Abstract

The integral equation of a “flat-roof resonator” is solved by the Fox and Li method of iteration in a number of particular cases.

Mode patterns, phase shifts, and power losses are derived. A good overall agreement is found with the approximate theory previously developed by Toraldo di Francia.

Some experimental tests carried out on a microwave model give a further confirmation of the theoretical predictions.

© 1966 Optical Society of America

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Figures (12)

Fig. 1
Fig. 1

The flat-roof resonator.

Fig. 2
Fig. 2

The diamond cavity.

Fig. 3
Fig. 3

Normal cross section of the infinite strip roof resonator.

Fig. 4
Fig. 4

Fluctuation of the field amplitude at χ = 0.554 vs. the number of transits.

Fig. 5
Fig. 5

Relative amplitude and phase distributions of the dominant mode for three flat-roof resonators.

Fig. 6
Fig. 6

Intensity patterns for three flat-roof resonators. Dashed lines correspond to Toraldo’s treatment; continuous lines correspond to the iterative computations.

Fig. 7
Fig. 7

Half-intensity widths vs. roof angle for three values of the mirror apertures. Dashed curves have been derived from Toraldo’s theory.

Fig. 8
Fig. 8

Power loss per transit vs. the roof angle for three values of mirror aperture.

Fig. 9
Fig. 9

Phase shift per transit vs. the roof angle for three mirror apertures. Dashed lines have been derived from Toraldo’s theory.

Fig. 10
Fig. 10

The microwave model of a flat-roof resonator.

Fig. 11
Fig. 11

The flat-roof mirror.

Fig. 12
Fig. 12

(a) Measured and computed intensity patterns for the lowest order mode in the case α = 1°16′. (b) Measured and computed intensity pattern for the second-order mode, in the case α = 1°16′.

Equations (3)

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σ m v m ( x 2 ) = e i π / 4 λ - a a v m ( x 1 ) e - i k ρ ρ d x 1
ρ = { ( x 1 - x 2 ) 2 + { d + δ a [ 2 a ( x 1 + x 2 ) ] } 2 { the upper sign holds for 0 x 2 a and 0 x 1 a the lower sign holds for - a x 2 0 and - a x 1 0 ( x 1 - x 2 ) 2 + { d + δ a [ 2 a ( x 1 - x 2 ) ] } 2 { the upper sign holds for - a x 2 0 and 0 x 1 a the lower sign holds for 0 x 2 a and - a x 1 0.
σ m = e i π / 4 λ - a a - a a e - i k ρ ρ v m ( x 1 ) v m ( x 2 ) d x 1 d x 2 - a a [ v m ( x 1 ) ] 2 d x 1

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