Abstract

A rigorous solution is obtained for the problem of radiation from an electric line charge that moves, at a constant speed, parallel to an electrically perfectly conducting grating. The relevant vectorial electromagnetic problem is reduced to a two-dimensional scalar one. With the aid of a Green’s-function formulation of the problem, an integral equation of the second kind for the surface current density on a single period of the grating surface is derived. This integral equation is solved numerically by a method of moments. Some numerical results pertaining to the radiation from a moving line charge above a sinusoidal grating are presented.

© 1973 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. J. Smith and E. M. Purcell, Phys. Rev. 92, 1069 (1953).
    [Crossref]
  2. W. W. Salisbury, U. S. Patent No.2 634 372, 7April1953.
  3. K. Ishiguro and T. Tako, Opt. Acta 8, 25 (1961).
    [Crossref]
  4. W. W. Salisbury, J. Opt. Soc. Am. 52, 1315 (1962).
  5. G. Toraldo di Francia, Nuovo Cimento 16, 61 (1960).
  6. A. Hessel, Can. J. Phys. 42, 1195 (1964).
    [Crossref]
  7. O. A. Tret’yakov, S. S. Tret’yakov, and V. P. Shestopalov, Radio Eng. Electron. Phys. 10, 1059 (1965).
  8. B. M. Bolotovskii and G. V. Voskresenkii, Usp. Fiz. Nauk 94, 377 (1968) [Sov. Phys.-Usp. 11, 143 (1968)].
  9. J. Lam, J. Math. Phys. 8, 1053 (1967).
    [Crossref]
  10. R. D. Hazeltine, M. N. Rosenbluth, and A. M. Sessler, J. Math. Phys. 12, 502 (1971).
    [Crossref]
  11. J. P. Bachheimer, C.R. Acad. Sci. (Paris) 268, 599 (1969).
  12. For extensive literature about the Rayleigh hypothesis in scattering by a periodic surface we refer to B. A. Lippmann, J. Opt. Soc. Am. 43, 408 (1953); R. Petit and M. Cadilhac, C.R. Acad. Sci. (Paris) 262, 468 (1966); R. F. Millar, Proc. Camb. Philos. Soc. 65, 773 (1969); M. Neviere and M. Cadilhac, Opt. Commun. 2, 235 (1970); R. F. Millar, Proc. Camb. Philos. Soc. 69, 217 (1971).
    [Crossref]
  13. C. W. Barnes and K. G. Dedrick, J. Appl. Phys. 37, 411 (1966).
    [Crossref]
  14. P. M. van den Berg, Appl. Sci. Res. 24, 261 (1971).
  15. A referee has drawn the author’s attention to an unpublished NDA Report (18-8) by B. A. Lippmann, in which the same Green’s function has been derived in connection with his “variational formulation of a grating problem.”
  16. The application of the surface impedance (Leontovich) boundary condition would introduce an extra parameter, viz., the frequency-dependent surface impedance. Since the surface-impedance boundary condition holds only in a limited frequency domain, we cannot represent the field quantities as Fourier integrals.
  17. Note that this Green’s function is the same as the one used by B. A. Lippmann in NDA Report No. 18-8.
  18. This phenomenon is just the “lateral wave” case of the “Wood anomalies” (Rayleigh wavelengths) observed with diffraction gratings [see J. E. Stewart and W. S. Gallaway, Appl. Opt. 1, 421 (1962)].
    [Crossref]
  19. P. M. van den Berg, Thesis, Delft University of Technology, The Netherlands (1971).
  20. P. M. van den Berg and O. J. Voorman, Appl. Sci. Res. 26, 175 (1972).
    [Crossref]
  21. H. Blok and G. Mur, Appl. Sci. Res. 26, 389 (1972).
    [Crossref]

1972 (2)

P. M. van den Berg and O. J. Voorman, Appl. Sci. Res. 26, 175 (1972).
[Crossref]

H. Blok and G. Mur, Appl. Sci. Res. 26, 389 (1972).
[Crossref]

1971 (2)

R. D. Hazeltine, M. N. Rosenbluth, and A. M. Sessler, J. Math. Phys. 12, 502 (1971).
[Crossref]

P. M. van den Berg, Appl. Sci. Res. 24, 261 (1971).

1969 (1)

J. P. Bachheimer, C.R. Acad. Sci. (Paris) 268, 599 (1969).

1968 (1)

B. M. Bolotovskii and G. V. Voskresenkii, Usp. Fiz. Nauk 94, 377 (1968) [Sov. Phys.-Usp. 11, 143 (1968)].

1967 (1)

J. Lam, J. Math. Phys. 8, 1053 (1967).
[Crossref]

1966 (1)

C. W. Barnes and K. G. Dedrick, J. Appl. Phys. 37, 411 (1966).
[Crossref]

1965 (1)

O. A. Tret’yakov, S. S. Tret’yakov, and V. P. Shestopalov, Radio Eng. Electron. Phys. 10, 1059 (1965).

1964 (1)

A. Hessel, Can. J. Phys. 42, 1195 (1964).
[Crossref]

1962 (2)

1961 (1)

K. Ishiguro and T. Tako, Opt. Acta 8, 25 (1961).
[Crossref]

1960 (1)

G. Toraldo di Francia, Nuovo Cimento 16, 61 (1960).

1953 (2)

Bachheimer, J. P.

J. P. Bachheimer, C.R. Acad. Sci. (Paris) 268, 599 (1969).

Barnes, C. W.

C. W. Barnes and K. G. Dedrick, J. Appl. Phys. 37, 411 (1966).
[Crossref]

Blok, H.

H. Blok and G. Mur, Appl. Sci. Res. 26, 389 (1972).
[Crossref]

Bolotovskii, B. M.

B. M. Bolotovskii and G. V. Voskresenkii, Usp. Fiz. Nauk 94, 377 (1968) [Sov. Phys.-Usp. 11, 143 (1968)].

Dedrick, K. G.

C. W. Barnes and K. G. Dedrick, J. Appl. Phys. 37, 411 (1966).
[Crossref]

Gallaway, W. S.

Hazeltine, R. D.

R. D. Hazeltine, M. N. Rosenbluth, and A. M. Sessler, J. Math. Phys. 12, 502 (1971).
[Crossref]

Hessel, A.

A. Hessel, Can. J. Phys. 42, 1195 (1964).
[Crossref]

Ishiguro, K.

K. Ishiguro and T. Tako, Opt. Acta 8, 25 (1961).
[Crossref]

Lam, J.

J. Lam, J. Math. Phys. 8, 1053 (1967).
[Crossref]

Lippmann, B. A.

Mur, G.

H. Blok and G. Mur, Appl. Sci. Res. 26, 389 (1972).
[Crossref]

Purcell, E. M.

S. J. Smith and E. M. Purcell, Phys. Rev. 92, 1069 (1953).
[Crossref]

Rosenbluth, M. N.

R. D. Hazeltine, M. N. Rosenbluth, and A. M. Sessler, J. Math. Phys. 12, 502 (1971).
[Crossref]

Salisbury, W. W.

W. W. Salisbury, J. Opt. Soc. Am. 52, 1315 (1962).

W. W. Salisbury, U. S. Patent No.2 634 372, 7April1953.

Sessler, A. M.

R. D. Hazeltine, M. N. Rosenbluth, and A. M. Sessler, J. Math. Phys. 12, 502 (1971).
[Crossref]

Shestopalov, V. P.

O. A. Tret’yakov, S. S. Tret’yakov, and V. P. Shestopalov, Radio Eng. Electron. Phys. 10, 1059 (1965).

Smith, S. J.

S. J. Smith and E. M. Purcell, Phys. Rev. 92, 1069 (1953).
[Crossref]

Stewart, J. E.

Tako, T.

K. Ishiguro and T. Tako, Opt. Acta 8, 25 (1961).
[Crossref]

Toraldo di Francia, G.

G. Toraldo di Francia, Nuovo Cimento 16, 61 (1960).

Tret’yakov, O. A.

O. A. Tret’yakov, S. S. Tret’yakov, and V. P. Shestopalov, Radio Eng. Electron. Phys. 10, 1059 (1965).

Tret’yakov, S. S.

O. A. Tret’yakov, S. S. Tret’yakov, and V. P. Shestopalov, Radio Eng. Electron. Phys. 10, 1059 (1965).

van den Berg, P. M.

P. M. van den Berg and O. J. Voorman, Appl. Sci. Res. 26, 175 (1972).
[Crossref]

P. M. van den Berg, Appl. Sci. Res. 24, 261 (1971).

P. M. van den Berg, Thesis, Delft University of Technology, The Netherlands (1971).

Voorman, O. J.

P. M. van den Berg and O. J. Voorman, Appl. Sci. Res. 26, 175 (1972).
[Crossref]

Voskresenkii, G. V.

B. M. Bolotovskii and G. V. Voskresenkii, Usp. Fiz. Nauk 94, 377 (1968) [Sov. Phys.-Usp. 11, 143 (1968)].

Appl. Opt. (1)

Appl. Sci. Res. (3)

P. M. van den Berg and O. J. Voorman, Appl. Sci. Res. 26, 175 (1972).
[Crossref]

H. Blok and G. Mur, Appl. Sci. Res. 26, 389 (1972).
[Crossref]

P. M. van den Berg, Appl. Sci. Res. 24, 261 (1971).

C.R. Acad. Sci. (Paris) (1)

J. P. Bachheimer, C.R. Acad. Sci. (Paris) 268, 599 (1969).

Can. J. Phys. (1)

A. Hessel, Can. J. Phys. 42, 1195 (1964).
[Crossref]

J. Appl. Phys. (1)

C. W. Barnes and K. G. Dedrick, J. Appl. Phys. 37, 411 (1966).
[Crossref]

J. Math. Phys. (2)

J. Lam, J. Math. Phys. 8, 1053 (1967).
[Crossref]

R. D. Hazeltine, M. N. Rosenbluth, and A. M. Sessler, J. Math. Phys. 12, 502 (1971).
[Crossref]

J. Opt. Soc. Am. (2)

Nuovo Cimento (1)

G. Toraldo di Francia, Nuovo Cimento 16, 61 (1960).

Opt. Acta (1)

K. Ishiguro and T. Tako, Opt. Acta 8, 25 (1961).
[Crossref]

Phys. Rev. (1)

S. J. Smith and E. M. Purcell, Phys. Rev. 92, 1069 (1953).
[Crossref]

Radio Eng. Electron. Phys. (1)

O. A. Tret’yakov, S. S. Tret’yakov, and V. P. Shestopalov, Radio Eng. Electron. Phys. 10, 1059 (1965).

Usp. Fiz. Nauk (1)

B. M. Bolotovskii and G. V. Voskresenkii, Usp. Fiz. Nauk 94, 377 (1968) [Sov. Phys.-Usp. 11, 143 (1968)].

Other (5)

W. W. Salisbury, U. S. Patent No.2 634 372, 7April1953.

A referee has drawn the author’s attention to an unpublished NDA Report (18-8) by B. A. Lippmann, in which the same Green’s function has been derived in connection with his “variational formulation of a grating problem.”

The application of the surface impedance (Leontovich) boundary condition would introduce an extra parameter, viz., the frequency-dependent surface impedance. Since the surface-impedance boundary condition holds only in a limited frequency domain, we cannot represent the field quantities as Fourier integrals.

Note that this Green’s function is the same as the one used by B. A. Lippmann in NDA Report No. 18-8.

P. M. van den Berg, Thesis, Delft University of Technology, The Netherlands (1971).

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 (10)

Fig. 1
Fig. 1

Cross section of the configuration.

Fig. 2
Fig. 2

Dispersion diagram showing the interaction of the moving line charge with the reflection grating. For example, the n = −1 wave radiates when ω2 <ω <ω1.

Fig. 3
Fig. 3

Domain to which Green’s theorem is applied.

Fig. 4
Fig. 4

Moving line charge above a sinusoidal grating.

Fig. 5
Fig. 5

(a) Domain of radiating waves, (b) and (c) radiated intensity RmRm* of the Smith–Purcell radiation of order −m (m = 1, 2, 3, 4), (d) the factor exp{2iγ0(z0zmax)} as a function of D0 (c0/υ0 = 4; h/D = 0.1 and 0.5, respectively).

Fig. 6
Fig. 6

(a) Domain of radiation waves, (b) and (c) radiated intensity RmRm* of the Smith–Purcell radiation of order −m (m = 1, 2, 3, 4, 5), (d) the factor exp{2iγ0(z0zmax)} as a function of D0 (c0/υ0 = 2; h/D = 0.1 and 0.5, respectively).

Fig. 7
Fig. 7

(a) Domain of radiating waves, (b) and (c) radiated intensity RmRm* of the Smith–Purcell radiation of order −m (m= 1, 2, 3, 4, 5), (d) the factor exp{2iγ0(z0zmax)} as a function of D0 ( c 0 / υ 0 = 2; h/D = 0.1 and 0.5, respectively).

Fig. 8
Fig. 8

Radiated intensity R−1R−1* in the (−1)st order radiating wave as a function of the angle of emergence θ−1 (c0/υ0 = 4; h/D = 0.1, 0.25, 0.5, 0.75, and 1.0, respectively).

Fig. 9
Fig. 9

Radiated intensity R−1R−1* in the (−1)st order radiating wave as a function of the angle of emergence θ−1 (c0/υ0 = 2; h/D = 0.1, 0.25, 0.5, 0.75, and 1.0, respectively).

Fig. 10
Fig. 10

Radiated intensity R−1R−1* in the (−1)st order radiating wave as a function of the angle of emergence θ−1 ( c 0 / υ 0 = 2; h/D = 0.1, 0.25, 0.5, 0.75, and 1.0, respectively).

Tables (1)

Tables Icon

Table I Results pertaining to (4∊0/q2)Wm, in which Wm is defined by Eq. (44).

Equations (45)

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

E i ( x , z , t ) = ( 2 π ) 1 ε i ( x , z ; ω ) exp ( i ω t ) d ω , H i ( x , z , t ) = ( 2 π ) 1 H i ( x , z ; ω ) exp ( i ω t ) d ω ,
E i ( x , z , t ) = π 1 Re [ 0 ε i ( x , z ; ω ) exp ( i ω t ) d ω ] , H i ( x , z , t ) = π 1 Re [ 0 H i ( x , z ; ω ) exp ( i ω t ) d ω ] .
× H i + i ω 0 ε i = J , × ε i i ω μ 0 H i = 0 ,
J ( x , z ; ω ) = J ( x , z , t ) exp ( i ω t ) d t .
J ( x , z , t ) = q υ 0 δ ( x υ 0 t , z z 0 ) i x ,
J ( x , z ; ω ) = q exp ( i α 0 x ) δ ( z z 0 ) i x ,
α 0 ω / υ 0 = k 0 c 0 / υ 0 with k 0 = ω ( ε 0 μ 0 ) 1 2 ,
H i = U i i y , ε i = ( i ω 0 ) 1 [ z U i i x x U i i z + J x i x ] .
( x 2 + z 2 + k 0 2 ) U i = q exp ( i α 0 x ) z δ ( z z 0 ) .
U i ( x , z ; ω ) = 1 2 q sign ( z z 0 ) exp { i α 0 x + i γ 0 | z z 0 | } ,
γ 0 = i ( α 0 2 k 0 2 ) 1 2 with ( α 0 2 k 0 2 ) 1 2 0 ,
E r E E i , H r H H i .
E r ( x , z , t ) = π 1 Re [ 0 ε r ( x , z ; ω ) exp ( i ω t ) d ω ] , H r ( x , z , t ) = π 1 Re [ 0 H r ( x , z ; ω ) exp ( i ω t ) d ω ] .
× H r + i ω 0 ε r = 0 , × ε r i ω μ 0 H r = 0 .
n × ( ε i + ε r ) = 0 on Λ ,
H r = U r i y .
ε r = ( i ω ε 0 ) 1 [ z U r i x x U r i z ] ,
( x 2 + z 2 + k 0 2 ) U r = 0 .
n · U = n x x U + n z z U = 0 on Λ ,
U r ( x , z ; ω ) = n = U n r ( ω ) exp ( i α n x + i γ n z ) , when z max < z < ,
α n α 0 + 2 π n / D ( α 0 = ω / υ 0 ) , γ n ( k 0 2 α n 2 ) 1 2 with Re ( γ n ) 0 and Im ( γ n ) 0 .
α n = k 0 sin ( θ n ) , γ n = k 0 cos ( θ n ) ,
sin ( θ n ) = c 0 / υ 0 + n λ 0 / D ( n < 0 ) ,
| α n / k 0 | = | c 0 / υ 0 + n λ 0 / D | 1 .
ε r ( x , z ; ω ) = n = ε n r ( ω ) exp ( i α n x + i γ n z ) , when z max < z < , H r ( x , z ; ω ) = n = H n r ( ω ) exp ( i α n x + i γ n z ) ,
ε n r = ( γ n / ω ε 0 ) U n r i x ( α n / ω ε 0 ) U n r i z , H n r = U n r i y .
G = n = ( i / 2 γ n D ) exp { i α n ( x P x ) + i γ n | z P z | } ,
U P r = L U ( n · G ) d s when P above L ,
U n r ( ω ) = ( i / 2 γ n D ) L U ( n · ) exp ( i α n x i γ n z ) d s ( n = 0 , ± 1 , ± 2 , ) .
1 2 U P + P L U ( n · G ) d s = U P i when P on L ,
W = t 1 t 1 + D / υ 0 q υ 0 E x r ( υ 0 t , z 0 , t ) d t ,
E x r ( x , z , t ) = π 1 × Re [ n = 0 E x , n r exp ( i α n x + i γ n z i ω t ) d ω ] .
W = ( q υ 0 / π ) Re [ n = t 1 t 1 + D / υ 0 { exp ( i 2 π n υ 0 t / D ) × 0 E x , n r exp ( i γ n z 0 ) d ω } d t ] .
W = ( q D / π ) Re [ 0 ( γ 0 / ω ε 0 ) U 0 r exp ( i γ 0 z 0 ) d ω ] .
1 2 ε n r × H n r * = 1 2 U n r U n r * k n / ( ω ε 0 ) ,
2 U + k 0 2 U = 0 when z z 0 .
S { U * 2 U U 2 U * } d S = c { U * ( n · U ) U ( n · U * ) } d s
x 1 x 1 + D { U * z U U z U * } d x = 0 when z max < z < z 0 .
U = 1 2 q exp { i α 0 x i γ 0 ( z z 0 ) } + n = U n r exp ( i α n x + i γ n z ) when z max < z < z 0 .
real γ n U n r U n r * γ n = q Re [ U 0 r γ 0 exp ( i γ 0 z 0 ) ] ,
W = ( D / π ) 0 real γ n U n r U n r * ( γ n / ω ε 0 ) d ω ,
W = ( 2 D / π ) 0 radiating waves 1 2 ( ε n r × H n r * ) · i z d ω .
W = m = 1 W m ,
W m ( 2 / 0 ) m / ( c 0 / υ 0 + 1 ) m / ( c 0 / υ 0 1 ) U m r U m r * × cos ( θ m ) d ( D / λ 0 ) .
U m r ( ω ) = 1 2 q exp { i γ 0 ( z 0 z max ) } R m ( ω ) ( m = 1 , 2 , 3 , ) .