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Pre-wave zone effect in transition and diffraction radiation: Problems and solutions
Physics Letters A 345 (2005) 428–438 www.elsevier.com/locate/pla
Pre-wave zone effect in transition and diffraction radiation: Problems and solutions P.V. Karataev 1 KEK, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba 305-0801, Ibaraki-ken, Japan Received 2 April 2005; accepted 12 July 2005 Available online 20 July 2005 Communicated by V.M. Agranovich
Abstract Transition radiation (TR) and diffraction radiation (DR) appearing as a result of dynamic polarization of medium has widely been used for electron beam diagnostics during the last few years. A lot of techniques for electron beam diagnostics imply description of these phenomena assuming that the radiated area of the target is negligibly small in comparison with the radiation spot in the detector plane (far-field approximation). However, for high-energy electrons this area may reach a macroscopic dimension. In this Letter the general theory in the pre-wave zone is presented. Two new approaches for rejecting the pre-wave zone effect are described and analyzed. By installing a thin lens in the optical path of the measurement system or by developing a concave target, the pre-wave zone effect can be reduced or even rejected. 2005 Elsevier B.V. All rights reserved. PACS: 41.60.-m; 41.75.-i Keywords: Transition radiation; Diffraction radiation; Far-field approximation; Pre-wave zone effect
1. Introduction When a charged particle crosses an interface between two media with different dielectric constants (in ideal case when an electron crosses a vacuum—ideal conductor boundary) it induces currents changing in time [1] at the boundary. Those currents give rise in radiation called transition radiation (TR). TR has a tendency to propagate in two main directions: along the particle trajectory—forward transition radiation (FTR) and along the direction of specular reflection from the boundary—backward transition radiation (BTR). BTR has been widely used for different purposes because it allows registering the radiation at fine background conditions. The BTR application E-mail address: [email protected] (P.V. Karataev). 1 Tel.: +81 298 645715; fax: +81 298 640321.
0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.07.027
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range is very broad: from transversal beam parameter measurements in optical wavelength range [2,3] upto the bunch length measurement and generation of intense radiation in mm and sub-mm wavelength range [4,5]. During the last few years the electron beam energies in the modern accelerators was incredibly increased. As a result new problems appeared, which were considered as negligibly small for moderate relativistic electron energies. One of them is the pre-wave zone effect in transition radiation. In simple words the pre-wave zone is the distance from the target, where the contribution of the radiation source size into the BTR spot size in the detector plane is significant and cannot be neglected. Afterwards, this effect has been considered in details by many theoreticians using different approaches [6–10]. However, all of them agreed that in order use the far-field zone approximation (the target radiating area is negligibly small), the distance from the target to detector must be larger than the parameter γ 2 λ, where γ is the charged particle Lorentz-factor and λ is the radiation wavelength. Later the pre-wave zone effect was observed experimentally [11]. For example, at SLAC FFTB [2] the parameter γ 2 λ = 1.8 km, which is really difficult to achieve. Moreover, when measuring the long wavelength transition radiation (where the BTR is coherent, i.e. the radiation wavelength is comparable to or smaller than the electron bunch length) [4,5] the pre-wave zone effect could be significant at lower energies. Recently diffraction radiation (DR) appearing when the charged particle moves in the vicinity of a medium with impact parameter smaller than γ λ/2π (effective electron field radius) has been suggested as a possible tool for non-invasive beam diagnostics [12–16]. DR is a relative effect to TR, because it is also produced as a result of dynamic polarization of medium. The backward diffraction radiation (BDR) in optical wavelength range has been measured and applied for transversal beam size diagnostics [17–20]. The BDR in mm and sub-mm wavelength range has been applied for non-invasive bunch length measurements [21,22]. However, it has similar problems with the pre-wave zone effect as TR. So far nobody really knows how to deal with this effect. Some of them prefer to take into account the pre-wave zone effect [22] when comparison of the experiment with the theory is performed. However, the theory is very complicated for practical use. Moreover, in some cases the sensitivity of the far-field BTR and BDR to the electron beam parameters is higher. This Letter describes the classical backward transition and diffraction radiation theory in far-field zone and in the pre-wave zone. The choice of backward radiation was made because in this case it is not necessary to take into account the self-field of the electron. It has been shown that the far-field theory is just a particular case of the pre-wave zone theory. Two possible ways for pre-wave zone suppression are considered. By installing a lens in the optical path or by choosing a concave target it might be possible to describe the BTR and BDR characteristics using the formulas obtained from the far-field theory.
2. Far-field approach In this Letter I shall use the classical theory of backward transition radiation (BTR) and backward diffraction radiation (BDR) based on Huygen’s principle of plane wave diffraction [1]. In this theory the particle field in introduced as superposition of its pseudo-photons. When the particle interacts with the target surface, the pseudophotons are scattered from it converting into real ones and propagate in the direction of specular reflection. The difference from the plane wave is that the electron field strength depends on the distance from the particle. Let an electron move along the z-axis which is perpendicular to an ideally conducting target plane. At the zero time moment (z/β = 0, where β = v/c is the electron velocity in units of the light velocity) the electron crosses the boundary. Each point of the target surface can be represented as an elementary source. In this case two polarization components of TR field can be represented as a superposition of the waves from all elementary sources at certain distance: 1 eiϕ l i dys dxs . = (x , y ) E Ex,y (1) s s x,y r 4π 2
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This is the general equation for calculating BTR and BDR characteristics. Here x and y indexes indicate two polarization components perpendicular to each other; xs = ρs sin ψs and ys = ρs cos ψs are the coordinates of the particle field (or the position of an arbitrary elementary source at the target surface), ρs and ψs are the radius i represents the amplitude and azimuthal angle of the particle field respectively in polar coordinate system; Ex,y of an arbitrary elementary source at the target surface determined by the coordinates xs and ys ; ϕ determines the phase advance of the photons emitted by each elementary source, and r is the distance from an arbitrary elementary source on the target to the observation point. One should note that in principle there is a longitudinal, z, polarization component. However, in a relativistic case it is much smaller than the transversal one and can be neglected. i can be represented as the Fourier transform of the incident particle field [1]: The amplitude Ex,y kx,y exp[i(kx xs + ky ys )] k ie iek cos ψs i Ex,y (xs , ys ) = (2) K1 ρs . dkx dky = πγ sin ψs γ 2π 2 kx 2 + ky 2 + k 2 γ −2 Here kx,y are the components of the electron pseudo-photon wave vector, k = 2π/λ is the modulus of the radiation wave vector, λ is the BTR wavelength, γ is the charged particle Lorentz-factor, K1 is the first order McDonald function (modified Bessel function), e is the electron charge. Throughout the Letter the system of units h = me = c = 1 is used. In the far-field zone the distance from the target to the observation point is assumed to be so large that it is possible to introduce the radiation field as a superposition of the plane waves of different amplitude emitted by each elementary source of the target. In other words, it is possible to use Fraunhofer diffraction theory [23]. In this case the phase, ϕ, can be represented as
ϕ = −(r ρs ) = −xs kx − ys ky .
(3)
Here kx ≈ kθx and ky ≈ kθy are the components of the BTR field and θx and θy are the observation angles measured from the direction opposite to the particle trajectory [1]. When the observation angle differs from zero the path of the photons propagating from the target to the detector increases, which causes the phase delay. Because of that the phase (3) is negative. Substituting Eqs. (2) and (3) in Eq. (1) we have the general expression for calculating BTR and BDR characteristics in far-field zone kx,y exp[i((kx − kx )xs + (ky − ky )ys )] 1 l dkx dky dys dxs . Ex,y = (4) 8π 4 a kx 2 + ky 2 + k 2 γ −2 Here a ≈ |r| is the distance from the target to the observation plane. This equation has been used in many papers [1,12–20] for deriving equations describing BTR and BDR characteristics in far-field zone from different target types. One might be able to see the beauty of this approach. It allows to obtain very simple equations, which are flexible to other mathematical transformations like convolution with Gaussian distribution (to take into account transversal electron beam dimensions or angular divergence) or averaging over the detector angular and energy acceptance, etc. For example, integrating over xs and ys (target surface) from minus infinity to infinity (infinite target) one may obtain a multiplication of two delta functions like 4π 2 δ(kx − kx )δ(ky − ky ). Afterwards the integration over the electron field wave vector can be done trivially, and Eq. (4) is transformed into the expression for BTR fields from a particle crossing an infinite vacuum-ideal conductor interface TR Ex,y (θx , θy ) =
θx , θy ie . 2 2 2π ak θx + θy2 + γ −2
(5)
The BTR spectral angular distribution can be calculated as 2 2 θx2 + θy2 d 2 W TR α = 4π 2 k 2 a 2 ExTR + EyTR = 2 2 . dω dΩ π (θx + θy2 + γ −2 )2
(6)
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The middle part of Eq. (6) is a conventional equation for calculating spatial-spectral distribution, i.e. the power distribution per unit space interval (solid angle in far-field zone) and per unit photon energy interval. In the future I shall reference on this equation when dealing with the spatial-spectral characteristics. One may see that for an ideal conductor BTR is an azimuthally symmetric (crater like) distribution independent of the photon energy with TR = αγ 2 /4π 2 , where α is the fine structure the characteristic polar angle of γ −1 and with maximum intensity Imax constant. However, in a real experiment it is possible to use this equation for the BTR photon energies smaller than the plasmon energy of the target material, which is about 10 eV for a metallic foil. In [24] the author derived the same expression by applying an ultrarelativistic approximation to the solution of Maxwell equations; however, the approach presented in this section is much simpler. Therefore, in the rest of the Letter I shall lead all the considerations to this simple approach.
3. Pre-wave zone approach The far-field approach presented in the previous section is applicable if the distance from target to the observation point is very large. The question is: how large should it be? In this section I shall try to answer this question from the points of views of physics, mathematics and common sense. When the particle crosses the boundary, it induces radiation source, which size is equal to the electron field radius. It is well known that the effective electron field radius is treated as γ λ/2π [1] (see Fig. 1). At larger distances from the particle the electron field amplitude is significantly suppressed. Each point of the target surface emits radiation with divergence of order of γ −1 . Therefore, in order to reduce the contribution from the radiation source into the radiation spot size it is necessary to choose the distance from the target obeying the following condition γλ γ 2λ a ⇒a . γ 2π 2π
(7)
The radiation spatial-spectral distribution distortion happens at the distances (a) comparable or smaller than the parameter γ 2 λ/2π . This peculiarity is called as pre-wave zone effect. Let us discuss the mathematical approach. To perform the calculation it is necessary to change the phase advance (ϕ) in Eq. (1). From simple geometry the phase advance of the photons propagating from the target to the observation point is determined by the following relation eik|r | exp(ik a 2 + (xs − ξ )2 + (ys − η)2 ) eiϕ = = . (8) |r | |r | a 2 + (xs − ξ )2 + (ys − η)2
Fig. 1. Geometry of the BTR production at a finite distance (a) from the target. Here r1 , r2 , . . . , rN are the arbitrary vectors of the photons propagating from different target points into the same point in the detector plane.
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Here ξ = ρl cos ψl and η = ρl sin ψl are the coordinates of the observation plane and their polar representation. Actually the parameters (ξ − xs )/a and (η − ys )/a determine the angles of the photon emission by an arbitrary elementary source. In ultrarelativistic case those parameters are of order of γ −1 1. Therefore, Eq. (8) can be significantly simplified
k
eika k 2 k 2 eiϕ 2 2 ≈ exp x + ys − (xs ξ + ys η) + ξ +η . (9) |r | a 2a s a 2a Eq. (9) coincides with the one obtained in [7,8] where the authors considered the spatial resolution of optical transition radiation. However, the authors did not consider the pre-wave zone criterion in details from the viewpoint of mathematics. Moreover, Eq. (8) represents the phase relations in general form, which is more convenient for understanding of the approach. The third term in the square brackets is unimportant, because it independent of the integrals. One may see that the second term is very similar to Eq. (3). Actually this term is responsible for the Fraunhofer diffraction [23]. Therefore, in order to achieve the far-field approach it is necessary to take out the first term. Actually this term (where the target surface coordinates are involved with square powers) is responsible for the Fresnel diffraction [23]. In other words, the first term represents the first order Fresnel corrections to the far-field approximation. This term can be neglected if it is much smaller than unity. The coordinates xs and ys determine the position of an elementary source at the target, which is determined by the electron field radius. At the distances from the particle trajectory larger than γ λ/2π the intensity of the radiation sources is very small and can be neglected. Making the substitution xs = ys = γ λ/2π the far-field zone condition is
k γ 2 λ2 γ 2λ γ 2λ k 2 xs + ys2 = 1 ⇒ a . = (10) 2a a 4π 2 2πa 2π The condition (10) wholly coincides with Eq. (7). Therefore, if the condition (10) is fulfilled, the distance is so large that the parameters ξ/a → θx and η/a → θy , and Eq. (9) is transformed into Eq. (3), i.e. the far-field approach was derived from the pre-wave zone approach, which is a more general one. In [9] the author performed the considerations on the pre-wave zone effect in transition radiation. The author used a different approach; however, the same condition (10) was derived. Substituting Eq. (9) in (1) and using Eq. (6) one may calculate the BTR spatial-spectral characteristics in prewave zone. One should note that in the pre-wave zone Eq. (6) represents the spatial-spectral distribution, because the photons with different angles arrive in the same observation point (see, for instance, Fig. 1). The spatial distribution is transformed in to the angular distribution in the far-field zone only. It is necessary to keep in mind that the integration limits for the target surface must be much larger than the electron field radius. Otherwise, the finite outer target dimensions may cause an additional distortion [15]. Fig. 2 illustrates the calculated BTR spatial-spectral distribution in the pre-wave zone. It is obvious that the BTR characteristics differ from the far-field approach. These
Fig. 2. (a) represents the BTR spatial-spectral distribution calculated for different distances from the target: a = 10γ 2 λ/2π —solid line, a = 2γ 2 λ/2π —dashed line, a = γ 2 λ/2π —dashed–dotted line; (b) represents the BTR peak position as a function of the distance from the target.
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calculations coincide with the ones presented in [9] as well. Therefore both approaches are applicable for BTR calculation. The advantage of this approach is that it is possible to calculate diffraction radiation characteristics simply defining the integration region on the target surface. Many people are confused about the condition (10) because it coincides with the radiation formation zone distance condition, which has a completely different meaning. If an electron emits two photons at the distance comparable to or smaller than the radiation formation zone length, those two photons interfere. In other words, at the distance from the radiation origin along the particle trajectory much larger than the radiation formation zone distance the electron field and the radiation photon are completely separated. The radiation formation zone is determined by the following relation [1] Lf =
1 λ 1 βλ ≈ . 2π (1 − β cos θ ) π (γ −2 + θ 2 )
(11)
In (11) θ 2 = θx2 + θy2 is the polar observation angle, which is of order of γ −1 along the particle trajectory for FTR. Therefore, if the distance from the target along the particle trajectory a Lf ≈ γ 2 λ/2π the radiation photons and the electron field are completely separated. This condition coincides with the far-field condition (10). However, for BTR the observation angles reach large values. For example, for the case considered in this Letter the radiation propagate in the direction opposite to the particle trajectory. That means that the angle θ = π , and therefore the electron field and radiation photons are separated at the distance a λ/4π . Therefore, one should keep in mind that in spite of the radiation formation zone distance and the far-field zone distance are described by similar relations they determine completely different processes.
4. Two ways for suppression of the pre-wave zone effect In some cases, where the BTR or BDR is used for beam diagnostics, the pre-wave zone effect suppression is rather critical (for example, beam size measurement with optical diffraction radiation [17–20]). In the pre-wave zone the interference of the radiation from different parts of the target is infringed. In that case the sensitivity of the effect to the beam parameters becomes weaker. Therefore, in this section I shall consider two possible ways allowing to suppress or even reject the pre-wave zone effect. 4.1. Propagation of the BTR through a thin lens Fig. 3 illustrates a simple geometry of the photon propagation through a thin lens. If the target is placed in the back focal plane, all parallel waves moving from the target must be led to the same point in the detector plane. In this case the radiation spot in the detector plane should be proportional to the radiation angular distribution
Fig. 3. Geometry of the photon propagation through a thin lens.
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with the angles determined by the following relations: θx,y = x, y/f , where f is the lens focus, x = ρd cos ψd and y = ρd sin ψd are the coordinates of the detector plane and their polar representation. Let us derive the model describing this BTR geometry from the viewpoint of wave optics. The BTR fields on the lens surface frontal to the target can be described using Eq. (1) and the phase advance, ϕ, determined by Eq. (9). In this case the variables ξ and η are the coordinates of the lens surface and a is the target-to-lens distance. According to [23] a thin lens causes a phase delay, which is a function of its coordinates. The element of the BTR fields on the lens surface opposite to the target can be represented as
ik 2 l+ l 2 ξ +η . Ex,y = Ex,y exp − (12) 2f The BTR fields in the detector plane can be represented as iϕd l+ e Ex,y = (13) dξ dη × Ex,y . rd In Eq. (13) ϕd is the phase advance of the photons propagating from the lens to the detector and rd is the distance, which the photons overpass from the lens to the detector. In analogy with Eq. (8) the phase relations could be represented as
ik eik|rd | exp(ik f 2 + (x − ξ )2 + (y − η)2 ) ik 2 1 eiϕd 2 = = ξ + η − (ξ x + ηy) . ≈ exp (14) |rd | |rd | f 2f f f 2 + (x − ξ )2 + (y − η)2 The phase terms independent of the integrals are omitted for simplicity. The full equation for calculating the BTR characteristics behind a thin lens can be represented in the form 1 i Ex,y = (15) Ex,y (xs , ys )G(xs , ys ) dys dxs , 4π 2 af where k i 2a (xs2 +ys2 )
k
−ikξ( xs + x ) −ikη( ys + y )
a f e a f dξ dη × ei 2a (ξ +η ) e k i k (x x+y y) i k [ξ −(xs + fa x)]2 +i 2a [η−(ys + fa y)]2 =e f s s . dξ dη × e 2a
G(xs , ys ) = e
2
2
(16)
If we assume that the lens is infinite the double integral part in Eq. (16) is reduced to a constant and the factor G is i2a i fk (xs x+ys y) e . G∞ = (17) πk Substituting Eq. (17) in (15) one may obtain the expression similar to Eq. (4) in Section 2, i.e. similar to the Fraunhofer diffraction case. That means that a simple lens can reject the pre-wave zone effect. Here, an approximation of an infinite lens was used. Of cause, a real lens has a definite size, and diffraction on it can significantly distort the BTR spatial-spectral distribution. Therefore, at the end of this section I shall introduce a simple condition where the infinite lens approximation can be used. Obviously, the lens dimension must be larger than the radiation source size. Otherwise, the diffraction of the photons from the source tails may significantly distort the BTR spatial distribution. Furthermore, each radiation source emits radiation with divergence of order of γ −1 . Therefore, the condition for an infinite lens dimension can be formulated as Rl
γλ a + . γ 2π
(18)
Here, Rl is the lens radius. In Eq. (18) the first term takes into account the effect of the BTR spot increasing as a function of the distance from the source, and the second term takes into account the source size. For example,
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Fig. 4. (a) BTR spatial distribution in the far-field zone calculated for γ = 2500, a = 10 m, and three radiuses of the lens: Rl = 10a/γ —dashed line, Rl = 2a/γ —solid line, and Rl = a/γ —dashed–dotted line; (b) BTR spatial distribution in the pre-wave zone calculated for a = 1 m, Rl = 5 mm, and three electron energies: γ = 10000—solid line, γ = 25000—dashed line; γ = 40000—dash-dotted line. The calculation parameters: λ = 500 nm, f = 100 mm.
for KEK-ATF [17–20] (γ = 2500, λ = 500 nm, a = 1 m) the lens radius should be Rl 0.9 mm, which could be easily achieved. It is much easier to perform numerical calculations in the polar coordinate system. The BTR fields on the detector placed in the back focal plane of a thin lens with finite dimensions can be derived by substitution of the polar coordinates in Eqs. (2), (15) and (16). I omit the simple mathematical procedure here. The result is
k k k ik 2 iek 2 cos ψd TR 2 ρs ρl ρs J1 ρl ρd exp ρ + ρl . dρs ρs K1 dρl ρl J1 Ex,y = − 2πγ af sin ψd γ a f 2a s (19) Here, J1 is the Bessel function of the first order. The integration over the lens radius (over ρl ) should be performed from 0 to Rl . The spatial-spectral distribution can be calculated using Eq. (6). Fig. 4 illustrates the applicability of the condition (18). If the lens is in far-field zone (a γ 2 λ/2π) the first term in inequality (18) is dominant (Fig. 4(a)), however, in the strong pre-wave zone (a γ 2 λ/2π), the second term is dominant (Fig. 4(b)). If the target-to-lens distance is comparable to γ 2 λ/2π , both terms in Eq. (8) make comparable contributions. Nevertheless, Fig. 4(b) (solid line) illustrates that in spite of the fact that the lens-detector system is placed in a strong pre-wave zone, the BTR distribution has the same form as in far-field zone without the lens. 4.2. The BTR from a concave target The approach presented above is applicable for optical wavelengths. However, it is not applicable for long wavelengths (mm or sub-mm wavelength range). Moreover, even in optical wavelength range the aberrations in a real lens could be severe. Therefore, before making an experiment those conditions must be seriously analyzed. For the long wavelength region it is possible to use a concave target. In this case the target may work similar to the lens. The BTR calculation from a concave target is a little more complicated because a longitudinal component of the target appears; i.e. the electron field interacts with different target parts in different time. The calculation geometry is represented in Fig. 5. As before to calculate the BTR from a concave mirror it is necessary to change the exponential term containing the phase, ϕ, in Eq. (1). Let the electron cross the plane (xs , ys ) at the time equaled to zero ik|r|−ik a β exp[ik (a − a)2 + (xs − ξ )2 + (ys − η)2 − ik a eiϕ e β ] (20) , = = |r| |r| (a − a)2 + (xs − ξ )2 + (ys − η)2
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Fig. 5. Geometry of the BTR production from a concave (spherical) target.
where a = Rt −
Rt2
− xs2 − ys2 = Rt 1 −
x2 + y2 1− s 2 s Rt
.
(21)
Assuming that the radiating area of the concave target is much smaller than the radius of the curvature (i.e. Rt γ λ/2π ), then Eqs. (20) and (21) can be significantly simplified: eia 1 ik eiϕ ik 2 2 ≈ exp −ika + 1 + (xs − ξ ) + (ys − η) (22) |r| a β 2a 2a and a =
xs2 + ys2 . 2Rt
(23)
Substituting Eq. (23) into (22), involving ultrarelativistic case (1/β + 1 ≈ 2) and omitting the phase terms independent of the integrals in Eq. (1) we have 2
1 ik 1 1 eiϕ 2 − (xs ξ + ys η) . ≈ exp ik xs + ys − (24) |r| a 2a Rt a From Eq. (24) it is obvious that if the detector is placed in far-field zone, i.e. 1/2a ≈ 0, in this case there is no difference between concave and convex target, i.e. between Rt and −Rt . Both of them will lead to broadening of the BTR spatial distribution. The idea of this section is to obtain the approach allowing to calculate BTR and BDR in the pre-wave zone using the equations similar to the Fraunhofer diffraction theory. For that purpose it is necessary to take out the phase term depending on the squared coordinates of the electron field. It can be easily achieved by choosing the distance from the target equaled to a half of the target curvature radius (a = Rt /2). In this case Eq. (24) is transformed to the equation similar to Eq. (3), where the analogues of the angular variables θx and θy are ξ/a and η/a, respectively. It could be very useful for the reader if I present some calculation showing the efficiency of the pre-wave zone effect suppression using a concave target. By substituting Eq. (24) in (1), applying the polar coordinate system the BTR fields can be represented as
k k 1 iek 2 cos ψl 1 TR =− dρs ρs K1 . ρs J1 ρl ρs exp ikρs2 − Ex,y (25) 2πγ a sin ψl γ a 2a Rt Using Eq. (6) one may calculate the BTR spatial-spectral characteristics from a concave target in the pre-wave zone.
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Fig. 6. (a) BTR spatial distribution calculated for three radiuses of the target curvature determined as: 1/Rt = 1/2a—solid line, 1/Rt = 1/2a ± 0.2 × 2π/γ 2 λ—dashed line, 1/Rt = 1/2a ± 0.2 × 2π/γ 2 λ—dashed–dotted line; (b) peak position of the spatial distribution as a function of the radius of the target curvature. The calculation parameters: γ = 2500, λ = 500 nm, a = 0.5 m.
Fig. 6(a) represents an example of the BTR spatial distribution calculated in the pre-wave zone. One may see that if the Rt = 2a, the spatial distribution fully coincides with the one in far-field zone presented in Fig. 2(a) (solid line). However, further increasing or decreasing of the target curvature leads to broadening of the BTR spatial distribution. Fig. 6(b) shows the dependence of the BTR maximum position as a function of target curvature. It is seen that the strongest possible focusing of the BTR spatial distribution is up to γ −1 peak polar angle. In [6] the author has presented a different model for calculating the BTR and BDR from a parabolic target. A different condition for strongest focusing was approximated. However, a parabolic target focuses in one direction only. That means that focusing in one direction it might be possible to obtain the distribution narrower than γ −1 in one direction, but in the perpendicular direction the spatial distribution will be much broader than γ −1 , and the minimal solid angle remains unchanged.
5. Conclusion This Letter is devoted to the pre-wave zone effect, the problems related to it and possible solutions of the problems. The basic characteristics of the BTR in the pre-wave zone have been described in rather simple way. The electron energy in the accelerators incredibly increases the year after year. The pre-wave zone effect becomes more and more critical. Therefore it is necessary to find a solution of this problem. One of the solutions is to use the new theory accounting the pre-wave zone effect. However, this is not the best way because the theory is a lot more complicated. Moreover, when using the BTR or BDR for electron beam diagnostics in some cases the far-field radiation is much more sensitive to the beam parameters because in the pre-wave zone the interference of the radiation from different parts of the target is infringed. The models presented in this Letter are suitable for calculating backward transition radiation (BTR) and backward diffraction radiation (BDR) including the pre-wave zone effect. Two new theoretical models for calculating BTR and BDR characteristics at the detector plane placed in the back focal plane of a thin lens and from a concave target were presented. All the calculations have been presented for BTR; however, it is possible to perform the calculation for BDR as well. It is just necessary to choose the proper limits in the integral over the target surface. The analysis of the models have shown that it might be possible to achieve the same condition inside the pre-wave zone by either putting a lens in the optical path of the measurement system or by developing a concave target. Both solutions have their advantages and disadvantages. A real lens is usually not a thin one. In some cases the wavelength aberrations could be rather severe. Moreover, it can be used in optical wavelength range only. The concave target requires a very precise technology for the target preparation, because there are a lot of demands for the target quality [6,17]. However, there are no wavelength aberrations in the concave target.
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Nevertheless, in some cases the usage of one of the method can significantly simplify an experiment performance. Moreover, all models presented in [1,12–20] are valid in the pre-wave zone if one of the methods for pre-wave zone suppression is applied. For example, at KEK ATF extraction line (γ = 2500) our experimental group performed an experiment on investigation of Optical Diffraction Radiation as a possible tool for non-invasive electron beam diagnostics [17–20]. We achieved the sensitivity to the beam size as small as 14 µm. However, to achieve such success in our experiment we increased the target-to-detector distance up to 3.56 m, because the parameter γ 2 λ/2π = 0.3–0.7 m in optical wavelength range. By using one of the methods presented above it might be possible to reduce the distance from target to detector and develop a very compact device for beam diagnostics. In that case the expenses for the diagnostic equipment can be reduced.
Acknowledgements I would like to thank Professor Junji Urakawa for stimulating support of my work. I would like to thank Professor Alexander Potylitsyn and Doctor Gennady Naumenko for systematic discussion and useful criticism on the topics presented in the Letter. I acknowledge the Japanese Society of the Promotion of Science (JSPS) for partial support.
References [1] M.L. Ter-Mikaelyan, High Energy Electromagnetic Processes in Condensed Media, Wiley–Interscience, New York, 1972. [2] P. Catravas, W.P. Leemans, E. Esarey, et al., in: Proceedings of the 1999 Particle Accelerator Conference, New York, 1999, p. 2111. [3] M. Ross, S. Anderson, J. Frisch, et al., in: Proceedings of 10th Beam Instrumentation Workshop, BIW02, Upton, NY, 2002, SLAC-PUB9280. [4] Yu. Shibata, S. Hasebe, T. Takahashi, et al., Phys. Rev. E 49 (1994) 785. [5] C. Settacorn, PhD thesis, Stanford University, USA, SLAC-Report-576, August 2001. [6] G.A. Naumenko, Nucl. Instrum. Methods B 227 (2005) 87. [7] M. Castellano, V. Verzilov, Phys. Rev. ST Accel. Beams 1 (1998) 062801. [8] M. Castellano, A. Cianchi, G. Orlandi, V.A. Verzilov, Nucl. Instrum. Methods A 435 (1999) 297. [9] V.A. Verzilov, Phys. Lett. A 273 (2000) 135. [10] N.F. Shul’ga, S.N. Dobrovol’sky, Phys. Lett. A 259 (1999) 291, and references therein. [11] M. Castellano, V. Verzilov, L. Catani, et al., Phys. Rev. E 67 (2003) 015501. [12] A.P. Potylitsyn, Nucl. Instrum. Methods B 201 (2003) 161. [13] A.P. Potylitsyn, N.A. Potylitsyna, Russ. Phys. J. 4 (2000) 321. [14] R.B. Fiorito, D.W. Rule, Nucl. Instrum. Methods B 173 (2001) 67. [15] P. Karataev, S. Araki, R. Hamatsu, H. Hayano, T. Muto, G. Naumenko, A. Potylitsyn, N. Terunuma, J. Urakawa, Nucl. Instrum. Methods B 227 (2005) 198. [16] Y. Fukui et al., in: Proceedings of 2003 Particle Accelerator Conference, Portland, Oregon USA, 2003, p. 2709. [17] P. Karataev, PhD thesis, Tokyo Metropolitan University, 2004. [18] P. Karataev, S. Araki, R. Hamatsu, H. Hayano, T. Hirose, T. Muto, G. Naumenko, A. Potylitsyn, J. Urakawa, Nucl. Instrum. Methods B 201 (2003) 140. [19] P. Karataev, S. Araki, R. Hamatsu, et al., Phys. Rev. Lett. 93 (2004) 244802. [20] P. Karataev, S. Araki, R. Hamatsu, H. Hayano, T. Muto, G. Naumenko, A. Potylitsyn, N. Terunuma, J. Urakawa, Nucl. Instrum. Methods B 227 (2005) 158. [21] B. Feng, M. Oyamada, F. Hinode, et al., Nucl. Instrum. Methods A 475 (2001) 492. [22] M. Castellano, V.A. Verzilov, L. Catani, et al., Phys. Rev. E 63 (2001) 056501. [23] M. Born, E. Wolf, Principles of Optics, sixth ed., Pergamon Press, New York, 1980. [24] A.P. Potylitsyn, Nucl. Instrum. Methods B 145 (1998) 169.
Pre-wave zone effect in transition and diffraction radiation: Problems and solutions P.V. Karataev 1 KEK, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba 305-0801, Ibaraki-ken, Japan Received 2 April 2005; accepted 12 July 2005 Available online 20 July 2005 Communicated by V.M. Agranovich
Abstract Transition radiation (TR) and diffraction radiation (DR) appearing as a result of dynamic polarization of medium has widely been used for electron beam diagnostics during the last few years. A lot of techniques for electron beam diagnostics imply description of these phenomena assuming that the radiated area of the target is negligibly small in comparison with the radiation spot in the detector plane (far-field approximation). However, for high-energy electrons this area may reach a macroscopic dimension. In this Letter the general theory in the pre-wave zone is presented. Two new approaches for rejecting the pre-wave zone effect are described and analyzed. By installing a thin lens in the optical path of the measurement system or by developing a concave target, the pre-wave zone effect can be reduced or even rejected. 2005 Elsevier B.V. All rights reserved. PACS: 41.60.-m; 41.75.-i Keywords: Transition radiation; Diffraction radiation; Far-field approximation; Pre-wave zone effect
1. Introduction When a charged particle crosses an interface between two media with different dielectric constants (in ideal case when an electron crosses a vacuum—ideal conductor boundary) it induces currents changing in time [1] at the boundary. Those currents give rise in radiation called transition radiation (TR). TR has a tendency to propagate in two main directions: along the particle trajectory—forward transition radiation (FTR) and along the direction of specular reflection from the boundary—backward transition radiation (BTR). BTR has been widely used for different purposes because it allows registering the radiation at fine background conditions. The BTR application E-mail address: [email protected] (P.V. Karataev). 1 Tel.: +81 298 645715; fax: +81 298 640321.
0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.07.027
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range is very broad: from transversal beam parameter measurements in optical wavelength range [2,3] upto the bunch length measurement and generation of intense radiation in mm and sub-mm wavelength range [4,5]. During the last few years the electron beam energies in the modern accelerators was incredibly increased. As a result new problems appeared, which were considered as negligibly small for moderate relativistic electron energies. One of them is the pre-wave zone effect in transition radiation. In simple words the pre-wave zone is the distance from the target, where the contribution of the radiation source size into the BTR spot size in the detector plane is significant and cannot be neglected. Afterwards, this effect has been considered in details by many theoreticians using different approaches [6–10]. However, all of them agreed that in order use the far-field zone approximation (the target radiating area is negligibly small), the distance from the target to detector must be larger than the parameter γ 2 λ, where γ is the charged particle Lorentz-factor and λ is the radiation wavelength. Later the pre-wave zone effect was observed experimentally [11]. For example, at SLAC FFTB [2] the parameter γ 2 λ = 1.8 km, which is really difficult to achieve. Moreover, when measuring the long wavelength transition radiation (where the BTR is coherent, i.e. the radiation wavelength is comparable to or smaller than the electron bunch length) [4,5] the pre-wave zone effect could be significant at lower energies. Recently diffraction radiation (DR) appearing when the charged particle moves in the vicinity of a medium with impact parameter smaller than γ λ/2π (effective electron field radius) has been suggested as a possible tool for non-invasive beam diagnostics [12–16]. DR is a relative effect to TR, because it is also produced as a result of dynamic polarization of medium. The backward diffraction radiation (BDR) in optical wavelength range has been measured and applied for transversal beam size diagnostics [17–20]. The BDR in mm and sub-mm wavelength range has been applied for non-invasive bunch length measurements [21,22]. However, it has similar problems with the pre-wave zone effect as TR. So far nobody really knows how to deal with this effect. Some of them prefer to take into account the pre-wave zone effect [22] when comparison of the experiment with the theory is performed. However, the theory is very complicated for practical use. Moreover, in some cases the sensitivity of the far-field BTR and BDR to the electron beam parameters is higher. This Letter describes the classical backward transition and diffraction radiation theory in far-field zone and in the pre-wave zone. The choice of backward radiation was made because in this case it is not necessary to take into account the self-field of the electron. It has been shown that the far-field theory is just a particular case of the pre-wave zone theory. Two possible ways for pre-wave zone suppression are considered. By installing a lens in the optical path or by choosing a concave target it might be possible to describe the BTR and BDR characteristics using the formulas obtained from the far-field theory.
2. Far-field approach In this Letter I shall use the classical theory of backward transition radiation (BTR) and backward diffraction radiation (BDR) based on Huygen’s principle of plane wave diffraction [1]. In this theory the particle field in introduced as superposition of its pseudo-photons. When the particle interacts with the target surface, the pseudophotons are scattered from it converting into real ones and propagate in the direction of specular reflection. The difference from the plane wave is that the electron field strength depends on the distance from the particle. Let an electron move along the z-axis which is perpendicular to an ideally conducting target plane. At the zero time moment (z/β = 0, where β = v/c is the electron velocity in units of the light velocity) the electron crosses the boundary. Each point of the target surface can be represented as an elementary source. In this case two polarization components of TR field can be represented as a superposition of the waves from all elementary sources at certain distance: 1 eiϕ l i dys dxs . = (x , y ) E Ex,y (1) s s x,y r 4π 2
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This is the general equation for calculating BTR and BDR characteristics. Here x and y indexes indicate two polarization components perpendicular to each other; xs = ρs sin ψs and ys = ρs cos ψs are the coordinates of the particle field (or the position of an arbitrary elementary source at the target surface), ρs and ψs are the radius i represents the amplitude and azimuthal angle of the particle field respectively in polar coordinate system; Ex,y of an arbitrary elementary source at the target surface determined by the coordinates xs and ys ; ϕ determines the phase advance of the photons emitted by each elementary source, and r is the distance from an arbitrary elementary source on the target to the observation point. One should note that in principle there is a longitudinal, z, polarization component. However, in a relativistic case it is much smaller than the transversal one and can be neglected. i can be represented as the Fourier transform of the incident particle field [1]: The amplitude Ex,y kx,y exp[i(kx xs + ky ys )] k ie iek cos ψs i Ex,y (xs , ys ) = (2) K1 ρs . dkx dky = πγ sin ψs γ 2π 2 kx 2 + ky 2 + k 2 γ −2 Here kx,y are the components of the electron pseudo-photon wave vector, k = 2π/λ is the modulus of the radiation wave vector, λ is the BTR wavelength, γ is the charged particle Lorentz-factor, K1 is the first order McDonald function (modified Bessel function), e is the electron charge. Throughout the Letter the system of units h = me = c = 1 is used. In the far-field zone the distance from the target to the observation point is assumed to be so large that it is possible to introduce the radiation field as a superposition of the plane waves of different amplitude emitted by each elementary source of the target. In other words, it is possible to use Fraunhofer diffraction theory [23]. In this case the phase, ϕ, can be represented as
ϕ = −(r ρs ) = −xs kx − ys ky .
(3)
Here kx ≈ kθx and ky ≈ kθy are the components of the BTR field and θx and θy are the observation angles measured from the direction opposite to the particle trajectory [1]. When the observation angle differs from zero the path of the photons propagating from the target to the detector increases, which causes the phase delay. Because of that the phase (3) is negative. Substituting Eqs. (2) and (3) in Eq. (1) we have the general expression for calculating BTR and BDR characteristics in far-field zone kx,y exp[i((kx − kx )xs + (ky − ky )ys )] 1 l dkx dky dys dxs . Ex,y = (4) 8π 4 a kx 2 + ky 2 + k 2 γ −2 Here a ≈ |r| is the distance from the target to the observation plane. This equation has been used in many papers [1,12–20] for deriving equations describing BTR and BDR characteristics in far-field zone from different target types. One might be able to see the beauty of this approach. It allows to obtain very simple equations, which are flexible to other mathematical transformations like convolution with Gaussian distribution (to take into account transversal electron beam dimensions or angular divergence) or averaging over the detector angular and energy acceptance, etc. For example, integrating over xs and ys (target surface) from minus infinity to infinity (infinite target) one may obtain a multiplication of two delta functions like 4π 2 δ(kx − kx )δ(ky − ky ). Afterwards the integration over the electron field wave vector can be done trivially, and Eq. (4) is transformed into the expression for BTR fields from a particle crossing an infinite vacuum-ideal conductor interface TR Ex,y (θx , θy ) =
θx , θy ie . 2 2 2π ak θx + θy2 + γ −2
(5)
The BTR spectral angular distribution can be calculated as 2 2 θx2 + θy2 d 2 W TR α = 4π 2 k 2 a 2 ExTR + EyTR = 2 2 . dω dΩ π (θx + θy2 + γ −2 )2
(6)
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The middle part of Eq. (6) is a conventional equation for calculating spatial-spectral distribution, i.e. the power distribution per unit space interval (solid angle in far-field zone) and per unit photon energy interval. In the future I shall reference on this equation when dealing with the spatial-spectral characteristics. One may see that for an ideal conductor BTR is an azimuthally symmetric (crater like) distribution independent of the photon energy with TR = αγ 2 /4π 2 , where α is the fine structure the characteristic polar angle of γ −1 and with maximum intensity Imax constant. However, in a real experiment it is possible to use this equation for the BTR photon energies smaller than the plasmon energy of the target material, which is about 10 eV for a metallic foil. In [24] the author derived the same expression by applying an ultrarelativistic approximation to the solution of Maxwell equations; however, the approach presented in this section is much simpler. Therefore, in the rest of the Letter I shall lead all the considerations to this simple approach.
3. Pre-wave zone approach The far-field approach presented in the previous section is applicable if the distance from target to the observation point is very large. The question is: how large should it be? In this section I shall try to answer this question from the points of views of physics, mathematics and common sense. When the particle crosses the boundary, it induces radiation source, which size is equal to the electron field radius. It is well known that the effective electron field radius is treated as γ λ/2π [1] (see Fig. 1). At larger distances from the particle the electron field amplitude is significantly suppressed. Each point of the target surface emits radiation with divergence of order of γ −1 . Therefore, in order to reduce the contribution from the radiation source into the radiation spot size it is necessary to choose the distance from the target obeying the following condition γλ γ 2λ a ⇒a . γ 2π 2π
(7)
The radiation spatial-spectral distribution distortion happens at the distances (a) comparable or smaller than the parameter γ 2 λ/2π . This peculiarity is called as pre-wave zone effect. Let us discuss the mathematical approach. To perform the calculation it is necessary to change the phase advance (ϕ) in Eq. (1). From simple geometry the phase advance of the photons propagating from the target to the observation point is determined by the following relation eik|r | exp(ik a 2 + (xs − ξ )2 + (ys − η)2 ) eiϕ = = . (8) |r | |r | a 2 + (xs − ξ )2 + (ys − η)2
Fig. 1. Geometry of the BTR production at a finite distance (a) from the target. Here r1 , r2 , . . . , rN are the arbitrary vectors of the photons propagating from different target points into the same point in the detector plane.
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Here ξ = ρl cos ψl and η = ρl sin ψl are the coordinates of the observation plane and their polar representation. Actually the parameters (ξ − xs )/a and (η − ys )/a determine the angles of the photon emission by an arbitrary elementary source. In ultrarelativistic case those parameters are of order of γ −1 1. Therefore, Eq. (8) can be significantly simplified
k
eika k 2 k 2 eiϕ 2 2 ≈ exp x + ys − (xs ξ + ys η) + ξ +η . (9) |r | a 2a s a 2a Eq. (9) coincides with the one obtained in [7,8] where the authors considered the spatial resolution of optical transition radiation. However, the authors did not consider the pre-wave zone criterion in details from the viewpoint of mathematics. Moreover, Eq. (8) represents the phase relations in general form, which is more convenient for understanding of the approach. The third term in the square brackets is unimportant, because it independent of the integrals. One may see that the second term is very similar to Eq. (3). Actually this term is responsible for the Fraunhofer diffraction [23]. Therefore, in order to achieve the far-field approach it is necessary to take out the first term. Actually this term (where the target surface coordinates are involved with square powers) is responsible for the Fresnel diffraction [23]. In other words, the first term represents the first order Fresnel corrections to the far-field approximation. This term can be neglected if it is much smaller than unity. The coordinates xs and ys determine the position of an elementary source at the target, which is determined by the electron field radius. At the distances from the particle trajectory larger than γ λ/2π the intensity of the radiation sources is very small and can be neglected. Making the substitution xs = ys = γ λ/2π the far-field zone condition is
k γ 2 λ2 γ 2λ γ 2λ k 2 xs + ys2 = 1 ⇒ a . = (10) 2a a 4π 2 2πa 2π The condition (10) wholly coincides with Eq. (7). Therefore, if the condition (10) is fulfilled, the distance is so large that the parameters ξ/a → θx and η/a → θy , and Eq. (9) is transformed into Eq. (3), i.e. the far-field approach was derived from the pre-wave zone approach, which is a more general one. In [9] the author performed the considerations on the pre-wave zone effect in transition radiation. The author used a different approach; however, the same condition (10) was derived. Substituting Eq. (9) in (1) and using Eq. (6) one may calculate the BTR spatial-spectral characteristics in prewave zone. One should note that in the pre-wave zone Eq. (6) represents the spatial-spectral distribution, because the photons with different angles arrive in the same observation point (see, for instance, Fig. 1). The spatial distribution is transformed in to the angular distribution in the far-field zone only. It is necessary to keep in mind that the integration limits for the target surface must be much larger than the electron field radius. Otherwise, the finite outer target dimensions may cause an additional distortion [15]. Fig. 2 illustrates the calculated BTR spatial-spectral distribution in the pre-wave zone. It is obvious that the BTR characteristics differ from the far-field approach. These
Fig. 2. (a) represents the BTR spatial-spectral distribution calculated for different distances from the target: a = 10γ 2 λ/2π —solid line, a = 2γ 2 λ/2π —dashed line, a = γ 2 λ/2π —dashed–dotted line; (b) represents the BTR peak position as a function of the distance from the target.
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calculations coincide with the ones presented in [9] as well. Therefore both approaches are applicable for BTR calculation. The advantage of this approach is that it is possible to calculate diffraction radiation characteristics simply defining the integration region on the target surface. Many people are confused about the condition (10) because it coincides with the radiation formation zone distance condition, which has a completely different meaning. If an electron emits two photons at the distance comparable to or smaller than the radiation formation zone length, those two photons interfere. In other words, at the distance from the radiation origin along the particle trajectory much larger than the radiation formation zone distance the electron field and the radiation photon are completely separated. The radiation formation zone is determined by the following relation [1] Lf =
1 λ 1 βλ ≈ . 2π (1 − β cos θ ) π (γ −2 + θ 2 )
(11)
In (11) θ 2 = θx2 + θy2 is the polar observation angle, which is of order of γ −1 along the particle trajectory for FTR. Therefore, if the distance from the target along the particle trajectory a Lf ≈ γ 2 λ/2π the radiation photons and the electron field are completely separated. This condition coincides with the far-field condition (10). However, for BTR the observation angles reach large values. For example, for the case considered in this Letter the radiation propagate in the direction opposite to the particle trajectory. That means that the angle θ = π , and therefore the electron field and radiation photons are separated at the distance a λ/4π . Therefore, one should keep in mind that in spite of the radiation formation zone distance and the far-field zone distance are described by similar relations they determine completely different processes.
4. Two ways for suppression of the pre-wave zone effect In some cases, where the BTR or BDR is used for beam diagnostics, the pre-wave zone effect suppression is rather critical (for example, beam size measurement with optical diffraction radiation [17–20]). In the pre-wave zone the interference of the radiation from different parts of the target is infringed. In that case the sensitivity of the effect to the beam parameters becomes weaker. Therefore, in this section I shall consider two possible ways allowing to suppress or even reject the pre-wave zone effect. 4.1. Propagation of the BTR through a thin lens Fig. 3 illustrates a simple geometry of the photon propagation through a thin lens. If the target is placed in the back focal plane, all parallel waves moving from the target must be led to the same point in the detector plane. In this case the radiation spot in the detector plane should be proportional to the radiation angular distribution
Fig. 3. Geometry of the photon propagation through a thin lens.
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with the angles determined by the following relations: θx,y = x, y/f , where f is the lens focus, x = ρd cos ψd and y = ρd sin ψd are the coordinates of the detector plane and their polar representation. Let us derive the model describing this BTR geometry from the viewpoint of wave optics. The BTR fields on the lens surface frontal to the target can be described using Eq. (1) and the phase advance, ϕ, determined by Eq. (9). In this case the variables ξ and η are the coordinates of the lens surface and a is the target-to-lens distance. According to [23] a thin lens causes a phase delay, which is a function of its coordinates. The element of the BTR fields on the lens surface opposite to the target can be represented as
ik 2 l+ l 2 ξ +η . Ex,y = Ex,y exp − (12) 2f The BTR fields in the detector plane can be represented as iϕd l+ e Ex,y = (13) dξ dη × Ex,y . rd In Eq. (13) ϕd is the phase advance of the photons propagating from the lens to the detector and rd is the distance, which the photons overpass from the lens to the detector. In analogy with Eq. (8) the phase relations could be represented as
ik eik|rd | exp(ik f 2 + (x − ξ )2 + (y − η)2 ) ik 2 1 eiϕd 2 = = ξ + η − (ξ x + ηy) . ≈ exp (14) |rd | |rd | f 2f f f 2 + (x − ξ )2 + (y − η)2 The phase terms independent of the integrals are omitted for simplicity. The full equation for calculating the BTR characteristics behind a thin lens can be represented in the form 1 i Ex,y = (15) Ex,y (xs , ys )G(xs , ys ) dys dxs , 4π 2 af where k i 2a (xs2 +ys2 )
k
−ikξ( xs + x ) −ikη( ys + y )
a f e a f dξ dη × ei 2a (ξ +η ) e k i k (x x+y y) i k [ξ −(xs + fa x)]2 +i 2a [η−(ys + fa y)]2 =e f s s . dξ dη × e 2a
G(xs , ys ) = e
2
2
(16)
If we assume that the lens is infinite the double integral part in Eq. (16) is reduced to a constant and the factor G is i2a i fk (xs x+ys y) e . G∞ = (17) πk Substituting Eq. (17) in (15) one may obtain the expression similar to Eq. (4) in Section 2, i.e. similar to the Fraunhofer diffraction case. That means that a simple lens can reject the pre-wave zone effect. Here, an approximation of an infinite lens was used. Of cause, a real lens has a definite size, and diffraction on it can significantly distort the BTR spatial-spectral distribution. Therefore, at the end of this section I shall introduce a simple condition where the infinite lens approximation can be used. Obviously, the lens dimension must be larger than the radiation source size. Otherwise, the diffraction of the photons from the source tails may significantly distort the BTR spatial distribution. Furthermore, each radiation source emits radiation with divergence of order of γ −1 . Therefore, the condition for an infinite lens dimension can be formulated as Rl
γλ a + . γ 2π
(18)
Here, Rl is the lens radius. In Eq. (18) the first term takes into account the effect of the BTR spot increasing as a function of the distance from the source, and the second term takes into account the source size. For example,
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Fig. 4. (a) BTR spatial distribution in the far-field zone calculated for γ = 2500, a = 10 m, and three radiuses of the lens: Rl = 10a/γ —dashed line, Rl = 2a/γ —solid line, and Rl = a/γ —dashed–dotted line; (b) BTR spatial distribution in the pre-wave zone calculated for a = 1 m, Rl = 5 mm, and three electron energies: γ = 10000—solid line, γ = 25000—dashed line; γ = 40000—dash-dotted line. The calculation parameters: λ = 500 nm, f = 100 mm.
for KEK-ATF [17–20] (γ = 2500, λ = 500 nm, a = 1 m) the lens radius should be Rl 0.9 mm, which could be easily achieved. It is much easier to perform numerical calculations in the polar coordinate system. The BTR fields on the detector placed in the back focal plane of a thin lens with finite dimensions can be derived by substitution of the polar coordinates in Eqs. (2), (15) and (16). I omit the simple mathematical procedure here. The result is
k k k ik 2 iek 2 cos ψd TR 2 ρs ρl ρs J1 ρl ρd exp ρ + ρl . dρs ρs K1 dρl ρl J1 Ex,y = − 2πγ af sin ψd γ a f 2a s (19) Here, J1 is the Bessel function of the first order. The integration over the lens radius (over ρl ) should be performed from 0 to Rl . The spatial-spectral distribution can be calculated using Eq. (6). Fig. 4 illustrates the applicability of the condition (18). If the lens is in far-field zone (a γ 2 λ/2π) the first term in inequality (18) is dominant (Fig. 4(a)), however, in the strong pre-wave zone (a γ 2 λ/2π), the second term is dominant (Fig. 4(b)). If the target-to-lens distance is comparable to γ 2 λ/2π , both terms in Eq. (8) make comparable contributions. Nevertheless, Fig. 4(b) (solid line) illustrates that in spite of the fact that the lens-detector system is placed in a strong pre-wave zone, the BTR distribution has the same form as in far-field zone without the lens. 4.2. The BTR from a concave target The approach presented above is applicable for optical wavelengths. However, it is not applicable for long wavelengths (mm or sub-mm wavelength range). Moreover, even in optical wavelength range the aberrations in a real lens could be severe. Therefore, before making an experiment those conditions must be seriously analyzed. For the long wavelength region it is possible to use a concave target. In this case the target may work similar to the lens. The BTR calculation from a concave target is a little more complicated because a longitudinal component of the target appears; i.e. the electron field interacts with different target parts in different time. The calculation geometry is represented in Fig. 5. As before to calculate the BTR from a concave mirror it is necessary to change the exponential term containing the phase, ϕ, in Eq. (1). Let the electron cross the plane (xs , ys ) at the time equaled to zero ik|r|−ik a β exp[ik (a − a)2 + (xs − ξ )2 + (ys − η)2 − ik a eiϕ e β ] (20) , = = |r| |r| (a − a)2 + (xs − ξ )2 + (ys − η)2
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Fig. 5. Geometry of the BTR production from a concave (spherical) target.
where a = Rt −
Rt2
− xs2 − ys2 = Rt 1 −
x2 + y2 1− s 2 s Rt
.
(21)
Assuming that the radiating area of the concave target is much smaller than the radius of the curvature (i.e. Rt γ λ/2π ), then Eqs. (20) and (21) can be significantly simplified: eia 1 ik eiϕ ik 2 2 ≈ exp −ika + 1 + (xs − ξ ) + (ys − η) (22) |r| a β 2a 2a and a =
xs2 + ys2 . 2Rt
(23)
Substituting Eq. (23) into (22), involving ultrarelativistic case (1/β + 1 ≈ 2) and omitting the phase terms independent of the integrals in Eq. (1) we have 2
1 ik 1 1 eiϕ 2 − (xs ξ + ys η) . ≈ exp ik xs + ys − (24) |r| a 2a Rt a From Eq. (24) it is obvious that if the detector is placed in far-field zone, i.e. 1/2a ≈ 0, in this case there is no difference between concave and convex target, i.e. between Rt and −Rt . Both of them will lead to broadening of the BTR spatial distribution. The idea of this section is to obtain the approach allowing to calculate BTR and BDR in the pre-wave zone using the equations similar to the Fraunhofer diffraction theory. For that purpose it is necessary to take out the phase term depending on the squared coordinates of the electron field. It can be easily achieved by choosing the distance from the target equaled to a half of the target curvature radius (a = Rt /2). In this case Eq. (24) is transformed to the equation similar to Eq. (3), where the analogues of the angular variables θx and θy are ξ/a and η/a, respectively. It could be very useful for the reader if I present some calculation showing the efficiency of the pre-wave zone effect suppression using a concave target. By substituting Eq. (24) in (1), applying the polar coordinate system the BTR fields can be represented as
k k 1 iek 2 cos ψl 1 TR =− dρs ρs K1 . ρs J1 ρl ρs exp ikρs2 − Ex,y (25) 2πγ a sin ψl γ a 2a Rt Using Eq. (6) one may calculate the BTR spatial-spectral characteristics from a concave target in the pre-wave zone.
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Fig. 6. (a) BTR spatial distribution calculated for three radiuses of the target curvature determined as: 1/Rt = 1/2a—solid line, 1/Rt = 1/2a ± 0.2 × 2π/γ 2 λ—dashed line, 1/Rt = 1/2a ± 0.2 × 2π/γ 2 λ—dashed–dotted line; (b) peak position of the spatial distribution as a function of the radius of the target curvature. The calculation parameters: γ = 2500, λ = 500 nm, a = 0.5 m.
Fig. 6(a) represents an example of the BTR spatial distribution calculated in the pre-wave zone. One may see that if the Rt = 2a, the spatial distribution fully coincides with the one in far-field zone presented in Fig. 2(a) (solid line). However, further increasing or decreasing of the target curvature leads to broadening of the BTR spatial distribution. Fig. 6(b) shows the dependence of the BTR maximum position as a function of target curvature. It is seen that the strongest possible focusing of the BTR spatial distribution is up to γ −1 peak polar angle. In [6] the author has presented a different model for calculating the BTR and BDR from a parabolic target. A different condition for strongest focusing was approximated. However, a parabolic target focuses in one direction only. That means that focusing in one direction it might be possible to obtain the distribution narrower than γ −1 in one direction, but in the perpendicular direction the spatial distribution will be much broader than γ −1 , and the minimal solid angle remains unchanged.
5. Conclusion This Letter is devoted to the pre-wave zone effect, the problems related to it and possible solutions of the problems. The basic characteristics of the BTR in the pre-wave zone have been described in rather simple way. The electron energy in the accelerators incredibly increases the year after year. The pre-wave zone effect becomes more and more critical. Therefore it is necessary to find a solution of this problem. One of the solutions is to use the new theory accounting the pre-wave zone effect. However, this is not the best way because the theory is a lot more complicated. Moreover, when using the BTR or BDR for electron beam diagnostics in some cases the far-field radiation is much more sensitive to the beam parameters because in the pre-wave zone the interference of the radiation from different parts of the target is infringed. The models presented in this Letter are suitable for calculating backward transition radiation (BTR) and backward diffraction radiation (BDR) including the pre-wave zone effect. Two new theoretical models for calculating BTR and BDR characteristics at the detector plane placed in the back focal plane of a thin lens and from a concave target were presented. All the calculations have been presented for BTR; however, it is possible to perform the calculation for BDR as well. It is just necessary to choose the proper limits in the integral over the target surface. The analysis of the models have shown that it might be possible to achieve the same condition inside the pre-wave zone by either putting a lens in the optical path of the measurement system or by developing a concave target. Both solutions have their advantages and disadvantages. A real lens is usually not a thin one. In some cases the wavelength aberrations could be rather severe. Moreover, it can be used in optical wavelength range only. The concave target requires a very precise technology for the target preparation, because there are a lot of demands for the target quality [6,17]. However, there are no wavelength aberrations in the concave target.
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Nevertheless, in some cases the usage of one of the method can significantly simplify an experiment performance. Moreover, all models presented in [1,12–20] are valid in the pre-wave zone if one of the methods for pre-wave zone suppression is applied. For example, at KEK ATF extraction line (γ = 2500) our experimental group performed an experiment on investigation of Optical Diffraction Radiation as a possible tool for non-invasive electron beam diagnostics [17–20]. We achieved the sensitivity to the beam size as small as 14 µm. However, to achieve such success in our experiment we increased the target-to-detector distance up to 3.56 m, because the parameter γ 2 λ/2π = 0.3–0.7 m in optical wavelength range. By using one of the methods presented above it might be possible to reduce the distance from target to detector and develop a very compact device for beam diagnostics. In that case the expenses for the diagnostic equipment can be reduced.
Acknowledgements I would like to thank Professor Junji Urakawa for stimulating support of my work. I would like to thank Professor Alexander Potylitsyn and Doctor Gennady Naumenko for systematic discussion and useful criticism on the topics presented in the Letter. I acknowledge the Japanese Society of the Promotion of Science (JSPS) for partial support.
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