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18-819C Homework
创建日期：2024-04-16 16:08:39
所属分类：物理Physical
标签： 18-819C
Focusing a collimated Gaussian beam

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1. Focusing a collimated Gaussian beam (40 points)

In the following, let’s consider a collimated HeNe laser (emission wavelength of 633 nm) with a beam
diameter of 1 mm (2w0).
(1) What’s the beam divergence of the input beam? Note that in practice the term of “beam
divergence” typically refers to 2?, i.e., the total angular spread (5 points)
(2) Let’s assume the lens has f=10 mm . Calculate the Rayleigh distance zR (on the left side of the lens)
and see whether f << zR (5 points)
(3) Apply the thin-lens formula for Gaussian beams for s=100 mm and calculate the magnification of
the beam diamter after going through the lens and the new beam divergence (10 points)
(4) Suppose you want to minimize the spot size for a fixed s=100 mm and you are allowed to choose
different focal length (f). Is there an optimal f for this purpose? If yes please explain why and
calculate f and the minimize spot (20 points)

2. Thin-lens formula (30 points)

We will take a further look at the modified thin-lens formula in this problem:
(1) In geometric (ray) optics, if z=f, what would be z’? (5 points)
(2) For Gaussian beams, if z=f, what would be z’? (5 points)
(3) For a Gaussian beam with 2w0=4.5 ?m and z=f=10 mm, what would be the beam diameter and
divergence after the lens? (10 points)
(4) If you have a lens with f=10 mm, and you need to use it to reduce the input Gaussian beam waist
by a factor of 5. What is the appropriate z for this purpose? (10 points)

3. Confined modes in a slab waveguide with metal cap (30 points)

Let’s consider a slab waveguide structure shown above. It consists of a core layer with index of n1 and
thickness of h. It has a lower-index cladding layer (index n2) in the bottom and a metal layer on top. Here
we only consider modes that are confined in y with electric field pointing at the x direction (propagation
direction is z)
(1) What is the boundary condition between the metal layer and the waveguide core? (hint: electric field
inside the metal layer is zero) (10 points)
(2) Write down possible solutions for the confined mode (assuming its propagation constant is ?) with
undetermined coefficients in each layer (10 points)
(3) Now apply the boundary conditions and obtain an equation that could determine the propagation
constant. You don’t have to solve this equation if there is no analytical solution (10 points)

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