Single mode fibers Is it possible to focus white light into a single mode optical fiber? I thought no because in order to focus it should be a solution to the Helmholtz equation but I am not too sure about it. Also since the equation features the wave number, shouldn't it be impossible? 
 A: "White" light is light of multiple frequencies, usually a broad band of different frequencies ranging from ~($390nm$ to $700nm$) though white on your computer monitor consists of only three different frequencies (red green and blue).
It is possible to send multiple frequencies down an optical fiber. However the fiber will act like a prism and diffract the light. Information can be sent down fibers but white light is never used since the faster frequencies reach the end of the fiber before the slower ones meaning that your pulses will broaden and overlap. 
Graded index fiber can be used to counteract this. As it slows down the faster frequencies. Effectively acting like an achromatic doublet.
Depending on your range of frequencies what is single mode fiber for one frequency may be multimode for another. There are other ways to couple light into a fiber than just lens focusing it such as fiber couplers.
A: The answer is Yes. However the losses will be wavelength-dependent, you will likely get a "ugly" superposition of modes at the output and a dispersed signal as user288447 stated.
You are correct, the mode for each wavelength need to be solution of the Helmholtz equation. For a dielectric waveguide, there is no absolute cut-off wavelength (all wavelength can propagate), but there is a single-mode cutoff (see Normalize Frequency parameter V).
A good reference for all things fibers and laser related is http://www.rp-photonics.com
A: Are you concerned that because the properties of the focussed light does not have the same properties (e.g. wavevector) of the fiber mode that coupling is not possible?
It is possible to focus white light into a single mode fiber.  The main criterion is that the angle of convergence of the light must fit inside the angle of acceptance of the fiber.  The efficiency of the transfer of energy will not be 100%, as there will be reflections at the surface, some light will fall outside of the acceptance angle, some light will fall outside the core region.  But some will propagate in the fiber.
A: a need for coupling a white light into a SM fibre had just emerged to me. I remembered vaguely, that there is probably no better means to couple extended source, than stupid butt-coupling. So I took one FT030 SM fiber patchcord and simply put one end near to cellphone photo illumination LED, and some amount (exceptionally high to my poor expectations) passed through: a cone illuminating white canvas from some 10-20cm of distance was visible.
So, the answer is: yes.
A: The answer depends very much on the white light and you will always get some coupling. In general the coupling is very low, simply by the second law of thermodynamics applied to the phase space of light: it becomes the law of increase of Optical Grasp (oka Étendue). Light at one wavelength in a one moded optical fiber at one wavelength is essentially zero entropy light: there is only one possible bound (i.e. propagating indefinitely away from the fiber input) configuration of light, so its entropy $k\,\log \Omega =0$ since $\Omega=1$. Light from an extended source, even when restricted to one wavelength, will have present highly aberrated wavefront to a collector lens, and thus the Strehl ratio at the fiber input is very small. Not so for a non-extended source. White light from a powerful point source (i.e. with diameter much less than the Airy disk diameter for the numerical aperture of the collector system) will have an very high coupling efficiency into the fiber. Light from a white laser likewise, for the same reasons.
An interesting problem is the coupling of sunlight into optical fibers: I have been thinking a bit about this for space exploration purposes. Interestingly, the maximum light you can couple into each mode of a fiber is independent from the distance from the Sun:   as you move away from the Sun, you can increase the collection area to compensate. The reason is that the Sun subtends a nonzero angle at the collection point, so off-axis waves can’t couple efficiently into the fibre. You can increase the collection area, but the aberration introduced by the tilt of the off axis waves thwarts any increase in coupling. So, for each distance from the Sun, there is a maximum effective collection area possible. As we move away from the Sun, its angle of subtense decreases, and lets you increase the collection area to compensate for the loss if intensity.
The power that can be coupled into a single mode fiber under any circumstances in the frequency interval $[\nu,\nu+\Delta\nu]$ is approximately constant (independent of fiber modeshape and distance from the Sun) and equal to:
$$P(\nu) = \lambda^2\,\epsilon(\nu) \Delta\nu$$
Where:
$$\epsilon(\nu) = \frac{2\,\pi\,h}{c^2}\frac{\nu^3}{e^{\frac{h\,\nu}{k\,T_\odot}}-1}$$
is the spectral intensity, by the Planck law, in watts per metre squared per hertz emitted from the surface of the Sun. Hence, the total power that can be gathered into a single mode fiber from the Sun is:
$$P_{tot}=\frac{\pi^3\,k^2\,T_\odot^2}{3\,h}$$
which, for $T_\odot = 6000{\rm K}$ works out to be about 11 microwatts, as long as the fibre can be thought of as single moded over the whole range of frequencies where the Sun emits significant power.
