Abstract

A new lateral shear interferometer based on the Ronchi grating is proposed. The normal Ronchi test is impaired by multiple beam interference (Talbot effect). Furthermore, the fringe pattern is somewhat restricted by fringe number and orientation. The new shear interferometer suppresses the Talbot effect by spatial filtering and by using a second grating enables arbitrary fringe orientation and number.

© 1981 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing (Wiley, New York, 1979).
  2. A. W. Lohmann, D. E. Silva, Opt. Commun. 2, 413 (1971).
  3. B. R. Hunt, J. Opt. Soc. Am. 69, 393 (1979).
  4. G. Schulz, Opt. Acta 20, 141 (1973).
  5. D. Malacara, A. Cornejo, Bol. Inst. Tonantzintla 1, 193 (1974).
  6. A. Cornejo, H. Altamirano, M. V. R. K. Murty, Bol. Inst. Tonantzintla 2, 313 (1978).
  7. J. D. Briers, Opt Technol. 1, 196 (1969).
  8. The original paper was not accessible but only a short description by S. Flügge, “Uber die Prüfung optischer Systeme nach Inter-ferenzmethoden von V. Ronchi,” Z. Instrumentenkd. 46, 209 (1926).
  9. V. Ronchi, Riv. d'ottica edi mecc. di pree. 2, 4 (1923).
  10. J. Schwider, Appl. Opt. 19, 4233 (1980).
  11. D. Malacara, A. Cornejo, Appl. Opt. 10, 679 (1971).
  12. R. Barakat, J. Opt. Soc. Am. 59, 32 (1969).
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  14. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  15. O. Bryngdahl, A. W. Lohmann, J. Opt. Soc. Am. 58, 1 (1968);J. Schwider, “Applications de l'Holographie,” C. R. de Symposium International, Besancon, 1970;M. Marquet, M. Novaro, M. Raynaud, “Applications de l'Holographie,” C. R. de Symposium International, Besancon, 1970.
  16. It should be noted that a grating having an amplitude transparency of t = cos(ωx0) would also fulfill the condition. The fabrication of such a grating, however, is a serious technological problem.
  17. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).
  18. This treatment enables a deeper insight into the fringe formation and furnishes an explanation for the fringe shift caused by axial movements of grating G2.

1980 (1)

1979 (1)

1978 (1)

A. Cornejo, H. Altamirano, M. V. R. K. Murty, Bol. Inst. Tonantzintla 2, 313 (1978).

1974 (1)

D. Malacara, A. Cornejo, Bol. Inst. Tonantzintla 1, 193 (1974).

1973 (1)

G. Schulz, Opt. Acta 20, 141 (1973).

1971 (2)

A. W. Lohmann, D. E. Silva, Opt. Commun. 2, 413 (1971).

D. Malacara, A. Cornejo, Appl. Opt. 10, 679 (1971).

1969 (2)

R. Barakat, J. Opt. Soc. Am. 59, 32 (1969).

J. D. Briers, Opt Technol. 1, 196 (1969).

1968 (1)

1926 (1)

The original paper was not accessible but only a short description by S. Flügge, “Uber die Prüfung optischer Systeme nach Inter-ferenzmethoden von V. Ronchi,” Z. Instrumentenkd. 46, 209 (1926).

1923 (1)

V. Ronchi, Riv. d'ottica edi mecc. di pree. 2, 4 (1923).

Altamirano, H.

A. Cornejo, H. Altamirano, M. V. R. K. Murty, Bol. Inst. Tonantzintla 2, 313 (1978).

Barakat, R.

R. Barakat, J. Opt. Soc. Am. 59, 32 (1969).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Briers, J. D.

J. D. Briers, Opt Technol. 1, 196 (1969).

Bryngdahl, O.

Cornejo, A.

A. Cornejo, H. Altamirano, M. V. R. K. Murty, Bol. Inst. Tonantzintla 2, 313 (1978).

D. Malacara, A. Cornejo, Bol. Inst. Tonantzintla 1, 193 (1974).

D. Malacara, A. Cornejo, Appl. Opt. 10, 679 (1971).

Flügge, S.

The original paper was not accessible but only a short description by S. Flügge, “Uber die Prüfung optischer Systeme nach Inter-ferenzmethoden von V. Ronchi,” Z. Instrumentenkd. 46, 209 (1926).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hunt, B. R.

Lohmann, A. W.

Malacara, D.

D. Malacara, A. Cornejo, Bol. Inst. Tonantzintla 1, 193 (1974).

D. Malacara, A. Cornejo, Appl. Opt. 10, 679 (1971).

D. Malacara, Optical Shop Testing (Wiley, New York, 1979).

Murty, M. V. R. K.

A. Cornejo, H. Altamirano, M. V. R. K. Murty, Bol. Inst. Tonantzintla 2, 313 (1978).

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Ronchi, V.

V. Ronchi, Riv. d'ottica edi mecc. di pree. 2, 4 (1923).

Schulz, G.

G. Schulz, Opt. Acta 20, 141 (1973).

Schwider, J.

Silva, D. E.

A. W. Lohmann, D. E. Silva, Opt. Commun. 2, 413 (1971).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Appl. Opt. (2)

Bol. Inst. Tonantzintla (2)

D. Malacara, A. Cornejo, Bol. Inst. Tonantzintla 1, 193 (1974).

A. Cornejo, H. Altamirano, M. V. R. K. Murty, Bol. Inst. Tonantzintla 2, 313 (1978).

J. Opt. Soc. Am. (3)

Opt Technol. (1)

J. D. Briers, Opt Technol. 1, 196 (1969).

Opt. Acta (1)

G. Schulz, Opt. Acta 20, 141 (1973).

Opt. Commun. (1)

A. W. Lohmann, D. E. Silva, Opt. Commun. 2, 413 (1971).

Riv. d'ottica edi mecc. di pree. (1)

V. Ronchi, Riv. d'ottica edi mecc. di pree. 2, 4 (1923).

Z. Instrumentenkd. (1)

The original paper was not accessible but only a short description by S. Flügge, “Uber die Prüfung optischer Systeme nach Inter-ferenzmethoden von V. Ronchi,” Z. Instrumentenkd. 46, 209 (1926).

Other (6)

D. Malacara, Optical Shop Testing (Wiley, New York, 1979).

It should be noted that a grating having an amplitude transparency of t = cos(ωx0) would also fulfill the condition. The fabrication of such a grating, however, is a serious technological problem.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

This treatment enables a deeper insight into the fringe formation and furnishes an explanation for the fringe shift caused by axial movements of grating G2.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

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Figures (11)

Fig. 1
Fig. 1

Principal scheme of the extrafocal Ronchi test using a cos-type grating.

Fig. 2
Fig. 2

Talbot effect for an extrafocal Ronchi test due to chromatic and spherical aberrations of the lens under test.

Fig. 3
Fig. 3

Nonlinearly processed and filtered version of Fig. 2.

Fig. 4
Fig. 4

Principal scheme of the single sideband Ronchi test: G1,2, gratings; F1, spatial filter; x,y and ξ,η, coordinates in object and Fourier space, respectively.

Fig. 5
Fig. 5

Demonstration of the elimination of the Talbot effect by spatial filtering: left, intrafocal Ronchi test showing the Talbot effect; right, intrafocal Ronchi test, Talbot interference suppressed by spatial filtering.

Fig. 6
Fig. 6

Total scheme of the single sideband setup used in the Fresnel approximation.

Fig. 7
Fig. 7

Single sideband setup for the testing of microscope objectives. Version with laser illumination is in brackets.

Fig. 8
Fig. 8

Single sideband test of microscope achromat 0.50/25 × with He–Ne laser illumination.

Fig. 9
Fig. 9

Filtered white light illumination for single sideband Ronchi test: green filter light, λ = 550 nm.

Fig. 10
Fig. 10

Interferogram produced in single sideband Ronchi test with removed spatial filter 2 (see Fig. 6): green filter light, λ = 550 nm.

Fig. 11
Fig. 11

Interferogram produced in single sideband Ronchi test with all spatial filters removed from the ray path, Talbot effect in the central region: green filter light, λ = 550 nm.

Equations (37)

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u ( x , y ) + exp j [ k 2 z 1 ( x 0 2 + y 0 2 ) ] [ 1 + cos ω x 0 ] × exp j { k 2 z 2 [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } d x 0 d y 0 ,
u 0 ( x , y ) exp j k 2 z 2 ( x 2 + y 2 ) exp j k ( 1 2 z 1 + 1 2 z 2 ) ( x 0 2 + y 0 2 ) * exp j k z 2 ( x 0 x + y 0 y ) d x 0 d y 0 , u 1 ( x , y ) exp j k 2 z 2 ( x 2 + y 2 ) exp j k ( 1 2 z 1 + 1 2 z 2 ) ( x 0 2 + y 0 2 ) * exp j [ k z 2 ( x x 0 + y y 0 ) + ω x 0 ] d x 0 d y 0 , u 1 ( x , y ) exp j k 2 z 2 ( x 2 + y 2 ) exp j k ( 1 2 z 1 + 1 2 z 2 ) ( x 0 2 + y 0 2 ) * exp j [ k z 2 ( x x 0 + y y 0 ) + ω x 0 ] d x 0 d y 0 .
α = k ( 1 2 z 1 + 1 2 z 2 )
π α exp ( j π 4 ) exp ( j t 2 4 α ) = + exp ( j α x 2 ) exp ( j t x ) d x .
u 0 ( x , y ) exp j k 2 ( z 1 + z 2 ) ( x 2 + y 2 ) , u 1 ( x , y ) u 0 ( x , y ) exp j k ω 2 z 2 α x exp j ω 2 4 α , u 1 ( x , y ) u 0 ( x , y ) exp + j k ω 2 z 2 α x exp j ω 2 4 α .
i ( x , y ) = a + b { cos [ 2 π z 2 z 2 + z 1 ( z 1 z 2 x g λ z 1 2 g 2 ) ] + cos [ 2 π z 2 z 2 + z 1 ( z 1 z 2 x g + λ z 1 2 g 2 ) ] } + c cos [ 2 π z 2 z 2 + z 1 ( z 1 z 2 2 x g ) ] ,
i ( x , y ) = a + 2 b cos 2 π [ ( z 2 2 2 + z 1 ) ( z 1 z 2 x g ) ] cos [ 2 π ( z 2 z 1 + z 2 ) ( λ z 1 2 g 2 ) ] + c cos [ 4 π ( z 2 z 1 + z 2 ) ( z 1 z 2 x g ) ] .
cos ( z 1 z 1 + z 2 2 π λ z 1 2 g 2 )
a + 2 b cos { 2 π z 2 z 1 + z 2 ( z 1 z 2 x g ) }
ξ ̅ 1 = z 1 λ g 1 ,
g 1 z 1 g 2 z 2 .
x s z 1 + z 2 = ξ ̅ 1 z 2 , x s = ξ ̅ 1 ( 1 + z 1 z 2 ) .
ξ ̅ 1 = z 1 λ g 1 ,
x s = λ g 1 z 1 ( 1 + z 1 z 2 ) .
u ( x 1 , y 1 ) = t ( x 1 , y 1 ) exp [ j k 2 z 1 ( x 1 2 + y 1 2 ) ] .
t ( x 1 , y 1 ) = 1 + cos 2 π ν 1 x 1 .
u ( ξ 1 , η 1 ) = exp ( j k z 1 ) j λ z 1 t ( x 1 , y 1 ) exp [ j k z 1 ( ξ 1 x 1 + η 1 y 1 ) ] * exp [ j k 2 z 1 ( ξ 1 2 + η 1 2 ) ] d x 1 d y 1 .
u ( ξ 1 , η 1 ) = exp ( j k z 1 ) j λ z 1 exp [ j k 2 z 1 ( ξ 1 2 + η 1 2 ) ] * [ δ ( ξ 1 λ z 1 , η 1 λ z 1 ) + 1 2 δ ( ξ 1 λ z 1 ν 1 , η 1 λ z 1 ) + 1 2 δ ( ξ 1 λ z 1 + ν 1 , η 1 λ z 1 ) ] ,
δ ( ξ 1 λ z 1 + ν 1 , η 1 λ z 1 )
u ( x 2 , y 2 ) = exp [ j k ( z 1 + z 2 ) ] λ 2 z 1 z 2 exp [ j k 2 z 2 ( x 2 2 + y 2 2 ) ] * + exp j k 2 ( 1 z 1 + 1 z 2 ) ( ξ 1 2 + η 1 2 ) [ δ ( ξ 1 λ z 1 , η 1 λ z 1 ) + 1 2 δ ( ξ 1 λ z 1 ν 1 , η 1 λ z 1 ) ] exp 2 π j ( x 2 λ z 2 ξ 1 + y 2 λ z 2 η 1 ) d ξ 1 d η 1 .
+ δ ( x , y ) F ( x , y ) dxdy = F ( o , o ) , + δ ( x a , y ) F ( x , y ) dydy = F ( a , o ) ,
u ( x 2 , y 2 ) = z 1 z 2 exp j k ( z 1 + z 2 ) exp [ j k 2 z 2 ( x 2 2 + y 2 2 ) ] * [ 1 + 1 2 exp j k 2 ( 1 z 1 + 1 z 2 ) ( λ z 1 ν 1 ) 2 exp 2 π j x 2 z 2 z 1 ν 1 ] .
exp [ j k 2 f ( x 2 2 + y 2 2 ) ] .
u ( ξ 2 , η 2 ) = exp j k ( z 1 + z 2 + z 3 ) j λ z 3 z 1 z 2 exp j k 2 z 3 ( ξ 2 2 + η 2 2 ) * + exp j [ k 2 ( x 2 2 + y 2 2 ) ( 1 z 2 + 1 z 3 1 f ) ] * [ 1 + 1 2 exp j k 2 ( 1 z 1 + 1 z 2 ) ( λ z 1 ν 1 ) 2 exp ( 2 π j z 1 z 2 ν 1 x 2 ) ] * [ 1 + cos 2 π γ 2 ( x 2 cos φ + y 2 sin φ ) ] * exp j 2 π ( ξ 2 λ z 3 x 2 + η 2 λ z 3 y 2 ) d x 2 d y 2 ,
1 z 2 + 1 z 3 1 f = 0.
u ( ξ 2 , η 2 ) = exp [ j k ( z 1 + z 2 + z 3 ) ] j λ z 2 z 1 z 3 exp [ j k ( ξ 2 2 + η 2 2 ) 2 z 3 ] { δ ( ξ 2 λ z 3 , η 2 λ z 3 ) + 1 2 δ ( ξ 2 λ z 3 ν 2 cos φ , η 2 λ z 3 ν 2 sin φ ) + 1 2 δ ( ξ 2 λ z 3 + ν 2 cos φ , η 2 λ z 3 + ν 2 sin φ ) + 1 2 exp [ j k 2 ( 1 z 1 + 1 z 2 ) ( λ z 1 ν 1 ) 2 ] [ δ ( ξ 2 λ z 3 + z 1 z 2 ν 1 , η 2 λ z 3 ) + 1 2 δ ( ξ 2 λ z 3 + z 1 z 2 ν 1 ν 2 cos φ , η 2 λ z 3 ν 2 sin φ ) + 1 2 δ ( ξ 2 λ z 3 + z 1 z 2 ν 1 + ν 2 cos φ , η 2 λ z 3 + ν 2 sin φ ) ] } .
δ ( ξ 2 λ z 3 + ν 2 cos φ , η 2 λ z 3 + ν 2 sin φ ) , δ ( ξ 2 λ z 3 + z 1 z 2 ν 1 , η 2 λ z 3 )
u ( x 2 , y 3 ) = exp [ j k ( z 1 + z 2 + z 3 + z 4 ) ] λ 2 z 1 z 2 z 3 z 4 + exp [ j k 2 z 3 ( ξ 2 2 + η 2 2 ) ] * [ 1 2 δ ( ξ 2 λ z 3 + ν 2 cos φ , η 2 λ z 3 + ν 2 sin φ ) + 1 2 exp j k 2 ( 1 z 1 + 1 z 2 ) ( λ z 1 ν 1 ) 2 δ ( ξ 2 λ z 3 + z 1 z 2 ν 1 , η 2 λ z 3 ) ] * exp j k 2 z 4 [ ( x 3 ξ 2 ) 2 + ( y 3 η 2 ) 2 ] d ξ 2 d η 2 ,
u 01 exp ( j k i = 1 4 z i ) z 2 z 4 z 1 z 3 exp [ j k 2 z 4 ( x 3 2 + y 3 2 ) ] exp [ j k 2 z 3 z 4 ( z 3 + z 4 ) λ 2 ν 2 2 ] * exp [ j k λ z 3 z 4 ( x 3 ν 2 cos φ + y 3 ν 2 sin φ ) ] , u 10 exp ( j k i = 1 4 z i ) z 2 z 4 z 1 z 3 exp [ j k 2 z 4 ( x 3 2 + y 3 2 ) ] * exp [ j k 2 z 1 z 2 ( z 1 + z 2 ) λ 2 ν 1 2 ] exp [ j k 2 z 3 z 4 ( z 3 + z 4 ) λ 2 ( z 1 z 2 ν 1 ) 2 ] * exp [ j k λ z 3 z 4 ( x 3 z 1 z 2 ν 1 ) ] .
i ( x 3 , y 3 ) = a + b cos 2 π { z 3 z 4 [ ( z 1 z 2 ν 1 ν 2 cos φ ) x 3 + y 3 ν 2 sin φ ] ϕ } ,
ϕ = π λ [ z 3 z 4 ( z 3 + z 4 ) ν 2 2 z 1 z 2 ( z 1 + z 2 ) ν 1 2 z 3 z 4 ( z 1 z 2 ) 2 ( z 3 + z 4 ) ν 1 2 ] .
z 1 z 2 ν 1 ν 2 cos φ = 0 ,
z 3 z 4 ν 2 sin φ .
λ ν 2 z 2 ( z 1 + z 2 z 2 ) z 2 δ z z 1 cos φ ,
z 2 δ z z 1
tan ψ = z 1 z 1 + z 2 δ z 2 δ z tan φ ,
z 1 z 1 + z 2 δ z z 2 δ z

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