Engineering seems to have a lot of "magic numbers" – numbers used in equations with no explanations of where they come from. I REALLY do not like magic numbers because years from now some other engineer will be staring at this equation and asking the same question I just did – "Where did this number come from?" I encountered one this morning and I thought it would be worthwhile to document where it came from.
The magic number is "6.07." I saw a web site that recommended dividing the spectral width of a laser by 6.07 when performing some basic calculations. This blog describes where the number comes from.
Side-Mode Suppression Ratio and Spectral Width Definitions
I was looking at at laser specification and it contained two numbers of interest: Side-Mode Suppression Ratio (SMSR) and spectral width. SMSR is a measure of the wavelength purity of a laser's output. It is defined as the ratio of the power in the dominant mode to the power in the strongest side mode. Spectral width ( is the wavelength range between the points at the SMSR level. Figure 1 illustrates how the measurement is made.
Figure 2 shows the spectrum for a laser with a 45 dB SMSR (Source: Gipo).
Laser Spectral Modeling
We often model the output spectrum of a laser using the normal curve, which makes sense when there is a main lobe and minimal side-lobe structure, like Figure 2. A normal curve is characterized by three numbers:
- peak value (A)
- mean wavelength ()
- standard deviation ()
The SMSR is used to estimated the standard deviation of the normal curve that best models the laser spectrum.
There are two common ways to measure the spectral width.
- Full-Width Half Maximum (FWHM – Generally used only with FP lasers)
This approach measures the pulse width at the 50% power points.
- Side Lobe Suppression Ratio (SMSR – Generally used only with DFB lasers)
This approach measured the pulse width from the points that are 20 dB down.
These approaches are designed to make measurement simple, but they do not directly give you a standard deviation. A conversion is required. Since I am most interested DFB lasers, I will focus on converting an SMSR-based measurement to a standard deviation.
where is one of the two wavelengths where the spectral amplitude is down -20 dB from the peak. We define the as the difference between the two values. Figure 3 illustrates the measurement.
So to convert to , simply divide the value by 6.07 as shown below.
The post shows that the standard deviation of a laser's spectrum can be computed by dividing that laser's spectral width (i.e. the wavelength range between points 20 dB down from a laser's spectral peak) by 6.07.