 Solution

ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2021/1/b/solution

M = (1 - 1/ f 01) .

The complex beam parameter transformation gives

q_{2} = q_{1} / (1 - q_{1} / f)

, hence

\frac{1}{q _{1}} - \frac{1}{q _{2}} = \frac{1}{f} .

The sign is also obtained by multiplying the incident wave amplitude by the thin lens phase

exp [- ik ρ^{2} / (2 f)]

z = 0

and put

a = π w_{0}^{2} / λ

, the incident Rayleigh range. Since the incident beam waist lies there,

q_{1} = - ia

. Immediately after the thin lens,

q_{2} = \frac{- ia}{1 + ia / f} = - \frac{f a ^{2}}{f ^{2} + a ^{2}} - i \frac{f ^{2} a}{f ^{2} + a ^{2}} .

Subsequent free propagation adds

z

, so write

q (z) = z - z_{f} - i z_{R}^{'}

. Comparing real and imaginary parts gives

z_{f} = \frac{f a ^{2}}{f ^{2} + a ^{2}}, z_{R}^{'} = \frac{f ^{2} a}{f ^{2} + a ^{2}}, w_{f}^{2} = \frac{λ z _{R}^{'}}{π} .

For a positive focal length, the new beam waist lies downstream but closer than the geometrical focus. At fixed

f, λ

w_{0} \to \infty

z_{f} \to f, z_{R}^{'} \sim \frac{f ^{2} λ}{π w _{0}^{2}} \to 0, w_{f} \sim \frac{f λ}{π w _{0}} .

This is the formal unbounded-aperture scalar limit; eventually finite aperture and the paraxial approximation limit the extrapolation.

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