Abstract

The theory of Wood’s anomalous diffraction gratings, which was developed some years ago, has been reexamined in order to visualize its physical meaning. Each wave diffracted by a grating is identified through the component of its “wave vector” tangential to the grating. Surface waves similar to those found in total internal reflection are included (§2). The amplitudes of these waves can be calculated by successive approximations (§3). One feature of the anomalies is connected with the infinite dispersion of spectra at grazing emergence (§4). Emphasis is put on the existence of polarized quasi-stationary waves which represent an energy current rolling along the surface of a metal (§5). These waves can be strongly excited on the surface of metallic gratings under critical conditions depending also on the profile of the grooves; secondary interference phenomena arise then in the observed spectra (§6). The connection of the quasi-stationary surface waves with the wireless ground waves is discussed (§7). A general formulation is introduced to discuss the significance of the approximation used (Appendix).

© 1941 Optical Society of America

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References

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  1. R. W. Wood, Phil. Mag. 4, 393 (1902); Phil. Mag. 23, 310 (1912); Phys. Rev. 48, 928 (1935); J. Strong, Phys. Rev. 49, 291 (1936).
    [Crossref]
  2. Rayleigh, Phil. Mag. 14, 60 (1907).
    [Crossref]
  3. U. Fano, Ann. d. Physik 32, 393 (1938).
    [Crossref]
  4. This phenomenon is closely analogous to the so-called “Dellen” in the diffraction of molecular rays (discovered by Stern and Frisch), whose explanation in connection with superficial adsorption phenomena was first given by Lennard-Jones and Devonshire (Nature 137, 969 (1936), Proc. Roy. Soc. A158, 253 (1937)) and then further developed (reference 3).
    [Crossref]
  5. Rayleigh, Proc. Roy. Soc. A79, 399 (1907).
    [Crossref]
  6. Reference 2, Appendix 1. See also a paper by T. Westerdijk, Ann. d. Physik 36, 696 (1939).
  7. The zero-order approximation (ζ(x)=0) determines Ap+(0) and Ao−(0)) according to Fresnel’s formulae.
  8. The balance of intensity is determined by interference affecting only the central image which, within this approximation, acts as an infinitely powerful bank on which the various diffracted waves draw independently of one another. Here, as in the case of the “Dellen” (reference 4), to consider that anomalous dark bands are due to a “lack” of radiation arising from strong absorption by other processes is equivalent to assuming that the approximation method diverges.
  9. The equation determining the proper value of the momentum is:n2ξn++ξn-=n2(ko2-ξt2)12+(n2ko2-ξt2)12=0,if the magnetic vector is parallel to the surface, and:ξn++ξn-=(ko2-ξt2)12+(n2ko2-ξt2)12=0,if the electric vector is parallel to the surface.
  10. The apparent spectral width of the bands might not be due entirely to absorption of superficial waves within the metal. Second-order diffraction, i.e., the interaction with other diffracted waves, is also effective as an absorption (see Appendix).
  11. Rayleigh, Phil. Mag. 14, 60 (1907).
    [Crossref]
  12. See discussion by R. W. Wood, Phys. Rev. 48, 928 (1935).
    [Crossref]
  13. U. Fano, Phys. Rev. 50, 573 (1936).
    [Crossref]
  14. H. Weyl, Ann. d. Physik 60, 481 (1919).
    [Crossref]
  15. See reference 14, p. 404. When the magnetic vector is parallel to the grating, it is:αp+(1)(x)=exp [2πi(px/δ+kpn+ζ(x))];αp+(2)(x)=n2(kpn+-kptdζ/dx)αp+(1)(x);αp-(1)(x)=-exp [2πi(px/δ-kpn-ζ(x))];αp-(2)(x)=-(kpn-+kptdζ/dx)αp-(1)(x);β(1)(x)=-exp (-2πikon+ζ(x));β(2)(x)=-n2(kon++kotdζ/dx)β(1)(x).kpn+=(ko2-kpt2)12;0⩽Arg (kpn+)<π;kpn-=(n2ko2-kpt2)12;0⩽Arg (kpn-)<π.
  16. exp (2πikpn+ζ(x))=∑qgq(p) exp (2πiqx/δ)exp (-2πikpn-ζ(x))=∑qhq(p) exp (2πiqx/δ)ap′1,p+=gp′-p(p);         ap′1,p=hp′-p(p)ap′2,p+=n2[kpn+-kptkpn+p′-pδ]gp′-p(p)ap′2,p-=[kpn--kptkpn-p′-pδ]hp′-p(p)bp′1=-g-p′(0)*;         bp′2=n2[kon+-kotkon+p′δ]g-p′(0)*.
  17. Cn′n is determined by an algebraic formula in the case of a finite system of linear equations.
  18. Rayleigh’s method involves expansion of exponentials in powers of the depth of the grooves ζ(x), or better, of its ratio to the transversal wave-length.
  19. See, e.g., the article by A. Sommerfeld in Ph. Frank-R. von Mises, Die Differentialgleichungen der Mathematischen Physik (Vieweg, Braunschweig, 1935), second edition, Vol. II, p. 876.
  20. One can see that the additional damping to be added to γ is approximately proportional to the square of the depth of the grooves.
  21. The influence of the grating on the energy traveling within a superficial wave is not too close, however, because the wave stretches up to an appreciable distance from the grating into vacuum.
  22. When it is possible to bring the matrix Cn′n into a diagonal form, represented by a set of values γm, that is, to solve the homogeneous system:∑ncn′nxn(m)=γmxn′(m),         (∑n∣xn(m)∣2=1),the matrix Cn′n is given by the formula:Cn′n=∑mxn′(m)xn(m)/γm.Wood’s anomalies are represented by anomalously small values of one of the quantities γm. Within Rayleigh’s first approximation the γm are factors of the 2×2 determinants of diagonal squares of cn′n. It is easy to discuss the order of magnitude of the quantities xn(m) at different stages of Rayleigh’s approximation.

1939 (1)

Reference 2, Appendix 1. See also a paper by T. Westerdijk, Ann. d. Physik 36, 696 (1939).

1938 (1)

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

1936 (2)

This phenomenon is closely analogous to the so-called “Dellen” in the diffraction of molecular rays (discovered by Stern and Frisch), whose explanation in connection with superficial adsorption phenomena was first given by Lennard-Jones and Devonshire (Nature 137, 969 (1936), Proc. Roy. Soc. A158, 253 (1937)) and then further developed (reference 3).
[Crossref]

U. Fano, Phys. Rev. 50, 573 (1936).
[Crossref]

1935 (1)

See discussion by R. W. Wood, Phys. Rev. 48, 928 (1935).
[Crossref]

1919 (1)

H. Weyl, Ann. d. Physik 60, 481 (1919).
[Crossref]

1907 (3)

Rayleigh, Phil. Mag. 14, 60 (1907).
[Crossref]

Rayleigh, Proc. Roy. Soc. A79, 399 (1907).
[Crossref]

Rayleigh, Phil. Mag. 14, 60 (1907).
[Crossref]

1902 (1)

R. W. Wood, Phil. Mag. 4, 393 (1902); Phil. Mag. 23, 310 (1912); Phys. Rev. 48, 928 (1935); J. Strong, Phys. Rev. 49, 291 (1936).
[Crossref]

Devonshire,

This phenomenon is closely analogous to the so-called “Dellen” in the diffraction of molecular rays (discovered by Stern and Frisch), whose explanation in connection with superficial adsorption phenomena was first given by Lennard-Jones and Devonshire (Nature 137, 969 (1936), Proc. Roy. Soc. A158, 253 (1937)) and then further developed (reference 3).
[Crossref]

Fano, U.

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

U. Fano, Phys. Rev. 50, 573 (1936).
[Crossref]

Lennard-Jones,

This phenomenon is closely analogous to the so-called “Dellen” in the diffraction of molecular rays (discovered by Stern and Frisch), whose explanation in connection with superficial adsorption phenomena was first given by Lennard-Jones and Devonshire (Nature 137, 969 (1936), Proc. Roy. Soc. A158, 253 (1937)) and then further developed (reference 3).
[Crossref]

Rayleigh,

Rayleigh, Phil. Mag. 14, 60 (1907).
[Crossref]

Rayleigh, Proc. Roy. Soc. A79, 399 (1907).
[Crossref]

Rayleigh, Phil. Mag. 14, 60 (1907).
[Crossref]

Sommerfeld, A.

See, e.g., the article by A. Sommerfeld in Ph. Frank-R. von Mises, Die Differentialgleichungen der Mathematischen Physik (Vieweg, Braunschweig, 1935), second edition, Vol. II, p. 876.

von Mises, Ph. Frank-R.

See, e.g., the article by A. Sommerfeld in Ph. Frank-R. von Mises, Die Differentialgleichungen der Mathematischen Physik (Vieweg, Braunschweig, 1935), second edition, Vol. II, p. 876.

Westerdijk, T.

Reference 2, Appendix 1. See also a paper by T. Westerdijk, Ann. d. Physik 36, 696 (1939).

Weyl, H.

H. Weyl, Ann. d. Physik 60, 481 (1919).
[Crossref]

Wood, R. W.

See discussion by R. W. Wood, Phys. Rev. 48, 928 (1935).
[Crossref]

R. W. Wood, Phil. Mag. 4, 393 (1902); Phil. Mag. 23, 310 (1912); Phys. Rev. 48, 928 (1935); J. Strong, Phys. Rev. 49, 291 (1936).
[Crossref]

Ann. d. Physik (3)

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

Reference 2, Appendix 1. See also a paper by T. Westerdijk, Ann. d. Physik 36, 696 (1939).

H. Weyl, Ann. d. Physik 60, 481 (1919).
[Crossref]

Nature (1)

This phenomenon is closely analogous to the so-called “Dellen” in the diffraction of molecular rays (discovered by Stern and Frisch), whose explanation in connection with superficial adsorption phenomena was first given by Lennard-Jones and Devonshire (Nature 137, 969 (1936), Proc. Roy. Soc. A158, 253 (1937)) and then further developed (reference 3).
[Crossref]

Phil. Mag. (3)

R. W. Wood, Phil. Mag. 4, 393 (1902); Phil. Mag. 23, 310 (1912); Phys. Rev. 48, 928 (1935); J. Strong, Phys. Rev. 49, 291 (1936).
[Crossref]

Rayleigh, Phil. Mag. 14, 60 (1907).
[Crossref]

Rayleigh, Phil. Mag. 14, 60 (1907).
[Crossref]

Phys. Rev. (2)

See discussion by R. W. Wood, Phys. Rev. 48, 928 (1935).
[Crossref]

U. Fano, Phys. Rev. 50, 573 (1936).
[Crossref]

Proc. Roy. Soc. (1)

Rayleigh, Proc. Roy. Soc. A79, 399 (1907).
[Crossref]

Other (12)

The zero-order approximation (ζ(x)=0) determines Ap+(0) and Ao−(0)) according to Fresnel’s formulae.

The balance of intensity is determined by interference affecting only the central image which, within this approximation, acts as an infinitely powerful bank on which the various diffracted waves draw independently of one another. Here, as in the case of the “Dellen” (reference 4), to consider that anomalous dark bands are due to a “lack” of radiation arising from strong absorption by other processes is equivalent to assuming that the approximation method diverges.

The equation determining the proper value of the momentum is:n2ξn++ξn-=n2(ko2-ξt2)12+(n2ko2-ξt2)12=0,if the magnetic vector is parallel to the surface, and:ξn++ξn-=(ko2-ξt2)12+(n2ko2-ξt2)12=0,if the electric vector is parallel to the surface.

The apparent spectral width of the bands might not be due entirely to absorption of superficial waves within the metal. Second-order diffraction, i.e., the interaction with other diffracted waves, is also effective as an absorption (see Appendix).

See reference 14, p. 404. When the magnetic vector is parallel to the grating, it is:αp+(1)(x)=exp [2πi(px/δ+kpn+ζ(x))];αp+(2)(x)=n2(kpn+-kptdζ/dx)αp+(1)(x);αp-(1)(x)=-exp [2πi(px/δ-kpn-ζ(x))];αp-(2)(x)=-(kpn-+kptdζ/dx)αp-(1)(x);β(1)(x)=-exp (-2πikon+ζ(x));β(2)(x)=-n2(kon++kotdζ/dx)β(1)(x).kpn+=(ko2-kpt2)12;0⩽Arg (kpn+)<π;kpn-=(n2ko2-kpt2)12;0⩽Arg (kpn-)<π.

exp (2πikpn+ζ(x))=∑qgq(p) exp (2πiqx/δ)exp (-2πikpn-ζ(x))=∑qhq(p) exp (2πiqx/δ)ap′1,p+=gp′-p(p);         ap′1,p=hp′-p(p)ap′2,p+=n2[kpn+-kptkpn+p′-pδ]gp′-p(p)ap′2,p-=[kpn--kptkpn-p′-pδ]hp′-p(p)bp′1=-g-p′(0)*;         bp′2=n2[kon+-kotkon+p′δ]g-p′(0)*.

Cn′n is determined by an algebraic formula in the case of a finite system of linear equations.

Rayleigh’s method involves expansion of exponentials in powers of the depth of the grooves ζ(x), or better, of its ratio to the transversal wave-length.

See, e.g., the article by A. Sommerfeld in Ph. Frank-R. von Mises, Die Differentialgleichungen der Mathematischen Physik (Vieweg, Braunschweig, 1935), second edition, Vol. II, p. 876.

One can see that the additional damping to be added to γ is approximately proportional to the square of the depth of the grooves.

The influence of the grating on the energy traveling within a superficial wave is not too close, however, because the wave stretches up to an appreciable distance from the grating into vacuum.

When it is possible to bring the matrix Cn′n into a diagonal form, represented by a set of values γm, that is, to solve the homogeneous system:∑ncn′nxn(m)=γmxn′(m),         (∑n∣xn(m)∣2=1),the matrix Cn′n is given by the formula:Cn′n=∑mxn′(m)xn(m)/γm.Wood’s anomalies are represented by anomalously small values of one of the quantities γm. Within Rayleigh’s first approximation the γm are factors of the 2×2 determinants of diagonal squares of cn′n. It is easy to discuss the order of magnitude of the quantities xn(m) at different stages of Rayleigh’s approximation.

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Figures (7)

Fig. 1
Fig. 1

Showing the parameters which characterize a metallic reflection grating.

Fig. 2
Fig. 2

Analysis of a spectrum in different orders. The wave-lengths λR3(6500A) and λR4(4875A) represent the limits of the spectra of third and fourth order respectively; they appear in every spectrum as Rayleigh’s wave-lengths.

Fig. 3
Fig. 3

Construction of the normal momentum component of the diffracted waves. The construction cannot be applied to the waves of third or higher order, thus showing that these waves have imaginary normal momentum components, i.e. that they are superficial waves.

Fig. 4
Fig. 4

Schematic path of a light wave progressing within a glass plate between a metal and a vacuum.

Fig. 5
Fig. 5

Amplitude of vibration of a standing wave in the various layers of a set of three media: vacuum, glass, metal.

Fig. 6
Fig. 6

Geometrical representation of Rayleigh’s condition:

Fig. 7
Fig. 7

Rayleigh’s approximation is accurate when the important elements of both matrices cnn and Cnn are concentrated within the set of “diagonal squares.”

Equations (14)

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ζ ( x ) = p = - ζ p exp ( 2 π i p x / δ ) ;
A B - O B = p λ .
p , s α p s ( 1 ) ( x ) A p s = β ( 1 ) ( x ) I ,             p , s α p s ( 2 ) ( x ) A p s = β ( 2 ) ( x ) I ,
α p + ( 1 ) ( x ) = p a p 1 , p + exp ( 2 π i p x / δ )
p , s a p 1 , p s A p s = b p 1 I ;             p , s a p 2 , p s A p s = b p 2 I .
n c n n x n = y n ,
x n = n C n n y n ,
exp ( 2 π i k p n + ζ ( x ) ) = 1 + 2 π i k p n + ζ ( x ) - g q ( p ) = δ q + 2 π i k p n + ζ q - ;             h q ( p ) = δ q - 2 π i k p n - ζ q - ; δ q = 1 when q = 0 ;             δ q = 0 when q 0.
n2ξn++ξn-=n2(ko2-ξt2)12+(n2ko2-ξt2)12=0,
ξn++ξn-=(ko2-ξt2)12+(n2ko2-ξt2)12=0,
αp+(1)(x)=exp[2πi(px/δ+kpn+ζ(x))];αp+(2)(x)=n2(kpn+-kptdζ/dx)αp+(1)(x);αp-(1)(x)=-exp[2πi(px/δ-kpn-ζ(x))];αp-(2)(x)=-(kpn-+kptdζ/dx)αp-(1)(x);β(1)(x)=-exp(-2πikon+ζ(x));β(2)(x)=-n2(kon++kotdζ/dx)β(1)(x).kpn+=(ko2-kpt2)12;0Arg(kpn+)<π;kpn-=(n2ko2-kpt2)12;0Arg(kpn-)<π.
exp(2πikpn+ζ(x))=qgq(p)exp(2πiqx/δ)exp(-2πikpn-ζ(x))=qhq(p)exp(2πiqx/δ)ap1,p+=gp-p(p);         ap1,p=hp-p(p)ap2,p+=n2[kpn+-kptkpn+p-pδ]gp-p(p)ap2,p-=[kpn--kptkpn-p-pδ]hp-p(p)bp1=-g-p(0)*;         bp2=n2[kon+-kotkon+pδ]g-p(0)*.
ncnnxn(m)=γmxn(m),         (nxn(m)2=1),
Cnn=mxn(m)xn(m)/γm.