to determine the group velocity of surface plasmons on flat gold and silver surfaces. © 2007 Optical Society .... The results of such data processing are shown in Fig. 3, where the ... metric phase recovery procedures. These findings open a way ...
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Surface plasmon interferometry: measuring group velocity of surface plasmons Vasily V. Temnov and Ulrike Woggon Experimentelle Physik IIb, Universität Dortmund, Otto-Hahn Strasse 4, D-44221 Dortmund, Germany
José Dintinger, Eloise Devaux, and Thomas W. Ebbesen Laboratoire des Nanostructures, ISIS, Université Louis Pasteur, 8 allée Monge, BP 70028, 67083 Strasbourg, France Received December 11, 2006; revised February 15, 2007; accepted February 18, 2007; posted February 23, 2007 (Doc. ID 77942); published April 17, 2007 Optical transmission spectroscopy on metal films with slit–groove pairs is conducted. Spectra of the light transmitted through the slit exhibit Fabry–Perot-type interference fringes due to surface plasmons propagating between the slit and the groove. The spectral dependence of the period of interference fringes is used to determine the group velocity of surface plasmons on flat gold and silver surfaces. © 2007 Optical Society of America OCIS codes: 240.6680, 120.3180, 260.3910, 130.3120.
The investigation of the generation, propagation, and manipulation of propagating electromagnetic surface modes (surface plasmons or surface plasmon polaritons) on nanostructured metal–dielectic interfaces attracts a considerable research interest motivated by the possibility of designing highly integrated optical devices. Launching and decoupling of surface plasmons on flat surfaces by using arrays of subwavelength holes has been demonstrated by several groups.1,2 The dependence of surface plasmon propagation length on frequency was investigated both in thin metal films3 and with more complicated microscale metal stripes.4 Very recently surface plasmon propagation and interference effects on noble metal surfaces containing pairs of subwavelength slits5 or slit–groove pairs6–8 have attracted significant interest. Alternative concepts calling into question the validity of the surface plasmon model were developed6,7 and controversially debated.8,9 Another intriguing study by Bai and co-workers10 reports a big discrepancy between the experimentally determined group velocity on flat silver surface and the value expected from tabulated optical constants.11 In this Letter we apply broadband optical transmission spectroscopy to nanostructured metallic surfaces, measure surface-plasmon-mediated interference patterns in the spectral domain, and determine the group velocity of surface plasmons. We further discuss the feasibility of such structures for surfaceplasmon-based interferometry. Our samples consist of 200 nm thick gold or silver layers on glass substrates, with a 40 nm thin titanium layer between them (see Fig. 1). The role of a titanium layer in our experiments is to improve the adhesion between noble metal and glass and suppress the partial transmission of incident light through the grooves, although it may also help to suppress surface plasmon propagation on the metal– glass interface in other experiments.5 The focused ion beam was used to manufacture slit–groove pairs on a metal–air interface. Both the slit and the groove had 0146-9592/07/101235-3/$15.00
a width of 100 nm and a length of 30 m. The grooves had a depth of 50 nm. The distance d between the slit and the groove was varied between 4.5 and 50 m. The slits were illuminated through the substrate by white light from a halogen lamp, and transmission spectra were recorded by an imaging spectrometer. By use of a microscope objective 共20⫻, NA= 0.4兲, the sample surface was imaged with magnification M = 14 on the entrance slit of an imaging spectrometer (asymmetrical Czerny–Turner configuration, focal length f = 460 mm, NA= 0.07). Under these imaging conditions we observed a clear spatial separation of light transmitted through the slit from light scattered from the groove on the CCD chip of the spectrometer. Therefore the interference of light components transmitted trough the slit and scattered from the groove was avoided. White light was focused on the slits to a spot of about 20 m (FWHM) in diameter. Changing the focusing conditions of white light had little influence on the transmission spectra of the slits. The polarization state of white light was controlled by a polarizer. For electrical field polarization perpendicular to the slit, distinct periodic modulations in the spectrum are observed. Figure 2 shows such oscillation in the trans-
Fig. 1. Experimental geometry for optical transmission spectroscopy on metal films with slit–groove nanostructures. © 2007 Optical Society of America
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Fig. 2. Transmission spectra through the slits for different slit–groove distances d = 4.5, 9.5, 19.2 m. Spectra for 9.5 and 19.2 m are vertically displaced for clarity. Inset, absolute value of the Fourier transform of the spectrum for d = 19.2 m on a logarithmic scale (see text for details).
mission spectra of gold samples for different values of slit–groove distances d = 4.5, 9.5, 19.2 m. The spectral period of the oscillations becomes smaller for larger d. For polarization parallel to the slit no oscillations were observed. These observations are in agreement with recent results of Schouten and co-workers,5 who observed similar oscillations in the spectrum of light transmitted through pairs of slits. Physical interpretation of these oscillations invokes generation of propagating surface plasmons between the slits and their interference with directly transmitted light. In our slit–groove geometry, surface plasmons are excited at the slit, propagate toward the groove, are partially reflected by the groove, propagate back to the slit, and are rescattered by the slit in free-space radiation, which interferes with light directly transmitted through the slit. The validity of this scenario, which is visualized in Fig. 1, will be confirmed by the analysis of our experimental data on gold and silver samples. The light intensity I共兲 transmitted through the slit is given by I共兲 = E1共兲2 + E2共兲2 + 2E1共兲E2共兲cos共⌽共兲兲,
共1兲
with E1共兲 and E2共兲 being the amplitudes of directly transmitted and surface-plasmon-mediated contributions, respectively. The phase factor ⌽共兲 is given by ⌽共兲 = 2ksp共兲d + ⌽0共兲,
共2兲
where ksp共兲 denotes the wave vector of the surface plasmon and ⌽0共兲 represents an additional unknown phase shift due to reflection from the groove and rescattering of surface plasmons in free-space radiation. For slit–groove distances d much larger than the optical wavelength, the first term 2ksp共兲d becomes the dominant contribution to the phase, since ⌽0共兲 is shown to be of the order of .5,6 It can be seen in Fig. 2 that the transmission spectrum for d = 19.2 m consists of rapid sinelike modulation superimposed on a slowly varying background contribu-
tion. This type of signal with a high carrier-frequency modulation is common in optical interferometry and is well suited to extract the phase ⌽共兲 by using a Fourier-transform algorithm.12,13 The existence of a rapid sinelike modulation leads to the appearance of two symmetric sidebands in the Fourier transform of the signal (see inset in Fig. 2). One of these sidebands can be isolated by applying a filter in Fourier space; after that, the filtered signal is subjected to an inverse Fourier transform. The angle of this complex signal is equal to the desired phase ⌽共兲 of the oscillating term. However, the phase is determined with an intrinsic ambiguity of an arbitrary constant phase factor 2m (m is an integer number). That is why the described procedure does not allow a direct determination of the dispersion relation of surface plasmons. Instead we calculate the derivative d⌽共兲 d
= 2d
冉
dksp共兲 d
冊
+
d⌽0共兲 d
,
共3兲
where the first term is proportional to slit–groove distance d and, therefore, is expected to dominate for large d. After neglecting the second term d⌽0共兲 / d (to be discussed below) we obtain an expression for the group velocity of surface plasmons: vgr =
d dksp
冉 冊 d⌽
= 2d
d
−1
.
共4兲
The results of such data processing are shown in Fig. 3, where the experimentally determined group velocity for gold and silver samples is compared with theoretical values for a flat metal surface based on optical constants tabulated by Johnson and Christy.11 A distinct decrease of surface plasmon group velocity at higher photon energies around 2 eV is accurately reproduced. For both noble metals a good agreement between the experimental and theoretical curves is obtained, which validates the surface plasmon concept to explain the observed oscillations. The analysis of data sets for different values of slit–groove distance in the range between 4.5 and 50 m for both metals provides the same results. Small wiggles in spectral dependencies of the group velocity in Fig. 3 represent an artifact of the applied Fouriertransform algorithm. At small slit–groove distances, as in the case of d = 4.5 m, the period of spectral oscillations ⌬ is large, and the Fourier-transform algorithm fails to isolate sidebands in the Fourier domain in a correct way. Instead a simple approximate procedure of counting minima and maxima provides better results. The group velocity appears to be directly proportional to the spectral oscillation period ⌬: vgr = 2d共⌬/2兲,
共5兲
which represents a discrete approximation of expression (4). The observed independence of the group velocities calculated by approximate expressions (4) and (5) on slit–groove distance d justifies that the derivative d⌽0共兲 / d in Eq. (3) can indeed be neglected under
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where ␦⌽ stands for phase sensitivity. Assuming d = 20 m, = 800 nm, and a realistic value of ␦⌽ ⬃ 0.01,13 we conclude that minor changes in the refractive index of the surface plasmon of the order of ␦nsp ⬃ 10−4 can be measured. The concept of surfaceplasmon-based interferometry may be useful, for example, to study nonequlibrium electronic excitations in metals15 or the properties of optical emitters put on metallic surface. In summary, broadband optical spectroscopy has been applied to study the transmission of slit–groove nanostructures in films of noble metals. Distinct periodic modulations of transmitted light can be attributed to surface plasmon excitation and propagation between the slit and the groove. A detailed analysis of these oscillations allows us to obtain the group velocity of surface plasmons, in a good agreement with theoretical values for a flat metal surface based on tabulated optical constants. Slit–groove type nanostructures seem to be promising candidates for surface-plasmon-based interferometry on metal surfaces. Stimulating discussions with Marco Allione and Cyriaque Genet are gratefully acknowledged. Fig. 3. (Color online) Spectral dependence of the group velocity of surface plasmons on plane gold and silver surfaces. Theoretical curves for phase velocity are presented for comparison.
our experimental conditions, i.e., d . The neglected term may become important in experiment with much smaller d ⬃ , when an accurate determination of ⌽0 at a single excitation wavelength was demonstrated.6 The results of our measurements prove that the observed spectral oscillations in a relatively broad spectral range between 1.3 and 2.1 eV can be attributed to surface plasmons in gold and silver. The corresponding phase of surface-plasmon-mediated contribution of the transmitted signal can be extracted with high accuracy by using conventional interferometric phase recovery procedures. These findings open a way to investigate physical phenomena on metal surfaces by surface plasmon interferometry in the spectral domain. Indeed, the differential interferometric measurements (such as, for example, time-resolved interferometric measurements13,14) do not require knowledge of the absolute values of the phase, but offer the possibility to accurately compare phase functions themselves. In case of a surface-plasmon-based interferometer in the slit–groove configuration with distance d, the minimum detectable change ␦nsp in the refractive index nsp = 共 / 2兲Re共ksp兲 of the surface plasmon would be
␦nsp =
4d
␦⌽,
共6兲
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