignment. Before this can be done the losses calculated in percent must be converted into decibels. There are two differences between single mode fiber coupling and the tolerancing of lens alignment that we did in class.  First, the change in loss due to a small perturbation is not linear for single mode fiber coupling.  Second, the loss mechanisms are not decoupled.  A change in the angle of the two fibers will lead to a loss due to this tilt but it will also create a lateral offset.  We will conduct a back of the envelope calculation for the alignment of two single mode fibers to get a combined loss of only 10% (0.46 dB) illustrating each of these issues.

1. Decibels vs. Percent Loss

The losses due to each perturbation can not simply be added to determine the total system loss.  The proper way to calculate the system loss is to convert each system losses into a percent transmission and then multiply the transmission factors together.  For example consider only 3 loss mechanisms that contribute a loss of 20% each.  The percent transmission for the system is.  The system loss would be 49%.  If we simply added the 3 losses together we would get an incorrect result, 60% loss.  This becomes and issue when calculating the root sum square, RSS, of all of the tolerances to get the total system loss.   The RSS assumes that the figures of merit can be added.

One way to get around this is to convert all of the losses into decibels.  The conversion equation is:

Appendix B contains a table of example conversions.  When the losses are in decibels the system loss is calculated by summing the loss factors.  For example if there are 3 loss mechanisms which each contribute a loss of 20% (0.97 dB) then the system loss is 0.97dB + 0.97dB + 0.97dB = 2.91 dB.  Converting back to percent loss we get a system loss of 49%.  This is what we calculated before by multiplying transmission.  Now that we have a merit function that gives the correct result when summed together we can use the RSS to calculate the total system tolerance.

2. Non-linearity

In order to develop a tolerance that allows us to get a loss less than 0.46 db (10% loss) we start by guessing at the appropriate tolerances.  This is shown in Table 1.

Table 1.  First Guess at Tolerances.

In class, we then assumed that for small changes in the misalignments the results will scale linearly.  Using this assumption we get the results shown on the left side of Table 2.  This appears to be an acceptable solution.  However, if we go back and calculate the actual loss for these tolerances we get the results shown in the right side of the Table 2.  Rather than a loss of 0.46 dB (10%) we get a loss of 0.61dB (14.5%).

Table 2.  Assuming Linearity and Actual Loss

Table 3 shows a possible solution where the results were the actual loss for each misalignment was calculated.

Table 3.  One Possible Solution with a Loss Less than 0.46 dB

3. Correlated Misalignments

The Root Sum Square can only be used to calculate the system loss when the tolerances are uncorrelated.  With single mode fibers the misalignments are correlated.  For example, if there is a longitudinal displacement as the fiber is tilted there is not only a loss due to the tilt of the fiber, but also a loss due to the lateral displacement that occurs from the tilt.  In addition, this displacement increases as the longitudinal displacement increases.  Also consider the case where the fiber is both longitudinally displaced and laterally displaced.  If the fibers are tilted so that the beam hits the core of the other fiber then the coupling will actually go up.  But, if the fiber is tilted by the same amount in the other direction then the loss will be drastically higher. Additionally, the light out of a fiber is divergent.  As the longitudinal displacement is increased the size of the beam on the other fiber increases.  If there is a mode field diameter mismatch then the coupling will actually increase when the longitudinal displacement is such that the mode field diameter matches the size of the beam. Possibly the best way to deal with the correlated nature of the tolerances is to do a Monte Carlo analysis where all of the parameters are allowed to vary randomly to determine the statistical chance of a failure for a given set of tolerances. There is an equation listed in the reference which combines the effect of all of the tolerances into one equation which can be used for this Monte Carlo analysis.  It appears as if it assumes that one fiber rotates about the core of the other fiber.  If this is the case then as the longitudinal displacement is increased there isn’t a lateral shift.  If the fiber rotates about its own fiber core and a lateral shift is created then the lateral displacement x can be replaced with x + z·tan(θ) to include this in the equation.

λ is the wavelength, n0 is the index outside of the fiber, w1 is the mode field radius of one fiber, and w2 is the mode field radius of the other fiber.

4. Other Considerations

In addition to the misalignments discuss there are a few other issues to consider.  First if there is dirt or scratches on either fiber then even the best alignment won’t create very good coupling.  It is important to properly clean the fibers and protect them from dirt and scratches.  Additionally, as the wavelength changes the coupling efficiency may also change.

## Summary and Conclusions

The sensitivities for single mode fiber coupling has been shown including the loss due to lateral misalignments, angular misalignments, longitudinal displacements and a mode field diameter mismatch.  It has been shown that the fiber is very sensitive to lateral and angular misalignments.  A back of the envelope tolerancing has been shown as well as the importance of converting the loss into decibels and considering the non-linear nature of single mode fiber coupling.  This back of the envelope tolerancing can be used to get an idea of the tolerancing requirements, but a more rigorous method like a Monte Carlo analysis should be used because the misalignments are correlated.

## References

Appendix A.  Basic Single Mode Fiber Properties.

A single mode fiber is designed such that only one fiber mode travels down the fiber.  The output of a single mode fiber is a gaussian profile.  Because the gaussian profile doesn’t have a well defined size, the width of the gaussian is defined to be the point where the intensity has decreased to 1/e2 of its maximum value.  This is the mode field radius, ω0.  The mode field radius increases fairly rapidly from the fiber tip.

Figure 11.  Single mode fiber parameters

This size of the mode field radius varies with wavelength as shown in Table 4.

Table 4.  Typical Single Mode Field Radius

Appendix B.  Conversion between Percent Loss and Decibels

The following show the conversion from Percent Loss, L%, to Loss in Decibels, LdB and from Decibels back to Percent Loss. Also shown are a few example conversions.

Table 5.  Loss in Percent, L%, to Loss in dB, LdB

Table 5.  Loss in Percent, L%, to Loss in dB, LdB

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