The spectral-angular density of electromagnetic energy flux through the small detector in the "pre-wave zone" caused by relativistic electron in "forward" direction in the case of normal electron transition through a thin metallic transverse-bounded target is investigated. We show that detected intensity can be very differ from the spectral-angular density of radiation as well it is measured in the "wave zone" as it is measured in "pre-wave zone" by infinite detector. This distortion is depended from the detector placing in the "pre-wave zone" and from the ratio between transversal size of target and effective transversal diameter of radiation formation region

Dobrovolsky, S.N.

Shul’ga, N.F.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

LONG-WAVELENGTH TRANSITION RADIATIONBY RELATIVISTIC ELECTRONS IN A “PRE-WAVE ZONE”S.N. Dobrovolsky, N.F. Shul'gaNational Science Center “Kharkov Institute of Physics and Technology”, Kharkov, UkraineThe spectral-angular density of electromagnetic energy flux through the small detector in the "pre-wave zone" caused by relativistic electron in "forward" direction in the case of normal electron transition through a thin metallic transverse-bounded target is investigated. We show that detected intensity can be very differ from the spectral-angu-lar density of radiation as well it is measured in the "wave zone" as it is measured in "pre-wave zone" by infinite de-tector. This distortion is depended from the detector placing in the "pre-wave zone" and from the ratio between transversal size of target and effective transversal diameter of radiation formation region.PACS:  41.60.-m, 41.90.+e, 41.75.Ht.1. INTRODUCTIONThe transition radiation (TR) from relativistic elec-trons is widely used in last years in various experimental investigations. For example, the long-wavelength TR is explored  for  diagnostic  of  electron's  beams and  as  a source of quasi-monochromatic long-wavelength waves [1-4]. The thin metallic plates are usually used for pro-duction  the  long-wavelength  transition  radiation  from the ultra-short relativistic electron's bunches. Transition radiation in the "forward" direction is formed over long distance for the case of high-energy electrons and mil-limeter waverange. This distance ("coherence length" or "pre-wave zone") is determined in the order of value by the relation λγ 22≈L  , where γ  is the Lorentz-factor of electron and λ  is the radiated wave length. It can reach the tens of meters for relativistic electrons and millime-ter waverange. So, if the detector is placed in the "pre-wave zone", the strong interference between own elec-trons fields and radiation fields is observed [5,6]. This well-known effect was investigated theoretically and ex-perimentally for case when both detector  and metallic plate have infinite diameter. Recently the investigation of new effects connected with finite diameter of targets and detector was present-ed. So, the effect of targets diameter influence on the TR intensity was investigated [7-9]. The influence of detec-tor diameter on the TR intensity was studied in [10] for "backward"  TR  in  small-angle  approximation.  It  has been shown in [7-10] that the TR intensity is suppressed in compare with classical case of infinite diameter of tar-get and detector, when the target or detector diameter is less or equal to γ λ . Such effects can be very important for  analysis  of  modern  experiments  concerned  to  the millimeter  and  sub-millimeter  TR  of  relativistic  ultra-short electron bunches.In  the  present  paper  we will  investigate  the  long-wavelength "forward" TR from relativistic electron on a thin metallic transversally bounded target. We will ana-lyze the spectral-angular distribution of electromagnetic energy flux through the small detector within "formation zone". 2. THE TRANSITION RADIATION AND OWN ELECTRON ELECTROMAGNE-TIC FIELDWe will consider the case of normally electron pas-sage through the center of thin metallic disc with radius a . We assume that disk is fairly thin, i. e. the thickness of target is less than radiated wavelength  λ , but much more than "penetration length" of electron field into a metal. Let's the electron is moved along the OZ axis. We will interest by the energy flux through the small plate (detector), which placed at the distance r  from the tar-get center. Firstly, we should calculate the electromag-netic  fields,  which  are  formed  after  electron  passage through the  target.  The  summary electric  field  in  for-ward (concerning electrons passage) direction is consist-ed  of  the  own  electron  field  ),()( te rE  and  TR  field ),( trE′ :),(),(),( )( ttt e rErErE ′+=+ .            (1)The Fourier component of the ),()( te rE  field with re-spect to time is determined by the expression( ) ( ) −=vizvKvivKvcvee ωγω ργγω ρργωω exp1201vρrE , (2)PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, 121-124. 121where  e - is an electron's charge, v=v ,  ρ is transver-sal  coordinate  ( zeρr ⋅+= z ,  ze  is  unit  vector  of  OZaxis), 0K  and 1K  are McDonald's functions of zero and first kind.In the relativistic case (when  1> >γ ) the electron's own field can be considered as transversal. It can be rep-resented with accuracy of 1−γ  by expression( ) ( ) ≈vizvKvee ωγω ργωρωexp2 12ρrE .  (3)We  assume  that  target  is  an  ideal  thin  metallic screen, therefore  the part  of  own electron field  which fall  into  target  is  fully  reflected  and  the  part  which missed the target is diffracted onto space 0>z . So, we can write for  summary electric  field and TR field the next conditions)()()()( )()( rErErE ee a ωωω ρ −Θ=′+  , 0=z , (4))()()( )( rErE ea ωω ρ−Θ−=′ , 0=z ,where  )(xΘ - is a Heaviside step function: 1)( =Θ x , if x≥0, and 0)( =Θ x , if x < 0. The "forward" radiation field ),( trE′  is propagating in positive direction of OZ axis as a wave packet of free electromagnetic wavesNow, by using the equations (3) and (5) we obtain the expression for the "forward" and "backward" radia-tion fields in the point zeρr ⋅+= z( ) −±××+⋅−=′ ∫∞2202212exp),()(2)(χωγωχγωχχ ρχχρωcizvvaaFJdve ρrE,          (5)where ∫ += a vKJdvvF 0 1122 )()( γω ρχ ρρργ χωγωχ . (6)The 1J  is Bessel function of first kind. The formula (5) is describing the "forward" TR field which appeared af-ter relativistic electron passage through a thin metallic disk.  This  expression  is  correct  under  all  distances greater than radiated wavelength from the target.3. SPECTRAL-ANGULAR DENSITYOF ELECTROMAGNETIC ENERGY FLUXNow let's consider the electromagnetic energy flux that crossed over the entire observation time through the small detecting plate that is perpendicular to the vector r  and  placed  at  the  distance  r  from the  target.  We should calculate the value of Poynting vector. The flux of the Poynting vector is given by∫ ×= dtdcS σpi 2)(4 nHE ,    (7)where r/rn =  and σ2d  is an elementary area with nor-mal vector n . Here the integration is over the surface of detecting plate and the time of observation. We express the  field  H  through  E  and  expand  these  fields  into Fourier integrals by time. After that, we can write sum-mary energy flux in the "forward" direction on a unit of frequency and on the unit of solid angle by expression 2)(22),(),(4zzcrdddS e ρEρE ωωpiω′+=Ω ,   0>ω .       (8) Here ϕϑϑ ddd sin=Ω  ( ϑ  and ϕ  is the polar and the azimuthal angles). We can calculate the transition radia-tion intensity and energy flux through the detecting plate by using the equations (5). So, the spectral-angular den-sity of electromagnetic flux is followed2)()(2242cos rade SSycedddS−⋅=Ωϑpiβω , (9)where ( ) [ ]1111)( expsin −−−= βϑ γγ iqyKS e  ,∫  −+⋅=−1022212)( 1exp),()sin( xiqxuxFxyJdxxS radγγϑβ ,∫+=uwxJwKdwwxxuxF0112)()(1)(),( γγγγ ,1−⋅⋅= cry ω , ϑcosyq = , 1)( −⋅⋅⋅= γω cau , 1−⋅= cvβ .The expression (9) describes the energy flux in "for-ward" direction through the small detecting plate that is seemed on  the  solid  angle  Ωd  and  placed  into  point zeρr ⋅±= z . In other words, it is a spatial distribution of electromagnetic energy flux for waves with the frequen-cy  ω .  It  is  obviously that  spectral-angular  density of electromagnetic energy flux is composed from the term of energy flux of own electron field ("electron's term"), from the  term of  energy flux of  TR  field  ("radiation term") and the interference term. The terms of equation (9) depends on distance from the target and on target's diameter. Further we will analyze the angular distribu-tion  of  spectral-angular  density  of  "forward"  electro-magnetic energy flux for various distances from the tar-get.4. DISCUSSIONLet's consider the angular distribution of electromag-netic energy flux (9). While  the  transversal  size  of  target  is  essentially greatly  than  transversal  diameter  of  formation  zone  (1> >u ), the function F  is near to unit. This case corre-sponds to  the  problem of transition radiation by rela-122tivistic  electron  on  the  infinite  plate.  So,  for  the  dis-tances γ λ>z  we can to simplify the formula (9)2)(0)(022222cos rade SSycedddS+=Ωϑγpiβω ,     (10)where ( )111)(0 sin −−= ϑ γβ yKS e ,∫∞ −−+=0222112)(02cosexp1)sin(uyiuuyJduuS radγϑϑγ .The term of radiation field energy flux in the  equa-tion (10) depend on the value 22cosγϑy. It is the ratio be-tween the longitudinal  detector-target  distance and the longitudinal  "coherence  length"  ("pre-wave  zone") ωγ cl 22= . Such dependence for )(0redS  from parameter lz /  is caused by the small diameter of detector.  Let's explain this fact in detail. Toward this end we consider the  spectral-angular  density of  electromagnetic  energy "forward" flux in case when the detector is a transversal-ly-infinite plate. It represented by the following expres-sion [7]2)()(22222sinsin rade SScedddS∞∞−∞−+=Ω γϑϑpiω,     (11)where( )vizS e ωexp)( =∞,   ( )cizS rad ωϑcosexp)( =∞.The module of )(radS∞ doesn't depend on target-detector distance, whereas the module of )(0radyS  sufficiently de-pends on this distance. Let’s note that the spectral-angu-lar density of TR does not depend from the distance to the target in the classical case when the detector is an in-finite plate. Following reason caused this situation. The fields of TR are formed by the radiation from the target surface with λ γ  effective diameter. So, the interference between radiation from various elementary radiators on the target's surface is appeared. Thanks to this interfer-ence the term )(0radS  depends on the target-detector dis-tance.  Within the limits of "pre-wave zone" (i.e., when λγ 22≤z ) only some part of whole radiated field is fall into detector. The radiation source is not dot-like in this case.  In  the  "wave  zone"  (i.e.,  when  λγ 22> >z )  the module of  )(0radyS  doesn’t depends from the target-de-tector distance. The radiation source is seemed as a dot-like source in this case. The above-considered effect oc-curred when the detector is small plate and it is absents when the detector is an infinite plate.The whole spectral-angular distribution (10) has also the target-detector distance dependence that is caused by the interference term. It is the interference between own electron field and TR fields, which is sufficient within the "coherence length" ωγ cl 22= . The graphs on Fig. 1 show the angular distribution of electromagnetic energy flux (10) under various distances from the target. The formula (10) showed that the detec-tor  is  placed  closely  to  the  electron's  trajectory  (i.e., when  λ γρ ≤ )  the own electron's  field gives the main contribution  to  the  summary  electromagnetic  energy flux. This  situation corresponds to  the small-angle re-gion on Fig. 1. If the detector is placed far from the elec-tron's trajectory (i.e., when λ γρ ≥ ) the main contribu-tion in the whole electromagnetic energy flux is caused by the TR fields. This case corresponds to the great-an-gle region on Fig. 1.Fig. 1. Angular  distribution  of  spectral-angular  density of electromagnetic energy flux in "forward" di-rection  for  electron  with  30=γ .  Solid  curve  corre-sponds case when  50=y , dotted curve -  300=y  and  dashed curve - 3000=y . W(Θ)=cπ2e-2(dS/dωdΩ) and Θ is marked in a degree scaleThe angular distribution on Fig. 1 have showed that if the detector is placed in the "pre-wave zone" ( 50=y , 300=y ), the summary electromagnetic flux is strongly oscillate  in  the  large-angles  region.  These  oscillations are caused by the interference both between own elec-tron's field and TR fields, and radiation fields from the various elementary radiators on the target's surface. Both this interference effects are decreased under increasing of target-detector distance. So, within the "wave zone" (3000=y ) these effects are diminished.Now consider the case when the target's diameter is near or less than  λ γ .  In this case the function  F  has strong oscillations under 1≈u  and 1< <F  under 1< <u . The suppression of "radiation term" in the equation (9) is occurred.  The additional distortion of summary elec-tromagnetic energy flux through the small detector will be observed.Fig. 2 shows the angular dependence of electromag-netic energy flux (9) in the "pre-wave zone" for various diameters of target. The detected summary spectral-an-gular electromagnetic energy "forward" flux is strongly distorted at the large angles when target diameter is near to the value λ γ .5. CONCLUSIONSWe have considered the long-wavelength transition radiation  from relativistic  electron  on  a  thin  metallic ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.Серия: Ядерно-физические исследования (36), с. 3-6.123disk in the "pre-wave zone".  The  general  formula for spectral-angular density of electromagnetic energy flux through the small detector in the "forward" direction is obtained. We have analyzed the spectral-angular distri-bution of "forward" electromagnetic energy flux through the small detector. Fig. 2. Angular  distribution  of  spectral-angular  density of electromagnetic energy flux in "forward" di-rection for electron with  30=γ  on the finite-diameter  target.  Dotted  curve  corresponds  to  the  case  when 5.0=u , solid curve - 2=u . The distance from the tar-get 300=y , W(Θ)=cπ2e-2(dS/dωdΩ) and Θ is marked in  a degree scaleIt is shown that within the limits of "pre-wave zone" the interference between radiation fields from various el-ementary radiators and interference between own elec-tron field and whole radiation field are occurred. Both the  spectral-angular  density  of  whole  electromagnetic energy flux and spectral-angular density of TR is depend on distance from the target. We have analyzed the angular distribution of electro-magnetic energy flux in the "pre-wave zone". The strong oscillation of intensity of electromagnetic energy flux is observed in the  intermediate  range  of  observation an-gles. For small angles (when detector is placed nearly from the electron trajectory) and for big angles (when detector is placed at the length much more than λ γ  from the electron trajectory) the own electron's field energy flux and the TR energy flux is dominated correspond-ingly. For the case when the target diameter is equal or less than transversal size of "pre-wave zone"  λ γ  the suffi-cient suppression of TR field contribution to the whole electromagnetic energy "forward" flux is observed.A  number  of  experiments  concerning  long-wave-length  TR  of  relativistic  electron  bunches  were  per-formed in the last years [1-4,6,11,12].  For this experi-ments  conditions  (electrons  with  10010 ÷≈γ ,  long-wavelength  TR  with  .101 cm÷≈λ )  the  longitudinal  (λγ 22≈l )  and  transversal  ( λ γ≈⊥l )  size  of  "pre-wave zone" are the macroscopic values and can be compared with distance to the detecting setup and transversal tar-get's  size.  So,  the  considered  interference  effects  can play important role in such experiments.REFERENCES1. Y. Shibata,  K. Ishi,  T. Takahashi  et  al.  Coherent transition radiation in the far-infrared region // Phys.  Rev. E. 1994, v. 49, №1, p. 785-793.2. U. Happek,  A.J. Sievers  and  E.B. Blum.  Observa-tions  of  coherent  transition radiation  //  Phys.  Rev.  Lett. 1991, v. 67, №21, p. 2962-2965.3. X. Artru,  R. Chehab,  K. Honkavaara,  A. Variola. Resolution power of optical transition radiation: the-oretical  considerations  //  Nucl.  Instr.  and  Meth.  B 1998, v. 145, №1-2, p. 161-168.4. V.V. Kaplin,  S.R. Uglov.  Original  and  diffracted transition radiation for diagnostic of relativistic elec-tron beams  // Nucl. Instr. and Meth. B. 1998, v. 145, №1-2, p. 253-259.5. L.S. Wartski, S. Roland, J. Lasalle et al. Interference phenomenon in  optical  transition  radiation  and  its application to  particle beam diagnostics and multi-ple-scattering measurements //  J. Appl. 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Long-wavelength transition radiation by relativistic electrons in a pre-wave zone

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Long-wavelength transition radiation by relativistic electrons in a pre-wave zone

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)