Abstract

Various sets of equations describe the geometric optics of the corneal lens of backswimmer, Notonecta glauca.

© 1989 Optical Society of America

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References

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  1. R. Schwind, “Zonation of the Optical Environment and Zonation in the Rhabdom Structure Within the Eye of the Backswimmer, Notonecta glauca,” Cell Tissue Res. 232, 453–63 (1983).
    [CrossRef]
  2. R. Schwind, “Sehen unter und über Wasser, Sehen von Wasser; Das Sehsystem eines Wasserinsektes,” Naturwissenschaften 72, 343–352 (1985).
    [CrossRef]
  3. R. Schwind, “Geometrical Optics of the Notonecta Eye: Adaptations to Optical Environment and Way of Life,” J. Comp. Physiol. 140, 59–68 (1980).
    [CrossRef]

1985 (1)

R. Schwind, “Sehen unter und über Wasser, Sehen von Wasser; Das Sehsystem eines Wasserinsektes,” Naturwissenschaften 72, 343–352 (1985).
[CrossRef]

1983 (1)

R. Schwind, “Zonation of the Optical Environment and Zonation in the Rhabdom Structure Within the Eye of the Backswimmer, Notonecta glauca,” Cell Tissue Res. 232, 453–63 (1983).
[CrossRef]

1980 (1)

R. Schwind, “Geometrical Optics of the Notonecta Eye: Adaptations to Optical Environment and Way of Life,” J. Comp. Physiol. 140, 59–68 (1980).
[CrossRef]

Schwind, R.

R. Schwind, “Sehen unter und über Wasser, Sehen von Wasser; Das Sehsystem eines Wasserinsektes,” Naturwissenschaften 72, 343–352 (1985).
[CrossRef]

R. Schwind, “Zonation of the Optical Environment and Zonation in the Rhabdom Structure Within the Eye of the Backswimmer, Notonecta glauca,” Cell Tissue Res. 232, 453–63 (1983).
[CrossRef]

R. Schwind, “Geometrical Optics of the Notonecta Eye: Adaptations to Optical Environment and Way of Life,” J. Comp. Physiol. 140, 59–68 (1980).
[CrossRef]

Cell Tissue Res. (1)

R. Schwind, “Zonation of the Optical Environment and Zonation in the Rhabdom Structure Within the Eye of the Backswimmer, Notonecta glauca,” Cell Tissue Res. 232, 453–63 (1983).
[CrossRef]

J. Comp. Physiol. (1)

R. Schwind, “Geometrical Optics of the Notonecta Eye: Adaptations to Optical Environment and Way of Life,” J. Comp. Physiol. 140, 59–68 (1980).
[CrossRef]

Naturwissenschaften (1)

R. Schwind, “Sehen unter und über Wasser, Sehen von Wasser; Das Sehsystem eines Wasserinsektes,” Naturwissenschaften 72, 343–352 (1985).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Path of a ray of light parallel to the axis of the ommatidium in the corneal lens and the crystalline cone of the backswimmer.

Fig. 2
Fig. 2

Theoretically calculated aspheric curve for the corneal lens of Notonecta glauca.

Fig. 3
Fig. 3

Interference micrograph of a section through a corneal facet of Notonecta glauca (courtesy of Rudolf Schwind).3 The bell shaped (dark) aspheric layer can be seen well.

Equations (12)

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f 1 ( x 1 ) = 0 , f 2 ( x 2 ) = c ( x 2 / r ) 2 .
x = x 1 = tan ( α β ) [ f 1 ( x 1 ) + a y ( x ) ] ,
x 2 = x tan ( ω δ + α β ) [ y ( x ) + b + c f 2 ( x 2 ) ] ,
L + f 2 ( x 2 ) = x 2 tan ( θ ν ) ,
tan ( α β ) = f 1 ( x 1 ) + f 1 ( x 1 ) n 1 / n 2 [ 1 + f 1 ( x 1 ) ] 1 / 2 [ 1 f 1 ( x 1 ) ( n 1 / n 2 ) 1 + f 1 ( x 1 ) ] 1 / 2 1 + f 1 ( x 1 ) n 1 / n 2 [ 1 + f 1 ( x 1 ) ] 1 / 2 [ 1 f 1 2 ( x 1 ) ( n 1 / n 2 ) 2 1 + f 1 ( x 1 ) ] 1 / 2
tan δ = tan ( α β ) + y ( x ) 1 tan ( α β ) y ( x ) ,
tan ( ω δ ) = tan δ n 2 / n 3 [ 1 + tan 2 δ ] 1 / 2 [ 1 ( tan δ n 2 / n 3 ) 2 1 + tan 2 δ ] 1 / 2 tan δ 1 + tan δ n 2 / n 3 [ 1 + tan 2 δ ] 1 / 2 [ 1 ( tan δ n 2 / n 3 ) 2 1 + tan 2 δ ] 1 / 2 ,
tan ( θ ν ) = tan η n 3 / n 4 ( 1 + tan 2 η ) 1 / 2 [ 1 ( tan η n 3 / n 4 ) 2 1 + tan 2 η ] 1 / 2 f 2 ( x 2 ) 1 + f 2 ( x 2 ) tan η n 3 / n 4 ( 1 + tan 2 η ) 1 / 2 [ 1 ( tan η n 3 / n 4 ) 2 1 + tan 2 η ] 1 / 2 ,
tan η = tan ( ω δ + α β ) + f 2 ( x 2 ) 1 tan ( ω δ + α β ) + f 2 ( x 2 ) ,
P 12 x 2 12 + P 10 x 2 10 P 9 x 2 9 + P 8 x 2 8 + P 7 x 2 7 + P 6 x 2 6 P 5 x 2 5 + P 4 x 2 4 + P 3 x 2 3 + P 2 x 2 2 P 1 x 2 P 0 = 0 ,
y 4 [ 1 n 2 2 n 3 2 ( 1 + z 2 ) ] + 2 z y 3 + y 2 ( 1 n 2 2 n 3 2 ) ( 1 + z 2 ) + 2 z y + z 2 = 0 ,
k 1 = 1 + 4 c 2 L 2 / r 4 + 4 c L / r 2 , k 2 = 4 c 4 / r 8 , k 3 = 4 c 2 / r 4 + 8 c 3 L / r 6 , g 1 = ( n 3 L / n 4 ) 2 , g 2 = ( n 3 c / n 4 r 2 ) 2 ( 1 n 3 2 / n 4 2 ) k 3 , g 3 = 2 c L ( n 3 / n 4 r ) 2 + ( 1 n 3 2 / n 4 2 ) k 1 , g 4 = ( 1 n 3 2 / n 4 2 ) k 2 , h 1 = 4 c x ( y + b + c ) / r 2 2 x , h 2 = 1 + 4 c 2 ( y + b + c ) 2 / r 4 4 c ( y + b + c ) / r 2 , h 3 = 4 c 2 / r 4 8 c 3 ( y + b + c ) / r 6 , h 4 = 4 c 4 / r 8 , h 5 = 4 c 2 x / r 4 , Q 1 = 4 c 2 x / r 4 , Q 2 = ( y + b + c ) 2 , Q 3 = c 2 / r 4 , Q 4 = 2 c ( y + b + c ) / r 2 + 4 c 2 x 2 / r 4 , Q 5 = 4 c x ( y + b + c ) / r 2 , P 12 = h 4 g 4 , P 10 = Q 3 k 2 h 4 g 2 + h 3 g 4 , P 9 = Q 1 k 2 + h 5 g 4 , P 8 = Q 3 k 3 + Q 4 k 2 h 3 g 2 + h 4 g 3 + h 2 g 4 , P 7 = h 5 g 2 + h 1 g 4 Q 1 k 3 Q 5 k 2 , P 6 = Q 4 k 3 + Q 2 k 2 + Q 3 k 1 h 2 g 2 h 4 g 1 + h 3 g 3 + x 2 g 4 , P 5 = h 1 g 2 + h 5 g 3 + Q 5 k 3 + Q 1 k 1 , P 4 = Q 2 k 3 + Q 4 k 1 h 3 g 1 x 2 g 2 + h 2 g 3 P 3 = h 1 g 3 + h 5 g 1 Q 5 k 1 , P 2 = Q 2 k 1 h 2 g 1 + x 2 g 3 , P 1 = h 1 g 1 , P 0 = x 2 g 1 , z tan ( ω δ ) = x x 2 y + b + c c ( x 2 / r ) 2 .

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