# Young's double slit experiment with two glass slabs

Suppose I place two glass slabs with different thickness and refractive indices in a YDSE setup. I was told that if $$(\mu_1-1)t_1=(\mu_2-1)t_2$$ then there would be no change in the interference pattern, as compared to the one without any glass slab. Why isn't this true for $$\mu_1 t_1=\mu_2 t_2$$ i.e. equating their geometrical paths rather than the difference of geomterical and optical paths? Since the only difference caused presence of glass slab is that the ray travels a distance $$\mu t$$ rather than $$t$$.

• You can mark the question as answered if you think it is. Commented Apr 18 at 21:45

For each path you started with $$\dfrac t\lambda_{\rm air}$$ wavelengths in a thickness $$t$$ of air.

Replacing the air with a medium of refractive index $$\mu$$ means that in a thickness $$t$$ there are now $$\dfrac{t}{\lambda_{\rm air}/\mu} = \dfrac{\mu t}{\lambda_{\rm air}}$$ wavelengths.

For there to be no change in the pattern the change in the number of waves must be the same for both paths, ie $$\dfrac{\mu_1 t_1}{\lambda_{\rm air}} -\dfrac {t_1}{ \lambda_{\rm air}} = \dfrac{\mu_2 t_2}{\lambda_{\rm air}} -\dfrac {t_2}{ \lambda_{\rm air}}$$ and the relationship follows.

• What does "$\frac{t}{\lambda_{air}}$ wavelength in a thickness $t$ of air" mean? What exactly is $\frac{t}{\lambda_{air}}$? Commented Apr 18 at 18:31
• $t$ is a thickness of air and when divided by the wavelength in air gives the number of wavelengths in that thickness $t$. Because the wavelength in the medium is shorter by a factor of $\mu$ there are more wavelength in an equal thickness of the medium. This was in general and then I applied the idea to your two paths through medium $\mu_1$ and medium $\mu_2$. Commented Apr 18 at 20:46

That is because for the thinner slab the wave travels a distance t$$_2$$-t$$_1$$ trough vacuum. Equality of the phases requires $$\mu_1 t_1 +t_2-t_1 = \mu_2 t_2$$ which immediately gives the correct result.

For no change in interference pattern the sufficient condition is just that the central bright is observed at central line(as the other pattern would automatically follow and turn out to be the same)

Hence, we want path difference at the centre to be zero