Abstract

A new two-stage axiconic reflector called a reflaxicon is discussed. This device consists essentially of a primary conical mirror and a larger secondary conical mirror located coaxially with respect to the primary mirror. The secondary mirror is truncated at such a location that the inner diameter of the secondary mirror (the diameter of the hole) exceeds the base diameter of the primary mirror. The function of this device is to convert a solid light beam (such as a Gaussian intensity distribution laser beam) into a hollow one in an essentially lossless manner. The solid (or Gaussian) light beam is incident upon the outside of the primary mirror and is reflected therefrom to the inside of the secondary mirror, from which it is in turn reflected in the direction desired, such as to a target or receiver. Proper selection of the half-angle of the secondary cone, in comparison with the half-angle of the primary cone, will result in a converging, parallel, or diverging beam. Some modifications to the basic reflaxicon and potential applications in laser bore sighting and collimation are also discussed. In addition, a holographic technique using reflaxicons is considered.

© 1973 Optical Society of America

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References

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  1. C. W. Martin, U.S. Patent2,457,253 (1948).
  2. J. H. McLeod, J. Opt. Soc. Am. 44, 592 (1954).
    [CrossRef]
  3. J. H. McLeod, J. Opt. Soc. Am. 50, 166 (1960).
    [CrossRef]

1960 (1)

1954 (1)

J. Opt. Soc. Am. (2)

Other (1)

C. W. Martin, U.S. Patent2,457,253 (1948).

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Figures (13)

Fig. 1
Fig. 1

Cross section of reflaxicon. The angles denoted by θ1 and θ2 are half-angles of the inner and outer mirrors, respectively.

Fig. 2
Fig. 2

Path of light ray through reflaxicon. The incoming ray is at the distance r1 from the axis of the system. The outgoing ray is incident upon the target plan a distance r3 from the axis. This figure defines the symbols used in Eqs. (1) through (6).

Fig. 3
Fig. 3

Expanded view of portion of the reflaxicon. This figure defines the symbols used in Eqs. (9) through (19).

Fig. 4
Fig. 4

Illustration of the back focal distance D0b of a focusing reflaxicon.

Fig. 5
Fig. 5

Reflaxicon with θ1 = θ2 < 45°. This choice of angle results in a lateral displacement of the two mirrors with respect to one another.

Fig. 6
Fig. 6

Reflaxicon with θ1 = θ2 > 45°. This system resembles a conic Cassegrain telescope.

Fig. 7
Fig. 7

Cross sections of four diverging reflaxicons. The circles in the fourth drawing represent the limits of the focal region of the light beam on the reflaxicon’s axis.

Fig. 8
Fig. 8

Effects of a reflaxicon on the intensity distribution of a square-wave (as a function of radius) light-beam input. The ordinate in each case (before passage and after passage) is the beam intensity. The zeros shown in the figure are on the axis of the system. The outer radius of the inner mirror is r1 (which equals the inner radius of the outer mirror). The outer radius of the outer mirror is r2.

Fig. 9
Fig. 9

Effects of a reflaxicon on a Gaussian intensity distribution (as a function of radial position) light beam. The dashed lines depicted in the before-passage case show where the corresponding input beam distribution of Fig. 8 would be placed if it were superimposed on this figure.

Fig. 10
Fig. 10

Application of reflaxiconic beam diverter as a drill sight marker. This application can be used to illuminate with visible light a site to be drilled (such as with a CO2 laser) on an extended specimen where the use of TV would be impractical.

Fig. 11
Fig. 11

Application of focusing reflaxiconic beam diverter to offline holography. The illustration shows use in transmission holography, but this device can also be used in reflection holography. The recording film is shown here at the front focal distance of the reflaxicon [defined in Eqs. (23) and (24)].

Fig. 12
Fig. 12

Illustration of use of focusing reflaxicon to read the hologram formed by the system shown in Fig. 11. A denotes the laser, B is the reflaxicon, and C denotes the holographic film placed at the front focal distance of the reflaxicon (which is the location of the film shown in Fig. 13).

Fig. 13
Fig. 13

Reflaxiconic beam collimator. This device couples the reflaxiconic beam divergence corrector, shown in the box in this figure, with an unmodified reflaxicon to remove the nondiffraction limited beam divergence of an incident light beam (such as that from a laser) reflectively.

Equations (26)

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x 1 = r 1 cot θ 1 .
x 2 = ( h - r 1 ) cot 2 θ 1 .
z = y tan ϕ .
z = y tan 2 ( θ 1 - θ 2 ) .
z = h - r 3 ,
y = ( h - r 3 ) cot 2 ( θ 1 - θ 2 ) .
D = x 1 + x 2 + y .
D = r 1 cot θ 1 + ( h - r 1 ) cot 2 θ 1 + ( h - r 3 ) cot 2 ( θ 1 - θ 2 ) .
x 3 = d / ( cot σ - cot w ) .
x 3 = d / ( tan 2 θ 1 - tan θ 2 ) .
p = x 3 tan θ 2 ,
h = r 2 + p .
p = d / ( tan 2 θ 1 - tan θ 2 ) cot θ 2 .
k = a + x 1 .
c + d = r 2 - r 1 ,
c = a tan 2 θ 1 .
p = ( r 2 - r 1 - c ) / ( tan 2 θ 1 - tan θ 2 ) cot θ 2 ,
p = ( r 2 - r 1 - a tan 2 θ 1 ) / ( tan 2 θ 1 - tan θ 2 ) cot θ 2 .
h = r 2 + { [ r 2 - r 1 - ( k - r 1 cot θ 1 ) tan 2 θ 1 ] / ( tan 2 θ 1 - tan θ 2 ) cot θ 2 } .
D = r 1 cot θ 1 + ( h - r 1 ) cot 2 θ 1 + ( h - r 3 ) cot 2 ( θ 1 - θ 2 ) ,
D 0 = r 1 cot θ 1 + ( h 0 - r 1 ) cot 2 θ 1 + h 0 cot 2 ( θ 1 - θ 2 ) ,
h 0 = r 2 + { [ r 2 - r 1 - ( k - r 1 cot θ 1 ) tan 2 θ 1 ] / ( tan 2 θ 1 - tan θ 2 ) cot θ 2 } .
D 0 f = h f cot 2 θ 1 + h f cot 2 ( θ 1 - θ 2 ) ,
h f = r 2 + [ ( r 2 - k tan 2 θ 1 ) / ( tan 2 θ 1 - tan θ 2 ) cot θ 2 ] .
I a ( r ) = I b ( r ) { ( r a 2 - r b 2 ) sin θ 1 / [ ( r 1 + r a ) 2 - ( r 1 + r b ) 2 ] sin θ 2 } ,
I a ( r ) = I b ( r 0 ) [ r a + r b ) sin θ 1 / ( 2 r 1 + r a + r b ) sin θ 2 ] .

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