DEVELOPMENT OF THE THEORY OF DIFFRACTION RADIATION FROM SURFACES WITH FINITE CONDUCTIVITY
K. O. Kruchinin and D. V. Karlovets UDC 537.8
A solution of the problem on diffraction radiation arising at inclined passage of a point-like charged particle near the thin rectangular screen of finite sizes, finite conductivity, and frequency dispersion is found. The problem on diffraction radiation arising at inclined passage of a charge through a rectangular slit in a conductive screen is solved. For zero slit width, the results obtained completely coincide with the corresponding results of the theory of transition radiation at arbitrary values of the dielectric permittivity and angle of particle incidence on the screen surface. In addition to the diffraction mechanism, expressions obtained for the radiation intensity describe the Cherenkov radiation mechanism.
Diffraction radiation (DR) arises when a charged particle passes near an optical inhomogeneity, for example, when a charge moves in vacuum near a conductive screen [1]. The given phenomenon is physically related to Cherenkov’s radiation and transition radiation, because it arises due to dynamic polarization of atoms in the medium upon exposure to the fast charged particle field. Particle energy losses on polarization radiation are usually considered negligibly small in comparison with the total particle energy; therefore, the particle motion can be considered uniform and rectilinear. Ideas of DR application for non-invasive diagnostics of accelerator beams [2] are discussed.
Until recently, theoretical DR investigations have been limited mainly to a consideration of ideally conductive surfaces with application of methods of plane wave diffraction theory. These methods are approximate for the problem on diffraction of a non-transverse wave (that is, a charge field) [3]. In [4, 5], some DR problems have been solved for the case when the dielectric permittivity of the screen slightly differs from unity: ( ) 1 1ε ω − , for example, for frequencies ω that exceeded the plasma frequency. In the present work, a method is developed that allows the DR characteristics to be calculated for targets of finite sizes and finite conductivity. In particular, radiation is considered arising at inclined passage of a particle near a rectangular screen and through a slit in the screen of finite dielectric permittivity. It is essential that when the slit width tends to zero, the expression for the radiation intensity completely coincides with the result of the theory of transition radiation for arbitrary incidence angle.
DIFFRACTION RADIATION FROM THE RECTANGULAR SCREEN
Let us consider the problem on radiation arising at inclined passage of the point-like charged particle with
energy 2 2/ 11/E mc =γ = −β , where /v cβ = is the relative particle velocity, near the rectangular screen having finite dielectric permittivity and frequency dispersion (see Fig. 1).
National Research Tomsk Polytechnic University, Tomsk, Russia, e-mail: [email protected]; [email protected]. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 10–16, January, 2012. Original article submitted February 2, 2011.
To determine the radiation characteristics, we take advantage of the method of polarization currents described in details in [6]. The essence of the method is that the polarization current is induced in the medium by the charge moving uniformly and rectilinearly whose density in the non-magnetic medium has the form
( )pol0pol pol( , ) ( ) + ( )ω = σ ωj r E E j . (1)
Here 0E is the field of the charge moving uniformly and rectilinearly in vacuum, and polE is the field of polarization
radiation. The conductivity of the medium (target) is linearly related to the dielectric permittivity: ( ) ( ( ) 1) / 4 iσ ω = ε ω − ω π . Using the given approach, the radiation field in the far field zone can be obtained as a solution
of the vacuum Maxwell equations with polarization current on the right side [6]:
(
p)
o 0 '/
3l 2( , ) ' (( ' )) ,T
ci
V
i ri e r ec r
dε ω ω
−× σ ω ωπ
ω = ∫ krH r k E r . (2)
Here the wave vector in the medium is / ( )cω ε= ωk e [7] and / r=e r . We note that consideration of the second term on the right side of Eq. (1) has led only to the replacement of the vacuum wave number / cω in the exponent by
( ) / cε ω ω . Integration in Eq. (2) is performed only over the region occupied by the polarization current (over the target volume in Fig. 1). In the examined case, Eq. (2) can be rewritten as follows (the screen size along the x axis is assumed infinite):
' 'p( ) /
o
0
l ' ' ( ) ( )2( , ) , ', ', y zc d h a ik
i
x
r
h
y ik zdz di e y zc r
y k eε ω ω +
− −× σ ωπ
ω = ω∫ ∫H k Er . (3)
The Fourier components of the particle field entering into Eq. (3) can be found from the total Fourier transform of the field
2 2
22 2 2 ( )( , )
2ieq q q q
cq
c⎛ ⎞ ⎛ ⎞ω ω
ω − δ ω−⎜ ⎟ ⎜ ⎟π ω⎝
= ⋅ ⋅⎠ ⎝ ⎠
− ⋅vE v v , (4)
Fig. 1. Scheme of DR generation from the screen.
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taking advantage of the inverse Fourier transform, where sin , cos0, v vα= − αv , q is the momentum variable, ( )qδ ω− ⋅v is the Dirac delta-function. As a result, we obtain the Fourier field component of the charge moving
uniformly and rectilinearly (the frequency dependence of the dielectric permittivity is implied below):
2sign( t
0 12
2 1 2
sin cos' ) 1 ( ) ) '' cos ' sian sign( t nn a
sign( tan
sign( tan
1( , ', ', ) , sin )2 1 ( )
cos 1 ( ) , cos )sin 1 ( )
x
x xx
x x
z y z e y z yiz iyv vv v
iek y z e i y zv e
e i y z e
e e e e
−
−
ω α ω αω ω′ ′ ′ ′− + α +ε βγ − + αα − αγ
′ ′ω = − βγ ε −γ α + + απ + ε βγ
′ ′× α + ε βγ γ α + + α α + ε βγ
×
E
21 ( ).
xe+ε βγγ
(5)
Here the functions with signs give “+” if the target is located below the particle trajectory and “–” if it is above the trajectory. In the case under consideration, the screen is located below the charge trajectory for backward radiation (see Fig. 1). Substituting Eq. (5) into Eq. (3), we obtain for the radiation field
( )
1 2
2 2
sincos 1 ( )/
pol1 2
cos 1 ( ) sin cos 1 ( )
2 2
1( , ) ( 1)4 cos sin 1 ( )
1
1 ( ) cos 1 ( )
z x
x y x
id e i ei r c c
z x
a e i e i h ev v
x x y
e e ec r e i e
e e
e e i e i
−ω α⎛ ⎞β α− ε + +ε βγ⎜ ⎟ε ω βγ⎝ ⎠
−
ω ω− α +ε βγ + βγ ε + γ α − α +ε βγ
γ γ
βγ −ω = ε ε −
π α −β ε + γ α + ε βγ
⎛ ⎞⎜ ⎟−⎜ ⎟⎝ ⎠×
+ ε βγ α + ε βγ + βγ ε + γ
H r h
( )sin
,sin
yih ev ceω ω⎛ ⎞− α+ ε⎜ ⎟
⎝ ⎠
α
(6)
where we have used the following designations:
( ) ( )
( ) ( )
1 2 1 2
1 2 2 1
cos sin 1 ( ) sin cos 1 ( ) ,
cos sin 1 ( ) , cos 1 ( ) sin .
x y x z
z x x x y x
i e e i e e
e i e e i e e e
− −
− −
= γ α + α + ε βγ + γ α − α + ε βγ
βγ ε − γ α − α + ε βγ α + ε βγ − γ α −βγ ε
h (7)
This formula determines the total field of diffraction radiation in the medium. The components of the vector e of backward radiation entering into the above relations are expressed through the polar angle mθ in the medium as follows: sin sin ,sin cos , cos m m m= θ φ θ φ − θe . To determine the radiation field in vacuum, it is impossible to use directly the Fresnel refraction laws, because for good conductors, radiating dipoles are concentrated near the interface, and the field near the surface does not correspond to the wave zone. To this end, the reciprocity theorem [7]
pol(vac) (vac) pol( ) ( )m m=⋅ ⋅E d E d (8)
can be used, where pol(vac)E is the thought-after radiation field in vacuum created by the dipole with moment d located in the medium, and pol( )mE is the radiation field in the medium created by the same dipole located in vacuum far from the interface. In our case, we consider that the dipole moment d is perpendicular to the interface, that is, is directed along the z axis. We neglect the wave refraction by the screen edges; to this end, we consider the screen to be thin: d a<< . Here we also note that possible multiple re-reflections of waves inside of the plate are disregarded, which is valid only for conductive media. Solving Eq. (8), for energy density dW radiated into unit element of the solid angle dΩ in unit frequency range dω we obtain (see [6] for more details)
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( )2 2 2 22 22 pol pol2 pol(vac)2 H E
d W crcr f H f Hd d ⊥= = + εω Ω ε
E . (9)
Here
pol polpol pol pol 2 pol pol 2cos sin , ( ) ( sin cos )x y z x yH H H H H H H⊥ = φ− φ = + φ+ φ (10)
are components of magnetic field (6) perpendicular and parallel to the plane of wave incidence on the interface, and Hf and Ef are the Fresnel coefficients for one infinite interface. To determine the radiation intensity in vacuum using Eq. (9), the radiation angles in the medium must be expressed through the angles in vacuum [7]:
21 sin sin ,sin cos , sin .= θ φ θ φ − ε − θε
e
This expression is written for backward radiation; the negative
z-component corresponds to it. As a result, we obtain the expression for the spectral and angular density of backward diffraction radiation:
Here coshχ = α is the shortest distance from the screen edge to the particle trajectory (the impact parameter). The formula for forward DR cannot be obtained by simple replacement of the sign before the particle velocity by analogy, for example, with transition radiation (TR) (see [6] for more details). In this case, the problem is complicated by the fact that the particle trajectory can intersect the screen in this position. The radiation field is then determined from Eq. (3),
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with the only difference that integration over z is performed in the limits from –d to 0 and as before, radiation enters the vacuum through the plane z = 0.
Let us consider the case when the particle does not intersect the screen. In this case, the condition cotd h< α is satisfied. Calculating field (3) and substituting it into the formula for the intensity, we obtain the expression for the spectral and angular density of forward DR in the case when the particle does not intersect the screen:
Here the impact parameter is related to the distance to the coordinate origin h as follows: cos sinh bχ = α − α , because we must consider that radiation enters the vacuum through the plane z = 0. Unlike backward DR given by Eq. (11), the expression for forward DR contains Cherenkov’s radiation in addition to diffraction one. The former has the intensity
pole where ( )2 1 2Re cos sin sin 1 ( sin sin ) 0i −α −β ε − θ + γ α + βγ θ φ = . As 0α→ , this expression is transformed
into the well-known Cherenkov condition written in terms of vacuum variables. Setting 0α = in formulas (11) and (12), we obtain expressions completely coinciding with those derived in [6].
Going to the limit of an ideal conductor ( ε → ∞ ), we obtain the expression for the backward DR intensity from ideally conductive half-plane:
( )2 2
2 2 2 2 2 22 2 1 cos sin (1 sin sin ) 2 sin cos sin
4
ad W e
d d c
→∞
ε→∞
⎡= −β θ+β α − θ φ + β α φ θ⎣ω Ω π γ
( ) (12 2 2 2 2 2 2 2 2 21 ( sin sin ) sin (sin cos sin sin ) cos (cos sin cos )−⎤× + βγ θ φ θ α φ−βγ θ φ + γ α θ+ θ φ⎦
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( ) ( ))2
12 1 ( sin sin )
2 2 2 2 21 ( sin sin ) cos sin ( sin sin ) .veω
−χ + βγ θ φγ× + βγ θ φ + θ α + βγ θ φ (13)
This formula coincides with the formula obtained in [8] by another method. Let us consider some special features of radiation from the target of finite dielectric permittivity. For
transparent substance at small incidence angle α , given that the Cherenkov condition is satisfied, the DR intensity is small, and Cherenkov’s radiation gives the main contribution.
With increasing incidence angle, the DR intensity starts to increase, and the Cherenkov radiation maximum is displaced toward larger angles (see Fig. 2).
DIFFRACTION RADIATION FROM THE SLIT IN A RECTANGULAR SCREEN
The problem close to that considered above is the determination of radiation arising at inclined passage of a charged particle through the slit of width 2h in the rectangular screen of arbitrary conductivity. This problem was considered previously in [8] only in the approximation of ideal conductivity. An analysis for the case of the target having arbitrary conductivity and finite sizes was not performed.
Generally, when the particle trajectory can intersect the screen, by analogy with Eq. (4), we obtain for the radiation field
Here we have used the following designations: 1 2cos sin 1 ( )z xe i e−= α −β ε + γ α + ε βγ , sinyi e iξ = − βγ ε − γ α
2cos 1 ( )xe− α + ε βγ , ( )( )1cos 1 sin cosy ze e−Δ = α +β ε α − ε α , and Θ is a unit step function.
Generally, the expression for the spectral and angular radiation density is sufficiently cumbersome, and we do not write it. For concrete calculations, it is better to use formula (14) and any standard mathematical code. For zero slit width ( 0)b → and infinite screen sizes ( , )a d→∞ →∞ , Eq. (14) substituted into formula (9) for the intensity yields the well-known expression
( )( )( )
( )(
)
2 2 2 222 2 22
22 2 2 22
2 2 2 2
2 22 2 2
2
cos cos 1 1 sin sin cos cos sin
11 sin sin cos cos cos sincos sin
1 cos sin cos sin sin cos sin
1sin cos sin cos sin scos sin
d W ed d c
−
−
β= α θ ε − +β α θ φ+β α ε − θ
ω Ω π
⎡× +β α θ φ −β α θ θ ⎢
⎢ ε θ + ε − θ⎣
× +β α ε − θ −β α +β α θ φ θ
+β α α θ φ ε − θ + βθ+ ε − θ
( )2in cos sin cos i ,s nα⎤⎥⎥⎥⎦
α θ θ φ
(15)
derived by Pafomov [9] using another method. We note that formula (14) is valid in the most general case of particle passage through the slit. However, bearing in mind practical applications, we now consider the case when the particle does not intersect the screen. In our case, this corresponds to the condition cotd h< α at which field (14) assumes the following form:
( )
* **
/pol
2
*
* *
1( , ) 14 1 ( )
1 1 1 1 .
i r c
x
a h a hid idv v v vv v
e ec r e
e e e e e e
ε ω
ω ω ω ωω ω− ξ − ξ − ξ − ξ−γ γ γ γ
βγω = ε ε −
π + ε βγ
⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟× − − + − −
⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟ξ ξ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠
H r
h h (16)
In the limit of ideal conductivity ( Imε →∞ ), expression (16) substituted into formula (9) for the intensity yields the result exactly coinciding with that obtained in [7] by another method.
Since the examined problem is similar to that considered above, the intensity of DR from the slit has the characteristic pattern with a minimum in the direction of particle motion. Consideration of finite dielectric permittivity leads to the fact that the curve of angular DR distribution becomes oscillatory in character for media with weak absorption. In the limit of ideal conductivity, the curve has the characteristic pattern with two maxima at angles 1~ −θ γ .
This work was supported in part by the Special Federal Program of the Ministry of Education and Science of the Russian Federation “Scientific and Pedagogical Personnel of Innovative Russia” for 2009–2013, item No. 1.3.1 (State Contract No. P1199, Code “NK-653P”).
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