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An analogue system displaying beam optics
NUCLEAR
INSTRUMENTS
AND
M E T H O D S 50
0967)
153-t56:
~',
NORTH-HOLLAND
PUBLISHING
CO.
AN ANALOGUE SYSTEM DISPLAYING BEAM OPTICS R F. R U M P H O R S T ,
M A A
SONNEMANS,
H. A R N O L D
and L A Ch
KOERTS
Inst:tuut coor Kernphystsch Onderzoek, Amsterdam, The Netherland~ Received 26 O c t obe r 1966 An a p p a r a t u s designed for fast wsual inspection of particle trajectories in b eam t r a n s p o r t s~,stems ts described It ~s based on an electrical a n a l o g u e of the thin lens approx~matmn The umt
solves contmuousl> for ~mage formaUon m both the horizontal and ~erttcal planes stmultaneously, d~splaymg the c o r r e s p o n d i n g pa ruc l e trajectories e~ ery 40 msec
1. Principle The trajectory o f a particle through a thin magnetic lens system is readily described by two ray diagrams, showing the projections of the particle path on the horizontal plane and the vertical plane through the principal axis of the system. Each of these projected particle paths or ray diagrams are conventionally described by classical ray optics using lenses of appropriate strength. Simple electrical analogues are used m the apparatus to simulate classical thin lens optics. The simplest electrical analogue is the four terminal network of fig. l a. The corresponding graph shows the voltage (V) along a reststor as a function of resistance (R) measured from one terminal of the resistor. This graph will be interpreted as the ray dtagram for undisturbed trajectories, the R-ax~s corresponding to the distance along the prmcipal axts and the V-axis to distances from the principal axis. The ~lope of the trajectory ~s then proportional to dV,.dR = t, the current through the resistor. Thin lenses acting on trajectories are represented by the presence of a current branch point P in the cor-
responding networks shown in fig. l b c. The voltage pattern along the reststance in the presence of a current branch point is also given in fig. I b, c. They show qualitattvely the similarity with the ray diagrams for a defocuslng and focusing thin lens, respectively. In order to derive the correspondance quantitatively we write the lens formula' (1/L~)+(I,'~') :
(l)
where s and s' are the object and image distances, J the focal length. For the network we can write, referring to fig. lb,c ll-[-12nt-13 :
(2)
0,
where we choose the currents positwe if they flo~ away fi'om the branch point P. If the voltage at P is 1o we can write" il = (dV/dR)]l = ~o.,'R,: t 2 = (dl,.,'dR)[ 2 = l,o.,R,,,
l 3 = l,o/R j,
where R,, R,, and R / a r e the effecttve resistances seen
~R
a
1,'[,
I~R
b
~R
C
Fig. I. The voltage alo ng the resistor c o n n e c t i n g the upper t e r m i n a l s ms a f u n c h o n of the resistance measured a l o n g t ha t resistor to the left t e r m i n a l This dependence of voltage (V) on resistance (RI is shown in graphical form for three cases These graphs resemble trajectories of thin lens ray optics
153
R.F. RUMPHORST et al.
154
from P m the three branches carrying the currents 4, t2 and t3, respectively. Substitution into eq. (2) and division by Vo results in: (I/R,) + (l/R,,) =
-
l/Ry.
/
(3)
This is in complete analogy with the Fens formula. This result illustrates the following facts: 1. There is proportionality between Rs and s, Rs, and s' and - R I and t, all featuring one proportionality constant r. 2. The value of I% is irrelevant for the analogy. 3. The resistances may have to be negative for complete analogy. Introduction of a negative resistance value is accomplished by taking the internal resistance value of a voltage source. The focal length ts given by Rj (fig. I b, c) and the sign of ~3 determines whether we are dealing w~th a focusing or a defocusmg lens. In the analogue the deflecting power of a divergent lens is therefore fully determined by the value of Ry in fig. lb, while a focusing lens (fig. lc) is represented by a voltage source with a certain internal resistance R,. Since each magnetic element acting on charged particles is treated by introducing two optical systems, the lnterdependance of the two corresponding electrical analogue systems has to be defined. Since a quadrupole lens for example has two focal lengths of opposite sign acting in perpendicular planes through the beam axis, the analogue circuit of a quadrupole must consist of a resistance tap and a voltage tap. The relation between them follows from beam optics
A
B
"HORIZONTAL" VOLTAGE DIAGRAM
A
C
Rc
CONTROLLED 7L----]
IVOLTAGE SOOICEI
AMPLIFIER /
O
J Rc
O Fig. 2 B l o c k d l a g r a m of a voltage source with internal resistance Rl = R c + R$, i nde pe nde nt o f the load Such a voltage source is used as a resistor with preset negative resistance in the networks.
for K = k!(Bp): floe • {K½ sin ( K "r- L)} - 1,
(4)
Jdefoc, = - {K~ sxnh (K ~. L ) } - 1,
(5)
where L is the effective length of the pole faces, k the field-gradient and Bp the magnetic rigidity of the particles. Series expansion m powers of the quantity (K ~" L), neglecting third order terms, gives for difference in focal lengths . Iffoe.
1--Ifdef
I =
½L.
(6)
Translation of eq. (6) into the analogue picture results in
R , - R f = R c.
c
(7)
D
"VERTICAL" VOLTAGE DIAGRAM
B
HORIZONTAL
D
- vk,XAAA/x/,q~-VER~ICAI-
VERTICAL
a
..~ DIFFERENCE
b
F~g 3 E x a m p l e s o f the two electrically identical n e t w o r k s s~mulatlng the projections of a particle trajectory on two p e r p e n d i c u l a r planes The particle starts from the optical axm, traverses a system of three q u a d r u p o l e lenses and intersects the principal axis again. The circu lar netwo rk (b) has the a d v a n t a g e that the voltage d i a g r a m s can be c onve ni e nt l y m e a s u r e d with a r o t a t i n g pick-off c ont a c t
AN ANALOGUE
SYSTEM
DISPLAYING
B EA M O P T I C S
155
O* 180" i i i
0° .~
TRIGGERSIGNAL
O*
SHAPER
O* -
180"
O* 180" O* W
AMPLIFIER
~
"
1
1-
INVERTER
~0~
180
TRIGGERSIGNAL 1 ~ 0 " o
SHAPER
ADD
]
180"- O* J
~ i
F~g 4
SAWTOOTH
f I_~ 180" O*
Sch ematic d i a g r a m s h o w i n g the a n a l o g u e n e t w o r k and the w a y the v o l t a g e a l ong the resistor ring ~s me a s ure d m a d e v i s i b l e on the oscilloscope.
The value of R L is a constant depending only on the effective length of the pole faces of the quadrupole lens. It is required that the voltage element used has an internal resistance R, which depends on the value of R I as shown in relation (7). Such a voltage source is drawn schematically in fig. 2. In fig. 3 and 4 a battery symbol is used to designate these special voltage sources. Three types of 4-terminal networks have been introduced, i.e. focusing elements, defocusing elements and drift spaces. A complete system can be assembled by coupling the appropriate building blocks together into two systems: one for the horizontal plane and one for the vertical plane (fig. 3). The two resulting networks are interdependent, first of all since each voltage element should fulfill relation (7) and furthermore since the resistance between the current branch points (which is proportional to the distances between the magnets) should be equal for the horizontal and the vertical plane. A measurement of the voltage pattern from A to B and C to D (fig. 3) is made appreciably easier by changing the configuration to that of fig 3b which ~s electrically ~dent~cal, but allows a rotating pick-off contact to be used.
and
trlcally on the ring. The ring is divided into two half rings which wdl correspond to the horizontal and vertical trajectories, respectively. A rotating brush contact (approximately 1500 rpm) picks off the voltages along the ring. This signal is ultimately supplied to the vertical input terminal of an oscilloscope. A sawtooth generator delivering the horizontal deflection voltage is triggered when the brush contact passes the end of each half ring. By periodic voltage invers]on, synchromzed by the same trigger signals, the two resulting ~mages for both halves 1.e. the horizontal and vertical trajectories are displayed
2. Construetion
The umt is shown schematically m fig. 4. The voltage patternls developed over a ring. This ring ]s essenually a switch with 420 consecutive contacts. Neighbouring contact points are interconnected w~th resistances of 22 O accurate to 0.2°o. The resistance tap and the associated voltage tap of a deflecting element are rotatable around the center over 180" and set dmme-
Fig 5 This p h o t o g r a p h shows the s ol ut i on of a system of two q u a d r u p o l e lenses m the thin lens approMmat~on, by t a k i n g the focusing lengths equal m bot h p e r p e n d i c u l a r planes. If the three relevant &s t a nc e s are 8, 4 and 9 m the s ol ut i on Is calculated to be J 1 - - ~ 3 . 6 3 m and J 2 - - T 3 7 8 m The e x p e r i m e n t a l l y de t e rmi ne d values are ]1 -- T 3 64 and ]2 = ~ 3 78 m
156
R.V. RUMPHORST et al.
below each other on the CRT (fig. 5). Tile values of R I corresponding to the defocusing focal lengths of the lenses are set up w~th helipots of 10 k(2 within an accuracy of 10 £2. The corresponding values of R, are automatically set by the self-regulating voltage elements. The umt is provided with ten elements which can represent the focusing properties of quadrupole lenses or other kinds of magnets. The plane of focusing of the elements can be chosen w~th switches. A solution to a simple problem is gwen in fig. 5 as photographed from the CRT display Although it was stated above that the analogy is independent of the voltage on the network, it should be realized that this is only true if the conditions for the existence of the analogy are fulfilled. In other words the resistances should ha~e values corresponding to a solution, if this is not the case, the self-regulating voltage source will e~ther collaps to very small voltage or give the highest voltage obtainable. Characteristic of a solution is therefore tile fact that the picture on the screen is stable at fimte values. The circuit operates in this sense as an analogue computer forcing the operator to provide solvable problems.
3. Discussion
The results available indicate that the umt ~s capable of solving the xmage formation conditions of any deflecting system within one percent of the exact values of the focal lengths, described by the lens formula. The apparatus has proved itself useful m providing good mtlUal conditions for computer programs, reducing the number of iterations needed for exact calculation and therefore saving computer time. Moreover, the flexibility of the unit is of value for fast visual Inspection of various beam optical systems for beam transport.
We are indebted to messrs. R Bregman, N. Hoetmer and H. Schwebke who made this unit to the exacting specifications, and to Dr. F. Udo for valuable discussions. This work is part of the research program of the Institute for Nuclear Physics Research (IKO), made possible by financial support from the Foundation for Fundamental Research on Matter (FOM) and the Netherlands Organization for Pure Scientific Research (ZWO).
INSTRUMENTS
AND
M E T H O D S 50
0967)
153-t56:
~',
NORTH-HOLLAND
PUBLISHING
CO.
AN ANALOGUE SYSTEM DISPLAYING BEAM OPTICS R F. R U M P H O R S T ,
M A A
SONNEMANS,
H. A R N O L D
and L A Ch
KOERTS
Inst:tuut coor Kernphystsch Onderzoek, Amsterdam, The Netherland~ Received 26 O c t obe r 1966 An a p p a r a t u s designed for fast wsual inspection of particle trajectories in b eam t r a n s p o r t s~,stems ts described It ~s based on an electrical a n a l o g u e of the thin lens approx~matmn The umt
solves contmuousl> for ~mage formaUon m both the horizontal and ~erttcal planes stmultaneously, d~splaymg the c o r r e s p o n d i n g pa ruc l e trajectories e~ ery 40 msec
1. Principle The trajectory o f a particle through a thin magnetic lens system is readily described by two ray diagrams, showing the projections of the particle path on the horizontal plane and the vertical plane through the principal axis of the system. Each of these projected particle paths or ray diagrams are conventionally described by classical ray optics using lenses of appropriate strength. Simple electrical analogues are used m the apparatus to simulate classical thin lens optics. The simplest electrical analogue is the four terminal network of fig. l a. The corresponding graph shows the voltage (V) along a reststor as a function of resistance (R) measured from one terminal of the resistor. This graph will be interpreted as the ray dtagram for undisturbed trajectories, the R-ax~s corresponding to the distance along the prmcipal axts and the V-axis to distances from the principal axis. The ~lope of the trajectory ~s then proportional to dV,.dR = t, the current through the resistor. Thin lenses acting on trajectories are represented by the presence of a current branch point P in the cor-
responding networks shown in fig. l b c. The voltage pattern along the reststance in the presence of a current branch point is also given in fig. I b, c. They show qualitattvely the similarity with the ray diagrams for a defocuslng and focusing thin lens, respectively. In order to derive the correspondance quantitatively we write the lens formula' (1/L~)+(I,'~') :
(l)
where s and s' are the object and image distances, J the focal length. For the network we can write, referring to fig. lb,c ll-[-12nt-13 :
(2)
0,
where we choose the currents positwe if they flo~ away fi'om the branch point P. If the voltage at P is 1o we can write" il = (dV/dR)]l = ~o.,'R,: t 2 = (dl,.,'dR)[ 2 = l,o.,R,,,
l 3 = l,o/R j,
where R,, R,, and R / a r e the effecttve resistances seen
~R
a
1,'[,
I~R
b
~R
C
Fig. I. The voltage alo ng the resistor c o n n e c t i n g the upper t e r m i n a l s ms a f u n c h o n of the resistance measured a l o n g t ha t resistor to the left t e r m i n a l This dependence of voltage (V) on resistance (RI is shown in graphical form for three cases These graphs resemble trajectories of thin lens ray optics
153
R.F. RUMPHORST et al.
154
from P m the three branches carrying the currents 4, t2 and t3, respectively. Substitution into eq. (2) and division by Vo results in: (I/R,) + (l/R,,) =
-
l/Ry.
/
(3)
This is in complete analogy with the Fens formula. This result illustrates the following facts: 1. There is proportionality between Rs and s, Rs, and s' and - R I and t, all featuring one proportionality constant r. 2. The value of I% is irrelevant for the analogy. 3. The resistances may have to be negative for complete analogy. Introduction of a negative resistance value is accomplished by taking the internal resistance value of a voltage source. The focal length ts given by Rj (fig. I b, c) and the sign of ~3 determines whether we are dealing w~th a focusing or a defocusmg lens. In the analogue the deflecting power of a divergent lens is therefore fully determined by the value of Ry in fig. lb, while a focusing lens (fig. lc) is represented by a voltage source with a certain internal resistance R,. Since each magnetic element acting on charged particles is treated by introducing two optical systems, the lnterdependance of the two corresponding electrical analogue systems has to be defined. Since a quadrupole lens for example has two focal lengths of opposite sign acting in perpendicular planes through the beam axis, the analogue circuit of a quadrupole must consist of a resistance tap and a voltage tap. The relation between them follows from beam optics
A
B
"HORIZONTAL" VOLTAGE DIAGRAM
A
C
Rc
CONTROLLED 7L----]
IVOLTAGE SOOICEI
AMPLIFIER /
O
J Rc
O Fig. 2 B l o c k d l a g r a m of a voltage source with internal resistance Rl = R c + R$, i nde pe nde nt o f the load Such a voltage source is used as a resistor with preset negative resistance in the networks.
for K = k!(Bp): floe • {K½ sin ( K "r- L)} - 1,
(4)
Jdefoc, = - {K~ sxnh (K ~. L ) } - 1,
(5)
where L is the effective length of the pole faces, k the field-gradient and Bp the magnetic rigidity of the particles. Series expansion m powers of the quantity (K ~" L), neglecting third order terms, gives for difference in focal lengths . Iffoe.
1--Ifdef
I =
½L.
(6)
Translation of eq. (6) into the analogue picture results in
R , - R f = R c.
c
(7)
D
"VERTICAL" VOLTAGE DIAGRAM
B
HORIZONTAL
D
- vk,XAAA/x/,q~-VER~ICAI-
VERTICAL
a
..~ DIFFERENCE
b
F~g 3 E x a m p l e s o f the two electrically identical n e t w o r k s s~mulatlng the projections of a particle trajectory on two p e r p e n d i c u l a r planes The particle starts from the optical axm, traverses a system of three q u a d r u p o l e lenses and intersects the principal axis again. The circu lar netwo rk (b) has the a d v a n t a g e that the voltage d i a g r a m s can be c onve ni e nt l y m e a s u r e d with a r o t a t i n g pick-off c ont a c t
AN ANALOGUE
SYSTEM
DISPLAYING
B EA M O P T I C S
155
O* 180" i i i
0° .~
TRIGGERSIGNAL
O*
SHAPER
O* -
180"
O* 180" O* W
AMPLIFIER
~
"
1
1-
INVERTER
~0~
180
TRIGGERSIGNAL 1 ~ 0 " o
SHAPER
ADD
]
180"- O* J
~ i
F~g 4
SAWTOOTH
f I_~ 180" O*
Sch ematic d i a g r a m s h o w i n g the a n a l o g u e n e t w o r k and the w a y the v o l t a g e a l ong the resistor ring ~s me a s ure d m a d e v i s i b l e on the oscilloscope.
The value of R L is a constant depending only on the effective length of the pole faces of the quadrupole lens. It is required that the voltage element used has an internal resistance R, which depends on the value of R I as shown in relation (7). Such a voltage source is drawn schematically in fig. 2. In fig. 3 and 4 a battery symbol is used to designate these special voltage sources. Three types of 4-terminal networks have been introduced, i.e. focusing elements, defocusing elements and drift spaces. A complete system can be assembled by coupling the appropriate building blocks together into two systems: one for the horizontal plane and one for the vertical plane (fig. 3). The two resulting networks are interdependent, first of all since each voltage element should fulfill relation (7) and furthermore since the resistance between the current branch points (which is proportional to the distances between the magnets) should be equal for the horizontal and the vertical plane. A measurement of the voltage pattern from A to B and C to D (fig. 3) is made appreciably easier by changing the configuration to that of fig 3b which ~s electrically ~dent~cal, but allows a rotating pick-off contact to be used.
and
trlcally on the ring. The ring is divided into two half rings which wdl correspond to the horizontal and vertical trajectories, respectively. A rotating brush contact (approximately 1500 rpm) picks off the voltages along the ring. This signal is ultimately supplied to the vertical input terminal of an oscilloscope. A sawtooth generator delivering the horizontal deflection voltage is triggered when the brush contact passes the end of each half ring. By periodic voltage invers]on, synchromzed by the same trigger signals, the two resulting ~mages for both halves 1.e. the horizontal and vertical trajectories are displayed
2. Construetion
The umt is shown schematically m fig. 4. The voltage patternls developed over a ring. This ring ]s essenually a switch with 420 consecutive contacts. Neighbouring contact points are interconnected w~th resistances of 22 O accurate to 0.2°o. The resistance tap and the associated voltage tap of a deflecting element are rotatable around the center over 180" and set dmme-
Fig 5 This p h o t o g r a p h shows the s ol ut i on of a system of two q u a d r u p o l e lenses m the thin lens approMmat~on, by t a k i n g the focusing lengths equal m bot h p e r p e n d i c u l a r planes. If the three relevant &s t a nc e s are 8, 4 and 9 m the s ol ut i on Is calculated to be J 1 - - ~ 3 . 6 3 m and J 2 - - T 3 7 8 m The e x p e r i m e n t a l l y de t e rmi ne d values are ]1 -- T 3 64 and ]2 = ~ 3 78 m
156
R.V. RUMPHORST et al.
below each other on the CRT (fig. 5). Tile values of R I corresponding to the defocusing focal lengths of the lenses are set up w~th helipots of 10 k(2 within an accuracy of 10 £2. The corresponding values of R, are automatically set by the self-regulating voltage elements. The umt is provided with ten elements which can represent the focusing properties of quadrupole lenses or other kinds of magnets. The plane of focusing of the elements can be chosen w~th switches. A solution to a simple problem is gwen in fig. 5 as photographed from the CRT display Although it was stated above that the analogy is independent of the voltage on the network, it should be realized that this is only true if the conditions for the existence of the analogy are fulfilled. In other words the resistances should ha~e values corresponding to a solution, if this is not the case, the self-regulating voltage source will e~ther collaps to very small voltage or give the highest voltage obtainable. Characteristic of a solution is therefore tile fact that the picture on the screen is stable at fimte values. The circuit operates in this sense as an analogue computer forcing the operator to provide solvable problems.
3. Discussion
The results available indicate that the umt ~s capable of solving the xmage formation conditions of any deflecting system within one percent of the exact values of the focal lengths, described by the lens formula. The apparatus has proved itself useful m providing good mtlUal conditions for computer programs, reducing the number of iterations needed for exact calculation and therefore saving computer time. Moreover, the flexibility of the unit is of value for fast visual Inspection of various beam optical systems for beam transport.
We are indebted to messrs. R Bregman, N. Hoetmer and H. Schwebke who made this unit to the exacting specifications, and to Dr. F. Udo for valuable discussions. This work is part of the research program of the Institute for Nuclear Physics Research (IKO), made possible by financial support from the Foundation for Fundamental Research on Matter (FOM) and the Netherlands Organization for Pure Scientific Research (ZWO).