IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-9, NO. 4, DECEMBER 1973
678
One-sided Fluxes
- A Magnetic Curiosity?
J. C. MALLINSON
Abstract-It is shown that a previously unknown class of constantor zero.Suppose the magnetization is the supermagnetization patterns exists in planar structures which have position of two sinusoids in quadrature
the unique property that all the flux escapes from one surface
m, = mo sin k x
my = mo cos kx
m, = 0
(1)
with none leaving the other side. A simple case is a constant
amplituderotatingvectormagnetizationwherethe
senseof
where k is the wavenumber, 271/h.We wish to solve the boundrotation dictates which surface has no flux. More complicated ary value problem for the scalar potentials and fields above
magnetization patterns are elucidated. The likelihood that the
and below the sheet.
one-sidedfluxphenomenonoccurspartiallyin
thenormal
Thepotentialswithinthesheetobey
Poisson’s equation
write, the contact-printing and the print-through processes of
V2 @inside = mo k cos kx
tape recording is discussed. It is concluded that significant im(24
provementsintaperecordingperformancewouldensue
if
and Laplace’s equation above and below,
means could be foundto enhance the effect.
(2b)
@above = 0
v2 $’below
INTRODUCTION
= 0.
(2c)
Since the particular solution of (2a)
is -(molk) COS k x , the genThis paper is concerned with the symmetry of themagnetic
eral solutions are of the form
flux emanating from tapes,
disks, and other planar magnetic
structures. It is shown theoretically that a previously unknown
@above e-ky + B e+ky cos kx
(3 4
class of magnetization patterns exists which has the remarkable property that all the flux escapes from one surface with
$‘inside = C e-b’ + D e+ky - 9 COS kX
(3b)
k
none leaving the other side. Such one-sided’ fluxes are often
quitesurprisingandinconsistentwith“intuitive”concepts.
$’below = E e-ky + F e+ky
cos kx .
In the first part of the paper the
magnetic theory is de(3c)
veloped. Two proofs aregiven that aone-sidedfluxoccurs
when the magnetization is a constant amplitude rotating vec- They are subject to six boundary conditions. The fields and
tor, the sense of rotation dictating which surface has no flux. potentials must goto zero as y becomes infinite; that is,
The theory is thenexpanded to includegeneral twodimen@above = @below = 0,
when y = m, respectively.
(4a)
sionalmagnetizationpatterns;
it transpiresthatin principle
aninfinitesetof
one-sided fluxmagnetizationpatternsexThetangential fields mustmatch
onthesheetupperand
ists. Anexample of athree-dimensionalmagnetization
patlower surfaces, thus
tern is introduced. Finally, the physical realizability of such
one-sided fluxes is discussed.
@above= @inside
when y = o
(4b)
Thesecondpartofthepaperdealswithpracticalsituations in which the one-sided flux phenomena may occur. In
@below= @inside3
wheny =-d.
(4c)
recordingtapesapartialeffectmayoccur
inthe writing
process, and as Daniel[l] recently pointedout,thecontact
The normal flux density must be continuous on the upper and
printingandprint-through
processes. Littledirectevidence
lower surfaces, thus
for,one-sidedfluxeshasbeenreported.Until
such time as
deliberate efforts are made to exploit the effect, it remains a
-- @above =
+mocos k x , when y = 0
(4d)
magnetic curiosity.
3.
I
{
j
t
I
9
a
aY
I. THEORETICAL CONSIDERATIONS
-?-!hi&
aY
a h i d e + mo cos k x ,
--a @below- -aY
aY
when y = 4. (4e)
Rotating Magnetization VectorCase
We consider a planar structure, of thickness d, lying in the
x , z plane. The upper andlowersurfacesare
at y = 0 and
4, respectively. Discussion is restricted to two-dimensional
magnetization patterns inwhich the z component is either
The reader may verify that the solutions consonant withthese
restrictions are
@above = 0
@inside =-
Manuscript received October 12, 1972; revised March 2 , 1973.
The author is with the Ampex Corporation, Redwood City, Calif.
94063.
$below
m0
k
=?
(54
(eky - 1)COS k X
(1 - e k d ) eky cos k x .
(5b)
(5c)
MALLINSON: ONE-SIDED FLUXES - A MAGNETIC CURIOSIm?
679
The remarkable fact thus emerges that the scalar potential
above the sheet is identically zero everywhere. Thus no flux
emerges from the sheet’s upper surface. All the flux emerges
from the lowersurface.
It may be shown that if the sense of magnetization rotation
is reversed, so that, for example,
m, = mo sin k x
(64
my = - mo cos kx
(6b)
m, = 0
(6c)
rnx
= sin kx
my ‘COS kK
m=mx+my
the potentials then become
Gibove =
@Aside =
(1 - e-kd) e-kY cos kx
(e-k @+Y)
- 1)cos k x
(74
(7b)
Fig. 1. (a) Flux from longitudinallymagnetized tape. (b) Flux from
perpendicularlymagnetizedtape.
(c) Flux from rotating vector
magnetized tape.
adding this y component, the sheet then contains
a number
of superimposed clockwise rotatingmagnetizationvectors,
all of whichyield
one-sided fluxesseparately.Obviously
and no flux escapes from the lower surface.
their
sum
is
also
a
one-sided
flux.
When themagnetization, viewed inthe negativez-direcThe
mathematical
transformation
which changes the phase
tion, rotates clockwise the flux is all below the sheet (Eq. 5);
of
all
harmonics
by
R/Z
i
s
called
the
Hilbert transform and
whenanticlockwise, theflux is all above thesheet (Eq. 7).
is
defined
by
[
21
Thesefindingsmay
well restuncomfortablywiththe
ca
reader.Considerable
insight andperhapsamorephysical
understanding of thesituationmaybeprovidedbythe
?(x)
71
f(x‘) (x-X’)-l dx‘
following second proof. Consider the sheet to be longitudinally magnetizedaccording to m, = mo sin kx only. Figure l a
showsschematicallythismagnetization,
the -V*M volume
where it is understood that at x’= x the Cauchy principal valpolesinducedandthesymmetry
of the fluxes leaving the ue is to bz taken. Extensive tables of Hilbert transformpairs,
upper and lower surfaces. Now suppose the sheet is mag- f (x) and f ( x ) , exist. [ 31
netizedperpendicularlyaccording
to my = mo cos kx only,
We see, therefore,thatfor
every conceivable x comFigure Ib shows this magnetization, the Men surface poles inponent there exists a particular y component which produces
duced and the symmetry of the outside fluxes. In Fig. ICwe
a one-sided flux,The
converse is also true, so thatwhen
show the result of the superposition of the two magnetbaeither
tions; again it becomes clear that a clockwise rotation of the
magnetization vector produces a
one-sided flux with no flux
crosskgtheupper
surface.Notice
that, while thefluxes
from the x and y components add in phase below the tape,
theyare R outof phaseandcancelcompletely
above the
tape. The point of view may be takenthat,with
regard to
a one-sided flux ensues.
theupper region, the surfacepolesand
the volumepoles
The readerwithinterestsincommunication
theory will
cancel each other exactly.
notethe
similaritybetween
the generalone-sided
fluxcriterion given
above
andcertainotherknown
“one-sided”
Extension to General Magnetization Patterns
results. Writing the Fourier transform pairs
as f ( x ) + + F ( kwe
)
Suppose now that one component of the magnetization is may cite the causality criterion,[2]
specified and we pose the question, “is it possible to find the
f(x),
when x 2 0
other magnetization component so that a one-sided flux enF ( k ) i &k) -+
(11)
sues?” In principle, the answer is always affirmative.
0,
whenx < O
Suppose that the x component is an arbitrary, real function of x, m, = f(x),and we seek the matching y component
and its dual, thesingle sideband criterion,
whichyields the one-sided flux. The function f(x) contains
certain sine andcosineharmonics.Clearly
the y component
sought must have just the same amplitude harmonics shifted
in phase by n/2, so that the sines become cosines, etc. Upon
96elow = 0
(74
=‘J
-
IEEE TRANSACTIONS O N MAGNETICS, DECEMBER 1973
680
The similaritymay
pressed in
as
be formalized bywritingtheresultsex-
where the operator i denotes n / 2 spatial rotation of the magnetization
vector
and
the
symbol
represents
the
linear
transformationconnectingthemagnetizationand
the flux
outside the sheet.
The linearity of Maxwell’s equations permits an extension
ofthe above two-dimensional (z invariant)results to three
dimensions.Suppose
we have,separately,twosheets
magnetized according to
-!!//h\\
- xy ,
-I
- x2+1
x - $+I
Fig. 2. Two magnetization patternsyieldingone-sidedfluxes.
flux exists abovethe planes.
As an example of a
sider
No
realizable Hilbert transform pair con-
n
m, = 0
and
mz = f2
(2).
(14c)
Clearly both yield one-sided fluxes. Furthkrmore,they may
beaddedtogether
so that we may conclude that the threedimensional patterns
A diagram of the corresponding- pair
of vector magnetiza_
tionpatterns is shownin Fig. 2. The readermay test his
‘‘intuition’’ andspeculatewhetherthefluxesemergefrom
the upper orlower surfaces.
As an example of impermissible functions, let us consider
the integrals of the pair just given
x
g(x) =$
f(x) dx = tan-’ x
alsogives a one-sided flux for arbitrary a and 0. It maybeHere
g ( x ) fails condition (18) and $(x), being unbounded, is
shown thatmore
generalthree-dimensionalmagnetization
not physically realizable.
patterns giving one-sided fluxes exist.
Reverting- to the two-dimensional cases we conclude with
a discussion ofphysicalrealizability.
We have provedthat,
11. PRACTICAL MANIFESTATIONS
foranarbitraryfunction
f ( x ) , a one-sidedfluxfollows
if
theinanAd out-of-plane components of magnetizationare
The Normal Writing Process
f ( x ) and f ( x ) . Inorder to bephysicallyrealizable the magIn magnetic recording the medium (tape or
disk) is magnitude of the magnetization vector cannot exceed the saturanetized
by
subjecting
it
to
the
fringing
field
of agapped
tion value m, anywhere, thus
writehead. It is usuallyassumed that thisproducesapreV(.)l2 + t i c X ) I 2 G d
(16) dominantly longitudinal (in plane) magnetization pattern. The
fluxes recoverable in the read head are, at the limits of long
and shortwavelengths, given by [41
Thecondition
placesconstraintsupon
thz allowablefunctions; it is necessary that both f ( x ) and f(x) remainfinite
(long wavelengths) (21a)
@+- ” ~ K M Rd
everywhere. Whereas in principleany bounded f ( x ) may be
P+1
matchedwithitsHilberttransform,physicalrealizability
limits f ( x ) to functionswithboundedHilbert
transforms.
(short
wavelengths)
(21b)
@+- 2P 2 pkM R e-kQ
For continuous functions, the Hilbert transforms are bounded
P+1
providing
where p is the read head permeability and a is the read headIf(x)I < m,
for all x
(17)
to-tape
spacing. If
the
means
could
be
found
to add to this
Iongitudinal magnetization an equal, and properly phased, perf ( ~ )-+ 0,
as x + ? m.
(18)
pendicular
(out-of-plane)
component,
the
above
fluxes
would
-
MALLINSON: ONE-SIDED FLUXES - A MAGNETIC CURIOSITY?
LAh4lM IN PARTICULAR
,NOTETHIS
.
68 1
.
.
.
.
.
.
.
.
.
Fig. 3. Actual magnetheon pattern recorded on tape (after Tjaden
and Leyten[S]).
bedoubled.For
become
high permeabilityheadsthelimitsthen
Fig. 4. Contact printing (and, with upper tape inverted, print-through)
configuration.
@ + ~ T M Rd
8s MI(
@ + -e-ka
k
j
(long
wavelengths)
(short
wavelengths).
(22b)
How can this be achieved? Presently, no well-defined techniques suggest themselves. As hasbeenemphasized
by repeated investigations, it is very difficult t o conceive of practical write heads whose fringing field is substantially different
from the norm.
Similarly, the scopeformakingtapes
with
radically different properties seems to be rather limited.
ArationalstartingpointforinquiryappearsinTjaden
andLeyten’s work.[5]In Fig. 3, copiedfromTjadenand
Leyten, we showavectormagnetizationpattern
measured
in the Philips 5000:l scale-up recorder. The reader will note
that not only is there
evidence of a rotating magnetization
vector, but also that it is rotating in the correct
sense t o increase the read head flux! With our current deplorable lack
of understanding of the full vector writing process, not
even
an intuitive explanation of this finding can
be given.Nevertheless, experiments might well be undertaken
t o determine
which factors promote the magnetization rotation.
The Contact Printing Process
Incontactprintingthe
slave tape is magnetized bythe
fringing field of a prerecorded master tape.
The process may
beenhancedbyeithermagneticofthermalexcitations;
comprehensivetheoreticalanalysesare
available forboth
methods. [ 61
Consider the master tape fringing
field. It may be shown
from (5) and (7) and is clear from Fig..
1 that the fringing
field appears to rotate whenobserved in a plane n o p a l t othe
tapeplane. The senseof rotation(asdefined
in Part I) is
clockwiseaboveandanticlockwisebelow
themastertape.
Thisremainstruewithout
regard tothe vectordirection,
magnitude or phaseof the master tape magnetization.
Aftercontactprinting
hasoccurred,
we might naively
expect,therefore,the
slave tapemagnetization to rotate in
the same direction as themastertape,field.Thusa
slave
tape placed below the master
tape would have a magnetization vector rotating anticlockwise and vice versa for the slave
tape above. It is intriguing to note that, in accordance with
the one-sided flux rules deduced in Part I, thesesenses of rotationofthe
slave tapemagnetizationaresuchthat
all their
fluxes appear on the master tape side only as shown in Fig. 4.
Inreality,however,twofactorsspoilthe
simplescheme
justpostulated.First,the
slave tapemagnetizationactually
acquired is proportional t o the
difference
between
the
mastertapefieldandthe
slave tape’s owndemagnetizing
field. Second, most tapes are magnetically anisotropic so that
theanhystereticorthermoremanantsusceptibilities
are
not equal in the
x and y directions. For both reasons, the
x and y magnetizationcomponentsare
not usuallyequal.
Thecompletetwo-component,cross-coupledproblemhas
beensolved.[7]Theresultsshowthatthefluxesonthe
mastertapeside
of the slave tapeareproportionaltothe
following factors:
(x)
{
xx
+ l +xxy y
1) (x, + 1) - 1
d ( x x + 1) (x, + 1) + 1
4
X
X +
(long
wavelengths)
(23a)
(shortwavelengths).(23b)
x,
= 0 and
Uponcomparingthesefluxes
€or two cases,
0, it is seen that the addition of y components of magnetization always increases the slave tape flux. Thus, although
generally the x and y components of the slave tape magnetization are not equal, they are
always in phasequadrature. A
fiaction of the slave tape magnetization always rotates.
Rotation of the slave total magnetization vector can only occur
in the limit of short wavelengths for isotropic (X, =
tape.
l
i no doubt exist in this case it
Although one-sided fluxes w
should be borne in mind that, due
to spacing loss effects at
short wavelengths, in no case would any flux emerge from the
reverse (away from master tape) side of the slave tape.
X, #
x,)
The Print-Through Process
Print-through is the unintentional contact printing which
occurs
when
reels
of
prerecorded
tape
are
stored.
The
physical process is analogous t o contact printing and consequently all thetheoreticalapparatusofcontactprinting
is
directlyapplicable
t o print-through. A majordifferenceis
thattheadjacentlayers
oftapeareseparatedbythe
relatively thick tape base 6Ln so that print-through at short wavelengths is precluded. Furthermore, the
values of susceptibilities
IEEE TRANSACTIONS ON MAGNETICS, DECEMBER 1973
682
pertainingare so low that,atlong
wavelengths, very little
“shearing”of
the y componentoccursand,therefore,the
limitingexpression(Eq.23a)
given forcontactprintingat
long wavelengths becomes
xx
*xy -
(25)
Here the + sign refers to the master tape side of the slave ;ind
vice versa. As discussed above, it is clear that, while the x and
y components are not usually equalinmagnitude,they
are
in phase quadrature.Onlyafractionof
the magnetization
rotates and onlya fraction of the fluxis one-sided.
Experimentally,inaccordwith(25),
it is foundthat a
larger print-through signal is alwaysmeasured onthetape
layerswhichfaced
the master. Fortape reels woundoxide
side in, the “pre-prints” always exceed’
the “post-prints.” In
standard YFe203 tapes,a 3-4 dBdifference is measured at
wavelengths of 5-10 mils, which is morethancanbe
accounted for by spacing loss effects: clearly a fraction of the
flux is one-sided. Incobaltdoped
y F e 2 0 3 “high energy”
tapes, the predominance of arandomcubicmagnetocrystalline energyrenders the tapesalmostmagneticallyisotropic
and, satisfyihgly enough,much larger differences of 8-9 dB
have been observed; in this
case about one half of the printthrough flux must be one-sided.
Next, we consider theprint-throughexpectedfor
aperfectly
isotropic
( X , = X , ) tape.Inthis
case the
printthrough
magnetizations
are full
vector
rotating
and
the
fluxes one-sided so thatnoprint-through
signal is expected
forthetape
layersfacingawayfrom
the mastertape. Dependentuponthewindingmethod(oxidein
or out), isotropic tapes would have the special property of having either
no “pre-prints” or not “post-prints.”
Finally,weremarkupontheintriguingpossibilitythat
tapes completely without print-through are conceivable. Suppose that the tape is isotropic and has been prerecorded with
a rotating vector magnetization
so that its flux is one-sided.
No matter which way this tape is wound, no print-through can
occur. If the “pre-print” is zero due to isotropy of the tape,
the “post-print” is zerodue tothe one-sided mastertape
flux and vice versa.
CONCLUSIONS
It has been demonstrated theoretically that there exists a
class of two-dimensional magnetization patterns which yield
one-sided fluxes. It is hoped that the analogy between this effect and certain well-known results in communications theory
may stimulate further theoretical work.
Some evidence exists that one-sided fluxesexistin
both
the normal write and the print-through
processes of tape recording, If the effects could be enhanced, significant improvements in tape recorder performance would ensure. Not only
might the signal-to-noise ratio be increased but also, perhaps
more importantly, print-through might beeliminated.
Finally,wemay
pointoutthat
it is possible that otlesided fluxes occur, and may
be detectable, in a wide variety
of other magnetic structures. Thus there may
be magnetization
patterns whichproducenoBitterpatternat
all orperhaps
thereexist“bubble”structureswithanamolously
small (or
large) detectable fluxes.
REFERENCES
Daniel, E. D., “Tape Noise in Audio Recording,” A.E.S. JournaZ,
vol. 2 0 , 2 , pp. 92-99, March 1972.
Papoulis, A., “The Fourier Integral andIts
Applications,”
McGraw-Hill, p. 198,1962.
Erdelyi, “Tables of Integral Transforms,” vol. 2, McGraw-Hill,
pp. 243-262, 1954.
Wallace, R. L., “The Reproduction of Magnetically Recorded
Signals,” Bell Syst. Tech, J., pp. 1145-1123, October 1951.
Tjaden, D. L. A. and Leyten, J., “A 5000:l Scale Model of the
Magnetic RecordingProcess,”
Philips Tech. Review, vol. 25,
pp. 319-329,1964.
Mallinson, J. C., Bertram, H. N. and Steele, C. W., “A Theory
of Contact Printing,” IEEE Truns. Magn., vol. MAG-7, pp. 524527, Sept. 1971.
Bertram, H. N., “Imaging Effects
in
Anisotropic Contact
Duplication,” AIP C m f Proc., vol. 10, pp. 1440-1444, 1973.