Abstract

We treat gratings as arrays of lines or elements rather than as infinite periodic structures. Since we cannot calculate accurately the diffraction pattern of a single rectangular-profile line, we have constructed a simple model which assumes Fraunhofer diffraction from the various parts of the line. Three measured parameters, functions of the angle of incidence, determine the entire diffraction pattern with good accuracy except for large angles of diffraction. The radiance either of the single-line or of a grating of any number of lines is then given by the triple product of the squares of the incident amplitude, the single-line diffraction amplitude (or the effective single-line pattern) and the array factor. Spectral radiances in S polarization calculated for a set of 20-line rectangular-profile gratings are in good agreement with measured values. Even anomalous behavior is accurately predicted.

© 1969 Optical Society of America

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References

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  1. Rayleigh, Proc. Roy. Soc. (London) A79, 399 (1907); Phil. Mag. 14, 60 (1907).
  2. W. C. Meecham and C. W. Peters, J. Appl. Phys. 28, 216 (1957).
    [CrossRef]
  3. R. F. Millar, Can. J. Phys. 39, 81 (1961).
    [CrossRef]
  4. A. Hessel and A. A. Oliner, Appl. Opt. 4, 1275 (1965).
    [CrossRef]
  5. C. H. Palmer and F. W. Phelps, J. Opt. Soc. Am. 58, 1184 (1968).
    [CrossRef]
  6. H. A. Rowland, Phil. Mag. [5] 35, 397 (1893).
  7. R. D. Hatcher and J. H. Rohrbaugh, J. Opt. Soc. Am. 46, 104 (1956).
    [CrossRef]
  8. R. P. Madden and J. Strong, Concepts of Classical Optics (W. H. Freeman & Co., San Francisco, 1958), Appendix P.
  9. V. Twersky, IRE Trans. Antennas Propagation AP-4, 330 (1956).
    [CrossRef]
  10. L. N. Zadoff and R. D. Hatcher, Ann. N. Y. Acad. Sci. 93, 301 (1962).
  11. I. Lazar and L. A. DeAcetis, Appl. Opt. 7, 1609 (1968); Am. J. Phys. 36, 830 (1968).
    [CrossRef] [PubMed]
  12. C. H. Palmer, in Symposium on Quasi-Optics, Jerome Fox’ Ed. (Polytechnic Press, Brooklyn, N. Y., 1964).
  13. C. H. Palmer, F. C. Evering, and F. M. Nelson, Appl. Opt. 4, 1271 (1965).
    [CrossRef]

1968 (2)

1965 (2)

1962 (1)

L. N. Zadoff and R. D. Hatcher, Ann. N. Y. Acad. Sci. 93, 301 (1962).

1961 (1)

R. F. Millar, Can. J. Phys. 39, 81 (1961).
[CrossRef]

1957 (1)

W. C. Meecham and C. W. Peters, J. Appl. Phys. 28, 216 (1957).
[CrossRef]

1956 (2)

V. Twersky, IRE Trans. Antennas Propagation AP-4, 330 (1956).
[CrossRef]

R. D. Hatcher and J. H. Rohrbaugh, J. Opt. Soc. Am. 46, 104 (1956).
[CrossRef]

1907 (1)

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

1893 (1)

H. A. Rowland, Phil. Mag. [5] 35, 397 (1893).

DeAcetis, L. A.

Evering, F. C.

Hatcher, R. D.

L. N. Zadoff and R. D. Hatcher, Ann. N. Y. Acad. Sci. 93, 301 (1962).

R. D. Hatcher and J. H. Rohrbaugh, J. Opt. Soc. Am. 46, 104 (1956).
[CrossRef]

Hessel, A.

Lazar, I.

Madden, R. P.

R. P. Madden and J. Strong, Concepts of Classical Optics (W. H. Freeman & Co., San Francisco, 1958), Appendix P.

Meecham, W. C.

W. C. Meecham and C. W. Peters, J. Appl. Phys. 28, 216 (1957).
[CrossRef]

Millar, R. F.

R. F. Millar, Can. J. Phys. 39, 81 (1961).
[CrossRef]

Nelson, F. M.

Oliner, A. A.

Palmer, C. H.

C. H. Palmer and F. W. Phelps, J. Opt. Soc. Am. 58, 1184 (1968).
[CrossRef]

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

C. H. Palmer, in Symposium on Quasi-Optics, Jerome Fox’ Ed. (Polytechnic Press, Brooklyn, N. Y., 1964).

Peters, C. W.

W. C. Meecham and C. W. Peters, J. Appl. Phys. 28, 216 (1957).
[CrossRef]

Phelps, F. W.

Rayleigh,

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

Rohrbaugh, J. H.

Rowland, H. A.

H. A. Rowland, Phil. Mag. [5] 35, 397 (1893).

Strong, J.

R. P. Madden and J. Strong, Concepts of Classical Optics (W. H. Freeman & Co., San Francisco, 1958), Appendix P.

Twersky, V.

V. Twersky, IRE Trans. Antennas Propagation AP-4, 330 (1956).
[CrossRef]

Zadoff, L. N.

L. N. Zadoff and R. D. Hatcher, Ann. N. Y. Acad. Sci. 93, 301 (1962).

Ann. N. Y. Acad. Sci. (1)

L. N. Zadoff and R. D. Hatcher, Ann. N. Y. Acad. Sci. 93, 301 (1962).

Appl. Opt. (3)

Can. J. Phys. (1)

R. F. Millar, Can. J. Phys. 39, 81 (1961).
[CrossRef]

IRE Trans. Antennas Propagation (1)

V. Twersky, IRE Trans. Antennas Propagation AP-4, 330 (1956).
[CrossRef]

J. Appl. Phys. (1)

W. C. Meecham and C. W. Peters, J. Appl. Phys. 28, 216 (1957).
[CrossRef]

J. Opt. Soc. Am. (2)

Phil. Mag. [5] (1)

H. A. Rowland, Phil. Mag. [5] 35, 397 (1893).

Proc. Roy. Soc. (London) (1)

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

Other (2)

R. P. Madden and J. Strong, Concepts of Classical Optics (W. H. Freeman & Co., San Francisco, 1958), Appendix P.

C. H. Palmer, in Symposium on Quasi-Optics, Jerome Fox’ Ed. (Polytechnic Press, Brooklyn, N. Y., 1964).

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Figures (9)

Fig. 1
Fig. 1

Conventions used for the grating.

Fig. 2
Fig. 2

Comparison of measured (solid curves) and calculated (dots) single-line diffraction patterns for four angles of incidence. Grating FP-4 has d = 3.06λ, w = 0.75λ, h = 0.25λ. The arrows indicate angles of diffraction where the model is chosen to fit.

Fig. 3
Fig. 3

Comparison of three predicted single-line diffraction patterns for two angles of incidence. Profile FP-4 as in Fig. 2. Points are for 21-lines (△), 5-lines (×) and 1-line (○).

Fig. 4
Fig. 4

Values of the amplitude parameters A (●), B (×), and C (○) for four 20-line gratings of different slot depth.

Fig. 5
Fig. 5

Typical predicted single-line diffraction pattern for grating FP-6 at θi = −10°. The locations of the various spectral orders are indicated by the arrows.

Fig. 6
Fig. 6

Calculated (×) vs measured (●) spectral radiances for grating FP-6: d=4.11λ, w=0.75λ, h=0.13λ, N=20 lines. (Measurements could not be made for order −3 when θi>−32°.)

Fig. 7
Fig. 7

Calculated (×) vs measured (●) spectral radiances for grating FP-7: d,w, and N as in Fig. 6, h=0.25λ.

Fig. 8
Fig. 8

Calculated (×) vs measured (●) spectral radiances for grating FP-8: d, w, and N as in Fig. 6, h=0.38λ. Note: spectral order −2 (−2*) curves are shifted vertically by 5 db for clarity. Spectral order −1 radiances for θi> −32° used to calculate A, B and C so that theoretical points are omitted.

Fig. 9
Fig. 9

Calculated (×) vs measured (●) spectral radiances for grating FP-9: d, w and N as in Fig. 6, h = 0.5λ. Note: for spectral order −3 the signal was too small to measure except for the point shown.

Equations (24)

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E i = E 0 cos ( θ i ) exp [ i ( ω t + β x sin θ i ) ] ,
F x 1 x 2 E i exp [ - i β ( r - x sin θ ) ] d x .
F 1 = - d / 2 d / 2 A e i ν e i β p x d x - - ω / 2 ω / 2 A e i ν e i β p x d x ,
ν ω t - β r
p sin θ i + sin θ .
F 1 = A e i ν [ d sin ( π β p d / 2 π ) π β p d / 2 π - w sin ( π β p w / 2 π ) π β p w / 2 π ] .
F 1 = A e i ν [ d sin c ( X ) - w sin c ( Y ) ] ,
X β p d / 2 π = p d / λ
Y β p w / 2 π = p w / λ
F 2 = B e i ν w sinc ( Y ) .
F 3 = C e i ν w sinc ( Z ) ,
Z ( β w / 2 π ) sin θ = ( w / λ ) sin θ .
F = e i ν [ A d sinc ( X ) + ( B - A ) w sinc ( Y ) + C w sinc ( Z ) ] ,
F F * = [ A d sinc ( X ) + ( B - A ) w sinc ( Y ) + C w sinc ( Z ) ] 2 ,
G = [ 1 - exp ( i 2 π N d p / λ ) ] / [ 1 - exp ( i 2 π d p / λ ) ]
G G * = sin 2 ( π N d p / λ ) / sin 2 ( π d p / λ )
P N = F F * G G * .
P N ( 0 ) = [ A d + ( B - A ) W + C W sinc { - ( w / λ ) sin θ i } ] 2 N 2 .
sinc X = 0             n 0
sinc Y = sinc ( n w / d )
sinc Z = sinc [ ( w / λ ) sin θ n ] = sinc [ ( n w / d - ( w / λ ) sin θ i ] .
P N ( n ) = [ ( B - A ) w sinc ( n w / d ) + C w sinc { n w / d - ( w / λ ) sin θ i } ] 2 N 2
{ a 0 A + b 0 B + c 0 C = d 0 n = 0 a 1 A + b 1 B + c 1 C = d 1 n = n 1 a 2 A + b 2 B + c 2 C = d 2 n = n 2 .
{ a 0 = d - w b 0 = w c 0 = w sinc { ( - w / λ ) sin θ i } d 0 = [ P N ( O ) ] 1 2 / N a 1 = - w sinc ( n 1 w / d ) b 1 = + w sinc ( n 1 w / d ) c 1 = w sinc { n 1 w / d - ( w / λ ) sin θ i } d 1 = [ P N ( n 1 ) ] 1 2 / N