Abstract

A rigorous solution is obtained for the problem of radiation from an electric point charge that moves, at a constant speed, parallel to an electrically perfectly conducting grating. The relevant vectorial electromagnetic problem is reduced to two two-dimensional scalar ones. A Green’s-function formulation of the two problems is employed. For both cases, an integral equation of the second kind for the remaining unknown function is derived. This integral equation is solved numerically by a method of moments. Some numerical results for the radiation from a moving point charge above a sinusoidal grating are presented; in particular, the power losses of the point charge have been estimated.

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References

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  1. S. J. Smith and E. M. Purcell, Phys. Rev. 92, 1069 (1953)
    [CrossRef]
  2. P. M. van den Berg, J. Opt. Soc. Am. 63, 689 (1973).
    [CrossRef]
  3. G. Toraldo di Francia, Nuovo Cimento 16, 61 (1960).
    [CrossRef]
  4. J. Lam, J. Math. Phys. 8, 1053 (1967).
    [CrossRef]
  5. E. V. Avdeev and G. V. Voskresenskii, Radiotekh. Elektron. 11, 1419 (1966).
  6. B. M. Bolotovskii and G. V. Voskresenskii, Usp. Fiz. Nauk 94, 377 (1968) [Sov. Phys.-Usp. 11, 143 (1968)].
    [CrossRef]
  7. B. A. Lippmann, J. Opt. Soc. Am. 43, 408 (1953).
    [CrossRef]
  8. P. M. van den Berg, Appl. Sci. Res. 24, 261 (1971).
  9. E. Lalor, Phys. Rev. A 7, 435 (1973), presented a theory of three-dimensional Smith–Purcell radiation, based upon a combination of Green's-function theory and perturbation theory, valid as long as the wavelength is large compared with the local radius of curvature of the grating surface.
    [CrossRef]

1973

P. M. van den Berg, J. Opt. Soc. Am. 63, 689 (1973).
[CrossRef]

E. Lalor, Phys. Rev. A 7, 435 (1973), presented a theory of three-dimensional Smith–Purcell radiation, based upon a combination of Green's-function theory and perturbation theory, valid as long as the wavelength is large compared with the local radius of curvature of the grating surface.
[CrossRef]

1971

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

1968

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

1967

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

1966

E. V. Avdeev and G. V. Voskresenskii, Radiotekh. Elektron. 11, 1419 (1966).

1960

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

1953

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

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

Avdeev, E. V.

E. V. Avdeev and G. V. Voskresenskii, Radiotekh. Elektron. 11, 1419 (1966).

Bolotovskii, B. M.

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

Francia, G. Toraldo di

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

Lalor, E.

E. Lalor, Phys. Rev. A 7, 435 (1973), presented a theory of three-dimensional Smith–Purcell radiation, based upon a combination of Green's-function theory and perturbation theory, valid as long as the wavelength is large compared with the local radius of curvature of the grating surface.
[CrossRef]

Lam, J.

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

Lippmann, B. A.

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

Purcell, E. M.

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

Smith, S. J.

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

van den Berg, P. M.

P. M. van den Berg, J. Opt. Soc. Am. 63, 689 (1973).
[CrossRef]

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

Voskresenskii, G. V.

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

E. V. Avdeev and G. V. Voskresenskii, Radiotekh. Elektron. 11, 1419 (1966).

Other

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

P. M. van den Berg, J. Opt. Soc. Am. 63, 689 (1973).
[CrossRef]

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

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

E. V. Avdeev and G. V. Voskresenskii, Radiotekh. Elektron. 11, 1419 (1966).

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

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

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

E. Lalor, Phys. Rev. A 7, 435 (1973), presented a theory of three-dimensional Smith–Purcell radiation, based upon a combination of Green's-function theory and perturbation theory, valid as long as the wavelength is large compared with the local radius of curvature of the grating surface.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Geometry of the configuration. Trajectory of point charge q is at z0 above mean plane (z = 0) of grating.

Fig. 2
Fig. 2

The direction of emergence of a radiating wave of spectral order n.

Fig. 3
Fig. 3

Dispersion of diagram showing the interaction of the moving point charge with the reflection grating. A wave of spectral order n radiates when αn2 + β2k02.

Fig. 4
Fig. 4

Domain to which Green’s theorem is applied. Grating surface is indicated by the irregular profile.

Fig. 5
Fig. 5

Moving point charge above sinusoidal grating.

Fig. 6
Fig. 6

Radiated intensity R−1R−1* of the (−1)st-order radiation as a function of the angles of emergence η−1, ζ−1 (c0/υ0 = 4, h/D = 0.1).

Fig. 7
Fig. 7

Radiated intensity R−1R−1* of the (−1)st-order radiation as a function of the angles of emergence η−1, ζ−1 (c0/υ0 = 4, h/D = 0.5).

Fig. 8
Fig. 8

Radiated intensity R−1R−1* of the (−1)st-order radiation as a function of the angles of emergence η−1, ζ−1 (c0/υ0 = 4, h/D = 1.0).

Fig. 9
Fig. 9

Radiated intensity R−1R−1* of the (−1)st-order radiation as a function of the angles of emergence η−1, ζ−1 ( c 0 / υ 0 = 2, h/D = 0.1).

Fig. 10
Fig. 10

Radiated intensity R−1R−1* of the (−1)st-order radiation as a function of the angles of emergence η−1, ζ−1 ( c 0 / υ 0 = 2, h/D = 0.5).

Fig. 11
Fig. 11

Radiated intensity R−1R−1* of the (−1)st-order radiation as a function of the angles of emergence η−1, ζ−1 ( c 0 / υ 0 = 2, h/D = 1.0).

Tables (2)

Tables Icon

Table I Results for D(40/q2)W−1, in which W−1 is defined by Eqs. (43).

Tables Icon

Table II Results for DW−1/Ekin, in which W−1 is defined by Eqs. (43) and Ekin is defined by Eq. (47); q = 1.6 × 10−19 C, me = 9.1 × 10−31 kg, 0 = 8.854 × 10−12 F/m.

Equations (52)

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