Solution

Use the unfolded propagation convention after reflection. Free-space propagation and a concave spherical mirror have matrices

T (d) = (10 d 1), F (R) = (1 - 2/ R 01) .

The mirror is equivalent in this paraxial convention to a thin lens of focal length

R /2

. The excursion from the reference plane to mirror 1 and back is

S_{1} = T (L_{1}) F (R_{1}) T (L_{1}) = (1 - 2 L_{1} / R_{1} - 2/ R_{1} 2 L_{1} - 2 L_{1}^{2} / R_{1} 1 - 2 L_{1} / R_{1}) .

Likewise

S_{2} = T (L - L_{1}) F (R_{2}) T (L - L_{1})

. Multiplying

S_{2} S_{1}

, or using cyclic invariance of the trace to move the reference plane, gives

tr M = 2 - \frac{4 L}{R _{1}} - \frac{4 L}{R _{2}} + \frac{4 L ^{2}}{R _{1} R _{2}} = 4 g_{1} g_{2} - 2, g_{j} = 1 - \frac{L}{R _{j}} .

Consequently the strict criterion of a is

0 < g_{1} g_{2} < 1, g_{1} = 1 - L / R_{1}, g_{2} = 1 - L / R_{2} .

The trace is independent of

L_{1}

R_{1} = R_{2} = L

L_{1} = L /2

, each excursion is

(0 - 2/ L L /2 0)

and its square is

- I

. This gives

g_{1} g_{2} = 0

and the degenerate boundary exception already explained.

 Ancestors (12)

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