Abstract

The distribution of modulation frequencies has been determined for radiation chopped by passing chopper teeth, whose sides are radii of the chopper wheel, over a circular aperture. The calculation reported here establishes the relationship between the aperture radius-to-chopper wheel radius ratio and the number of notch-tooth pairs that will produce an amplitude of the fundamental frequency equal to the amplitude obtained for the ideal sinusoidal modulation of the radiation emanating from the same aperture. The rms value of the radiation modulated in the assumed manner is also discussed.

© 1959 Optical Society of America

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References

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  1. R. B. McQuistan, J. Opt. Soc. Am. 48, 63 (1958).
    [CrossRef]

1958 (1)

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Figures (6)

Fig. 1
Fig. 1

The angle subtended by the aperture, 2θ0, is less than the angle subtended by a notch or tooth, 2θB.

Fig. 2
Fig. 2

The angle subtended by the aperture, 2θ0, is greater than the angle subtended by a notch or tooth, 2θB.

Fig. 3
Fig. 3

The normalized amplitude C1/C1′ of the fundamental as a function of the aperture radius-to-chopper wheel radius ratio, z, for various number of notch-tooth pairs, n.

Fig. 4
Fig. 4

The number of notch-tooth pairs, n, as a function of the aperture radius-to-chopper wheel radius ratio, z, that will produce a fundamental equal to that produced by the ideal sinusoidal modulation of radiation emanating from the same aperture.

Fig. 5
Fig. 5

The normalized rms value of the radiation power, Prms/Prms, as a function of the aperture radius-to-chopper wheel radius ratio, z, for various numbers of notch-tooth pairs, n.

Fig. 6
Fig. 6

The number of notch-tooth pairs, n, as a function of the aperture radius-to-chopper wheel radius ratio, z, that will produce an rms value of the radiation equal to that produced by the ideal sinusoidal modulation of the radiation emanating from the same aperture.

Equations (33)

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A 1 = 0 ,             when             0 t 1 ω w [ θ B - θ 0 ] ,
A 2 = ρ d ρ d θ = 1 2 ρ 2 ] ρ l ρ u d θ = 2 R 2 - θ 0 - θ B + ω w t [ cos θ z ] [ 1 - ( sin θ z ) 2 ] 1 2 d θ = R 2 { π 2 + sin [ ω w t - θ B ] z [ 1 - sin 2 ( ω w t - θ B ) z 2 ] 1 2 + arc sin [ sin ( ω w t - θ B ) z ] } ,
1 ω w [ θ B - θ 0 ] t 1 ω w [ θ B + θ 0 ] , A 3 = π R 2 ,             when             1 ω w [ θ B + θ 0 ] t 1 ω w [ 3 θ B - θ 0 ] ,
A 4 = 2 R 2 - 3 θ B + ω w t θ 0 [ cos θ z ] [ 1 - ( sin θ z ) 2 ] 1 2 d θ = R 2 { π 2 - sin [ ω w t - 3 θ B ] z [ 1 - sin 2 ( ω w t - 3 θ B ) z 2 ] 1 2 - arc sin [ sin ( ω w t - 3 θ B ) z ] } ,
1 ω w [ 3 θ B - θ 0 ] t 1 ω w [ 3 θ B + θ 0 ] , A 5 = 0 ,             when             1 ω w [ 3 θ B + θ 0 ] t T .
C K = i = 1 5 C K i ,
C K i 1 T A i ( t ) exp { - K j ω t } d t ,
C K = - 4 R 2 K π sin [ K π 2 ] 0 θ 0 cos φ z × [ 1 - ( sin φ z ) 2 ] 1 2 cos p φ d φ ,
( π / 4 ) F [ - p / 2 , p / 2 , 2 , z 2 ] ,
C K = - R 2 K sin [ K π 2 ] F [ - p / 2 , p / 2 , 2 , z 2 ] .
C 0 = π R 2 / 2.
P ( t ) = π R 2 2 P 0 { 1 - 4 π K = 1 1 K sin [ K π 2 ] × F [ - K n 2 , K n 2 , 2 , z 2 ] cos K ω t } ,
A 1 = 2 R 2 - θ 0 - θ B + ω w t cos θ z [ 1 - ( sin θ z ) 2 ] 1 2 d θ + 2 R 2 θ B + ω w t - θ 0 cos θ z [ 1 - ( sin θ z ) 2 ] 1 2 d θ = R 2 { π + sin [ ω w t - θ B ] z [ 1 - sin 2 ( ω w t - θ B ) z 2 ] 1 2 + arc sin [ sin ( ω w t - θ B ) z ] - sin [ ω w t + θ B ] z × [ 1 - sin 2 ( ω w t + θ B ) z 2 ] 1 2 - arc sin [ sin ( ω w t + θ B ) z ] } ,
0 t 1 ω w ( θ 0 - θ B ) . A 2 = 2 R 2 - θ 0 - θ B + ω w t cos θ z [ 1 - ( sin θ z ) 2 ] 1 2 d θ = R 2 { π 2 + sin [ ω w t - θ B ] z [ 1 - sin 2 ( ω w t - θ B ) z 2 ] 1 2 + arc sin [ sin ( ω w t - θ B ) z ] } ,
1 ω w [ θ 0 - θ B ] t 1 ω w [ 3 θ B - θ 0 ] . A 3 = 2 R 2 - 3 θ B + ω w t - θ B + ω w t cos θ z [ 1 - ( sin θ z ) 2 ] 1 2 d θ = R 2 { sin [ ω w t - θ 0 ] z [ 1 - sin 2 ( ω w t - θ B ) z 2 ] 1 2 + arc sin [ sin ( ω w t - θ B ) z ] - sin [ ω w t - 3 θ B ] z × [ 1 - sin 2 ( ω w t - 3 θ B ) z 2 ] 1 2 - arc sin [ sin ( ω w t - 3 θ B ) z ] } ,
1 ω w [ 3 θ B - θ 0 ] t 1 ω w [ θ B + θ 0 ] . A 4 = 2 R 2 - 3 θ B + ω w t θ 0 cos θ z [ 1 - ( sin θ z ) 2 ] 1 2 d θ = R 2 { π 2 - sin ( ω w t - 3 θ B ) z [ 1 - sin 2 ( ω w t - 3 θ B ) z 2 ] 1 2 - arc sin [ sin ( ω w t - 3 θ B ) z ] } ,
1 ω w [ θ B + θ 0 ] t 1 ω w [ 5 θ B - θ 0 ] . A 5 = 2 R 2 - 3 θ B + ω w t θ 0 cos θ z [ 1 - ( sin θ z ) 2 ] 1 2 d θ + 2 R 2 - θ 0 - 5 θ B + ω w t cos θ z [ 1 - ( sin θ z ) 2 ] 1 2 d θ = R 2 { π - sin ( ω w t - 3 θ B ) z [ 1 - sin 2 ( ω w t - 3 θ B ) z 2 ] 1 2 - arc sin [ sin ( ω w t - 3 θ B ) z ] + sin ( ω w t - 5 θ B ) z × [ 1 - sin 2 ( ω w t - 5 θ B ) z 2 ] 1 2 + arc sin [ sin ( ω w t - 5 θ B ) z ] } ,
1 ω w [ 5 θ B - θ 0 ] t T .
C K = - R 2 K sin [ K π 2 ] F [ - p / 2 , p / 2 , 2 , z 2 ] .
C 0 = π R 2 2 .
C K = - R 2 K sin [ K π 2 ] F [ - p / 2 , p / 2 , 2 , z 2 ]
C 0 = π R 2 2 .
P ( t ) = π R 2 2 P 0 { 1 - 4 π K = 1 1 K F [ - p / 2 , p / 2 , 2 , z 2 ] × sin [ K π 2 ] cos K ω t } .
lim z 0 n { R 2 K F [ - p / 2 , p / 2 , 2 , z 2 ] } = 4 R r K 2 π J 1 [ K π R 2 r ] ,
P ( t ) = π R 2 2 P 0 { 1 - cos ω t }
C 0 = π R 2 / 2             C ± 1 = - π R 2 / 4
C ± K = 0 ,
C 1 = C 1 ,
F [ - n / 2 , n / 2 , 2 , z 2 ] = π / 4.
P rms = π R 2 2 P 0 ( 3 2 ) .
P rms = π R 2 2 P 0 { 1 + 8 π 2 [ K = 1 1 K 2 F 2 ( - p / 2 , p / 2 , 2 , z 2 ) sin 2 ( K π 2 ) ] } 1 2
P rms P rms = { 2 3 [ 1 + 8 π 2 K = 1 1 K 2 F 2 ( - p / 2 , p / 2 , 2 , z 2 ) sin 2 ( K π 2 ) ] } 1 2 .
( π 4 ) 2 = K = 1 1 K 2 F 2 [ - p / 2 , p / 2 , 2 , z 2 ] sin 2 [ K π 2 ] .