## Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited

Optics Express, Vol. 16, Issue 12, pp. 9073-9086 (2008)

http://dx.doi.org/10.1364/OE.16.009073

Acrobat PDF (395 KB)

### Abstract

The asymptotic closed-form solution to the fundamental diffraction problem of a linear horizontal Hertzian dipole radiating over the metallo-dielectric interface is provided. For observation points just above the interface, we confirm that the total surface near-field is the sum of two components: a long-range surface plasmon polariton and a short-range radiative cylindrical wave. The relative phases, amplitudes and damping functions of each component are quantitatively elucidated through simple analytic expressions for the entire range of propagation: near and asymptotic. Validation of the analytic solution is performed by comparing the predictions of a dipolar model with recently published data.

© 2008 Optical Society of America

## 1. Introduction

6. P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Physics **2**, 551–556 (2006). [CrossRef]

## 2. Derivation of the closed-form solution

8. 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). [CrossRef] [PubMed]

9. B. Ung and Y. Sheng, “Interference of surface waves in a metallic nanoslit,” Opt. Express **15**, 1182–1190 (2007). [CrossRef] [PubMed]

10. 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 (2007). [CrossRef]

11. L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano. Letts. **5**, 1399–1402 (2005). [CrossRef]

*w*of the given nano-object is smaller than a half-wavelength,

*w*<

*λ*/2, and if observation points are located at a distance

*x*from the source larger than

*x*>

*λ*/

*π*. The remaining discussion in this Section follows the contemporary mathematical treatment derived by R. E. Collin for the distinct case of a 3D vertical point dipole radiating over the earth’s surface [12

12. R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: some early and late 20th-century controversies,” IEEE Antennas Propag. Mag. **46**, 64–79 (2004). [CrossRef]

### 2.1 Expression of the Sommerfeld integral for a linear horizontal Hertzian dipole

*x*-axis and located in the half-space

*z*≥0 at a height

*h*from the interface separating two semi-infinite nonmagnetic and isotropic dielectrics (Fig. 1). The infinitesimal line dipole is described by its surface current density

*J*=

_{x}*δ*(

*x*)·

*δ*(

*z*-

*h*) normalized to unit electric moment. We will not cover here the details of the straightforward though lengthy derivation of Eq. (1), which can be found in textbooks [13

13. R. W. P. King, M. Owens, and T. T. Wu, *Lateral Electromagnetic Waves* (Springer-Verlag, New York, 1992). [CrossRef]

*E*,

_{x}*E*and

_{z}*H*fields (TM-polarization) in Maxwell’s curl equations then taking the inverse Fourier transform of the fields and choosing the appropriate Green’s function, one derives the Sommerfeld-type integral for the normal electric field in region 1 (

_{y}*z*≥0):

### 2.2 Asymptotic solution to the integral via the modified method of steepest descents

*E*

_{1z}=

*K*(

*I*

_{1}+

*I*

_{2}), with

*K*=

*ωμ*

_{0}/2

*π*, and

*I*

_{1}and

*I*

_{2}respectively corresponding to the first and second term inside the brackets. We first consider the integral

*I*

_{1}and impose

*γ*

_{1}and

*γ*

_{2}to have positive imaginary parts in order for the field to be bounded at infinity. In the corresponding first quadrant of the complex

*k*-plane, we discard the branch cut running from the point

*k*=

*k*

_{2}since its contribution - characterized by an attenuation factor exp(-Im{

*k*

_{2}}·

*x*) that drops to nearly zero within a quarter-wavelength distance

*x*- is negligible for the optical frequencies of interest compared to that of the branch cut running from

*k*=

*k*

_{1}. There is also a pole singularity in the denominator of

*I*

_{1}located at

*ε*=

_{p}*ε*

_{1}

*ε*

_{2}/(

*ε*

_{1}+

*ε*

_{2}) defines the SPP’s “effective permittivity”. The proper pole (+

*k*) lies very close to the branch point

_{p}*k*

_{1}and we will see later on that this aspect entails particular considerations. We make the simplification

*ε*

_{1}=1 such that

*k*

_{1}=

*k*

_{0}and perform the successive transformations

*k*=

*k*

_{0}sin

*α*,

*x*=

*R*

_{2}sin

*θ*and (

*z*+

*h*)=

*R*

_{2}cos

*θ*, where

*R*

_{2}is the distance from the mirror image of the dipole in region 2 to the observation point. These procedures enable one to express

*I*

_{1}as the integral over an angular spectrum of plane waves:

*α*=

*σ*+

*iη*defines the angle between the direction of propagation and the

*x*-axis. The saddle-point is found by setting the derivative of the argument in exponential equal to zero, thus yielding

*α*=

*θ*where

*θ*is real. The location of the pole in the

*α*-plane is given by

*k*=

_{p}*k*

_{0}sin

*α*. As a numerical example, at the excitation wavelength λ=852

_{p}*nm*and

*ε*

_{2}=-33.22+1.17

*i*(silver metal) we have

*α*=

_{p}*π*-Arcsin (

*k*/

_{p}*k*

_{0})=1.574-

*i*0.175 where Arcsin

*C*is then deformed into the steepest-descent contour (

*SDC*) whose path, cos(

*σ*-

*θ*)cosh

*η*=1, is shifted to pass through the saddle-point at

*θ*=

*π*/2 where the highest accuracy is assigned to observation points along the surface. Since the pole

*α*has been crossed by the path in the process, and is positioned below the

_{p}*SDC*and very close to the saddle-point (see Fig. 2), the pole is captured and its contribution must be accounted for with a residue [12

12. R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: some early and late 20th-century controversies,” IEEE Antennas Propag. Mag. **46**, 64–79 (2004). [CrossRef]

14. P. C. Clemmow, “A note on the diffraction of a cylindrical wave by a perfectly conducting half-plane,” Q. J. Mech. Appl. Math. **3**, 377–384 (1950). [CrossRef]

*τ*=2

*e*

^{iπ/4}sin((

*α*-

*θ*)2), which yields the relations

*I*

_{1}is rewritten:

*f*(

*τ*)=

*g*(

*τ*)/

*h*(

*τ*) and:

*θ*=

*π*/2, the classical steepest-descent method cannot be applied directly. To circumvent this problem, the procedure that we will employ here is a modified steepest-descent technique described in [12

12. R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: some early and late 20th-century controversies,” IEEE Antennas Propag. Mag. **46**, 64–79 (2004). [CrossRef]

*f*(

_{P}*τ*) and

*f*(

_{A}*τ*), by subtracting out the pole term from the main integrand:

*f*(

_{P}*τ*) is the pole term with residue constant

*A*. The remaining

*f*(

_{A}*τ*) is holomorphic up to the next singular point at

*τ*

^{2}=4

*i*which arises in the term

*dα*/

*dτ*present in

*g*(

*τ*). The function

*f*(

_{A}*τ*) is expanded in a Taylor series around the saddle-point

*τ*=0 (i.e.

*α*=

*θ*) where we keep only the first term and neglect the remaining higher even-order correction terms to obtain

*f*(

_{A}*τ*)≈

*f*(0)=

_{A}*f*(0)+

*A*/

*τ*. By evaluating the remainder’s upper bound, we estimated the typical maximum relative error associated with the truncation of the Taylor series at the first term to be below 12%. With the help of the relations

_{p}^{2}

*α*=(1+

_{p}*ε*)

_{2}^{-1}, we find the value of the constant

*A*:

*I*

_{1}in two parts,

*I*=

_{1}*I*+

_{P}*I*, where

_{A}*I*and

_{P}*I*respectively denote the pole and analytical contributions from

_{A}*f*(

_{P}*τ*) and

*f*(

_{A}*τ*). We first consider the analytical part:

*f*(0) has no dependence in

_{A}*τ*, it can be removed from the integrand. Upon evaluating the remaining kernel,

*P*(

*τ*) since it can be defined by different functions depending on whether the pole is located in the lower-half (Im{

*τ*}<0) or upper-half (Im{

_{p}*τ*}>0)

_{p}*τ*-plane. We can find a solution valid over the entire

*τ*-plane by first evaluating

*P*(

*τ*) for Im{

*τ*}>0 and then assuming Im{

_{p}*τ*}<0 in the resulting integral expression. In which case we obtain

_{p}*t*=-

*iχτ*, such that:

_{p}*P*(

*τ*), we refer the reader to the Appendix of [12

**46**, 64–79 (2004). [CrossRef]

*I*

_{2}, is an entire function; therefore the standard method of steepest descents can be applied to it. With the identity

*I*and

_{I}*I*, respectively designating the image and direct dipole contributions. Each part is then solved following the same procedure previously described for

_{D}*I*with the exception that the transformations

_{A}*x*=

*R*

_{1}sin

*θ*and (

*z*-

*h*)=

*R*

_{1}cos

*θ*are substituted in the single case of

*I*. The solutions for

_{D}*I*and

_{I}*I*then represent free-space cylindrical waves emanating from their respective image and direct dipole origins:

_{D}*E*-field is written

*E*

_{1z}=

*K*(

*I*+

_{P}*I*+

_{A}*I*+

_{I}*I*). Upon adding the terms

_{D}*I*and

_{A}*I*together and performing some algebraic manipulations, one obtains the following general closed-form expression:

_{I}*U*(

*R*

_{2},

*θ*), found in Eq. (18) and determined in Eq. (12), defines a complex envelope multiplying the SPP phasor.

## 3. Analysis of the surface near field

### 3.1 Reduction of the general solution for observation points along the surface

*θ*=

*π*2, we get

*x*=

*R*

_{2}sin

*θ*=

*R*

_{2}and the geometrical-optics field components [Eqs. (15)–(16)] vanish while the diffracted field components [Eqs. (17)–(18)] remain. The total near field along the interface then describes a composite surface wave (SW) created by two co-propagating vectors: a surface plasmon polariton (SPP) evanescent wave and a “boundary wave” (BW) having essentially a free-space cylindrical nature. The boundary wave lies in the geometrical shadow so as to compensate for the discontinuity in the geometrical-optics field across the planar interface [15]. We refrain from using the denomination “creeping wave”, which was previously chosen in [6

6. P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Physics **2**, 551–556 (2006). [CrossRef]

*k*=(

*k*-

_{p}*k*

_{0}) defines the difference between the evanescent and free-space wavevectors. We normalize the total surface wave field, which we denote

*E*, by taking out the factor

_{sw}*ωμ*

_{0}

*Ae*from Eqs. (17)–(18) and moving the √

^{iπ}*x*root from the denominator to the numerator in order to eliminate any sign ambiguity when

*x*<0:

*x*direction. An identical SW is excited in the -

*x*direction whose field is the complex conjugate:

*E*

^{-}

_{sw}(

*x*)=

*Ē*

^{+}

_{sw}(

*x*). We note that the SPP and BW are initially phase-shifted by

*π*/4. One may also notice that the expression of the BW involves the asymptotic form of the first-order Hankel function of the first kind,

*k*

_{0}

*x*>2. This outcome is not incidental because the steepest-descent method - which is asymptotically exact - was used to obtain the solution. Thus in principle, the BW could be written in “exact” form with the Hankel function,

*k*

_{0}

*x*>2.

### 3.2 Fresnel diffraction effects and the asymptotic propagation regime

*U*(

*x*) owing to Fresnel diffraction effects [Eq. (12) evaluated at

*θ*=

*π*/2]:

*erfc*term describes the fringe pattern generated by the process of coupling incident homogeneous light into the evanescent mode via scattering. As a note, Eq. (20) can equivalently be expressed using the conventional complex Fresnel integral,

*F*(

*ν*)=∫

^{ν}

_{0}exp(

*iπy*

^{2}/2)·

*dy*, instead of the

*erfc*:

*U*(

*x*)} evolves from

*U*(0)=0.5 at the origin before reaching the first peak at some distance

*x*

_{1}and then oscillates onward with a large distinctive period Δ

*L*=2

*π*/|Δ

*k*| to lower amplitudes before asymptotically growing towards ever higher amplitudes due to the positive imaginary component inside the argument of

*erfc*.

### 3.3 Characterization of the field at the origin and in the near-zone propagation regime

*x*=λ/8, yielding:

*x*≥λ/8 the real part is well-behaved; while for

*x*<λ/8 it rapidly diverges towards negative values to become singular at the limit

*x*→0

^{+}. By contrast, the SPP mode is well defined at

*x*=0:

*E*

^{+}

_{spp}(0)=

*U*(0)·

*e*

^{-iπ/2}. These previous remarks suggest that the SW field at the origin can be reasonably approximated by:

*π*|Δ

*k*|·

*x*)

^{1/2}in accordance with the 1/

*x*

^{1/2}damping predicted in [6

6. P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Physics **2**, 551–556 (2006). [CrossRef]

*U*(

*x*)}·exp(-Im{

*k*}

_{p}*x*). The “critical distance” where the moduli of the SPP and BW coincide is approximately located at

*x*=4

_{c}*π*/(67|Δ

*k*|). For the conditions λ=852

*nm*and

*ε*

_{2}=-33.22+1.17

*i*, the predicted value is

*x*=1.64

_{c}*µm*. For propagation lengths below the critical distance (0<

*x*<

*x*) the BW provides the dominant contribution whereas the long-range SPP is the main vector thereafter (Fig. 4). The 1/

_{c}*e*intensity decay lengths of the SPP and BW are respectively

*L*=1/(2·Im{

_{spp}*k*}) and

_{p}*L*=

_{bw}*e*/(4

*π*|Δ

*k*|). For the same previous conditions, the BW’s characteristic decay length

*L*=1.9

_{bw}*µm*is much shorter compared to the SPP’s

*L*=122

_{spp}*µm*. As a consequence, the effective value (

*k*=2

_{sw}*π*/

*λ*) of the composite SW’s wavevector (or effective index:

_{sw}*n*=

_{sw}*k*/

_{sw}*k*

_{0}) is significantly affected by the rapid decay of the BW inside the near-zone regime (0<

*x*<2

*L*) as shown in Fig. 4. The behavior of the near field depicted by the analytical solution in the source origin’s vicinity, brings a clear and cohesive physical explanation to the reported accounts of transient phenomena in the first few wavelengths of propagation [3

_{bw}3. 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,” Nat. Physics **2**, 262–267 (2006). [CrossRef]

4. G. Gay, O. Alloschery, B. Viaris 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**, 213901 (2006). [CrossRef] [PubMed]

7. L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julié, V. Mathet, and M. Mortier, “Near-field analysis of surface waves launched at nanoslit apertures,” Phys. Rev. Lett. **98**, 153902 (2007). [CrossRef] [PubMed]

10. 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 (2007). [CrossRef]

## 4. Dipolar model of the near-field interactions between nano-objects

*et al.*, [10

10. 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 (2007). [CrossRef]

### 4.1 Groove-slit transmission

*et al.*, [3

3. 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,” Nat. Physics **2**, 262–267 (2006). [CrossRef]

**2**, 551–556 (2006). [CrossRef]

*d*.

*x*-polarized plane wave of amplitude

*E*is scattered on both 100

_{i}*nm*-wide groove and slit as HEDs. The fraction of incident light that is coupled by the groove into a

*z*-polarized +

*x*-directed SW is determined by the multiplication factor

*βE*

_{sw}^{+}(

*d*) where

*β*is the amplitude-coupling coefficient and

*E*

_{sw}^{+}(

*d*) describes the field propagation of the excited SW. The diffraction-launched SW then impinges on the adjacent nano-object (i.e. slit) where it is partially reflected and re-radiated. The re-radiated

*x*-polarized component

*E*that is coupled into the slit is defined by the factor

_{x}^{sw}*βE*

_{sw}^{+}(0), where backconversion reciprocity (SW-to-radiation and vice-versa) is assumed for the coupling coefficient

*β*and the factor

*E*

_{sw}^{+}(0) accounts for the generation of a new SW along the slit’s left-wall. Indeed, as demonstrated in [9

9. B. Ung and Y. Sheng, “Interference of surface waves in a metallic nanoslit,” Opt. Express **15**, 1182–1190 (2007). [CrossRef] [PubMed]

*π*/2 phase-shift in magnitude. This intrinsic

*π*/2 phase-shift between the SW generated at the groove and the directly incident light is consistent with earlier experimental [3

3. 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,” Nat. Physics **2**, 262–267 (2006). [CrossRef]

4. G. Gay, O. Alloschery, B. Viaris 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**, 213901 (2006). [CrossRef] [PubMed]

**76**, 155418 (2007). [CrossRef]

*x*-polarized light normally incident on the slit similarly generates two SWs: one in the forward and another in the backward

*x*-direction. As described in [9

9. B. Ung and Y. Sheng, “Interference of surface waves in a metallic nanoslit,” Opt. Express **15**, 1182–1190 (2007). [CrossRef] [PubMed]

*x*-polarized component

*E*that interferes with the transmitted component

_{x}^{sw}*E*

_{0}in the slit. The complete process of interference at the slit, as described by this simple model, is expressed by

*I*=|

*E*+

_{x}^{sw}*E*

_{0}|

^{2}where

*E*=

_{x}^{sw}*βE*

_{sw}^{+}(0)·

*βE*·

_{i}*E*

_{sw}^{+}(

*d*) and

*E*

_{0}=

*t*

_{0}

*E*respectively denote the

_{i}*x*-polarized transmitted components relating to the left incident SW and the normal incident plane wave. The total transmitted intensity

*I*is a function of groove-slit distance

*d*, and is normalized with that without adjacent groove (

*I*

_{0}=|

*E*

_{0}|

^{2}) such as performed in [3

**2**, 262–267 (2006). [CrossRef]

16. M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bientsman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. **2**, 07022 (2007). [CrossRef]

**2**, 262–267 (2006). [CrossRef]

16. M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bientsman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. **2**, 07022 (2007). [CrossRef]

*nm*,

*ε*

_{1}=1, and

*ε*

_{2}=-33.22+1.17

*i*. The approximate values of the scattering and modal transmission coefficients, respectively

*β*=0.357 and

*t*

_{0}=1.40, are both calculated using the semi-analytical SPP generation model of Lalanne,

*et al.*, [17

17. P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett. **95**, 263902 (2005). [CrossRef]

### 4.2 Double-slit near field

*d*=10.44

*µm*and illuminated with λ=974.3

*nm*normal incident TM-polarized light, was recently measured through a scanning near-field optical microscope (SNOM) by Aigouy and co-workers [7

7. L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julié, V. Mathet, and M. Mortier, “Near-field analysis of surface waves launched at nanoslit apertures,” Phys. Rev. Lett. **98**, 153902 (2007). [CrossRef] [PubMed]

*E*-field at the interface, the patterns recorded from fluorescence emission are expected to scale with |

*E*|

_{z}^{4}. Modeling once more each nano-object (i.e. the slits) as a linear HED, the resulting near-field intensity pattern arises from the interference of two counter-propagating SWs,

*E*(

_{z}*x*)=

*E*

^{+}

_{sw}(

*x*+

*d*)+

*E*

^{-}

_{sw}(

*x*-

*d*), where the origin of the

*x*-axis is located at the half slit-to-slit separation distance. In Fig. 7 the near field between the slits,

*a*|

*E*(

_{z}*x*)|

^{4}+

*b*, is plotted in the vertical axis, where

*b*=0.33 is the background illumination offset taken from the original data and

*a*=0.031 is the best-fit gain factor. The decay, phase and pattern periodicity predicted by the analytical model, with

*ε*

_{1}=1 and

*ε*

_{2}=-44.05+3.24

*i*, closely match the experimental data. Clearly, the dipolar model is again fully consistent with the near-field structure depicted in real-world experiments.

**2**, 262–267 (2006). [CrossRef]

7. L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julié, V. Mathet, and M. Mortier, “Near-field analysis of surface waves launched at nanoslit apertures,” Phys. Rev. Lett. **98**, 153902 (2007). [CrossRef] [PubMed]

*x*<2

*L*). The corresponding results yielded for either investigation by the analytical dipolar model indicate that the asymptotic solution is fairly accurate inside this transient near-zone as expected from the accuracy range

_{bw}*k*

_{0}

*x*>2, or stated alternatively,

*x*>λ/

*π*(see Subsection 3.1).

## 5. Conclusion

*x*>λ/

*π*from the origin - demonstrates that the total surface near field is composed of two distinct components as previously evidenced in [6

**2**, 551–556 (2006). [CrossRef]

## Acknowledgments

## References and links

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4. | G. Gay, O. Alloschery, B. Viaris 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. |

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7. | L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julié, V. Mathet, and M. Mortier, “Near-field analysis of surface waves launched at nanoslit apertures,” Phys. Rev. Lett. |

8. | Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through slit apertures in metallic films,” Opt. Express |

9. | B. Ung and Y. Sheng, “Interference of surface waves in a metallic nanoslit,” Opt. Express |

10. | G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behavior of surface plasmon polaritons scattered at a subwavelength groove,” Phys. Rev. B. |

11. | L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano. Letts. |

12. | R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: some early and late 20th-century controversies,” IEEE Antennas Propag. Mag. |

13. | R. W. P. King, M. Owens, and T. T. Wu, |

14. | P. C. Clemmow, “A note on the diffraction of a cylindrical wave by a perfectly conducting half-plane,” Q. J. Mech. Appl. Math. |

15. | M. Born and E. Wolf, |

16. | M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bientsman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. |

17. | P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(240.6680) Optics at surfaces : Surface plasmons

(240.6690) Optics at surfaces : Surface waves

(260.3910) Physical optics : Metal optics

(310.2790) Thin films : Guided waves

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: April 8, 2008

Revised Manuscript: May 27, 2008

Manuscript Accepted: May 30, 2008

Published: June 4, 2008

**Citation**

Bora Ung and Yunlong Sheng, "Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited," Opt. Express **16**, 9073-9086 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-9073

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### References

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- R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic Waves (Springer-Verlag, New York, 1992). [CrossRef]
- P. C. Clemmow, "A note on the diffraction of a cylindrical wave by a perfectly conducting half-plane," Q. J. Mech. Appl. Math. 3, 377-384 (1950). [CrossRef]
- M. Born and E. Wolf, Principles of optics 7th ed, (Cambridge University Press, Oxford, 1999), Chap. 8.9.
- M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bientsman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, "Numerical analysis of a slit-groove diffraction problem," J. Eur. Opt. Soc. 2, 07022 (2007). [CrossRef]
- P. Lalanne, J. P. Hugonin, and J. C. Rodier, "Theory of surface plasmon generation at nanoslit apertures," Phys. Rev. Lett. 95, 263902 (2005). [CrossRef]

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