The invention discloses a wiggler used in synchrotron radiation sources and free electron lasers, where each pole is surrounded by at least two electromagnetic coils. The electromagnetic coils are energized with different amounts of current to provide a wide tunable range of the on-axis magnetic flux density, while preventing magnetic saturation of the poles.

BACKGROUND OF THE INVENTION 
Free electron lasers are able to produce very high power laser radiation in 
an efficient manner. In addition to their high efficiency and high power 
capability, they are attractive since they can be tunable over a wide 
spectrum from millimeter wavelengths to the x-ray region. Free electron 
lasers pass a relativistic electron beam through a spatially varying 
magnetic field called a wiggler, which wiggles the electrons in the 
electron beam. The wiggle of the electrons cause the electrons to radiate. 
If the proper phase is maintained by the electron beam, the radiation 
produced can amplify an existing electromagnetic field creating a laser 
beam. Tuning the wiggler so that the electrons emit light of a desired 
wavelength and so that the light can be efficiently extracted from the 
electron beam, comprises adjusting the magnetic field strength in the 
wiggler. 
There are other applications for wigglers, such as third generation 
synchrotron radiation rings. These synchrotron radiation rings employ 
wigglers to increase radiation power output and expand or customize the 
wavelength spectrum of the radiation produced. In the claims and 
specification the word "wiggler" will also include those wigglers called 
undulators producing narrowly peaked radiation spectrums. 
FIG. 1 is a schematic drawing of a relativistic electron beam passing 
through a wiggler to produce directed radiation. Alternating magnetic 
poles 12 are used to create an alternating magnetic field B.sub.w, which 
in the x-z plane is parallel and antiparallel to the y axis. An electron 
beam 10 is directed along the z axis through the alternating magnetic 
field. The magnetic field along the z axis is known as the on-axis 
magnetic field. The maximum magnetic field values along the z-axis defines 
the magnitude of B.sub.w The movement along the z direction through an 
alternating magnetic field causes the electron beam 10 to oscillate in the 
x direction causing a sinusoidal path 14. The oscillation of the 
relativistic electrons creates electromagnetic radiation 6 with a 
frequency which is a function of the electron energy, the oscillation 
frequency, and the magnetic field strength. The electromagnetic radiation 
is highly directional in the z direction. 
FIG. 2 is a cut away view of a wiggler used in the prior art. An upper 
magnetic yoke 16 is used to hold a plurality of upper magnetic poles with 
a first upper magnetic pole 18. A lower magnetic yoke 20 is used to hold a 
plurality of lower magnetic poles with a first lower magnetic pole 22. The 
magnetic poles and the magnetic yokes 16, 20 are made of a ferromagnetic 
material. Adjacent to the first upper magnetic pole are two permanent 
magnets 24. Adjacent to the first lower magnetic pole are two permanent 
magnets 26. 
The first upper magnetic pole 18 and the adjacent magnets 24 are used as a 
mandrel for a first upper electromagnetic coil 36. The first lower 
magnetic pole 22 and the adjacent magnets 26 are used as a mandrel for a 
first lower electromagnetic coil 38. 
The first upper electromagnetic coil 36 and the first lower electromagnetic 
coil 38 are wound so that as viewed from above looking down in the -y 
direction the currents in the first upper electromagnetic coil 36 and the 
first lower electromagnetic coil 38 flow in a clockwise direction. Current 
in the clockwise direction in the first upper electromagnetic coil 36 
creates in the first upper magnetic pole 18 a flux in the downward (-y) 
direction from the upper magnetic yoke 16 to the tip 28 of the first upper 
magnetic pole 18. Current in the clockwise direction in the first lower 
electromagnetic coil 38 creates in the first lower magnetic pole 22 a flux 
in the downward (-y) direction from the tip 30 of the first lower magnetic 
pole 22 to the lower magnetic yoke 20. This results in a net effect of a 
magnetic flux in a downward direction passing across the gap between the 
first upper magnetic pole 18 and the first lower magnetic pole 22. 
The two permanent magnets 24 adjacent to the first upper magnetic pole 18 
are oriented to place in the first upper magnetic pole 18 a net magnetic 
flux into the pole at the location of the first upper adjacent permanent 
magnets 24. This flux then travels in the general upward (+y) direction in 
the pole toward the upper magnetic yoke 16. The two permanent magnets 26 
adjacent to the first lower magnetic pole 22 are oriented to place in the 
first lower magnetic pole 22 a net magnetic flux out of the pole at the 
location of the first lower adjacent permanent magnets 26. This flux 
travels in the general upward (+y) direction in the pole from the lower 
magnetic yoke 20. The magnetic flux from the permanent magnets is induced 
in the poles but does not cross the gap between the first upper magnetic 
pole 18 and the first lower magnetic pole 22. 
A second upper pole has two adjacent permanent magnets 32. The second lower 
pole has two adjacent permanent magnets 34. The second upper magnetic pole 
and the adjacent magnets 32 are used as a mandrel for a second upper 
electromagnetic coil 35. The second lower magnetic pole and the adjacent 
magnets 34 are used as a mandrel for a second lower electromagnetic coil 
37. 
The second upper electromagnetic coil 35 and the second lower 
electromagnetic coil 37 are wound so that as viewed from above looking 
down in the -y direction the current in the second upper electromagnetic 
coil 35 and the second lower electromagnetic coil 37 flow in a 
counterclockwise direction. Current in the counterclockwise direction in 
the second upper electromagnetic coil 35 and current in the 
counterclockwise direction in the second lower electromagnetic coil 37 
create a flux in the upward (+y) direction. This results in a net effect 
of a magnetic flux in a upward direction passing across the gap between 
the second upper magnetic pole and the second lower magnetic pole. 
The two permanent magnets 32 adjacent to the second upper magnetic pole are 
oriented to place in the second upper magnetic pole a magnetic flux in the 
downward (-y) direction. The two permanent magnets 34 adjacent to the 
second lower magnetic pole are oriented to place in the second lower 
magnetic pole a magnetic flux in the downward (-y) direction. The magnetic 
flux from the permanent magnets is induced in the poles and does not cross 
the gap between the second upper magnetic pole and the second lower 
magnetic pole. 
Each pole and adjacent set of permanent magnet are used as a mandrel for an 
electromagnetic coil. The electromagnetic coils are used to induce a 
magnetic flux in the magnetic poles and thus in the gap between opposite 
poles giving rise to the alternating magnetic field which causes the 
electron beam spatial oscillations or "wiggles." Adjusting the current in 
the electromagnetic coil changes the magnitude of the magnetic field and 
thus allows the tuning of the wiggler to either 1) compensate for a 
decrease in electron beam energy along the z direction and thus maintain a 
resonance condition between the electron beam and the radiation being 
amplified over a larger spatial distance or 2) change the frequency of the 
electromagnetic radiation produced by an electron beam of a given energy 
passing between the tips of the magnetic poles of the wiggler. 
Steering coils 40 are wrapped around the upper magnetic yoke 16. The 
steering coils 40 provide a magnetic field used to make minor steering 
corrections of the electron beam as it passes through the wiggler. 
FIG. 3 is a graph of a hysteresis loop for an iron material. The 
magnetizing force H applied to the iron material is plotted along the 
abscissa, and the magnetic induction B induced in the iron is plotted 
along the ordinate. The slope of the curve forming the loop at a point on 
the curve is .mu.=.mu..sub.o .mu..sub.r, where .mu..sub.o is the 
free-space permeability. At B=0, for some iron .mu..sub.r =1,000. At point 
b, .mu..sub.r is close to one. At point b, the iron is magnetically 
saturated. At point b, an increase in the magnetizing force H, causes only 
a slight increase in the induced magnetism B in the iron. At points c and 
d, one side of the hysteresis loop goes from being approximately linear at 
B=0 to becoming significantly nonlinear. For a wiggler made of this iron, 
the sum of the magnetic fields in the iron pole induced by the permanent 
magnets and the electromagnetic coil around a pole is kept between H.sub.d 
and H.sub.c. This is practiced for two reasons. Beyond H.sub.c and H.sub.d 
the absolute value of .mu. decreases, decreasing the change in B for a 
unit change in H, thus making the change in H less efficient outside of 
the range. Secondly, the slope .mu. becomes variable, making B harder to 
predict outside of the range. In the claims and specification, applying 
summed magnetic fields in the pole outside of the range H.sub.d to H.sub.c 
will be considered a saturating magnetic flux density in the poles. 
The design of an iron-core electromagnetic wiggler pole is largely an 
exercise in simultaneously sufficiently limiting both the maximum magnetic 
flux density in the iron of the pole structure and the current density in 
the electromagnetic coils while satisfying system level requirements, 
minimizing cost and technical risks, etc. A wiggler must often attain the 
following three systems level goals: (1) high wiggler on-axis magnetic 
flux density (magnetic field), (2) low magnetic field errors (including 
those due to saturation of the poles), and (3) widely tunable range. 
Wiggler design features enabling the attainment of the first goal, e.g. 
larger electromagnetic coil currents and/or cross-sectional areas, tend to 
inhibit the attainment of the second due to the onset of magnetic 
saturation of the wiggler poles. Wiggler pole magnetic saturation also 
inherently limits the degree to which the first goal can be attained due 
to the leveling off of the slope beyond the saturation points. K. Halbach 
in "Some Concepts To Improve The Performance Of DC Electromagnetic 
Wigglers," Nuclear Instruments and Methods in Physics Research A250 
(1986) pp 115-119, North-Holland, Amsterdam describes the design which 
enables the attainment of much higher magnetic flux densities (while also 
maintaining a low level of magnetic field errors) in electromagnet 
wigglers by employing permanent magnets 24, 26, 32, 34 to put a reverse 
bias magnetic flux in the wiggler pole, without directly altering the 
wiggler's on-axis magnetic field. This allows the electromagnetic coil 
current (and thus on-axis magnetic flux density) to be increased to a 
higher level before the onset of wiggler pole magnetic saturation. 
FIG. 4a is a cross section of half a pole shown in FIG. 2 along cut lines 
4--4 with a graph of the magnetic flux density along the pole. The 
magnitude of the on-axis magnetic field B.sub.w is proportional to the 
magnetic scalar potential at the tip 28 of the pole (U(T)), so that 
EQU B.sub.w =qU(T), (1) 
where q is a constant. U(T) is proportional to the number of ampere-turns 
in the electromagnetic coil surrounding the coil. The scalar potential 
anywhere along the pole is given by 
EQU U(y)=U(T)(1-(y-T)/h), (2) 
where T is the value of y at the tip of the pole and h is the height of the 
electromagnet coil 36. The increment in electromagnet coil 36 induced 
magnetic flux entering (or leaving) the pole per unit vertical distance 
along the pole (.delta..PHI..sub.EM /.delta.y) is proportional (to first 
order) to the magnetic scalar potential at that location on the pole 
(U(y)). Thus from an electromagnetic coil, the induced magnetic flux in 
the pole which it surrounds is: 
##EQU1## 
where c is a constant and h is the height of the electromagnetic coils. 
Since .PHI..sub.EM (T) is proportional to U(T), 
EQU .PHI..sub.EM (y)=U(T)(k+c((y-T)-(y-T).sup.2 /2h), (3) 
where k is a constant. .PHI..sub.EM (y) is maximum at y=y.sub.base =T+h and 
has a value: 
EQU .PHI..sub.EMmax =.PHI..sub.EM (T+h)=U(T)(k+c(h/2))=cU(T)(k/c+h/2). (4) 
Thus, the magnetic flux density in the pole 18 is a function of both the 
number of ampere-turns in the electromagnetic coil 36 and the location of 
those ampere-turns in the electromagnetic coil 36 on the pole 18, while 
the on-axis magnetic flux density is a function to first order) of the 
number of ampere-turns only, irrespective of their location in the 
electromagnetic coil 36 along the pole 18. Equations 1-4 are depicted 
graphically in FIG. 4. With cU(y) plotted along the abscissa and y along 
the ordinate, the slope 41 of the shaded region shows how the scalar 
potential U varies as a function of y according to equation 2. Since 
B.sub.w is proportional to U(T), the on-axis magnetic flux density is 
proportional to the width of the base of the shaded region 42, and since 
B.sub.pole is proportional to .PHI..sub.pole, the electromagnet-induced 
pole magnetic flux density at any given y is proportional to the area of 
that portion of the shaded region 42 below that y according to equation 3. 
In particular, at the base, the maximum electromagnet-induced magnetic 
flux density is proportional to the area of the entire shaded region, 
according to equation 4. For simplicity, the proportionality constant 
between the area of the shaded region 42 and the maximum electromagnet 
induced pole magnetic flux density is set to 1 in the following examples. 
In an example of the requirements for certain iron wigglers the iron 
reaches its saturation point at .+-.14 kilo Gauss (kG). To avoid the 
saturation range, the absolute value of the sum of the magnetic flux 
density of the electromagnetic coil in the pole (EM) and the magnetic flux 
density of the permanent magnet in the pole (PM) everywhere within the 
pole must be less than or equal to 14 kG, denoted by the equation: 
-14.ltoreq.(EM+PM).ltoreq.14. In this example the permanent magnet induced 
a magnetic flux density at the base of the pole is -20 kG. Then to avoid 
saturation, the magnetic flux density at the base of the pole from the 
electromagnetic coil EM must fall in the range 6&lt;EM&lt;34. This means that 
the shaded region corresponding to the electromagnetic coil induced 
magnetic flux density at the pole base must have an area between 6 and 34. 
FIGS. 4b, c illustrate the range of on-axis magnetic flux densities and 
the corresponding range of the magnetic flux densities in the pole. In 
FIG. 4b the saturation limited maximum pole flux density, and thus the 
pole tip potential corresponding to the saturation-limited maximum on-axis 
magnetic flux density are shown. The distances over which the ampere-turns 
are applied is from T=2 to y.sub.base =6 so that h=4. Since the area of 
the shaded region is set equal to the maximum flux density in the pole, 
the area of the shaded region is 34 kG. Using the equation for the area of 
a triangle A=HB/2 and the equation for the area of a rectangle A=HB, where 
H refers to the height and B refers to the base of the triangle or 
rectangle. 34=4cU(T)/2+2cU(T). Therefore, cU(T)=8.5 as denoted along the 
abscissa. The value B.sub.w is proportional to the scalar potential at the 
tip (c.f. equation 1), and in this configuration, B.sub.w, max =8.5(q/c) 
is the maximum on-axis magnetic flux density (represented by the width of 
the base of the shaded region), given the maximum magnetic flux density of 
34 (represented by the area of the shaded region) and the geometry of the 
pole and electromagnet. 
In FIG. 4c the saturation limited minimum pole flux density and thus the 
pole tip potential corresponding to the saturation-limited minimum on-axis 
magnetic flux density are shown. Since the area of the shaded region is 
set equal to the minimum flux density in the pole, the area of the shaded 
region is 6 kG. From the equation for the area of the shaded region 
6=4cU(T)/2+2cU(T). Therefore, cU(T)=1.5, as denoted along the abscissa. In 
this configuration, B.sub.w, min =1.5(q/c) (represented by the width of 
the base of the shaded region) is the minimum on-axis magnetic flux 
density value, given the minimum pole electromagnet magnetic flux density 
of 6 (represented by the area of the shaded region) and the geometry of 
the pole and electromagnet. By decreasing the height h of the 
electromagnet coils, B.sub.w, max can be increased for a given B.sub.pole, 
max but then B.sub.w, min would also be increased. By increasing h, 
B.sub.w, min can be decreased for a given B.sub.pole, min but then 
B.sub.w, max would also be decreased. What would be desirable is a means 
to both increase B.sub.w, max and decrease B.sub.w, min thus increasing 
the tunable range of the wiggler. 
It should be noted that the slopes 41 of the boundary of the shaded region 
in FIGS. 4abc are proportional to .DELTA.y/.DELTA.U. Heat transfer 
limitations restrict the maximum allowable current density in the 
electromagnet coils. The coil current density, J.sub.coil is proportional 
to the rate of change of the magnetic scalar potential along the pole 
face, dU/dy. Therefore the heat transfer constraint limiting the magnitude 
of J.sub.coil effectively puts a lower bound on the slope 41 of the 
boundary of the shaded region. A vertical boundary (infinite slope) 
implies a zero current in the coil (as in FIG. 10 for example), while a 
physically impossible horizontal boundary would imply an infinite current 
in the coil. The important point is that there are two design constraints: 
a magnetic saturation constraint and a heat transfer contraint. In FIG. 
4b, then, maximum B.sub.w is attained by increasing coil current until 
either (1) the pole saturates (i.e. the area of the shaded region 42 is 
34) or (2) the slope 41 reaches its heat transfer limited maximum 
allowable value, whichever comes first. We have assumed this example is 
saturation limited. 
The base of the pole is the location where the electromagnets induce the 
greatest flux density in the pole. When the permanent magnets are used to 
apply a reverse bias flux in the pole, then the permanent magnet may 
induce an incremental flux in the pole at a specific location that exceeds 
the incremental flux induced by the electromagnet at that location and it 
is possible that pole saturation may first occur at a location other than 
the pole base. Thus, in general, one must insure that 
-14.ltoreq.EM+PM.ltoreq.14 for all pole locations "y". The illustrative 
example of FIG. 4 assumed the pole first saturated at the base, however 
the general principles for determining the tuning range, outlined above, 
are not restricted to this special case. 
FIG. 5 illustrates the change in the tunable range caused by adding the 
adjacent permanent magnets to the poles. The dashed line 141 indicates the 
range over which electromagnetic coils alone may induce flux into the pole 
without incurring saturation. The solid line 142 indicates the range over 
which electromagnetic coils may induce flux into a pole surrounded by 
adjacent reverse-biasing permanent magnets. The adjacent permanent magnets 
shift the range but the width of the range due to the electromagnets for a 
pole surrounded by adjacent permanent magnets 44 remains approximately 
equal to the width of the range due to the electromagnets alone 43. 
FIG. 6 illustrates another type of wiggler assembly used in the prior art 
as described by K. Halbach in "Some Concepts To Improve The Performance of 
DC Electromagnetic Wigglers" cited above. The apparatus shown here uses 
sheets of permanent magnets (laced magnets) 45 between the electromagnetic 
coil windings 46 in addition to the permanent magnets 48 adjacent to the 
poles 47 to further increase the attainable on-axis magnetic flux density 
beyond that attainable with the use of only adjacent permanent magnets. 
Since the electromagnetic flux density in the pole is correspondingly 
increased for the same amount of current in the electromagnet due to its 
vertical displacement on the pole so as to accommodate the laced permanent 
magnet, the minimum attainable on-axis magnetic flux density B.sub.w, min 
increases more than does the maximum on-axis magnetic flux density 
B.sub.w, max and thus the tunable range decreases. In the prior art the 
plurality of electromagnetic coils surrounding a pole were electrically 
connected so that the current through each coil along a pole is not 
independently controlled. It would be desirable to increase the tunable 
range of this apparatus. 
SUMMARY OF THE INVENTION 
It is an object of the invention to provide a wiggler with a high on-axis 
magnetic flux density, low magnetic field errors, and a widely tunable 
range. 
Additional objects, advantages and novel features of the invention will be 
set forth in part in the description which follows, and in part will 
become apparent to those skilled in the art upon examination of the 
following or may be learned by practice of the invention. The objects and 
advantages of the invention may be realized and attained by means of the 
instrumentalities and combinations particularly pointed out in the 
appended claims. 
Tunability enhancement, introduced herein, is a design feature enabling 
attainment to a much larger extent than heretofore possible of the three 
goals of high wiggler on-axis magnetic flux density, low magnetic field 
errors, and wide tunable range simultaneously. Alternatively, tunability 
enhancement allows attainment of much larger tuning ranges in accurate, 
high magnetic flux density electromagnetic wigglers than have been 
heretofore possible. 
The invention comprises the use of a plurality of spatially separated 
electromagnetic coils surrounding a single pole, wherein the currents in 
the individual coils are independently controllable to provide large 
tuning ranges while preventing magnetic flux density saturation of the 
poles. 
The inventive apparatus and method increases the tunable range of a wiggler 
whereby the current density of the electromagnetic coil as a function of 
coil location along the pole is varied spatially to exploit the 
differences in the functional dependencies of on-axis magnetic flux 
density, B.sub.w and pole magnetic flux density, B.sub.pole on the 
magnitude and pole location of applied ampere-turns. To increase the 
maximum attainable on-axis magnetic flux density most of the ampere-turns 
are put as close to the pole tip as possible so as to minimize the pole 
magnetic flux density and thus keep the pole from magnetically saturating 
in the forward direction until a higher on-axis magnetic flux density is 
reached. Similarly, to decrease the minimum attainable on-axis magnetic 
flux density most of the ampere-turns are put as far from the pole tip as 
possible so as to maximize the pole flux density and thus keep the pole 
from magnetically saturating in the reverse direction until a lower 
on-axis magnetic flux density is reached. 
Another important feature of the inventive apparatus and method is that it 
is very flexible in that it can accommodate large changes in design points 
with very little performance loss, where such changes would render a 
conventional wiggler totally useless.

DETAILED DESCRIPTION OF THE EMBODIMENTS 
FIG. 7 illustrates a cross-sectional view of a wiggler using an embodiment 
of the invention. A first upper pole 50 is supported by an upper magnetic 
yoke 52 and has a tip 51 which is at the end of the first upper pole 50 
that is furthest from the upper magnetic yoke 52. The first upper pole 50 
is surrounded by a first electromagnetic coil 54 and a second 
electromagnetic coil 56. The number of turns in the first electromagnetic 
coil 54 is of the same order of magnitude as the number of turns in the 
second electromagnetic coil 56 meaning that the number of turns in the 
first electromagnetic coil 54 is greater than half of the number of turns 
in the second electromagnetic coil 56, and that the number of turns in the 
second electromagnetic coil 56 is greater than at least half of the number 
of turns in the first electromagnetic coil 54. The first electromagnetic 
coil 54 is closer to the upper magnetic yoke 52 than the second 
electromagnetic coil 56, and the second electromagnetic coil 56 is closer 
to the tip 51 than the first electromagnetic coil 54. The first 
electromagnetic coil 54 is electrically connected to a first 
electromagnetic coil current source 53. The second electromagnetic coil 56 
is electrically connected to a second electromagnetic coil current source 
55. These current sources 53, 55 allow the first and second 
electromagnetic coils 54, 56 to be powered independently of each other. 
Across the electron beam axis A--A from the first upper pole 50 is a first 
lower pole 58 supported by a lower magnetic yoke 60. The first lower pole 
58 has a tip 61 which is at the end of the first lower pole 58 that is 
furthest from the lower magnetic yoke 60. The first lower pole 58 is 
surrounded by a first electromagnetic coil 62 and a second electromagnetic 
coil 64. The first electromagnetic coil 62 is closer to the lower magnetic 
yoke 60 than the second electromagnetic coil 64, and the second 
electromagnetic coil 64 is closer to the tip 61 than the first 
electromagnetic coil 62. The first electromagnetic coil 62 is electrically 
connected to the first electromagnetic coil current source 53. The second 
electromagnetic coil 64 is electrically connected to the second 
electromagnetic coil current source 55. These current sources 53, 55 allow 
the first and second electromagnetic coils 62, 64 to be powered 
independently of each other. A master controller 67 is used to control the 
current sources 53, 55. FIG. 8 is a cross-sectional view of the wiggler 
shown in FIG. 7 along cut lines 8--8. Adjacent to the first upper pole 50 
is a pair of permanent magnets 66. Adjacent to the first lower pole 58 is 
a pair of permanent magnets 68. Next to the first upper pole 50 is a 
second upper pole 74, which is supported by the upper magnetic yoke 52. 
The second upper pole 74 is surrounded by a first electromagnetic coil 76 
and a second electromagnetic coil 78. Adjacent to the second upper pole 74 
is a pair of permanent magnets, not shown. Across the magnetic axis A--A 
from the second upper pole 74 and next to the first lower pole 58 is a 
second lower pole 82, which is supported by the lower magnetic yoke 60. 
The second lower pole 82 is surrounded by a first electromagnetic coil 86 
and a second electromagnetic coil 84. Adjacent to the second lower pole 82 
is a pair of permanent magnets, not shown. 
FIGS. 9 and 10 illustrate how the inventive wiggler is useful in increasing 
the tunable range of a wiggler. FIG. 9 is a half pole shown in FIG. 8 with 
a graph of the magnetic scalar potential U along the pole 50. The 
attainable on-axis magnetic flux density can be inferred from the value of 
the scalar potential at the pole tip U(T) using Equation 1. The pole 
magnetic flux density due to the electromagnetic coils at any location "y" 
can be inferred from the area of the shaded region below that "y" value 
using Equation 3. Using the parameters given in the example of the prior 
art, for certain iron wigglers the iron reaches its saturation point at 
.+-.14 kG. To avoid the saturation range, the absolute value of the sum of 
the pole magnetic flux density due to the electromagnetic coils and the 
pole magnetic flux density due to the permanent magnet must be less than 
or equal to 14 kG along the entire pole length, ie: 
-14.ltoreq.(EM+PM).ltoreq.14. As in the previous example the magnetic flux 
density of the permanent magnets is -20 kG. To avoid saturation the pole 
magnetic flux density due to the electromagnetic coil EM must fall in the 
range 6.ltoreq.EM.ltoreq.34. This means that the shaded area 24 must have 
an area between 6 and 34. FIG. 9 illustrates an approximate B.sub.w, max 
obtainable if the first and second electromagnetic coils 54, 56 
surrounding the pole are independently controllable. In this example, the 
current in the second electromagnetic coil 56 is first increased from zero 
until it reaches its maximum heat transfer limited value or until the 
magnetic field in the pole reaches a point just before the pole starts to 
saturate whichever occurs first. In FIG. 9 we assume the heat transfer 
limited value of current in the second electromagnetic coil 56 is reached. 
Then the current in the first electromagnetic coil 54 is increased from 
zero until the magnetic field produced by the current reaches a point just 
before the pole starts to saturate. In this example, which is the same as 
the example in the prior art except that two independently controlled 
electromagnetic coils are used in place of the single electromagnetic coil 
used in the prior art, for a shaded region of 34 kG an obtainable magnetic 
flux density B.sub.w =10.8(q/c) kG, as shown in FIG. 9. This provides a 
magnetic flux density increase of 27% over the maximum obtainable magnetic 
flux density increase in the prior art. 
FIG. 10 is a half pole shown in FIG. 8 and a graph of the magnetic scalar 
potential U along the pole. FIG. 10 illustrates the minimum on-axis flux 
density attainable by the embodiment of the invention illustrated in FIGS. 
7 and 8 using two electromagnetic coils and where current sources 55 and 
53 independently provide current to the coils. Starting with both coil 
currents at zero, the current in the first coil 54 is increased until the 
pole becomes unsaturated (i.e. the area of the shaded region becomes 6) or 
until it reaches its maximum heat transfer-limited value, which ever 
occurs first. If the pole is first unsaturated, then the current in the 
second electromagnetic coil is left at zero as shown in FIG. 10. If the 
first coil 54 has reached its maximum heat transfer limited value, the 
current in the second electromagnetic coil 56 is increased until the pole 
becomes unsaturated. In this example as shown in FIG. 10 the minimum 
attainable on-axis magnetic flux density B.sub.w =1.2(q/c) kG representing 
a 20% reduction in the minimum on-axis magnetic flux density in the prior 
art with the same parameters. 
Therefore in this embodiment of the invention using two independently 
controllable coil currents the on-axis magnetic field can be increased by 
approximately 27% and decreased by approximately 20% over the prior art. 
This allows the wiggler to have a 37% wider tunable range, without causing 
the poles to experience magnetic saturation, thus reducing magnetic field 
errors. 
FIG. 11 illustrates an on-axis magnetic flux density which is attainable if 
the electromagnets have a polarity that is reversible. FIG. 11 illustrates 
a half pole shown in FIG. 8, with a graph of the magnetic scalar potential 
U along the pole 50. In this example the net shaded area will be equal to 
6, corresponding to the minimum allowable electromagnetic coil induced 
flux density of the pole which is 6 kG. The first electromagnetic coil 54 
generates a magnetic flux density so that its contribution to the magnetic 
flux density on-axis is 7.2(q/c) kG The area of the shaded region 84 from 
y=4 to y=6 forming a triangle is 7.2. The second electromagnetic coil 56 
generates a magnetic flux density so that its contribution to the magnetic 
flux density on-axis is -10.0. This forms two triangular shaded areas over 
the distance between y=2 to y=4. The first triangular shaded area 86 has 
an area of (1/2)(7.2)(1.44)=5.184. The second triangular shaded area 88 
has an area equal to (1/2)(-2.8)(2-1.44)=-0.784. This also yields a shaded 
rectangular area 90 equal to (-2.8)(2)=-5.6. The total area is 
7.2+5.184-0.784-5.6=6. In this configuration, the electromagnetic coil 
induced magnetic flux density in the pole is not less than 6 kG, 
preventing saturation, and yet an on-axis magnetic flux density of -2.8 is 
obtained. This represents a 94% increase in the tunable range over that of 
the prior art. 
In review, FIG. 9 illustrates how for a given pole-coil-permanent magnet 
geometry the invention increases the maximum attainable on-axis magnetic 
flux density by putting as much current as possible as close to the tip of 
the pole as possible. FIGS. 10 and 11 illustrate how for a given 
pole-coil-permanent magnet geometry the invention decreases the minimum 
attainable on-axis magnetic flux density by putting as much current as 
possible as far from the tip of the pole as possible. The combined effect 
is to greatly expand the tunable range of the wiggler. 
Therefore the inventive wiggler in this embodiment has a tunable range from 
-2.8(q/c) kG to 10.8(q/c) kG, with a width of 13.6(q/c)kG. The prior art 
under the same parameters may have a tunable range from 1.5(q/c) kG to 
8.5(q/c) kG, with a width of 7(q/c) kG. Therefore in this embodiment the 
invention provides a tunable range with a width that almost doubles the 
width of the tunability range of the prior art without introducing 
additional field errors. The increase in tuning range attained by the use 
of the invention is more dramatic for higher field wigglers employing 
laced permanent magnets. 
As long as the electromagnetic coils are adjusted to keep the magnetic flux 
density at all locations in the pole within the unsaturated range, the 
main limitation on the range of the magnetic flux density is the heat 
transfer constraint. An increase in current in the electromagnetic coils 
increases the amount of heat produced. The ability of the wiggler to 
remove the produced heat limits the current in the electromagnetic coils, 
thus limiting the magnetic field produced by the electromagnetic coils. 
Although the invention may be used with more than two independent 
electromagnetic coils around each pole, two independent electromagnetic 
coils provides the tunability range desired in current applications of the 
invention. 
FIG. 12 is a graph of the linear operating regime with the current density 
in the first electromagnetic coil 54 plotted along the ordinate and the 
current density in the second electromagnetic coil 56 plotted along the 
abscissa. In the prior art, the current density in the first 
electromagnetic coil 54 was always approximately equal to the current 
density in the second electromagnetic coil 56 since the prior art used 
either a single electromagnet around each pole or a plurality of 
electromagnetic coils which were not independently controlled. Therefore 
the range achieved by the prior art was along a diagonal line 70. Also 
shown are lines illustrating the magnetic saturation constraint 72 and 
lines illustrating the heat transfer constraint 74. Given these 
constraints, the prior art may only operate along line 70 between points 
76 and 78. The invention allows the flexibility to operate anywhere within 
the shaded region 80. The lines 82 shading the region 80 are lines of 
constant on-axis magnetic flux density, B.sub.w Thus for the invention, 
the on-axis magnetic flux density B.sub.w is maximized at operating point 
84 and minimized at operating point 86 (or operating point 88 if the coils 
do not have reverse polarity capability). The prior art yields a tunable 
range between 76 and 78, with the inventive improved tunable range being 
between 84 and 86 (or 88). 
The foregoing description of preferred embodiments of the invention have 
been presented for purposes of illustration and description. It is not 
intended to be exhaustive or to limit the invention to the precise form 
disclosed, and obviously many modifications and variations are possible in 
light of the above teaching. The embodiments were chosen and described in 
order to best explain the principles of the invention and its practical 
application to thereby enable others skilled in the art to best utilize 
the invention in various embodiments and with various modifications as are 
suited to the particular use contemplated. It is intended that the scope 
of the invention be defined by the claims appended hereto.