Time Dependence of the Intensity of Diffracted Radiation Produced by a Relativistic Particle Passing through a Natural or Photonic Crystal

0.27 MB·English

by  V. G. Baryshevsky

Time Dependence of the Intensity of
Diffracted Radiation Produced by a
Relativistic Particle Passing through a
Natural or Photonic Crystal
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V.G. Baryshevsky, A.A. Gurinovich
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J

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Abstract
]
h
p
The formulas which describe the time evolution of radiation produced by a rela-
-
c tivistic particle moving in a crystal are derived. It is shown that the conditions are
c
realizableunderwhichparametric(quasi-Cherenkov)radiation,transitionradiation,
a
. diffracted radiation of the oscillator, surface quasi-Cherenkov and Smith-Purcellra-
s
c diation last considerably longer than the time τ of the particle flight through the
i p
s crystal. The results of carried out experiments demonstrate the presence of addi-
y
h tionalradiationpeakappearingaftertheelectronbeamhasleftthephotoniccrystal.
p
[

1
v
Introduction
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6
1
4
. Atpresent, theprocessesofdiffractedradiationofphotonsbyrelativisticparti-
1
0 clespassing throughcrystals (naturalorartificial spatiallyperiodicstructures)
1 are intensively studied both theoretically and experimentally. Worthy of men-
1
tion are such types of diffracted radiation as parametric (quasi-Cherenkov)
:
v
radiation and diffracted radiation of a relativistic oscillator [1–3]. It should be
i
X
noted, however, that until now, theoretical and experimental analysis of radi-
r
a ation produced by a relativistic particle passing through a crystal has focused
on spectral-angular characteristics of radiation. Nevertheless, it was shown
in [5,6] that because of diffraction, photons produced through radiation in
crystals have group velocity vp , which is appreciably smaller than the veloc-
gr
ity v of a relativistic particle. As a result, the situation is possible in which
radiation from the crystal still continues after the particle has passed through
it [5,6]. This enables studying time evolution of the process of photon radi-
ation produced during the particle transmission through the crystal (natural
or photonic), or during the particle flight along the surface of such crystals. In
the present paper the formulas are derived, which describe the time evolution
of radiation produced by a relativistic particle moving in a crystal. It is shown
Preprint submitted to Elsevier January 24, 2011
that the conditions are realizable under which parametric (quasi-Cherenkov)
radiation, transition radiation, diffracted radiation of the oscillator, surface
quasi-Cherenkov and Smith-Purcell radiation last considerably longer than
the time τ of the particle flight through the crystal, i.e., much longer than
p
τ 10−9 s.
p
≤
1 Spectral-angular distribution of radiation produced by a particle
transmitted through a crystal
Let us first recall the conventional consideration of the radiation process in
crystals [1,8].
Both the spectral-angular density of radiation energy per unit solid angle W
~nω
and the differential number of emitted photons dN ω = 1/~ω W can be
~nω ~nω
·
easily obtained if the field E~(~r,ω) produced by a particle at a large distance
~r from the crystal is known [3]
W = er2 E~(~r,ω) 2, (1)
~nω
4π2
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
The vinculum here means averaging over all possible states of the radiating
system. In order to obtain E~(~r,ω), Maxwell’s equation describing the interac-
tion of particles with the medium should be solved. The transverse solution
can be found with the help of Green’s function of this equation, which satisfies
the expression:
ω2
G = G +G (εˆ 1)G, (2)
0 04πc2 −
G is the transverse Green’s function of Maxwell’s equation at εˆ = 1. It is
0
given, for example, in [19].
Using G, we can find the field we are concerned with
iω
E (~r,ω) = G (~r,~r′,ω) j (~r,ω)d3r′, (3)
n nl c2 0l
Z
where n,l = x,y,z, j (~r,ω) is the Fourier transformation of the e-th compo-
0l
nent of the current produced by a moving beam of charged particles (in the
linear field approximation, the current is determined by the velocity and the
trajectory of a particle, which are obtained from the equation of particle mo-
tion in the external field, by neglecting the influence of the radiation field on
2
the particle motion). Under the quantum-mechanical consideration the cur-
rent j shouldbeconsideredasthecurrent oftransitionoftheparticle-medium
0
system from one state to another.
According to [3,8], Green’s function is expressed at r through the
→ ∞
solution of homogeneous Maxwell’s equations E(−)(~r,ω) containing incoming
n
spherical waves:
eikr
limG (~r,~r′,ω) = esE(−)s∗(~r′,ω), (4)
nl r n ~kl
S
X
r
→ ∞
where ~es is the unit polarization vector, s 1,2, ~e1 ~e2 ~k.
− ⊥ ⊥
Iftheelectromagneticwaveisincident onacrystaloffinitesize,thenatr
→ ∞
eikr
E~(−)s(~r,ω) =~esei~k~r +const ,
k r
andonecanshowthat therelationbetween thesolutionE~(−)s andthesolution
k
of Maxwell’s equation E~(+)(~k,ω) describing scattering of a plane wave by the
target (crystal), is given by:
E~(−)s∗ = E~(+)s (5)
~k −~k
Using (3), we obtain
eikr iω
E (~r,ω) = es E(−)s∗(~r,ω)~j (~r′,ω)d3r′. (6)
n r c2 n ~k 0
S Z
X
As a result, the spectral energy density of photons with polarization s can be
written in the form:
ω2 2
Ws = E~(−)s∗(~r,ω)~j (~r,ω)d3r , (7)
~n,ω 4π2c2 ~k 0
(cid:12)Z (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
~j (~r,ω) = e(cid:12)iωt~j (~r,ω)dt = eQ eiωt~v((cid:12)t)δ(~r ~r(t))dt, (8)
0 0
−
Z Z
where eQ is the charge of the particle, ~v(t) and ~r(t) are the velocity and the
trajectory of the particle at moment t. By introducing (8) into (7) we get
e2Q2ω 2
dNs = E~(−)s∗(~r(t),ω)~v(t)eiωtd t. (9)
~n,ω 4π2~c3 ~k
(cid:12)Z (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3
Integration in (9) is carried out over the whole interval of the particle motion.
It should be noted that the application of the solution of a homogeneous
Maxwell’s equation instead of the inhomogeneous one essentially simplifies the
analysis of the radiation problem and enables one to consider various cases of
radiation emission taking into account multiple scattering.
Using equations (7)–(9), one can easily obtain the explicit expression for the
radiation intensity and that for the effect of multiple scattering on the process
under study [3,8,9].
Consider, for example, the PXR radiation. Let a particle moving with a uni-
formvelocity beincident onacrystal platewiththethickness Lbeing L L ,
c
where L = (ωq)−1/2 is the coherent length of bremsstrahlung q = θ2/4 a≪nd θ2
c
is the mean square angle of multiple scattering. The latter requirement allows
neglecting the multiple scattering of particles by atoms. A theoretical method
describing multiple scattering effect on the radiation process is given in [10].
According to (9), in order to determine the number of quanta emitted by a
particle passing through the crystal plate, one should first find the explicit
~(−)s ~(−)s
expressions for the solutions E . As was mentioned above, the field E
~k ~k
can be found from the relation E~(−)s = (E~(+)s)∗ if one knows the solution
~k −~k
E~(+)s describing the photon scattering by the crystal.
~k
In the case of two strong waves excited under diffraction (the so-called two-
beam diffraction case [11]), one can obtain the following set of equations for
determining the wave amplitudes (see [12]):
k2
1 χ∗ E~(−)sc χ∗ E~(−)s = 0
ω2 − − 0! ~k s −~τ ~kτ
k2
1 χ∗ E~(−)sc χ∗E~(−)s = 0. (10)
ω2 − − 0! ~kτ s ~τ ~k
~ ~
Here k = k + ~τ, ~τ is the reciprocal lattice vector, χ , χ are the Fourier
~τ 0 ~τ
components of the crystal susceptibility. It is well known that the crystal is
described by a periodic susceptibility (see, for example, [11]:
χ(~r) = χ exp(i~τ~r). (11)
~τ
~τ
X
c = ~es~es, where ~es(~es) are the unit polarization vectors of the incident and
s ~τ ~τ
diffracted waves, respectively.
The condition for the linear system (10) to be solvable leads to a dispersion
~
equation that determines the possible wave vectors k in a crystal. These wave
4
vectors are convenient to present in the form:
ω
~k =~k +~κ∗ N~, κ∗ = ε∗ ,
µs µs µs cγ µs
0
~
where µ = 1,2; N is the unit vector of a normal to the entrance crystal surface
which is directed into the crystal,
1 1
ε = [(1+β )χ β α ] [(1 β )χ +β α ]2
1(2)s 1 0 1 B 1 0 1 B
4 − ± 4 −
n
−1/2
+4β C2χ χ . (12)
1 s ~τ −~τ
o
α = (2~k~τ + τ2)k−2 is the off-Bragg parameter (α = 0 if the exact Bragg
B B
condition of diffraction is fulfilled),
~ ~
k γ k +~τ
γ =~n N~, ~n = , β = 0, γ =~n N~, ~n = .
0 γ · γ k 1 γ1 1 γτ · γτ ~k +~τ
| |
The general solution of (10) inside a crystal is:
2
E~(−)s(~r) = ~esA exp(i~k ~r)+~esA exp(i~k ~r) . (13)
~k µ µs τ τµ µsτ
µX=1h i
Associating these solutions with the solutions of Maxwell’s equations for the
vacuum area, one can find the explicit form of E~(−)s(~r) throughout the space.
~k
It is possible to discriminate several types of diffraction geometries, namely,
the Laue (a) and the Bragg (b) schemes are most well known.
(a) Let us consider the PXR in the Laue case.
In this case, the electromagnetic waves emitted by a particle in both the
forwardandthediffracteddirectionsleavethecrystalthroughthesamesurface
(k > 0,k +τ > 0), the z-axis is parallel to the normal N (where N is the
z z z
normal to the crystal surface being directed inside a crystal). By matching the
solutions of Maxwell’s equations on the crystal surfaces with the help of (10),
(12), (13), one can obtain the following expressions for the Laue case:
2 2
E~~k(−)s = ~es− ξµ0∗se−iγω0ε∗µsLei~k~r +e~τsβ1 ξµτs∗e−iγω0ε∗µsLei~kτ~rθ(−z)
 µX=1 µX=1 
   
2 2
+~es− ξµ0∗se−iγω0ε∗µs(L−z)ei~k~r +e~sτβ1 ξµτs∗e−iγω0ε∗µs(L−z)ei~kτ~r
 µX=1 µX=1 
   
 
5
θ(L z)θ(z)+~esei~k~rθ(z L), (14)
× − −
where
2ε χ
ξ0 = 2,1s − 0 ;
1,2s ∓2(ε ε )
2s 1s
−
c χ
ξτ = s −τ ;
1,2s ∓2(ε ε )
2s 1s
−
1, if z 0
θ(z) = ≥

0, if z < 0.

Substitution of (14) into (9) gives for the Laue case the differential number of
quanta of the forward directed parametric X-rays with the polarization vector
~e :
s
dd2ωNd0ΩLs = 4eπ2Q2~2cω3(~es~v)2(cid:12) ξµ0seicωγ0εµsLω 1~k~v − ω 1~k∗ ~v
(cid:12)(cid:12)µX=1,2 − − µs
[ei(ω−~kµ∗s~v)T 1] 2, (cid:12)(cid:12)(cid:12)   (15)
× −
(cid:12)
(cid:12)
where T = L/cγ is the(cid:12) particle time of flight; ~e [~k~τ]; ~e [~k~e ].
0 1 2 1
k k
One can see that formula (15) looks like the formula which describes the
spectral and angular distribution of the Cherenkov and transition radiations
in the matter with the index of refraction n = k /k = 1+κ /k .
µs zµs z µs z
~
The spectral angular distribution for photons in the diffraction direction k =
τ
~
k +~τ can be obtained from (15) by a simple substitution
~e ~e , ξ0 β ξτ ,
s → sτ µs → 1 µs
χ c
ξτ = τ s
1(2)s ±2(ε ε )
1s 2s
−
~ ~ ~ ~ ~
k k , k k = k +τ.
τ µs τµs µs
→ →
(b) Now let us consider PXR in the Bragg case. In this case, side by side with
theelectromagneticwaveemittedintheforwarddirection,theelectromagnetic
wave emitted by a charged particle in the diffracted direction and leaving
the crystal through the surface of the particle entrance can be observed. By
matching the solutions of Maxwell’s equations on the crystal surface with the
help of (10), (12), (13), one can get the formulas for the Bragg diffraction
schemes.
6
It is interesting that the spectral angular distribution for photons emitted in
the forward direction can be obtained from (15) by the following substitution,
ξ0 γ ,
µs → µs
2ε χ
γ0 = 2(1)s − 0 (16)
1(2)s (2ε2(1)s χ0) (2ε1(2)s χ0)eiγω0(ε2(1)s−ε1(2)s)L
− − −
Thespectralangulardistributionofphotonsemittedinthediffracteddirection
can be obtained from (15) by substitution
~es →~esτ, ~k →~kτ, kµs →~kµτs, ξµ0seiγω0εµsL → γµτs,
where
β c χ
γτ = 1 s τ . (17)
1(2)s −(2ε2(1)s χ0) (2ε1(2)s χ0)eiγω0(ε2(1)s−ε1(2)s)L
− − −
Letusnotethattheaboveformulasfullydescribeparametric(quasi-Cherenkov)
radiation in natural and photonic crystals and they certainly include that con-
tribution to radiation, which goes over to ordinary transition radiation, if the
radiation is considered outside the region of diffraction reflection. A descrip-
tion of diffracted radiation of a relativistic oscillator is given in [1,2] and the
reference therein.
Let ustakenoticeofthefactthatinphotoniccrystals built frommetal threads
with the diameter smaller than or comparable with λ, the value of χ(τ) is
practically independent on τ. As a result, it is possible to effectively excite
radiation in, e.g., the terahetrz range in a lattice with a period of several
millimeters.
When a particle travels in a vacuum near the surface of a spatially periodic
medium, new kinds of radiation arise [13,14] – surface parametric (quasi-
Cherenkov) X-ray radiation (SPXR) and surface DRO (see Figure 1). This
phenomenon takes place under the condition of uncoplanar surface diffraction,
first considered in [15].
The solution of Maxwell’s equation E~(+)(~r) in this case of uncoplanar surface
~k
diffraction was obtained in [15]. It was shown that the surface diffraction in
the two-wave case is characterized by two angles of total reflection (several
angles in the case of multi-wave diffraction [16]). The solution obtained in [16]
contains the component, which describes the state that damps with growing
distance from the surface of the medium, both within the material and in the
vacuum, and which describes a surface wave, i.e., a wave in which the energy
flux is directed along the boundary of the surface of a spatially periodic target
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k kτ
0
k
k 0 τ
0t k
τt
τ
k t
0t
k1 kτ1
photonic crystal
Figure 1. Surface diffraction of a radiated photon
(see review [17]). According to [15], this solution, which describes scattering
of a plane wave by the target under the surface diffraction geometry, can be
written in the form:
E~(+)s = e ei~k~r +A (~k,ω)ei~k1~r +B (~k,ω)ei~k2~r, (18)
~k s s s
~ ~ ~ ~ ~ ~ ~
where the wave vector in a vacuum k = (k ,k ), k = (k , k ), k =
t ⊥ 1 t ⊥ 2
−
(~k , ~k ), ~k = k2 k2 , ~k = ~k +~τ, ~k is the component of the wave
2t − 2⊥ | 2⊥| − 2t 2t t t
vector that is parallqel to the surface, ~τ is the reciprocal lattice vector, ω is
the photon frequency. The amplitudes A and B are given in [9,14]. Substi-
s s
tuting the solution E~(−)s = (E~(+)s)∗ into (3), we can find the spectral-angular
~k −~k
distribution of SPXR and DRO.
2 Time dependence of the intensity of radiation produced by a
particle transmitted through a crystal
TheintensityI(t)ofradiationproducedbyaparticlewhichhaspassedthrough
a crystal can be found with known intensity of the electric field E~(~r,t)) (mag-
netic field H~(~r,t)) of the electromagnetic wave, which is produced by this
particle [18],
c
I(t) = E~(~r,t) 2r2dΩ, (19)
4π| |
where r is the distance from the crystal, which is assumed to be larger than
the crystal size.
The field E~(~r,t) can be presented as an expansion in a Fourier series
1
E~(~r,t) = E~(~r,ω)e−iωtdω. (20)
2π
Z
8
According totheresults obtainedin[3,7,9],atalongdistancefromthecrystal,
the Fourier component can be written as follows:
E~(~r,t) = eikr iω es E~(−)s∗(~r′ω)~j(~r′,ω)d3r′. (21)
r c2 i ~k
s Z
X
where i = 1,2,3 (and correspond(s) to the coordinate axes x, y, z), es is the i-
i
component of the wave polarization vector ~es; s = 1,2; E~(−)s is the solution of
~k
Maxwell’s equations describing scattering of a plane wave with a wave vector
~k = k~r and the asymptotic of a converging spherical wave,
r
~j(~r,ω) = ~j(~r,t)eiωtdt (22)
Z
~j(~r,ω) = Q~v(t)δ(~r ~r(t)) is the is the current density of the particle with
−
charge Q, ~r(t) is the particle coordinate at time t.
The explicit form of the expressions E~(−)s describing diffraction of the elec-
tromagnetic wave in a crystal in the Laue and Bragg cases is given in [3,8,12]
(See Section 1).
Now let us take a closer look at the expression for the amplitude A(ω) of the
emitted wave:
As(ω) = iω E~(−)s∗(~r′,ω)~j(~r′,ω)d3r′. (23)
~k c2 ~k
Z
Using (22), (23) can be recast as follows
As(ω)= iω E~(−)s∗(~r′,ω)Q~v(t)δ(~r′ ~r(t))eiωtdtd3r′
~k c2 ~k −
Z
= iωQ E~(−)s∗(~r(t),ω)~v(t)eiωtdt (24)
c2 ~k
Z
~(−)s∗ ~(+)s ~(+)s
Recall that E = E , where the field E is the solution of Maxwell’s
~k −~k −~k
equations describing scattering by a crystal of a plane wave with wave vector
~
( k) and the asymptotics of a diverging wave at infinity. According to (24),
−
the radiation amplitude is determined by the field E~(−)s taken at point ~r(t) of
~k
particle location at time t and integrated over the time of particle motion.
Let us consider in more detail the constant motion of a particle in passing
through the crystal. In this case, parametric quasi-Cherenkov radiation can
appear [1,8], which includes, as a particular case, diffracted transition radia-
tion. The explicit formulas for the radiation amplitude in the case of two-wave
9
diffraction of photons in crystals for the Laue and Bragg geometries are given
in [3,8,12] (Section 1).
From (20), (21), and (23) follows that the expression for the electromagnetic
wave emitted by the particle passing through the crystal (natural or photonic)
can be presented in a form:
1
E~i(~r,t) = 2πr esi A~sk(ω)e−iω(t−rc)dω, (25)
s Z
X
i.e., E~ (~r,t) = 1 esAs(t r).
i r s i ~k − c
P
From(25)followsthatthetimedependenceoftheformofthepulseI(~r,t)(E~(~r,t))
of radiation generated by a particle passing through the crystal is determined
by the dependence of the radiation amplitude As(ω) on frequency. Accord-
~k
ing to the explicit expression for the radiation amplitudes given in [3,8,12],
the radiation amplitudes As(ω) can be presented as sums proportional to the
~k
amplitudes of diffraction reflection from the crystal and to the amplitude of
wave transmission through the crystal. For example, for the case of forward
parametric radiation in the Laue geometry
A~sk(ω)= cQ2(~es~v) ξµ0seiγω0εµsL
µ=1,2
X
1 1 ei(ω−(~k+κµsN~)~u)cLγ0 1 (26)
×"ω ~k~v − ω (~k +κµsN~)~v#(cid:20) − (cid:21)
− −
Thus, the time dependence of the from of the radiation pulse is determined
by the time dependence of the radiation amplitude As(t r).
~k − c
By way of example, let us consider the characteristics of the time dependence
of radiation produced by a particle passing through the crystal for a wave
packet passing through the crystal [2,5,6]
Let us consider the pulse of electromagnetic radiation passing through the
medium with the index of refraction n(ω). The group velocity of the wave
packet is as follows:
−1
∂ωn(ω) c
v = = , (27)
gr c∂ω ! n(ω)+ω∂n(ω)
∂ω
where c is the speed of light, ω is the quantum frequency.
10