Abstract

By use of simple theoretical considerations on the efficiency dependence of a wide class of commercially available gratings, we obtained an empiric equivalence rule. This rule makes it possible to create equivalence classes of gratings that among themselves show a variety of profiles but have, as long as only two orders are propagating, efficiency curves lying very close to each other. The equivalence rule is visualized by drawing a nomogram. The rule and the grating nomogram serve as a basis for finding new and explaining old properties of ruled, holographic, and lamellar gratings.

© 1980 Optical Society of America

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References

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  1. D. Maystre, R. Petit, M. Duban, J. Gilewicz, Nouv. Rev. Opt. 5, 79 (1974).
    [CrossRef]
  2. J. Chandezon, presented at Symposium Optique Hertzienne et Diélectriques, Marseille, 8 Sept. 1977.
  3. N. K. Sheridon, Appl. Phys. Lett. 12, 316 (1968).
    [CrossRef]
  4. M. C. Hutley, Opt. Acta 22, 1 (1975).
    [CrossRef]
  5. S. Johansson, L. E. Nilsson, K. Biedermann, K. Kleveby, in Applications of Holography and Optical Data Processing, E. Marom, A. A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977), pp. 521–530.
  6. E. G. Loewen, M. Neviere, D. Maystre, Appl. Opt. 15, 2937 (1976).
    [CrossRef] [PubMed]
  7. E. G. Loewen, M. Neviere, D. Maystre, Appl. Opt. 16, 2711 (1977).
    [CrossRef] [PubMed]
  8. I. J. Wilson, L. C. Botten, R. C. McPhedran, J. Opt. 8, 217 (1977).
    [CrossRef]
  9. D. Maystre, J. Opt. Soc. Am. 68, 490 (1978).
    [CrossRef]
  10. A. Roger, D. Maystre, Opt. Acta 26, 447 (1979).
    [CrossRef]
  11. R. Petit, Opt. Acta 14, 301 (1967).
    [CrossRef]
  12. D. Maystre, R. C. McPhedran, Opt. Commun. 12, 164 (1974).
    [CrossRef]
  13. R. C. McPhedran, D. Maystre, Opt. Acta 21, 413 (1974).
    [CrossRef]
  14. A. Marechal, G. W. Stroke, C. R. Acad. Sci. 249, 2042 (1959).
  15. D. Maystre, R. Petit, Nouv. Rev. Opt. 2, 115 (1971).
    [CrossRef]
  16. R. C. McPhedran, Ph.D. Thesis, U. Tasmania Hobart, Australia (1973).
  17. A. Wirgin, Thèse d'Etat, No. AO 1429, Paris (1967).
  18. D. Maystre, R. Petit, Opt. Commun. 5, 90 (1972).
    [CrossRef]
  19. E. G. Loewen, M. Neviere, D. Maystre, Appl. Opt. 18, 2262 (1979).
    [CrossRef] [PubMed]

1979 (2)

1978 (1)

1977 (2)

I. J. Wilson, L. C. Botten, R. C. McPhedran, J. Opt. 8, 217 (1977).
[CrossRef]

E. G. Loewen, M. Neviere, D. Maystre, Appl. Opt. 16, 2711 (1977).
[CrossRef] [PubMed]

1976 (1)

1975 (1)

M. C. Hutley, Opt. Acta 22, 1 (1975).
[CrossRef]

1974 (3)

D. Maystre, R. C. McPhedran, Opt. Commun. 12, 164 (1974).
[CrossRef]

R. C. McPhedran, D. Maystre, Opt. Acta 21, 413 (1974).
[CrossRef]

D. Maystre, R. Petit, M. Duban, J. Gilewicz, Nouv. Rev. Opt. 5, 79 (1974).
[CrossRef]

1972 (1)

D. Maystre, R. Petit, Opt. Commun. 5, 90 (1972).
[CrossRef]

1971 (1)

D. Maystre, R. Petit, Nouv. Rev. Opt. 2, 115 (1971).
[CrossRef]

1968 (1)

N. K. Sheridon, Appl. Phys. Lett. 12, 316 (1968).
[CrossRef]

1967 (1)

R. Petit, Opt. Acta 14, 301 (1967).
[CrossRef]

1959 (1)

A. Marechal, G. W. Stroke, C. R. Acad. Sci. 249, 2042 (1959).

Biedermann, K.

S. Johansson, L. E. Nilsson, K. Biedermann, K. Kleveby, in Applications of Holography and Optical Data Processing, E. Marom, A. A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977), pp. 521–530.

Botten, L. C.

I. J. Wilson, L. C. Botten, R. C. McPhedran, J. Opt. 8, 217 (1977).
[CrossRef]

Chandezon, J.

J. Chandezon, presented at Symposium Optique Hertzienne et Diélectriques, Marseille, 8 Sept. 1977.

Duban, M.

D. Maystre, R. Petit, M. Duban, J. Gilewicz, Nouv. Rev. Opt. 5, 79 (1974).
[CrossRef]

Gilewicz, J.

D. Maystre, R. Petit, M. Duban, J. Gilewicz, Nouv. Rev. Opt. 5, 79 (1974).
[CrossRef]

Hutley, M. C.

M. C. Hutley, Opt. Acta 22, 1 (1975).
[CrossRef]

Johansson, S.

S. Johansson, L. E. Nilsson, K. Biedermann, K. Kleveby, in Applications of Holography and Optical Data Processing, E. Marom, A. A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977), pp. 521–530.

Kleveby, K.

S. Johansson, L. E. Nilsson, K. Biedermann, K. Kleveby, in Applications of Holography and Optical Data Processing, E. Marom, A. A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977), pp. 521–530.

Loewen, E. G.

Marechal, A.

A. Marechal, G. W. Stroke, C. R. Acad. Sci. 249, 2042 (1959).

Maystre, D.

A. Roger, D. Maystre, Opt. Acta 26, 447 (1979).
[CrossRef]

E. G. Loewen, M. Neviere, D. Maystre, Appl. Opt. 18, 2262 (1979).
[CrossRef] [PubMed]

D. Maystre, J. Opt. Soc. Am. 68, 490 (1978).
[CrossRef]

E. G. Loewen, M. Neviere, D. Maystre, Appl. Opt. 16, 2711 (1977).
[CrossRef] [PubMed]

E. G. Loewen, M. Neviere, D. Maystre, Appl. Opt. 15, 2937 (1976).
[CrossRef] [PubMed]

R. C. McPhedran, D. Maystre, Opt. Acta 21, 413 (1974).
[CrossRef]

D. Maystre, R. C. McPhedran, Opt. Commun. 12, 164 (1974).
[CrossRef]

D. Maystre, R. Petit, M. Duban, J. Gilewicz, Nouv. Rev. Opt. 5, 79 (1974).
[CrossRef]

D. Maystre, R. Petit, Opt. Commun. 5, 90 (1972).
[CrossRef]

D. Maystre, R. Petit, Nouv. Rev. Opt. 2, 115 (1971).
[CrossRef]

McPhedran, R. C.

I. J. Wilson, L. C. Botten, R. C. McPhedran, J. Opt. 8, 217 (1977).
[CrossRef]

D. Maystre, R. C. McPhedran, Opt. Commun. 12, 164 (1974).
[CrossRef]

R. C. McPhedran, D. Maystre, Opt. Acta 21, 413 (1974).
[CrossRef]

R. C. McPhedran, Ph.D. Thesis, U. Tasmania Hobart, Australia (1973).

Neviere, M.

Nilsson, L. E.

S. Johansson, L. E. Nilsson, K. Biedermann, K. Kleveby, in Applications of Holography and Optical Data Processing, E. Marom, A. A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977), pp. 521–530.

Petit, R.

D. Maystre, R. Petit, M. Duban, J. Gilewicz, Nouv. Rev. Opt. 5, 79 (1974).
[CrossRef]

D. Maystre, R. Petit, Opt. Commun. 5, 90 (1972).
[CrossRef]

D. Maystre, R. Petit, Nouv. Rev. Opt. 2, 115 (1971).
[CrossRef]

R. Petit, Opt. Acta 14, 301 (1967).
[CrossRef]

Roger, A.

A. Roger, D. Maystre, Opt. Acta 26, 447 (1979).
[CrossRef]

Sheridon, N. K.

N. K. Sheridon, Appl. Phys. Lett. 12, 316 (1968).
[CrossRef]

Stroke, G. W.

A. Marechal, G. W. Stroke, C. R. Acad. Sci. 249, 2042 (1959).

Wilson, I. J.

I. J. Wilson, L. C. Botten, R. C. McPhedran, J. Opt. 8, 217 (1977).
[CrossRef]

Wirgin, A.

A. Wirgin, Thèse d'Etat, No. AO 1429, Paris (1967).

Appl. Opt. (3)

Appl. Phys. Lett. (1)

N. K. Sheridon, Appl. Phys. Lett. 12, 316 (1968).
[CrossRef]

C. R. Acad. Sci. (1)

A. Marechal, G. W. Stroke, C. R. Acad. Sci. 249, 2042 (1959).

J. Opt. (1)

I. J. Wilson, L. C. Botten, R. C. McPhedran, J. Opt. 8, 217 (1977).
[CrossRef]

J. Opt. Soc. Am. (1)

Nouv. Rev. Opt. (2)

D. Maystre, R. Petit, M. Duban, J. Gilewicz, Nouv. Rev. Opt. 5, 79 (1974).
[CrossRef]

D. Maystre, R. Petit, Nouv. Rev. Opt. 2, 115 (1971).
[CrossRef]

Opt. Acta (4)

A. Roger, D. Maystre, Opt. Acta 26, 447 (1979).
[CrossRef]

R. Petit, Opt. Acta 14, 301 (1967).
[CrossRef]

R. C. McPhedran, D. Maystre, Opt. Acta 21, 413 (1974).
[CrossRef]

M. C. Hutley, Opt. Acta 22, 1 (1975).
[CrossRef]

Opt. Commun. (2)

D. Maystre, R. Petit, Opt. Commun. 5, 90 (1972).
[CrossRef]

D. Maystre, R. C. McPhedran, Opt. Commun. 12, 164 (1974).
[CrossRef]

Other (4)

R. C. McPhedran, Ph.D. Thesis, U. Tasmania Hobart, Australia (1973).

A. Wirgin, Thèse d'Etat, No. AO 1429, Paris (1967).

S. Johansson, L. E. Nilsson, K. Biedermann, K. Kleveby, in Applications of Holography and Optical Data Processing, E. Marom, A. A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977), pp. 521–530.

J. Chandezon, presented at Symposium Optique Hertzienne et Diélectriques, Marseille, 8 Sept. 1977.

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Figures (11)

Fig. 1
Fig. 1

Profile of three types of commercial grating: (a) ruled grating with triangular profile; (b) holographic grating with sinusoidal profile; (c) symmetrical lamellar grating.

Fig. 2
Fig. 2

Grating nomogram for commercial gratings. All gratings located on a vertical line in the nomogram have the same fundamental; they thus have almost the same efficiency curve as long as only two orders are propagating. Each curve corresponds to a certain value A of the apex angle of a ruled grating. The characteristic point of a ruled grating (A,α) is the intersection of curve A with the horizontal line having ordinate α. The characteristic points of the sinusoidal and lamellar gratings are located on the two abscissas. The blaze angles and blaze wavelengths of a grating are the ordinates of intersection of the vertical line passing through the characteristic point with the two heavy curves labeled TM Blaze (full line A = 90°) and TE Blaze (dashed line). The first ordinate axis (α) gives the blaze angles in the Littrow mounting. The second ordinate axis (2 · sinα) gives the ratio of blaze wavelength to groove spacing in small-deviation mountings.

Fig. 3
Fig. 3

Efficiency of 1800-groove/mm infinitely conducting gratings located on a vertical line in the nomogram. We have represented the maximum of 100% predicted by the grating nomogram by arrows.

Fig. 4
Fig. 4

The same as Fig. 3 except: —Sinusoidal gratingH/d = 0.23- - -Lamellar gratingH/d = 0.1806·····Ruled grating  A = 90°, α = 69.5°- · -Ruled grating  A = 74°, α = 86°●Ruled grating  A = 120°, α = 28°.

Fig. 5
Fig. 5

The same as Fig. 3 except: —Sinusoidal gratingH/d = 0.4- - -Lamellar gratingH/d = 0.3142.·····Ruled grating  A = 90°, α = 41°- · -Ruled grating  A = 60°, α = 32°.

Fig. 6
Fig. 6

The same as Fig. 3 except: —Sinusoidal gratingH/d = 0.5- - -Lamellar gratingH/d = 0.3927- · -Ruled grating  A = 74°, α = 43°.

Fig. 7
Fig. 7

Same as Fig. 4 but for an 1800-groove/mm aluminum grating.

Fig. 8
Fig. 8

Positions of the 100% maximum in TM polarization on the efficiency curves in Littrow mountings for sinusoidal gratings. The full line is taken from the nomogram labeled TM Blaze.

Fig. 9
Fig. 9

Assuming that the two shaded beams leaving the grating have approximately the same amplitude and are phase shifted by about 180°, one finds that (a) if θ is small, (tanθ)/2 ≈ H/d; (b) if θ is large, (cotθ)/2 ≈ H/d. One may expect that the efficiency in the zero order is small, which gives a very good efficiency in the −1 order.

Fig. 10
Fig. 10

Use of grating nomogram to design a grating: A = 90°,α = 20.5°·····A = 120°,α = 20.5°- - -A = 120°,α = 28°.

Fig. 11
Fig. 11

Same as Fig. 10 except: A = 90°,α = 41°·····A = 60°,α = 41°- - -A = 60°,α = 32°.

Equations (22)

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f ( x ) = n = 1 b n sin nKx ,
b n = d ( n π ) 2 [ tan α tan ( A + α ) ] sin [ π n sin ( A + α ) cos α sin A ] ,
b 1 = H / 2 , b n = 0 , n 1 ,
b 1 = 2 H n π n odd , b n = 0 n even .
y = f ( x ) = b 1 sin Kx + b 2 sin 2 Kx .
y = f [ x + ( d / 2 ) ] = b 1 sin Kx + b 2 sin 2 Kx ;
E i ( b 1 , b 2 ) = E i ( b 1 , b 2 ) i .
y = f ( x ) = b 1 sin Kx b 2 sin 2 Kx
E 0 ( b 1 , b 2 ) = E 0 ( b 1 , b 2 ) .
E 0 ( b 1 , b 2 ) + E 1 ( b 1 , b 2 ) = 1 .
E i ( b 1 , b 2 ) = E i ( b 1 , b 2 ) = E i ( b 1 , b 2 ) = E i ( b 1 , b 2 ) .
E 1 ( 0 , b 2 ) 0 .
E 1 ( b 1 , b 2 ) = α 0 + α 1 b 1 2 + α 2 b 2 2 + α 3 b 1 4 + α 4 b 2 4 + α 5 b 1 2 b 2 2 + . . . .
E 1 ( b 1 , b 2 ) = α 1 b 1 2 + α 3 b 1 4 + α 5 b 1 2 b 2 2 .
E 0 ( b 1 , b 2 ) + E 1 ( b 1 , b 2 ) = R ,
H / d = 2 π 2 [ tan α tan ( A + α ) ] sin [ π sin ( A + α ) cos α sin A ] .
λ = 2 d sin θ .
H / d = 1 2 π ( tan θ + cot θ ) sin ( π cos 2 θ ) .
sin ( π cos 2 θ ) = sin [ π ( 1 1 + tan 2 θ ) ] sin [ π ( 1 tan 2 θ ) ] sin ( π tan 2 θ ) ,
H / d ( tan θ ) / 2 .
H / d ( cot θ ) / 2 .
( i ) A = 120 ° , α = 28 ° ; ( ii ) A = 120 ° , α = 32 ° .

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