Abstract

A description is given of the properties of plane diffraction gratings used in conical diffraction. Formulas are given for computing the direction of the diffracted orders. Experiments were performed to investigate the behavior of gratings used on conical diffraction mountings. Comparisons made with classical diffraction mountings show a significant increase in the efficiency of the −1 order. An empirical formula to predict the efficiencies of gratings used in conical diffraction mountings has been verified by the measurements.

© 1978 Optical Society of America

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References

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  1. G. H. Spencer and M. V. R. K. Murty, J. Opt. Soc. Am. 52672 (1962).
    [CrossRef]
  2. W. Werner (thesis, 1970).
  3. W. Werner, Appl. Opt. 16, 2078 (1977).
    [CrossRef] [PubMed]
  4. D. Maystre and R. Petit, Opt. Commun. 4, 97 (1971).
    [CrossRef]
  5. M. Neviere and M. Cadilhac, Opt. Commun. 4, 13 (1971).
    [CrossRef]
  6. D. Maystre and R. Petit, Opt. Commun. 5, 35 (1972).
    [CrossRef]
  7. R. Petit and D. Maystre, Rev. Phys. Appl. 7, 427 (1972).
    [CrossRef]
  8. D. Maystre, thesis, Marseille, 1974, No. CNRS: AO, 9545.
  9. W. R. Hunter, in Proceedings of the IVth International Conference on Vacuum Ultraviolet Radiation Physics, edited by E. E. Koch, R. Haensel, and C. Kunz (Pergaman-Vieweg, Hamburg, 1974).
  10. D. Maystre, Opt. Commun. 8, 216 (1973).
    [CrossRef]
  11. M. Neviere, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 65 (1974).
    [CrossRef]
  12. D. Maystre and R. Petit, J. Spectroscopical Soc. Jpn. 23, 61 (1974).
  13. P. Vincent, M. Neviere, and D. Maystre, Nucl. Instrum. Methods (to be published).
  14. D. Maystre and R. Petit, Nouv. Rev. Opt. 7, 165 (1976).
    [CrossRef]
  15. D. J. Michels, T. L. Mikes, and W. R. Hunter, Appl. Opt. 13, 1223 (1974).
    [CrossRef] [PubMed]
  16. W. R. Hunter, in Proceedings of the Xth Colloquium Spectroscopicum Internationale, edited by E. R. Lippincott and M. Margoshes (Spartan Books, Washington, D. C., 1963), p. 247.
  17. M. Detaille, Laboratoire d'Astronomie Spatiale, Traverse du Siphon, 13012 Marseille, France (private communication).

1977 (1)

1976 (1)

D. Maystre and R. Petit, Nouv. Rev. Opt. 7, 165 (1976).
[CrossRef]

1974 (3)

D. J. Michels, T. L. Mikes, and W. R. Hunter, Appl. Opt. 13, 1223 (1974).
[CrossRef] [PubMed]

M. Neviere, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 65 (1974).
[CrossRef]

D. Maystre and R. Petit, J. Spectroscopical Soc. Jpn. 23, 61 (1974).

1973 (1)

D. Maystre, Opt. Commun. 8, 216 (1973).
[CrossRef]

1972 (2)

D. Maystre and R. Petit, Opt. Commun. 5, 35 (1972).
[CrossRef]

R. Petit and D. Maystre, Rev. Phys. Appl. 7, 427 (1972).
[CrossRef]

1971 (2)

D. Maystre and R. Petit, Opt. Commun. 4, 97 (1971).
[CrossRef]

M. Neviere and M. Cadilhac, Opt. Commun. 4, 13 (1971).
[CrossRef]

1962 (1)

Cadilhac, M.

M. Neviere and M. Cadilhac, Opt. Commun. 4, 13 (1971).
[CrossRef]

Detaille, M.

M. Detaille, Laboratoire d'Astronomie Spatiale, Traverse du Siphon, 13012 Marseille, France (private communication).

Hunter, W. R.

D. J. Michels, T. L. Mikes, and W. R. Hunter, Appl. Opt. 13, 1223 (1974).
[CrossRef] [PubMed]

W. R. Hunter, in Proceedings of the Xth Colloquium Spectroscopicum Internationale, edited by E. R. Lippincott and M. Margoshes (Spartan Books, Washington, D. C., 1963), p. 247.

W. R. Hunter, in Proceedings of the IVth International Conference on Vacuum Ultraviolet Radiation Physics, edited by E. E. Koch, R. Haensel, and C. Kunz (Pergaman-Vieweg, Hamburg, 1974).

Maystre, D.

D. Maystre and R. Petit, Nouv. Rev. Opt. 7, 165 (1976).
[CrossRef]

D. Maystre and R. Petit, J. Spectroscopical Soc. Jpn. 23, 61 (1974).

D. Maystre, Opt. Commun. 8, 216 (1973).
[CrossRef]

D. Maystre and R. Petit, Opt. Commun. 5, 35 (1972).
[CrossRef]

R. Petit and D. Maystre, Rev. Phys. Appl. 7, 427 (1972).
[CrossRef]

D. Maystre and R. Petit, Opt. Commun. 4, 97 (1971).
[CrossRef]

D. Maystre, thesis, Marseille, 1974, No. CNRS: AO, 9545.

P. Vincent, M. Neviere, and D. Maystre, Nucl. Instrum. Methods (to be published).

Michels, D. J.

Mikes, T. L.

Murty, M. V. R. K.

Neviere, M.

M. Neviere, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 65 (1974).
[CrossRef]

M. Neviere and M. Cadilhac, Opt. Commun. 4, 13 (1971).
[CrossRef]

P. Vincent, M. Neviere, and D. Maystre, Nucl. Instrum. Methods (to be published).

Petit, R.

D. Maystre and R. Petit, Nouv. Rev. Opt. 7, 165 (1976).
[CrossRef]

M. Neviere, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 65 (1974).
[CrossRef]

D. Maystre and R. Petit, J. Spectroscopical Soc. Jpn. 23, 61 (1974).

R. Petit and D. Maystre, Rev. Phys. Appl. 7, 427 (1972).
[CrossRef]

D. Maystre and R. Petit, Opt. Commun. 5, 35 (1972).
[CrossRef]

D. Maystre and R. Petit, Opt. Commun. 4, 97 (1971).
[CrossRef]

Spencer, G. H.

Vincent, P.

M. Neviere, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 65 (1974).
[CrossRef]

P. Vincent, M. Neviere, and D. Maystre, Nucl. Instrum. Methods (to be published).

Werner, W.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Spectroscopical Soc. Jpn. (1)

D. Maystre and R. Petit, J. Spectroscopical Soc. Jpn. 23, 61 (1974).

Nouv. Rev. Opt. (2)

M. Neviere, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 65 (1974).
[CrossRef]

D. Maystre and R. Petit, Nouv. Rev. Opt. 7, 165 (1976).
[CrossRef]

Opt. Commun. (4)

D. Maystre, Opt. Commun. 8, 216 (1973).
[CrossRef]

D. Maystre and R. Petit, Opt. Commun. 4, 97 (1971).
[CrossRef]

M. Neviere and M. Cadilhac, Opt. Commun. 4, 13 (1971).
[CrossRef]

D. Maystre and R. Petit, Opt. Commun. 5, 35 (1972).
[CrossRef]

Rev. Phys. Appl. (1)

R. Petit and D. Maystre, Rev. Phys. Appl. 7, 427 (1972).
[CrossRef]

Other (6)

D. Maystre, thesis, Marseille, 1974, No. CNRS: AO, 9545.

W. R. Hunter, in Proceedings of the IVth International Conference on Vacuum Ultraviolet Radiation Physics, edited by E. E. Koch, R. Haensel, and C. Kunz (Pergaman-Vieweg, Hamburg, 1974).

P. Vincent, M. Neviere, and D. Maystre, Nucl. Instrum. Methods (to be published).

W. Werner (thesis, 1970).

W. R. Hunter, in Proceedings of the Xth Colloquium Spectroscopicum Internationale, edited by E. R. Lippincott and M. Margoshes (Spartan Books, Washington, D. C., 1963), p. 247.

M. Detaille, Laboratoire d'Astronomie Spatiale, Traverse du Siphon, 13012 Marseille, France (private communication).

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Figures (6)

FIG. 1
FIG. 1

Illustration of the directions of the spectral orders for conical diffraction. The wave vector of the incident wave is k and the wave vector of the zero order is k0.

FIG. 2
FIG. 2

Grating configuration with k parallel to the plane of the small facet (GMS mounting).

FIG. 3
FIG. 3

RWF mounting in classical diffraction.

FIG. 4
FIG. 4

Measured efficiency curves for a 600 g/mm, 2°4′ blazed gold grating. Curve 1, GMS mounting; curve 2, classical RWF mounting; curve 3, reflectance of gold for P polarization; curve 4, reflectance of gold for S polarization.

FIG. 5
FIG. 5

Same as Fig. 4, except that the grating has 1800 g/mm, is blazed at 9°19′, and is coated with gold.

FIG. 6
FIG. 6

Evolution of the efficiency curve when, starting from GMS mounting, one rotates the 1800 g/mm, 9°19′ blazed gold grating around Δ. Curve 1, GMS mount; curve 2, 30° rotation from GMS mount; curve 3, 60° rotation from GMS mount; curve 4, 90° rotation (classical RWF mounting).

Tables (4)

Tables Icon

TABLE I Comparison of a geometric factor τ, obtained from Eq. (13), to the ratio ρ of measured efficiencies in the classical RWF mount to the GMS mount for a gold grating with blaze angle 2°4′. [Note that ϕ (or θ″) is given by λ = 2d sinδ cosϕ.]

Tables Icon

TABLE II Comparison of a geometric factor τ, obtained from Eq. (13), to the ratio ρ of measured efficiencies in the classical RWF mount to the GMS mount for a gold grating with blaze angle 9°19′. [Note that ϕ (or θ″) is given by λ = 2d sinδ cosϕ.]

Tables Icon

TABLE III Comparison of a geometric factor τ, obtained from Eq. (13), to the ratio of measured efficiencies in the classical mount to a conical diffraction mount wherein the grating has been rotated through an angle ψ = 30° from the GMS mount. Note that τ = min [cos θ′/cos θ, cos θ/cos θ′], and can be calculated from values obtained from Eq. (14).

Tables Icon

TABLE IV Comparison of a geometric factor τ, obtained from Eq. (13), to the ratio of measured efficiencies in the classical mount to a conical diffraction mount wherein the grating has been rotated through an angle ψ = 60° from the GMS mount. Note that τ = min [cos θ′/cos θ, cos θ/cos θ′], and can be calculated from values obtained from Eq. (14).

Equations (31)

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Δ E d + k 2 E d = 0 ,             if y > g ( x )
n ( E i + E d ) = 0 ,             if y = g ( x )
lim y [ E d ] = n H n ( x , y , z ) ,
E i ( x + d , g ( x ) , z ) = E i ( x , g ( x ) , z ) exp ( i k α d ) .
n [ E i + v ] = n [ E i ( x , g ( x ) , z ) + E d ( x + d , g ( x ) , z ) exp ( - i k α d ) ] z , = n [ E i ( x , g ) x ) , z ) exp ( i k α d ) + E d ( x + d , g ( x ) , z ) ] exp ( - i k α d ) = n [ E i ( x + d , g ( x ) , z ) + E d ( x + d , g ( x ) , z ) ] exp ( - i k α d ) .
n [ E i + v ] = n [ E i ( x , g ( x ) , z ) + E d ( x , g ( x ) , z ) ] exp ( - i k α d ) ,
E d ( x + d , y , z ) exp ( - i k α d ) = E d ( x , y , z ) ,
E d ( x , y , z ) exp ( - i k γ z ) = f ( x , y ) ,
E d ( x , y , z ) = f ( x , y ) exp ( i k γ z ) .
f ( x , y ) exp ( - i k α x ) = n = - + f n ( y ) exp ( i n K x ) ,
E d ( x , y , z ) = f ( x , y ) exp ( i k γ z ) = exp ( i k γ z ) n = + f n ( y ) exp [ i ( n K + k α ) x ] .
d 2 f n / d y 2 + [ k 2 - k 2 γ 2 - ( n K + k α ) 2 ] f n = 0.
χ n = [ k 2 - k 2 γ 2 - ( n K + k α ) 2 ] 1 / 2
χ n = i { - [ k 2 - k 2 γ 2 - ( n k + k α ) 2 ] } 1 / 2 ,
f n ( y ) = B n exp ( i χ n y ) ,
E d ( x , y , z ) = n = - + B n exp [ i ( n K + k α ) x ] × exp ( i χ n y ) exp ( i k γ z ) .
u n x = n λ / d + α , u n y = [ 1 - γ 2 - ( n λ / d + α ) 2 ] 1 / 2 , u n z = γ .
E 1 = e 1 A exp ( i k u 1 · r ) ,             E 2 = - e 1 A exp ( i k u 2 · r ) ,
E = e 1 A exp [ i k ( - Y cos ϕ - Z sin ϕ ) ] - e 1 A [ exp i k ( Y cos ϕ - Z sin ϕ ) ]
E = - e 1 A 2 i sin [ ( 2 π / λ ) m d sin δ cos ϕ ] exp ( - i k z sin ϕ ) ,
θ + θ = 2 θ ,
δ = θ - θ ,
λ = 2 d sin δ cos θ .
u x = cos ξ sin θ , u y = - cos ξ cos θ , u z = sin ξ ,
v x = - sin δ ,             v y = cos δ ,             v z = 0.
- u · v = cos ξ sin θ sin δ + cos ξ cos θ cos δ = cos θ .
u - 1 , x = cos ξ sin θ - λ / d , u - 1 , y = [ cos 2 ξ - ( cos ξ sin θ - λ / d ) 2 ] 1 / 2 , u - 1 , z = sin ξ .
u - 1 , y = - ( cos ξ cos θ - 2 cos θ cos δ ) .
- 1 = R ( θ ) min [ cos θ cos θ , cos θ cos θ ] ,
τ = min [ cos θ cos θ , cos θ cos θ ]
cos θ = cos θ cos δ - sin θ sin δ sin ψ , cos θ = cos θ cos δ + sin θ sin δ sin ψ .