Abstract

One-dimensional diffraction-grating theory is developed in vector notation. First-order expressions are derived for the effects of the rotation of the grating about an arbitrary axis upon the diffracted beam. These results are applied to the three-grating interferometer whose characteristics are given in some detail. Generalization to moiré patterns or to any number of gratings with arbitrary separations is indicated.

© 1962 Optical Society of America

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References

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  1. C. Barus, Carnegie Inst. Wash. Publ. 149[Part I (1911); Part II (1912); Part III (1914); Publ. 229 (1915)].
  2. R. Kraushaar, J. Opt. Soc. Am. 40, 480 (1950).
    [Crossref]
  3. L. Marton, Electron Physics (Symposium, 1951), Natl. Bur. Standards Circular No. 527; Phys. Rev. 85, 1057 (1952).
    [Crossref]
  4. L. Marton, J. A. Simpson, and J. A. Suddeth, Phys. Rev. 90, 490 (1953).
    [Crossref]
  5. J. Arol Simpson, Rev. Sci. Instr. 25, 1105 (1954).
    [Crossref]
  6. Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).
    [Crossref]
  7. W. Ehrenberg and R. E. Siday, Proc. Phys. Soc. (London) B62, 8 (1949).
    [Crossref]
  8. L. Marton, J. A. Simpson, and J. A. Suddeth, Rev. Sci. Instr. 25, 1099 (1954).
    [Crossref]
  9. Z. Bay and H. Mendlowitz (unpublished), quoted in reference 10.
  10. F. G. Werner and D. R. Brill, Phys. Rev. Letters 4, 344 (1960).
    [Crossref]
  11. O. Rang, Z. Krist. 114, 98 (1960).
    [Crossref]
  12. C. Jonsson, Z. Physik 161, 954 (1961).

1961 (1)

C. Jonsson, Z. Physik 161, 954 (1961).

1960 (2)

F. G. Werner and D. R. Brill, Phys. Rev. Letters 4, 344 (1960).
[Crossref]

O. Rang, Z. Krist. 114, 98 (1960).
[Crossref]

1959 (1)

Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).
[Crossref]

1954 (2)

J. Arol Simpson, Rev. Sci. Instr. 25, 1105 (1954).
[Crossref]

L. Marton, J. A. Simpson, and J. A. Suddeth, Rev. Sci. Instr. 25, 1099 (1954).
[Crossref]

1953 (1)

L. Marton, J. A. Simpson, and J. A. Suddeth, Phys. Rev. 90, 490 (1953).
[Crossref]

1950 (1)

1949 (1)

W. Ehrenberg and R. E. Siday, Proc. Phys. Soc. (London) B62, 8 (1949).
[Crossref]

1911 (1)

C. Barus, Carnegie Inst. Wash. Publ. 149[Part I (1911); Part II (1912); Part III (1914); Publ. 229 (1915)].

Aharonov, Y.

Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).
[Crossref]

Arol Simpson, J.

J. Arol Simpson, Rev. Sci. Instr. 25, 1105 (1954).
[Crossref]

Barus, C.

C. Barus, Carnegie Inst. Wash. Publ. 149[Part I (1911); Part II (1912); Part III (1914); Publ. 229 (1915)].

Bay, Z.

Z. Bay and H. Mendlowitz (unpublished), quoted in reference 10.

Bohm, D.

Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).
[Crossref]

Brill, D. R.

F. G. Werner and D. R. Brill, Phys. Rev. Letters 4, 344 (1960).
[Crossref]

Ehrenberg, W.

W. Ehrenberg and R. E. Siday, Proc. Phys. Soc. (London) B62, 8 (1949).
[Crossref]

Jonsson, C.

C. Jonsson, Z. Physik 161, 954 (1961).

Kraushaar, R.

Marton, L.

L. Marton, J. A. Simpson, and J. A. Suddeth, Rev. Sci. Instr. 25, 1099 (1954).
[Crossref]

L. Marton, J. A. Simpson, and J. A. Suddeth, Phys. Rev. 90, 490 (1953).
[Crossref]

L. Marton, Electron Physics (Symposium, 1951), Natl. Bur. Standards Circular No. 527; Phys. Rev. 85, 1057 (1952).
[Crossref]

Mendlowitz, H.

Z. Bay and H. Mendlowitz (unpublished), quoted in reference 10.

Rang, O.

O. Rang, Z. Krist. 114, 98 (1960).
[Crossref]

Siday, R. E.

W. Ehrenberg and R. E. Siday, Proc. Phys. Soc. (London) B62, 8 (1949).
[Crossref]

Simpson, J. A.

L. Marton, J. A. Simpson, and J. A. Suddeth, Rev. Sci. Instr. 25, 1099 (1954).
[Crossref]

L. Marton, J. A. Simpson, and J. A. Suddeth, Phys. Rev. 90, 490 (1953).
[Crossref]

Suddeth, J. A.

L. Marton, J. A. Simpson, and J. A. Suddeth, Rev. Sci. Instr. 25, 1099 (1954).
[Crossref]

L. Marton, J. A. Simpson, and J. A. Suddeth, Phys. Rev. 90, 490 (1953).
[Crossref]

Werner, F. G.

F. G. Werner and D. R. Brill, Phys. Rev. Letters 4, 344 (1960).
[Crossref]

Carnegie Inst. Wash. Publ. (1)

C. Barus, Carnegie Inst. Wash. Publ. 149[Part I (1911); Part II (1912); Part III (1914); Publ. 229 (1915)].

J. Opt. Soc. Am. (1)

Phys. Rev. (2)

L. Marton, J. A. Simpson, and J. A. Suddeth, Phys. Rev. 90, 490 (1953).
[Crossref]

Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).
[Crossref]

Phys. Rev. Letters (1)

F. G. Werner and D. R. Brill, Phys. Rev. Letters 4, 344 (1960).
[Crossref]

Proc. Phys. Soc. (London) (1)

W. Ehrenberg and R. E. Siday, Proc. Phys. Soc. (London) B62, 8 (1949).
[Crossref]

Rev. Sci. Instr. (2)

L. Marton, J. A. Simpson, and J. A. Suddeth, Rev. Sci. Instr. 25, 1099 (1954).
[Crossref]

J. Arol Simpson, Rev. Sci. Instr. 25, 1105 (1954).
[Crossref]

Z. Krist. (1)

O. Rang, Z. Krist. 114, 98 (1960).
[Crossref]

Z. Physik (1)

C. Jonsson, Z. Physik 161, 954 (1961).

Other (2)

L. Marton, Electron Physics (Symposium, 1951), Natl. Bur. Standards Circular No. 527; Phys. Rev. 85, 1057 (1952).
[Crossref]

Z. Bay and H. Mendlowitz (unpublished), quoted in reference 10.

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of three-grating interferometer. Beam travels from left to right.

Fig. 2
Fig. 2

Definition of θ and Θ.

Fig. 3
Fig. 3

Definition of angles between K and coordinate axis.

Fig. 4
Fig. 4

Arrangement of gratings and beam designation in three-grating interferometer.

Fig. 5
Fig. 5

Illustration of effect of unequal grating separation. Shading indicates region of overlapping beams.

Equations (41)

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d · ( K 0 K i ) = 2 π n , n = 0 , 1 , 2 , ,
| K i | = | K 0 | = k = 2 π / λ ,
d · ( K i × K 0 ) = 0 ,
d ( sin θ 0 + sin Θ i ) = n λ ,
e = 1 η [ K i K i · d d 2 d ] ,
η = [ K i K i · d d 2 d ] .
K 0 = ( K 0 · d ) d 2 d + ( K 0 · e ) e .
K 0 2 = ( K 0 · d ) 2 d 2 + ( K 0 · e ) 2 = K i 2 ,
K 0 · d = 2 π n + K i · d ,
K 0 = ( 2 π n + K i · d ) d 2 d + [ K i 2 1 d 2 ( 2 π n + K i · d ) 2 ] 1 2 e ,
K 0 = d { 2 π n + K i · d d 2 1 η [ K i 2 1 d 2 ( 2 π n + K i · d ) 2 ] 1 2 K i · d d 2 } + K i { 1 η [ K i 2 1 d 2 ( 2 π n + K i · d ) 2 ] 1 2 } .
sin =
r = r + r × ε ,
d · ( K 0 K i ) = 2 π n ,
d · ( K i × K 0 ) = 0 ,
d = d + d × ε .
e = e [ 1 + ( K i · d ) ( K i · d × ε ) η 2 d 2 ] 1 η [ K i · d × ε d 2 d + ( K i · d ) d 2 e × ε ] ,
K 0 = K 0 + d ( K i · d × ε ) d 2 + d × ε ( 2 π n + K i · d ) d 2 e b ( ε ) + e ( K i · d ) ( K i · d × ε ) η 2 d 2 a η d × ε ( K i · d ) d 2 + d ( K i · d × ε ) d 2 ,
a = k [ 1 1 k 2 d 2 ( 2 π n + K i · d ) 2 ] 1 2 k [ 1 1 2 k 2 d 2 ( 2 π n + K i · d ) 2 ]
b ( ε ) = 1 k d 2 ( 2 π n + K i · d ) ( K i · d × ε ) ,
K i = k { x ˆ θ i δ i + y ˆ θ i + z ˆ ( 1 1 2 θ i 2 ) } ,
K i · d = k d θ i .
d × ε = z ˆ d
K i · d × ε = k d .
K 0 K 0 0.
K 0 K 0 = x ˆ 2 π n / d ,
K 0 = k { x ˆ θ i δ i + y ˆ ( θ i + ψ ) + z ˆ [ 1 ( θ i + ψ ) 2 ] }
ψ = n λ / d ,
K 0 = k { x ˆ ( θ i δ i + ψ ) + y ˆ ( θ i + ψ ) + z ˆ [ 1 1 2 ( θ i + ψ ) 2 ] } .
( K 2 I ) = k { x ˆ ( θ i δ i + ψ I ) + y ˆ ( θ i + ψ ) + z ˆ [ 1 1 2 ( θ i + ψ ) 2 ] } ,
K 2 II = k { x ˆ ( θ i δ i + ψ I ) + y ˆ ( θ i ) + z ˆ [ 1 1 2 ( θ i ) 2 ] } .
K 1 II = k { x ˆ θ i δ i + y ˆ ( θ i + ψ ) + z ˆ [ 1 1 2 ( θ i + ψ ) 2 ] } .
( K 1 III ) = k { x ˆ ( θ i δ i + ψ III ) + y ˆ θ i + z ˆ [ 1 1 2 θ i 2 ] } .
K 2 II = K 2 III .
Φ = ( K 1 III K 2 III ) · r ,
Φ = k ψ ( III I ) x ,
x s = d / ( III I ) .
Δ Φ = ( K 1 I · r 1 I + K 1 II · r 1 II ) ( K 2 I · r 2 I + K 2 II · r 2 II ) ,
Δ Φ III Δ = k Δ l [ θ i ψ + 1 2 ψ 2 ] ,
2 π k Δ l ( 1 2 ψ 2 ) ,
4 π / k ψ 2 = 2 d 2 / λ Δ l ,