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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 7 — Apr. 3, 2006
  • pp: 2909–2920
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Optical response of periodically modulated nanostructures near the interband transition threshold of noble metals

V. Halté, A. Benabbas, and J.-Y. Bigot  »View Author Affiliations


Optics Express, Vol. 14, Issue 7, pp. 2909-2920 (2006)
http://dx.doi.org/10.1364/OE.14.002909


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Abstract

We investigate the influence of the core d-electrons on the spectral optical response of arrays of sub-wavelength holes near the transition from the d-band to the Fermi level of noble metals (d→EF). Our model shows that, due to the dispersion of the metal dielectric function near d→EF, the first order peaks in the enhanced spectral transmission shift nonlinearly as a function of the period of the nanostructure. In addition, we outline in that spectral region an apparent large resonance which does not depend on the geometrical parameters of the nanostructure. It is shown to correspond to the transparency window resulting from the spectral superposition of the large absorption associated to the core d-electrons and high reflectivity due to the conduction electrons. The analysis is performed for gold, copper and silver nanostructures.

© 2006 Optical Society of America

We have studied the experimental transmission of several gold nanostructures with various periods. The samples are made on a 250 nm gold film, deposited on an Al2O3 substrate, milled with a Focused Ion Beam technique. The resulting nanostructures consist of squared grids of holes with a lattice constant a varying in the range 260-300 nm. For each array, the diameter d of the holes has been chosen such that the filling factor (f=πd24a2) is ~27%. Typical Scanning Electronic Microscopy images have been reported previously [24

24. A. Krishnan, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun 200, 1–7 (2001). [CrossRef]

]. The linear transmission spectra are measured by exciting the sample with a white light source in a quasi collimated configuration (Numerical Aperture = 0.1) and by imaging the nanostructures on the entrance slit of a spectrometer. The output numerical aperture is 0.3. These spectra are recorded by a liquid nitrogen cooled Coupled Charged Device camera. Figure 1 represents the linear zero order transmission at normal incidence, normalized to unity, of arrays with different periods. All spectra exhibit two main resonances which behave differently as a function of the period. The longer wavelength resonance (located at 647 nm for a=260 nm) shifts to longer wavelengths when the period of the array increases. Such spectral behavior has been shown to be characteristic of the surface plasmon mode associated to the metal/substrate interface [1–2

1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolf, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London) 391, 667–669 (1998). [CrossRef]

, 4

4. L. Salomon, F. Grillot, A. V. Zayats, and F. de Fornel, “Near-field distribution of optical transmission of periodic subwavelength holes in a metal film,” Phys. Rev. Lett. 86, 1110–1113 (2001). [CrossRef] [PubMed]

, 7–8

7. J. Gomez Rivas, C. Schotsch, P. Haring Bolivar, and H. Kurz, “Enhanced transmission of THz radiation through subwavelength hole,” Phys. Rev. B 68, 201306 (2003). [CrossRef]

, 24

24. A. Krishnan, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun 200, 1–7 (2001). [CrossRef]

]. Let us recall that the spectral position of the resonance is approximately given by [2

2. H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998). [CrossRef]

]: λmax=ai2+j2εI,IIIεMεI,III+εM, where (i, j) are integer mode indexes associated to the array. εI, III are respectively the dielectric functions of either the incident medium (air) or the substrate and εM is the one of the metal. In the present case, this long wavelength resonance corresponds to the (1, 0)SS peak on the substrate side. In contrast, the transmission has a more complex resonant behavior in the spectral range 500-600 nm. For the periods a=260 nm and a=280 nm, a fixed resonance at 515 nm is present. It apparently shifts to 530 nm when a increases. In fact, as will be shown in the following, this resonance is composed of the (1, 0)AS mode on the air side and a spectrally fixed resonance at 515 nm, associated to the transparency region of gold. When the period still increases [not shown on Fig. 1(a)], a shoulder corresponding to the (1, 1)SS mode on the substrate side occurs [2

2. H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998). [CrossRef]

]. For example it occurs at 580 nm for a period a=320 nm.

Fig. 1. Linear transmission spectra normalized to unity, of nanostructures with different periods a designed in a 250 nm thick gold film evaporated on an Al2O3 substrate.

To explain this complex behavior of the transmission in the 500-600 nm spectral region, we have modeled our nanostructures using an analytical model [25

25. A. Benabbas, V. Halté, and J.-Y. Bigot, “Analytical model of the optical response of periodically structured metallic films,” Opt. Express 13, 8730–8745(2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-22-8730. [CrossRef] [PubMed]

]. It consists in an approximate solution of Maxwell’s equations for a p-polarized electromagnetic wave illuminating a one-dimensional nanostructure [26–28

26. S. A. Damanyan and A. V. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: an analytical study,” Phys. Rev. B 67, 035424 (2003); [CrossRef]

]. The nanostructure consists in a periodically modulated metallic film of thickness h (region II). The period of the modulation is a. The nanostructure is surrounded by two media: air (region I with a dielectric function εI) and substrate (region III with a dielectric function εIII). In this method, based on the Rigorous Wave Coupled Analysis (RCWA), the field in the nanostructured region is developed in Bloch waves modes. The dielectric function is expressed in Fourier series as:

εII(x)=nεnexp(ingx)
(1)

where n is an integer, g = 2π/a is the period of the reciprocal lattice. The Fourier harmonics are given by:

{ε0=fεh+(1f)εm,forn=0ε0=(εhεm)sin(nπf)πnforn0
(2)

where εh is the dielectric function of the holes, εm the dielectric function of the metal and f the filling factor defined earlier.

To extract analytical expressions of the diffraction efficiencies, the developments have been truncated at first order. This allows us calculating the zero order (T 0, R 0) and first order (T 1, R 1) transmission and reflection coefficients, as well as the absorption defined as A = 1 - R 0 - 2R 1 - T 0 - 2T 1. The coefficients Tn and Rn are given by [25

25. A. Benabbas, V. Halté, and J.-Y. Bigot, “Analytical model of the optical response of periodically structured metallic films,” Opt. Express 13, 8730–8745(2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-22-8730. [CrossRef] [PubMed]

]:

Rn=rnrn*Re(klznk0εIcosθ)Tn=tntn*Re(εIkIIIznk0εIIIcosθ)
(3)

where tn and rn are the transmitted and reflected amplitudes of the diffracted fields and θ the angle of incidence of the light on the nanostructure. Let us stress that the one dimensional truncated RCW model does not allow retrieving the exact spectral position as well as the amplitudes of the nanostructure resonances. In addition, it is best suited for low filling factors [26

26. S. A. Damanyan and A. V. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: an analytical study,” Phys. Rev. B 67, 035424 (2003); [CrossRef]

]. Nevertheless, it has been shown to catch the main physical features involved in the transmission process of two-dimensional periodic nanostructures [28

28. A. M. Dykhne, A. K. Sarychev, and V. M. Shalaev, “Resonant transmittance through metal films with fabricated and light-induced modulation,” Phys. Rev. B 67, 195402 (2003). [CrossRef]

]. A more accurate description consists in solving numerically the 2D Maxwell equations with the proper boundary conditions [29

29. A. V. Kats and A. Yu. Nikitin, “Nonzeroth-Order anomalous optical transparency in modulated metal films owing to excitation of surface plasmon polaritons: an analytic approach,” JETP Lett. 79, 625 (2004). [CrossRef]

].

Let us focus now on the effect of the metal dielectric function. Figure 2 shows different complex dielectric functions of gold that we used to model the static response of our nanostructures. The black lines correspond to the dielectric function of gold εexp (real part (solid line); imaginary part (dashed line)) extracted from measurements of the linear reflectivity and transmission of a 29 nm gold film. In this procedure, we use a Fabry-Pérot-like model. This experimental dielectric function is close, although not perfectly superimposed, to the one tabulated (green circles) in ref [30

30. A. V. Kats and A. Yu. Nikitin, “Analytical treatment of anomalous transparency of a modulated metal film due to surface plasmon-polariton excitation,” Phys. Rev. B 70, 235412 (2004). [CrossRef]

]. The red triangles correspond to the real and imaginary part of the theoretical Lindhard dielectric function of gold εtheo, obtained with the self-consistent field theory [23

23. J. Lindhard, Kgl. Danske Videnskab, and Selskab, Mat.-fys. Medd. 28, 8 (1954).

, 31

31. E. Popov, M. Nevière, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000). [CrossRef]

]. It contains the two contributions associated to the conduction electrons (Drude-like dielectric function ε Drude) and to the d-band electronic states εinter from which optical interband transitions to the Fermi level may occur (d → EF threshold = 2.35 eV = 527 nm).

εDrude(ω)=14πNe2ħ2m1ħω(ħω+iγf)=1(ħωp)2ħω(ħω+iγf)
(4)
εinter(ω)=e2ħ2m2π2l,l'll'd3kPll'2(ħωl'l)2[fo(Ekl)fo(Ekl')]ħωl'lħω+γee
(5)

where N, e, m and fo are the density, the charge, the mass and the distribution function of the electrons in the metal. ħωp is the energy of the volume plasmon. γee and γf stand for the interband and intraband damping processes. Pll’ and ħωl’l are the matrix element and the energy of the transition between the bands l and l’. In the spectral region considered here, the interband dielectric function of gold is well described by a two-band model [32

32. D. W. Lynch and W. R. Hunter, in Handbook of optical constants of solids, E. D. Palik, ed. (Academic press, New York, 1991).

] corresponding to the d electrons [l=d in Eq. (5)] and the p conduction electrons [l’=p in Eq. (5)]. We have used the band parameters of B. R. Cooper et al. [33

33. H. Ehrenreich and M. H. Cohen, “Self-consistent field approach to the many-electron problem,” Phys. Rev. 115, 786 (1959). [CrossRef]

]. In practice, the imaginary part of the interband dielectric function is first computed and the real part is obtained by the Kramers-Kronig relations. Since our integration is performed in a limited frequency range, a constant has been added to Re[ε theo] so that it fits to Re[ε exp]. The inset of Fig. 2 shows the detailed dielectric functions near the interband threshold. In the following, we used ε theo to model the linear zero order transmission of the gold nanostructures.

Fig. 2. Spectral dielectric function of gold: tabulated values [30] (green circles) ε exp, determined experimentally on a 29 nm gold film (black lines) and ε theo calculated with the Lindhard formalism (red triangles). The closed symbols or solid line display the real part of these dielectric functions while the open ones or dashed line represent their corresponding imaginary parts.

Fig. 3. (a) Experimental linear transmission of a gold nanostructure with the following parameters: a = 300 nm, d = 177 nm, h = 250 nm, Al2O3 substrate. (b) Calculated zero order transmission spectra using the Lindhard dielectric function εtheo and the experimental one εexp.

In Fig. 4(a), we have represented the spectral positions of the maxima of transmission (closed symbols) and absorption (open symbols), calculated with our model, as a function of the period of the array. For each period, the diameter of the “holes” is modified so that the filling factor is kept constant (f ≈ 0.26) as required in the definition of the Fourier coefficients εn in Eq. (2). The other parameters of the gold nanostructures are: film thickness h = 110 nm, Al2O3 substrate with εIII= 3.1. The long wavelength resonance (1, 0)SS first shift linearly to the shorter wavelengths when the period decreases. Below a period of ~ 300 nm the maxima of both the transmission and absorption associated to this (1, 0)SS peak shifts sub-linearly to the blue and tends to saturate close to the interband threshold d → EF. In the spectral region 500-600 nm, the spectral behavior of the transmission is more complex. Two resonances are present. One corresponds to the (1, 0)AS order peak represented by the green circles. It is situated near 600 nm for a period of 550 nm. Like the (1, 0)SS peak it is blue shifted when the period decreases and ultimately also tends to saturate near the interband threshold as clearly seen in the inset of Fig. 4(a). In addition, for all periods, there is a fixed resonance, represented by the blue circles, situated slightly lower than the interband threshold i.e. at 510 nm for the transmission peak and at 490 nm for the associated absorption peak. This transmission peak results from the interplay of the large absorption above the interband threshold and the large reflectivity associated to the intraband transitions. The combination of these two effects manifests as an apparent transparency window. In Fig. 4(b) we have represented the transmission spectra of the nanostructures for the periods a = 450, 300 and 150 nm. For clarity purpose, a vertical offset has been added for each spectrum. In these spectra one can trace back the blue shift of both (1, 0)SS and (1, 0)AS. In addition, they spectrally converge towards the interband threshold. The large fixed resonance at ~510 nm also appears clearly. The spectral evolution of the two resonances (1, 0)SS and (1, 0)AS, when varying the period a, is characteristic of the surface plasmon dispersion observed in noble metal nanostructures. The fact that their maximum shifts sub-linearly and ultimately saturates to the interband threshold when the period decreases is due to the contribution of the d electrons to the real part of the dielectric function ε theo displayed in Fig. 2.

Fig. 4. (a) Spectral positions of the maxima in transmission (closed symbols) and absorption (open symbols) as a function of the period of the array of gold nanostructures, with a constant filling factor f = 0.26, a thickness h = 110 nm and ε III = 3.1. The red symbols correspond to the (1, 0)SS order peak. The blue symbols represent the transparency window and the green ones display the positions of the (1, 0)AS order peak. Inset: detailed view of the splitting of the resonance at ~510 nm. (b) Transmission spectra of the preceding nanostructures for different periods a. A vertical offset has been added for clarity. The metal dielectric function is ε theo.

To better see the relative importance between the core d electrons and the conduction p electrons, we have represented in Fig. 5(a) and 5(b) the same quantities as in Fig. 4(a) and 4(b) but considering only the real part of the Drude dielectric function. Both (1, 0)SS and (1, 0)AS resonances shift quasi-linearly to the blue, without saturation effect near 530 nm when the period decreases. In other words, it is the slight increase of Re(ε theo), due to the interband transitions, which induces the important nonlinear shift of these resonances when varying the period of the array. To check the consistency of this interpretation and the validity of the RCW model using the Lindhard dielectric function, we have analyzed the position of the (1, 0)SS order peak, obtained self-consistently from the dispersion relation λ=ai2+j2εI,IIIεM(λ)εI,III+εM(λ), as a function of the period of the nanostructure. The results are shown in Fig. 5(c) (red circles). In this figure we have also reported the position of the (1, 0)SS order peak obtained with our theoretical model, using the Lindhard dielectric function (green line). The two curves do not overlap because we have considered only the real part of the Lindhard dielectric function in the above dispersion relation. The blue line in Fig. 5(c) corresponds to the (1, 0)SS order peak positions using the Drude dielectric function in the theoretical model. It is completely unable to reproduce the correct behavior below 600 nm (a ≈ 300 nm), showing the predominance of the core d electrons in this spectral region.

Fig. 5. (a) Spectral positions of the maxima of transmission as a function of the period. The structures parameters are the same as in Fig. 4 but the dielectric function of gold does not include the interband transitions. (b). Corresponding transmission spectra for different periods. (c). Spectral positions of the (1, 0)SS peak determined by the surface plasmon dispersion relation given in the text (red circles), calculated with our model, using ε theo (green line) and the real part of the Drude-like dielectric function (blue line).

Fig. 6. Linear zero order transmission spectra of gold nanostructures with a = 300 nm, d = 180 nm, h = 110 nm, εIII = 3.1 for different dielectric functions εI.

To explore the pertinence of the above study for other noble metals, we have applied our model to the case of copper and silver. Figure 7(a) displays the calculated dielectric function of copper (red triangles) and the one given in ref [30

30. A. V. Kats and A. Yu. Nikitin, “Analytical treatment of anomalous transparency of a modulated metal film due to surface plasmon-polariton excitation,” Phys. Rev. B 70, 235412 (2004). [CrossRef]

] (green circles). Figure 7(b) shows the calculated transmission spectra using either εtheo or εtab in the case of a copper nanostructure with the parameters given in the figure caption. These spectra display essentially the same characteristics as in the case of the gold nanostructures. The longer wavelength resonance corresponds to the first order (1, 0)SS peak on the substrate side. The shorter wavelength resonance corresponds to the transparency window, situated near the interband threshold of copper (Ed→EF =2.1eV), resulting from the superposition of the strong absorption and the high reflectivity of copper. In Fig. 7(c), the (1, 0)SS order peak also displays a non linear shift with the period of the nanostructure with a saturation occurring near the interband threshold. Moreover, the second peak splits in two part, one corresponding to the (1, 0)AS peak on the air side, the other nearly independent of the period associated to the transparency window of copper.

Fig. 7. (a) Real (closed symbols) and imaginary parts (open symbols) of the dielectric function of copper calculated in the self-consistent field method (red triangles), or tabulated [30] (green circles). (b) Calculated transmission spectra with ε theo and ε tab of a copper nanostructure with a = 350 nm, d = 200 nm, h = 110 nm ε III = 2.25. (c) Spectral positions as a function of the period of the (1, 0)SS peak (red), (1, 0)AS peak (green), and the transparency window (blue).

Finally, we have also investigated the case of silver. Figure 8(a) displays the dielectric function tabulated in Ref. [30

30. A. V. Kats and A. Yu. Nikitin, “Analytical treatment of anomalous transparency of a modulated metal film due to surface plasmon-polariton excitation,” Phys. Rev. B 70, 235412 (2004). [CrossRef]

] (green circles), and the calculated one (red triangles). In the spectral region considered here, the dielectric function of silver is well described by a three band-model [34

34. R. Rosei, F. Antonangeli, and U. M. Grassano, “d bands position and width in gold from very low temperature thermomodulation measurements,” Surf. Science 37, 689 (1973). [CrossRef]

]. Following Rosei [35

35. B. R. Cooper, H. Ehrenreich, and H. R. Philipp, “Optical properties of noble metals. II,” Phys. Rev. 138, A494 (1965). [CrossRef]

], we have taken into account the interband transition from the filled d band to the Fermi level (Ed→EF =3.99eV) and the transition from the conduction band p to the empty band s (Ep→s = 3.85 eV). Figure 8(b) shows the transmission spectra of a silver nanostructure with a=400 nm, f=0.26, h = 110 nm and ε III = 2.25 using either ε theo or ε tab. The spectra exhibit the (1, 0)SS order peak on the substrate side at longer wavelengths and the (1, 0)AS order resonance on the air side located at 450 nm when using εtheo. As in the case of the other noble metals, an apparent resonance shows up in the interband region which corresponds to the transparency window of silver. A detailed view of the calculated zero order transmission in the interband region is presented in the inset of Fig. 8(b). Two peaks are present in this resonance. They correspond to the superposition of the high reflectivity due to intraband processes and the large absorption associated to each interband transition thresholds of silver. The peak located at 316 nm corresponds to the d→EF interband threshold and the one at 325 nm to the p→s interband one. In Fig. 8(c), the (1, 0) order peaks both exhibit a non linear shift and converge toward the p→s interband threshold when decreasing the period of the array. The spectral position of the transparency windows are, as previously mentioned for gold and copper, independent of the period.

Fig. 8. (a) Real (closed symbols) and imaginary parts (open symbols) of the dielectric function of silver calculated in the self-consistent field method (red triangles), or tabulated [30] (green circles). (b) Calculated transmission spectra with etheo and etab of a silverr nanostructure with a = 400 nm, f = 0.26, h = 110 nm ε III = 2.25. Inset: detailed view in the spectral region of the transparency windows. (c) Spectral positions as a function of the period of the (1, 0)SS peak (red), (1, 0)AS peak (green), and the transparency window associated to the absorption due to p→s transition (black) and due to d→EF transition (blue).

In conclusion, our analysis shows the crucial importance of the metallic dielectric function to predict the spectral behavior of the light transmission through noble metal nanostructures. It is particularly the case near the interband transitions threshold where the core electrons play a major role. Two main spectral features have been unraveled. The first one corresponds to the strong nonlinear shift, as a function of the lattice period, of the (1,0)SS and (1,0)AS peak positions when approaching the interband threshold. Secondly, a large resonance appears near the interband threshold of the noble metals. It is associated to a transparency window resulting from the superposition of the large absorption associated to the core d-electrons and the high reflectivity due to the conduction electrons.

Acknowledgments

We would like to thank Prof. T. W. Ebbesen and C. Genet for their careful reading of the manuscript. We thank A. Degiron for the samples preparation, as well as M. Albrecht, D. Acker and G. Versini for their technical support. This project has been carried out with the financial support of the “Centre National de la Recherche Scientifique” in France.

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A. M. Dykhne, A. K. Sarychev, and V. M. Shalaev, “Resonant transmittance through metal films with fabricated and light-induced modulation,” Phys. Rev. B 67, 195402 (2003). [CrossRef]

29.

A. V. Kats and A. Yu. Nikitin, “Nonzeroth-Order anomalous optical transparency in modulated metal films owing to excitation of surface plasmon polaritons: an analytic approach,” JETP Lett. 79, 625 (2004). [CrossRef]

30.

A. V. Kats and A. Yu. Nikitin, “Analytical treatment of anomalous transparency of a modulated metal film due to surface plasmon-polariton excitation,” Phys. Rev. B 70, 235412 (2004). [CrossRef]

31.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000). [CrossRef]

32.

D. W. Lynch and W. R. Hunter, in Handbook of optical constants of solids, E. D. Palik, ed. (Academic press, New York, 1991).

33.

H. Ehrenreich and M. H. Cohen, “Self-consistent field approach to the many-electron problem,” Phys. Rev. 115, 786 (1959). [CrossRef]

34.

R. Rosei, F. Antonangeli, and U. M. Grassano, “d bands position and width in gold from very low temperature thermomodulation measurements,” Surf. Science 37, 689 (1973). [CrossRef]

35.

B. R. Cooper, H. Ehrenreich, and H. R. Philipp, “Optical properties of noble metals. II,” Phys. Rev. 138, A494 (1965). [CrossRef]

36.

N. E. Christensen, “The band structure of silver and optical interband transitions,” Phys. Stat. Sol. B 54, 551 (1972). [CrossRef]

37.

R. Rosei, “Temperature modulation of the optical transitions involving the Fermi surface in Ag: Theory,” Phys. Rev. B 10, 474 (1974). [CrossRef]

38.

R. Rosei, C. H. Culp, and J. H. Weaver, “Temperature modulation of the optical transitions involving the Fermi surface in Ag: Experimental,” Phys. Rev. B 10, 484 (1974). [CrossRef]

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(240.0240) Optics at surfaces : Optics at surfaces
(240.6680) Optics at surfaces : Surface plasmons
(260.3910) Physical optics : Metal optics

ToC Category:
Optics at Surfaces

History
Original Manuscript: March 1, 2006
Revised Manuscript: March 20, 2006
Manuscript Accepted: March 20, 2006
Published: April 3, 2006

Citation
V. Halté, A. Benabbas, and J.-Y. Bigot, "Optical response of periodically modulated nanostructures near the interband transition threshold of noble metals," Opt. Express 14, 2909-2920 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2909


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References

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  28. A. M. Dykhne, A. K. Sarychev, and V. M. Shalaev, "Resonant transmittance through metal films with fabricated and light-induced modulation," Phys. Rev. B 67, 195402 (2003). [CrossRef]
  29. A. V. Kats, and A. Yu. Nikitin, "Nonzeroth-Order anomalous optical transparency in modulated metal films owing to excitation of surface plasmon polaritons: an analytic approach," JETP Lett. 79, 625 (2004). [CrossRef]
  30. A. V. Kats and A. Yu. Nikitin, "Analytical treatment of anomalous transparency of a modulated metal film due to surface plasmon-polariton excitation," Phys. Rev. B 70, 235412 (2004). [CrossRef]
  31. E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000). [CrossRef]
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  34. R. Rosei, F. Antonangeli, and U. M. Grassano, "d bands position and width in gold from very low temperature thermomodulation measurements," Surf. Science 37, 689 (1973). [CrossRef]
  35. B. R. Cooper, H. Ehrenreich, and H. R. Philipp, "Optical properties of noble metals. II," Phys. Rev. 138, A494 (1965). [CrossRef]
  36. N. E. Christensen, "The band structure of silver and optical interband transitions," Phys. Stat. Sol. B 54, 551 (1972). [CrossRef]
  37. R. Rosei, "Temperature modulation of the optical transitions involving the Fermi surface in Ag: Theory," Phys. Rev. B 10, 474 (1974). [CrossRef]
  38. R. Rosei, C. H. Culp, and J. H. Weaver, "Temperature modulation of the optical transitions involving the Fermi surface in Ag: Experimental," Phys. Rev. B 10, 484 (1974). [CrossRef]

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