While similar in effect to an interference patter type experiment, the actual physics behind the experiment in the article is subtly different. A 'Newtons Rings' type pattern emerges when the distance between the two (partially) reflective surfaces are a certain distance apart, coinciding with an integer value of wavelengths of the light involved. This can can, in theory, be any distance, as long as exact number of wavelengths fit inside. For example, standard interferometers can have distances as large as a centimetre, which is huge compared to the wavelength of visible light.
The effect described is based on the distance between two very reflective surfaces being smaller than the wavelength of light involved, thus preventing the light from travelling further down the waveguide. The taper on the waveguide means that as you go to shorter wavelengths of light, it can travel further, thus generating a 'trapped rainbow' of visible light inside the waveguide.
A key difference to note is that the fringe pattern generated by an interferometer type setup repeats itself as you increase/decrease the distance between the two reflective surfaces, so generating a series of lines or concentric circles. The setup with the 'trapped rainbow' will create a single rainbow pattern.