The following text is from Concepts of Physics by Dr. H.C.Verma, from the chapter on "Light Waves", for the number of fringes that will shift in a Young's double slit experiment (YDSE) when one of slits is covered by a transparent sheet of thickness $t$ made of a material of refractive index $\mu$:
When the light travels through a sheet of thickness $t$, the optical path travelled id $\mu t$, where $\mu$ is the refractive index. When one of the slits is covered by the sheet, air is replaced by the sheet and hence, the path changes by $(\mu-1)t$. >> One fringe shifts when the optical path changes by one wavelength. Thus the number of fringes shifted due to the introduction of the sheet is $\frac{(\mu-1)t}{\lambda}$. <<
Could anyone please explain what the author means by the statement marked by ">>" and "<<"? I understood the rest of the text. I thought when we introduce the transparent sheet, the entire interference pattern (all fringes) will be shifted on the screen. But according to the given statement, it seems only few fringes will shift their positions. If this is the case, how will the resultant interference pattern look like? Will there be huge regions of pure darkness and regions where shifted fringes coincide with existing fringes? (I don't think so)
There is no further explanation for this in my textbook. I think the Wikipedia article on Fringe Shift is some other phenomenon (reason: Michelson–Morley experiment is new to me and not introduced anywhere in my textbook).
Best Answer
The author means that the entire interference pattern shifts, and that the magnitude of the shift is by one fringe.
Similarly, if $(\mu-1)t=n\lambda$, the entire interference pattern will shift, and the magnitude of the shift will be by $n$ fringes.