Wednesday, 6 March 2019

homework and exercises - Transmission of Gaussian Beam Through Graded-Index Slab


The ABCD matrix of a glass graded-index slab with refractive index n(y)=n0(112α2y2) and length d is A=cos(αd), B=1αsin(αd), C=αsin(αd), D=cos(αd) for paraxial rays along the z axis. Usually, α is chosen to be sufficiently small so that α2y2<<1. A Gaussian beam of wavelength λ0, waist radius W0 in free space, and axis in the z direction enters the slab at its waist. How can I use the ABCD law to get an expression for the beam width in the y direction as a function of d?



Answer



The ABCD law can be used for Gaussian beam propagation using the complex beam radius q. Defining 1q=1Ri2kW2, R=R(z) being the radius of curvature of the beam and W=W(z) the halfwidth at point z and k=2π/λ0, the complex beam radius transforms as qAq+BCq+D. In your case (waist at the beginning of the medium, radius of curvature at the waist being infinite), so that q=ikW20/2 at the front of the medium.


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