Why is it possible to couple multiple wavelengths into a single mode optical fiber? I'm quite new to the mode theory, but as I understand, single mode fiber should only allow a single pattern of wavelength + polarization.
I'm assuming a non-modulated non-coherent light (a white LED, for example) coupled into single-mode fiber.
According to brief info I found, including this post, multiple modes of light can propagate through a single mode fiber but will experience losses,
Will energy of all coupled modes transfer into heat inside of a single-mode core due to interference? Or loss occurs only in context of data transfer, when light is modulated?
 A: Single-mode fibers are optical fibers that for a specific wavelength range of interest allow a "single" mode (actually two modes, given there's always two orthogonal polarization modes). For wavelengths above a threshold, there is no posible propagation through the fiber. Meanwhile, for a lower wavelength threshold, two propagating modes start to be possible, then a third, and so on... (x2 because of polarization).
This means that to say a fiber is single-mode you should specify for what wavelength, and one could argue that all fibers are single-mode for a certain wavelength range. The single-mode range is mainly dependent on the core size.
The wavelengths that cannot propagate still get diffracted by the fiber but never couple into the fiber nor reach the other end. So, in the case of white light, depending on the fiber, you can have a combination of wavelengths lost (not coupled), wavelengths propagating in a single mode and wavelengths propagating in more than one mode.
A: The single mode operation of optical fiber depends on V-number that relates to diameter, wavelength, and refractive index.
\begin{align}
\ V^2 &= \frac{2}{} (_1^2−_2^2)\\\\
 \end{align}

The optical fiber works best at 850nm and 1310-1550nm of spectrum because of minimum loss. All optical network compnents (amplifiers, couplers) are designed around that spectrum.
You can launch multiple modes of UV waves (small wavelength) into a single mode fiber but attenuation would be too high becuase of rayleigh scattering and UV absorption.
