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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 2 — Jan. 21, 2008
  • pp: 950–956
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Phase shifts and interference in surface plasmon polariton waves

J. Weiner  »View Author Affiliations


Optics Express, Vol. 16, Issue 2, pp. 950-956 (2008)
http://dx.doi.org/10.1364/OE.16.000950


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Abstract

A π phase shift between the incident wave and surface plasmon polariton (SPP) waves launched from a one-dimensional (slit or groove) subwavelength structure has been found in numerical simulations and invoked to explain recent measurements of optical transmission in slit arrays. Although groove launchers exhibit an overall phase shift that depends on the groove depth, it is shown here how magnetic field induction at the incident surface, and oscillating dipoles from the accumulated charge at the slit or groove edges on the entrance facet lead to an intrinsic π phase shift, independent of the groove or slit depth. Destructive interference between the π-shifted surface wave and the incident wave explains the observed transmission minima when the pitch of an array of slits becomes equal to an integer multiple of the SPP wavelength.

© 2008 Optical Society of America

1. Introduction

Over the past several years optical transmission through apertures and the generation of surface waves from subwavelength structures has been the subject of intensive study and lively debate. From these researches has gradually emerged a clearer understanding of the physics underlying some of these phenomena. The existence of transient “leaky” surface waves and long-lived surface plasmon polariton waves (SPPs), originating from the interaction between incident TM-polarized electromagnetic waves and one-dimensional (1-D) subwavelength slits and grooves at a metal-dielectric interface has now been definitely established through numerical simulation and direct measurement [1

1. A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Surface plasmon polaritons on metallic surfaces,” Opt. Express 15, 183–197 (2007). [CrossRef] [PubMed]

, 2

2. G. Gay, O. Alloschery, J. Weiner, H. J. Lezec, C. O’Dwyer, M. Sukharev, and T. Seideman, “Surface quality and surface waves on subwavelength-structured silver films,” Phys. Rev. E 75, 016612-1–4 (2007) and references cited therein. [CrossRef]

]. In periodic array structures the resonance excitation of delocalized, Bloch-state SPPs has been proposed [3

3. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission Resonances on Metallic Gratings with Very Narrow Slits” Phys. Rev. Lett. 83, 2845–2848 (1999). [CrossRef]

] to explain peaks in light transmission observed when the wavelength of light is tuned near integer multiples of the array pitch. Other theoretical studies have emphasized the role of wave interference rather than resonant excitation and have predicted transmission minima [4

4. Q. Cao and Lalanne Ph., “Negative of Surface Plasmons in the Transmission of Metallic Gratins with Very Narrow Slits,” Phys. Rev. Lett. 88, 057403-1–4 (2002). [CrossRef] [PubMed]

, 5

5. P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68, 125404-1–11 (2003). [CrossRef]

] where Bloch-state resonances predict maxima. Recent quantitative measurements [6

6. D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, “Quantitative Determination of Enhanced and Suppressed Transmission through Subwavelength Slit Arrays in Silver Films,” arXiv:0708.1886v2 [physics.optics] 15 Aug 2007.

] in 1-D slit arrays confirm that the transmission indeed reaches a minimum when the array pitch is equal to the SPP wavelength. These results are consistent with the interpretation that destructive interference between the incident wave and the SPP is responsible for the extinction. But if this interpretation is correct, it implies a definite phase relation between the incident wave and the scattered surface wave; specifically that these two waves are π out of phase.

Fig. 1. Orientation of orthogonal coordinate axes is shown along with a light wave TM polarized (H field aligned along the 0y axis). Incident and scattered E-field components are confined to the x-z plane. A subwavelength slit milled at the interface between the metal slab (m) and a dielectric (d) is also aligned along the y- axis.

2. Field relations

This section begins by following the development in the appendix of Raether [12

12. H. RaetherSurface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, Berlin, 1988).

]. In a source-free environment and in rationalized MKS units Faraday’s Law and Ampère’s Law are expressed as

×E=Bt
(1)
×H=Dt
(2)

with the constitutive relations

B=μ0μHandD=ε0εE
(3)

where ε 0, μ 0 are the permittivity and permeability of free space, and ε,μ are the dielectric constant and the “specific magnetic inductive capacity” [11

11. J. S. StrattonElectromagnetic Theory, (McGraw-Hill, New York, 1941).

]. In nonmagnetic media μ is essentially unity. The vector fields D, E are the displacement field and electric field, respectively, and H, B are the the magnetic field and magnetic induction, respectively. In TM polarization, with the magnetic field parallel to the y axis, plane-wave solutions to Eqs 1,2 are written

Hy(x,z,t)=H0exp[i(kxx+kzzωt)]
(4)
Ex(x,z,t)=Exexp[i(kxx+kzzωt)]
(5)
Ez(x,z,t)=Ezexp[i(kxx+kzzωt)]
(6)

where E0=Ex2+Ey2 and E0=Z0H0με with Z0=μ0ε0 , the impedance of free space. At the dielectric-metal boundary, field continuity conditions require,

Hyd=Hym
(7)
Exd=Exm
(8)
εdEzd=εmEzm
(9)

where Hd, Ed denote field components in the dielectric (z>0); Hm, Em of the metal (z<0); and εd,εm are the dielectric constants in the dielectric and metal, respectively.

From Eq. 2 in the dielectric half-space,

Hydz=Dxdt=εdExdt

Hydx=Dzdt=εdEzdt

and substituting from Eqs 4,5,6,

kzdεdωHyd=Exd
(10)
kxdεdωHyd=Ezd
(11)

Similarly in the metal half-space

kzmεmωHym=Exm
(12)
kxmεmωHym=Ezm
(13)

From condition 7 and Eqs 10, 12 at the dielectric-metal interface

HydHym=0
(14)
kzdεdHyd+kzmεmHym=0
(15)

and in order that Eqs 14, 15 be satisfied simultaneously

kzdεd=kzmεm
(16)

which expresses the relation between the wave propagation parameters perpendicular to the surface on each side of the interface. Similarly from Eqs. 7,11,13,

kxd=kxm=kx
(17)

showing that the propagation parameter parallel to the surface is continuous across the boundary. Furthermore, conservation of wave energy on both sides of the boundary requires

(kxd)2+(kzd)2=(ωc)2εd
(18)
(kxm)2+(kzm)2=(ωc)2εm
(19)

Using Eq. 16 to express the ratio kdz/kmz in terms of εd and εm, and dividing Eq. 18 by Eq. 19 to eliminate the kz components results in the well-known expression for kx at the boundary,

kx=(ωc)εdεmεd+εm
(20)

Further use of Eqs. 16, 18, 19 results in

kx=kzdεmεd
(21)

and since the metal dielectric constant is always negative,

kx=ikzdεmεd
(22)

Then from Eqs. 10

10. G. Gay, O. Alloschery, B. Viaris de Lesegno, and J. Weiner, “Surface Wave Generation and Propagation on Metallic Subwavelength Structures Measured by Far-Field Interferometry,” Phys. Rev. Lett. 96, 213901-1–4 (2006). [CrossRef] [PubMed]

and 11

11. J. S. StrattonElectromagnetic Theory, (McGraw-Hill, New York, 1941).

and at the surface

Ex=iεdεmEzd
(23)
Hy=cεd(εd+εm)εmEzd
(24)

Equations 23 and 24 show that at the surface Ex and Hy are in quadrature and in phase with Ez, respectively.

3. Relative phase between the incident wave and the surface wave

It is often said that subwavelength structures “launch” surface waves without specifying the physical origins of the launching process. From a physical optics point of view, the “launch” mechanism is diffraction of the incident wave, but it is doubtful that standard Fraunhofer diffraction theory can be legitimately applied at the subwavelength scale, and even the more general Kirchoff diffraction theory invokes field boundary conditions in the immediate vicinity of the structure that are only approximate [13

13. M. Born and E. WolfPrinciples of Optics, 6th edition, (Pergamon, Oxford, 1980).

]. Furthermore, surface plasmon polaritons always involve a metallic surface with high conductivity, and therefore it is more natural to seek an explanation in terms of charges and currents on the surface and in the immediate vicinity of the structure. It has been recently shown[9

9. G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behavior of surface plasmon polaritons scattered at a subwavelength groove,” Phys. Rev. B 76, 155418-1–8 (2007). [CrossRef]

] that a groove or slit “launch” site can be successfully modeled by replacing the structure with an oscillating charge dipole on the surface. The results of this model are in good agreement with simulations [2

2. G. Gay, O. Alloschery, J. Weiner, H. J. Lezec, C. O’Dwyer, M. Sukharev, and T. Seideman, “Surface quality and surface waves on subwavelength-structured silver films,” Phys. Rev. E 75, 016612-1–4 (2007) and references cited therein. [CrossRef]

,14

14. F. Kalkum, G. Gay, J. Weiner, H. J. Lezec, Y. Xie, and M. Mansuripur, “Surface-wave interferometry on single subwavelength slit-groove structures fabricated on Au films,” Opt. Express 15, 2613–261 (2007). [CrossRef] [PubMed]

] and measurements [15

15. G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave model,” Nature Phys. 264–267 (2006).

] of both the “near zone” transient surface wave and the “far zone” surface plasmon polariton. It therefore appears that under the influence of the incident wave charges accumulate at the slit or groove edges, oscillating at the optical frequency. This charge accumulation itself is the result of currents induced at the surface and within the skin depth of the metal by the standing wave established by the incident and reflected plane waves at the surface. The amplitude of the Hy-field at the surface is twice the amplitude of the incident wave, while the amplitude of the Ex-field is nearly zero. The top panel of Fig. 2 shows a numerical simulation of the intensity of the magnetic field standing wave. The simulation is for a two-slit structure milled in a silver layer with 100 nm widths and incident free-space wavelength λ 0=852 nm. The spatially standing wave Hy(z, t)=2H 0cos(k 0 z)exp(- 0 t) oscillates temporally at the optical frequency ω 0 and induces an emf within the skin depth of the metal according to the Faraday law of induction. The magnetic field intensity map clearly shows that the presence of the slits has little effect on the standing wave pattern, and therefore the structured surface can be considered essentially a plane, highly reflecting mirror. At z=0

Hy(z,t)=2H0exp(iω0t)
(25)

and the magnetic flux is given by

Φ(t)=2μ0H0exp(iω0t)dS=μ02H0(δp)exp(iω0t)
(26)

where S=δp, is the cross sectional area defined by the distance between the slits p and the skin depth δ, normal to the flux lines (see Fig. 2). Then according to Faraday’s Law the emf, oriented along the x direction, is given by

emfx=dΦ(t)dt=2μ0H0ω0(δp)exp[i(ω0tπ2)]
(27)

The linear current density, oriented along x and flowing perpendicular to the slit axes aligned along y, is given by

𝓘x=emfx𝓡=2μ0H0ω0(δp)𝓡exp[i(ω0tπ2)]
(28)

where 𝓡 is the resistivity of the metal in the skin depth. Here 𝓡 is taken to be real since the imaginary term in the wavelength range of interest (near IR) is very small compared to the real term. The linear charge density accumulated along the edge of each slit during a time t is

𝒬(t)=2μ0H0(δp)𝓡0tω0exp[i(ω0tπ2)dt]=2μ0H0(δp)𝓡exp[i(ω0tπ)]
(29)
Fig. 2. Top panel is a numerical simulation of an incident plane wave TM polarized and incident on a two-slit structure from the top. The color map indicates the amplitude of the magnetic field component (red highest amplitude, dark blue lowest amplitude) and shows the standing wave set up by the incident and reflected wave. Note that the standing H-field is maximum at the surface. Middle and lower panels show how the time-oscillating H-field induces current in the skin depth δ of the metal slit structure, resulting in charge accumulation at the slit edges with widths d. Middle panel: first optical half-cycle; lower panel: second optical half-cycle.

Finally the oscillating dipole density 𝒫 along the slit of width d is

𝒫x(t)=𝒬(t)d=2dμ0H0(δp)𝓡exp[i(ω0tπ)]
(30)

Comparison of Eq. 25 with Eq. 30 shows that the oscillating dipole at the slit is in antiphase with the incident field. The phase shift of π occurs through the sum of two steps: first, a phase shift of π/2 from the magnetic induction in the metal skin depth and a second phase shift of π/2 from the capacitive charge buildup at the slits. The induced emf within the skin depth cannot couple directly to the SPP wave because the propagation parameter kxk 0 associated with the magnetic field is always less than k SPP. In contrast the oscillating E-field at the slit is not harmonic but characterized by a band of propagation parameters δk≃2π/d where d is the slit width. In practice the surface is driven by this “broad-band” k-mode excitation and couples efficiently to the k SPP mode.

4. Conclusions

The lines of force of the E-field at the surface must converge at points of charge concentration, and therefore the Ez amplitude at the surface is correlated to and in phase with 𝒫(t). Thus the surface wave component Ez is in antiphase with the incident magnetic field Hy in agreement with the Green’s tensor analysis of Ref. [9

9. G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behavior of surface plasmon polaritons scattered at a subwavelength groove,” Phys. Rev. B 76, 155418-1–8 (2007). [CrossRef]

]. Field continuity at the surface and Eq. 23 shows that Ex is in quadrature with Ez. This result is in accord with the diffraction analysis of Ref. [7]. Similarly Eq. 24 shows that for the wave propagating on the surface H SPP y is in phase with E SPP z. Therefore the incident and surface Hy fields must be in antiphase, again in accord with the simulations of Ref. [8

8. O. T. A. Janssen, H. P. Urbach, and G. W.’t Hooft, “On the phase of plasmons excited by slits in a metal film,” Opt. Express 14, 11823–11832 (2006). [CrossRef] [PubMed]

] and the interpretation of the measurements in Ref. [6

6. D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, “Quantitative Determination of Enhanced and Suppressed Transmission through Subwavelength Slit Arrays in Silver Films,” arXiv:0708.1886v2 [physics.optics] 15 Aug 2007.

]. The present analysis therefore lends support to the interference model proposed in Ref. [6

6. D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, “Quantitative Determination of Enhanced and Suppressed Transmission through Subwavelength Slit Arrays in Silver Films,” arXiv:0708.1886v2 [physics.optics] 15 Aug 2007.

] to explain optical transmission minima through slit arrays when the pitch becomes equal to an integer number of SPP wavelengths. It is worthwhile noting that the π phase shift may not hold for surface waves launched by structures other than slits or grooves. In particular arrays of subwavelength holes, even if they are capacitively charged by induced currents, will drive the surface with cylindrical rather than plane waves as in the case of slits. The form of these waves are essentially half-integer Bessel functions, J n+1/2(r) and their asymptotic form has an extra phase shift of π/2 that would have to be taken into account.

References and links

1.

A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Surface plasmon polaritons on metallic surfaces,” Opt. Express 15, 183–197 (2007). [CrossRef] [PubMed]

2.

G. Gay, O. Alloschery, J. Weiner, H. J. Lezec, C. O’Dwyer, M. Sukharev, and T. Seideman, “Surface quality and surface waves on subwavelength-structured silver films,” Phys. Rev. E 75, 016612-1–4 (2007) and references cited therein. [CrossRef]

3.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission Resonances on Metallic Gratings with Very Narrow Slits” Phys. Rev. Lett. 83, 2845–2848 (1999). [CrossRef]

4.

Q. Cao and Lalanne Ph., “Negative of Surface Plasmons in the Transmission of Metallic Gratins with Very Narrow Slits,” Phys. Rev. Lett. 88, 057403-1–4 (2002). [CrossRef] [PubMed]

5.

P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68, 125404-1–11 (2003). [CrossRef]

6.

D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, “Quantitative Determination of Enhanced and Suppressed Transmission through Subwavelength Slit Arrays in Silver Films,” arXiv:0708.1886v2 [physics.optics] 15 Aug 2007.

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). [CrossRef] [PubMed]

8.

O. T. A. Janssen, H. P. Urbach, and G. W.’t Hooft, “On the phase of plasmons excited by slits in a metal film,” Opt. Express 14, 11823–11832 (2006). [CrossRef] [PubMed]

9.

G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behavior of surface plasmon polaritons scattered at a subwavelength groove,” Phys. Rev. B 76, 155418-1–8 (2007). [CrossRef]

10.

G. Gay, O. Alloschery, B. Viaris de Lesegno, and J. Weiner, “Surface Wave Generation and Propagation on Metallic Subwavelength Structures Measured by Far-Field Interferometry,” Phys. Rev. Lett. 96, 213901-1–4 (2006). [CrossRef] [PubMed]

11.

J. S. StrattonElectromagnetic Theory, (McGraw-Hill, New York, 1941).

12.

H. RaetherSurface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, Berlin, 1988).

13.

M. Born and E. WolfPrinciples of Optics, 6th edition, (Pergamon, Oxford, 1980).

14.

F. Kalkum, G. Gay, J. Weiner, H. J. Lezec, Y. Xie, and M. Mansuripur, “Surface-wave interferometry on single subwavelength slit-groove structures fabricated on Au films,” Opt. Express 15, 2613–261 (2007). [CrossRef] [PubMed]

15.

G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave model,” Nature Phys. 264–267 (2006).

OCIS Codes
(240.5420) Optics at surfaces : Polaritons
(240.6680) Optics at surfaces : Surface plasmons
(240.6690) Optics at surfaces : Surface waves

ToC Category:
Optics at Surfaces

History
Original Manuscript: December 6, 2007
Revised Manuscript: January 8, 2008
Manuscript Accepted: January 9, 2008
Published: January 11, 2008

Citation
J. Weiner, "Phase shifts and interference in surface plasmon polariton waves," Opt. Express 16, 950-956 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-950


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References

  1. A. R. Zakharian, J. V. Moloney, and M. Mansuripur, "Surface plasmon polaritons on metallic surfaces," Opt. Express 15, 183-197 (2007) and references cited therein. [CrossRef] [PubMed]
  2. G. Gay, O. Alloschery, J. Weiner, H. J. Lezec, C. O’Dwyer, M. Sukharev, and T. Seideman, "Surface quality and surface waves on subwavelength-structured silver films," Phys. Rev. E 75, 016612-1-4 (2007) and references cited therein. [CrossRef]
  3. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry," Transmission resonances on metallic gratings with very narrow slits" Phys. Rev. Lett. 83, 2845-2848 (1999). [CrossRef]
  4. Q. Cao and Ph. Lalanne, "Negative of surface plasmons in the transmission of metallic gratins with very narrow slits," Phys. Rev. Lett. 88, 057403-1-4 (2002). [CrossRef] [PubMed]
  5. P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, "Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures," Phys. Rev. B 68, 125404-1-11 (2003). [CrossRef]
  6. D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, "Quantitative determination of enhanced and suppressed transmission through subwavelength slit arrays in silver films," arXiv:0708.1886v2 [Physics.Optics] 15 Aug 2007.
  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). [CrossRef] [PubMed]
  8. O. T. A. Janssen, H. P. Urbach, and G. W. ’t Hooft, "On the phase of plasmons excited by slits in a metal film," Opt. Express 14, 11823-11832 (2006). [CrossRef] [PubMed]
  9. G. Leveque, O. J. F. Martin, and J. Weiner, "Transient behavior of surface plasmon polaritons scattered at a subwavelength groove," Phys. Rev. B 76, 155418-1-8 (2007). [CrossRef]
  10. G. Gay, O. Alloschery, B. Viaris de Lesegno, and J. Weiner, "Surface wave generation and propagation on metallic subwavelength structures measured by Far-Field Interferometry," Phys. Rev. Lett. 96, 213901-1-4 (2006). [CrossRef] [PubMed]
  11. J. S. Stratton, Electromagnetic Theory, (McGraw-Hill, New York, 1941).
  12. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, Berlin, 1988).
  13. M. Born and E. Wolf, Principles of Optics, 6th edition, (Pergamon, Oxford, 1980).
  14. F. Kalkum, G. Gay, J. Weiner, H. J. Lezec, Y. Xie, and M. Mansuripur, "Surface-wave interferometry on single subwavelength slit-groove structures fabricated on Au films," Opt. Express 15, 2613-261 (2007). [CrossRef] [PubMed]
  15. G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 264-267 (2006).

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