Abstract

A rigorous electromagnetic theory of the diffraction of light by blazed lamellar gratings has been developed. It is applied to calculate the diffracted power distribution of four gratings (R1R4) whose grooves have the following depths and widths (b,l) in units of grating period a: (0.433, 0.750), (0.333, 0.667), (0.250, 0.500), and (0.200, 0.400). All four gratings are theoretically blazed in the minus-first order for 30° incidence and for the wavelength equal to the grating period. The blaze is, however, broad band, and the minus-first-order theoretical power-conversion efficiency (P−1) for the most interesting grating, R3, exceeds 0.8, 0.9, 0.95, and 0.98 in the wavelength bands 0.84 a−1.49 a, 0.85 a−1.47 a, 0.87 a−1.43 a, and 0.98 a−1.31 a, respectively, for linearly polarized light. These surprising performances, which far exceed those obtainable with echelette gratings insofar as the incident light is linearly polarized, have been experimentally verified with the help of a microwave interference spectrometer, and point to the potential usefulness of blazed lamellar-reflection gratings in far-infrared spectroscopy.

© 1969 Optical Society of America

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References

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  1. F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw–Hill Book Co., New York, 1957).
  2. G. W. Stroke, in Handbuch der Physik Vol. 29, S. Flügge, Ed. (Springer-Verlag, Berlin, 1967).
  3. R. Petit, Compt. Rend. 258, 1429 (1964).
  4. Lord Rayleigh, Phil. Mag. 14, 60 (1907);Proc. Roy. Soc. (London) A79, 399 (1907).
  5. R. W. Wood, Proc. Phys. Soc. (London) 18, 396 (1902);Phil. Mag. 4, 396 (1902).
  6. R. P. Madden and J. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958).
  7. G. W. Stroke, Phys. Letters 5, 45 (1963).
    [Crossref]
  8. R. Petit, Rev. Opt. 45, 249, 353 (1966).
  9. G. W. Stroke, Rev. Opt. 39, 291 (1960).
  10. B. A. Lippmann, J. Opt. Soc. Am. 43, 408 (1953).
    [Crossref]
  11. W. C. Meecham, J. Acoust. Soc. Am. 28, 370 (1955).
    [Crossref]
  12. P. Bousquet, Japan J. Appl. Phys. 4, suppl. 1, 549 (1965).
  13. A. Wirgin, Rev. Opt. 43, 449 (1964);Rev. Opt. 44, 20 (1965).
  14. J. Pavageau, thesis, Ed. de la Rev. Opt., Paris (1966).
  15. R. Deleuil, Opt. Acta 16, 23 (1969).
    [Crossref]
  16. J. M. Proud, P. Tamarkin, and W. C. Meecham, J. Appl. Phys. 28, 1298 (1957).
    [Crossref]
  17. A. Hadni, E. Décamps, D. Grandjean, and C. Janot, Compt. Rend. 250, 2007 (1960).
  18. A. Wirgin, Compt. Rend. B262, 385 (1966).
  19. A. Wirgin, Compt. Rend. B262, 579 (1966).
  20. A. Wirgin, Compt. Rend. B262, 870 (1966).
  21. A. Wirgin, Compt. Rend. B262, 1032 (1966).
  22. A. Wirgin, Rev. Opt. 47, 333 (1968).
  23. A. Wirgin, French patent, BF No. prov. 159952, 19July1968.
  24. J. P. Chauvineau, L. Constanciel, A. Marraud, and R. Petit, Rev. Opt. 46, 417 (1967).
  25. R. Deleuil and F. Varnier, Compt. Rend. B267, 1074 (1968).
  26. C. H. Palmer, J. Opt. Soc. Am. 42, 268 (1952).
    [Crossref]
  27. C. H. Palmer, F. C. Evering, and F. C. Nelson, Appl. Opt. 4, 1271 (1965).
    [Crossref]
  28. R. W. Wood, Phil. Mag. 23, 315 (1912);Phys. Rev. 48, 928 (1935).
  29. A. Wirgin, French patent, BF No. prov. 179261, 19December1968.

1969 (1)

R. Deleuil, Opt. Acta 16, 23 (1969).
[Crossref]

1968 (2)

A. Wirgin, Rev. Opt. 47, 333 (1968).

R. Deleuil and F. Varnier, Compt. Rend. B267, 1074 (1968).

1967 (1)

J. P. Chauvineau, L. Constanciel, A. Marraud, and R. Petit, Rev. Opt. 46, 417 (1967).

1966 (5)

A. Wirgin, Compt. Rend. B262, 385 (1966).

A. Wirgin, Compt. Rend. B262, 579 (1966).

A. Wirgin, Compt. Rend. B262, 870 (1966).

A. Wirgin, Compt. Rend. B262, 1032 (1966).

R. Petit, Rev. Opt. 45, 249, 353 (1966).

1965 (2)

P. Bousquet, Japan J. Appl. Phys. 4, suppl. 1, 549 (1965).

C. H. Palmer, F. C. Evering, and F. C. Nelson, Appl. Opt. 4, 1271 (1965).
[Crossref]

1964 (2)

A. Wirgin, Rev. Opt. 43, 449 (1964);Rev. Opt. 44, 20 (1965).

R. Petit, Compt. Rend. 258, 1429 (1964).

1963 (1)

G. W. Stroke, Phys. Letters 5, 45 (1963).
[Crossref]

1960 (2)

G. W. Stroke, Rev. Opt. 39, 291 (1960).

A. Hadni, E. Décamps, D. Grandjean, and C. Janot, Compt. Rend. 250, 2007 (1960).

1957 (1)

J. M. Proud, P. Tamarkin, and W. C. Meecham, J. Appl. Phys. 28, 1298 (1957).
[Crossref]

1955 (1)

W. C. Meecham, J. Acoust. Soc. Am. 28, 370 (1955).
[Crossref]

1953 (1)

1952 (1)

C. H. Palmer, J. Opt. Soc. Am. 42, 268 (1952).
[Crossref]

1912 (1)

R. W. Wood, Phil. Mag. 23, 315 (1912);Phys. Rev. 48, 928 (1935).

1907 (1)

Lord Rayleigh, Phil. Mag. 14, 60 (1907);Proc. Roy. Soc. (London) A79, 399 (1907).

1902 (1)

R. W. Wood, Proc. Phys. Soc. (London) 18, 396 (1902);Phil. Mag. 4, 396 (1902).

Bousquet, P.

P. Bousquet, Japan J. Appl. Phys. 4, suppl. 1, 549 (1965).

Chauvineau, J. P.

J. P. Chauvineau, L. Constanciel, A. Marraud, and R. Petit, Rev. Opt. 46, 417 (1967).

Constanciel, L.

J. P. Chauvineau, L. Constanciel, A. Marraud, and R. Petit, Rev. Opt. 46, 417 (1967).

Décamps, E.

A. Hadni, E. Décamps, D. Grandjean, and C. Janot, Compt. Rend. 250, 2007 (1960).

Deleuil, R.

R. Deleuil, Opt. Acta 16, 23 (1969).
[Crossref]

R. Deleuil and F. Varnier, Compt. Rend. B267, 1074 (1968).

Evering, F. C.

Grandjean, D.

A. Hadni, E. Décamps, D. Grandjean, and C. Janot, Compt. Rend. 250, 2007 (1960).

Hadni, A.

A. Hadni, E. Décamps, D. Grandjean, and C. Janot, Compt. Rend. 250, 2007 (1960).

Janot, C.

A. Hadni, E. Décamps, D. Grandjean, and C. Janot, Compt. Rend. 250, 2007 (1960).

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw–Hill Book Co., New York, 1957).

Lippmann, B. A.

Madden, R. P.

R. P. Madden and J. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958).

Marraud, A.

J. P. Chauvineau, L. Constanciel, A. Marraud, and R. Petit, Rev. Opt. 46, 417 (1967).

Meecham, W. C.

J. M. Proud, P. Tamarkin, and W. C. Meecham, J. Appl. Phys. 28, 1298 (1957).
[Crossref]

W. C. Meecham, J. Acoust. Soc. Am. 28, 370 (1955).
[Crossref]

Nelson, F. C.

Palmer, C. H.

Pavageau, J.

J. Pavageau, thesis, Ed. de la Rev. Opt., Paris (1966).

Petit, R.

J. P. Chauvineau, L. Constanciel, A. Marraud, and R. Petit, Rev. Opt. 46, 417 (1967).

R. Petit, Rev. Opt. 45, 249, 353 (1966).

R. Petit, Compt. Rend. 258, 1429 (1964).

Proud, J. M.

J. M. Proud, P. Tamarkin, and W. C. Meecham, J. Appl. Phys. 28, 1298 (1957).
[Crossref]

Rayleigh, Lord

Lord Rayleigh, Phil. Mag. 14, 60 (1907);Proc. Roy. Soc. (London) A79, 399 (1907).

Stroke, G. W.

G. W. Stroke, Phys. Letters 5, 45 (1963).
[Crossref]

G. W. Stroke, Rev. Opt. 39, 291 (1960).

G. W. Stroke, in Handbuch der Physik Vol. 29, S. Flügge, Ed. (Springer-Verlag, Berlin, 1967).

Strong, J.

R. P. Madden and J. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958).

Tamarkin, P.

J. M. Proud, P. Tamarkin, and W. C. Meecham, J. Appl. Phys. 28, 1298 (1957).
[Crossref]

Varnier, F.

R. Deleuil and F. Varnier, Compt. Rend. B267, 1074 (1968).

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw–Hill Book Co., New York, 1957).

Wirgin, A.

A. Wirgin, Rev. Opt. 47, 333 (1968).

A. Wirgin, Compt. Rend. B262, 385 (1966).

A. Wirgin, Compt. Rend. B262, 579 (1966).

A. Wirgin, Compt. Rend. B262, 870 (1966).

A. Wirgin, Compt. Rend. B262, 1032 (1966).

A. Wirgin, Rev. Opt. 43, 449 (1964);Rev. Opt. 44, 20 (1965).

A. Wirgin, French patent, BF No. prov. 159952, 19July1968.

A. Wirgin, French patent, BF No. prov. 179261, 19December1968.

Wood, R. W.

R. W. Wood, Phil. Mag. 23, 315 (1912);Phys. Rev. 48, 928 (1935).

R. W. Wood, Proc. Phys. Soc. (London) 18, 396 (1902);Phil. Mag. 4, 396 (1902).

Appl. Opt. (1)

Compt. Rend. (7)

A. Hadni, E. Décamps, D. Grandjean, and C. Janot, Compt. Rend. 250, 2007 (1960).

A. Wirgin, Compt. Rend. B262, 385 (1966).

A. Wirgin, Compt. Rend. B262, 579 (1966).

A. Wirgin, Compt. Rend. B262, 870 (1966).

A. Wirgin, Compt. Rend. B262, 1032 (1966).

R. Deleuil and F. Varnier, Compt. Rend. B267, 1074 (1968).

R. Petit, Compt. Rend. 258, 1429 (1964).

J. Acoust. Soc. Am. (1)

W. C. Meecham, J. Acoust. Soc. Am. 28, 370 (1955).
[Crossref]

J. Appl. Phys. (1)

J. M. Proud, P. Tamarkin, and W. C. Meecham, J. Appl. Phys. 28, 1298 (1957).
[Crossref]

J. Opt. Soc. Am. (2)

B. A. Lippmann, J. Opt. Soc. Am. 43, 408 (1953).
[Crossref]

C. H. Palmer, J. Opt. Soc. Am. 42, 268 (1952).
[Crossref]

Japan J. Appl. Phys. (1)

P. Bousquet, Japan J. Appl. Phys. 4, suppl. 1, 549 (1965).

Opt. Acta (1)

R. Deleuil, Opt. Acta 16, 23 (1969).
[Crossref]

Phil. Mag. (2)

R. W. Wood, Phil. Mag. 23, 315 (1912);Phys. Rev. 48, 928 (1935).

Lord Rayleigh, Phil. Mag. 14, 60 (1907);Proc. Roy. Soc. (London) A79, 399 (1907).

Phys. Letters (1)

G. W. Stroke, Phys. Letters 5, 45 (1963).
[Crossref]

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

R. W. Wood, Proc. Phys. Soc. (London) 18, 396 (1902);Phil. Mag. 4, 396 (1902).

Rev. Opt. (5)

A. Wirgin, Rev. Opt. 43, 449 (1964);Rev. Opt. 44, 20 (1965).

A. Wirgin, Rev. Opt. 47, 333 (1968).

R. Petit, Rev. Opt. 45, 249, 353 (1966).

G. W. Stroke, Rev. Opt. 39, 291 (1960).

J. P. Chauvineau, L. Constanciel, A. Marraud, and R. Petit, Rev. Opt. 46, 417 (1967).

Other (6)

A. Wirgin, French patent, BF No. prov. 179261, 19December1968.

A. Wirgin, French patent, BF No. prov. 159952, 19July1968.

J. Pavageau, thesis, Ed. de la Rev. Opt., Paris (1966).

R. P. Madden and J. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958).

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw–Hill Book Co., New York, 1957).

G. W. Stroke, in Handbuch der Physik Vol. 29, S. Flügge, Ed. (Springer-Verlag, Berlin, 1967).

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Figures (10)

F. 1
F. 1

Echelette grating.

F. 2
F. 2

Lamellar grating.

F. 3
F. 3

Grating R1: b = 0.433a, l = 0.750a, θi = 30°. Theoretical diffraction efficiency (Pn) vs a/λ in the orders n = 0, −2 (top) and blazed order n = −1 (bottom) for the two polarization states E and E.

F. 4
F. 4

Grating R2: b = 0.333a, l = 0.667a, θi = 30°. Theoretical P0 (top) and P−1 (bottom) vs a/λ.

F. 5
F. 5

Grating R3: b = 0.250a, l = 0.500a, θi = 30°. Theoretical P0, P−2 (top) and P−1 (bottom) vs a/λ.

F. 6
F. 6

Grating R4: b = 0.200a, l = 0.400a, θi = 30°. Theoretical P0, P−2 (top) and P−1 (bottom) vs a/λ.

F. 7
F. 7

CA = calibrated attenuator, D = detector, DC = directional coupler, FI = ferrite isolator, G = galvanometer, K = klystron, PC = phase changer, PS = power supply, RC = reference cavity, SC = stabilizing circuit, VA = variable attenuator, μV = microvoltmeter.

F. 8
F. 8

Grating R2. P0 in E polarization (top) and E polarization (bottom). Dashed curve is theoretical and points are experimental.

F. 9
F. 9

Grating R3. P0 in E (top) and E (bottom) polarizations.

F. 10
F. 10

Grating R4. P0 in E (top) and E (bottom) polarizations.

Equations (36)

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sin θ n = sin θ i + n λ / a ; { | θ i | < π / 2 | sin θ n | 1 π / 2 θ n 3 π / 2 .
λ m = ( 2 a / m ) sin β cos ( β θ i ) .
λ m = ( 2 a / m ) sin θ i .
( Δ + k 2 ) U ( P ) = 0 , P S ,
1 2 ( 1 β ) U ( P ) + 1 2 ( 1 β ) n U ( P ) = 0 , P C ,
( r i k ) [ U ( P ) U i ( P ) A U i ( P * ) ] 0 ( r 1 2 ) ; r = ( x 2 + z 2 ) 1 2 , P S .
U i ( P ) = exp i k ( s i x ω i z ) ; { s i = sin θ i ω i = cos θ i | θ i | < π / 2 .
S ( P ) = S + ( P ) + n S n ( P ) ( n = n = ) ;
x n = x n a + l / 2 ,
U ( P ) = U i ( P ) + β U i ( P * ) + 1 / 2 1 / 2 [ 1 2 ( 1 β ) U ( P 0 ) × z F β + ( P ; P 0 ) 1 2 ( 1 + β ) F β + ( P ; P 0 ) z U ( P 0 ) ] d x ; P S + ,
U ( P n a ) = U ( P ) exp i k s i n a ,
P = P ( x , z ) , P * = P * = ( x , z ) , P 0 = P 0 ( x , 0 ) , P n a = P n a ( x + n a , z ) , F β + ( P ; P ) = n G β + ( P ; P n a ) exp i k s i n a , G β + ( P ; P ) = g ( P ; P ) + β g ( P ; P * ) , g ( P ; P ) = 1 4 i H 0 ( 1 ) [ k ( P ; P ) ] = 1 4 i H 0 ( 1 ) ( k R ) = g ( x x , z z ) , R = | [ ( x x ) 2 + ( z z ) 2 ] 1 2 | .
1 2 ( 1 β ) G β + ( P ; P 0 ) + 1 2 ( 1 + β ) z G β + ( P ; P 0 ) = 0 , x , x .
F β + ( P ; P ) = n ( 2 i k a ω n ) 1 exp i k s n ( x x ) × [ exp i k ω n | z z | + β exp i k ω n | z + z | ] ,
s n = s i + 2 n π / k a , ω n = + ( 1 s n 2 ) 1 2 .
U ( P ) = U i ( P ) + n A n exp i k ( s n x + ω n z ) ; P S + ,
A n = β δ n , 0 + 1 / 2 1 / 2 d x a [ 1 2 ( 1 β ) U ( P 0 ) + 1 2 ( 1 + β ) ( i k ω n ) 1 z U ( P 0 ) ] exp i k s n x ,
U ( P n ) = 0 1 [ G β ( P n ; P n 0 ) z U ( P n 0 ) U ( P n 0 ) z G β ( P n ; P n 0 ) ] d x n ; P n S n .
P n = P n ( x n , z ) , P n 0 = P ( x n , 0 ) , P n = P n ( x n , z ) , G β ( P n ; P n ) = m [ g ( x n x n + 2 m l , z z ) + β g ( x n + x n + 2 m l , z z ) + β g ( x n x n + 2 m l , z + z + 2 b ) + g ( x n + x n + 2 m l , z + z + 2 b ) ] ;
G β ( P n ; P n ) = m = ( 1 β ) / 2 m ( 2 k l γ m ) 1 [ cos k α m ( x n x n ) + β cos k α m ( x n + x n ) ] [ 1 2 ( 1 β ) sin k γ m ( z + b ) + 1 2 ( 1 + β ) cos γ m ( z + b ) ] exp i [ k b γ m + 1 4 π ( 1 + β ) ] ;
α m = m π / k l , γ m = ( 1 α m 2 ) 1 2 , 0 = 1 , m 0 = 2 .
U ( P n ) = m = ( 1 β ) / 2 n R m [ cos k ( α m x n γ m ( z + b ) ) + β cos k ( α m x n + γ m ( z + b ) ) ] ; P n S n ,
n R m = R m exp i k s i n a
P n = | A n | 2 ω n / ω i ,
n σ n P n = 1 .
U ( x , 0 + ) = U ( x , 0 ) U ( x , 0 ) = U ( P 0 ) z U ( x , 0 + ) = z U ( x , 0 ) z U ( x , 0 ) = z U ( P 0 ) ; n a l / 2 < x < n a + l / 2 , n ( z ± = lim 0 ( z ± | | ) ) .
[ 1 2 ( 1 β ) z + 1 2 ( 1 + β ) ] [ U ( P 0 ) 2 U i ( P 0 ) ] = 1 2 ( 1 β ) ( k 2 + d x 2 ) 1 / 2 1 / 2 d x U ( P 0 ) F β + ( P 0 + ; P 0 ) 1 2 ( 1 + β ) 1 / 2 1 / 2 d x F β + ( P 0 + ; P 0 ) z U ( P 0 ) ; | x | < l / 2 ,
2 1 / 2 1 / 2 d x l [ 1 2 ( 1 β ) sin k α j x 0 + 1 2 ( 1 + β ) cos k α j x 0 ] ( ) ; j = 1 2 ( 1 β ) , 1 2 ( 1 + β ) + 1 , ,
m = ( 1 β ) / 2 F j m R m = T j ; j = 1 2 ( 1 β ) , 1 2 ( 1 β ) + 1 , ,
T j = [ 1 2 ( 1 β ) 2 ω i γ j 1 + 1 2 ( 1 + β ) j ] C 0 j + sec k b γ j , F j m = δ j , m + ( l / a i ) sin k b γ m sec k b γ j × n [ 1 2 ( 1 β ) ω n 2 γ m + 1 2 ( 1 + β ) j γ m ω n ] C n m C n j + , C n m ± = 2 0 1 d ζ l exp ± i k s n ( ζ l / 2 ) × [ 1 2 ( 1 β ) i sin k α m ζ + 1 2 ( 1 + β ) cos k α m ζ ] .
A n = β δ n , 0 + m = ( 1 β ) / 2 H n m R m ; n = 0 , ± 1 , ± 2 , ,
H n m = ( l 2 a i ) [ 1 2 ( 1 β ) + 1 2 ( 1 + β ) γ m ω n ] C n m sin k b γ m .
FR = T
A β 1 = HR
m = ( 1 β ) / 2 M F j m ( M ) R m ( M ) = T j ; j = 1 2 ( 1 β ) , 1 2 ( 1 β ) + 1 , , M ,
A n ( M ) = β δ n , 0 + m = ( 1 β ) / 2 M H n m R m ( M ) ; n = 0 , ± 1 , , ± M ,