Abstract

The experimental and quantitative study of the measurement of the lateral spherical aberration with the shearing interferometer using Fourier imaging and the moiré method is given. When the aberration of a camera lens with small f/number(1.7) is measured with an optical arrangement employing the spherical wavefront as a reference at the observation plane, the distortion correction of the obtained fringes is necessary. However, the fringe pattern directly illustrating the lateral spherical aberration can be obtained by the proposed optical arrangement using the plane reference wavefront, without correcting the fringe distortion.

© 1975 Optical Society of America

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References

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  1. S. Yokozeki, T. Suzuki, Appl. Opt. 10, 1575 (1971).
    [CrossRef] [PubMed]
  2. S. Yokozeki, T. Suzuki, Appl. Opt. 10, 1690 (1971).
    [CrossRef] [PubMed]
  3. A. W. Lohmann, D. E. Silva, Opt. Commun. 2, 413 (1971).
    [CrossRef]
  4. D. E. Silva, Appl. Opt. 10, 1980 (1971).
    [CrossRef]
  5. A. W. Lohmann, D. E. Silva, Opt. Commun. 4, 326 (1972).
    [CrossRef]
  6. D. E. Silva, Appl. Opt. 11, 2613 (1972).
    [CrossRef] [PubMed]

1972

A. W. Lohmann, D. E. Silva, Opt. Commun. 4, 326 (1972).
[CrossRef]

D. E. Silva, Appl. Opt. 11, 2613 (1972).
[CrossRef] [PubMed]

1971

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Figures (7)

Fig. 1
Fig. 1

Optical arrangement with the spherical reference wavefront.

Fig. 2
Fig. 2

Spatial shift δx due to the aberration.

Fig. 3
Fig. 3

(a) Moiré pattern for a camera lens (focal length = 55 mm, f/1.7) obtained with a spherical reference wavefront arrangement (λ = 0.6328 × 10−3 mm, d = 0.25 mm, and zp = 2 × 55 mm). (b) Lateral spherical aberration of the camera lens obtained from (a) with the computed one.

Fig. 4
Fig. 4

Moiré fringe distortion of Fig. 3(a), together with the one computed by Eq. (14).

Fig. 5
Fig. 5

Wave aberration in the case of (a) the normal incident beam and (b) the reverse incident beam.

Fig. 6
Fig. 6

Optical arrangement with the plane reference wavefront.

Fig. 7
Fig. 7

Lateral spherical aberration of the camera lens measured with the plane reference wavefront arrangement, together with the computed one for λ = 0.6328 × 10−3 mm, d = 0.1 mm, zp = 200 mm, and z1 = −70 mm.

Equations (22)

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I ( x , y , z p ) = A o 2 + 2 A 2 ± 4 A o A cos ( 2 π d ) f z p f [ x z p f f z p w ( x , y , z p ) x ] + 2 A 2 cos ( 2 π d / 2 ) f z p f [ x z p f f z p w ( x , y , z p ) x ] ,
α 0 , y = cos α f / ( z p f ) sin α x + z p sin α w ( x , y , z p ) x + d sin α m ,
α = 0 , x = z p f 2 f z p d m + z p f 2 f z p z p w ( x , y , z p ) x ,
Δ N ( x , 0 , z p ) = z p d w ( x , 0 , z p ) x .
w ( x , 0,2 f ) w ( x , 0,0 ) ,
Δ t ( x , 0,0 ) = f { [ w ( x , 0,0 ) ] / ( x ) } f { [ w ( x , 0,2 f ) ] / ( x ) } = ( d / 2 ) Δ N ( x , 0,2 f ) .
δ x = ( z p / f ) | Δ t | = d Δ N ( x , 0,2 f ) .
Δ x = ( 2 λ z p ) / d
δ x < Δ x .
ϕ ( x , 0 , z p ) = [ ( 2 π ) / λ ] ( { x 2 / [ 2 ( z p f ) ] } g ( x ) ) ,
g ( x ) = [ x 4 / 8 ( z p f ) 3 ] [ x 6 / 16 ( z p f ) 5 ] + { ( 5 x 8 ) / [ 128 ( z p f ) 7 ] } + .
U ( x , 0 , z p ) = A o exp 2 π i λ [ x 2 2 ( z p f ) g ( x ) w ( x , 0 , z p ) ] + A exp 2 π i λ [ ( x a ) 2 2 ( z p f ) g ( x a ) w ( x a f z p , 0 , z p ) + λ 2 2 d 2 ( f + z g ) ] + A exp 2 π i λ [ ( x + a ) 2 2 ( z p f ) g ( x + a ) w ( x + a f z p , 0 , z p ) + λ 2 2 d 2 ( f + z g ) ] ,
I ( x , 0 , z p ) = A o 2 + 2 A 2 ± 4 A o A cos ( 2 π d ) f z p f { x z p f f [ z p w ( x , 0 , z p ) x + f g ( x ) x ] } + 2 A 2 cos ( 4 π d ) f z p f { x z p f f [ z p w ( x , 0 , z p ) x + f g ( x ) x ] } .
Δ N ( x , 0 , z p ) = 1 d [ z p w ( x , 0 , z p ) x + f g ( x ) x ] 1 d { z p w ( x , 0 , z p ) x + f [ x 3 2 ( z p f ) 3 3 x 5 8 ( z p f ) 5 + 5 x 7 16 ( z p f ) 7 ] } .
F ( x , y ) = ( A o 2 + 2 A 2 ) / 2 ± A o A cos [ ( 2 π ) / d ] { ( 1 cos α ) x + y sin α z p [ w ( x , y , z p ) ] / ( x ) } ,
α 0 , y = [ ( cos α 1 ) / ( sin α ) ] x + ( z p ) / ( sin α ) [ w ( x , y , z p ) ] / ( x ) + [ d / ( sin α ) ] m ,
α = 0 , z p { [ w ( x , y , z p ) ] / x } = m d ,
Δ N ( x , 0 , z p ) = ( z p / d ) [ w ( x , 0 , z p ) ] / ( x )
w ( x , 0 , z p ) w ( x , 0 , z 1 ) ,
Δ t ( x , 0 , z 1 ) = f w ( x , 0 , z 1 ) x f w ( x , 0 , z p ) x = f d z p Δ N ( x , 0 , z p ) .
δ x = ( z p z 1 ) { [ w ( x , 0 , z p ) ] / ( x ) } = [ ( z p z 1 ) / z p ] d Δ N ( x , 0 , z p ) ,
Δ x = ( 2 λ z p ) / d .

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