Abstract

The coupling of long and thin radiators with nonaxisymmetric reflectors is considered for the production of generally rectangular beams with very good uniformity and attractive collection efficiencies. Reflector profiles are derived analytically, and cases of practical interest are discussed. Numerical results show that medium size beams (0.2–1 sr) may be obtained with collection efficiencies of 40% to 70%. Such beams are perfectly uniform over one angle while over the other they have nonuniformities generally lower than ±10%. Limitations of the reflector dimensions and the effects of the radiator thickness are evaluated both analytically and by experimental observation.

© 1970 Optical Society of America

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Figures (9)

Fig. 1
Fig. 1

Cross section of radiator-reflector geometry with definition of the collection angle c and the azimuthal angle a. ROR′: reflector profile; S(w, 0): radiator; 0(0,0): vertex; R(x,y): reflecting point; RT: tangent; RN: normal.

Fig. 2
Fig. 2

Examples of reflector semiprofiles for uniform beam formation. Radiator at x* = 1, y* = 0.

Fig. 3
Fig. 3

Four reflector profiles with the same collection efficiency. The radiator is positioned at x* = 1, y* = 0 and the limiting value of the collection angle is 135°, in all cases. Reflectors Nos. 1 and 4 produce a 27° uniform beam, while Nos. 2 and 3 produce a uniform beam of 13.5°.

Fig. 4
Fig. 4

Normalized reflector depth vs azimuthal limit of the beam for several azimuthal efficiencies.

Fig. 5
Fig. 5

Normalized reflector aperture vs azimuthal limit of the beam for several azimuthal efficiencies.

Fig. 6
Fig. 6

Polar efficiency vs polar limit of the beam.

Fig. 7
Fig. 7

Departure from azimuthal uniformity vs azimuthal angle in the beam for two different radiator positions.

Fig. 8
Fig. 8

Observed azimuthal distribution of intensity in a relatively narrow beam. Configuration of case I, Table II.

Fig. 9
Fig. 9

Observed azimuthal distribution of intensity in a wide beam. Configuration of case II, Table II.

Tables (2)

Tables Icon

Table I Summary of Numerical Results for Uniform Bean Formation with High Efficiency

Tables Icon

Table II Design Data for Two Radiator-Reflector Configurations Resulting in the Distributions of Fig. 8 (Narrow Beam) and Fig. 9 (Wide Beam)

Equations (32)

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d E = cos b × d b × d a .
I ( b ) = ( P / π 2 ) cos b ,
I ( a , b ) = ( P / π 2 ) ( cos b ) A ( a ) ,
A ( a ) = 1 + | d c / d a | ,
a = k c .
g = ( π c ± a ) / 2
tan g = d y / d x ,
g = ( π / 2 ) m c , m = ( 1 k ) / 2 .
tan g = cot m c ,
tan c = y / ( w x ) ,
d x / ( w x ) = ( sec 2 c ) d c / ( tan c + cot m c ) .
d x / ( w x ) = [ tan c tan ( 1 m ) c ] d c .
1 x * = cos c [ cos ( 1 m ) c ] 1 / ( 1 m ) ,
y * = sin c [ cos ( 1 m ) c ] 1 / ( 1 m ) ,
0 < m 1 2 ,
1 2 m < 1 ,
a + c = π ,
c = π / ( 1 + k ) and a = k π / ( 1 + k ) .
A ( a ) = 1 + ( 1 / k ) = A , for 0 < a k π / ( 1 + k )
A ( a ) = 0 , for k π / ( 1 + k ) a < π .
E I d E = P ,
e a = ( a + c ) / π .
A ( a ) = 1 + ( 1 / k ) = A for 0 < a a ,
A ( a ) = 1 for a < a π c ,
A ( a ) = 0 for π c < a < π .
e b = ( 2 b + sin 2 b ) / π
e = e a e b .
I s = ( P / π 2 ) cos b [ 1 + ( d c / d a s ) ] ,
I s / I = k ( 1 + k ) 1 [ 1 + ( d c s / d a s ) ] .
a s a = c s c .
d c s / d a s = [ 1 ( 1 k ) ( d c / d c s ) ] 1
d c / d c s = [ ( y * ) 2 + ( w s * x * ) 2 ] × [ ( y * ) 2 + ( 1 x * ) 2 ] 1 × [ y * + ( 1 x * ) d y / d x ] × [ y * + ( w s * x * ) d y / d x ] 1 .

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