Diffraction-free nonevanescent nano-beams using the Fresnel-waveguide concept

0.21 MB·English

by  M. I. Mechler

Diffraction-free nonevanescent nano-beams using the Fresnel-waveguide concept
M.I. Mechler1, Z. Tibai2, S.V. Kukhlevsky2
1High-Field Terahertz Research Group, MTA-PTE, Ifju´sa´g u´. 6, 7624 P´ecs, Hungary and
2Institute of Physics, University of P´ecs, Ifju´sa´g u´. 6, 7624 P´ecs, Hungary
Strong intensity attenuation limits the use of conventional diffraction-free optical elements. We
show a possible solution to the exponential intensity attenuation limiting the use of Fresnel-type
diffraction-freenanometer-scaleopticsbyusingmaterialswithappropriatelychosenrefractiveindex.
Suchbeams may beapplied for technical and physical problems.
PACSnumbers: 41.20.Jb,42.25.Fx,42.25.Gy,42.79.Dj
3
1
0
I. INTRODUCTION This was the basic idea of Ruschin and Leizer [16].
2
In their paper they concentrated on Bessel-type beams

n Resolvingmicro- andnanoscopicobjects is a challeng- which are generally treated as propagating waves. They
a
J ingproblemandithasbeenanimportantfieldofresearch focused on the problem of generating evanescent Bessel-
6 in the last decades. Although a simple microscope is beams,thereforeintheirworktheyusedelectromagnetic
1 sufficient to magnify microscopic images, its resolution solutions of the form
capability is confined by the Abbe diffraction limit:
U (r,φ,z)=exp(iφm)J (αr)exp(−β′z), (4)
] m m
s
λ
c d= , (1) where β′ = α2−n2k2 is the propagation constant,
i 2nsin(θ) 0
t (r,φ,z)arecylindricalcoordinates,J istheBesselfunc-
p m
o whereλisthewavelength,nistherefractiveindexwhile tion. Whenepver β′ is real, i.e. α2 > n2k02, this func-
′
s. the angleof incidence is denotedby θ. Thereforediffrac- tion decreases exponentially; however,if β is imaginary,
c tion is an ultimate barrier in traditional microscopy. theexponentialpartdescribesapropagatingcomponent.
i Thiscanbeachievedforexamplebychangingtherefrac-
s Many solutions have been proposed to overcome this
y limit,thatis,toeliminateitseffect,suchasphotoactivat- tive index of the materials.
h Anotherimportantresultinthisfieldwaspublishedby
ablelocalizationmicroscopy[1](PALM),stimulatedemis-
p Yacob Ben-Aryeh[17]. This paper analyzes theoretically
sion depletion[2] (STED), surface plasmon polaritons[3]
[
the microsphere-based field conversion from evanescent
(SPPs), various nanostructures such as the recently dis-
1 coveredmicrospheres[4],orthediffraction-freewavesolu- to propagating waves [4]. Information on the nanostruc-
v tions[5–14]. Ultrashortdiffraction-freepulsescanalsobe ture of a nano-corrugated metallic thin film is included
2 intheevanescentwavesproducedbytheplaneEMwaves
produced[18,20,21]. Diffraction-freelightbeamscanbe
1 transmittedthroughthestructure. Thisinformationcan
definedasbeamswithnodiffractivebroadeningthrough-
7
be extracted from the evanescent wave by microspheres
outthe free-spacepropagation. This canbe expressedas
3
located above the surface that collect and convert them
. follows: for all x, y and z coordinates the averagetrans-
1 verse components I =S of the Poynting vector S of into propagatingwaves. The conversioneffect is also the
0 T T T consequence of the refractive index difference of the ma-
the beam must fulfill the following condition [15]:
3
terials.
1
: ∇·IT =0 (2) Nondiffractingbeamshavesomespecialfeatureswhich
v may be used in technical as well as physicalapplications
i
X In [18] we described a diffraction-free arrangement that suchasimaging,particleaccelerationorlightmotors[23–
consistsofanarrayofslitsilluminatedbyTM-waveswith 28]. Intheseapplications,however,theuseofnonevanes-
r
a periodically altering phase; however, it was also pointed cent beams would be preferred.
out that this diffraction-free feature implies an exponen- In our paper we show that the production of prop-
tial decrease in the intensity, i.e. behind the slit array agating (nonevanescent) diffraction-free nano-beams us-
evanescent beams can be observed. ingthe Fresnel-waveguideconceptisfeasible. Ourgeom-
Evanescent beams are described through an exponen- etry is similar to a simple grating; however, the phase
tial term: difference between the EM fields falling to the adjacent
subwavelength slits and the near-field treatment of the
U(x,y,z)=F(x,y,z)·exp(iβz), (3) problem distinguish our case from that of a simple grat-
ing. The paper is arranged as follows. In Section II we
where z is the coordinate along the propagation direc- present the basic idea and the results of our theoretical
tion,andβ is the wavevectorcomponentinthe direction analysis. Section III contains the results of our numeri-
of propagation. If β is purely imaginary, the beam be- cal simulations using a finite element analysis code. We
comesevanescent;however,ifthiswavevectorcomponent presentdifferentgeometriesandconcludethatcriticalre-
remains real, propagating waves are generated. fractive indices separating the evanescent and propagat-
2
ing regions depend on the transverse characteristic sizes k2 <0thebeamisevanescentwhileinthecaseofk2 >0
y y
of the geometries. Finally, in Section IV conclusions are a propagating beam can be obtained. This leads to the
drawn and some final remarks are made. followinginequalities (note thatall the quantities – n, a,
λ0 – are positive real values):
II. THEORY • evanescent beams are described by:
λ0
In [18] we showed that it is possible to produce n<
4a
diffraction-free nano-beams in the subwavelengthregime
using the Fresnel-waveguide concept. Throughout our
• propagating beams are described by:
analysis we modeled an array of one-dimensional slits
(Fig.1)ofwidth2ainascreenofthicknessbilluminated
λ0
byTM-polarizedplanewavesofwavelengthλ. Thephase n>
4a
φ of the wave falling on the m-th slit of the array was
m
adjusted as φ = m·π, m = 0,±1,±2... These phase
m We also note that this formula was derived for the ge-
jumps are very important as the phase difference of the
ometry in Figure 1 analyzed in our earlier paper [18]. A
fields distinguishes our problem from that of a grating.
more general formula can be written through the trans-
In our earlier paper we observed that the width of the
versecharacteristicdimensionofthegeometry,e.g.inour
central beam remains the same for great distances and −1
2π
thebeamshapeissinusoidinthedirectionperpendicular case Λ= , therefore
tothepropagationdirection. However,wealsopresented 4a
(cid:18) (cid:19)
thattheintensitydecreasesexponentially. Inthissection
we find a solution to this problem. n2π 2 1 2
k = − . (8)
The beam leaving the slit array propagates in homo- y
s λ0 Λ
geneousmaterial;inthiscasethepropagationcanbe de- (cid:18) (cid:19) (cid:18) (cid:19)
scribed by the Helmholtz equation and we may use the This deduction leads to the conclusion that in our ge-
scalar field U(~r,t): ometryitispossibletoachievediffraction-freeand prop-
agating waves by 1) increasing the refractive index 2)
∇2+k2 U(~r,t)=0, (5)
decreasingthewavelengthor3)increasingthetransverse
where ∇2 is the(cid:0)Laplacia(cid:1)n and k is the wavenumber. characteristic dimension of the geometry. In the follow-
ing section we show some simulation results for various
Owing to the geometry of the structure and the TM-
refractiveindicesandtwodifferentgeometries(transverse
polarization (z-invariance) of the incident light, in our
characteritic sizes).
casethereisnoz-dependency. Wealsodisregardtimede-
pendency. Moreover, U(x,y) can be split into two func-
tions, F(x) and G(y), i.e. variables can be separated.
III. NUMERICAL SIMULATIONS
Now we assume an exponential form for the y-
dependent part: G(y) ∼ exp(ik y), where k is the
y y
y-component of the wavevector (propagation constant). In order to verify our result, we performed finite ele-
Substituting this into (5) we get: ment analysis simulations for various geometries (trans-
versecharacteristicdimensions)andrefractiveindices. In
∂2 thefollowingswepresentsomeresultsofoursimulations.
F(x)=−(k2−k2)F(x). (6)
∂x2 y Thefirstsimulatedgeometrywasthesameweanalyzed
inSectionII(Fig.1). Inthiscaseweusedanarrayofslits
Considering that a harmonic function is expected in the
of width 2a in a screen of thickness b. It is illuminated
x direction: F(x) ∼ sin( k2−k2x), by solving (6) for fromregionIbyTMpolarizedfieldthatisperpendicular
y
a beam with harmonic shqape in the transverse direction to the arrayofslits, andthe phase ofwhichwaschanged
and exponential function in the longitudinal direction, by π in the adjacent slits, i.e. φm = m·π. In this case
one may obtain the following function: no distance was assumed between the individual slits,
thereforethisgeometryisquitehypothetic. Wewillshow
2πx a more realistic model hereinafter.
U(x,y)=F(x)·G(y)∼sin ·exp(ik y), (7)
4a y The aim ofthis paper wasto find a solutionto the ex-
(cid:18) (cid:19)
ponential decrease of the wave leaving the slit (in region
where ky = nλ20π 2− 24πa 2, and λ0 is the wavelength IleIaI)d.sTtoo pnroonveevatnheastcetnhte(sporluoptiaognatpinrogp)owseadveisnwSeecptliootnteIdI
r
in vacuum. Fr(cid:16)om t(cid:17)his la(cid:0)tter(cid:1)formula one may draw the the energy flow Sy along the y axis behind the screen at
conclusion that there are some situations (specific geo- x=0 for variousrefractiveindices (Fig. 2). For compar-
metric dimensions and wavelengths) in which the oth- ison the same quantity for a real material (germanium,
erwise evanescent waves become propagating, namely if n = 4.7096 @ λ = 800 nm [22]) was also plotted. In
3
y
high-refractive-index case their sign is the same which
results in a double peak.
In Figure 4 we present a more realistic geometry. In
this case the separation s between the slits is nonzero.
I.
2a Actually, the separation can be arbitrary, and still the
diffraction-free quality remains; as an example, in our
+ − b + − + II. simulations we used s = 4a. The increase of the sep-
aration affects the characteristic dimension: it becomes
x Λ= 4a+2s. The width of the diffraction-free beams also
III. 2π
changes: w = 2a+s. In fact, the former value is the
period of the slit system; the latter value is presented
in Figure 5. In the figure we show the energy flow S
y
along the x direction at y = 32a for n = 1.5 (evanes-
cent region) and n = 1.66 (nonevanescent region). As
it is expected from Eq. (7), the numerical model showed
FIG. 1: The original geometry used in [18]. In the present
that in order to produce propagating beams, increasing
article we used the following values: wavelength in vacuum
λ0 = 800 nm, slit width 2a = 80 nm, screen thickness the characteristic dimension two times allows us to de-
b=50 nm. The ”individual” slits are separated by perfectly creasetherefractiveindextwotimes. Itcanbeseenthat,
conducting material. The phase of the EM field illuminating similarly to Fig. 3, separate y axes must be assigned to
the slits changes periodically (in the figure it is indicated by the two cases because of the magnitude difference, the
theplus and minus characters) diffraction-freequalityofthe beamstill canbe observed,
and better localization can be observed for the propa-
gating case. However, the width of the beam is much
this case the characteristic dimension of the geometry is greater (w = 2a+s) than in Fig. 3, and the critical re-
4a fractive index is reduced to n ≈ 5/3 which corresponds
Λ = , therefore the critical refractive index of the to a characteristic dimension of Λ= 4a+2s.
2π 2π
(cid:18) (cid:19)
geometry that separatesthe evanescent and propagating For comparisonwe also show the energy flow Sy along
case can be formulated as n = λ0. Considering a wave- the y axis at x=0 for various refractiveindices (Fig. 6).
4a
length of λ0 = 800 nm and slit width 2a = 80 nm, the In this case it is not only possible to find a real material
critical refractive index separating the evanescent and withappropriaterefractiveindexbutthese materialsare
propagating cases is n = 5. Indeed, Figure 2 confirms liquidthereforetheycanbeusedinrealexperiments. As
this value. Note that the transition from evanescent to anexample,in the simulationwe usedcarbondisulphide
propagating waves is accompanied by an oscillating ef- (n = 1.6295, σ ≈ 0 @ λ = 800 nm). It is clear from the
fect, i.e. in the case of materials with refractive indices insetofFig.6thatwhileSy convergestozeroforn=1.5,
close to the critical value the y-component of the Poynt- thevaluetowhichSy convergesforCS2 differsfromzero,
ing vector S of the wave leaving the slit array oscillates i.e. it is possible to realize diffraction-free nonevanescent
y
while decreases; this is shown for n = 4.9 in the inset of beams with this geometry.
Fig.2. Theeffectisnotsoapparentforgermaniumpartly As a final remark we note that our results are in ac-
becauseitisfurtherfromthecriticalvalueandpartlybe- cordance with the Abbe-limit (1). If sinθ = 1, the ex-
cause the material has nonzero conductivity (σ =217). pression changes to d = λ/(2n). Substituting the values
To verify that the beam preserves its diffraction-free used for the simulations, we find that the Abbe-limit for
quality,inFig.3weplottedthe energyflowS alongthe n = 5/3 is d = 240 nm, and for n = 5 it is d = 80 nm.
y
x direction at y = 32a for n = 4.9 (evanescent region) These are exactly the transverse characteristic sizes we
andn=4.99(nonevanescentregion). Weonlypresentan usedforthetwocases. Consequently,thereisaclearlink
example of the beam shape at y = 32a, our calculations between the evanescence limit and the Abbe-limit.
showed, however, that the energy flow S of the beam Our results showed that varying the refractive index
y
preserves its shape in the region |y| > 2a, i.e. this phe- of the material in which the diffraction-free beams are
nomenoncannotbeattributedtoself-imaging. Although generated leads to nonevanescent beams. These non-
separateyaxesmustbeassignedtothetwocasesbecause diffractingbeamsshowsomeuniquepropertiesusefulfor
ofthe magnitude difference, the diffraction-freebehavior both physical and technical applications. Nondiffracting
canbe observedinbothcases;however,inthe propagat- beamsmaybefoundapplicationsinimagingasitwaver-
ing casethe originallysinglepeak becomes double which ifiedthatextremelylongfocaldepthscanbeproducedin
shows a better localization. This phenomenon, however, imaging realized with such beams[24]. Particle accelera-
cannot be observed in the E or H components from tion is another field of research in which diffraction-free
x z
which S is calculated. The only difference that can be beamsmaybeused;thesebeamscanserveasalternative
y
observed–apartfromtheintensityincrease–isachange accelerating fields owing to the strong longitudinal com-
in the sign of the E field. In the case of low refractive ponent ofthe electric field [26]. It is also possible to ma-
x
indices the sign of E and H are opposite while in the nipulateensemblesofparticlesinmultipleplanes[27,28]
x z
4

10

n=4
2
n=4 ) n=4.9
2
m Ge, n=4.7096
n=4.9 /
W
n=4.99 m0
) 5
2 (
m Ge, n=4.7096 y
S
/
W
-2
(

S y 0 -500 -1000 -1500
y (nm)
0
-5
0 -500 -1000 -1500
y (nm)
FIG. 2: Energy flow on the propagation axis of the central slit calculated behind the slit array shown in Fig. 1 for various
refractiveindices. Slitwidth2a=80nm,wavelength invacuumλ0 =800 nm,screen thicknessb=50nm. Forcomparison we
plotted thesame quantityfor a real material with high refractive index (germanium, n≈4.7096).
whichcanbe realizeddue to the self-reconstructionabil- nectsthewavelengthinvacuum,therefractiveindexand
ity of the nondiffracting beams. As a final example, we the characteristic size of the slit system. We presented
drawattentiontothepossibilityofrealizinglightmotors the results of finite element simulations for two slightly
drivenby the diffraction-free beam. However,extracting differentgeometriesandvariousrefractiveindicesaswell
this nondiffracting nonevanescent beam from the higher asrealmaterials. Ournumericalresultssupportthefind-
refractive index material to air or any lower refractive ings presented in Section II.
index one is a challenging problem.
Acknowledgments
IV. CONCLUSION
The authors gratefully acknowledge Prof. Shlomo
In our paper we show that the production of prop- Ruschin for his helpful comments and suggestions. This
agating (nonevanescent) diffraction-free nano-beams us- work was supported in part by the Hungarian ELI
ing the Fresnel-waveguide concept is feasible. The an- project (hELIos, ELI 09-1-2010-0013, ELIPSZTE), and
alytical expression derived for nonevanescent beams in theHungarian”SocialRenewalOperationalProgramme”
Section II, similarly to the well-known Abbe-limit, con- (TA´MOP 4.2.1./B-10/2/KONV-2010-0002).
5

0
-7
1,0x10

9
4. -1 9
= 9
n 0,0 4.
, =
2 ) n
m
,
W/ 2 m)
-2
( /
W
y
S
-7 (
-1,0x10 Sy
-3
-7
-2,0x10
-100 -50 0 50 100
x (nm)
FIG. 3: Energy flow Sy along the x direction at y =32a for refractive indices n=4.9 and n=4.99. Parameters: wavelength
λ0 = 800 nm, slit width 2a = λ0/10, screen thickness b = 50 nm. Transverse characteristic dimension of the geometry is
Λ = (cid:0)4a(cid:1). Note that, although this is only an example of the beam shape at y = 32a, our calculations showed, however,
2π
that the energy flow Sy of the beam preserves its shape in the region |y| > 2a in both cases, i.e. this phenomenon cannot be
attributed to self-imaging.
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7

-8
6,0x10
-8
4,0x10 0
-8
2,0x10
0,0

6
5 -8 6
1. -2,0x10 -2 .
1
=
=
n -8 n
, -4,0x10
) ,
2 )
m 2
-8 m
/ -6,0x10
W /
W
S (y -8,0x10-8 -4 (y
S
-7
-1,0x10
-7
-1,2x10
-7
-1,4x10 -6
-200 0 200
x (nm)
FIG. 5: Energy flow Sy along the x direction at y=32a for n=1.5 (evanescent region) and n=1.66 (nonevanescent region).
Parameters: wavelength λ0 = 800 nm, slit width 2a = λ0/10, screen thickness b = 50 nm, separation s = 4a. Characteristic
dimension: Λ= 4a+2s, critical refractive index n≈5/3.
2π
8

0,0

1,0
n=1.5 n=1.5
)
2 0,5 CS2, n=1.6295 n=1.64
m )
2
m
/ / n=1.65
W W 0,0
-0,2 m CS2, n=1.6295
(
y (y
S S -0,5

-1,0
0 -500 -1000 -1500
y (nm)
-0,4
0 -500 -1000 -1500
y (nm)
FIG.6: EnergyflowSy alongtheyaxisatx=0forrefractiveindicesn=1.5,1.64,1.65. Parameters: wavelengthλ0 =800nm,
slit width 2a=λ0/10, screen thickness b=50 nm, separation s=4a. Characteristic dimension: Λ= 4a2+π2s, critical refractive
indexn≈5/3. Forcomparison weplottedthesamequantityforarealmaterialwith moderatelyhighrefractiveindex(carbon
disulphide, n≈1.6295).