Abstract

The “dynamical” theory of gratings originally developed by Rayleigh and Voigt is applied to derive the intensity of the light diffracted in various directions by an imperfect grating of finite area. The problem is reduced to the numerical solution of a system of linear equations by an approximation method in which “ghosts” and high order spectra are treated as perturbations of the main spectra. Current elementary theories are then seen to yield merely order of magnitude estimates of the intensity of the ghosts caused by various grating imperfections.

© 1948 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E.g., C. M. Sparrow, Astrophys. J. 49, 65 (1919).
    [CrossRef]
  2. E.g., H. A. Rowland, Physical Papers (Johns Hopkins University Press, Baltimore, 1902), p. 525.
  3. E.g., P. Frank and R. V. Mises, Differentialgleichungen der Physik (Friedrich Vieweg and Sohn, Braunschweig, 1935), Vol. 2, p. 853 ff.
  4. Rayleigh, Proc. Roy. Soc. (A) 79, 399 (1907).
    [CrossRef]
  5. W. Voigt, Göttinger Nachrichten40 (1911).
  6. U. Fano, Ann. d. Physik 32, 393 (1938).
    [CrossRef]
  7. U. Fano, J. Opt. Soc. Am. 31, 213 (1941).
    [CrossRef]
  8. A detailed analytical treatment by this method has also been given by C. T. Tai, for the case in which the grating spacing is much shorter than the radiation wave-length, see , Cruft Laboratory, Harvard University, Cambridge, Massachusetts, Jan.15, 1948.
  9. This set includes waves reflected and transmitted in “complex directions,” i.e., waves confined to the proximity of the grating’s surface. See reference 7, p. 214. For a derivation of the composition of the diffracted light, see reference 6, p. 402.
  10. See reference 7, Appendix.
  11. See reference 1, p. 83. See also reference 14.
  12. An index applied to v or w means that these functions have the values corresponding to the value of u with the same index.
  13. In fact, if no assumption were made at this point, the calculation could be carried further, but more laboriously, taking into account interaction terms between the light diffracted from the grating and that coming from the rest of the metal surface. In that case rq and tq should still be considered as slowly varying functions of u.
  14. No specific reference is given by Sparrow (see references 1 and 11) for this type of result. A paper by Rayleigh (Phil. Mag. 37, 498 (1919) and Coll. Works, Vol.  6, p. 627) is pertinent to this question. A proof of (12) is outlined here. The coefficient ap can be written in the formap(v)=(1/N)∑m=0N-1(N/L)∫mL/N(m+1)L/Nexp[2πi(vζ(x)-px/L)]dx=(1/N)∑m=0bpm(v).If the ruling is perfect except that each groove may be displaced from its theoretical position by a small fraction δ of the spacing L/N and if p is a multiple of N,bpm=(ap)perf exp(-2πipδm/N),where (ap)perf is the value of ap in the absence of groove displacements. Assuming, finally, that all displacements δm are distributed as random errors with a r.m.s. value σ, the expected, or mean, value of ap is the product of (ap)perf and of the expected value of exp(−2πipδm/N), namely,∫-∞∞exp[-2πipδm/N-δm2/2σ2]dδm/(2π)12σ=exp(-2π2σ2p2/N2).
    [CrossRef]

1941 (1)

1938 (1)

U. Fano, Ann. d. Physik 32, 393 (1938).
[CrossRef]

1919 (2)

E.g., C. M. Sparrow, Astrophys. J. 49, 65 (1919).
[CrossRef]

No specific reference is given by Sparrow (see references 1 and 11) for this type of result. A paper by Rayleigh (Phil. Mag. 37, 498 (1919) and Coll. Works, Vol.  6, p. 627) is pertinent to this question. A proof of (12) is outlined here. The coefficient ap can be written in the formap(v)=(1/N)∑m=0N-1(N/L)∫mL/N(m+1)L/Nexp[2πi(vζ(x)-px/L)]dx=(1/N)∑m=0bpm(v).If the ruling is perfect except that each groove may be displaced from its theoretical position by a small fraction δ of the spacing L/N and if p is a multiple of N,bpm=(ap)perf exp(-2πipδm/N),where (ap)perf is the value of ap in the absence of groove displacements. Assuming, finally, that all displacements δm are distributed as random errors with a r.m.s. value σ, the expected, or mean, value of ap is the product of (ap)perf and of the expected value of exp(−2πipδm/N), namely,∫-∞∞exp[-2πipδm/N-δm2/2σ2]dδm/(2π)12σ=exp(-2π2σ2p2/N2).
[CrossRef]

1911 (1)

W. Voigt, Göttinger Nachrichten40 (1911).

1907 (1)

Rayleigh, Proc. Roy. Soc. (A) 79, 399 (1907).
[CrossRef]

Fano, U.

U. Fano, J. Opt. Soc. Am. 31, 213 (1941).
[CrossRef]

U. Fano, Ann. d. Physik 32, 393 (1938).
[CrossRef]

Frank, P.

E.g., P. Frank and R. V. Mises, Differentialgleichungen der Physik (Friedrich Vieweg and Sohn, Braunschweig, 1935), Vol. 2, p. 853 ff.

Mises, R. V.

E.g., P. Frank and R. V. Mises, Differentialgleichungen der Physik (Friedrich Vieweg and Sohn, Braunschweig, 1935), Vol. 2, p. 853 ff.

Rayleigh,

No specific reference is given by Sparrow (see references 1 and 11) for this type of result. A paper by Rayleigh (Phil. Mag. 37, 498 (1919) and Coll. Works, Vol.  6, p. 627) is pertinent to this question. A proof of (12) is outlined here. The coefficient ap can be written in the formap(v)=(1/N)∑m=0N-1(N/L)∫mL/N(m+1)L/Nexp[2πi(vζ(x)-px/L)]dx=(1/N)∑m=0bpm(v).If the ruling is perfect except that each groove may be displaced from its theoretical position by a small fraction δ of the spacing L/N and if p is a multiple of N,bpm=(ap)perf exp(-2πipδm/N),where (ap)perf is the value of ap in the absence of groove displacements. Assuming, finally, that all displacements δm are distributed as random errors with a r.m.s. value σ, the expected, or mean, value of ap is the product of (ap)perf and of the expected value of exp(−2πipδm/N), namely,∫-∞∞exp[-2πipδm/N-δm2/2σ2]dδm/(2π)12σ=exp(-2π2σ2p2/N2).
[CrossRef]

Rayleigh, Proc. Roy. Soc. (A) 79, 399 (1907).
[CrossRef]

Rowland, H. A.

E.g., H. A. Rowland, Physical Papers (Johns Hopkins University Press, Baltimore, 1902), p. 525.

Sparrow, C. M.

E.g., C. M. Sparrow, Astrophys. J. 49, 65 (1919).
[CrossRef]

Tai, C. T.

A detailed analytical treatment by this method has also been given by C. T. Tai, for the case in which the grating spacing is much shorter than the radiation wave-length, see , Cruft Laboratory, Harvard University, Cambridge, Massachusetts, Jan.15, 1948.

Voigt, W.

W. Voigt, Göttinger Nachrichten40 (1911).

Ann. d. Physik (1)

U. Fano, Ann. d. Physik 32, 393 (1938).
[CrossRef]

Astrophys. J. (1)

E.g., C. M. Sparrow, Astrophys. J. 49, 65 (1919).
[CrossRef]

Göttinger Nachrichten (1)

W. Voigt, Göttinger Nachrichten40 (1911).

J. Opt. Soc. Am. (1)

Phil. Mag. (1)

No specific reference is given by Sparrow (see references 1 and 11) for this type of result. A paper by Rayleigh (Phil. Mag. 37, 498 (1919) and Coll. Works, Vol.  6, p. 627) is pertinent to this question. A proof of (12) is outlined here. The coefficient ap can be written in the formap(v)=(1/N)∑m=0N-1(N/L)∫mL/N(m+1)L/Nexp[2πi(vζ(x)-px/L)]dx=(1/N)∑m=0bpm(v).If the ruling is perfect except that each groove may be displaced from its theoretical position by a small fraction δ of the spacing L/N and if p is a multiple of N,bpm=(ap)perf exp(-2πipδm/N),where (ap)perf is the value of ap in the absence of groove displacements. Assuming, finally, that all displacements δm are distributed as random errors with a r.m.s. value σ, the expected, or mean, value of ap is the product of (ap)perf and of the expected value of exp(−2πipδm/N), namely,∫-∞∞exp[-2πipδm/N-δm2/2σ2]dδm/(2π)12σ=exp(-2π2σ2p2/N2).
[CrossRef]

Proc. Roy. Soc. (A) (1)

Rayleigh, Proc. Roy. Soc. (A) 79, 399 (1907).
[CrossRef]

Other (8)

E.g., H. A. Rowland, Physical Papers (Johns Hopkins University Press, Baltimore, 1902), p. 525.

E.g., P. Frank and R. V. Mises, Differentialgleichungen der Physik (Friedrich Vieweg and Sohn, Braunschweig, 1935), Vol. 2, p. 853 ff.

A detailed analytical treatment by this method has also been given by C. T. Tai, for the case in which the grating spacing is much shorter than the radiation wave-length, see , Cruft Laboratory, Harvard University, Cambridge, Massachusetts, Jan.15, 1948.

This set includes waves reflected and transmitted in “complex directions,” i.e., waves confined to the proximity of the grating’s surface. See reference 7, p. 214. For a derivation of the composition of the diffracted light, see reference 6, p. 402.

See reference 7, Appendix.

See reference 1, p. 83. See also reference 14.

An index applied to v or w means that these functions have the values corresponding to the value of u with the same index.

In fact, if no assumption were made at this point, the calculation could be carried further, but more laboriously, taking into account interaction terms between the light diffracted from the grating and that coming from the rest of the metal surface. In that case rq and tq should still be considered as slowly varying functions of u.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

Disposition of the elements of the matrix cnn. The large elements are confined within the shadowed areas.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

n = - c n n x n = d n ,
A ( x , z ) exp ( - 2 π i ν t ) in air ( z ζ ( x ) ) , B ( x , z ) exp ( - 2 π i ν t ) in the metal ( z ζ ( x ) ) .
A ( x , ζ ) = B ( x , ζ ) , K ( A z - A x d ζ d x ) z = ζ = ( B z - B x d ζ d x ) z = ζ ,
exp [ 2 π i ( u x + v z ) ]             and             exp [ 2 π i ( u x - w z ) ] ,
v = v ( u ) = ( k 2 - u 2 ) 1 2 ,             w = w ( u ) = ( n 2 k 2 - u 2 ) 1 2 .
0 arg ( v ) π / 2 ,             0 arg ( w ) π / 2.
A ( x , z ) = I exp [ 2 π i ( u 0 x - v 0 z ) ] + - d u R ( u ) exp [ 2 π i ( u x + v z ) ] , B ( x , z ) = - d u T ( u ) exp [ 2 π i ( u x - w z ) ] ,
- d u exp ( 2 π i u x ) × [ R ( u ) exp ( 2 π i v ζ ) - T ( u ) exp ( - 2 π i w ζ ) ] = - I exp [ 2 π i ( u 0 x - v 0 ζ ) ] . - d u exp ( 2 π i u x ) × [ K ( v - u d ζ / d x ) R ( u ) exp ( 2 π i v ζ ) + ( w + u d ζ / d x ) T ( u ) exp ( - 2 π i w ζ ) ] = K ( v 0 + u 0 d ζ / d x ) I exp [ 2 π i ( u 0 x - v 0 ζ ) ] .
exp [ 2 π i v ζ ( x ) ] = 1 - P ( x ) + P ( x ) p = - a p ( v ) exp ( 2 π i p x / L ) , ( d ζ / d x ) exp [ 2 π i v ζ ( x ) ] = P ( x ) p = - a p ( v ) ( p / v L ) exp ( 2 π i p x / L ) ,
a p ( v ) = ( 1 / L ) 0 L exp [ 2 π i ( v ζ ( x ) - p x / L ) ] d x .
- exp [ 2 π i ( u - u ) x ] d x = δ ( u - u ) , - f ( u ) δ ( u - u ) d u = f ( u ) , - exp [ 2 π i ( u - u ) x ] P ( x ) d x = 0 L exp [ 2 π i ( u - u ) x ] d x = exp [ i π ( u - u ) L ] sin [ π ( u - u ) L ] π ( u - u ) = Δ ( u - u ) ,
[ R ( u ) - T ( u ) ] - - [ R ( u ) - T ( u ) ] Δ ( u - u ) d u + p = - - [ a p ( v ) R ( u ) - a p ( - w ) T ( u ) ] × Δ ( u - u - p / L ) d u = - I [ δ ( u - u 0 ) - Δ ( u - u 0 ) + p a p ( - v 0 ) Δ ( u - u 0 - p / L ) ] ; [ K v R ( u ) + w T ( u ) ] - - [ K v R ( u ) + w T ( u ) ] Δ ( u - u ) d u + p = - - [ K ( v - u p / v L ) a p ( v ) R ( u ) + ( w - u p / w L ) a p ( - w ) T ( u ) ] × Δ ( u - u - p / L ) d u = I { K v 0 [ δ ( u - u 0 ) - Δ ( u - u 0 ) ] + p K ( v 0 - u 0 p / v 0 L ) × a p ( - v 0 ) Δ ( u - u 0 - p / L ) } .
R ( u ) = ρ [ δ ( u - u 0 ) - Δ ( u - u 0 ) ] + q = - r q Δ ( u - u 0 - q / L ) , T ( u ) = τ [ δ ( u - u 0 ) - Δ ( u - u 0 ) ] + q = - t q Δ ( u - u 0 - q / L ) .
- Δ ( u - u ) Δ ( u - u ) d u = Δ ( u - u ) ,
- R ( u ) Δ ( u - u ) d u = q r q Δ ( u - u 0 - q / L ) ,
( ρ - τ ) [ δ ( u - u 0 ) - Δ ( u - u 0 ) ] + q ( r q - t q ) Δ ( u - u 0 ( q ) ) - q ( r q - t q ) Δ ( u - u 0 ( q ) ) + p q [ a p ( v 0 ( q ) ) r q - a p ( - w 0 ( q ) ) t q ] × Δ ( u - u 0 ( p + q ) ) = - I [ δ ( u - u 0 ) - Δ ( u - u 0 ) + p a p ( - v 0 ) Δ ( u - u 0 ( p ) ) ] ; ( K v 0 ρ + w 0 τ ) [ δ ( u - u 0 ) - Δ ( u - u 0 ) ] + q ( K v 0 ( q ) r q + w 0 ( q ) t q ) Δ ( u - u 0 ( q ) ) - q ( K v 0 ( q ) r q + w 0 ( q ) t q ) Δ ( u - u 0 ( q ) ) + p q [ K ( v 0 ( q ) - u 0 ( q ) p / v 0 ( q ) L ) a p ( v 0 ( q ) ) r q + ( w 0 ( q ) - u 0 ( q ) p / w 0 ( q ) L ) a p ( - w 0 ( q ) ) t q ] × Δ ( u - u 0 ( p + q ) ) = I K { v 0 [ δ ( u - u 0 ) - Δ ( u - u 0 ) ] + p ( v 0 - u 0 p / v 0 L ) a p ( - v 0 ) Δ ( u - u 0 ( p ) ) } .
ρ - τ = - I ;             K v 0 ρ + w 0 τ = I ;
q [ a p - q ( v 0 ( q ) ) r q - a p - q ( - w 0 ( q ) ) t q ] = - I a p ( - v 0 ) ; q [ K ( v 0 ( q ) - ( p - q ) u 0 ( q ) / v 0 ( q ) L ) a p - q ( v 0 ( q ) ) r q + ( w 0 ( q ) - ( p - q ) u 0 ( q ) / w 0 ( q ) L ) a p - q ( - w 0 ( q ) ) t q ] = I K ( v 0 - p u 0 / v 0 L ) a p ( - w 0 ) .
x 2 q = r q ;             x 2 q + 1 = t q ; c 2 p , 2 q = a p - q ( v 0 ( q ) ) ;             c 2 p , 2 q + 1 = - a p - q ( - w 0 ( q ) ) ; c 2 p + 1 , 2 q = K [ v 0 ( q ) - ( p - q ) u 0 ( q ) / v 0 ( q ) L ] × a p - q ( v 0 ( q ) ) ; c 2 p + 1 , 2 q + 1 = [ w 0 ( q ) - ( p - q ) u 0 ( q ) / w 0 ( q ) L ] × a p - q ( - w 0 ( q ) ) ; d 2 p = - I a 2 p ( - v 0 ) ; d 2 p + 1 = K ( v 0 - p u 0 / v 0 L ) I .
a p ~ exp [ - ( p / N ) 2 ] ,
ap(v)=(1/N)m=0N-1(N/L)mL/N(m+1)L/Nexp[2πi(vζ(x)-px/L)]dx=(1/N)m=0bpm(v).
bpm=(ap)perfexp(-2πipδm/N),
-exp[-2πipδm/N-δm2/2σ2]dδm/(2π)12σ=exp(-2π2σ2p2/N2).