Th ee-dimensional supe esolu ion in me ama e ial slab lenses: Expe imen and heo y
F. Mesa
Depa men de Física Aplicada I. Uni e sidad de Se illa, 41012-Se illa, Spain
M. J. F ei e, R. Ma qués, and J. D. Baena
Depa men de Elec ónica y Elec omagne ismo, Uni e sidad de Se illa, 41012-Se illa, Spain
共Recei ed 29 Sep embe 2005; published 27 Decembe 2005兲
This pape p esen s a heo e ical and expe imen al s udy on he iabili y o ob aining wo- and h ee-
dimensional supe esolu ion 共i.e., esolu ion o e coming he di ac ion limi o all di ec ions in space兲by
means o me ama e ial slab lenses. Al hough he sou ce ield canno be ac ually ep oduced a he back side o
he lens wi h supe esolu ion in all space di ec ions, he ma ching capabili ies o me ama e ial slabs does make
possible he de ec ion o images wi h h ee-dimensional supe esolu ion. This imaging akes place because o
he coupling be ween he e anescen space ha monic componen s o he ield gene a ed a bo h he sou ce and
he de ec o .
DOI: 10.1103/PhysRe B.72.235117 PACS numbe 共s兲: 42.30.Wb, 41.20.Jb, 73.20.M , 78.20.Ci
I. INTRODUCTION
I is well known om he ea ly wo ks o Veselago1 ha a
slab made o a le -handed medium will ocus he elec o-
magne ic ene gy coming om a poin sou ce o ano he poin
loca ed a he opposi e side o he slab. An expe imen al
con i ma ion o his ocusing o ene gy has been epo ed by
Houck e al.2Subsequen wo ks3–5 ha e shown ha , unde
some ci cums ances, me ama e ial lenses made o le -
handed media can p oduce images a ce ain planes wi h a
esolu ion beyond he classical di ac ion limi , o “supe -
esolu ion imaging” 共SRI兲. This SRI has been a ibu ed o an
ampli ica ion, inside he lens, o he e anescen space Fou-
ie ha monics 共SFHs兲coming om he sou ce.3I has been
also discussed6 ha his p ocess gi es ise o ields ha decay
exponen ially om he lens owa d he image, which causes
supe esolu ion o ake place only in planes pa allel o he
slab in e aces. In he di ec ion pe pendicula o he lens, a
s ong decay o he ield is obse ed, and hus, a h ee-
dimensional 共3D兲pic u e o he sou ce canno be di ec ly
ob ained om he ield dis ibu ion a he back side o he
lens. In o he wo ds, supe esolu ion in he ans e se di ec-
ions is ob ained a he p ice o a d as ic loss o esolu ion in
he longi udinal di ec ion. This ac has been expe imen ally
co obo a ed by ecen ly epo ed ield measu emen s,4
whe e he ield g ow h om he image plane o he “supe
lens” can be clea ly app ecia ed. O he expe imen al esul s
also lead o he same conclusion, showing ha images “o
ini e dep h”5canno be di ec ly ob ained om ield measu e-
men s in SRI expe imen s.
Following he seminal pape o Pend y,3o he supe eso-
lu ion de ices ha e been p oposed making use o slabs o
nega i e pe mi i i y,7,8 e i e slabs,9coupled plana
pola i on- esonan me asu aces,10 and a pai o coupled
magne oinduc i e su aces.11 In spi e o he di e en physi-
cal na u e o hei cons i u i e elemen s, all hese s uc u es
p esen se e al common cha ac e is ics. In his way, he
abo e well-de ined amily o imaging de ices can be cha -
ac e ized by: 共i兲 he p esence o plana slabs, 共ii兲 he dis ance
be ween he sou ce and he image planes is always 2d共whe e
dis he slab wid h兲, and 共iii兲 he imaging p ope ies a e based
on he ampli ica ion o e anescen SFHs inside he slab.
Consequen ly, he imaging p ope ies o Pend y’s le -handed
slab lens a e also expec ed o be sha ed by hese de ices.6
Ne e heless, some expe imen al esul s ecen ly epo ed by
some o he au ho s11 ha e sugges ed he possibili y o ob-
aining supe esolu ion in he longi udinal di ec ion in one o
he de ices men ioned abo e. In hese expe imen s, a 3D
map o a poin like sou ce 共i.e., a sou ce o subwa eleng h
size兲was ob ained a he image side o he lens. In o he
wo ds, h ee-dimensional supe esolu ion imaging 共3D-SRI兲
was ob ained. The aim o he p esen con ibu ion is o p o-
ide he gene al heo y unde lying his 3D-SRI. I will be
shown ha 3D-SRI o poin like sou ces is ac ually a gene al
p ope y o he a o emen ioned amily o de ices, p o ided
ha he app op ia e de ec ion p ocedu e is ollowed.
II. ANALYSIS
The main di e ence be ween SRI and s anda d imaging
p ocesses is ha , in he o me , he in o ma ion o he image
o ma ion is ca ied ou by e anescen SFHs,3which can
suppo subwa eleng h in o ma ion, unlike he p opaga i e
SFHs esponsible o he con en ional imaging, which can
only anspo o e wa eleng h in o ma ion. Since e anescen
ields canno ca y powe and any image measu emen e-
qui es some powe ansmission, i is appa en ha he de-
ec ion p ocedu e has o somehow “c ea e” a a eling powe
lux. This combined e ec should hen pe u b, subs an ially,
he ields a ound he de ec o in a way simila o ha ound
in he unneling e ec . As is well known, he unneling o
powe is due o he exci a ion o a pai o e anescen elec-
omagne ic wa es, whose in e e ence gi es ise o a non a-
nishing lux o powe . “Pe ec unneling” o powe in a
wa eguide illed by a me ama e ial has been ecen ly e-
po ed by some o he au ho s.12 In his wo k i is shown ha
maximum unneling o powe in a se up wi h iden ical inpu
and ou pu wa eguides is achie ed when he ou pu is placed
a a dis ance om he inpu equal o ha om he sou ce o
PHYSICAL REVIEW B 72, 235117 共2005兲
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he image in a me ama e ial supe lens. This sugges s ha a
simila e ec could ake place in a me ama e ial supe lens,
p o ided ha a de ec o iden ical o he sou ce is used o he
measu emen s. In he ollowing, how o ake ad an age o
such an e ec in o de o ob ain 3D-SRI wi h me ama e ial
slabs will be shown.
The p esen s udy s a s wi h he canonical p oblem o he
o ma ion o images by a le -handed slab o hickness d
cha ac e ized by
⑀
/
⑀
0=
␮
/
␮
0=−1+i
␦
, whe e
␦
Ⰶ1 accoun s
o he necessa y losses ac o o a oid he di e gence o
ield in eg als.13–15 In ou s udy, and ollowing a usual p o-
cedu e in mic owa e SRI expe imen s, he sou ce will be an
an enna 共speci ically, a loop an enna兲whose plane is loca ed
pa allel o he slab in e aces as shown in Fig. 1. The ield
beyond he lens will be scanned by an ou pu an enna ha
plays he ole o de ec o . The sou ce employed he e is
equi alen o an homogeneous su ace dis ibu ion o mag-
ne ic dipoles gi en by
Ms=
再
I0e−i
␻
,inside he loop
0, ou side,
冎
共1兲
whe e Ideno es he ampli ude o he imposed ime-ha monic
cu en in he loop. The compu a ion o he longi udinal mag-
ne ic ield Hza he image side o he slab 共z⬎⌬+d兲can be
ca ied ou by means o he ollowing double-in e se Fou ie
ans o m:
Hz共x,y,z兲=1
4
␲
2
冕冕
dkxdkyG
˜
共kx,ky;z兲M
˜
s共kx,ky兲ei共kxx+kyy兲,
共2兲
whe e G
˜
共kx,ky;z兲is he Fou ie ans o m o he G een’s
unc ion o he s uc u e unde s udy and M
˜
s共kx,ky;z兲, he
Fou ie ans o m o he spa ial su ace dis ibu ion o mag-
ne ic dipoles. A e applying he duali y p inciple o exp es-
sion 共6兲in Re . 15 and aking ⌬=d/2, he Fou ie ans o m
o he G een’s unc ion is ound o be
G
˜
共kx,ky;z兲=−共kx
2+ky
2兲2
␤
e−i
␤
0共z+d兲
␻
兵共
␤␮
0−
␤
0
␮
兲2ei
␤
d−共
␤␮
0+
␤
0
␮
兲2e−i
␤
d其,
共3兲
whe e
␤
=冑
␻
2␧
␮
−kx
2−ky
2
␤
0=冑
␻
2␧0
␮
0−kx
2−ky
2.
Since Ez=0 in he p esen case, all he emaining ield com-
ponen s can be deduced om Hz.15
The nume ical compu a ion o 共2兲p o ides he map de-
pic ed in Fig. 2 o he magni ude o Hzin he 共x−z兲plane a
he back side o he lens. I should be no ed ha , gi en ha
he ope a ion equency is 3 GHz 共␭0=100 mm兲and ha d
and ⌬a e aken much less han he ee-space wa eleng h,
he ay model app oxima ion canno be employed he e o
ob ain he ields in he conside ed egion 共in o he wo ds, we
a e dealing wi h nea ields and he e o e in a SRI si ua ion兲.
Acco ding o p e ious discussions, Fig. 2 shows ha he ield
magni ude has a s ong decay along he zdi ec ion, so ha
no in o ma ion abou he loca ion o he sou ce in he longi-
udinal di ec ion can be ex ac ed om he ield dis ibu ion.
A mo e de ailed pic u e o he ield dis ibu ion a wo planes
z=共2±
␦
兲dis plo ed in Fig. 3 共wi h
␦
se o 0.3兲and com-
pa ed o he ield ampli ude nea he sou ce a z=±
␦
d共 he
ield dis ibu ions a z=
␦
dand z=−
␦
da e iden ical兲.Asis
expec ed om he p ope ies o he me ama e ial slab, he
ield dis ibu ion a he z=共2+
␦
兲dplane is almos iden ical o
ha a he z=
␦
dplane, hus con i ming ha he z=
␦
dand he
z=共2+
␦
兲dplanes a e, in ac , conjuga e planes. On he con-
a y, he ield dis ibu ion a he z=共2−
␦
兲dplane subs an-
FIG. 1. Geome y o a me ama e ial lens. The sou ce is a loop
an enna loca ed a z=0. The de ec o is also an iden ical loop an-
enna. The lens is o med ei he by a le -handed me ama e ial slab
o by ano he plana de ice ha p oduces SRI h ough ampli ica ion
o e anescen ha monics. The ou pu an enna can e en ually be
loaded wi h mic owa e esis o s in o de o minimize ield
pe u ba ion.
FIG. 2. 共Colo online兲Mapo log
10兩Hz兩co esponding o he
ield gene a ed a he back side o he lens by a sou ce loop an enna
shown in Fig. 1. The slab is made o a le -handed medium o
hickness d=4 mm wi h
⑀
/
⑀
0=
␮
/
␮
0=−1+i0.001, and is sepa a ed
by a dis ance ⌬=d/2 om he sou ce. The sou ce is conside ed, o
simplici y, a ci cula loop an enna o adius 1=5 mm. The wi e
adius is o 0.2 mm. The ope a ion equency is 3 GHz.
MESA e al. PHYSICAL REVIEW B 72, 235117 共2005兲
235117-2
ially di e s in magni ude om he ield dis ibu ion a he
z=−
␦
dplane, nea he sou ce. These ac s co obo a e he
a o emen ioned discussions: he supe esolu ion in he ans-
e se di ec ions 共 he la e al dimensions o he sou ce and he
image a e app oxima ely one- en h o he wa eleng h兲is
compensa ed by an almos comple e loss o esolu ion in he
longi udinal di ec ion.
The p oblem o he measu emen o he image in he SRI
se up unde s udy is nex conside ed. In he mic owa e
ange, he image de ec ion is pe o med by measu ing he
ansmission coe icien be ween a sou ce an enna and a e-
cei ing an enna, which is scanned in he image side o he
lens.2,4,5 Fo simplici y, he ecei ing 共o ou pu 兲an enna is
assumed o be iden ical o he inpu an enna employed as
sou ce. The ansmission coe icien is measu ed by connec -
ing he inpu an enna o a wa e gene a o ia a wa eguide,
and he ou pu one o a de ec o h ough ano he iden ical
wa eguide 共mo e de ails o his measu emen se up a e e-
po ed in Re . 11兲. In his app oach, he me ama e ial slab
should be iewed as a ma ching de ice,16 whose ansmis-
sion coe icien 共o S21 in he usual mic owa e e minology兲
is gi en by17
=2Z12Z0
共Z11 +Z0兲共Z22 +Z0兲−Z12
2,共4兲
whe e Zij a e he elemen s o he impedance ma ix o he
sys em o med by he wo an ennas and he le -handed slab,
and Z0is he cha ac e is ic impedance o he inpu and ou pu
wa eguide. In o de o measu e a supe esolu ion image, he
size o he loop an ennas has o be smalle han he ee-
space wa eleng h, which addi ionally would make he eal
pa o Zij 共 he adia ion esis ance17兲be negligible wi h e-
spec o i s imagina y pa ; namely, Zij⯝−i
␻
Lij, whe e Lij is
he induc ance ma ix o he sys em. The diagonal e ms o
he induc ance ma ix co espond o he induc ances o a
single-loop an enna aced o he le -handed slab. Howe e ,
in he “pe ec lens” con igu a ion conside ed he e, he slab
does no a ec he ields a ound he sou ce3and, he e o e,
L11=L22 ⬅L, whe e Lis he sel -induc ance o he loops in
ee space. The nondiagonal e ms o he induc ance ma ix
accoun o he mu ual induc ance o he loop an ennas in he
p esence o he slab, namely, L12 =L21⬅M. Thus, he ans-
mission coe icien in 共4兲can be w i en as
=2i
␻
MZ0
␻
2共L2−M2兲+2i
␻
LZ0−Z0
2.共5兲
Since Ldoes no change wi h he posi ion o he an ennas,
he spa ial dependence in 共5兲comes only om M. The de-
pendence o 兩 兩wi h he eac ances X11=X22 =−
␻
Land X12
=X21=−
␻
Mdeduced om 共5兲is shown in Fig. 4. I can be
seen how he maximum ansmission 共兩 兩⬇1兲is achie ed
when X11⯝X12, excep o e y small alues o he a io
X11/Z0. This e ec is illus a ed in Fig. 5, whe e he alues o
X12 and X11, which co esponds o a maximum o he ans-
mission coe icien , a e plo ed. The e o e, excep when
␻
L
ⰆZ0, he ansmission coe icien eaches i s maximum a
hose poin s whe e M⬇L共兩 兩⬇1i LⲏZ0兲. Since bo h he
inpu and ou pu an ennas a e iden ical, and he ield a he
sou ce plane is ep oduced a he image plane 共see Fig. 3兲,
FIG. 3. Plo o he log10兩Hz兩a di e en z-planes o Fig. 1. The
slab and he sou ce loop an enna a e as in Fig. 2.
FIG. 4. Plo o he modulus o he ansmission coe icien
=S21 as a unc ion o he modulus o he eac ances X11=X22=
−
␻
Land X21=−
␻
M.
FIG. 5. Plo o he alues o he eac ances X11=X22=−
␻
Land
X112=X21=−
␻
M o which he modulus o he ansmission coe -
icien 共5兲 eaches a maximum.
THREE-DIMENSIONAL SUPERRESOLUTION IN…PHYSICAL REVIEW B 72, 235117 共2005兲
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he condi ion M⬇Lis expec ed o be sa is ied a ound he
poin 共x,y,z兲=共0,0,2d兲. In consequence, a maximum o he
ansmi ed powe should be de ec ed a ound his pa icula
poin , and hus, i would appea as an “e ec i e ocusing
poin ” o he pe ec lens.
In o de o show he abo e expec ed e ec s in a p ac ical
si ua ion, he magni ude o he ansmission coe icien has
been nume ically compu ed o he con igu a ion unde
s udy. The impedance ma ix has been calcula ed by impos-
ing known cu en s, Ii=1,0 A, in he loops, and hen com-
pu ing he co esponding sel and mu ual induc ances. The
compu a ion o he sel -induc ance is an s anda d elec o-
magne ic p oblem, and he mu ual induc ance is ob ained
a e compu ing he lux o he magne ic ield gi en by 共2兲
ac oss he su ace o he ou pu an enna. The ansmission
coe icien is inally de e mined om 共4兲, assuming Z0
=50 ⍀. In Fig. 6, a map o he compu ed ansmission coe -
icien o a sys em composed o wo iden ical lossless loop
an ennas is shown. I can be obse ed ha his igu e sub-
s an ially di e s om Fig. 2; in pa icula , a clea maximum
o 兩 兩can be obse ed in Fig. 6 in he neighbo hood o he
image a 共0,0,2d兲. These esul s clea ly show he di e ence
be ween he ansmi ed powe and he ield dis ibu ion in
he absence o he ou pu an enna, and also how 3D-SRI can
be ob ained when he app op ia e de ec o is used.
Le us now conside he measu emen o he unpe u bed
ield. Fo his pu pose, he de ec o should be designed o
a ec he ield dis ibu ion as li le as possible. I could be
closely achie ed by loading he ou pu an enna wi h an ad-
di ional high esis ance, which will signi ican ly educe he
cu en induced a he ou pu an enna, as well as i s gene a ed
ield. In Fig. 7 he compu ed alues o he ansmission co-
e icien along he z-axis o Fig. 1 a e shown o di e en
esis ances, R, loading he ou pu an enna. As is expec ed, he
cu e o R=0 ⍀shows a maximum nea he loca ion o he
image, which hus appea s as a “ ocusing poin ” o he lens,
a z⬇2d.共The small di e ence be ween he ac ual loca ion
o his maximum wi h espec o z=2dcan be a ibu ed o
he no e y high alue o he a io
␻
L/Z0⬇8 in he case
unde s udy兲. As is also expec ed, he cu es o he highes
alues o R esemble he beha io o 兩Hz兩in Fig. 2 o he
unpe u bed con igu a ion.
Ano he ob ious s a egy o educing he pe u ba ion o
he ield by he measu emen is o educe he adius o he
ou pu an enna o Fig. 1. This will educe bo h L2共 he induc-
ance o he ou pu an enna兲and Mwi hou changing L1共 he
induc ance o he inpu loop兲. In he limi when
␻
L2,
␻
M
ⰆZ0,共4兲 educes o
⬇2i
␻
M
i
␻
L1−Z0
.共6兲
Since bo h L1and Z0do no depend on he loca ion o he
ou pu , he measu ed ansmission coe icien u ns ou o be
p opo ional o M, i.e. o he magne ic ield Hza he loca ion
o he ou pu . Thus, he analyzed expe imen al se up is ap-
p op ia e o he de ec ion o he ield o he unpe u bed
sys em o med by he inpu an enna and he lens. The mag-
ni ude o he ansmission coe icien along he zaxis o Fig.
1 is plo ed in Fig. 8 o se e al alues o he adius o he
ou pu loop. The cu es co esponding o 2艌4 mm clea ly
show ha he loca ion o he de ec ed ansmission maximum
app oaches he lens in e ace as he adius o he ou pu loop
is educed. In he limi o small adius 共 2=1 mm兲, he p e-
iously p edic ed mono onic inc ease o he ield owa d he
igh -hand-side lens in e ace is obse ed. Figu e 8 also
shows ha he sensi i i y o he expe imen al se up o small
di e ences be ween he shapes o he inpu and ou pu an-
ennas is no e y high. As i can be eadily seen om he
cu es co esponding o 2=5 and 4.5 mm, he ansmission
peak is loca ed e y close o 2d o mode a e de ia ions o 2
om i s op imum alue 2= 1.
III. EXPERIMENT
In o de o show he applica ion o he abo e heo y o
p ac ical de ices, he expe imen al demons a ion o 3D-SRI
ecen ly epo ed by some o he au ho s11 will now be eex-
amined in ligh o he abo e conside a ions. Al hough he
FIG. 6. 共Colo online兲Map o he magni ude in decibels o he
ansmission coe icien , 兩S21兩, be ween he inpu and ou pu an en-
nas o Fig. 1. The s uc u al pa ame e s a e as in Fig. 2. The ou pu
an enna is iden ical o he inpu one.
FIG. 7. Magni ude o he compu ed ansmission coe icien
along he Zaxis be ween he inpu and ou pu an ennas o Fig. 1 o
di e en esis ances loading he ou pu an enna. The s uc u al pa-
ame e s a e as in Fig. 2.
MESA e al. PHYSICAL REVIEW B 72, 235117 共2005兲
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unde lying physics o he de ice analyzed he e is no ex-
ac ly he same as ha o he le -handed pe ec lens,11 bo h
sys ems show he same p ocess o ampli ica ion o e anes-
cen modes and a e equi alen in p ac ice o he p esen
pu poses. Thus, expe imen al esul s plo ed in Fig. 9 show
he loca ion o he maximum o he ansmission coe icien
along he zaxis o he magne oinduc i e lens11 o di e en
alues o he esis ance loading he ou pu loop. The expe i-
men al se up used o ob ain hese esul s is desc ibed in de-
ail in Re . 11. The only di e ence is ha he ou pu an enna
is now loaded by di e en mic owa e esis o s o ob ain he
esul s shown in Fig. 9. These esul s show he same ype o
beha io as ha epo ed in Fig. 7; namely, he loca ion o a
maximum o he magni ude o he ansmission coe icien
in he neighbo hood o z⬇2d o iden ical ou pu and inpu
an ennas, and a simila displacemen o he maximum o he
ansmission coe icien as he adius o he ou pu an enna
dec eases. The ag eemen be ween he heo y and he expe i-
men s can be conside ed as a alida ion o he p esen heo y.
I also shows ha he ideas de eloped in he p e ious analy-
sis a e e y gene al, and ha , ac ually, hey a e applicable o
any supe esolu ion lens based on he ampli ica ion and am-
pli ude es o a ion o FHs along he de ice.
IV. CONCLUSIONS
Supe esolu ion in me ama e ial supe lenses has been in-
es iga ed. As is well known, his imaging is p ima ily due o
he ampli ica ion o e anescen modes inside he lens. As a
consequence, supe esolu ion in he planes pa allel o he
slab is una oidably compensa ed by a d as ic loss o esolu-
ion in he di ec ion pe pendicula o he slab. Howe e ,
since e anescen modes do no ca y powe , any physical
de ec ion 共namely, a measu emen 兲o he image a he back
side o he lens will equi e he exis ence o some amoun o
ansmi ed powe om he sou ce o he de ec o , which can
signi ican ly a ec he o iginal ield dis ibu ion. I his las
e ec is aken in o accoun , he seconda y ields gene a ed by
he p esence o he de ec o should be conside ed. Following
his app oach, i has been shown ha he ansmission coe -
icien be ween he sou ce and he de ec o can be inc eased
conside ably. P o ided ha he app op ia e de ec o is em-
ployed 共 ypically, a lossless ou pu an enna iden ical o he
inpu an enna兲, his maximum will occu in a neighbo hood
o he image, hus esul ing in supe esolu ion imaging also
in he di ec ion pe pendicula o he slab. Howe e , i should
be emphasized ha o ob aining his 3D-SRI, some p e ious
knowledge o he sou ce is necessa y in o de o design he
de ec o p ope ly. Thus, a gene al conclusion a ises om he
analysis: Supe esolu ion in me ama e ial supe lenses is al-
ways incomple e; i he dis ance om he sou ce o he lens is
known, he shape and cha ac e is ics o he sou ce can be
eco e ed wi hou unce ain y om he analysis o he ield
dis ibu ion a he image side o he lens. Con e sely, i he
shape and cha ac e is ics o he sou ce a e known, i is pos-
sible o design an app op ia e de ec o in o de o localize
he sou ce spa ially om he alues o he ansmission co-
e icien be ween he sou ce and he de ec o . Howe e , i is
impossible o de e mine, simul aneously, by means o a
me ama e ial supe lens, he loca ion, shape, and cha ac e -
is ics o an unknown sou ce wi h a esolu ion o e coming he
di ac ion limi . We eel ha he epo ed analysis and ex-
pe imen s, as well as he conclusions a ising om hem, will
be o impo ance in he design o me ama e ial supe esolu-
ion de ices.
ACKNOWLEDGMENT
This wo k has been suppo ed by he Spanish Minis y o
Educa ion and Science by p ojec Con ac No. TEC2004-
04249-C02-02.
FIG. 8. Magni ude o he compu ed ansmission coe icien
along he zaxis be ween he inpu and ou pu an ennas o Fig. 1 o
se e al alues o he adius o he de ec o 共 2兲. The s uc u al pa-
ame e s a e as in Fig. 2.
FIG. 9. Plo s o he measu ed magni ude o he ansmission
coe icien be ween he inpu and ou pu an ennas o Fig. 1 when
he slab is subs i u ed by he magne oinduc i e lens epo ed in Re .
11. 共F equency=3 GHz兲.
THREE-DIMENSIONAL SUPERRESOLUTION IN…PHYSICAL REVIEW B 72, 235117 共2005兲
235117-5

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