Asymptotic solutions for transmitted Gaussian beam through a plane dielectric interface

We have derived the asymptotic solutions for the transmitted Gaussian beam observed in the rarer medium when the Gaussian beam is incident on a plane dielectric interface from the denser medium. We have confirmed the validity and the applicable range of the proposed asymptotic solutions by comparing with the reference solution calculated numerically. We have shown that, as observed in the reflection of the beam, the beam shift is also appeared in the transmitted beam.