Abstract

In the introduction we have generalized the definition of isoplanatism and established the corresponding representation for centered systems. In Part I we explain the theory for the meridional bundles, applying the general differential law for two independent wave surfaces. The angle of field may have any finite magnitude. After having found Eq. (35), we transform it conveniently, so that there appear in it only the intrinsic variables which are introduced for the bundles under consideration and the total aberration of aperture, obtaining the condition (42) in invariant form. This is a necessary and sufficient condition.

The explained theory is valid for optical systems more general than centered systems, viz., for those which have only a single plane of symmetry, in which the isoplanatic, linear element can have any direction.

Though we defer the general discussion of the conditions to a later paper in which the theory will be extended to extra-meridional bundles, we make a first analysis of formula (42) deducing from it (Sections II.9–17) some characteristic laws. The extension of the validity of the form of some conditions, for instance those of Herschel and Lihotzky-Staeble, for general values of fields, shows the principal ray is an axis of bundle even more important than the optical axis in the centered systems.

In the Appendix we shall present some comments on main aspects of the theory.

© 1949 Optical Society of America

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References

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  1. M. Di Jorio, Ottica 1, 3 (1947).
  2. M. Di Jorio, Ottica 1, 13 (1947).
  3. M. Di Jorio, Ottica 1, 3 (1947).
  4. M. Herzberger, Zeits. f. Physik 53, 237 (1929).
    [Crossref]
  5. M. Di Jorio, Ottica 1, 3 (1947).
  6. M. Di Jorio, Ottica 1, 52 (1947).
  7. M. Di Jorio, Ottica 1, 3 (1947).
  8. M. Di Jorio, Ottica 1, 13 (1947).
  9. G. Giotti, Lezioni di Ottica Geometrica (Zanichelli, Bologna, 1931), p. 381.

1947 (7)

M. Di Jorio, Ottica 1, 3 (1947).

M. Di Jorio, Ottica 1, 13 (1947).

M. Di Jorio, Ottica 1, 3 (1947).

M. Di Jorio, Ottica 1, 3 (1947).

M. Di Jorio, Ottica 1, 52 (1947).

M. Di Jorio, Ottica 1, 3 (1947).

M. Di Jorio, Ottica 1, 13 (1947).

1929 (1)

M. Herzberger, Zeits. f. Physik 53, 237 (1929).
[Crossref]

Di Jorio, M.

M. Di Jorio, Ottica 1, 3 (1947).

M. Di Jorio, Ottica 1, 13 (1947).

M. Di Jorio, Ottica 1, 3 (1947).

M. Di Jorio, Ottica 1, 3 (1947).

M. Di Jorio, Ottica 1, 52 (1947).

M. Di Jorio, Ottica 1, 3 (1947).

M. Di Jorio, Ottica 1, 13 (1947).

Giotti, G.

G. Giotti, Lezioni di Ottica Geometrica (Zanichelli, Bologna, 1931), p. 381.

Herzberger, M.

M. Herzberger, Zeits. f. Physik 53, 237 (1929).
[Crossref]

Ottica (7)

M. Di Jorio, Ottica 1, 3 (1947).

M. Di Jorio, Ottica 1, 13 (1947).

M. Di Jorio, Ottica 1, 3 (1947).

M. Di Jorio, Ottica 1, 3 (1947).

M. Di Jorio, Ottica 1, 52 (1947).

M. Di Jorio, Ottica 1, 3 (1947).

M. Di Jorio, Ottica 1, 13 (1947).

Zeits. f. Physik (1)

M. Herzberger, Zeits. f. Physik 53, 237 (1929).
[Crossref]

Other (1)

G. Giotti, Lezioni di Ottica Geometrica (Zanichelli, Bologna, 1931), p. 381.

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Figures (6)

F. 1
F. 1

Fundamental representation in the object-space in tangential isoplanatism.

F. 2
F. 2

Fundamental representation in the image-space in tangential isoplanatism.

F. 3
F. 3

Illustration of the calculation of BL/dlp and cosβ′.

F. 4
F. 4

Special case in which the tangential image Sp, S ¯ p is normal to the principal ray p′.

F. 5
F. 5

The condition of tangential isoplanatism by assuming as point of departure the two superposed isoplanatic congruences. The differential general law, for two independent waves, deducted in this special case. In the figure there is the representation of the image-space.

F. 6
F. 6

The condition of tangential isoplanatism by assuming as point of departure the two superposed isoplanatic congruences. The differential general law for two independent waves, deducted in this special case. In the figure there is the representation of the object-space.

Equations (109)

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( S S p ) p = ( M ¯ M ¯ ) b ¯ = ( M ¯ S ¯ ) b ¯ + ( S ¯ S ¯ p ) b ¯ + ( S ¯ p M ¯ ) b ¯
( S ¯ S ¯ p ) b ¯ = ( S ¯ S ¯ p ) p ¯ .
( S ¯ S ¯ p ) p ¯ ( S S p ) p = n s ¯ n s ¯ ,
n s ¯ = n sin ( Ω γ ) d l ,
n s ¯ = n sin ( Ω γ p ) d l p .
( S ¯ S ¯ p ) p ¯ ( S S p ) p = n sin ( Ω γ p ) d l p n sin ( Ω γ ) d l ,
L S ¯ p = B S p
( B ¯ L ) ā = n s ¯ n s ¯ ,
B L = n s ¯ n s ¯ n .
( S ¯ B ¯ ) ā = ( S B ) a .
h = B C = B H B ¯ L .
( h ) = n B L cos β + n sin ( Ω γ ) d l n sin ( Ω γ p ) d l p ,
( S A ) a = ( h ) .
= 90 ° ( Ω γ + ω ) ,
( S A ) a = n sin ( Ω γ + ω ) d l .
n [ sin ( Ω γ ) sin ( Ω γ + ω ) ] d l = n [ sin ( Ω γ p ) B L d l p cos β ] d l p .
n [ sin ( Ω γ ) ( 1 cos ω ) cos ( Ω γ ) sin ω ] d l = n [ sin ( Ω γ p ) B L d l p cos β ] d l p .
δ = ω + ,
δ p = 90 ° ( Ω γ p ) ,
δ ω = ω + β .
L ¯ S ¯ p = L S ¯ p = B S p .
L L ¯ = L ¯ S ¯ p · d Ω = ( t p t ω ) d Ω .
( B L ) 2 = ( d l p ) 2 + ( t p t ω ) 2 ( d Ω ) 2 2 ( t p t ω ) cos ( L L ¯ , B L ¯ ) d l p d Ω ,
( L L ¯ , B L ¯ ) = [ 90 ° ( δ p 1 2 d Ω ) ] .
( B L ) 2 = ( d l p ) 2 + ( t p t ω ) 2 ( d Ω ) 2 2 ( t p t ω ) sin δ p d l p d Ω .
d l p / d Ω = ( t p T p ) / sin ( δ p d Ω ) ,
d Ω = ( sin δ p / t p T p ) d l p ,
( B L ) 2 ( d l p ) 2 = ( t p T p ) 2 + ( t p t ω ) 2 sin 2 δ p 2 ( t p T p ) ( t p t ω ) sin 2 δ p ( t p T p ) 2 .
( B L ) 2 ( d l p ) 2 = ( t p T p ) 2 + ( t ω 2 t p 2 + 2 t p T p 2 t ω T p ) sin 2 δ p ( t p T p ) 2 .
( B L ) 2 ( d l p ) 2 = ( t p T p ) 2 ( t p T p ) 2 sin 2 δ p + ( t ω T p ) 2 sin 2 δ p ( t p T p ) 2 .
B L d l p = [ ( t p T p ) 2 cos 2 δ p + ( t ω T p ) 2 sin 2 δ p ] 1 2 | t p T p | ,
B L d l p = | t ω T p | | t p T p | .
B L / d l p = ( t ω T p ) / ( t p T p ) .
cos β = cos δ ω cos ω + sin δ ω sin ω .
B L / d l p = sin ( B L ¯ , L L ¯ ) / sin ( B L ¯ , L L ¯ ) ,
( B L ¯ , L L ¯ ) = 90 ° ( δ p 1 2 d Ω ) , ( B L , L L ¯ ) = 90 ° ( δ ω 1 2 d Ω ) .
B L / d l p = cos δ p / cos δ ω .
cos δ ω = | t p T p | · cos δ p [ ( t p T p ) 2 cos 2 δ p + ( t ω T p ) 2 sin 2 δ p ] 1 2 .
sin δ ω = ± ( t ω T p ) sin δ p [ ( t p T p ) 2 cos 2 δ p + ( t ω T p ) 2 sin 2 δ p ] 1 2 .
t p T p | t p T p |
sin δ ω = ( t p T p ) ( t ω T p ) sin δ p | t p T p | · [ ( t p T p ) 2 cos 2 δ p + ( t ω T p ) 2 sin 2 δ p ] 1 2 .
cos β = t p T p | t p T p | ( t p T p ) cos δ p cos ω + ( t ω T p ) sin δ p sin ω [ ( t p T p ) 2 cos 2 δ p + ( t ω T p ) 2 sin 2 δ p ] 1 2 ,
( B L / d l p ) · cos β ,
B L d l p · cos β = cos δ p cos ω + t ω T p t p T p sin δ p sin ω ,
( B L / d l p ) = ( cos δ p / cos δ ω )
cos β = cos δ ω cos ω + sin δ ω sin ω
( B L / d l p ) · cos β = cos δ p cos ω + cos δ p tan δ ω sin ω .
cos δ ω = ( d l p / B L ) cos δ p .
N L d l p = | t ω T p | | t p T p |
sin δ ω = ( N L / B L ) · sin δ p ,
sin δ ω = | t ω T p | | t p T p | d l p B L sin δ p .
sin δ ω = t ω T p t p T p d l p B L sin δ p .
tan δ ω = t ω T p t p T p tan δ p .
B L d l p · cos β = cos δ p cos ω + t ω T p t p T p · sin δ p sin ω
B L d l p cos β = t ω T p t p T p sin ω ,
B L d l p · cos β = sin ( Ω γ p ) cos ω + t ω T p t p T p cos ( Ω γ p ) sin ω ,
n [ sin ( Ω γ ) ( 1 cos ω ) cos ( Ω γ ) sin ω ] d l = n [ sin ( Ω γ p ) ( 1 cos ω ) t ω T p t p T p × cos ( Ω γ p ) sin ω ] d l p .
( t ω T p ) / ( t p T p ) .
t ω t p .
t ω T p t p T p = 1 + t ω t p t p T p ,
n [ sin ( Ω γ ) ( 1 cos ω ) cos ( Ω γ ) sin ω ] d l = n [ sin ( Ω γ p ) ( 1 cos ω ) cos ( Ω γ p ) sin ω ] d l p n t ω t p t p T p cos ( Ω γ p ) sin ω d l p .
n t ω t p t p T p cos ( Ω γ p ) sin ω d l p .
Δ n [ sin ( Ω γ p ) ( 1 cos ω ) cos ( Ω γ p ) sin ω ] d l p = n t ω t p t p T p cos ( Ω γ p ) sin ω d l p ,
Δ = 0 .
Δ n [ sin ( Ω γ p ) ( 1 cos ω ) cos ( Ω γ p ) sin ω t ω t p t p T p cos ( Ω γ p ) sin ω ] d l p = 0 .
n [ cos δ ( 1 cos ω ) sin δ sin ω ] d l = n [ cos δ p ( 1 cos ω ) t ω T p t p T p sin δ p sin ω ] d l p ,
Δ n [ cos δ p ( 1 cos ω ) sin δ p sin ω t ω t p t p T p sin δ p sin ω ] d l p = 0 .
cos δ p ( 1 cos ω ) sin δ p sin ω = cos δ p cos ( δ p ω ) = cos δ p cos
Δ n ( cos δ p cos t ω t p t p T p sin δ p sin ω ) d l p = 0 .
d l p , δ p , ω , , t ω , t p , T p
Ω γ = 90 ° δ ; Ω γ p = 90 ° δ p δ ω = ; δ p ω = } .
Δ 1 , k + 1 = 0 , *
Δ 1 , i + 1 = 0
Δ i + 1 , k + 1 = 0 .
cos = cos δ p cos ω + sin δ p sin ω
cos = cos δ p + sin δ p d ω .
Δ n sin δ p d ω d l p = 0 .
d l p d l = n n sin δ sin δ p d ω d ω ,
d l p d l = n n cos ( Ω γ ) cos ( Ω γ p ) d ω d ω .
Δ n cos ( Ω γ p ) d ω d l p = 0 .
n sin δ p d ω d l p ; n cos ( Ω γ p ) d ω d l p
n cos d l n cos d l = constant .
n cos ( δ ω ) d l n cos ( δ ω ) d l = constant .
constant = n cos δ d l n cos δ d l .
n [ cos ( δ ω ) cos δ ] d l = n [ cos ( δ ω ) cos δ ] d l .
n ( 1 cos ω ) d l = ± n ( 1 cos ω ) d l p .
Δ n ( 1 cos ω ) d l p = 0 .
( B L / d l p ) cos β
β + d ω = γ p + ( 90 ° Ω )
cos β = sin [ ( Ω γ p ) + d ω ] = sin ( Ω γ p ) + cos ( Ω γ p ) d ω .
n cos ( Ω γ ) d ω d l = n cos ( Ω γ p ) d ω d l p
n ( 1 cos ω ) d l = ± n ( 1 cos ω ) d l p
h = B H B ¯ L .
n ( 1 + t ω t p t p T p ) sin ω d l = n ( cos δ p cos t ω t p t p T p sin δ p sin ω ) d l p
n ( 1 + t ω t p t p T p ) sin ω d l = ± n ( 1 + t ω t p t p T p ) sin ω d l p ,
Δ n t ω T p t p T p sin ω d l p = 0 .
n sin ω d l = n t ω T p t p T p sin ω d l p .
n sin ω d y = n x ω X p x p X p sin ω d y p
± n d ω d l p = n sin δ p d ω d l p ,
Δ n d ω d l p = 0 ,
Δ n ω 2 d l p = 0 ,
I x = ( d x p / d x ) = n / n ( ω / ω ) 2 ,
( T / t ) + ( T / t ) = 1 ,
T = n r cos 2 φ n cos φ n cos φ ; T = n r cos 2 φ n cos φ n cos φ .
Longitudinal magnification = ( d t / d t ) = [ T T / ( t T ) 2 ] > 0
( S ¯ B ¯ ) ā = ( S B ) a .
( S ¯ B ¯ ) a = ( A C ) a ,
( S ¯ B ¯ ) ā = ( S B ) a ( S A ) a + ( B C ) a .
( S A ) a = ( h ) .