SYNCHROTRON RADIATION and its interferometry
at cebaf beam lines*
P. Chevtsov, Jefferson Lab, Newport News, VA 23606, USA
INTRODUCTION
Two first synchrotron light interferometers (SLI) have recently been built at Jefferson Lab. Each SLI is a valuable beam diagnostic instrument for the CEBAF accelerator. It is a non-invasive, multi-element measurement device consisting of a CCD video camera, a video image processing system, and a set of stepper-motors to adjust the positions of optics parts and diffraction slits. The main functions of the SLI elements are automated with the use of new distributed control software.
Figure 1: The experimental Hall A beam line at Jefferson Lab with the installed synchrotron light interferometer.
The SLI design at Jefferson Lab is a “classic” wave front division interferometer that uses polarized quasi-monochromatic synchrotron light [1]. It has a 3-D structure, with major elements placed on two horizontal levels that are parallel to the ground plane (see Fig.1). Limited space and relatively high radiation in the accelerator tunnel strongly influence this SLI design and implementation.
1. SYNCHROTRON RADIATION and some its PROPERTIES
Synchrotron radiation (SR) is emitted from relativistic charged particles when they are traveling on curved paths. The dipole magnets of the charged particle accelerators are the most common generators of the SR.
Because of strong relativistic effects (factor b»1), the synchrotron radiation is emitted in a very narrow cone in the forward direction tangent to the particle orbit (Fig. 2). For example, for 5 GeV electrons, a typical opening angle of the SR is ~10-4. In the other words, each relativistic electron traveling in a magnetic field looks like a moving flashlight giving off synchrotron light in front of itself.
The SR has a wide energy spectrum, from infrared to g rays, including visible light. In the visible part of the energy spectrum, the properties of the SR are independent of the particle energy and depend only on the radius of the trajectory curvature R. For example, the RMS opening angle of the visible SR emitted by electrons in basic bending magnets of CEBAF beam transfer lines (R=40 m) is ~10-3.
Figure 2: Angular distribution of synchrotron radiation.
2. SLI and its basic ELEMents
The synchrotron light generated by the electron beam in a dipole magnet (a blue element at the center of Fig.1) is extracted through a quartz window by a mirror installed in a vacuum chamber. Two additional adjustable mirrors guide the synchrotron light through the SLI optical system. One of them is remotely controlled with the use of the RS-232 serial interface. The main task of this mirror is to send light on the CCD head through a long (~5 m) plastic pipe, diffraction slits, and all SLI optical components (a narrow band pass filter, a polarization filter, and a video camera objective lens), in the direction opposite to the direction of the electron beam. The CCD and optical components are placed in an optical box. A double slit assembly with a predefined set of distances between slits and small slit openings is located right in the front of the camera objective. The assembly is moved in horizontal and vertical directions by the remotely (RS-232) controlled stepper-motors.
The SLI video camera is the STV digital integrating video system from the Santa Barbara Instrument Group [2]. The camera has its own control box with the RS-232 interface to an external computer. Its quantum efficiency is high (~60-70%) and the pixel size is very small (7.4´7.4 mm). An electronic cooling system keeps CCD dark currents extremely low. The exposure time of the camera can gradually be changed from 0.001 seconds to 10 minutes. The CCD camera is connected to an image processor. The SLI image processor is Datacube’s Maxvideo MV200 board [3] that is the basic video image processing system for beam diagnostic applications at Jefferson Lab.
In the SLI, the synchrotron light generated by the electron beam produces an interference pattern. With a Gaussian beam profile approximation, we can easily calculate the RMS beam size from the visibility (contrast) of the interferogram [4]. The SLI control and data processing software created at Jefferson Lab [7] automates the main functions of the synchrotron light interferometer and performs all necessary mathematical calculations.
Figure 3: Technical drawing of the beam line at location 3C12 (Hall C line).
3. geometry of sli locations
The Hall A and Hall C beam transport lines at Jefferson Lab are built with the use of a set of “standard” magnet structures that consist of a dipole magnet with a bending radius of about 40 meters (m) following several correctors and a quadrupole magnet. The technical drawing represented in Fig. 3 shows such structures at the high dispersion location 3C12. All sizes and distances provided in this figure are in meters. The transverse sizes of the vacuum chamber are a 7.62x2.54 cm rectangle inside the dipole magnet that is 3 m long, a 2.54 cm diameter circle in areas between dipoles (these areas are 1.37 m long), and a 5.08 cm diameter circle in connecting sections.
The main synchrotron light extraction (“in-vacuum”) mirror (3.81x2.54 cm rectangle, aluminized) of the SLI is located in a short distance (0.18+0.11 m) after the dipole magnet and is looking approximately at the center of the magnet, at the design orbit of the electron beam.
It appears that in conditions of high dispersion locations the SLI is very sensitive to the position of this mirror relatively to the beam trajectory. At some positions, the SLI can see synchrotron light generated not only in the dipole magnet closest to it (we shall call it magnet B) but also in the upstream dipole (magnet A). As a result, the SLI interference picture becomes a superposition of interferograms from multiple sources. This makes the direct interpretation of the visual SLI data more complicated and requires either the modification of the SLI design to get rid of reflection sources or the development of special mathematical SLI data models that work well even in “multi-source” cases.
Figure 4.
It is easy to calculate that if the arc ABC (see Fig. 4) is the design beam trajectory in the dipole magnet with bending radius R=40 m, then it is only 0.7 mm longer than the 3 m chord of this arc ADC (Fig. 3).
Such small differences in the lengths of circular arcs representing the beam trajectory and their chords make it possible to create a simplified SLI geometry model. This model allows us to get not only a good qualitative picture of the synchrotron light behavior but also very good estimates for the locations of synchrotron light sources at high dispersion areas that can be obtained with the use of more precise and complicated models. At the same time, the model is so simple that problems of the synchrotron light behavior in the accelerator beam lines can be good exercises in calculus and geometry optics for high school students. The basic SLI geometry problems and their solutions are presented in the Appendix to this paper. In the main part of the paper, we will reference our Appendix as it is necessary.
4. 2-D sli geometry model
We shall assume that the electron beam moves very closely to the design orbit and examine the propagation of the maximum intensity of visible synchrotron light generated by the beam. In these conditions, all synchrotron light emitted by the beam is described by a light ray emitted from the design beam trajectory and we can use the approximations of geometry optics to study the propagation of this ray in the accelerator. The structure of the vacuum chamber and beam steering components at high dispersion locations allows us to further simplify the model and consider the propagation of the synchrotron light ray only in the horizontal midplane of the beam line. The SLI geometry model becomes two-dimensional.
Define how the vacuum chamber and the design beam trajectory at location 3C12 can be described in the frames of our geometry model.
Begin with the beam trajectory that is represented by a solid red curve in Fig. 5. Two basic arcs C1C and DD1 of the circles with radii R=40 m and centers at points OA1 and OB1 connected by the straight line segment CD that is 2.19 m long (0.18+0.21+1.37+0.24+0.19 – see Fig. 3). The trajectory then continues along the line segment D1D2 that is tangent to the arc DD1 at D1. The length of each basic arc is 3 m. The end points of the segment CD together with OA1 and OB1 form the vertices of a rectangle with dimensions 40x2.19 m. CD is tangent to both arcs at points C and D, and so the beam trajectory in our model is a smooth curve.
Figure 5.
Switch to the walls or boundaries of the vacuum chamber represented by solid blue curves in Fig. 5.
Shorten the vacuum chamber with a rectangular cross-section in the area of the magnets by 1 cm at the beam entrance (left) side. As it was already mentioned above, in conditions of the circular shapes with big radii and relatively small arc angles it will not have any noticeable impact on the results. At the same time, this will bring significant elements of symmetry into the combination of the beam trajectory and boundaries of the vacuum chamber inside the magnet. Consider, for example, the area of magnet B shown in detail in Fig. 6. The walls of the vacuum chamber become the arcs B23B26 and B25B28 of concentric circles with center at the point OB2. The last one is the intersection point of the lines B24B21 and B27B22. The lines B24B21 and B27B22 are perpendicular to the beam trajectory at B24 and B27 respectively where the beam enters and exits the chamber (the distances |B24D| and |D1B27| are equal to 0.18 m). From Fig. 6, is not difficult to calculate that if OB1B21 is perpendicular to B24B21 and |B21OB1|=0.18, then |B21OB1|=4.7977. Thus, the radii of the walls of the vacuum chamber inside the magnet are Rb=|B24OB2|+d=44.7977+0.0381=44.8358 (the external wall) and |B23OB2|=Rb-2d=44.7596 m (the internal wall).
Figure 6.
The walls of the vacuum chamber between dipoles are represented (from left to right) by two segments of 0.21 meters each, two of 1.37 m, and two of 0.24+0.01 m which are symmetrical with respect to the beam trajectory and separated from it by 2.54 cm, 1.27 cm, and again 2.54 cm. A few additional segments connect the end points of the mentioned above elements to complete the boundaries of the vacuum chamber.
In the SLI, we observe synchrotron light by positioning the “in-vacuum” mirror very closely (~1 cm) to the beam orbit, at the distance L=|D11D0|=0.11 m from D11. To simplify the alignment of all “out-of-vacuum” SLI components, the center of the mirror must be placed along the straight line D0D00 (that is perpendicular to the beam trajectory D1D2) and positioned to extract synchrotron light from the vacuum region in the direction of D0D00 or very close to it (see Fig. 5).
5. Synchrotron light behavior at location 3C12
Apply our SLI geometry model and the approximations of geometrical optics to determine the behavior of synchrotron light in the accelerator beam line at location 3C12. We remind that in the frames of this model, synchrotron light is only generated when electrons move along arcs C1C and DD1. The light is emitted tangentially to the electron trajectory and is reflected from the vacuum chamber walls in accordance with the classic law of reflection in optics.
Consider the first synchrotron light ray generated by the beam in magnet A and entering magnet B without reflection on its way. We shall call this ray the first direct synchrotron light ray from magnet A entering magnet B. This ray goes right above, almost through the point P that is the lower rightmost point of the 2.54 cm diameter vacuum chamber between dipole magnets (see Fig. 7).
Figure 7.
For convenience, we introduce the right-handed Cartesian coordinate system (z, x) shown in Fig. 7. The origin O of the system is placed in the point D. The axis Ox is perpendicular to CD, goes along DOB1 and is pointing upward (Ox). The axis Oz is directed along CD and is pointing to the right. The coordinates of all points in this system are provided in meters. For example, the point P has coordinates (zP=-0.43, xP=-0.0127), OA1 (zA1=-2.19, xA1=40), OB1 (zB1=0, xB1=40).
With the use of the solution of Problem 2 and equation (A2.1) of Appendix, we calculate that the first direct synchrotron light ray from magnet A entering magnet B is emitted by electrons at the point Pa on the arc C1C with coordinates zPa-zA1=-0.268 and xPa=0.001. The angle between the axis Oz and this ray PaP is defined from the next equation:
tan j » -0.0067
The ray PaP (brown in Fig. 8) then reaches the wall of the vacuum chamber inside magnet B and reflects from it at the point Pb as from any metal surface.
Define how this ray travels farther inside magnet B and beyond it. Use equation (A8) of Appendix to calculate the coordinates (zPb, xPb) of the reflection point Pb. It gives us zPb-zOA2»1.1888 and xPb=-0.0223. Then on the basis of equation (A10) we find that the reflected ray has the slope kb=tan(y)»0.0597 in our coordinate system.
Note than the total bending angle of magnet B a=0.075 is bigger than y=0.0597. This means that the beam trajectory inside the magnet has a point Db from which electrons emit synchrotron light rays at the same angle y to the axis Oz. In other words, the light rays reflected from Pb (brown in Fig. 8) and emitted from Db (green in Fig. 7) are parallel to each other.
Figure 8.
The coordinates (zd, xd) of Db are easily calculated:
zDb = R sin(y) » 2.38,
xDb = R - R cos(y) » 0.071.
These values and the solution of Problem 6 of Appendix show that the ray reflected from Pb goes lower than the ray emitted from Db and the immediately give us the distance between these rays: