In the application of lens antenna, what is the relationship between the refractive index of the lens and the radiation pattern？
It would be clearer if you specified the type of source (for example, plane or spherical wave), the shape of the lens, and its size relative to the wavelength. See, for example, Balanis' book "Antenna theory. Analysis and design"
There is no relationship between the refractive index of a lens and its radiation pattern, by itself.
The refractive index of the lens is used together with the shape of the lens to modify the wave-fronts of the the beam incident on the lens, to produce the required wave-fronts at the exit of the lens. The radiation pattern is the Fourier transform of the field pattern exiting the lens.
A lens can be made with varying refractive index across its volume, e.g. a Luneburg lens, in which case the radiation pattern is a function of the volume variation of refractive index.
If a lens gives a particular radiation pattern and the refractive index was changed (perhaps by filling it with a different fluid) it would give a different radiation pattern.
The far-field pattern is the Fourier transform of the aperture distribution. This means the 2D Fourier transform of the amplitude and phase pattern across the aperture. Use the refractive index distribution to calculate the phase and amplitude distribution, then do the Fourier transform. There is an additional factor of the cosine of the angle which is called the obliquity factor, which takes account of the reduced projected width of the aperture at wide angles.
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