Modeling of Fiber Amplifiers and Lasers
This is part 2 of a tutorial on modeling of fiber amplifiers and lasers from Dr. Paschotta. The tutorial has the following parts:
1: Introduction, 2: Optical channels, 3: Power propagation or field propagation, 4: The laser-active ions, 5: Continuous-wave operation of amplifiers and lasers, 6: Amplifying and generating short pulses, 7: Ultrashort pulses, 8: Using home-made software or a commercial product?
Part 2: Optical Channels
Within a quantitative model for the fiber amplifier or fiber laser, we somehow need to describe the light propagating in the fiber. How that should be done in detail, depends very much on the circumstances.
In most cases, we are dealing with different light waves at substantially different wavelengths – for example, the pump wave and a signal wave. In more complicated cases, we may have multiple pump and signal waves, and there may also be light from amplified spontaneous emission (ASE) (see section 4 of our fiber amplifier tutorial).
Although one could in principle describes the whole light field as a whole, it is normally very sensible to distinguish some number of optical channels, as we call them. In an amplifier model, we may have
- one or several pump channels,
- one or several signal channels, and
- typically between 10 and 100 ASE channels.
We split ASE light into multiple channels with different wavelengths, normally using equidistant wavelength values. Each ASE channel represents some narrow wavelength region within which properties like photon energy and transition cross-sections are approximately constant. Of course, there are cases where ASE can be neglected altogether – for example, when the amplifier gain is too low for ASE to be significant and one is not interested in that low-level ASE.
Pump and signal channels are often considered as monochromatic. For these, one usually does not take into account spontaneous emission. In case of a broadband signal, one can of course again use an array of channels with different wavelengths.
As an example, Figure 1 shows the optical channels chosen for a simple model of an erbium-doped fiber amplifier. The ASE channels span the wavelength range from 1520 nm to 1600 nm with a spacing of 5 nm. The two signal channels are also in that wavelength range, but are treated separately: we consider signal inputs only for these, whereas spontaneous emission is taken into account only for the ASE channels.
Another important aspect is the propagation direction. Each optical channel describes light which is either propagating from the left side to the right side (let's call it the forward direction) or vice versa (backward direction).
In many cases, light at some wavelength is propagating in both directions. For example, one may have a double-pass amplifier or a linear laser resonator. At one or both ends of the active fiber, light may be reflected, and we describe this quantitatively with some reflectivity, which can of course depend on the optical channel (usually due to the wavelength dependence). In many cases, it does not matter whether that reflection occurs directly at the fiber end (e.g., due to Fresnel reflection of a bare end) or in some larger distance (e.g., with a loop reflector made with additional passive fiber); only the reflectivities are relevant, i.e., how much light comes back.
Over all, an optical channel is characterized by
- its wavelength (which may be the center wavelength of a narrow wavelength region),
- its bandwidth (for ASE channels, but not for monochromatic pump or signal channels),
- its propagation direction (forward or backward), and
- its spatial properties in the fiber – to be discussed in the next section.