Nea - ield enhanced imaging by a magne ized e i e slab
R. Ma qués,a)F. Mesa, and F. Medina
Depa amen o de Elec ónica y Elec omagne ismo, Facul ad de Física, Uni e sidad de Se illa,
A . Reina Me cedes s/n, 41012 Se illa, Spain
(Recei ed 3 May 2004; accep ed 17 No embe 2004; published online 4 Janua y 2005)
This le e demons a es he nea - ield imaging enhancemen a mic owa e equencies o
wo-dimensional sou ces by a e i e slab magne ized o sa u a ion. I is shown ha his e ec is
based on he non ecip ocal ampli ica ion o magne os a ic su ace wa es (MSSW)ac oss he e i e
slab. The inclusion o losses in ou analysis has also made i possible o p o e his e ec o ealis ic
y ium i on ga ne e i e samples. Fo e i e slabs o wid h d, he esolu ion a he image plane (a
a dis ance 2d om he sou ce)is be e han he esolu ion in ai a a dis ance do he sou ce, which
leads o an equi alen ai leng h o he e i e slab less han ze o. Since he cons i u i e pa ame e s
o sa u a ed e i es depend on he ex e nal magne izing ield, he ope a ion equency o he
p oposed imaging de ices can be uned by a ying his biasing ield. © 2005 Ame ican Ins i u e o
Physics.[DOI: 10.1063/1.1852091]
The p edic ion o he pe ec lens e ec in a le -handed
slab1has ce ainly g own an inc easing in e es on he ocus-
ing p ope ies o ma e ials wi h nega i e pa ame e s. In ad-
di ion o ocusing a in e media e- ield dis ances,2i has been
shown ha imp o ed nea - ield imaging can be achie ed by
means o media wi h only nega i e pe mi i i y, such as me -
als a op ical equencies.1,3 This beha io has been ound o
be a consequence o he coupling o su ace plasmons a bo h
slab in e aces.1Spin wa es wi h a beha io simila o su -
ace plasmons in me als can be ound in e imagne ic slabs
a mic owa e equencies. This ac sugges s he possibili y
o ob aining some kind o nea - ield imaging enhancemen
by employing hese ma e ials. Ex emely low-loss e i e
slabs a e p esen ly a ailable as mic owa e ma e ials. Since
losses s ongly deg ade he image enhancemen ,4 his las
p ope y o e i es can esul in a signi ican imp o emen o
he imaging. Ano he ad an age would be he unabili y o
he equency o ope a ion, which could be achie ed by
a ying he in ensi y o he ex e nal magne izing ield.
Cubic low-loss so e i es, such as y ium i on ga ne s
(YIGs), a e commonly manu ac u ed o di e se applica ions
in mic owa e echnology. When hey a e magne ized o sa u-
a ion by an ex e nal s a ic magne ic ield H0=H0z
ˆ. i s mag-
ne ic p ope ies a e well desc ibed by he so-called Polde
enso 5
=
冢
−i
0
i
0
001
冣
,共1兲
wi h
=1+
m
h
h
2−
2;
=
m
h
2−
2,共2兲
whe e
h=
␥
H0and
m=4
␥
Ms(
␥
is he gy omagne ic a-
io, and Msis he magne iza ion o sa u a ion). Magne ic
losses can be accoun ed o by he subs i u ion
h→
h
+共i/2兲共⌬H/H0兲
, whe e ⌬His he esonance linewid h.
Magne os a ic o nonexchange spin wa es a e well known
solu ions o he elec omagne ic wa e equa ion in magne ic
media cha ac e ized by Eq. (1). They a e called magne o-
s a ic because mos o hei ene gy is associa ed wi h he
magne ic ield. The e o e, he elec ic ield can be neglec ed
in a i s o de o app oxima ion and ⵜ⫻H⬇0. Thus,
H⯝−ⵜ
m, whe e
mis he magne os a ic po en ial, which
mus sa is y he magne os a ic wa e equa ion,ⵜ·
·ⵜ
m
=0, o
冋
冉
2
x2+
2
y2
冊
+
2
z2
册
m=0. 共3兲
Magne os a ic su ace wa es (MSSWs)a e hose speci ic so-
lu ions o he magne os a ic wa e Eq. (3) ha a e con ined o
he su ace o he e i e.5Fo ideal lossless e i es, hey
appea when
+
=−1, o equi alen ly o a equency
=
h+1
2
m. In his case, he magne os a ic po en ial akes he
o m
m=
0exp共−i
+iky±kx兲, decaying exponen ially a
bo h sides o he e i e in e ace [see Fig. 1(a)]. The e o e,
MSSWs a e simila o su ace plasmons in many aspec s.
Howe e , due o he aniso opy o he magne ized e i e,
hey only p opaga e a he speci ic di ec ion de ined by he
wa e ec o k=kk
ˆ, wi h k
ˆ=H
ˆ0⫻n
ˆ[see Fig. 1(a)], bu no in
he opposi e one. This beha io u ns ou o be a di ec con-
sequence o he iola ion o he Lo en z ecip oci y heo em
in magne ized e i es.5
a)Elec onic mail: [email p o ec ed]
FIG. 1. (a)Rep esen a ion o he ampli ude o a MSSW a a e i e in e ace
(solid cu e). The di ec ions o he magne izing ield H0, he p opaga ion
uni ec o k
ˆ, and he su ace no mal n
ˆa e illus a ed. (b)Rep esen a ion o
he ampli ude o a MSSW in a quasipe ec nea - ield e i e ocusing
de ice.
APPLIED PHYSICS LETTERS 86, 023505 (2005)
0003-6951/2005/86(2)/023505/3/$22.50 © 2005 Ame ican Ins i u e o Physics86, 023505-1
Le us now conside he ex ension o he solu ion o Eq.
(3)ske ched in Fig. 1(a) o a e i e slab o ini e wid h das
in Fig. 1(b). The solu ion ske ched in Fig. 1(a)isaphysical
solu ion, decaying a bo h sides o he e i e in e ace. How-
e e , he e is also a ma hema ical solu ion o Eq. (3) ha
exponen ially g ows a bo h sides o he e i e in e ace and
p opaga es along he di ec ion k
ˆ=n
ˆ⫻H
ˆ0(see Re . 5). This
la e solu ion can be imposed a he le in e ace o he slab
and combined wi h he o me one a he igh in e ace o
gi e he solu ion ske ched in Fig. 1(b). This ield dis ibu ion
shows exac ly he beha io needed o he onse o he pe -
ec lens e ec ,1and becomes physical i he app op ia e
sou ce is loca ed a x=0. In ac , he ampli ude o he
MSSWs a x=0[see Fig. 1(b)] is es o ed a x=2d o all he
wa e numbe s k. Nex , i will be conside ed a wo-
dimensional ime-ha monic magne ic ield sou ce loca ed a
x=0, a a dis ance x0⬍d om he le slab in e ace. I dand
x0a e bo h much smalle han he ee-space wa eleng h
associa ed wi h he sou ce equency, we a e in he ealm o
he magne os a ic app oach. Thus, he magne ic ield p o-
duced by he ea lie sou ce can be exp essed as he ollowing
Fou ie in eg al o magne os a ic wa es (p opaga ing a bo h
di ec ions along he yaxis and decaying owa ds he e i e
slab):
m共x,y兲⬇ 1
2
冕
−⬁
⬁
˜
m
共
k
兲
eiky−kx−i
dk,共4兲
whe e
˜
m共k兲is he Fou ie ans o m o
m共y兲. Following
he ea lie heo y, a e se ing up a sou ce equency equal o
he equency o he MSSWs
共
=
h+1
2
m
兲
, he ampli ude o
all he magne os a ic wa es p opaga ing along he posi i e y
di ec ion will be es o ed a a dis ance 2d om he sou ce
[see Fig. 1(b)]. On he con a y, he magne os a ic wa es
p opaga ing along he nega i e ydi ec ion will be e lec ed
by he slab, and ha dly ep oduced a he igh -hand side o
he slab. Assuming a nea ze o ampli ude o hese la e
magne os a ic wa es a x=2d[see Fig. 1(b)], he magne o-
s a ic ield ampli ude a x=2dwill be app oxima ely gi en
by
m共2d,y兲⬇
冕
0
⬁
˜
m
共
k
兲
eiky−i
dk共5兲
o
m共2d,y兲⬇1
2
m
共
0,y
兲
+i
2
冕
−⬁
⬁
m共0,u兲
y−udu,共6兲
whe e
m共0,y兲is he magne os a ic po en ial a he sou ce
plane, and he second in eg al e m can be iden i ied as he
Hilbe ans o m o his la e quan i y.6I , o ins ance, he
magne ic ield dis ibu ion a he sou ce plane is gi en by a
del a unc ion, he co esponding magne ic ield a
x=2d— he image plane—will be a del a unc ion plus he
Hilbe ans o m o his unc ion ( he 1/y unc ion). This
example illus a es how he e i e slab beha es as a quasip-
e ec nea - ield ocusing de ice.
Up o now only ideal lossless e i es ha e been consid-
e ed in he ame o he magne os a ic app oach. Howe e ,
ac ual e i es a e lossy, which subs an ially a ec s he ocus-
ing p ope ies o he e i e slab.4Mo eo e , he magne o-
s a ic app oxima ion is s ic ly alid only in he limi d→0.
In o de o ha e a comple e pic u e o he a o emen ioned
e ec s in ac ual e i e slabs, an exac elec omagne ic analy-
sis has been also pe o med. This analysis accoun s o bo h
p opaga ing and e anescen wa es in he Fou ie expansion
o he ields [a a di e ence o (4-6)]. I also accoun s o
magne ic losses by in oducing a non anishing esonance
linewid h ⌬Hin he Polde enso (1). Assuming a gi en
dis ibu ion o he ans e se- o-xmagne ic ield,
H 共0,y兲exp共−i
兲, a he sou ce plane 共x=0兲, he magne ic
ield a he image plane 共x=2d兲can be exp essed as he
ollowing spec al-domain in eg al:
H 共2d,y兲=1
2
冕
−⬁
⬁
关
T
共
k;
兲兴
·H
˜
共
0,k
兲
eiky−i
dk,共7兲
whe e H
˜
共0,k兲is he Fou ie ans o m o H 共0,y兲, and
关T共k;
兲兴 is a spec al-domain ans e ma ix. This ans e
ma ix is ob ained a e sol ing he Maxwell equa ions wi h
he app op ia e bounda y condi ions a he sou ce plane, a
he ai - e i e in e aces, and o x→⬁(Somme eld adia-
ion condi ion). Thus, assuming an exp关i共ky−
兲兴 phase ac-
o common o all he ields, he ans e se- o-x ields can be
ela ed by X共x,k兲=关A共k兲兴·X共0,k兲,7,8 whe e a ay X
=共Ey,Ez,Hy,Hz兲and 关A兴is a 共4⫻4兲ma ix ha only de-
pends on he cha ac e is ics and dimensions o he e i e and
ai laye s. The combina ion o he ea lie ela ion wi h he
ee space bounda y condi ion a he image plane, leads o
he ollowing exp ession o he ans e ma ix:
关T兴=兵关Z0兴−关A11兴关A21兴−1其−1 ·兵关A12兴−关A11兴
⫻关A21兴−1关A22兴其,共8兲
wi h 关Aij兴being he 共2⫻2兲subma ices o ma ix 关A兴and
关Z0兴 he ee-space impedance ma ix. Following he a o e-
men ioned p ocedu e, he Tyy e m o he ans e dyad o a
ypical low-loss YIG slab has been compu ed, and he esul s
a e shown in Fig. 2.
The ans e ma ix in ai , a a dis ance 2d om he
sou ce plane, is also shown in Fig. 2. The ans e ma ix o
he e i e slab is nea ly uni y o posi i e alues o k. This
beha io is main ained up o a maximum alue o k, which
inc eases o hinne slabs. This esul is consis en wi h he
epo ed heo y ( he hinne he slab, he be e he magne o-
s a ic app oxima ion wo ks). I clea ly sugges s a signi ican
imp o emen o he esolu ion in he p esence o he e i e
slab. In o de o illus a e his la e poin , he image o an
imposed magne ic ield a he sou ce, Hy共0,y兲, is shown in
FIG. 2. Magni ude o he yy elemen o he ans e ma ix as a unc ion o
he no malized wa e ec o k/k0共k0=
/c兲 o a e i e slab as ha shown in
Fig. 1(b), wi h x0=d/2 and cons i u i e pa ame e s
⑀
=12, 4
Ms=1200 G,
and ⌬H=1 Oe. The ex e nal magne izing ield is H0=520 Oe, and he op-
e a ion equency is =3.1351 GHz. Cu es o d=0.1 and 0.3 mm a e
shown (solid lines). Fo compa ison pu poses, he alues o he ans e
unc ions in ai a a dis ance 2do he sou ce a e also shown (dashed lines)
023505-2 Ma qués, Mesa, and Medina Appl. Phys. Le . 86, 023505 (2005)
Fig. 3. The imposed magne ic ield is uni y a wo hin s ips
(ma ked in he igu e)and ze o o he wise. A signi ican en-
hancemen o he image h ough he e i e can be clea ly
seen. As usual, he image h ough he e i e has been com-
pa ed wi h he alue o he Hy ield in ai , a a dis ance 2d
om he sou ce. This compa ison shows ha he equi alen
ai leng h o he e i e is less han d. In addi ion, he image
h ough he e i e has been also compa ed wi h he image
h ough ai , a a dis ance d om he sou ce. This la e com-
pa ison is o g ea in e es , since i shows ha he equi alen
ai leng h o he e i e slab is less han ze o.
Nea - ield imaging using MSSWs in e i e slabs has
se e al ad an ages o e o he me hods. One o hem is he
educed le el o losses, which p o ides a signi ican im-
p o emen o he imaging.4In ac , i is possible o ind
comme cial YIG slabs p o iding a ios ⌬H/H0o a ew hun-
d ed hs. Ano he in e es ing ad an age o he p esen p o-
posal is unabili y. Resolu ion enhancemen in slabs made o
ma e ials wi h nega i e pa ame e s is es ic ed o a single
equency ( ha p o iding he adequa e alues o he cons i-
u i e pa ame e s). The e o e, de ices made wi h his kind o
ma e ials will be usually es ic ed o a single equency o
ope a ion. Howe e , since he cons i u i e pa ame e s o e -
i es s ongly depends on he ex e nal magne izing ield, he
equency o ope a ion o imaging e i e de ices can be
uned by a ying he magni ude o his ex e nally applied
ield. The p ice o pay o hese ad an ages is ha , owing o
he aniso opy inhe en o magne ized e i es, he a o emen-
ioned o e - esolu ion can be only achie ed in he plane pe -
pendicula o he magne izing ield [ ha shown in Fig. 1(b)].
Ano he inhe en limi a ion o ocusing de ices based on he
epo ed e ec is ha hey could only ope a e in he nea -
ield limi , wi h e i e slabs o wid h dⱗ0.010, whe e 0is
he ee space wa eleng h.
This wo k has been suppo ed by he Spanish Minis y
o Science and Technology and FEDER unds (P ojec Nos.
TEC2004-03214 and TEC2004-04249-C02-02).
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FIG. 3. The image o an imposed magne ic ield a he sou ce plane 共x
=0兲o Fig. 1(b)gi en by Hy=1 in he s ips and ze o o he wise (do ed
line). The alue o he Hy ield a he image plane 共x=2d兲is shown (solid
line). The images in ai a a dis ance x=d(do ed line), and a a dis ance
x=2d(do -dashed line)a e also shown. Cons i u i e pa ame e s o he e -
i e a e as in Fig. 2, wi h d=0.1 mm.
023505-3 Ma qués, Mesa, and Medina Appl. Phys. Le . 86, 023505 (2005)
