high-frequency response of subwavelength-structured metals in the petahertz domain

High-frequency response ofsubwavelength-structured metals in the

petahertz domain

J. Weiner1∗ and Frederico D. Nunes2

1IFSC/CePOF, Universidade de Sao Paulo, Avenida Trabalhador Sao-carlense,400-CEP13566-590 Sao Carlos SP, Brazil

2Grupo de Engenharia da Informacao, Universidade Federal de Pernambuco, Recife, Brazil∗[email protected]

Abstract: Electromagnetic plane waves, incident on and reflecting from adielectric-conductor interface, set up a standing wave in the dielectric withthe B-field adjacent to the conductor. It is shown here how the harmonictime variation of this B-field induces an E-field and a conduction current J c

within the skin depth of a real metal; and that at frequencies in the visibleand near-infrared range, the imaginary term σ i of the complex conductivityσ = σr + iσi dominates the optical response. Continuity conditions of theE-field through the surface together with the in-quadrature response of theconductivity determine the phase relation between the incident E-M fieldand Jc. If slits or grooves are milled into the metal surface, a displacementcurrent in the dielectric gap and oscillating charge dipoles at the structureedges are established in quadrature phase with incident field. These dipolesradiate into the aperture and launch surface waves from the edges. They arethe principle source of light transmission through the apertures.

© 2008 Optical Society of America

OCIS codes: (240.5420) Polaritons; (240.6680) Surface Plasmons; (240.6690) Surface waves

References and links1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) 424,

824–830 (2003).2. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys.

Reports-Review Section Of Phys. Lett. 408, 131–314 (2005).3. E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science 311, 189–193

(2006).4. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).5. F. J. G. de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1290

(2007).6. M. M. J. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic

gratings,” Phys. Rev. B 66, 195,105 (2002).7. H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission

through subwavelength hole arrays,” Opt. Express 12, 3629–3651 (2004).8. H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731

(2008).9. A. R. Zakharian, M. Mansuripur, and J. V. Moloney, “Transmission of light through small elliptical apertures,”

Opt. Express 12, 2631–2648 (2004).10. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through slit apertures in

metallic films,” Opt. Express 12, 6106–6121 (2004).

#103320 - $15.00 USD Received 27 Oct 2008; revised 3 Dec 2008; accepted 3 Dec 2008; published 9 Dec 2008

(C) 2008 OSA 22 December 2008 / Vol. 16, No. 26 / OPTICS EXPRESS 21256

11. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through a periodic array ofslits in a thick metallic film,” Opt. Express 13, 4485–4491 (2005).

12. N. Engheta, “Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials,” Science 317,1698–1702 (2007).

13. J. Weiner, “Phase shifts and interference in surface plasmon polariton waves,” Opt. Express 16, 950–956 (2008).14. G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, “Surface wave generation and propagation

on metallic subwavelength structures measured by far-field interferometry,” Phys. Rev. Lett. 96, 213,901 (2006).15. D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, “Quantitative determination of optical transmission through

subwavelength slit arrays in Ag films: Role of surface wave interference and local coupling between adjacentslits,” Phys. Rev. B 77, 115411 (2008).

16. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, chap. 3, pp. 109–120 (Cambridge UniversityPress, 1995).

17. C. Kittel, Introduction to Solid State Physics, chap. 6, p. 150, 7th ed. (John Wiley & Sons Inc, New York, 1996).18. N. W. Ashcroft and N. D. Mermin, Solid State Physics, chap. 1, p. 10 (Thomson Learning, 1976).19. P. B. Johnson and R. W. Christy, “Optical Constants of Noble Metals,” Phys. Rev. B 6, 4370–4379 (1972).20. E. Palik and G. Ghosh, eds., The Electronic Handbook of Optical Constants of Solids (Academic, New York,

1999).21. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons Inc, 1999).22. R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312

(2002).23. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through periodic arrays of

sub-wavelength slits in metallic hosts,” Opt. Express 14, 6400–6413 (2006).24. G. Leveque, O. J. F. Martin, and J. Weiner, “Transient behavior of surface plasmon polaritons scattered at a

subwavelength groove,” Phys. Rev. B 76, 155,418 (2007).25. B. Ung and Y. L. Sheng, “Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals

revisited,” Opt. Express 16, 9073–9086 (2008).

1. Introduction

The generation of surface waves and optical transmission around and through subwavelengthstructures has been the object of an avalanche of publications over the past decade, and re-views of some of this work have started to appear [1, 2, 3, 4, 5]. Most studies have approachedthe subject either by emphasizing periodic array properties (delocalized optical Bloch statesand energy bands) [2, 5] or have couched their analysis in terms of physical optics (diffraction,reflection, transmission) [6, 7, 8]. The interpretation of numerical solutions to the Maxwellequations, governing the electromagnetic response of a structured metal film to incident ra-diation, has also provided insight into charge, current, near- and far-field distributions in andaround subwavelength apertures [9, 10, 11]. In a similar vein electromagnetic analysis of “pro-totypical” subwavelength objects (spheres, rods, squares, etc.) has led to the development ofa plasmonic circuit theory of lumped elements, guiding the engineering design of functionaldevices [12].

This electromagnetic point of view has been recently applied to the problem of light trans-mission through subwavelength slits milled into thin metal films [13]. This study concluded thatcurrents, induced within the metal skin depth by a time-varying standing wave at the surface,produce oscillating charge dipoles at the slit edges, and that these dipoles are primarily respon-sible for transmission through the slit as well as the launch of surface waves at the dielectric-metal interface. The overall layout of the situation considered here and in Ref. [13] is shown inFig. 1, and the disposition of fields and structured metal surface is shown in Fig. 2. Panel (A)of Fig. 2 shows a numerical simulation of the standing wave above a silver metal film in whichtwo slits have been milled. The B-field is adjacent to metal surface. Panel (B) is an enlargeddiagram of the interface region, showing the optical half-cycle in which the B-field points alongthe positive y direction. The purpose of the present study is to explore the consequences of apetahertz driving field incident on the surface for the phase and magnitude of the conductioncurrent in the metal, the displacement current and fields present in the slit gap, and the trans-mission of light through the slits separated by a distance close to the surface plasmon polariton



Fig. 1. Basic layout showing metal layer with two slits of length l and width d separatedby distance p. Dimensions are those typical of experiments, for example, in Refs. [14], [15]with l ∼ 10−20 μm, d ∼ 50−100 nm, p ∼ λspp. Metal layer thickness ∼ 200−300 nmwith skin depth δ � 25 nm. Plane waves incident normal to the top surface are polarized inTM mode, penetrate to depth δ and reflect from the surface.

wave length.

2. Complex permittivity and conductivity

We start with Maxwell’s equations in a real metal conductor with no free charges present.

∇ ·E = 0

∇ ·B = 0

∇×E = −∂B∂ t

Faraday-Maxwell (1)

∇×B = με∂E∂ t

+ μJc Ampere-Maxwell (2)

where E,B are the electric and magnetic induction fields respectively; μ ,ε the permeabilityand permittivity of the material; and the conduction current J c is given by Jc = σE with σ thematerial conductivity. Equations (1), (2) can be decoupled by first applying the curl operationto the Faraday-Maxwell equation,

∇× (∇×E) = ∇(∇ ·E)−∇2E = ∇×(−∂B

∂ t

)= −∂ (∇×B)

∂ t

= −με∂ 2E∂ t2 − μσ

∂E∂ t

which together with ∇ ·E = 0 yields

∇2E = με∂ 2E∂ t2 + μσ

∂E∂ t

(3)



Fig. 2. Incident light on a structured metal surface. Panel (A) shows a numerical simulationof the standing wave field set up by the incident and reflected light in TM mode as indicatedin Fig. 1. Dark blue rectangle with two slits is a silver layer. Above the surface and in the slitgaps B-field amplitude is color coded with red maximum and blue minimum. Note that theB-field predominates near the surface. Panel (B) is an enlarged schematic of the interfaceregion, showing the optical half-cycle in which the B-field points in the +y direction, theskin depth δ , the distance p between slits of width d, and the induced charges at the topedges of the slits.

Similarly applying the curl to the Ampere-Maxwell equation results in

∇2B = με∂ 2B∂ t2 + μσ

∂B∂ t

(4)

Equations (3), (4) are wave equations describing the propagation of electromagnetic waves inthe metal. A standard approach in the analysis of these waves is to assume plane-wave solutionsof the form

Em(r,t) = E0mei(km·r−ωt) (5)

Bm(r,t) = B0mei(km·r−ωt) (6)

where the subscript m indicates a field in the metal, E0m, B0m denote complex amplitudes, andkm is the complex propagation parameter in the metal. Equations (5), (6) can be consideredplane-wave basis functions from which wave solutions of arbitrary form propagating in themetal can be constructed. The amplitude distribution of these plane wave components is calledthe “angular spectrum” [16] of the field. Substituting these solutions back into Eqs. (3), (4) re-sults in an expression for the propagation parameter in terms of the permeability, complex



permittivity, and complex conductivity of the metal.

k2m = μεω2 + iμσω (7)

where the complex quantities are defined as

km = km1 + ikm2 ε = εr + iεi σ = σr + iσi (8)

Substituting and gathering real and imaginary terms,

k2m =

(μεrω2 − μσiω

)+ i

(μεiω2 + μσrω

)k2

m1 − k2m2 =

(μεrω2 − μσiω

)(9)

2km1km2 =(μεiω2 + μσrω

)(10)

and writing the real and imaginary parts of the permittivity in terms of the corresponding di-electric constants, εr = ε0ε ′ and εi = ε0ε ′′ we have

k2m1 − k2

m2 = μ0ε0ω2(

ε ′ − σi

ε0ω

)= k2

0

(ε ′ − σi

ε0ω

)= β 2 (11)

2km1km2 = μ0ε0ω2(

ε ′′ +σr

ε0ω

)= k2

0

(ε ′′ +

σr

ε0ω

)= γ2 (12)

where the standard relations μ0ε0 = 1/c2 and k0 = ω/c have been used.Equations (11), (12) can be separated to find expressions for the real and imaginary propaga-

tion parameters, km1,km2

km1 = ± β√2

√√√√1±

√1+

(γβ

)4

(13)

km2 = ± β√2

√√√√−1±√

1+(

γβ

)4

(14)

The complex propagation parameter km can also be expressed in terms of a complex index ofrefraction, n.

km = k0n = k0(n1 + in2) (15)

with the usual relation between refractive index and dielectric constant, n 2 = ε1 + iε2. Againequating real and imaginary terms results in expression for n 1,n2 in terms of ε1,ε2

n1 =√

ε1

2

√√√√1±√

1+(

ε2

ε1

)4

n2 =√

ε1

2

√√√√−1±√

1+(

ε2

ε1

)4

and consequently

km1 = k0

√ε1

2

√√√√1±

√1+

(ε2

ε1

)4

(16)

km2 = k0

√ε1

2

√√√√−1±√

1+(

ε2

ε1

)4

(17)



Comparing Eqs. (16), (17) to (13), (14) shows that

ε1 =(

ε ′ − σi

ε0ω

)(18)

ε2 =(

ε ′′ +σr

ε0ω

)(19)

Equations (18), (19) separate the metal dielectric constant into two parts: first, ε ′ and ε ′′, repre-senting the dispersive and absorptive response of the metal excluding the conduction electronsand second, σi,r/ε0ω , expressing the conducting electrons contribution to the dielectric con-stant. The next step is to model the conducting electron motion by a driven, damped harmonicoscillator.

3. Damped harmonic oscillator model for conduction electrons

We posit that the equation of motion of the conduction electrons is governed by harmonicacceleration issuing from an electromagnetic field propagating in the metal and polarized alongthe x direction.

med2xdt2 + meΓ

dxdt

= eE(x,t) = eE0me−iωt

where me is the electron mass, Γ a phenomenological damping constant, and eE 0me−iωt thedriving force of an E-M field in the metal with amplitude E 0m and frequency ω . Then, droppingthe common harmonic time factor, the solutions for position, velocity and acceleration are

x = − 1me (ω2 + iΓω)

eE0mdxdt

=iω

me (ω2 + iΓω)eE0m

d2xdt2 =

ω2

me (ω2 + iΓω)eE0m

The conduction current density is then given by

Jc = eNedxdt

= eNeω

me (Γω − iω2)eE0m

The electron density Ne is related to the bulk plasmon frequency ω p by

ω2p =

e2Ne

meε0(20)

and therefore the magnitude of the conduction current density can be written

Jc =ω2

p

(Γ− iω)ε0E0m

From the standard constitutive relations

Jc = σE =σε0

ε0E (21)

and so the expression for the complex conductivity within the damped harmonic oscillatormodel can be written,

σε0

=Γ(

Γ2

ω2p+ ω2

ω2p

) + iω(

Γ2

ω2p+ ω2

ω2p

) (22)

Compare the damping rate Γ to the bulk plasmon frequency ω p. From Eq. (20) we can determineωp if we know the conduction electron density of the metal. A typical “good” metal such as



silver exhibits an electron density Ne � 5.85×1028 m−3 [17], and therefore ω p � 1.4×1016 s−1.The damping rate Γ due to radiative loss and phonon collisions is typically ∼ 10 14 s−1 [18].Therefore Γ << ωp, and in the visible and near-infrared spectral region (ω � 10 15 s−1), Eq. (22)can be written

σε0

� Γ(ω2

ω2p

) + iω(ω2

ω2p

) = Γω2

p

ω2 + iω2

p

ω(23)

This last expression for the conductivity shows that at relatively low frequencies (RF, mi-crowave, etc.) the real term dominates, and the conductivity is “ohmic”. As the frequency in-creases into the optical range, however, the electrons can no longer follow the driving field inphase, and the imaginary term dominates. At ω ∼ 1015 s−1, the real term is only about 5% of theimaginary term. We can therefore, within the harmonic oscillator model, write Eqs. (18), (19)as

ε1 =

(ε ′ − ω2

p

ε0ω2

)ε2 =

(ε ′′ +

Γω2p

ε0ω3

)(24)

If we consider the metal response, excluding the conduction electrons, to behave essentially asa lossless dielectric material,we can estimate ε ′ and ε ′′ as 1 and 0, respectively. This assump-tion is equivalent to the Drude model for metals. Taking ω = 2.0× 10 15 s−1 (λ0 = 942 nm)as a specific example, we calculate ε1 = −47 and ε2 = 2.4. We can compare these values forthe dispersive and absorptive parts of the dielectric constant to measured optical constants forsilver at λ0 = 942 nm. Johnson and Christy [19] report the complex index of refraction in themetal as n = η + iκ , and from interpolation of their data we find η = 0.04 and κ = 6.964 atλ0 = 942 nm. Converting these data to the equivalent dielectric constants yields ε 1 = −48 andε2 = 0.6. We see that the large, negative real value for ε1 can be interpreted as arising fromthe in-quadrature component of the complex conductivity. With η = 0.04 the measured dataresults in an absorptive dielectric constant about one fourth the value calculated from the as-sumed damping constant Γ = 1014 s−1. However, from another commonly cited data set [20]we find η = 0.2,κ = 6.7 with corresponding dielectric constants ε1 = −45,ε2 = 2.6. Dielec-tric constants derived from the harmonic oscillator model are in good agreement with both thedispersive and absorptive terms from this second data set. Table 1 summarizes the comparisonbetween measured data and the harmonic oscillator model(HOM) at ω = 2.0×10 15 s−1, while

Table 1. Comparison of dielectric constants for silver derived from two frequently citeddata sets and the damped harmonic oscillator model (HOM), ω = 2.0×1015 s−1. The realand imaginary terms of the index of refraction η,κ are equivalent to n1,n2 used here.

η(n1) κ(n2) ε1 ε2

Johnson and Christy [19] 0.04 6.964 −48 0.6

Palik and Ghosh [20] 0.2 6.7 −45 2.6

HOM [Eq. 24] −47 2.4

Fig. 3 plots the real and imaginary permittivities from published data and compares them tothe HOM in the petahertz range of frequencies. It follows from Eqs. (21), (23) that the conduc-tion current is in quadrature with the E-field propagating in the metal. In order to determinethe phase of the current density with respect to the incident E-M field we have to examine thecontinuity conditions at the surface.

Fig. 3. Left and right panels show the real and imaginary parts respectively of the per-mittivity for Ag metal over the petahertz frequency range. Plotted blue triangles are from[19], red circles, [20]. Black curve is the harmonic oscillator model (HOM), Eq. 24, withωp = 1.4×1016 rad s−1 and Γ = 1×1014 rad s−1. The model agrees well with data overthe range 1 ≤ ω ≤ 6×1015 rad s−1.

4. Continuity conditions

4.1. Amplitude and phase relations

Figure 4 shows the orientation of the E- and B-fields incident and reflected normal to the metalsurface.

E0I + E0R = E0T (25)

B0I − B0R = B0T =1c

(E0I − E0R

)=

km

ωE0T (26)

Where E0, B0 denote complex amplitudes, E0 = E0eiϕE , B0 = B0eiϕB and the subscripts I,R,Tindicate incident, reflected and transmitted fields, respectively. The complex propagation para-meter of the wave in the metal is km and km/ω is the phase velocity of propagation there. As inEq. (8) we have

km = km1 + ikm2 = |km|eiϕ and |km| =√

k2m1 + k2

m2 with ϕ = tan−1 km2

km1(27)

and from Eq. (26) the relative phase of the transmitted B-field to the transmitted E-Field is givenby

B0T

E0T= ei(ϕBT −ϕET ) =

|km|ω

eiϕ (28)

The relation between the complex refractive index and the propagation parameter in the metalis

n = n1 + in2 =km

k0=

cω

(km1 + ikm2) (29)

Eliminating E0R from Eqs. (25), (26)

E0T =(

21+ n

)E0I =

2n

(1

1/n+1

)E0I =

2n

(1− 1

n+

1n2 −·· ·

)E0I � 2

nE0I (30)



Fig. 4. (A) Orientation of E-M fields propagating along +z. (B) Orientation of E-M fieldspropagating along −z. (C) Orientation of E-M fields for normal reflection at x− y planeof incidence. Applying the continuity conditions upon reflection at the surface reverses thedirection of the E-field and leaves the B-field orientation unchanged, consistent with thePoynting vector condition S = 1/μ0 (E×B).

Writing out the phases explicitly and using Eqs. (28), (29) we have

E0T

E0Iei(ϕET −ϕEI) → 2k0

|km|e−iϕ =

2k0

|km|e−i(ϕBT −ϕET )

Now from Table I we know that, |ε1|>> |ε2|; from Eqs. (16), (17), (29) |km2|>> |km1|; |n2|>>|n1|. Then from Eqs. (27), (30)

ϕ = tan−1 km2

km1� π

2ϕBT −ϕET � π

2and

E0T

E0I→ 2k0

|km|e−iπ/2 (31)

To a quite good approximation the transmitted B-field lags the transmitted E-field by π/2(Eq. 28), and the transmitted E-field leads the incident E-field by π/2 (Eq. 31). Furthermorefrom Eqs. (28), (31), and the relation E 0I = cB0I we find

B0T � 2B0I (32)

The amplitude of the transmitted B-field at the boundary is about twice the incident B-field andthe two B-fields are in phase.

Eliminating E0T between Eqs. (25), (26) results in

E0R =(

1− n1+ n

)E0I and lim

n>>1E0R →−E0I

and, as expected, the reflected wave is π out of phase with the incident wave.To summarize the amplitude and phase relations at the dielectric/metal boundary,

• The B-field in the metal lags the E-field in the metal by π/2.

• The E-field incident at the surface lags the E-field in the metal by π/2 and the amplitudeis attenuated by a factor of 2k0/|km|.

• The amplitude of the B-field in the metal is about twice the incident amplitude and thephase is continuous across the boundary.

• The reflected E-field is π out of phase with respect to the incident E-field.

It follows from Eqs. (21), (23), (31) that in the petahertz frequency domain the conduction cur-rent Jc is in phase with the incident E-M field.



4.2. Comparison of field amplitudes and energy densities at the interface

In order to assess the relative importance of the E- and B-fields at the interface we investigatefield amplitudes and energy densities on the dielectric and metal sides. It is a standard textbookresult [21] that the components of an electromagnetic field parallel to a material interface arecontinuous through it. From the continuity conditions Eqs (30), (32) the relative amplitudes ofthe B- and E-fields at the interface are given by

|B0T ||E0T | =

|B(i)0 |

|E(i)0 |

� 2 |B0I|(2|E0I |

n2

) (33)

where the superscript (i) indicates the field amplitudes at the interface. The E-field at the inter-face is attenuated from the incident E-field by n2, and from Table 1 we see that at the examplefrequency ω = 2×1015 s−1 the B-field is greater than the E-field by a factor of 6.7.

On the dielectric side of the interface the field energy density averaged over an optical cycled is given by

d=12

[12

ε0

∣∣∣E(i)0

∣∣∣2 +12

1μ0

∣∣∣B(i)0

∣∣∣2]

(34)

where we have assumed the vacuum dielectric. Substituting from Eq. (33) and using the relationB0 = E0/c,

d=12

ε0 |E0|2︸︷︷︸incident

[2

n22

+2

](35)

Thus on the dielectric side of the interface the energy density of the standing-wave field is abouttwice the incident field, and the E- and B-fields contribute 2% and 98% respectively.

In contrast to the dielectric side, the field energy density penetrating into an electrically andmagnetically dispersive and absorptive metal, averaged over an optical cycle is [22]

m=[

ε0

4

(ε1 +

2ωε2

Γe

)|E0T |2 +

μ0

4

(μ1 +

2ωμ2

Γh

)|H0T |2

]e−2k2mz

where H0T = B0T /μ0 is the magnetic field amplitude in the metal, μ1,μ2 are the real and imag-inary terms of the magnetic permeability, and Γh is the magnetic absorptive damping constant.We take the common case of metals with nondispersive, nonabsorptive magnetic properties andwrite

m=[

ε0

4

(ε1 +

2ωε2

Γ

)|E0T |2 +

μ0

4|H0|2

]e−2k2mz

Using the damped harmonic oscillator to model the frequency dependence ε(ω),

ε(ω) = ε1 + iε2 with ε1 = 1− ω2p

ω2 , ε2 =Γω2

p

ω3

the energy density in the metal is

m=

[ε0

4

(1+

ω2p

ω2

)|E0T |2 +

14μ0

|B0T |2]

e−2k2mz

and from the continuity relations

|E0T |2 � 4

n22

|E0I |2 and |B0T |2 =∣∣∣B(i)

0I

∣∣∣2 � |2B0I|2 � 4c2 |E0I|2

The energy density can be written as the product of the incident energy density and the sum ofthe separate contributions from the E- and B-fields in the metal.

m=12

ε0 |E0I|2︸︷︷︸incident

[(1+

ω2p

ω2

)2

n22

+2

]e−2k2mz

and using the values in Table I we find that at z = 0

m=12

ε0 |E0I|2 [2.2+2]

Thus on the metal side of the interface the energy density of the E- and B-fields contribute aboutequally. It is worth noting that while the field amplitudes are continuous through the interface,the field energy densities are not. The energy density jumps by about a factor of 2 from thedielectric to the metal side of the interface.

5. E-M fields in the presence of a slit array

5.1. Induced E-M fields with no slits

Reference [13] discussed the phase of the oscillating charge dipole and surface wave generatedat a slit or groove in the metal with the aim of explaining the π-out-of-phase relation betweenthe incident and surface waves. In that study, however, the resistivity R, the inverse of theconductivity σ , was assumed to be real. At optical frequencies, however, Eq. (23) shows thatthe imaginary term of the complex conductivity predominates, and therefore the amplitudesand phase of the fields formed in the metal and in slits must be reexamined. In Section 2 weposited a damped harmonic plane wave solution to the Helmholtz equation in the metal as thepoint of departure and worked outward through the continuity conditions at the metal-dielectricinterface to determine amplitude and phase relations between fields in the metal and incidentfields in the dielectric. In this section, following Ref. [13], we start from the dielectric side andposit a plane wave incident on the interface, setting up a standing wave there; and, workinginward through the continuity conditions, determine the fields set up within the skin depth ofthe metal. From this point of view the E-field and conduction current in the metal arise frominduction by the standing-wave B-field present at the surface. Figure 5 shows the standing wavewith the B-field adjacent to the surface. The B-field at the surface, with approximately twicethe amplitude of the incident wave, attenuates exponentially in +z direction into the metal toa level characterized by the penetration depth (20-30 nm at optical frequencies). The magneticfield is time-harmonic, and the integral form of the Faraday-Maxwell law (Eq. 1) can be usedto calculate the E-field in the metal.

∮A1

(∇×E(1)

T

)·nA1 dA1 =

∮C1

E(1)T ·ds1 = −

∮A1

(∂B(0)

T

∂ t

)·nA1 dA1 (36)

The right hand side of Eq. (36) represents the integral of the time derivative of the magneticflux penetrating the surface A1 shown in Fig. 5. This time-dependent magnetic flux induces anelectric field in the skin depth represented by the integral of the curl expression on the left handside of Eq. (36). By virtue of Stokes’ theorem this integral over an area can be converted intoan integral over the circuit C1 also shown in Fig. 5. The integrals over both sides of Eq. (36) canbe easily carried out. The B-field time derivative is

−∂B(0)T

∂ t= iωB(0)

T e−iωt

Fig. 5. Panel (A): Front view of incident plane wave setting up a standing wave at thesurface of a smooth, featureless metal slab. Red amplitude indicates B-field maxima, blueshows E-field maxima. Panel (B) Diagram of the field metal surface shown in panel (A) foran optical half-cycle with EI pointing along +x and BI pointing along +y. The skin depthin z is indicated as δ , and p is some arbitrary length along x. The area A1 = pδ and contourC1 are used for the integration of Eq. (36). The conduction current induced in the metal, Jc,is in phase with the incident E-field.

and because the B-field amplitude is assumed constant along x (plane wave) and exponentially

attenuated along z, we can write B(0)T (x,z) = B(0)

T e−km2z; and from the continuity conditions

B(0)T � 2BI where BI is the incident B-field. Integrating over A1 on the right hand side and C1

over the left hand side of Eq. (36) yields,

E(1)T = −i

ωB(0)T

km2= −i

c2BI

n2= −i

2EI

n2(37)

This result for E(1)T is consistent with the field obtained from the continuity conditions (Eq. 30).

The conduction current in the metal, the product of the electric field and the complex conduc-tivity,

Jc = σE(1)T � σi

2EI

n2(38)

is also consistent with the continuity conditions at the surface; the conduction current in the

metal is in phase with the incident field. The B-field in the metal B(1)T can be obtained by

substituting E(1)T into Eq. (2) and integrating over an appropriate area and contour or more

directly from Eq. (28).

B(1)T = −μ0σiE

(1)T

n2k2m� μ0σi2EI

n22k0

�(ωp

ω

)2 2EI

n22c

(39)

In the last term on the right the expression for σ i from harmonic oscillator model Eq. (23) hasbeen substituted.



5.2. E-M fields with a slit array

Introduction of discontinuities in the metal surface such as slits, grooves, and holes unleashes awide spectrum of scattered waves in the space above, below and along the metal surface. A care-ful and quantitative analysis of the Bloch modes populated in a one-dimensional periodic arrayof slits in a silver slab has been carried out by Xie et al. [23]. In general many Bloch modes ofthe periodic array within the metal and within the slits must be populated to match the incidentand reflected modes at the surface. Therefore, for an arbitrary slit periodicity, the description oftransmission through the array does not lend itself to a simple physical model. However, in thespecial case when the separation between the slits approaches an integer number of λ spp wavelengths, Xie et al. [23] have shown that only the normally reflected propagating mode and thesurface plasmon polariton (spp) modes, evanescent along z but guided and propagating alongthe surface, remain. At these special slit positions two counterpropagating spps form a standingwave at the surface, superposing with the standing wave arising from the incident and reflectedpropagating plane wave. The spatial distributions of these two standing waves are shown inFig. 6. Conduction currents Jp,Jspp, appear in the metal skin by Faraday induction from theseB-field standing waves present at the surface. The conduction currents charge the slit edgessimilar to a conventional parallel plate capacitor, and the displacement current, J slit, in the slitgap gives rise to an E-field and B-field, oscillating in quadrature with the incident plane wave.Consider first the slit fields induced by the plane standing wave.

5.2.1. Fields generated in one slit

From the condition of current density continuity across the slit gap, ∇ ·J = 0,

Jp = Jslit = ε0εslit∂Eslit/∂ t (40)

and using Eqs. (38), (23) we can easily find the E-field in the slit induced by the plane wave,

Epslit = i

(ωp

ω

)2 2EI

|εslit|n2(41)

We see that this slit field is in quadrature with the incident E-field. The dielectric constant inthe gap εslit for subwavelength slit widths d will be somewhat greater than unity, determinedby field continuity conditions for this metal-insulator-metal geometry along x. We can nowcalculate the B-field in the slit by integrating the Ampere-Maxwell equation (Eq. 2) over theappropriate contour and area, C2,A2 and using Eq. (40).

∮C2

Bpslit

μ0·ds2 =

∮A2

Jslit ·nA2 dA2

=∮

A2Jp ·nA2 dA2 =

∮A2

(ε0εslit

∂Eslit

∂ t

)·nA2 dA2

so that

Bpslit = i

μ0σi2EI

n2kslit= i

(ωp

ω

)2 2EI

n2 |nslit|c (42)

where nslit =√

εslit. We see that this B-field in the slit is also in quadrature with the incidentE-M field, and therefore these fields can be associated with the propagating mode in the slitgap. Consider now the fields on the surface and in the slits induced by the spp standing wave.

5.2.2. Fields generated by an spp standing wave in a slit array

When the slits are separated by λspp, the counterpropagating spps launched at the slit sites forma standing wave between them, the E-field of which is illustrated in Fig. 6 (B). This E-field



Fig. 6. Panel (A): Standing plane wave above the two-slit structure, showing transmissionthrough the slits at some arbitrary slit separation. Red (blue) color shows maximum B-field(E-field) amplitudes. Note that propagating fields within the slits set up Fabry-Perot-likecavities, the characteristics of which depend on the slit width, metal film thickness, andmetal permittivity. The simulation shown here is for 100 nm slit in a silver film of thickness� 350 nm. Panel (B): A diagram of the area between the slits separated by λspp, showingthe two superimposed standing-wave B-field amplitudes, one from the incident plane wave,the other from the spp standing wave. The E-field of the spp standing wave is indicated asa thick black arrow (Espp) pointing from positive to negative charge at the two slit edges.The induced current density in the metal from the plane standing wave is denoted Jpw andfrom the spp standing wave Jspp.

“loop,” a half sine wave originating at the positive charge at the right slit and terminating on thenegative charge on the left slit, oscillates at the incident frequency and in phase with the dipolesat the slits. The standing wave spp B-field, aligned along y and supported by J spp runningalong x in the metal, must oscillate in quadrature with this E-field and therefore in quadraturewith the dipoles at the slits. But Eqs. (41), (42) show that the dipoles themselves oscillate inquadrature with the incident E-M field; and therefore, the standing wave spp B-field is π out ofphase with the incident E-M field. The standing wave spp B-field opposes the standing waveB-field at the surface. A Green’s tensor analysis [24] of a line dipole oriented just above a metalsurface confirms the quadrature phase relation between the slit dipoles and the incident field.The induced current in the metal from the standing wave spp J spp therefore opposes the inducedcurrent from the plane standing wave Jpw. The two opposing conduction currents result in anet conduction current Jnet(x) = Jmax

spp cos[(kspp/2)(x−λspp/2)]−Jpw. If Jmaxspp > Jpw, then, as x

moves away from the midpoint and J spp(x) decreases from its maximum value, there will be twonull points where the two B-fields cancel and Jnet = 0. Using results from Table 1 of Ref. [23]it is easy to show that these null points for that study are near x = λ spp/4 and x = 3λspp/4.Points of null current are points of maximum charge. The overall effect of establishing thestanding wave spp is to shift the charge centers from the slit edges to the one-quarter andthree-quarter points along the surface between the slits. The slit corners transform from dipolesources of transmission to sites of maximum E-and B-field in phase with the reflected wave. Atpetahertz frequencies the physical slits present almost negligible capacitive impedance to theshifted field. The analysis of Ref. [23] confirms that the critical slit separation of λ spp results in



a near-extinction of the transmission mode through the slits and a corresponding maximum inreflection mode. This transmission extinction at slit separations of an integral number of λ spp

has been confirmed by precise measurements [15]. Numerical field simulations of Ref. [11] (seeespecially the right-hand column of Fig. 3 of that study) are consistent with the conclusion thatat the spp standing wave condition the charge centers shift away from the slit sites to the quarterpoints between the slits.

6. Conclusions

We have shown here how the complex dielectric constant of a metal can be separated into twoparts: one that depends primarily on the conduction electrons and the other that depends onlyon the core. By describing the response of the conduction electrons to an external driving fieldwith a damped harmonic oscillator model we have identified a separate frequency dependencefor the real and imaginary parts of the complex conductivity. The low-frequency real part ofthe conductivity corresponds to the familiar ohmic response in which the electron flow in themetal is controlled by collisional diffusion. However, at petahertz frequencies (visible and near-infrared) the imaginary term dominates, corresponding to ballistic electron motion. Applicationof the E-M field continuity conditions through the metal surface shows that the conductioncurrent density within the metal penetration depth is in phase with the external driving field.We have also shown that although the field amplitudes and energy density on the dielectric sideof the interface are dominated by the B-field, on the metal side, due to dispersion of the material,the electric and magnetic field components contribute nearly equally to the energy density. Theenergy density is discontinuous through the boundary. Integration of the Faraday-Maxwell andAmpere-Maxwell equations, starting from the standing-wave B-field at the dielectric side ofthe interface, determines E- and B-fields within the metal as well as the displacement currentdensity, E- and B-fields in the slit gap. The fields in the gap oscillate in phase with the cornercharge dipoles and in quadrature with the incident E-M field. For arrays of slits, a periodicity ofλspp sets up an spp standing wave and results in near-extinction of transmission through the slits.There exists now a significant body of evidence supporting the view that surface wave responseto optical excitation of subwavelength metallic structures can be modeled by oscillating chargedipoles at the dielectric-metallic interface [25], [24]. The launch of surface waves originatingfrom these structures is often couched in the language of diffraction [25] or more generally“scattering,” [24]. Here we take the view that the principal origin of the dipole charging currentis the illuminated planar surface in the vicinity of the metal discontinuity and that this currentis induced by standing-wave, time-harmonic E-M fields at the surface. If this view is correct,the “effective source” of E-M fields in the subwavelength opening is the oscillating dipoleinduced at the aperture entrance; and direct illumination of the structure aperture by the incidentplane wave contributes little to optical transmission through the opening. These ideas must beconfirmed and quantified by careful experiments.

Acknowledgments

We are greatly indebted to valuable discussions with H. J. Lezec, M. Mansuripur, and technicalassistance from G. Ribeiro, M. Bezerra, and T. Vasconcelos. J. W. gratefully acknowledgesthe hospitality of the Center for Research in Photonics and Optics, supported by FAPESP andIFSC/USP-SC, and a visiting fellowship from the CNPq.



high-frequency response of subwavelength-structured metals in the petahertz domain

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