Abstract

A general algebraic theory based on first and third order equations has been developed for systems composed of two separated spherical mirrors. The name inverse cassegrainian systems has been provisionally assigned by the author to this general arrangement. One subgroup is the aplanatic Schwarzschild mirror system. The theory not only includes previously known examples of this subgroup, but has revealed the existence of others. In addition, nonaplanatic systems of major potential interest are discussed. These include systems of predecided geometrical arrangement, telecentric systems, inside-out systems, parfocal systems, and others. Means of extending the theory to include nonspherical surfaces are discussed.

© 1968 Optical Society of America

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References

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  1. D. S. Grey, J. Opt. Soc. Amer. 39, 723 (1949).
    [CrossRef]
  2. K. Schwarzschild, Untersuchungen zur Geometrische Optik, II—Theorie der Spiegeltelescope, Royal Observatory at Gottingen1905.
  3. C. R. Burch, Proc. Phys. Soc. London 59, 41 (1947).
    [CrossRef]
  4. B. Jurek, Optik 18, 413 (1961).
  5. P. Erdos, J. Opt. Soc. Amer. 49, 877 (1959).
    [CrossRef]
  6. O. N. Stavroudis, J. Opt. Soc. Amer. 57, 741 (1967).
    [CrossRef]
  7. J. P. C. Southall, Mirrors, Prisms and Lenses (Macmillan Company, New York, 1936).
  8. E. T. Whittaker, The Theory of Optical Instruments (Cambridge University Press, Cambridge, 1907).
  9. C. G. Wynne, Proc. Phys. Soc. B62, 772 (1949).
  10. H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, Oxford, 1950).

1967

O. N. Stavroudis, J. Opt. Soc. Amer. 57, 741 (1967).
[CrossRef]

1961

B. Jurek, Optik 18, 413 (1961).

1959

P. Erdos, J. Opt. Soc. Amer. 49, 877 (1959).
[CrossRef]

1949

D. S. Grey, J. Opt. Soc. Amer. 39, 723 (1949).
[CrossRef]

C. G. Wynne, Proc. Phys. Soc. B62, 772 (1949).

1947

C. R. Burch, Proc. Phys. Soc. London 59, 41 (1947).
[CrossRef]

Burch, C. R.

C. R. Burch, Proc. Phys. Soc. London 59, 41 (1947).
[CrossRef]

Erdos, P.

P. Erdos, J. Opt. Soc. Amer. 49, 877 (1959).
[CrossRef]

Grey, D. S.

D. S. Grey, J. Opt. Soc. Amer. 39, 723 (1949).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, Oxford, 1950).

Jurek, B.

B. Jurek, Optik 18, 413 (1961).

Schwarzschild, K.

K. Schwarzschild, Untersuchungen zur Geometrische Optik, II—Theorie der Spiegeltelescope, Royal Observatory at Gottingen1905.

Southall, J. P. C.

J. P. C. Southall, Mirrors, Prisms and Lenses (Macmillan Company, New York, 1936).

Stavroudis, O. N.

O. N. Stavroudis, J. Opt. Soc. Amer. 57, 741 (1967).
[CrossRef]

Whittaker, E. T.

E. T. Whittaker, The Theory of Optical Instruments (Cambridge University Press, Cambridge, 1907).

Wynne, C. G.

C. G. Wynne, Proc. Phys. Soc. B62, 772 (1949).

J. Opt. Soc. Amer.

D. S. Grey, J. Opt. Soc. Amer. 39, 723 (1949).
[CrossRef]

P. Erdos, J. Opt. Soc. Amer. 49, 877 (1959).
[CrossRef]

O. N. Stavroudis, J. Opt. Soc. Amer. 57, 741 (1967).
[CrossRef]

Optik

B. Jurek, Optik 18, 413 (1961).

Proc. Phys. Soc.

C. G. Wynne, Proc. Phys. Soc. B62, 772 (1949).

Proc. Phys. Soc. London

C. R. Burch, Proc. Phys. Soc. London 59, 41 (1947).
[CrossRef]

Other

K. Schwarzschild, Untersuchungen zur Geometrische Optik, II—Theorie der Spiegeltelescope, Royal Observatory at Gottingen1905.

J. P. C. Southall, Mirrors, Prisms and Lenses (Macmillan Company, New York, 1936).

E. T. Whittaker, The Theory of Optical Instruments (Cambridge University Press, Cambridge, 1907).

H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, Oxford, 1950).

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Figures (20)

Fig. 1
Fig. 1

Condensing system for simultaneous emission and absorption spectra of hot gases.

Fig. 2
Fig. 2

Generalized arrangement for inverse cassegrainian systems.

Fig. 3
Fig. 3

First and third order representation of Inca system.

Fig. 4
Fig. 4

Figure 3 shown in reverse order.

Fig. 5
Fig. 5

Aplanatic system, μ = 0, root number 1.

Fig. 6
Fig. 6

Aplanatic system, μ = 0, root number 2.

Fig. 7
Fig. 7

Aplanatic system, μ = −7, root number 1.

Fig. 8
Fig. 8

Aplanatic system, μ = −7, root number 2.

Fig. 9
Fig. 9

Aplanatic system, μ = −5, root number 1.

Fig. 10
Fig. 10

Aplanatic system, μ = +7, root number 1.

Fig. 11
Fig. 11

Aplanatic system, μ = +7, root number 2.

Fig. 12
Fig. 12

Spherically corrected system, d/s1 = 0.8, μ = −1.

Fig. 13
Fig. 13

Four spherically corrected systems, d/s1 = 0.8, μ = −4.

Fig. 14
Fig. 14

Spherically corrected inside-out system, μ = −5.

Fig. 15
Fig. 15

Spherically corrected inside-out system, no obscuration.

Fig. 16
Fig. 16

Spherically corrected inside-out system, μ = 0.

Fig. 17
Fig. 17

Spherically corrected parfocal systems, μ = −5.

Fig. 18
Fig. 18

Spherically corrected parfocal systems, μ = +5.

Fig. 20
Fig. 20

Spherically corrected telecentric systems, μ = −5.

Tables (5)

Tables Icon

Table I Aplanatic (Schwarzschild) Systems, μ Negative, Root Number 1

Tables Icon

Table II Aplanatic (Schwarzschild) Systems, μ Positive, Root Number 1

Tables Icon

Table III Spherically Corrected Systems for, μ = −4 with d/s1 = 0.8

Tables Icon

Table IV Telecentric Systems, μ Negative, Root Number 2

Tables Icon

Table V Obscuration Factor ϕ = 2.5. Negative

Equations (112)

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s 1 = A 1 M 1 ,             s 1 = A 1 M 1 ,             s 2 = A 2 M 2 ,             s 2 = A 2 M 2 ,
α = h 2 / h 1 .
S 1 = 2 R 1 - S 1
s 1 = 1 / ( 2 R 1 - S 1 ) , s 2 = s 1 - d = [ 1 - d ( 2 R 1 - S 1 ) ] / ( 2 R 1 - S 1 )
S 2 = ( 2 R 1 - S 1 ) / [ 1 - d ( 2 R 1 - S 1 ) ] .
S 2 = 2 R 2 - S 2 = 2 R 2 - ( 2 R 1 - S 1 ) / [ 1 - d ( 2 R 1 - S 1 ) ] .
α = h 2 / h 1 = s 2 / s 1 = ( s 1 - d ) / s 1 = 1 - d S 1 = 1 - d ( 2 R 1 - S 1 ) .
S 1 = 2 R 1 - S 1 S 2 = ( 2 R 1 - S 1 ) / α S 2 = 2 R 2 - ( 2 R 1 - S 1 ) / α } .
μ = [ 2 R 2 α - ( 2 R 1 - S 1 ) ] / S 1 .
R 1 = κ S 1 , R 2 = λ S 1 ,
μ = 2 α λ - ( 2 κ - 1 ) .
α λ = [ μ + ( 2 κ - 1 ) ] / 2 = κ + ( μ - 1 ) / 2 } .
γ = μ / α
s 2 / s 1 = α / μ .
κ ( κ - 1 ) 2 = α 2 λ [ α λ - ( 2 κ - 1 ) ] 2 ,
κ ( κ - 1 ) 2 = α ( α λ ) [ α λ - ( 2 κ - 1 ) ] 2 .
λ [ κ α + ( κ - 1 ) ] = κ ( 2 κ - 1 ) .
α = 1 - d ( 2 R 1 - S 1 ) = 1 - d S 1 ( 2 κ - 1 ) .
d / s 1 = ( 1 - α ) / ( 2 κ - 1 ) .
α λ = [ 2 λ + ( 2 κ - 1 ) ] / 2.
λ = μ / 2.
ϕ = h 3 / h 1 .
h 1 = - s 1 ω 1 , h 3 = - ( A 1 M 2 ) ω 2 = - ( A 1 A 2 + A 2 M 2 ) ω 2 = - ( d + s 2 ) ω 2 .
ϕ = ( ω 2 / ω 1 ) ( d + s 2 ) / s 1 = μ [ ( d / s 1 ) + ( s 2 / s 1 ) ] ,
= μ [ ( d / s 1 ) + ( α / μ ) ]
ϕ = α + μ ( d / s 1 ) . Alternately , from Eq . ( D ) ϕ = α + μ ( 1 - α ) / ( 2 κ - 1 ) . }
ϕ * = [ 1 - ( d / s 1 ) ] / α = ( 2 κ + α - 2 ) / α ( 2 κ - 1 ) } .
α λ = κ + ( μ - 1 ) / 2 ,
( κ - 1 ) 2 κ = α ( α λ ) [ α λ - ( 2 κ - 1 ) ] 2 ,
λ [ κ α + ( κ - 1 ) ] = κ ( 2 κ - 1 ) .
μ = 0
α λ = κ - 1 / 2.
α = ( κ - 1 ) 2 κ / ( κ - 1 / 2 ) 3 .
λ = ( κ - 1 / 2 ) 4 / ( κ - 1 ) 2 κ .
8 κ 3 - 16 κ 2 + 8 κ - 1 = 0.
4 κ 2 - 6 κ + 1 = 0 ,
κ = [ 3 ± ( 5 ) 1 2 ] / 4.
κ = [ 3 + ( 5 ) 1 2 ] / 4 = 1.30902 ,
α = 0.236068 λ = 3.42705 d / s 1 = 0.472136 ϕ * = 2.236068 r 1 / s 1 = 0.763932 r 2 / s 1 = 0.291796 } .
κ = [ 3 + ( 5 ) 1 2 ] / 4 = [ ( 5 ) 1 2 - 1 ] / 4 [ ( 5 ) 1 2 - 2 ] α = ( 5 ) 1 2 - 2 λ = [ ( 5 ) 1 2 + 1 ] / 4 [ ( 5 ) 1 2 - 2 ] d / s 1 = 2 [ ( 5 ) 1 2 - 2 ] ϕ * = ( 5 ) 1 2 r 1 / s 1 = 4 [ ( 5 ) 1 2 - 2 ] / [ ( 5 ) 1 2 - 1 ] r 2 / s 1 = 4 [ ( 5 ) 1 2 - 2 ] / [ ( 5 ) 1 2 + 1 ] }
r 1 : r 2 : d = [ ( 5 ) 1 2 + 1 ] : [ ( 5 ) 1 2 - 1 ] : 2.
κ = [ 3 - ( 5 ) 1 2 ] / 4 = 0.190983 ,
α = - 4.23607 ϕ * = - 2.23607 λ = 0.0729490 r 1 / s 1 = 5.23607 d / s 1 = - 8.47214 r 2 / s 1 = 13.7082 } .
α λ = κ - 4 , α = ( κ - 1 ) 2 κ / ( κ - 4 ) ( κ + 3 ) 2 ,
κ 3 + 5 κ 2 - 6 κ - 36 = 0.
κ 2 + 2 κ - 12 = 0.
κ 3 - 9 κ 2 + 8 κ + 48 = 0.
κ 2 - 5 κ - 12 = 0 ,
8 κ 2 - 4 κ - 5 = 0 ,
1.6 κ 4 - 11.3 κ 2 + 1.45 κ + 10.125 = 0.
α = μ / [ μ - ( 2 κ - 1 ) ] .
α = - 2 ( κ - 1 )
α = - μ .
α λ = 2 κ - 1.5.
s 1 = d + s 2 .
α / μ + ( 1 - α ) ( 2 κ - 1 ) = 1 ,
α λ = κ - 3.
- α / 5 + ( 1 - α ) ( 2 κ - 1 ) = 1 ,
α = - 5 ( κ - 1 ) / ( κ + 2 ) .
κ 2 - κ - 5 = 0.
α λ = κ + 2 ,
α = 5 ( κ - 1 ) / ( κ - 3 ) .
2 κ 2 - 2 κ - 15 = 0 ,
10 λ = κ + 2 ,
10 / 5 + ( 1 - 10 ) / ( 2 κ - 1 ) = 1 ,
α λ = κ + ( μ - 1 ) / 2 ,
( κ - 1 ) 2 κ = α α λ [ α λ - ( 2 κ - 1 ) ] , 2
λ = μ / 2.
2 κ 4 + κ 3 - 32 κ 2 + 29 κ + 72 = 0.
d / s 1 = S 1 / 2 R 1 = 1 / 2 κ .
α = 1 / 2 κ .
16 κ 4 - 40 κ 3 + 28 κ 2 - 6 κ + 1 + 0.
α λ = κ - 1 / 2 ,
α = 2 ( κ - 1 ) / [ 2 ϕ ( α λ ) - 1 ] ,
( 4 κ - 1 ) κ [ ϕ ( 2 - 1 ) - 1 ] = ( 2 κ - 1 ) 3 .
q = ( h 1 ω 1 / μ 2 ) { κ ( κ - 1 ) 2 - α α λ [ α λ - ( 2 κ - 1 ) ] 2 } .
q * = h 1 ω 1 { κ ( κ - 1 ) 2 - α α λ [ α λ - ( 2 κ - 1 ) ] 2 } .
α λ = κ - 3 ,
λ = - 2.5 ,
α = - 2 ( κ - 3 ) / 5.
2 κ 4 + κ 3 - 32 κ 2 + 29 κ + 72 = 25.
x = y 2 / 2 r + ( 1 - 2 ) y 4 / 8 r 3 ;
x = R y 2 / { 1 + [ 1 - ( 1 - 2 ) R 2 y 2 ] 1 2 } .
x = ( Q h 2 ) 2 J ,
Q = ν ( R - S ) = ν ( R - S ) ,
J = S / ν - S / ν .
Q 1 = R 1 - S 1
J 1 = 2 R 1
x 1 = 2 R 1 h 1 4 ( R 1 - S 1 ) 2 .
Q 2 = - [ R 2 α - ( 2 R 1 - S 1 ) ] / α .
J 2 = - 2 R 2 .
x 2 = - 2 R 2 α 2 h 1 4 [ R 2 α - ( 2 R 1 - S 1 ) ] 2 .
Σ x = x 1 + x 2 = 2 h 1 4 { R 1 ( R 1 - S 1 ) 2 - R 2 α 2 [ R 2 α - ( 2 R 1 - S 1 ) ] 2 } .
Σ x = 2 h 1 4 S 1 3 { κ ( κ - 1 ) 2 - α 2 λ [ α λ - ( 2 κ - 1 ) ] 2 } .
κ ( κ - 1 ) 2 = α 2 λ [ α λ - ( 2 κ - 1 ) ] 2 or             κ ( k = 1 ) 2 = α α λ [ α λ - ( 2 κ - 1 ) ] 2 } .
Σ x = 2 h 1 ω 1 3 { κ ( κ - 1 ) - λ α 2 [ α λ - ( 2 κ - 1 ) ] 2 } .
q = Σ x / 2 μ 2 ω 1 2 ,
q = h 1 ω 1 μ 2 { κ ( κ - 1 ) 2 - α α λ [ α λ - ( 2 κ - 1 ) ] 2 } .
Y = - ( G + 1 / Q h 2 ) ,
G = Σ ( d i , i + 1 / ν i , i + 1 h i h i + 1 ) ,
coma = x Y = - Q h 2 J [ G ( Q h 2 ) + 1 ] .
x 1 Y 1 = - 2 R 1 h 1 2 ( R 1 - S 1 ) .
G 2 = - d / h 1 / h 2 = - d / α h 1 2 ,
x 2 Y 2 = 2 R 2 α h 1 2 [ R 2 α - ( 2 R 1 - S 1 ) ] × { α [ R 2 d - ( 2 R 1 - S 1 ) ] - R 2 α ( 2 R 1 - S 1 ) } .
R 1 ( R 1 - S 1 ) ( 2 R 1 - S 1 ) = α 2 R 2 × [ α R 2 - ( 2 R 1 - S 1 ) ] { [ α R 2 - ( 2 R 1 - S 1 ) ] - R 2 } .
B = α R 2 - ( 2 R 1 - S 1 ) ,
R 1 ( R 1 - S 1 ) ( 2 R 1 - S 1 ) = α 2 R 2 B ( B - R 2 ) .
R 1 ( R 1 - S 1 ) 2 = R 2 α 2 B 2 .
B ( 2 R 1 - S 1 ) = ( R 1 - S 1 ) ( B - R 2 ) .
[ α R 2 - ( 2 R 1 - S 1 ) ] ( 2 R 1 - S 1 ) = ( R 1 = S 1 ) × [ ( α - 1 ) R 2 - ( 2 R 1 - S 1 ) ] .
[ α λ - ( 2 κ - 1 ) ( 2 κ - 1 ) = ( κ - 1 ) [ ( α - 1 ) λ - ( 2 κ - 1 ) ] .
λ [ α κ + ( κ - 1 ) ] = κ ( 2 κ - 1 ) .

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