Calculation Of Fields On The Beam Axis Of The Delta Undulator

Transcription

1 LCLS-TN-3- Calculation Of Fields On The Beam Axis Of The Delta Undulator Zachary Wolf SLAC April, 20 Abstract It is not possible to do Hall probe measurements on the beam axis of the Delta undulator since no transverse probe motion is possible. The elds at the measurement location must be used to calculate the elds on the beam axis. This note discusses the method used to do the calculation. Introduction Undulators are typically characterized by measuring with a Hall probe on their magnetic center line, which becomes the beam axis. The Hall probe is on stages and it is moved to the magnetic center line. The beam axis then becomes the Hall probe path, and the undulator is ducialized to this axis. The measurements are done on the beam axis, so all quantities calculated from the measurements apply to the beam axis. These techniques do not work for the Delta undulator. The probe position is set by a guide tube and the probe follows the guide tube in the transverse directions. It is not possible to move the Hall probe to the beam axis, rather, the elds on the beam axis must be calculated from the measured elds. This note discusses the calculations. LCLS technical note LCLS-TN-3-2 presented a measurement plan for the Delta undulator. Technical note LCLS-TN further discussed the Hall probe array measurements. These notes lay out the procedure for measuring the Delta undulator and determining the beam axis. This note details the procedure for taking the measured eld at the measurement location, and calculating the eld on the beam axis. The Delta undulator measurements proceed as follows. The Hall probe array follows a curved path through the undulator and measures the three eld components on this path. This is illustrated in gure. The path of the probes is found relative to a straight line by a laser system that measures transverse position changes. The probes are found relative to ducials at the two ends of the undulator using high gradient ducialization magnets. The probe positions at the two ends of the undulator de ne a line and the probe position is calculated relative to this line using the laser measurements. We take this line to be the axis of the measurement coordinate system. In the gure, y is the position of probe relative to the coordinate system axis. The measurements from the array of Hall elements in the probe allow us to calculate the position of the magnetic center relative to the probe. This is denoted by y c y in the gure. The subscript "c" refers to the magnetic center, and the subscript "" refers to Hall probe in the probe package. Probe is the probe mounted on the axis of the probe assembly. To make the calculation of the probe position Work supported in part by the DOE Contract DE-AC02-76SF0055. This work was performed in support of the LCLS project at SLAC. 2 Z. Wolf, "A Magnetic Measurement Plan For The Delta Undulator", LCLS-TN-3-, March, Z. Wolf, "Hall Probe Array Measurements Of The Delta Undulator", LCLS-TN-3-9, November, 203.

2 Figure : The eld is measured on the probe path. The probe path is know relative to the coordinate system axis which is de ned by the probe positions at the two ends of the undulator. The magnetic center position is calculated from the measurements. A linear t to the magnetic centers de nes the beam axis. The measured elds and the functional form of the elds allows the elds to be calculated on the beam axis. relative to the magnetic center, the functional form of the elds must be speci ed. All parameters in the functional form must be calculated from the measurements. The way the functional form is determined is that Maxwell s equations are used to calculate the form of the terms in a series expansion of the eld. We assume that near the magnetic center, there is a largest term at the fundamental longitudinal frequency and that it dominates the series expansion. This term is what we use for the functional form of the eld. An expansion in the transverse coordinates is made assuming the transverse coordinates are close enough to the magnetic center that a second order expansion is adequate. We approximate the form of the transverse expansion as the form of the elds. Basically, we are doing a second order expansion in the transverse coordinates of the elds, and we use the Maxwell s equation solution to guide the form of the quadratic. As long as we are close enough to the magnetic center for the second order expansion to be accurate, this technique should provide an adequate parameterization of the elds. The coe cients in the quadratic expansion are parameters that must be determined from the measurements. Once we know the position of the magnetic center and the parameters in the functional form of the elds, we can calculate the elds at an arbitrary point, in particular, we can calculate the elds on the beam axis. We t the magnetic centers with a line to determine the beam axis. We ducialize this axis. We then calculate the elds on this line to determine the undulator characteristics. 2 Calculation Of The Fields On The Beam Axis From The Measurements In our analysis of the elds using a second order expansion in the transverse directions, the elds had the form B i = B i0 f i (x; y)g i (z) () where i = x; y, or z; f is a function which depends on the polarization mode and whose form is given explicitly below for the polarization modes discussed; and g is a function giving the longitudinal 2

3 behavior of the eld. The eld component measured by probe is B i = B i0 f i (x x c ; y y c )g i (z) (2) where (x c ; y c ) are the coordinates of the magnetic center, (x ; y ) are the coordinates of probe, and i = x, y, or z. The eld amplitude at the magnetic center is then B i0 = B i f i (x x c ; y y c )g i (z) (3) For the main eld component, we nd the eld on the beam axis by substituting the beam axis coordinates to the function f i. The eld on the beam axis is given by B ib = B i0 f i (x b x c ; y b y c )g i (z) () where (x b ; y b ) are the coordinates of the beam axis. the beam axis are given by By combining these equations, the elds on B ib = B i f i (x x c ; y y c ) f i(x b x c ; y b y c ) (5) The measured eld B i and all quantities in the function f i are know, so the eld on the beam axis is determined. For the main eld component, f i is of order, so there is no chance that the denominator will go to zero. For the smaller eld components, the functions f i can go to zero, so the procedure used above will not work. Instead we correct the measured elds by using the Taylor series expansion in the transverse coordinates. In particular, B ib = B i0 f i (x b x c ; y b y c )g i (z) X ' B i + (x b x + (y b n= n B i j (6) This procedure keeps the measured eld as the rst term and adds corrections based on the derivatives of the transverse coordinates in the functional form of the elds. By keeping the measured eld, the e ect of small error elds, from magnetization direction errors for instance, which are not part of the predicted form of the eld, are included. 3 Fields On The Beam Axis In The Di erent Undulator Modes In this section, we detail the calculations for the elds on the beam axis. From LCLS-TN-3-9, we know the position of the probe relative to the magnetic center, x x c, y y c. We also know the parameters k x and k y which characterize the transverse behavior of the elds. Finally, we know the beam axis position (x b, y b ) and the magnetic center position (x c, y c ) in the measurement coordinate system. In the following, we let ex = x x c, where x is the x-position where the eld is determined, and x c is the x-position of the magnetic center. Similarly, ey = y y c, where y is the y-position where the eld is determined, and y c is the y-position of the magnetic center. So ex = x x c, ey = y y c for probe, and ex b = x b x c, ey b = y b y c for the position of the beam axis relative to the magnetic center. 3

4 3. Linear Polarization Vertical Field Mode Consider the linear polarization vertical eld mode. The elds are B x = 0 k x sinh (k x ex) sinh (k y ey) cos (k u z) (7) B y = 0 k y cosh (k x ex) cosh (k y ey) cos (k u z) (8) B z = 0 k u cosh (k x ex) sinh (k y ey) sin (k u z) (9) Expanding to second order in k x ex and k y ey, the elds become The elds measured by probe are The elds on the beam axis are B x = 0 k x k x exk y ey cos (k u z) (0) B y = 0 k y + 2 (k xex) (k yey) 2 cos (k u z) () B z = 0 k u k y ey sin (k u z) (2) B x = 0 k x k x ex k y ey cos (k u z) (3) B y = 0 k y + 2 (k xex ) (k yey ) 2 cos (k u z) () B z = 0 k u k y ey sin (k u z) (5) B xb = 0 k x k x ex b k y ey b cos (k u z) (6) B yb = 0 k y + 2 (k xex b ) (k yey b ) 2 cos (k u z) (7) B zb = 0 k u k y ey b sin (k u z) (8) Consider rst the main eld component B y. We nd B yb given B y as B yb = B y + 2 (k xex b ) (k yey b ) 2 = + 2 (k xex ) (k yey ) 2 This can be approximated as B yb = B y + 2 (k xex b ) (k yey b ) 2 2 (k xex ) 2 2 (k yey ) 2 (9) (20) B y on the beam axis is given by B y as measured by probe, plus corrections for the di erence between the beam axis position and the position of probe. Now consider B x. Its derivative with respect to x is where we have used B y ' 0 k y cos (k u z). order that we are working B x = 0 k x k x k y ey cos (k u z) = B y k 2 xey (2) This approximation does not change the equation B x = 0 k x k x exk y cos (k u z) = B y k 2 xex (22)

5 The mixed second derivative is B x = 0 k x k x k y cos (k u z) Using the Taylor series expansion, we nd B x at the beam axis as All other terms in the expansion are zero. = B y k 2 x (23) B xb ' B x + (x b B xj + (y b B xj + (x b x ) (y b B xj (2) Inserting values for the derivatives, we nd B xb ' B x + B y k 2 x [ey (x b x ) + ex (y b y ) + (x b x ) (y b y )] (25) Using the fact that (x b x ) = (ex b ex ) with a similar equation for y, this becomes B xb ' B x + B y k 2 x [ey (ex b ex ) + ex (ey b ey ) + (ex b ex ) (ey b ey )] (26) This expression gives B x on the beam axis in terms of the measured elds at the probe position and in terms of quantities we know from the Hall probe array measurement analysis. Now consider B z. Its derivative with respect to x is zero, and with respect to y is The Taylor series expansion of the B z = 0 k u k y sin (k u z) B y (28) B zb = B B yj (ey b ey ) (29) where we used y b y = ey b ey and B B y which is seen from the form of the elds, and is also known since r B = 0. We know B z from the measurements, and we B yj since we know B y as a function of z. Both ey b and ey are known from the measurements with the Hall probe array. In summary, the elds on the beam axis are given by B xb = B x + B y kx 2 [ey (ex b ex ) + ex (ey b ey ) + (ex b ex ) (ey b ey )] (30) B yb = B y + 2 (k xex b ) (k yey b ) 2 2 (k xex ) 2 2 (k yey ) 2 (3) B zb = B B yj (ey b ey ) (32) 3.2 Linear Polarization Horizontal Field Mode Consider the linear polarization horizontal eld mode of the undulator. the eld expansion are The fundamental terms in B x = 0 k x cosh (k x ex) cosh (k y ey) cos (k u z) (33) B y = 0 k y sinh (k x ex) sinh (k y ey) cos (k u z) (3) B z = 0 k u sinh (k x ex) cosh (k y ey) sin (k u z) (35) 5

6 Expanding to second order in k x ex and k y ey, the elds become B x = 0 k x + 2 (k xex) (k yey) 2 cos (k u z) (36) B y = 0 k y k x exk y ey cos (k u z) (37) B z = 0 k u k x ex sin (k u z) (38) Using the same techniques as for the vertical eld mode, the elds on the beam axis are given by B xb = B x + 2 (k xex b ) (k yey b ) 2 2 (k xex ) 2 2 (k yey ) 2 (39) B yb ' B y + B x k 2 y [ey (ex b ex ) + ex (ey b ey ) + (ex b ex ) (ey b ey )] (0) B zb = B B xj (ex b ex ) () 3.3 Circular Polarization Right Hand Mode The elds in the circular polarization right hand mode are B x = 2 p 2 0k u cosh p2 k u (ex + ey) cos (k u z) 2 p 2 0k u cosh p2 k u (ex ey) sin (k u z) (2) B y = 2 p 2 0k u cosh p2 k u (ex + ey) cos (k u z) + 2 p 2 0k u cosh p2 k u (ex ey) sin (k u z) (3) B z = 2 0k u sinh p2 k u (ex + ey) sin (k u z) 2 0k u sinh p2 k u (ex ey) cos (k u z) () Expanding to rst order, the elds become B x = B y = B z = 2 p 2 0k u cos (k u z) 2 p 2 0k u sin (k u z) (5) 2 p 2 0k u cos (k u z) + 2 p 2 0k u sin (k u z) (6) 2 0k u p2 k u (ex + ey) sin (k u z) 2 0k u p2 k u (ex ey) cos (k u z) (7) These expressions can be simpli ed by using the following identities. cos(x) + sin(x) = p 2 sin + x cos(x) sin(x) = p 2 sin x (8) (9) 6

7 With these identities, the elds become B x = 2 0k u sin k u z B y = 2 0k u sin + k uz B z = 2 0kuex 2 sin uz + k + 2 0kuey 2 sin Note that B x and B y have no x or y dependence. are the same as the measured elds. k u z (50) (5) (52) This means that the elds on the beam axis B xb = B x (53) B yb = B y (5) For B z, the eld on the beam axis is obtained from the rst order Taylor expansion in the transverse coordinates. Carrying out the derivatives, we nd B zb ' B B zj (ex b ex B zj (ey b ey ) (55) B zb = B z + 2 0ku 2 sin + k uz (ex b ex ) + 2 0ku 2 sin k u z (ey b ey ) (56) = B z B y k u (ex b ex ) + B x k u (ey b ey ) (57) In summary, we nd B xb = B x (58) B yb = B y (59) B zb = B z B y k u (ex b ex ) + B x k u (ey b ey ) (60) 3. Circular Polarization Left Hand Mode The elds in the circular polarization left hand mode are B x = B y = B z = 2 p 2 0k u cosh + 2 p 2 0k u cosh 2 p 2 0k u cosh 2 p 2 0k u cosh 2 0k u sinh + 2 0k u sinh p2 k u (ex + ey) p2 k u (ex p2 k u (ex + ey) p2 k u (ex p2 k u (ex + ey) p2 k u (ex cos (k u z) ey) sin (k u z) (6) cos (k u z) ey) sin (k u z) (62) sin (k u z) ey) cos (k u z) (63) 7

8 Expanding to rst order and using the identities given above, the elds become B x = 2 0k u sin + k uz B y = 2 0k u sin k u z B z = 2 0kuex 2 sin k u z 2 0kuey 2 sin uz + k Using the methods given above for circular right hand polarization, we nd for circular left hand polarization Conclusion (6) (65) (66) B xb = B x (67) B yb = B y (68) B zb = B z + B y k u (ex b ex ) B x k u (ey b ey ) (69) The elds on the beam axis were calculated from the measured elds and the functional form of the transverse behavior of the elds. In particular, the eld on the beam axis was given by the measured eld plus a correction term calculated from the functional form of the elds. This procedure keeps the e ect of eld errors from imperfections such as magnetization direction errors of the magnet blocks which are present in the measured eld. Acknowledgements I am grateful to Heinz-Dieter Nuhn and Yurii Levashov for many discussions about this work. 8