Single-mode and multi-mode optical fiber

From a practical point of view single-mode:

upside: can go further without a repeater. In multi-mode optical fiber, different modes travel at different speeds, and start interfering with each other at some point
downside: lower bandwitdh, because we can fit less modes into it

As such, typical applications are:

single-mode optical fiber: longer distance communications across buildings and cities
multi-mode optical fiber: shorter distance communications e.g. within a single data center

Then there are some more hardcore threads actually pondering about specific cost trade-offs:

From a mathematical point of view:

multi-mode: en.wikipedia.org/w/index.php?title=Optical_fiber&oldid=1229833804#Multi-mode_fiber:
Fiber with large core diameter (greater than 10 micrometers) may be analyzed by geometrical optics. Such fiber is called multi-mode fiber. In a step-index multi-mode fiber, rays of light are guided along the fiber core by total internal reflection.
Figure 2. Source.
single-mode: en.wikipedia.org/w/index.php?title=Optical_fiber&oldid=1229833804#Single-mode_fiber:
Fiber with a core diameter less than about ten times the wavelength of the propagating light cannot be modeled using geometric optics. Instead, it must be analyzed as an electromagnetic waveguide structure, according to Maxwell's equations as reduced to the electromagnetic wave equation. As an optical waveguide, the fiber supports one or more confined transverse modes by which light can propagate along the fiber. Fiber supporting only one mode is called single-mode.
Figure 3. Source.

Another difference is that single-mode fiber usually uses lasers as the light soruce, while multi-mode fiber usually uses LED:

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