Experimental determination of the dispersion ... - OSA Publishing

660

J. Opt. Soc. Am. B / Vol. 27, No. 4 / April 2010

Pshenay-Severin et al.

Experimental determination of the dispersion relation of light in metamaterials by white-light interferometry Ekaterina Pshenay-Severin,1,* Frank Setzpfandt,1 Christian Helgert,1 Uwe Hübner,2 Christoph Menzel,3 Arkadi Chipouline,1 Carsten Rockstuhl,3 Andreas Tünnermann,1 Falk Lederer,3 and Thomas Pertsch1 1

Institute of Applied Physics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany 2 Institute of Photonic Technology, Albert-Einstein-Strasse 9, 07745 Jena, Germany 3 Institute of Condensed Matter Theory and Solid State Optics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany *Corresponding author: [email protected] Received December 3, 2009; accepted January 12, 2010; posted January 20, 2010 (Doc. ID 120877); published March 9, 2010

We present a method to experimentally measure the complex reflection and transmission coefficients of optical waves at metamaterials under normal incidence. This allows us to determine their pertinent dispersion relation without resorting to numerical simulations. For this purpose we employ a spectrometer and a white light interferometer for amplitude and phase measurements, respectively. To demonstrate the reliability of the method, it is applied to two referential metamaterial geometries, namely, the fishnet and the double-element structure. Involved aspects of the phase measurements as well as the accuracy of the method are discussed. © 2010 Optical Society of America OCIS codes: 160.3918, 120.5050, 290.3030.

1. INTRODUCTION Characterization of optical material properties is a pivotal problem in experimental physics. Thus far, nearly all methods developed for this purpose address only the dielectric properties, since natural materials do not exhibit magnetism in the optical domain. However, with the availability of metamaterials (MMs) this restriction was lifted. The interest in such MMs was stipulated by novel opportunities to control light propagation [1], which are permitted by their extraordinary dispersive properties. One of the most fascinating consequences is negative refraction [2]. Nowadays MMs are, in most cases, made of periodically arranged nanostructured unit cells, which may be termed meta-atoms. The common technology for the fabrication of MMs is based on top-down techniques such as electron beam lithography. On one hand, this permits for a large degree of flexibility in designing suitable meta-atoms. On the other hand, it renders the fabricated MMs to be essentially planar structures, called meta-films. Nevertheless, bulk MMs can be fabricated by stacking such metafilms [3]. To integrate a MM into a functional device, it is essential to know how it affects light propagation. Previously, this was described in terms of effective material parameters as the effective permittivity ␧共␻兲 and the effective permeability ␮共␻兲 [4,5]. However, it turned out that the assignment of effective material parameters, which depend only on frequency and not on the wave vector, is usually not feasible because of the mesoscopic nature of optical MMs and the resulting strong spatial dispersion [6]. There is a clear trade-off; artificial magnetic properties, which arise as a result of electric quadupole or mag0740-3224/10/040660-7/$15.00

netic dipole resonances of plasmon polaritons in metaatoms, are accompanied by this spatial dispersion because their existence requires a minimum meta-atom size, which is only a few times less than the wavelength. But in order to describe light propagation in an arbitrary medium it suffices to solve the respective eigenvalue problem resulting in eigenfunctions (modes), which have to obey a dispersion relation relating the wave vector components to the frequency ␻ = ␻共kx , ky , kz兲. Of course, this dispersion relation will be governed by the intrinsic material dispersion and, most notably, by the meta-atom shape [7,8]. It has been shown that in most cases the periodic optical MM, although spatially dispersive, may be described as an effective homogeneous medium [9]. This has important consequences and facilitates the description of light propagation. If the dispersive properties are dominated by the lowest-order Bloch mode, it can be replaced by the eigenmode of a homogeneous medium, i.e., a plane wave. For a given frequency, the medium can then be characterized by an effective wave vector, which constitutes an effective wave parameter. If the dispersion relation is a sphere, or if one is only concerned with a definite transverse wave vector component (angle of incidence), it is even possible to introduce an effective refractive index as is usually done in the MM literature. The effective index can be retrieved from the complex transmission t共␻兲) and the reflection 共r共␻兲兲 coefficients [10]. Because both quantities are complex, information on their amplitude and their phase are required to be known. Since experimental access to both quantities is difficult to achieve, the usual retrieval algorithm consists of a © 2010 Optical Society of America


combination of experimental and numerical simulation results. As numerical tools, very often the Fourier modal method [11] or the finite-difference time-domain method [12] are used for a MM with a given geometry. If a sufficient coincidence of the measured and simulated transmittance 共兩t共␻兲兩2兲 and reflectance 共兩r共␻兲兩2兲 is achieved in a broad spectral range, the numerical data for both complex coefficients can be used for the retrieval of the effective wave parameters. This approach requires only spectral measurements of the transmittance and reflectance. However, in order to achieve a sufficient agreement between simulations and measurements, the adjustment of the simulation model parameters is required, which is a timeconsuming procedure. Moreover, computational requirements for some MMs such as amorphous structures [13] are fairly demanding due to the lack of periodicity. The necessity of phase measurements for the comprehensive experimental characterization of MMs has been discussed in literature [14,15]. However, there are only a few works addressing direct measurements of the MMs effective optical parameters [15–17]. Though these experiments successfully demonstrated the negative refractive index of MMs, they can hardly be used as a routine technique. The method proposed in [16] and used in [18] requires an additional structuring of the investigated samples and results in an extremely high phase-noise level, most notably for small phase shifts. A time-domain interferometric technique was used in [17] for group and phase velocity measurements. However, the configuration of the Fabry–Perot interferometer used in the experiments does not allow a correct treatment of the interface effects between the MM film and the substrate. The authors of [15] performed interferometric phase measurements in both transmission and reflection using polarization and walk-off interferometers and a set of tunable diode lasers. The used method demonstrated high accuracy and sensitivity of the phase measurements. However, the measurements were performed in narrow spectral ranges, while a broad spectral range is needed for the MMs characterization. In the experiments described in this contribution we implement a white-light Fourier-transform spectral interferometer [19] for broadband 共1.1 ␮m – 1.7 ␮m兲 phase measurements to determine the dispersion relation of MMs at normal incidence k = k共␻兲 by purely experimental means, where k is the complex wavenumber. An effective refractive index can be formally introduced as neff共␻兲 = k共␻兲 / k0 with k0 = ␻ / c. The experimental setup is a Jamin–Lebedeff interferometer modified for measurements in transmission and reflection under normal incidence. With this technique, a precision of the phase measurements of about 0.02 rad was achieved that results in an accuracy of the refractive-index definition for investigated structures of about 4% in the real and imaginary parts.

2. RETRIEVAL OF THE EFFECTIVE CONSTANTS To retrieve the complex wave number, the equations for the complex transmission and reflection coefficient of an

Vol. 27, No. 4 / April 2010 / J. Opt. Soc. Am. B

661

effectively homogenous slab sandwiched between a substrate (s) and a cladding (c) have to be inverted [10]: 1 k = k0neff =

d

冋冉 arccos

ks共1 − r2兲 + kct2 t共ks共1 − r兲 + kc共1 + r兲兲

+ 2m␲

冊册

. 共1兲

Here, d is the thickness of the MM layer, and ks and kc are the wave numbers in the substrate and cladding, respectively. Thus, for the experimental definition of the effective wave number (or refractive index) the complex transmission and reflection coefficients have to be measured. Whereas the measurements of the transmittance and reflectance are a standard problem solved by spectroscopy, broadband phase measurements require more demanding interferometric techniques. In the following section we describe the procedure for the phase information retrieval from the experimental data.

3. PHASE MEASUREMENTS: BACKGROUND The measurements of the complex transfer function of MM samples were performed by white-light Fouriertransform spectral interferometry [19]. In this type of spectral interferometry experiment, a time delay 共␶兲 between two beams is introduced in an interferometer with two arms of different geometrical length. The complex transfer functions of the sample arm, denoted as Tsam共␻兲, and the reference arm, denoted as Tref共␻兲, contain an amplitude and a phase term and are given by Tref共␻兲 = Aref共␻兲exp关i␸ref共␻兲兴 Tsam共␻兲 = Asam共␻兲exp关i␸sam共␻兲兴.

共2兲

The fields in each arm of the interferometer can be written as follows: Eref共␻兲 = Aref共␻兲exp关i␸ref共␻兲兴Ein共␻兲, Esam共␻兲 = Asam共␻兲exp关i共␸sam共␻兲 + ␸MM共␻兲 + ␻␶兲兴Ein共␻兲.

共3兲

Here Ein共␻兲 is the amplitude of the incident field and ␸ref共␻兲 is the phase delay in the reference arm that is induced due to the dispersive optical elements. The phase delay in the sample arm can be separated into three terms. The term ␸sam共␻兲 is the phase delay due to the dispersive optical elements placed in the sample arm; the term ␸MM共␻兲 is the phase delay in the MM sample (which is to be retrieved); and the term ␻␶ corresponds to the geometrical length difference between the two arms. Interference of the reference and the sample arm fields gives rise to a measured optical signal in the frequency domain that is I共␻兲 = 兩Esam共␻兲 + Eref共␻兲兩2 = 兩Esam共␻兲兩2 + 兩Eref共␻兲兩2 + 2兩Esam共␻兲兩兩Eref共␻兲兩cos共␸sam共␻兲 + ␸MM共␻兲 + ␻␶ − ␸ref共␻兲兲.

共4兲

The argument of the cos function in the interference term contains the phase delay of the MM sample ␸MM共␻兲.

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Extraction of the interference term can be easily achieved in the time domain. The signal in the time domain S共t兲 obtained by Fourier transformation of I共␻兲 is * 共− t兲 + E 共t兲丢 E* 共− t兲 S共t兲 = FT关I共␻兲兴 = Esam共t兲丢 Esam ref ref * 共− t兲 + E* 共␶ − t兲丢 E 共t兲. + Esam共t + ␶兲丢 Eref ref sam

共5兲

In Eq. (5) the first two terms are the autocorrelation functions of the individual fields, centered at t = 0. The third and fourth terms are the correlation functions of two fields centered at t = ␶ and t = −␶, respectively (selected experimental results are shown for illustrative purposes in Fig. 1). If the time delay ␶ is sufficiently large, the correlation term does not overlap with the autocorrelation terms and can be extracted by applying a finite time window:

冦

0, t ⬍ ␶ − ␦␶

冧

rect共t, ␶, ␦␶兲 = 1, ␶ − ␦␶ ⱕ t ⱕ ␶ + ␦␶ . 0, t ⱖ ␶ + ␦␶

共6兲

In the experiment, the time delay ␶ has to be chosen, on the one hand, sufficiently large to provide separation of the correlation functions for a proper filtering. On the other hand, it should be small enough to facilitate data acquisition. Path differences in excess lead to small periods of the interference pattern in the frequency domain and result in an increase of the required resolution. Applying the inverse Fourier transform to the interference pattern after filtering with Eq. (6) gives the soughtafter interference term in the frequency domain, denoted here by the subscript “int.” Eint共␻兲 = FT−1关rect共t, ␶, ␦␶兲S共t兲兴.

共7兲

Fig. 1. (Color online) (a) Measured interference pattern in the wavelength domain. (b) The signal as shown in (a) in the time domain after Fourier transformation; rectangle depicts the filter.

For MMs, where the thickness d is in the order of only a few hundreds of nanometers, the measured phase ␸MM can be safely assumed to lie between −␲ ⬍ ␸MM ⬍ ␲. For the retrieval of the effective constants, as outlined in Eq. (1), this implies that we assume that m = 0. However, for thicker MMs made of a larger number of functional layers, m ⫽ 0 has to be considered. In this case, the choice of an appropriate value of m has to be based on an effective refractive index measured in the low-frequency limit where the magnetic properties disappear.

The argument of the interference term is ⌬␸ = ␸sam共␻兲 + ␸MM共␻兲 − ␸ref共␻兲

共8兲

and represents the phase delay between the signals passing the two arms of the interferometer. The phase difference as given in Eq. (8) not only contains ␸MM共␻兲 but also the phase difference introduced by the dispersive elements in both arms of the interferometer. Therefore a reference measurement is required for the extraction of ␸MM共␻兲. For the reference measurement, the MM sample is replaced by a known material, for instance, air in the case of the transmission measurement or a mirror in the case of the reflection measurement. If ␸ref.sam. is the phase delay due to the known reference object, the phase difference measured and retrieved in the reference measurement is ⌬␸ref.sam. = ␸sam共␻兲 + ␸ref.sam.共␻兲 − ␸ref共␻兲.

共9兲

If the dispersion and thickness of the reference sample are known, ␸MM共␻兲 can be extracted from Eq. (8) and Eq. (9):

␸MM共␻兲 = ⌬␸ − ⌬␸ref.sam. + ␸ref.sam.共␻兲.

共10兲

It has to be noted that the retrieved phase is not an absolute phase, since it has an ambiguity of 2␲. Additionally a priori information about the sample under consideration would be required in order to remove the ambiguity.

4. PHASE MEASUREMENTS: EXPERIMENT A. Experimental Setup The interferometric setup for the phase measurements is shown in Fig. 2. The interferometer represents a Jamin– Lebedeff scheme modified for simultaneous measurements in transmission and reflection, and was also used in [15]. The interferometric measurements were performed with the supercontinuum light source SuperK Versa from KOHERAS (spectral bandwidth 0.4 ␮m – 1.7 ␮m). The broadband generation in this source is based on supercontinuum generation in an optical fiber [20]. The light from the supercontinuum light source is delivered by a fiber to the setup. After a collimator and a polarizer (P1), the linear polarized light passes a calcite beam displacer (B1), where the beam is divided into two parallel, orthogonally polarized beams with a displacement of 4 mm. These two beams represent the two arms of the interferometer. In the configuration for the transmission phase measurements the beams traverse the sample and then pass an achromatic half-wave plate (L1), which rotates the polarization in each beam by 90°. Therefore two beams are recombined in the second beam displacer (B2). As the beams in the interferometer arms are orthogonally polarized, the interference between them can be obtained using an another linear polarizer (P2). Finally, the light is collected


B. Experimental Results Two MM samples were investigated: the first is a fishnet structure [21] [Fig. 3(a)] and the second is a doubleelement structure comprising a cut-wire pair and a continuous wire element [22] [Fig. 3(b)]. The structures were fabricated by a standard electron-beam lithography technology combined with a lift-off process. The fishnet structure consisted of two dAu = 20 nm thick Au layers separated by a dMgO = 40 nm layer of MgO. The period of the

Fig. 3. Investigated structures. (a) SEM image of the fishnet structure. (b) SEM image of the double-element structure.

φT [rad]

into a photonic crystal (PhC) endlessly single-mode fiber and is measured with an optical spectrum analyzer. Phase measurements of the reflected signal are performed in the same setup. However, for this case, the signal reflected by a sample is routed by a nonpolarizing beamsplitter cube to a part of the setup for reflection measurements. Upon reflection at the beamsplitter the beams pass a half-wave plate (L2), a calcite beam displacer (B3), and a linear polarizer (P3). The light is collected in a PhC, which is connected to the same optical spectrum analyzer. The constant path difference 共␶兲 between the arms of the interferometer was introduced by a delay element (D). It consists of two BK7 glass blocks of the length of 22 mm and 20 mm, respectively. The optical path difference is 2 mm⫻ 共nBK7 − 1兲 ⬇ 1 mm, that corresponds to a period of the interference pattern of ⬇2.2 nm in the spectral domain. At the same time, such a time delay is sufficient to separate an autocorrelation signal and a delayed signal in the time domain [Fig. 1(b)]. In the experiment, the thick blocks 共20 mm兲 were chosen in order to prevent an influence of the interference signal occurring due to multiple reflections in the delay element.

φR [rad]

Fig. 2. (Color online) Interferometric setup. P1,P2, P3, polarizers; B1,B2,B3, beam displacers; L1,L2, half-wave plates; D, delay element. The insert shows the configuration of the experimental chip.

A [%]

T, R [%]


100 80 60 40 20 0 0.8 100 80 60 40 20 0 0.8 2 1.5 1 0.5 0 −0.5 −1 0.8 −1 −1.5 −2 −2.5 −3 −3.5 −4 0.8

663

(a)

1

1.2

1.4

1.6

1.8

2

(b)

1

1.2

1.4

1.6

1.8

2

(c)

1

1.2 1.4 1.6 wavelength [µm]

1.8

2

(d)

1


1.8

2

Fig. 4. (Color online) Results for the fishnet structure. Solid curves correspond to measured data. Dotted curves correspond to simulated data. (a) Black curves (blue online) are transmittances; gray curves (red online) are reflectances. (b) Absorption. (c) Phase of the transmitted amplitude: solid curve is the normalized measured phase, dotted curve is the simulated data, crossed curve is the phase from the amplitude measurements. (d) Phase of the reflected amplitude: solid curve is the normalized measured phase, dotted is the simulated data, crossed line is the phase from the amplitude measurements.

structure was 500 nm in both directions. The fishnet is defined by a rectangular hole with a size of Wx = 180 nm ⫻ Wy = 380 nm. The double-element structure consisted of two dAu = 40 nm thick Au layers separated by a dMgO = 30 nm layer of MgO. The length of the cut wires was L = 270 nm, and their width was Wc = 140 nm. The width of the wire element was Wl = 150 nm. The period of the structures was 550 nm in both directions. All structures were fabricated on a 1 mm SiO2 substrate. These structures have been primarily studied for identifying spectral domains exhibiting a negative refraction. The test sample arrangement is shown in Fig. 2. The MM sample 共2 ⫻ 2 mm2兲 is situated in the middle of the chip. The field above is the pure substrate serving as a reference for the transmission measurements. The field below is a nonstructured multilayer system corresponding to that of the MM sample and is used as the reference for the reflection measurements. The phase delay R ␸ref.mirror occurring upon reflection at the nonstructured

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multilayer system was calculated with great accuracy using standard multilayer matrix routines. Spectroscopic measurements of the transmittance and reflectance were performed under normal incidence with the spectrometer Perkin Elmer Lambda 950 in the wavelength range from 0.4 ␮m to 2.5 ␮m. Phases were measured in the wavelength region from 1.1 ␮m to 1.7 ␮m where the negative refractive index is expected. Here we demonstrate results for the polarization [Figs. 3(a) and 3(b)] that allows the excitation of the antisymmetric resonance at ⬇1.5 ␮m for both structures. Relevant resonances for the opposite polarization occur in a spectral domain not accessible by the current experimental setup. Simulations of the fabricated structures were performed by the FMM [11]. In the simulations the refractive index of MgO was assumed to be 1.72. We assumed a dispersive permittivity for gold as documented in the literature [23].

T, R [%]

100

(a)

80 60 40 20 0 0.8

1

1.2

1.4

1.6

1.8

2

100

(b)

A [%]

80 60 40 20

φR [rad]

φT [rad]

0 0.8 2 1.5 1 0.5 0 −0.5 −1 0.8 −1 −1.5 −2 −2.5 −3 −3.5 −4 0.8

1

1.2

1.4

1.6

1.8

2

(c)

1

1.2

1.4

1.6

1.8

2

(d)

1


1.8

2

Fig. 5. (Color online) Results for the double-element structure. Solid curves correspond to measured data. Dotted curves correspond to simulated data. (a) Black (blue online) curves are transmittances; gray (red online) curves are reflectances. (b) Absorption. (c) Phase of the transmitted amplitude: solid curve is the normalized measured phase, dotted curve is the simulated data, crossed curve is the phase from the amplitude measurements. (d) Phase of the reflected amplitude: solid curve is the normalized measured phase, dotted curve is the simulated data, crossed curve is the phase from the amplitude measurements.

wavelength [µm] Fig. 6. (Color online) Effective refractive index of (a) the fishnet structure and (b) the double-element structure. Real parts n⬘ are represented with dark curves, imaginary parts n⬙ are shown with gray (red) curves. Solid curves correspond to measured data; dashed curves correspond to simulated data.

The measured and simulated transmittance and reflectance as well as the phase of the transmitted and reflected fields are presented in Fig. 4 for the fishnet structure and in Fig. 5 for the double-element structure. Since the coincidence of the simulated and measured transmittance and reflectance for the fishnet structure (Fig. 4) is not perfect, some quantitative deviations of the phases are expected, though their qualitative behavior is similar. In the spectral region where an antisymmetric resonance is expected (fishnet structure at about 1.45 ␮m, doubleelement structure at about 1.38 ␮m) the phase in transmission has a dip corresponding to a decreasing refractive index. The phase jump of ␲ in the reflection corresponds to vanishing reflection, due to impedance matching between the MM and air. It turned out that an additional uncertainty in the measured phase between the sample and the reference arm is most detrimental to the accuracy of the phase measurements. This is introduced by the shift of the sample and due to variations in the thickness of the substrate. Therefore, a correction for this offset was mandatory. To do so, we take advantage of the fact that magnetic properties well off the resonance disappear. In this spectral region the refractive index can be defined using the standard techniques from a measured transmittance 共兩t兩2兲 and reflectance 共兩r兩2兲 [24]. With this index the expected absolute phase can be retrieved and compared with the measured values. The obtained difference is used for final data adjustment. These reference measurements were performed in the long wavelength region, where the antisymmetric (magnetic) resonance does not appear. The central wavelength of this resonance was 1.45 ␮m for the fishnet structure and 1.38 ␮m for the double-element structure, respectively. Its bandwidth can be estimated from the absorpTable 1. Root Mean Square Errors ␴兩t兩

␴兩r兩

␴arg t

␴arg r

0.01

0.007

0.02

0.02



665

Table 2. Partial Derivatives for the Fishnet Structure ␭ 关␮m兴

⳵n⬘ ⳵兩t兩

⳵n⬘ ⳵兩r兩

⳵n⬘ ⳵ arg t

⳵n⬘ ⳵ arg r

⳵n⬙ ⳵兩t兩

⳵n⬙ ⳵兩r兩

⳵n⬙ ⳵ arg t

⳵n⬙ ⳵ arg r

Fishnet structure Double-element structure

0.14 0.40

0.17 0.44

2.52 1.56

0.69 0.66

4.88 2.66

2.18 1.73

0.06 0.24

0.05 0.16

Partial derivatives for the fishnet structure at the wavelength ␭exp = 1.45 ␮m and for the double-element structure at ␭exp = 1.38.

␴=

⳵f共x兲

i=1

⳵xi

␴ xi

冊

2

,

共11兲

it can be calculated for the real and the imaginary parts of the refractive index. The quantities ␴兩t兩, ␴arg t, ␴arg r, ␴兩r兩 are the root-mean-squared errors of the corresponding measured quantities

冑兺 n

␴x =

¯ − x i兲 2 共x

i=1

n

,

共12兲

which depend on the wavelength. The values of ␴ given in Table 1 represent maximum observable ones in the region of the wavelength of interest 共1.1 ␮m – 1.7 ␮m兲. The values of the partial derivatives for the fishnet and double-element structure at the wavelengths for the minimum of n⬘ are given in Table 2. The values for the derivatives in Table 2 were obtained from the experimental data. The partial derivatives for the fishnet structure obtained from the simulations are shown in Fig. 7. The real parts of the respective derivatives are depicted by black lines (blue online) , and the imaginary parts by the gray ones (red online) . It is seen that, in the region where the negative refraction is expected, the real part of the refrac-

∂n′/∂|t|, ∂n′′/∂|t|

(a)

8 6 4 2 0 0.8

1

1.2

1.4

1.6

1.8

2

6 ∂n′/∂|r|, ∂n′′/∂|r|

冑兺冉 n

10

(b) 4 2 0 0.8

∂n′/∂arg(t), ∂n′′/∂arg(t)

C. Accuracy of the Method The complex refractive index 共n = n⬘ + in⬙兲 is an indirectly measured quantity depending on the modulus of the transmission coefficient 兩t兩, the phase of the transmission coefficient arg t, the modulus of the reflection coefficient 兩r兩, and the phase of the reflection coefficient arg r. Following the definition of the root-mean-square error of indirectly measured quantities,

tive index depends mostly on the phases of the transmission and reflection coefficients. The dependence on the absolute values of t and r is weak. On the contrary, the imaginary part of the refractive index depends more on the absolute values of t and r than on their phases. This makes questionable the standard retrieval procedure, which is based only on the amplitude measurements. This issue, namely, the necessity of phase measurements for the correct refractive index retrieval was discussed in [25]. Taking the accuracy of the transmittance and reflectance measurements from Table 1 and the experimental values for the derivatives we obtain ␴n⬘ = 0.04 and ␴n⬙ = 0.07 for the fishnet structures at ␭ = 1.45 ␮m and ␴n⬘ = 0.03 and ␴n⬙ = 0.04 for the double-element structures at ␭ = 1.38 ␮m.

∂n′/∂arg(r), ∂n′′/∂arg(r)

tion measurements. The FWHM amounts to 0.135 ␮m and 0.185 ␮m for the fishnet and the double-element structure, respectively. It implies that, for wavelengths larger than 1.6 ␮m, both MMs can be described by an ef1/2 where ␧eff follows from a fective refractive index neff = ␧eff simple effective medium approach. In Figs. 4(c) and 4(d) and Figs. 5(c) and 5(d) the crossed curves depict the transmission and the reflection phases obtained from the transmittance and reflectance measurements. By using Eq. (1) the dispersion relation (or the effective refractive index) of the MM structures can be determined. Results for the fishnet and the double-element structure are shown in Fig. 6(a) and Fig. 6(b), respectively. These results agree with values documented in literature. It should be stressed again, that, in contrast to the previously reported results, they were exclusively obtained from experimental data only.

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(d) 4 2 0 0.8

1


1.8

2

Fig. 7. (Color online) Partial derivatives of n with respect to (a) 兩t兩 , (b) 兩r兩 , (c)arg t, and (d)arg r. Real part n⬘ is represented by black (blue online) curves, imaginary part n⬙is shown by gray (red online) curves.

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5. CONCLUSIONS We have presented an experimental method for the determination of the dispersion relation in optical MMs for normal incidence. We implemented the white-light Fouriertransform spectral interferometry [19] for broadband 共1.1 ␮m – 1.7 ␮m兲 phase measurements in transmission and reflection. The experimental setup for the phase measurements was a Jamin–Lebedeff interferometer modified for measurements in the transmission and in the reflection under normal incidence. For the transmittance and the reflectance measurements we used a PerkinElmer Lambda 950 spectrometer. The effective refractive indices of the investigated MMs have been calculated by using a standard retrieval algorithm [16] using measured complex transmission and reflection coefficients. For both structures investigated, this technique provides an accuracy of about 4% with respect to both the real and imaginary parts of the refractive index. The method was applied to a fishnet and a doubleelement MM. The measured refractive index was n = −0.97± 0.04+ i共1.76± 0.07兲 at ␭ = 1.45 ␮m for the fishnet structure, and n = 0.16± 0.03+ i共0.24± 0.04兲 at the ␭ = 1.38 ␮m for the double-element structure.

Pshenay-Severin et al. 8.

9.

10. 11. 12. 13.

14. 15.

16.

ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from the German Federal Ministry of Education and Research (Metamat) and the State of Thuringia (Proexzellenz Mema).

17. 18.

19.

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