Abstract

In this paper the influence of the linear heterogeneity on the ray path in some reflecting prisms is considered. It is stated that all the prisms discussed here have some focusing properties. The focal length and focus coordinates are calculated as the functions of magnitude and direction of refractive index gradient.

© 1968 Optical Society of America

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References

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  1. Z. Bodnar, Pomiary Automat. Kontr. 99, 35 (1963).
  2. Z. Bodnar, F. Ratajczyk, Appl. Opt. 4, 351 (1965).
    [CrossRef]
  3. Z. Bodnar, F. Ratajczyk, Appl. Opt. 4, 181 (1965).
    [CrossRef]
  4. Z. Bodnar, F. Ratajczyk, Pomiary Automat. Kontr. 11, 175 (1965)[sic].

1965 (3)

1963 (1)

Z. Bodnar, Pomiary Automat. Kontr. 99, 35 (1963).

Bodnar, Z.

Z. Bodnar, F. Ratajczyk, Appl. Opt. 4, 181 (1965).
[CrossRef]

Z. Bodnar, F. Ratajczyk, Appl. Opt. 4, 351 (1965).
[CrossRef]

Z. Bodnar, F. Ratajczyk, Pomiary Automat. Kontr. 11, 175 (1965)[sic].

Z. Bodnar, Pomiary Automat. Kontr. 99, 35 (1963).

Ratajczyk, F.

Appl. Opt. (2)

Pomiary Automat. Kontr. (2)

Z. Bodnar, Pomiary Automat. Kontr. 99, 35 (1963).

Z. Bodnar, F. Ratajczyk, Pomiary Automat. Kontr. 11, 175 (1965)[sic].

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Figures (33)

Fig. 1
Fig. 1

Ray path through the plane parallel plate made of linearly heterogeneous glass.

Fig. 2
Fig. 2

Decomposition of gradient G its components in a rectangular singly reflecting prism.

Fig. 3
Fig. 3

Path of the ray through the prism development into a plane parallel plate.

Fig. 4
Fig. 4

Ray path in the plane z = const in the development.

Fig. 5
Fig. 5

An auxiliary figure for the calculation of the prism foci in the XY plane.

Fig. 6
Fig. 6

Position of the foci in a prism dependent on the directional angle γ of the Gxy component.

Fig. 7
Fig. 7

Product fGxy and quotient X/b relative to the angle γ in a polar coordinate system.

Fig. 8
Fig. 8

Decomposition of G into its components in a rectangular doubly reflecting prism.

Fig. 9
Fig. 9

Ray path in the plane z = const of the prism development.

Fig. 10
Fig. 10

An auxiliary figure for calculation of the focus position in the XY plane.

Fig. 11
Fig. 11

Graph of the product fGxy in polar coordinates and the plot of the focus positions in the XY plane, dependent upon the Gxy component direction.

Fig. 12
Fig. 12

Decomposition of gradient G into components in a pentagonal prism.

Fig. 13
Fig. 13

Ray path in the plane z = const of the prism development into a plane parallel plate.

Fig. 14
Fig. 14

An auxiliary figure for the calculation of focus positions in the XY plane.

Fig. 15
Fig. 15

Approximate focus position distribution in a pentagonal prism dependent upon the Gxy component direction in the XY plane.

Fig. 16
Fig. 16

Plot of the product fGxy and quotient X/b vs the gradient directional angle γ in a polar coordinate system.

Fig. 17
Fig. 17

Decomposition of vector G into its components in a Dove prism.

Fig. 18
Fig. 18

Ray path in the z = const plane.

Fig. 19
Fig. 19

An auxiliary figure for the calculation of the focus positions for a Dove prism in the XY plane.

Fig. 20
Fig. 20

Approximate focus position vs the Gxy component direction in the XY plane.

Fig. 21
Fig. 21

Plot of the product fGxy and quotient X/b vs the gradient directional angle γ in a polar coordinate system.

Fig. 22
Fig. 22

Decomposition of the vector G into its components in a roof prism.

Fig. 23
Fig. 23

Incidence point of the ray on the roof prism.

Fig. 24
Fig. 24

Development of the roof prism into the plane parallel plate seen along the roof edge.

Fig. 25
Fig. 25

Decomposition of the gradient G into its components in the roof prism before the first reflection.

Fig. 26
Fig. 26

Decomposition of the gradient G into its components in the first mirror reflection in the roof prism.

Fig. 27
Fig. 27

Components of the gradient G in the second mirror reflection in the roof prism.

Fig. 28
Fig. 28

X,Y,Z, coordinate system related to the exit plane of the roof prism.

Fig. 29
Fig. 29

An auxiliary figure for the calculation of focus positions in the XY plane in the roof prism.

Fig. 30
Fig. 30

An auxiliary figure for the calculation of focus positions in the YZ plane in the roof prism.

Fig. 31
Fig. 31

Cross hair made through a homogeneous liquid Dove prism.

Fig. 32
Fig. 32

Cross hair made through a liquid Dove prism with a gradient of the index of refraction.

Fig. 33
Fig. 33

As in Fig. 32 after defocusing of the horizontal cross hair.

Equations (58)

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x = ( n / 2 G x ) [ exp ( y G x / n ) - 1 ] 2 ,
d x / d y = x y = exp ( 2 y G x / n ) - exp ( y G x / n ) .
x y = y G x / n .
1 x y = ( b - a ) G x / n ,
1 y z = ( b - a ) G z / n .
2 x y = a G x / n = - a G y / n ,
2 y z = a G z / n
x y = [ b G x - a ( G x + G y ) ] / n ,
y z = b G z / n
θ x y = b G x - a ( G x + G y ) , θ y z = b G z .
Δ θ x y = - ( G x + G y ) Δ a
Y = [ cos ( θ x y + Δ θ x y ) cos θ x y ] / ( G x + G y ) .
Y = f = ( G x + G y ) - 1 .
X = b ( G x - G y ) / 2 ( G x + G y ) .
Z = b G z / ( G x + G y ) .
X = ( b / 2 ) tan ( 45° - γ ) ,
Y = f = [ G ( 2 ) 1 2 cos δ cos ( 45° - γ ) ] - 1 ,
Z = b tan δ / [ ( 2 ) 1 2 cos ( 45° - γ ) ] .
θ x y = l 2 ( G x + G y ) / [ ( 2 ) ] 1 2 ,
Δ θ x y = - ( 2 ) 1 2 ( G x + G y ) Δ l 1
X = 0 ,
Y = - [ ( 2 ) 1 2 ( G x + G y ) ] - 1 ,
Z = - b G z / ( G x + G y ) .
X = 0 ,
Y = - [ 2 G cos δ cos ( 45° - γ ) ] - 1 ,
Z = - b tan δ / [ ( 2 ) 1 2 cos ( 45° - γ ) ] .
θ x y = ( 2 b + { [ ( 2 ) 1 2 - 2 ] / ( 2 ) 1 2 } a tan 22.5° ) ( G x + G y ) ,
θ y z = [ ( 2 ) 1 2 + 2 ] b G z .
Δ θ x y = { [ ( 2 ) 1 2 - 2 ] / ( 2 ) 1 2 } tan 22.5° ( G x + G y ) Δ a ;
X = - 2 ( 2 ) 1 2 b / { [ ( 2 ) 1 2 - 2 ] tan 22.5° } ,
Y = - ( 2 ) 1 2 / { [ ( 2 ) 1 2 - 2 ] tan 22.5° ( G x + G y ) } ,
Z = - ( 2 ) 1 2 [ ( 2 ) 1 2 + 2 ] b G z / { [ ( 2 ) 1 2 - 2 ] tan 22.5° ( G x + G y ) } .
X = - 2 ( 2 ) 1 2 b / { [ ( 2 ) 1 2 - 2 ] tan 22.5° } ,
Y = - { [ ( 2 ) 1 2 - 2 ] G tan 22.5° cos δ cos ( 45° - γ ) } - 1 ,
Z = - ( [ ( 2 ) 1 2 + 2 ] b tan δ / { [ ( 2 ) 1 2 - 2 ] tan 22.5° cos ( 45° - γ ) } ) .
θ x y = [ k / sin ( 45° - β ) ] [ - ( 2 ) 1 2 h ( G x + G y ) cos ( 45° - β ) + H ( G x sin β + G y cos β ) ] ,
k = ( n 2 - sin 2 α ) 1 2 / n cos α ,
θ y z = H G z / sin ( 45° - β ) ,
Δ θ x y = - ( 2 ) 1 2 k ( G x + G y ) cot ( 45° - β ) Δ h ,
X = H tan ( 45° - β ) [ ( G y - G x ) / 2 ( G y + G x ) ] ,
Y = tan ( 45° - β ) / ( 2 ) 1 2 k ( G x + G y ) ,
Z = H G z / [ ( 2 ) 1 2 k ( G x + G y ) cos ( 45° - β ) ] ,
X = - ( H / 2 ) tan ( 45° - β ) tan ( 45° - γ ) ,
Y = tan ( 45° - β ) / [ 2 k G cos δ cos ( 45° - γ ) ] ,
Z = H tan δ / [ 2 k cos ( 45° - β ) cos ( 45° - γ ) ] ,
k = C E / C G and k = J F / E F ;
G x = G x , G y = ( 2 ) - 1 2 ( G y - G z ) , G z = ( 2 ) - 1 2 ( G y + G z ) .
l 2 = 2 k h ( 1 - k ) ,
G x = G z , G y = ( 2 ) - 1 2 ( G y - G x ) , G z = ( 2 ) - 1 2 ( G x + G y ) .
l 3 = h [ 1 - k k - 2 k ( 1 - k ) ] ,
G x = - G x , G y = ( 2 ) - 1 2 ( G y + G x ) , G z = ( 2 ) - 1 2 ( G y - G z ) .
θ x y = 2 b G x - h G x + 2 ( 2 ) 1 2 a G z , θ y z = ( 2 ) - 1 2 [ 2 ( 2 ) 1 2 a G x + h G y + 2 b G z - h G z ] ,
Δ θ x y = 2 ( 2 ) 1 2 G z Δ a
Δ θ y z = ( 2 ) 1 2 G z Δ b ,
X = ( h - 2 b ) G x / 2 ( 2 ) 1 2 G z ,
Y = f = - [ 2 ( 2 ) 1 2 G z ] - 1 ,
Y = f = - [ ( 2 ) 1 2 G z ] - 1 ,
Z = [ h ( G z - G y ) - 2 ( 2 ) 1 2 a G x ] / 2 G z ,

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