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 Z. Bay and H. Mendlowitz (unpublished), quoted in reference 10.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

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

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

Z. Bay and H. Mendlowitz (unpublished), quoted in reference 10.

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

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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 ,