Diffraction-limited resolution of the optical transition radiation monitor

Jg$L&

Nuclear Instruments

and Methods in Physics Research

A 372 (1996) 344-348

NUCLEAR INSTRUMENTS 8 METHODS IN PHYSICS RESEARCH SectIon A

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ELSEVIER

Diffraction-limited

resolution of the optical transition radiation monitor ’ V.A. Lebedev *

Cnrnitumus Elrctrm

Beam Accclemmr

Received

Facility.

1X.W

Jrfirrso~

Aw..

17 July 1995; revised form received

Newport

16 October

News, VA 2.7606. USA

1995

Abstract The space resolution of the optical transition radiation (OTR) monitor is discussed. It was found that for an ultrarelativistic electron beam the space resolution does not depend on beam energy but long tails are created in the image which make a measurement of the beam halo difficult.

1. Introduction During the last several years the profile measurement of an electron beam using optical transition radiation (OTR) has been developed in many laboratories [ 11.In contrast to other laboratories, the beam size at the Continuous Electron Beam Accelerator Facility (CEBAF) is very small (g-50 mm for both transverse sizes) and the question of the ultimate monitor resolution is very important. Although the theory of transition radiation was developed about 50 years ago [2]. the discussion is still continuing about this diffraction limit [3,4]. It is well known that for an ultrarelativistic beam the angular divergence of the radiation is inversely proportional to the Lorentz factor, 0% I l-y. Then. using a standard estimate for the microscope resolution, Ap=A/f?, we obtain that resolution becomes worse with energy increase. Ap=I\y. Thus. for the maximum

CEBAF energy of 4 GeV we obtain that the resolution for visible light should be about a few millimeters. On the other hand it was predicted in Refs. [3,4] that the resolution is not significantly different from standard diffraction limits, Ap=AI@, where 0, is the angle subtended by the focusing lens from the point of radiation. Thus. this article is devoted to the calculation of the OTR intensity in the image plane (the plane where OTR radiation is focused by an ideal lens).

2. Electric field in the image plane Consider a simple model which allows us to do a correct calculation of the phenomenon. The scheme is shown in Fig. I. An electron with ultrarelativistic energy, E=mc’y, y >> I, comes normally onto a perfect conductor. The emerging transition radiation propagates from the entrance point (.r=O, v=O) and is focused on the screen S by an ideal lens with the focus distance F and the diameter D. As usual. distances from the lens to the object and its image are bound up by the equation

The expression for Fourier components of fhe radiation field are well known [5,h]. For the ultrarelativistic case we need to consider only small angles and the equation can be simplified so that the electric field is’: ’ In comparison with Ref. [S] we Insert here a factor have the following definition of Fourier harmonics:

Fig. I. Layout of measurement. ’ Work supported by 84ER40 150. *Corresponding author: [email protected]. Elsevier Science B.V. SSDI 0 16%9007(95

U.S.

DOE

contract

#DE-ACOS-

E(r,r) = dw dq, dq, i

Tel.

)01400-4

+ I

804

349

7114,

e-mail

I E,,$)

= ~

(h)’

E,,,V(s)r-“9’-‘““,

dr dv d: E(r,t)e”“-

““‘.

I /(3-r) to

V.A. Lehedev

I Nrtrl. lmr.

and Merh. in Ph.vs. Res. A 372 (1996) .W-SJ8

34s

(2) Here q is the transverse

component

of the wave vector k.

(6) we finally have

O=qlk.

Our task is to find an electromagnetic field on the screen S. To do it we will use a standard procedure. First. we need to find an image of each Fourier harmonic of the beam radiation on the screen S and then we must find a sum of all harmonics. Fig. 2 shows a picture of wave propagation. A plane wave radiated at an angle 19will be focused at the focal plane of the lens in a point with y-coordinator equal to F tg 8. The phase advance of the wave between the origin of the coordinate frame (point 0) and the focus point 0, is the same for all rays of this wave and is equal to

t.4

F a cos 0 + ~ cos e >

*

=

. B<<‘l.

(7)

The first three addenda do not have a dependence on q and we will omit them as a common phase shift for all waves. To understand how the wave amplitude is changed we use conservation of energy. Consider a wave with amplitude E, incident on the lens. Then a round spot of radius rP on the lens will be imaged on the screen with radius ‘C

‘F

b-F F

(8)

- “,’

Equalling energy fluxes on the lens and screen.

=k(

Note that advance in calculation. focus point v=p cos 4,

PO -ycoscp

Efrf = Et-l-;,

we omitted here the common for all rays phase the lens, which does not affect the result of An additional phase advance between the and a point on the screen with coordinates :=p sin 4 is

(b - F)” + (F ~ F tan 0 )’ + :‘.

(4)

Adding phases from Eqs. (3) and (4) and expanding the result for small 0 we obtain the dependence of phase on the screen for the initially plane wave described by Eq.

(9)

we obtain Es=5

(IO)

Combining Eqs. (2). (7) and (IO) and summing Fourier components we obtain a wave amplitude at a point located on the screen at distance p from the axis E,o(p) = -i-

I 4neH q dq, dq, (2n)‘V IIL, W(Y_’ + 8’) .s

(2):

(11) Here we omit the common phase. + p2 - 2FpO cos 4 2(b -F)

1,

(12)

H<
Substituting parameter CI from Eq. (I) and taking into account that the amplification of the image is,

The integration over dy, dq, can be replaced by the integration over angles 4 and 13of a spherical coordinate frame with the axis directed along the x-axis so that dq, dy, = y dq d4 = k”sin 6 d(sin 8) d4 = k’B de d&. (13) Owing to symmetry. the electric field has only a radial component and we can replace q/q in Eq. ( I I) by cos 4. Thus, summing up contributions of all plane waves radiated into the solid angle subtended by the lens from the point of radiation we obtain from Eq. (I 1)

‘I.a

S

Fig. 2. Scheme of wave propagation.

kp0 cos c$

v

(14)

Here Q,,is the angle subtended by the lens from the point

of radiation, 6’,,=(Dl(2n). and D is the lens diameter. Strictly speaking, we can do it only for the case when the approximation of Fraunhofer diffraction is justified. As can be seen later. the full size of the region which emits the radiation is about dr= yA: then we can put down the condition of the Fraunhofer diffraction (1 >>g

A

= Ay’.

(15)

In reality it is not strictly required because the main part of the radiation is nevertheless concentrated in much smaller size and Eq. ( 13) will work quite well for smaller distances than are determined by Eq. (15). The integral over 4 in Eq. ( 14) can be calculated with use of the formula

of the function @‘(r,tl) for f/=10. 100 and 1000 and its comparison with the square of the function F(r) of Eq. (22) are shown in Fig. 3. One can see that for small I’. r-50.219 the function @(r,B) can be approximated with very good accuracy by the function F(r). This means that for an ultrarelativistic beam an intensity of the main part of radiation, piO.2yH,,lk. is proportional to F’(kpH,,) and exponentially dies beyond this interval. Thus for an ultrarelativistic beam the radiation intensity is proportional to 1Ip’ in the range I << kp << yH,, and the full intensity of the radiation can be easily calculated with logarithmic accuracy

PC,,=

P)~TP dp = $ln( I 0 s
yH,,). yH,, >> I

4. Discussion and we finally obtain. (17) where J,(z) is the Bessel function of the first order.

We will determine the FWHM resolution of the OTR monitor as a diameter of the circle at half of the full

‘-

3. Spectral power in the image plane We will use the general definition of the spectral power for radiation so that the power flux (the Poynting vector) is equal to S=

= S<,,(w)do. I_.

(18)

For a process which consists of pulses randomly distributed in time the spectral power of the radiation is equal to S<”= +

IE;J.

(19)

where E, is a Fourier spectrum of one pulse, and ri is the average number of pulses for one second. Substituting I?,, from Eq. (I 7) we finally obtain the spectral power for the point beam with current I,,

I

I

O1.. II

1 !

I

!

I

I

!

I

!

I

I

where we introduce the function

(21) For the case r << -9 or, what is the same, kp << y, we can neglect I in the denominator and we obtain that

I - J,,(r) r<
(22)

where J,,(u) is the Bessel function of the zero order. A plot

Fig. 3. Dependence @‘(.r.z) upon .r for := IO, 100 and 1000 and comparison with F’(x) (bottom plot).

V;A. Lehedev

intensity. Then the resolution radiation (V= I ) is

related

/ Nucl. Instr. and Meth. in Phvs. Res. A 372 (1996) 344-348

to the point

A FWHM= I .44hlB,,. ye,, >> I,

of

(25) Nevertheless, unlike from the point source, the OTR has a dark spot in the center and a large halo which, in reality, carries much larger energy than the central spot. To understand the above results we consider a simple model. For an ultrarelativistic particle the electric field is compressed along the particle velocity, so that the field width at distance Y in the transverse direction is about ply. For large enough wavelengths and ultrarelativistic beam all this field will be reflected from the perfect conductor with, in first approximation, no change in the field shape and intensity. Because a size of the OTR light source is much larger than the wavelength we can consider radiation from each point independently. For p << y/l the field width is smaller than the wavelength and we can consider it as the S-function

r=p+.k

eycrcp2

+

yz(_y

_

v,,t) ,,()t)Z)vZ

p

from

the place

where

the particle

hits the

(24)

where A is the wavelength at which the radiation is registered. A comparison of intensities for OTR and the point light source is shown in Fig. 4. One can see that the spot size for OTR is about 3 times larger than for the ideal point source (microscope resolution), which intensity is described by the well known formula

E@J) =

distance screen,

341

=

p.x-

VJ),

(26)

Taking the Fourier transform of Eq. (26) we obtain the Fourier component of the field radiated by a particle at

This equation is justified at distances A << p << yA. The Fourier component will exponentially decrease at larger distance where the field width is larger than the wavelength. Thus, one can see that the OTR is generated at a region with size about yA and, consequently, its image has to have the same size multiplied by magnification V. Diffraction on the lens aperture limits the field of the image for distances p”VAlO,, and generates the radial oscillation in the intensity, but a general behavior of fields described by Eqs. (27) and (17) is similar. Really, substituting Eqs. (21) and (22) into Eq. ( 17) we obtain

=+( 1 -J,,(7)),kp <
Ecu

which is in agreement with Eq. (27), taking into account the magnification and the changes due to diffraction on the lens aperture. As can be seen from the described picture the result will not be changed for the case of non-normal incidence. It is explained by the fact that the field of an ultrarelativistic electron is very close to a field of the free electromagnetic wave so that this field reflects the same way as light does. Note also that a small decrease in reflectivity for a nonperfect conductor will cause a decrease of intensity but will not cause a change in the shape of power distribution on the screen. Finally we can conclude that for an ultrarelativistic beam the resolution of the OTR monitor does not depend on energy. The same as for the ideal point source, it is determined by the wavelength of the light and the angle subtended by the lens from the point of radiation. But unlike the ideal point source, the OTR has a large halo which carries the main part of the radiation power and which grows with energy as yh.

Acknowledgments I would like to thank J.-C. Denard, G. Krafft, R. Legg and M. Tiefenback for useful discussions and the help in editing of this article.

References

Fig. 4. Comparison of intensity distributions for the OTR and the ideal point source related to point of radiation, .r=kpB,,.

[I] L. Wartsky, Thesis, Univ. de Paris-Sud, Centre d’Orsay (19761; J. Bosser et al., Optical Transition Radiation Proton Beam Monitor, CERN/SPS/84-17.

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