## Introduction

I had a request for an example of how the output power of a laser varies with temperature. We call this parameter tracking error. Tracking error varies from part to part and the manufacturers simply put a bound on this variation (e.g. ±1.5 dB). The variation is a function of temperature, but the variation is not consistent between parts.

The specific request I received was to provide an example of a part’s tracking error for temperatures above room temperature. I have monitor current versus optical power and temperature data, and I need optical power versus temperature at fixed monitor currents. I ended up using Mathcad to beat my data into a form that makes generating plots easy.

This post will assume some knowledge of the electrical characteristics of a laser, which I covered in this post.

## Background

Optical communication systems assume that the light power coupled onto a fiber from laser’s front facet and the light emitted from the laser’s rear facet (and measured by the monitor photodiode) are linearly related. Tracking error is a measure of the maximum deviation of this relationship from linearity. These deviations are primarily due to mismatches in the thermal coefficient of expansion for the materials that make up the optical modules. In order to plot tracking error versus temperature, we need to precisely define what tracking error is. Equation 1 presents the formal definition.

 Eq. 1 $\displaystyle \text{Tracking Error }={{\left. \max \left( 10\cdot \log \left( \frac{{{P}_{Output}}\left( T \right)}{{{P}_{Output}}\left( 25{}^\circ C \right)} \right) \right) \right|}_{{{I}_{Monitor}}\text{ constant}}}$

where

• POutput is the laser output power
• T is the ambient temperature that is varying over a range from -40 °C to 85 °C
• IMonitor, the current from the monitor photodiode, is held at a constant value.

Tracking error is the largest source of variation that we see in a laser’s output power. It is also not predictable in any way that I have seen — you cannot compensate for it like other temperature-dependent parameters.

## Analysis

### My Data

Our laser power feedback control systems work by maintaining a constant average monitor current over temperature. Figure 1 shows a plot of the data I have. The data always comes to me in the form of an Excel worksheet, and it shows monitor current versus optical power at various temperatures. What I need is optical power versus temperature at a fixed monitor current.

Figure 1: Monitor Current Versus Optical Power and Temperature.

### Approach

Here is my approach for analyzing this data:

• I insert an Excel component into Mathcad and insert the data into the Excel component.
• I perform a two-dimensional curve-fit using cubic splines.
• I now plot my output power versus temperature for various monitor currents.

### Interpolation

Figure 2 shows a screenshot from Mathcad of my data and interpolation approach. I used a Mathcad program to actually process the data. I find the programs simple to develop and easy to use.

Figure 2: Test Data Component and Interpolation Program.

### Optical Power Versus Temperature at Fixed Monitor Currents

Figure 3 shows the final result of my work. While the plot is boring, it is what I needed to answer the question.

Figure 3: Laser Tracking Error Versus Temperature.

## Conclusion

I thought this was a good, practical example illustrating two-dimensional interpolation applied in an actual application. I often need to present data in a way that differs from how the data was originally gathered. Interpolation allows me to obtain the data I need with minimal effort.