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ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2019/2/c/solution

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Past exam of the laser science and quantum information processing course of the University of Oxford / 2019 / 2 / c / Solution by

Codex  0 2026-10-25

Let

a = L /2

, so each mirror in the symmetric confocal optical resonator has radius

R = L = 2 a

. At mirror 1, an incident beam whose waist is at the centre has

q_{in} = a + i z_{R}

, with distances measured along its incoming direction. Reflection gives

q_{refl} = \frac{q _{in}}{1 - q _{in} / a} = - a + i \frac{a ^{2}}{z _{R}} .

After propagating the distance

a

back to the centre, its parameter is purely imaginary. Thus the reflected waist is also at the cavity centre and

z_{R}^{'} = \frac{a ^{2}}{z _{R}}, z_{R} z_{R}^{'} = (\frac{L}{2})^{2} .

The incident and reflected waist radii need not agree. With axial position

z

measured from the centre, their radii obey

w^{2} (z) = \frac{λ}{π} (z_{R} + \frac{z ^{2}}{z _{R}}), w^{'2} (z) = \frac{λ}{π} (\frac{a ^{2}}{z _{R}} + \frac{z ^{2} z _{R}}{a ^{2}}) .

At either mirror,

z^{2} = a^{2}

and the two spot sizes agree, as reflection requires. Between the mirrors they generally have different waists while still reproducing themselves after a full round trip.

Figure 1.
Incident and reflected Gaussian envelopes in a symmetric confocal resonator
. Two unequal waist radii at the same central plane have reciprocal Rayleigh ranges $z_{R} z_{R}^{'} = (L /2)^{2}$ . The envelopes meet at both mirrors. The matched fundamental mode is included for comparison. Details.

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