Abstract

A general analysis of an n-grating interferometer under various conditions of illumination is presented, where n = 1, …, 4. Conditions for fringe localization and effects of misalignment are given. The lesser-known phenomenon of the imaging of a grating by a second grating is described from which the fringe-forming capacity of multiple-grating interferometers stems; this can occur regardless of the coherence of the source.

© 1975 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. J. Chang, Ph.D. Thesis, The University of Michigan (1974), available from University Microfilms, Ann Arbor, Michigan.
  2. E. Leith, B. Chang, R. Alferness, S. Case, J. Opt. Soc. Am. 64, 558 (1974).
  3. C. Barus, Carnegie Inst. Wash. Publ. 149 [Part 1 (1911); Part 2 (1912); Part 3 (1914); Publ. 229 (1915)].
  4. J. C. Wyant, Appl. Opt. 13, 200 (1974).
    [CrossRef] [PubMed]
  5. V. Ronchi, Riv. Ottica Meccan Presis 1, 9 (1923).
  6. V. Ronchi, Appl. Opt. 3, 437 (1963).
    [CrossRef]
  7. R. Kraushavar, J. Opt. Soc. Am. 40, 480 (1950).
    [CrossRef]
  8. J. R. Sterrett, J. R. Erwin, NACA TN2827 (1952).
  9. F. J. Weinberg, N. B. Wood, J. Sci. Instrum. 36, 227 (1959).
    [CrossRef]
  10. L. Marton, Phys. Rev. 85, 1057 (1952).
    [CrossRef]
  11. L. Marton, J. A. Simpson, J. A. Suddeth, Phy. Rev. 90, 490 (1953).
    [CrossRef]
  12. J. A. Simpson, Rev. Sci. Instrum. 25, 1105 (1954).
    [CrossRef]
  13. H. Medlowitz, J. A. Simpson, J. Opt. Soc. Am. 52, 520 (1962).
    [CrossRef]
  14. U. Bose, M. Hart, Acta. Cryst. A24, 240 (1968).
  15. E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 57, 975 (1967).
    [CrossRef]
  16. M. Kato, T. Suzuki, J. Opt. Soc. Am. 59, 303 (1969).
    [CrossRef]
  17. O. Bryngdahl, A. Lohmann, J. Opt. Soc. Am. 60, 281 (1970).
    [CrossRef]
  18. E. N. Leith, B. J. Chang, Appl. Opt. 8, 1957 (1973).
    [CrossRef]
  19. B. J. Chang, Opt. Commun. 9, 357 (1973).
    [CrossRef]
  20. B. J. Chang, J. Opt. Soc. Am. 64, 552 (1974).
  21. A. Lohmann, in Proc. ICO Conf. Opt. Instr., K. J. Habbell, Ed. (Chapman and Hall, London, 1961), Vol. 54.
  22. A. R. Maddox, R. C. Binder, Appl. Opt. 8, 2191 (1969).
    [CrossRef] [PubMed]
  23. A. W. Lohmann, D. E. Silva, Opt. Commun. 2, 413 (1971).
    [CrossRef]
  24. S. Mallick, Opt. Acta. 19, 747 (1972).
    [CrossRef]
  25. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  26. H. F. Talbot, Philos. Mag. 9, 401 (1836).
  27. We do not consider here the effect of rotational error about the other two axes that result in the grating planes not being parallel. That effect will be considered in a subsequent paper.

1974 (3)

E. Leith, B. Chang, R. Alferness, S. Case, J. Opt. Soc. Am. 64, 558 (1974).

B. J. Chang, J. Opt. Soc. Am. 64, 552 (1974).

J. C. Wyant, Appl. Opt. 13, 200 (1974).
[CrossRef] [PubMed]

1973 (2)

1972 (1)

S. Mallick, Opt. Acta. 19, 747 (1972).
[CrossRef]

1971 (1)

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

1970 (1)

1969 (2)

1968 (1)

U. Bose, M. Hart, Acta. Cryst. A24, 240 (1968).

1967 (1)

1963 (1)

1962 (1)

1959 (1)

F. J. Weinberg, N. B. Wood, J. Sci. Instrum. 36, 227 (1959).
[CrossRef]

1954 (1)

J. A. Simpson, Rev. Sci. Instrum. 25, 1105 (1954).
[CrossRef]

1953 (1)

L. Marton, J. A. Simpson, J. A. Suddeth, Phy. Rev. 90, 490 (1953).
[CrossRef]

1952 (1)

L. Marton, Phys. Rev. 85, 1057 (1952).
[CrossRef]

1950 (1)

1923 (1)

V. Ronchi, Riv. Ottica Meccan Presis 1, 9 (1923).

1836 (1)

H. F. Talbot, Philos. Mag. 9, 401 (1836).

Alferness, R.

E. Leith, B. Chang, R. Alferness, S. Case, J. Opt. Soc. Am. 64, 558 (1974).

Barus, C.

C. Barus, Carnegie Inst. Wash. Publ. 149 [Part 1 (1911); Part 2 (1912); Part 3 (1914); Publ. 229 (1915)].

Binder, R. C.

Bose, U.

U. Bose, M. Hart, Acta. Cryst. A24, 240 (1968).

Bryngdahl, O.

Case, S.

E. Leith, B. Chang, R. Alferness, S. Case, J. Opt. Soc. Am. 64, 558 (1974).

Chang, B.

E. Leith, B. Chang, R. Alferness, S. Case, J. Opt. Soc. Am. 64, 558 (1974).

Chang, B. J.

B. J. Chang, J. Opt. Soc. Am. 64, 552 (1974).

E. N. Leith, B. J. Chang, Appl. Opt. 8, 1957 (1973).
[CrossRef]

B. J. Chang, Opt. Commun. 9, 357 (1973).
[CrossRef]

B. J. Chang, Ph.D. Thesis, The University of Michigan (1974), available from University Microfilms, Ann Arbor, Michigan.

Erwin, J. R.

J. R. Sterrett, J. R. Erwin, NACA TN2827 (1952).

Goodman, J. W.

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

Hart, M.

U. Bose, M. Hart, Acta. Cryst. A24, 240 (1968).

Kato, M.

Kraushavar, R.

Leith, E.

E. Leith, B. Chang, R. Alferness, S. Case, J. Opt. Soc. Am. 64, 558 (1974).

Leith, E. N.

Lohmann, A.

O. Bryngdahl, A. Lohmann, J. Opt. Soc. Am. 60, 281 (1970).
[CrossRef]

A. Lohmann, in Proc. ICO Conf. Opt. Instr., K. J. Habbell, Ed. (Chapman and Hall, London, 1961), Vol. 54.

Lohmann, A. W.

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

Maddox, A. R.

Mallick, S.

S. Mallick, Opt. Acta. 19, 747 (1972).
[CrossRef]

Marton, L.

L. Marton, J. A. Simpson, J. A. Suddeth, Phy. Rev. 90, 490 (1953).
[CrossRef]

L. Marton, Phys. Rev. 85, 1057 (1952).
[CrossRef]

Medlowitz, H.

Ronchi, V.

V. Ronchi, Appl. Opt. 3, 437 (1963).
[CrossRef]

V. Ronchi, Riv. Ottica Meccan Presis 1, 9 (1923).

Silva, D. E.

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

Simpson, J. A.

H. Medlowitz, J. A. Simpson, J. Opt. Soc. Am. 52, 520 (1962).
[CrossRef]

J. A. Simpson, Rev. Sci. Instrum. 25, 1105 (1954).
[CrossRef]

L. Marton, J. A. Simpson, J. A. Suddeth, Phy. Rev. 90, 490 (1953).
[CrossRef]

Sterrett, J. R.

J. R. Sterrett, J. R. Erwin, NACA TN2827 (1952).

Suddeth, J. A.

L. Marton, J. A. Simpson, J. A. Suddeth, Phy. Rev. 90, 490 (1953).
[CrossRef]

Suzuki, T.

Talbot, H. F.

H. F. Talbot, Philos. Mag. 9, 401 (1836).

Upatnieks, J.

Weinberg, F. J.

F. J. Weinberg, N. B. Wood, J. Sci. Instrum. 36, 227 (1959).
[CrossRef]

Wood, N. B.

F. J. Weinberg, N. B. Wood, J. Sci. Instrum. 36, 227 (1959).
[CrossRef]

Wyant, J. C.

Acta. Cryst. (1)

U. Bose, M. Hart, Acta. Cryst. A24, 240 (1968).

Appl. Opt. (4)

J. Opt. Soc. Am. (7)

J. Sci. Instrum. (1)

F. J. Weinberg, N. B. Wood, J. Sci. Instrum. 36, 227 (1959).
[CrossRef]

Opt. Acta. (1)

S. Mallick, Opt. Acta. 19, 747 (1972).
[CrossRef]

Opt. Commun. (2)

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

B. J. Chang, Opt. Commun. 9, 357 (1973).
[CrossRef]

Philos. Mag. (1)

H. F. Talbot, Philos. Mag. 9, 401 (1836).

Phy. Rev. (1)

L. Marton, J. A. Simpson, J. A. Suddeth, Phy. Rev. 90, 490 (1953).
[CrossRef]

Phys. Rev. (1)

L. Marton, Phys. Rev. 85, 1057 (1952).
[CrossRef]

Rev. Sci. Instrum. (1)

J. A. Simpson, Rev. Sci. Instrum. 25, 1105 (1954).
[CrossRef]

Riv. Ottica Meccan Presis (1)

V. Ronchi, Riv. Ottica Meccan Presis 1, 9 (1923).

Other (6)

C. Barus, Carnegie Inst. Wash. Publ. 149 [Part 1 (1911); Part 2 (1912); Part 3 (1914); Publ. 229 (1915)].

J. R. Sterrett, J. R. Erwin, NACA TN2827 (1952).

B. J. Chang, Ph.D. Thesis, The University of Michigan (1974), available from University Microfilms, Ann Arbor, Michigan.

We do not consider here the effect of rotational error about the other two axes that result in the grating planes not being parallel. That effect will be considered in a subsequent paper.

A. Lohmann, in Proc. ICO Conf. Opt. Instr., K. J. Habbell, Ed. (Chapman and Hall, London, 1961), Vol. 54.

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Diffraction by a grating. S is the source plane, with polar coordinates; Lc is the colliminating lens of focal length F; G1 is the grating; H is the observation plane; k ¯ s is the propagation vector of light emitted from an element at; s ¯; f ¯ 1 is the spatial frequency vector of G1; x ̂, Ŷ, and are the direction vectors; and the sign of angles α and ω is positive in the counterclockwise direction.

Fig. 2
Fig. 2

Imaging of a grating by a second grating in the illumination of limited coherence.

Fig. 3
Fig. 3

One-grating interferometers. M1 and M2 are mirrors, and H is the observation plane.

Fig. 4
Fig. 4

Generalized model for a three-grating interferometer. G1 is the grating used as a beam splitter, and G2r(G2o) is the grating used as beam deflector for beam r (beam o). The lines of the grating G1 are parallel to the y axis, while those of G2r(G2o) make an angle αr(αo) with the y axis. The spatial frequencies of G1 and G2 are f1 and f2.

Fig. 5
Fig. 5

A well-aligned four-grating interferometer; S is the ordinary tungsten lamp, G is a grating, and T is the test object, where each grating has 200 l/mm. C is the camera.

Fig. 6
Fig. 6

Interference patterns formed in all-grating interferometer shown in Fig. 5 (a) without test object; (b) with test object that is a temperature field around a heated metal.

Fig. 7
Fig. 7

Fringe visibility vs Δz for a fixed slit width (1.0 mm), f = 330 l/mm.

Fig. 8
Fig. 8

Half-width at half-maximum (HWHM) vs grating frequency (o) for fixed slit width (2.0 mm); HWHM vs slit width (X) for fixed grating frequency (f = 330 l/mm).

Fig. 9
Fig. 9

Fringes formed 2.5 mm off the plane of localization with slit source (∼ 0.5 mm × 500 mm) aligned (a) parallel to grating lines of grating interferometer; (b) perpendicular to grating lines. Grating frequency f = 330 l/mm.

Fig. 10
Fig. 10

Fringe visibility vs rotation error αr for fixed grating frequency (300 /mm) and fixed slit width (0.285 mm).

Fig. 11
Fig. 11

Fringes formed at the plane of localization for rotation error αr 0.5 × 10−3 rad with slit source (∼ 0.5 mm × 1000 mm) aligned (a) parallel to lines of first grating of the interferometer; (b) perpendicular to lines of first grating. Grating frequency f = 330 l/mm.

Tables (2)

Tables Icon

Table I Summary of Sufficient Conditions for Fringe Types of One-Grating interferometers

Tables Icon

Table II Summary of Sufficient Conditions for Fringe Types of Three-Grating Interferometers

Equations (45)

Equations on this page are rendered with MathJax. Learn more.

k ¯ s = k ( x ̂ sin θ cos ω + Ŷ sin θ sin ω + cos θ ) ,
θ = tan 1 ( l / F ) l / F if l F .
f ¯ 1 = f 1 ( x ̂ cos α + Ŷ sin α ) ,
T 1 ( r ¯ 1 ) = n = N N A n exp ( j 2 π n f ¯ 1 r ¯ 1 ) ,
V z = 0 ( r ¯ 1 , s ¯ , λ ) = [ S ( θ , ω , λ ) ] 1 / 2 exp ( j k ¯ s r ¯ 1 ) ,
V z = 0 + ( r ¯ 1 , s ¯ , λ ) = V z = 0 ( r ¯ 1 , s ¯ , λ ) T 1 ( r ¯ 1 ) .
V z = z 1 ( r ¯ 1 , s , λ ) ,
V z = z 1 ( r ¯ 1 , s ¯ , λ ) = n = N N A n [ S ( θ , ω , λ ) ] 1 / 2 exp ( j 2 π z 1 / λ ) exp { j π z 1 [ ( P + n C α ) 2 + ( q + n S α ) 2 ] } exp { j 2 π [ ( P + n C α ) x + ( q + n S α ) y ] } ,
I z = z 1 ( r ¯ ) = s λ | V z = z 1 | 2 d λ d s ,
s
Δ θ θ Δ θ and λ 0 Δ λ λ λ 0 + Δ λ .
V z = d 1 ( x 2 , θ , λ ) = n = N N A n [ S ( θ , λ ) ] 1 / 2 exp [ j π d 1 λ ( n f 1 + sin θ / λ ) 2 ] exp [ j 2 π ( n f 1 + sin θ / λ ) x 2 ] .
V z = d 1 + ( x 2 , θ , λ ) = T 2 ( x 2 ) V z = z 1 ( x 2 , θ , λ ) ,
T 2 ( x 2 ) = m = M M B m exp ( j 2 π m f 2 x 2 ) .
V z = z 2 ( x , θ , λ ) = n = N N m = M M A n B m [ S ( θ , λ ) ] exp ( j 2 π z 2 / λ ) exp [ j π d 1 ( n f 1 + sin θ / λ ) 2 ] exp [ j π d 2 ( n f 1 + m f 2 + sin θ / λ ) 2 ] exp [ j 2 π ( n f 1 + m f 2 + sin θ / λ ) x ] ,
I z = z 2 ( x ) = Δ θ Δ θ λ 0 Δ λ λ 0 + λ | V z = z 2 | 2 d λ d θ .
I 2 d 1 + z ( x ) n = N N A n 2 + n = N N n = N n n N A n A n * W n , n ( Δ θ , Δ λ , 2 d 1 + z 1 ) exp [ j π λ 0 ( n 2 n 2 ) ( 2 d 1 + z 1 ) f 1 2 ] exp [ j 2 π ( n n ) f 1 x ] ,
W n , n ( Δ θ , Δ λ , 2 d 1 + z 1 ) = sinc [ ( n 2 n 2 ) ( 2 d 1 + z 1 ) f 1 2 Δ λ ] sinc [ 2 ( n n ) z 1 f 1 Δ θ ] ,
I z = z 1 ( x ) n = N N | A n | 2 + n = N N n = N N n n A n A n * W n , n ( Δ θ , Δ λ , z 1 ) exp [ j π z 1 λ 0 ( n 2 n 2 ) f 1 2 ] exp [ j 2 π ( n n ) f 1 x ] ,
W n , n ( Δ θ , Δ λ , z 1 ) = sinc [ ( n 2 n 2 ) z 1 f 1 2 Δ λ ] sinc [ 2 ( n n ) z 1 f 1 Δ θ ] .
2 λ 0 d 1 f 1 2 = J , J = 1 , 2 , . . . .
I z = 2 d 1 + z 1 ( x ) 1 + sinc ( 2 z 1 f 2 Δ θ ) cos ( 2 π f 2 x ) ,
I z = d 1 ( x ) I 1 + W 1 cos [ 2 π ( δ r δ o ) f 1 x ] ,
I 1 = S ( θ , ω , λ ) θ d θ d ω d λ ,
W 1 = | S ( θ , ω , λ ) exp [ j ϕ ( θ , ω , λ ) ] θ d θ d ω d λ | ,
ϕ ( θ , λ ) = π [ 2 d 1 f 1 ( δ r δ o ) cos ω sin θ + d 1 f 1 2 ( δ r 2 δ o 2 ) ] .
V z = d 1 o ( r ¯ , s ¯ , λ ) = C 1 exp ( j 2 π d 1 λ ) [ S ( θ , ω , λ ) ] 1 / 2 exp { j π ( d 1 d 2 o ) λ [ ( P + δ 1 o f 1 ) 2 + q 2 ] } exp { j π d 2 o λ [ ( P + δ 1 o f 1 + δ 2 o C α o ) 2 + ( q + δ 2 o S α o ) 2 ] } exp { j 2 π [ ( P + δ 1 o f 1 + δ 2 o C α o ) x + ( q + δ 2 o S α o ) y ] } ,
V z = d 1 r ( r ¯ , s ¯ , λ ) = C 1 exp ( j 2 π d 1 λ ) [ S ( θ , ω , λ ) ] 1 / 2 exp { j π ( d 1 d 2 r ) λ [ ( P + δ 1 r f 1 ) 2 + q 2 ] } exp { j π d 2 r λ [ ( P + δ 1 r f 1 + δ 2 r C α r ) 2 + ( q + δ 2 r S α r ) 2 ] } exp { j 2 π [ ( P + δ 1 r f 1 + δ 2 r C α r ) x + ( P + δ 2 r S α r ) y ] } ,
I z = d 1 ( r ¯ , s ¯ , λ ) = 2 | C 1 | 2 S ( θ , ω , λ ) + 2 | C 1 | 2 Re S ( θ , ω , λ ) exp [ j ϕ ( θ , ω , λ ) ] exp [ j 2 π ( f x x + f y y ) + j ϕ o ] ,
λ = λ λ o , λ o is the center wavelength ; f x = ( δ 1 o δ 1 r ) f 1 + ( δ 2 o C α o δ 2 r C α r ) = ( δ 1 o δ 1 r ) f 1 + ( δ 2 o cos α o + δ 2 r cos α r ) f 2 ; f y = ( δ 2 o S α o δ 2 r S α r ) = ( δ 2 o sin α o δ 2 r sin α r ) ; ϕ o = π λ o [ d 1 ( δ 1 r 2 δ 1 o 2 ) f 1 2 + ( d 2 r δ 2 r 2 d 2 o δ 2 o 2 ) f 2 2 + 2 ( d 2 r δ 1 r δ 2 r cos α r d 2 o δ 1 o δ 2 o cos α o ) f 1 f 2 ] ,
λ = λ λ o , λ 0 is the center wavelength ; f x = ( δ 1 o δ 1 r ) f 1 + ( δ 2 o C α o δ 2 r C α r ) = ( δ 1 o δ 1 r ) f 1 + ( δ 2 o cos α o + δ 2 r cos α r ) f 2 ; f y = ( δ 2 o S α o δ 2 r S α r ) = ( δ 2 o sin α o + δ 2 r sin α r ) ; ϕ o = π λ o [ d 1 ( δ 1 r 2 δ 1 o 2 ) f 1 2 + ( d 2 r δ 2 r 2 d 2 o δ 2 o 2 ) f 2 2 + 2 ( d 2 r δ 1 r δ 2 r cos α r d 2 o δ 1 o δ 2 o cos α o ) f 1 f 2 ] ,
ϕ ( θ , ω , λ ) = π [ d 1 ( δ 1 r 2 δ 1 o 2 ) f 1 2 + ( d 2 r δ 2 r 2 d 2 o δ 2 o 2 ) f 2 2 + 2 ( d 2 r δ 1 r δ 2 r cos α r d 2 o δ 1 o δ 2 o cos α o ) f 1 f 2 ] λ + 2 π { [ d 1 ( δ 1 r δ 1 o ) f 1 + ( d 2 r δ 2 r cos α r d 2 o δ 2 o cos α o ) f 2 ] cos ω + ( d 2 r δ 2 r sin α r d 2 o δ 2 o sin α o ) f 2 sin ω } sin θ .
I z = d 1 ( r ¯ ) I 2 + W 2 cos 2 π ( f x x + f y y ) + ϕ o ,
I 2 = S ( θ , ω , λ ) θ d θ d ω d λ ,
W 2 = | S ( θ , ω , λ ) [ exp j ϕ ( θ , ω , λ ) ] θ d θ d ω d λ | .
FV = W 2 / I 2 .
S ( x s , y s , λ ) = 1 { D X 2 x s D X 2 D Y 2 y s D Y 2 Δ λ λ Δ λ = 0 otherwise ,
x s = F tan θ cos ω F sin θ cos ω , y s = F tan θ sin ω F sin θ cos ω ,
FV = | sinc { f 2 Δ λ [ d 1 d 2 o + d 2 r ( 1 2 cos α r ) ] } | | sinc [ f D X F ( d 1 + d 2 r cos α r + d 2 o cos α o ) ] | | sinc [ f D Y F ( d 2 r sin α r + d 2 o sin α o ) ] | .
FV = | sinc { f 2 Δ λ [ 2 d 2 r ( 1 cos α r ) + Δ z ( 1 2 cos α r ) ] } | | sinc { f D X F [ d 2 o ( cos α o 1 ) + d 2 r ( cos α r 1 ) + Δ z ( cos α o + cos α r 1 ) ] } | | sinc { f D Y F [ d 2 o sin α o + d 2 r sin α r + Δ z ( sin α o + sin α r ) ] } | .
FV = | sinc ( f 2 Δ λ Δ z | | sinc ( f D X Δ z F ) | .
FV = | sinc [ f D Y F ( d 2 o α o + d 2 r α r ) ] | .
{ | α o | = | α r | = α ; f 2 = 2 f 1 cos α ( in practice , f 2 > f 1 ) ; δ 1 r = ± 1 and δ 2 r = 1 ; δ 1 o = 1 and δ 2 o = ± 1 .
{ d 1 f 1 2 ( δ 1 r 2 δ 1 o 2 ) + f 2 2 ( d 2 r δ 2 r 2 d 2 o δ 2 o 2 ) + 2 ( d 2 r δ 1 r δ 2 r cos α r d 2 o δ 1 o δ 2 o cos α o ) f 1 f 2 = 0.
{ d 1 ( δ 1 r δ 1 o ) f 1 ( d 2 r δ 2 r cos α r d 2 o δ 2 o cos α o ) f 2 = 0 ; d 2 δ 2 r sin α r d 2 o δ 2 o sin α o = 0 .

Metrics