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Di et al. Vol. 24, No. 9 /September 2007 /J. Opt. Soc. Am. B 2349

Transmitted behavior of the light waves througha horn-opened single nanoslitin finite-width metallic films

Si Di,1 Juan Liu,1,3 Ben-yuan Gu,2,* Bin Hu,1 Xiao-dong Sun,1 Shang-qi Wang,1 and Yun Liu1

1School of Electronics and Information Engineering, Beijing Jiaotong University, Beijing, 100044, China2Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China

3E-mail: juanliu@center.njtu.edu.cn*Corresponding author: guby@aphy.iphy.ac.cn

Received October 12, 2006; revised April 17, 2007; accepted June 25, 2007;posted July 11, 2007 (Doc. ID 75955); published August 23, 2007

Transmission properties of light waves through a horn-opened subwavelength-size single slit in finite-widthhomogeneous or inhomogeneous metallic films are studied by using the boundary integral method. We calcu-late transmission spectra in the visible wavelength regime, display the optical field intensity distributions, anddemonstrate the directions of the energy flow. The results show that the transmission spectrum crucially de-pends on the physical and geometrical parameters of samples, for instance, the split angle in the horn-openedregion of the slit, the width and depth of the slit, and the metallic materials surrounding the slit, etc. Twopeaks in the transmission spectrum are observed for the nanoslit with a horn opening in the homogeneousmetallic film, which originate from the metallic film and air slit, respectively. However, there are three peaksfor the nanoslit in the hybrid metallic films consisting of two metallic films on either side of the slit, stitchedtogether. Two peaks correspond to two different metallic materials, and the other one corresponds to the airslit, for the slit surrounded by two metallic materials. An equivalent structure is proposed to give an approxi-mate description of the transmitted behavior. © 2007 Optical Society of America

OCIS codes: 310.6860, 240.6680, 220.4000, 240.0310.

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. INTRODUCTIONince the pioneering work of Ebbesen et al. in 1998 [1],he extraordinary optical transmission (EOT) phenom-non of light waves through nanohole arrays in thick me-allic films has attracted the interest of researchers due tots significant potential for use with electroluminescentevices, integrated photonics devices, nanofabricationrocessing, biological sensors, and so on [2–7]. On the the-retical side, there is no united understanding and inter-retation of this novel phenomenon so far [8–11]. One ofhe popular explanations is that this phenomenon may bettributed to the excitation of the surface plasmon polari-ons (SPPs) propagating on the surface of the metalliclm and the coupling interaction between the SPPs andhe light wave. It is observed that the transmission spec-rum is crucially influenced by the physical and struc-ural parameters of the nanoslit in the metallic films.indberg et al. [12] have revealed the relationship be-ween the optical transmission and the length or theidth of the single subwavelength aperture in metallms. In practical fabrications, the exact rectangle shapet the edges of the nanoslit in metallic film is hardly re-lized because of its nanometer scale of the slit or the dif-culty of the exact collimation of conventionally focused

on beam lithography. Smolyaninov et al. representedlasmon-polaritons on the surface of a pseudosphere [13].he horn-opened shape of the single slit in metallic films,ore or less, may be closer to the real case, and the horn-

pening metallic slit might enhance the SPPs or EOT. The

0740-3224/07/092349-8/$15.00 © 2

etailed analysis of the transmission behavior of lightaves through this kind of structure is an interesting

opic. In this paper, we carry out the detailed calculationsf the transmission spectrum in the horn-opened singlelit in pure or hybrid metallic films. We explore the influ-nce of the physical and structural parameters of samplesn the transmission spectrum. It is observed that therexist two peaks in the transmission spectrum for theanoslit opening in homogeneous metallic films, whichriginate from the metallic films and air slit, respectively.he peak value, as well as the peak position and line-idth of the transmission peak depend on the structuralarameters. However, there are three peaks for theanoslit in the hybrid metallic films consisting of two me-allic films on either side of the slit, stitched together. Twoeaks correspond to two different metallic materials andhe other one corresponds to the air slit. We also demon-trate the corresponding field intensity distribution andhe direction of energy flow. We indicate that these spec-ra can be obtained in a high approximation from thequivalent averaged structures, i.e., the stepped shapelit. Two resonant cavities are formed; the obtained re-ults can be well understood.

This paper is organized as follows: in Section 2 theoundary integral method is briefly described. The trans-ission spectra in the horn-opened nanoslit in the metal-

ic films for various parameters are presented and ana-yzed in Section 3. Finally, the conclusions are drawn inection 4.

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2350 J. Opt. Soc. Am. B/Vol. 24, No. 9 /September 2007 Di et al.

. BOUNDARY INTEGRAL METHODhe model structure of a horn-opened single slit in finite-idth metallic films is sketched in Fig. 1. This system cane regarded as the construction of two pieces of metallichin films with a cutting corner, face-to-face arrangement,eparated by a spacing of d. Each piece of the metalliclm (located at the left or right side of the slit) has aidth of l and a thickness of h. The section of the film is

aid on the xy plane, the z is perpendicular to the sectionf the films, the extension of the sample along the z direc-ion is infinite, and therefore, the structures consideredre two dimensional (2D). The geometrical entry shape ofhe horn-opened nanoslit is described by two parameters,and b, as remarked in Fig. 1. The whole space is parti-

ioned into three regions; we mark them as Region 1metal, V1), Region 2 (metal, V2), and Region 3 (air, V3).he metal materials in Regions 1 and 2 can be different.We assume that a beam of the p-polarized [in-plane

lectric field E= �Ex ,Ey ,0� or z-direction magnetic field= �0,0,Hz�] plane wave with a wavelength �0 and wave

ector k is normally launched onto the nanosilt along theaxis. In this 2D system, the Maxwell equations govern-

ng the propagation of electromagnetic (EM) waves inhree regions read as [14]

�2�1tot�r� + k1

2�1tot�r� = 0, r � V1,

�2�2tot�r� + k2

2�2tot�r� = 0, r � V2,

�2�3tot�r� + k3

2�3tot�r� = − f�r�, r � V3, �1�

here k1,2,3= �2�n1,2,3� /�0 denotes the wave number inegion 1, 2, or 3, respectively; r denotes the position vec-

or of a given point in space; �tot�r� stands for the totaleld, being Hz�r� for the p polarized. f�r� describes the in-ident field. These equations can be solved with the use ofhe boundary integral method (BIM). The correspondingoundary integral equations can be derived by applyingreen’s second identity [14]; they read as

ig. 1. Schematic of a horn-opened nanoslit opening in metalliclms. A p-polarized (transverse magnetic) plane wave is nor-ally incident on the nanoslit.

�tot�r1� =�C1

�G1�r1,r����tot�r��

�n− �tot�r��

�G1�r1,r��

�n �dl,

r1 � V1,

�tot�r2� =�C2

�G2�r2,r����tot�r��

�n− �tot�r��

�G2�r2,r��

�n �dl,

r2 � V2,

�tot�r3� = −�C1+C2

�G3�r3,r����tot�r��

�n

− �tot�r���G3�r3,r��

�n �dl + �inc�r3�, r3 � V3, �2�

here G1,2,3�r1,2,3 ,r��=H0�2��k1,2,3�r1,2,3−r��� are the 2D

reen functions, and H0�2� is the zero-order Hankel func-

ion of the second kind. C1 and C2 denote the surface con-our of Regions 1 and 2, respectively. n denotes the out-ard normal of C1 or C2. �inc�r3� is the incident field inegion 3. r1,2,3 denotes the position vector of a given point

the observation point) in Regions 1, 2, or 3. r� representshe source point at the boundaries C1,2. Thus, we can de-ive the equations to completely determine the boundaryelds by moving the observation point from the interior ofegions 1, 2, or 3 to the boundaries; the equations govern-

ng the boundary fields read as

�1 −�

2���tot�rs� +�

C1

��tot�r���G1�rs,r��

�n

− G1�rs,r����tot�r��

�n �dl = 0,

�1 −�

2���tot�rs� +�

C2

��tot�r���G2�rs,r��

�n

− G2�rs,r����tot�r��

�n �dl = 0,

�

2��tot�rs� +�

C1+C2

�G3�rs,r����tot�r��

�n

− �tot�r���G3�rs,r��

�n �dl = �inc�rs�, �3�

here rs and r� denote the positions of the observationnd source points at the boundary, respectively. Thehase � represents the outward angle subtended by theoundary at the point rs=r�, at which the singularity ofhe contour integral occurs. As soon as the boundary fieldsre determined, the total field distribution over the wholepace can then be evaluated by Eq. (2).

We now turn to consider the transmission of the-polarized incident light wave through the samples. Theransmission T is given by

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Di et al. Vol. 24, No. 9 /September 2007 /J. Opt. Soc. Am. B 2351

T

�−�l+d�/2

�l+d�/2

Sntotdx

�−d/2

d/2

Snincdx

, �4�

here Sntot and Sn

inc are the normal components of theime-averaged Poynting vectors of the total field and thencident field, respectively. d is the width of the slit, and ls the width of metal films. The integral in the nominatorf Eq. (4) is the incident power. The units of transmit-ance are 100% =1. The transmittance used in this papereans actually normalized transmission, which stands

or a ratio of the transmitted power to the incident power.y calculating Eq. (4), we can study the transmissionroperties of the light waves through the horn-openedingle nanoslit in the metallic films for various structuralarameters quantitatively.

. NUMERICAL RESULTSn numerical simulations, the structural parameters arehosen as follows: the width of nanoslit sets d=40 nm, thehickness of metal films is h=200 nm, and the width ofetal films is l=2.0 �m. The material in Regions 1 and 2

s silver with the wavelength-dependent refractive index15]. Region 3 is in the air with a refractive index of 1.0.he SPPs penetration depth into air is less than 600 nmver the considered wavelength range according to [16].herefore, in the calculations of the transmission spec-rum, we choose the observation (or the collecting fieldignal) plane at y=−700 nm so that the surface wave cane excluded.The transmission spectra of light waves through the

orn-opened single nanoslit in homogeneous metallic filmor various values of parameter a (marked in the plots),emaining a=b, are displayed in Figs. 2(a)–2(c). It islearly seen that there are two peaks in the transmissionpectrum; appearing at the wavelengths at 530 nmpeak 1—the short wavelength peak) and 650 nm (peak—the long wavelength peak), respectively. It is notedhat the transmission of the horn-shaped nanoslit com-ared to the rectangular-shaped nanoslit is much en-anced, from 3.44 �a=0 nm� to 7.63 �a=90 nm� for therst peak, and from 3.13 �a=0 nm� to 6.47 �a=60 nm� foreak 2. Peak 1 is mainly associated with the metallic film;n contrast, peak 2 is mainly concerned with the airanoslit, as discussed later. These two transmissioneaks crucially depend on the shapes of the entry horn-pened nanoslit. It is apparent that peaks 1 and 2 bothhift toward the short wavelength region when increasinghe parameter a from 0 to 120 nm, while peak 2 movesaster than peak 1. The linewidth of the two peaks in theransmission spectrum is broadened as increasing a. Re-arding the peak values, their variations with a are de-icted in Fig. 3. The solid curve corresponds to peak 1,ashed curve corresponds to peak 2. Both curves are firstaised and then declined with the increase of a. The maxi-um transmissions appear at 90 nm for peak 1 and at

0 nm for peak 2. We also investigated the case of maxi-

al cutting a=b=200 nm; that is, the tapered slit, andbtained a similar trend of the transmitted spectrum inig. 3.

ig. 2. Transmission spectrum for a horn-opened nanoslit open-ng in the pure silver films when a normally incident, p-polarizedlane wave impinges upon the sample. The parameters are cho-en as: fixed a=b and varying the values of a: (a) a=b=0, 20, 40,0 nm, (b) a=b=60, 80, 90 nm, and (c) a=b=90, 100, 110,20 nm.

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Fi=1u

2352 J. Opt. Soc. Am. B/Vol. 24, No. 9 /September 2007 Di et al.

To confirm the nature of the two resonant transmissioneaks, we calculate the transmission spectrum for theingle nanoslit in a gold film, replacing the previous silverlm. The shape of the horn-opened slit rectangular with=b=0 nm and d=40 nm. The results are displayed inig. 4. It is evident that there still are two peaks in theransmission spectrum for the gold sample. However, theriginal of peak 1 (at 530 nm) for the silver film ishifted to peak 1� (at 560 nm) for the gold film and thealue of peak 1� is much smaller than that of peak 1 inhe silver film sample. But peak 2 for the gold film onlyas a little redshift, and its peak value is decreased a

ittle, compared with peak 2 in the silver film. It mani-ests that peak 2 indeed is mainly concerned with the slit.eak 1 or 1� is mainly dominated by the material of metallms.We now turn to consider the influence of other shapes

f the horn-opened single nanoslit in the pure silver filmn the transmission spectrum by changing parameter b.

ig. 3. Variations of peak transmission as a function of the pa-ameter a in the same sample as in Fig. 2. The solid curve cor-esponds to peak 1 (the short-wavelength peak at 530 nm),ashed curve to peak 2 (the long-wavelength peaks at 650 nm).

ig. 4. Transmission spectrum for a nanoslit with horn-openedntry shape, opening in the metallic films of silver or gold, whenp-polarized plane wave is normally incident upon the sample.he geometrical parameters are chosen as: a=b=0 nm and d40 nm. The solid curve corresponds to the transmission spec-

rum in the nanoslit opening in the silver film, the dotted curveo that in the nanoslit opening in the gold film.

igs. 5(a) and 5(b) display the transmission spectra forifferent values of parameter b (marked its value in eachlot), fixed a=40 nm. This means that the inclinationngle at the horn-opened entrance of the sample is varied,nstead of the fixed 45° considered in Fig. 2. It is observedhat there still exist two peaks in the transmission spec-rum. The linewidth of the peaks is broadened with thencrease of b. Their peak values significantly depend onarameter b, as shown in Fig. 6. The solid curve corre-ponds to peak 1 (at 530 nm), dashed curve to peak 2 (at650 nm). It is seen from Fig. 5 that the two peaks ex-ibit a similar varying tendency; they are first increasednd then decreased; both reach a maximum transmissiont b=80 nm.To clearly understand the resonant transmission in the

orn-opened single nanoslit in the metallic film, we con-ider an equivalent averaged structure consisting of atepped slit, as drawn by the dashed line in Fig. 7. Thentry width of the stepped slit is given by w=a /2 and thetep depth is b. Therefore, the stepped nanoslit now isonstructed by two slits, which are joined together; one iswide slit with a total width of �2w+d� at the upper po-

ig. 5. Transmission spectrum for a horn-opened nanoslit open-ng in the silver film with the geometric parameters as: fixed a40 nm and varying value of b: (a) b=20, 50, 80 nm (b) b=80,10, 140 nm. The p-polarized plane wave normally impingespon the sample.

slt

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Fht=20 nm and b=40 nm.

F2ae=htPt

FtFwl

Fi

Di et al. Vol. 24, No. 9 /September 2007 /J. Opt. Soc. Am. B 2353

ition and another is a narrow slit with a width of d at theower position. The explanation of the other symbols ishe same as in Fig. 1.

The transmission spectra of the exact horn-openedingle silt in the metallic film, and the approximated av-raged structure are both illuminated in Fig. 8. It is ap-arent that the transmission spectrum of the horn-pened nanoslit with a=b=40 nm is similar to that of thetepped slit with w=20 nm and b=40 nm. In addition, theptical field intensity distributions and the directions ofhe energy flow at the resonant transmission peak656 nm� for the exact horn-opened slit and its equivalenttepped slit are illustrated in Figs. 9(a) and 9(b), respec-ively. The arrow stands for the direction of the time-veraged Poynting vector, that is, the direction of the av-rage energy flow. It is obvious that the optical intensityistribution and the direction of the energy flow of bothodels exhibit a similar pattern except for the entry re-

ion of the samples. We found that the equivalent steppedlit approximation of the horn-shaped slit is the best casehen a=b, and the equivalent step slit approximation

till holds true when a�b. In fact, the simplified steppedanoslit can be considered as two rectangular resonantavities join up, which have been extensively studied [17].

ig. 6. Variations of the peak transmission with the increase ofhe parameter b, fixed a=40 nm. The sample is the same as inig. 5. The solid curve corresponds to peak 1 (the short-avelength peak, at 530 nm), the dashed curve to peak 2, the

ong-wavelength peak, at 650 nm).

ig. 7. Schematic of a nanoslit opening in the metallic film andts equivalent averaged step-shaped slit.

ig. 8. Comparison of the transmission spectrum of the exactorn-opened nanoslit (solid curve) when a=b=40 nm to that ofhe equivalent averaged stepped slit (dotted curve) when w

ig. 9. Intensity distributions of total magnetic field plotted in a56 gray-level representation for peak 2 at 656 nm: (a) for the ex-ct horn-opened nanoslit with a=b=40 nm, and (b) for itsquivalent averaged structure of the stepped nanoslit with w20 nm and b=40 nm. Bright regions correspond to the areas ofigh field intensity, and dark regions to the areas of low field in-ensity. The white arrows denote the direction of time-averagedoynting vector. The contours of the structures are indicated byhe solid line.

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2354 J. Opt. Soc. Am. B/Vol. 24, No. 9 /September 2007 Di et al.

As is well known, the SPPs can be excited when the in-ident light wave is in TM mode so that the electric fields perpendicular to the nanoslit and its boundary, and theadiation light wave by SPPs is traveled only on the metal

ig. 10. Intensity distributions of total magnetic field plotted in128 gray-level representation for peak 2: (a) for 648 nm when=b=60 nm, (b) for 635 nm when a=b=90 nm, and (c) for34 nm when a=b=120 nm. Bright regions indicate the areas ofigh field intensity, and dark regions correspond to the areas of

ow field intensity. The solid lines indicate the contours of thetructures.

urface. The theory of surface plasmon generation at aanoslit aperture was reported recently [18,19]. The modef the excited field should match the eigenmodes of thepper and lower resonant cavities. The total field throughhe nanoslit is a sum of the incident field and the excitedeld. The SPPs are considerably enhanced with the in-rease of the values of a and b due to the increase of thenteraction extension on the surface. This is the reasonhat the two transmission peaks are first increased whenncreasing a and b as shown in Fig. 3. The short-avelength transmitted peak (peak 1) starts to decline at=b=90 nm and the long-wavelength transmitted peak

peak 2) starts at a=b=60 nm, which can be attributed tolarge field extension of the eigenmode of the upper rect-

ngular resonant cavity, much larger than the width ofhe lower slit when further increasing a and b. Therefore,he intrinsic mode of the upper cavity does not match thatf the lower cavity. The mode size (i.e., the field distribu-ion extension) of the long-wavelength eigenmode is big-er than that of the short-wavelength eigenmode [20];his is the reason that the declination of the transmissionf peak 2 with a large mode size starts at a�60 nm; onhe contrary, the declination of the transmission of peak 1

ig. 11. Transmission spectrum for a horn-opened nanoslitpening in the hybrid metallic films, i.e., the slit surrounded byhe silver and gold films on either side of it. The parameters ares follows: fixed a=b, and varying parameter a: (a) a=b=0, 20,0, 60 nm and (b) a=b=60, 80, 100, 120 nm. A p-polarized planeave normally impinges upon the sample.

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F=b

Di et al. Vol. 24, No. 9 /September 2007 /J. Opt. Soc. Am. B 2355

ith a small mode size (compared with that of peak 2)tarts at a larger value of a�90 nm than that of peak 2.

We also compute the optical field intensity distribu-ions of peak 2 for three different shapes of the horn-pened slit, as shown in Figs. 10(a)–10(c): (a) for a=b60 nm, (b) a=b=90 nm, and (c) a=b=120 nm. The re-ectivity for three cases is (a) R=0.889, (b) R=0.910, andc) R=0.943, as shown in Fig. 10. It is clearly seen thatost of the incident light energies is transmitted and

eaches the lower slit when a=b=60 nm [see Fig. 10(a)],ut only a part of the incident light energies can flow andrrive at the lower slit when a=b=90 nm; however, in thease of a=b=120 nm, it corresponds to a much largerpen degree at the entry of the slit [see Fig. 10(b)]. Most ofhe incident light energies are reflected back on the inci-ent region and only a very small part of the light ener-ies is allowed to pass through the upper slit and arrivet the lower slit [see Fig. 10(c)].To further investigate the influence of the metallic ma-

erial on the transmission behavior, we now consider ansymmetric structure. We replace silver with gold in Re-ion 2 and stay with the silver in Region 1. Thus, theample is a horn-opened nanoslit in the hybrid metalliclms, consisting of silver and gold films on either side ofhe slit, stitched together. We calculate the transmissionpectrum in the case of a=b for various values of a, as

ig. 12. Same as Fig. 11 except for different parameters as a40 nm and varying parameter b: (a) for b=20, 50, 80 nm (b) for=80, 110, 140 nm.

hown in Fig. 11(a): (a) for a= �0,20,40,60� nm and (b) for= �60,80,100,120� nm. It is observed that three peaksppear in the transmission spectrum, in which two peaksxist in the short-wavelength region, as clearly seen inhe inset of Fig. 11, marked as peaks 1 and 1�. Peak 1 isoncerned with silver and is now located at 520 nm, alight blueshift from the previous peak 1 (at 530 nm) inhe pure silver film, and the transmitted peak value isuch less than that in the pure silver film; while peak 1�

s associated with the gold film and located at 547 nm, alight blueshift from the original peak 1 (at 560 nm) inhe pure gold film. These changes might be attributed tohe SPPs’s coupling of different metallic materials in Re-ions 1 and 2 for the asymmetric hybrid structure. Peak 2orresponds to the nanoslit and is located at 660 nm.

Next, we calculate the transmission spectrum of theorn-opened nanosilt in the hybrid metallic films with a40 nm for different values of b, as shown in Fig. 12(a)nd 12(b): (a) for b= �20,50,80� nm and (b) for b�80,110,140� nm. Peaks 1 and 1� (as shown in the insetf Fig. 12) are associated with the silver and gold films,espectively, and they exhibit a similar varying tendencyo the structure with a=b.

. CONCLUSIONSe investigated the transmission properties of lightaves through a horn-opened nanoslit in finite-width ho-ogeneous or inhomogeneous metallic films for different

eometric and physical parameters with the use of theoundary integral method. The transmission spectrumrucially depends on the entry splitting shapes of theorn-opened nanoslit, the width and the depth of the slit,nd the metallic materials surrounding the slit. We calcu-ated the transmission spectra in the visible wavelengthegime, displayed the optical field intensity distributions,nd demonstrated the directions of the energy flow. Webserved that two peaks exist in the transmission spec-rum for the nanoslit in the homogeneous metallic film,hich are attributed to the contributions from the metal-

ic film and the air slit, respectively. However, there arehree peaks for the horn-opened nanoslit in the hybridetallic films, consisting of two metallic films on either

ide of the slit, stitched together. Two peaks correspond towo different metallic materials, and the other one corre-ponds to the air slit, for the slit surrounded by two me-allic materials. An equivalent average structure is sug-ested to approximately describe the transmissionharacteristics of the samples. Our findings should enables to adjust the geometric and physical parameters of theorn-opened nanoslit in the metallic films so that we canngineer the transmission spectrum to match variousanophotonics applications.

CKNOWLEDGMENTShis work was supported by the National Basic Researchrogram of China (grant 2006CB302901) and the Beijingatural Sciences Foundation (grant 4042023).

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2356 J. Opt. Soc. Am. B/Vol. 24, No. 9 /September 2007 Di et al.

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