Abstract

The output irradiance of a typical unstable resonator with circular symmetry has a central obscuration and is peaked near the obscuration. A method is presented for designing two-mirror optical systems to convert this beam into a beam of arbitrary obscuration ratio and more uniform irradiance. A method is also given for analyzing the alignment sensitivity of such a system. An example shows that useful irradiance redistributions can be achieved with alignment sensitivities comparable to ordinary Cassegrain systems.

© 1981 Optical Society of America

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References

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  1. D. F. Cornwell, J. Opt. Soc. Am. 69, 1456 (1979).
  2. D. N. Mansell, T. T. Saito, Opt. Eng. 16, 355 (1977).
    [CrossRef]
  3. J. W. Ogland, Appl. Opt. 17, 2917 (1978).
    [CrossRef] [PubMed]
  4. P. W. Scott, W. H. Southwell, J. Opt. Soc. Am. 69, 1456 (1979).

1979 (2)

D. F. Cornwell, J. Opt. Soc. Am. 69, 1456 (1979).

P. W. Scott, W. H. Southwell, J. Opt. Soc. Am. 69, 1456 (1979).

1978 (1)

1977 (1)

D. N. Mansell, T. T. Saito, Opt. Eng. 16, 355 (1977).
[CrossRef]

Cornwell, D. F.

D. F. Cornwell, J. Opt. Soc. Am. 69, 1456 (1979).

Mansell, D. N.

D. N. Mansell, T. T. Saito, Opt. Eng. 16, 355 (1977).
[CrossRef]

Ogland, J. W.

Saito, T. T.

D. N. Mansell, T. T. Saito, Opt. Eng. 16, 355 (1977).
[CrossRef]

Scott, P. W.

P. W. Scott, W. H. Southwell, J. Opt. Soc. Am. 69, 1456 (1979).

Southwell, W. H.

P. W. Scott, W. H. Southwell, J. Opt. Soc. Am. 69, 1456 (1979).

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

P. W. Scott, W. H. Southwell, J. Opt. Soc. Am. 69, 1456 (1979).

D. F. Cornwell, J. Opt. Soc. Am. 69, 1456 (1979).

Opt. Eng. (1)

D. N. Mansell, T. T. Saito, Opt. Eng. 16, 355 (1977).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Schematic optical layout. A ray entering at radius r2 in [b, R2] exits at radius r1 in [a, R1], b > R1.

Fig. 2
Fig. 2

A typical irradiance output of an unstable resonator chosen for the example in the text.

Fig. 3
Fig. 3

Sag calculated for the primary mirror as a function of radius for the example with a = 1.25 cm.

Fig. 4
Fig. 4

Sag calculated for the secondary mirror as a function of radius for example with a = 1.25 cm.

Fig. 5
Fig. 5

Sag calculated for the primary mirror as a function of radius for example with a = 0.5 cm.

Fig. 6
Fig. 6

Sag calculated for the secondary mirror as a function of radius for example with a = 0.5 cm.

Fig. 7
Fig. 7

Output irradiance calculated geometrically from the input irradiance of Fig. 2 and example with a = 1.25 cm.

Fig. 8
Fig. 8

Output irradiance calculated geometrically from the input irradiance of Fig. 2 and example with a = 0.5 cm.

Fig. 9
Fig. 9

RMS wave front error in waves at λ = 3.8 μm, corrected for overall tilt and focus for a tilted primary mirror. The dashed line is for a two-parabola Cassegrain system with magnification b/a, and the solid line is for the irradiance redistribution optics example with a = 1.25 cm.

Fig. 10
Fig. 10

RMS wave front error in waves at λ = 3.8 μm, corrected for overall tilt and focus for a tilted primary mirror. Essentially the same curve applies to the two-parabola Cassegrain system and the irradiance redistribution example with a = 1.25 cm.

Fig. 11
Fig. 11

RMS wave front error in waves at λ = 3.8 μm, corrected for overall tilt and focus for a decentered secondary mirror. Essentially the same curve applies to the two-parabola Cassegrain system and the irradiance redistribution example with a = 1.25 cm.

Fig. 12
Fig. 12

RMS wave front error in waves at λ = 3.8 μm, corrected for focus, for a longitudinal displacement of the secondary mirror. Essentially the same curve applies to the two-parabola Cassegrain system and the irradiance redistribution example with a = 1.25 cm.

Tables (2)

Tables Icon

Table I Interpolation Error for the Quintic Interpolation Described in the Test

Tables Icon

Table II Misalignment Summary for the Example Described in the Text

Equations (20)

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G ( r 2 ) + { [ G ( r 2 ) F ( r 1 ) L ] 2 + ( r 2 r 1 ) 2 } 1 / 2 + F ( r 1 ) = l 0 ,
G ( r 2 ) + F ( r 1 ) = ( r 2 r 1 ) 2 2 ( l 0 + L ) ( b a ) 2 2 ( l 0 + L ) ·
d G ( r ) d r | r = r 2 = d G d r 2 = d F d r 1 = d F ( r ) d r | r = r 1 ·
d G d r 2 = r 2 r 1 l 0 + L = d F d r 1 ·
a r 1 = f ( r 2 ) I 1 ( r ) r d r = b r 2 I 2 ( r ) r d r ,
a R 1 I 1 ( r ) r d r = b R 2 I 2 ( r ) r d r .
G ( r 2 ) = 1 l 0 + L b r 2 [ r f ( r ) ] d r ;
R ( r 1 ) = 1 l 0 + L a r 1 [ f 1 ( r ) r ] d r ,
I 0 = R 1 2 a 2 ( 1 + b c ) ( R 2 2 b 2 ) 2 c 3 ( R 2 3 b 3 ) ·
r 1 = f ( r 2 ) = [ a 2 + I 0 ( 1 + b c ) ( r 2 2 b 2 ) 2 I 0 c 3 ( r 2 3 b 3 ) ] 1 / 2 ,
G ( r 2 ) = 1 l 0 + L b r 2 r [ a 2 + I 0 ( 1 + b c ) ( r 2 b 2 ) 2 I 0 c 3 ( r 3 b 3 ) ] 1 / 2 d r .
I 1 ( r 1 ) r 1 d r 1 = I 2 ( r 2 ) r 2 d r 2 , I 1 ( r 1 ) r 1 d r 1 = I 2 ( r 2 ) r 2 d r 2 .
I 1 ( r 1 ) = r 2 r 1 d r 2 d r 1 I 2 ( r 2 ) = I 1 ( r 1 ) I 2 ( r 2 ) I 2 ( r 2 ) | r 2 = f 1 ( r 1 ) ·
a = 1.25 cm , R 1 = 5 cm , b = 5 cm , R 2 = 15 cm , c = 0 .092, L = 40 cm ;
H ( X ) = a 0 + a 1 X + a 2 X 2 + a 3 X 3 + a 4 X 4 + a 5 X 5 ,
X = r r n r n + 1 r n ,
H ( X ) = a 1 + 2 a 2 X + 3 a 3 X 2 + 4 a 4 X 3 + 5 a 5 X 4 ,
H ( X ) = 2 a 2 + 6 a 3 X + 12 a 4 X 2 + 20 a 5 X 3 .
a 0 = H ( 0 ) , a 1 = H ( 0 ) , a 2 = H ( 0 ) / 2 , a 3 = 10 [ H ( 1 ) H ( 0 ) ] 4 H ( 1 ) 6 H ( 0 ) + H ( 1 ) / 2 3 H ( 0 ) / 2 , a 4 = 15 [ H ( 1 ) H ( 0 ) ] + 7 H ( 1 ) + 8 H ( 0 ) H ( 1 ) + 3 H ( 0 ) / 2 , a 5 = 6 [ H ( 1 ) H ( 0 ) ] 3 H ( 1 ) 3 H ( 0 ) + [ H ( 1 ) H ( 0 ) ] / 2 .
H ( r r n r n + 1 r n )

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