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THz surface plasmon jump between two metal edges
Optics Communications 277 (2007) 33–39 www.elsevier.com/locate/optcom
THz surface plasmon jump between two metal edges M. Nazarov a, J.-L. Coutaz a
b,*
, A. Shkurinov a, F. Garet
b
Department of Physics and International Laser Center, M.V. Lomonosov Moscow State University, Moscow 119992, Russia b LAHC, University of Savoie, 73 376 Le Bourget du Lac Cedex, France Received 6 October 2006; received in revised form 4 May 2007; accepted 6 May 2007
Abstract We excite surface plasmon in the THz frequency range at a metal surface using a diffraction grating coupler. Then the excited surface plasmon propagates along the flat metal surface, scatters into free space at the sample edge and then couples onto the flat surface of a second device similar to the emitting one. Using THz time-domain spectroscopy, we study plasmon propagation and its coupling processes in time and frequency domains. We measure a surface plasmon propagation length, at a flat air–aluminum interface, smaller than expected. We report for the first time a high coupling efficiency of the THz surface plasmon field between the two grating devices separated by a several centimeters-thick air gap. 2007 Elsevier B.V. All rights reserved. PACS: 73.20.Mf; 84.40.x; 42.79.Dj; 42.25.Fx Keywords: Surface plasmon; Terahertz time-domain spectroscopy; Diffraction grating
1. Introduction Recently, confined terahertz (THz) guided waves at metal surfaces and especially THz surface plasmons (SP) have been the subject of numerous works, involving long distance propagation [1–3], field confinement in quantum cascade structures [4], grating excitation [5–9], etc. The reason for this intense activity is given by numerous foreseen applications such as biosensing and surface spectroscopy. The SP field is concentrated at the metal surface, which enhances the experimental sensitivity to any surface phenomenon. Moreover, THz SP waves are expected to propagate over tens of cm as their encounter losses about 106 times smaller than in the visible. Indeed metals behave practically as ideal conductors in the far infrared, and their electromagnetic response is well described by the Drude theory. The real part of the metal permittivity e could be as high as 104–105, while its imaginary part is even 10–50 *
Corresponding author. Tel.: +33 479 75 87 50; fax: +33 479 75 87 42. E-mail address: [email protected] (J.-L. Coutaz).
0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.05.005
times higher [10,11]. Practically, the SP field does not penetrate the metal because of the high metal permittivity, while it spreads over centimeters in the air, which explains the low SP propagation losses calculated in the THz domain. Nevertheless, SP dumping in THz range, stronger than expected, were noticed in our previous works [7,12], and by number of other groups [3,13,14]. In a recent study of THz SP propagation over a flat aluminum sheet [3], Tae-In Jeon and Grischkowsky have reported a propagation length of 43.5 cm at 0.4 THz instead of 38 m as calculated with an electromagnetic theory using the value of the aluminum complex dielectric constant at THz frequencies found in the literature. Consequently, the measured evanescent extent of the SP field in air is also smaller (7.6 mm) than the theoretical one (21.5 cm). A tentative explanation of this discrepancy by Tae-In Jeon and Grischkowsky is based on the difficulty of exciting a ‘‘true’’ SP mode at the metal surface in their experimental configuration. When arriving at the edge of a flat metal substrate, SP is scattered into the free space out of the metal surface. SP scattering at an edge of metal surface has already been
M. Nazarov et al. / Optics Communications 277 (2007) 33–39 THz receiver
THz emitter
z
SP
x
device B
SP device A
Fig. 1. (a) Picture of the grating coupler; (b) schematics of the coupling experiment.
reported in the infrared [15], and more recently in the far infrared [13,14], by the team of Zhizhin, who excited SP at a metal surface using a razor blade. The interferences pattern resulting from the superposition of the field scattered in air by the razor blade and of the SP field scattered at the metal edge was measured in the far field region. Zhizhin et al. have shown that, due to the large SP field extent in the air, the SP scattered field remains rather well collimated. Moreover they were able to estimate the order of magnitude of the metal permittivity in the THz range. Scattering of THz SP from metal wire was also recently studied by Mittleman et al. [1]. In this paper, we use a grating coupler engraved in a limited area of a flat metal sample to excite THz SP [12]. We propose to study SP as it propagates along the flat metal surface and couples through an air gap or a rectangular step to another similar device facing the emitting one. In the air gap, SP is scattered at the first device edge and this scattered field excites the SP mode of the second device (see Fig. 1). The SP that propagates on the second device is diffracted to free space using second grating as out coupler. We can shift both devices relative to each other and thus study the scattered SP field distribution in space. Due to the large SP field extent in the air, the scattered field remains rather well collimated and its map shape does not vary strongly, thus we expect a high coupling efficiency to the receiving metal structure located at a proper position. We report on such an SP–SP coupling experiment using a THz time-domain set up. From this study, we expect a better knowledge of fundamental properties of THz SP propagation at a metal surface and scattering at a metal edge, with anticipated applications in thin film THz spectroscopy. 2. Experiment The grating device is made of a flat 1-cm thick duralumin sheet (94% aluminum, 5% copper and a weak amount of magnesium and manganese) on which a diffraction grating is manufactured using a milling machine (Fig. 1a). The grating periodicity is K = 1.6 mm, the grating grooves are almost rectangular with a filling factor of nearly 50%. The grating area (1.5 · 2.5 cm2) is located at 3 cm from the duralumin sheet edge. This edge corresponds to the intersection of the flat horizontal and vertical sides of
the sheet, which form a p/2 angle. Both SP emitting (A) and SP receiving (B) devices (see Fig. 1b) are similar except the groove depth (95 lm – emitter and 120 lm – detector). The SP emitter groove depth is smaller to reduce diffraction losses of SP propagating along the grating, while the SP detector groove depth is larger to get a good outcoupling efficiency. Nevertheless, THz reflection spectra of devices A and B are almost identical. The experiment is performed using a classical THz timedomain set up that was already described in [16,17]. The THz beam is p-polarized (transverse magnetic) and its spectrum spreads up to 4 THz. The THz beam is made parallel by a large aperture silicon lens: at the device location, the THz beam diameter is / = 1.5 cm at 0.5 THz. The THz detector position could be adjusted to record the signal reflected by the SP-emitting device A, or the signal diffracted by the SP-receiving device B, both signals being focused onto the THz detector by a second large aperture silicon lens. Device B can be moved in directions x and z (Fig. 1b) using micrometer translation stages. 3. Results and discussion Fig. 2 shows the THz temporal waveform reflected by the first grating – device A – when the angle of incidence is set to h = 39. The reflected waveform is composed of a fast dipolar signal, corresponding to the reflection of the incident pulse, followed by ripples spreading over tens of ps, which arise from the excited SP field that is progressively diffracted by the grating during its propagation over the metal. This ringing signal can roughly be fitted by an exponential function with 12 ps decay time. By timewindowing the temporal waveform with a supergaussian tt0 m window (i.e. the time-window function is eð Dt Þ , where t0 is window center position, 2 Æ Dt is the time-window width, and m is an even integer – here t0 = 29 ps, Dt = 25 ps, m = 16) as shown in Fig. 2, the directly reflected signal is eliminated and the signature of coupled SP can be extracted [7]. The signal detected after the second device
1 0.75
Normalized amplitude
34
0.5 0.25 0 -0.25 -0.5 -0.75 -10
0
10
20
30
40
50
60
70
Time delay (ps) Fig. 2. Temporal waveform of the THz signal reflected by the first grating – device A (continuous line). The dashed curve indicates the time-window used to extract the excited SP spectrum.
M. Nazarov et al. / Optics Communications 277 (2007) 33–39
ki þ m
2p c ¼ k SP $ sin h þ m ¼ nSP K Kf
ð1Þ
where k i ¼ 2pf sin h is the tangential component ofthe incic pffiffiffiffiffi ffi 2pf e dent wavevector, k SP ¼ c nSP and nSP ¼ Re (e is the eþ1
incident 25
0.5 0 -0.5
radiated A
radiated B
0.5 0 0.2
0.4
0.6
0.8
1
Frequency (THz) Fig. 4. Spectra of the incident and reflected signals (upper part of the graph), and of the time-delayed ripples in the reflected waveform and of the signal diffracted by the second grating (lower part of the graph). Vertical unit is the same for all curves, but the scale of the lower part has been enlarged to facilitate the reading.
tered in air and then coupled to a similar SP on the second grating. The excited SP at 0.35 THz is not detected, because this SP is propagating in the direction opposite device B, due to the negative m = 3 order of diffraction. Complementary information on the spectral range around the SP excitation at 0.475 THz is given in Fig. 5. We plot the coefficient of power reflection at grating A together with powers radiated by gratings A and B. These radiated signals are the same as plotted in Fig. 4, but normalized in order to facilitate the comparison of their shapes. The amplitude of the reflectivity curve at SP frequencies is not equal to zero mostly because of a weak coupling efficiency due to a not-optimized grating groove depth and shape. We see that the absorption line in the reflected spectrum exhibits a rather large frequency width, 36 GHz FWHM, that we attribute to the limited area of enlighten grating grooves [12]. Indeed, the frequency resolution Df for grating
0.25
1. 2 reflected radiated A radiated B
0.2
1 0. 8
0.15 0. 6 0.1 0. 4 0.05 0 0.3
-1
reflected
0 1
Power reflection coefficient
1
12.5
0. 2
0.35
0.4
0.45
0.5
0.55
Normalized SP power
Normalized amplitude (A.U.)
dielectric constant of duralumin and Re( ) indicates the real part) are respectively the SP wavevector and effective refractive index. The experimental values lead to nSP 1 (±0.02), without enough accuracy to determine the dielectric constant e of aluminum. We can only assert that jej is much larger than 1. The signature of the coupled SP, i.e. the spectrum of the time-windowed waveform, reveals a strong peak at the SP excitation frequency (0.475 THz). This confirms that energy missing in the reflected spectrum around 0.475 THz is coupled to the SP wave. Besides the main peak at 0.475 THz, only the m = 3 excited SP signature is clearly seen (at 0.35 THz), the other m-line dips are too weak to be observed. The spectrum of the signal detected after the second grating occurs at the same frequency as the excited SP, proving that this detected signal originates from the first SP excited at grating A, which is scat-
37.5
Power (A. U.)
is shown in Fig. 3. Its temporal shape is rather well fitted by a sinus wave with a period of 2 ps limited by a gaussian envelop of width s = 18 ps. Fig. 4 shows the spectra of: (i) the incident signal (as reflected by a flat duralumin mirror), (ii) the signal reflected by grating A, (iii) the time-windowed reflected signal, i.e. the signal radiated by grating A, and (iv) the signal diffracted by grating B. The spectra are obtained by Fourier-transforming the temporal waveforms over a time window (70 ps) much longer than the signal decay time. We check that no disturbing pulses resulting from retroreflections in the set up appear in this time window. The lines at 0.557 and 0.750 THz seen in the incident spectrum are due to THz absorption by water vapor. The reflection curve exhibits a strong absorption line at frequency f = 0.475 THz induced by SP coupling through the diffraction order m = +1 of the grating, as well as a less-pronounced line at f = 0.35 THz (m = 3). The frequency position of these lines obeys the following wavevector conservation:
35
0 0.6
Frequency (THz) -40
-30
-20
-10
0
10
20
30
Time delay (ps) Fig. 3. Temporal waveform radiated from the second grating – device B.
Fig. 5. Spectrum of the reflection coefficient of the THz power and spectra of the signal amplitudes radiated by grating A (i.e. spectrum of the timewindowed reflected temporal waveform drawn in Fig. 2) and by grating B. The radiated signal spectra are normalized to 1 at their peak value.
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M. Nazarov et al. / Optics Communications 277 (2007) 33–39
diffraction is equal to f/N where f is the frequency and N the number of illuminated grooves [16]. In this work, the illuminated length of the grating is //cos h = 19 mm and the period is 1.6 mm, thus about 12 grooves are illuminated, giving Df = 42 GHz at f = 0.5 THz. Actually, as SP propagate over the grating even outside the enlighten area, additional grooves contribute to the diffraction process, which decreases the observed Df value as compared to the one calculated with //cos h. Note that the expected metal losses are almost negligible and would have lead, for an optimized configuration and an ideal plane wave excitation, to a SP bandwidth narrower than the frequency resolution of the set up (6 GHz). The peak around 0.475 THz of the signal radiated by grating A (see curve ‘‘radiated A’’ in Fig. 5) is wider than the corresponding SP line in the reflected spectrum, because we loss a part of the SP energy radiated at early times after excitation by time-windowing the signal in order to get the radiated power. If we shift the time-window closer to the main detected temporal pulse, the SP radiated peak becomes narrower, but the contrast of the peak becomes weaker due to the contribution of the directly reflected energy in the radiated spectrum. The peak at 0.475 THz recorded at device B is narrower (26 GHz FWHM) than the same peak radiated by device A (36 GHz) (Fig. 5), which is consequence of a plasmon ringing decay time at device A (12 ps) shorter than the SP signal at device B (29 ps FWHM). This can be understood, in the time-domain, as follows: the signal radiated by grating B starts to grow when the SP pulse reaches the first border of grating B, and it ends up when the last part of the SP pulse reaches the second border of grating B. As both grating are equal in size, the detected signal should be roughly twice longer than the coupled one, as observed. Note that, contrary to the case of the signal radiated by device A, no artifact due to time-windowing appears for grating B, as we Fourier transform the whole temporal waveform radiated by device B. We determine the grating coupling efficiency and then the coupling efficiency between the two devices as follows. The absorption line due to SP excitation (see Fig. 5), observed in the reflection spectrum, results from two types of losses: (a) the one due to SP excitation as the coupled energy misses in the reflected signal; this occurs around 0.475 THz, (b) losses due to scattering and diffraction towards other diffraction orders, which take place over the whole frequency range. At the boundaries of the SP absorption line, i.e. at 0.45 and 0.51 THz, the reflected energy is around 15% of the incident one. This value corresponds to losses induced by scattering and diffraction at these frequencies. Thus we assume that, if no SP would have been excited, only 15% of the incoming energy would have been reflected in the range 0.45–0.51 THz. Then we integrate the actual reflected signal, i.e. exhibiting the SP absorption line as drawn in Fig. 5, over the range 0.45–0.51 THz. We obtain 9% of the incident energy which is reflected in this frequency range. The difference 15–9% = 6% corresponds to the
amount of energy coupled to the SP. The cause for such small efficiencies (reflection and SP coupling) originates in the large grating period: diffraction orders from m = 4 to m = 1 at 0.5 THz propagate in air, carrying away a lot of energy. The energy coupled to the SP is diffracted by the grating while SP propagates along the grating surface before it reaches the flat part of the duralumin sample. This diffracted signal contributes to a part of the ripples observed after the main reflected signal in the temporal waveform (Fig. 2). To evaluate the amount of this diffracted SP energy, we perform another experiment [12] in which grating A is used to excite the SP, and grating B allows us to measure the SP signal transmitted by a third similar grating C located in between the two others, all of them being engraved on the same duralumin substrate. The so-measured diffraction losses of grating C is about 50%, which means that half the incoming SP energy is diffracted in free space by grating C. In the present experiment, grating A is illuminated at its center, thus the percentage of SP energy lost by diffraction before reaching the flat area is less than 50%. Giving a better estimate of this value is difficult, as the THz signal diffracted out far from the illumination area cannot reach the detector because of geometrical reasons [12]. Thus we estimate the energy I a0 of the SP that starts propagating over the flat duralumin surface to be between 3% (minimum) and 6% (maximum) of the incident energy in the range 0.45–0.51 THz. When putting the second device in contact with the first one, the SP propagates over 6 cm of flat duralumin before reaching the second grating and being diffracted towards the detector. The detected energy ID (integration of ‘‘radiated B’’ signal of Fig. 4 over the range 0.45–0.51 THz) is 0.8% of the incident one I0 (integration of ‘‘incident’’ signal of Fig. 4 over the same range). The out-coupling efficiency of the grating B is about 50%, thus SP energy absorbed after propagation over the 6-cm flat aluminum surface is within the range 47–73%. The corresponding attenuation coefficient a (for field amplitude) is 0.026 < a < 0.063 cm1, i.e. 0.22 < a < 0.54 dB/cm. To get rid of the coupling efficiency problem, we performed a complementary experiment in which an additional 2 cmthick bar of duralumin was inserted in between the two devices, making 8-cm long the total SP propagation length
IR= 0.15 I
Io
ID= 0.5 Ib
o
9%
max 50 % 6%
Ia= 0.06 Io
a
a'
Ia > Ia’ > 0.5 Ia
b
Ib= Ia' e-αL
Fig. 6. Estimation of the coupling and diffraction efficiencies as well as the propagation and diffraction losses when the two devices are in contact.
M. Nazarov et al. / Optics Communications 277 (2007) 33–39
between the two gratings, and we compared the results with the 6-cm propagation case. We obtain a = 0.048 cm1, which confirms our first evaluation. This is comparable with a = 0.023 cm1 reported by Tae-In Jeon and Grischkowsky [3]. The properties of the field radiated at the metal edge are first investigated by measuring the amplitude of the signal coupled to the second device versus the distance x between the two devices (see Fig. 1b). The two devices are set at the same elevation (z = 0). For each distance x, we record the waveform of the signal diffracted by the second device (Fig. 3) and we plot the peak value (at 0.475 THz) of its spectrum (Fig. 4) versus x. The experimental data, normalized at 1 for x = 1 mm, are presented in Fig. 7 for x varying from 0.1 mm up to 25 mm, i.e. from 0.17 · k to 41 · k. Thus the near field regime (x smaller than a few wavelengths) as well as the far field regime (x > 10 Æ k) are studied. As expected, the signal decreases with x but the coupling efficiency remains high since 30% of the energy is coupled for x = 10 mm. The log plot shows a linear behavior in the far field (i.e. for x > 6 mm 9 Æ k) while it saturates for small values pffiffiof ffi x. In the far field, the signal varies as 1:72x0:42 / 1= x, which is the expected behavior of a spherical wave radiated in the far field by a dipolar or punctual source. Let us notice that plotting the maximum of the temporal waveform instead of the spectrum peak value gives the same x-dependence. We use a simple Huygens scalar model to fit the experimental data. We assume that the incoming SP field ESP(0, z) at x = 0 is exponentially decreasing in the air (E(0, z) / ez/d, where d is the decay length of SP in air) and is almost negligible in metal because of the large dielectric constant of the metal. Each point along z in the x = 0 plane serves as a point-source that radiates in free space a spherical wave whose weight is proportional to the amplitude ESP(0, z) of the incoming SP field in z. Then we calculate the radiated field
Erad(x 5 0, z) at any location (x 5 0, z) in free space by adding all the contributions of the points of the x = 0 plane. This gives us the radiated field map in a plane x 5 0. Finally, we define the coupling efficiency g(x) as the overlapping integral of the radiated R þ1 field and the SP field at the second device gðxÞ / 1 Erad ðx; zÞESP ðzÞdz. The single unknown parameter is the decay length d of SP in air. The fits are plotted in Fig. 7 for different values of d. For d = 0.5 mm (±10%), the agreement between theory and experiment is very good above x = 1 mm, in particular in the far field regime. For smaller values of x (0.1 and 0.6 mm), the calculated curves are 10% weaker than the measured data. The crudity of the scalar Huygens theory could easily explain this discrepancy. In particular, for small values of x, back and forth reflections in between the two devices, as well as evanescent waves, may play an important role. The decay of the SP field in the z-direction is studied by putting the two devices in contact (x = 0) and then varying the difference of elevation z between them. Fig. 8 represents the peak signal at 0.475 THz detected after the second device versus z. The signal is maximum when the two devices are on the same elevation, i.e. the SP propagates from one device to the other one without encountering geometrical discontinuity. The signal decreases strongly with the difference of elevation. For weak signals (jzj < 0.5 mm), the results are not symmetrical versus z: when the first device is higher than the second one, the signal is stronger. We model these results using the coupling efficiency defined previously, which writes here Z þ1 gðx ¼ 0; zÞ / ESP ð0; z0 ÞESP ð0; z0 þ zÞdz0 ð2Þ 1
Here again, the only unknown parameter is the decay length d of SP in air. Fig. 6 shows the theoretical curves calculated for d = 0.5, 1 and 1.5 mm. For small z, experimental points are well fitted by the theoretical curves for
1.2
1.2
0.8 0.6
0.2 0 0.1
measured d=0.5 mm d=1 mm d=1.5 mm
1
Normalized amplitude
Amplitude (A. U.)
1
0.4
37
d=0.3 mm d=0.5 mm d=1 mm d=1.5 mm measured 1
x (mm)
0.8 0.6 0.4 0.2 0
10
Fig. 7. Peak amplitude of the detected signal versus the distance x between the two devices. The open circles represent experimental data, while the curves are calculated for different values of the SP decay length d in air.
-0.2
-2
-1
0
z (mm)
1
2
Fig. 8. Peak amplitude of the detected signal versus the elevation z between the two devices. The continuous lines are calculated. The drawings show the relative positions of the two devices for positive and negative values of z.
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M. Nazarov et al. / Optics Communications 277 (2007) 33–39
d = 0.5 mm. This confirms the previous value of d deduced from the x-dependence of the SP–SP coupling. For jzj > 0.5 mm, the crude overlap-integral model is no more valid. Indeed, it does not take into account scattering effects, and also excitation of SP along the vertical face of the devices. From the SP decay length d in air and the SP propagation attenuation coefficient a, we can deduce the value of the dielectric constant e of aluminum at 0.475 THz (let us notice that values at other frequencies may be studied by varying the incidence angle – see relation (1)). The x-component of the wavevector of the air–metal SP writes rffiffiffiffiffiffiffiffiffiffiffi x e k SP ¼ ð3Þ c eþ1 where e = er + jei is the metal complex dielectric constant and x = 2pf is the angular frequency. In the far-infrared, the value of e is large, and thus we can take an approached expression of kSP: x 1 1 k SP c 2e x er x ei 1 2 ð4Þ ¼ þj c c 2ðe2r þ e2i Þ 2ðer þ e2i Þ The attenuation coefficient a of the field amplitude is equal to the imaginary part of kSP: a ¼ Iðk SP Þ ¼
x ei 2 c 2ðer þ e2i Þ
The SP decay length d in air is given by 1 c pffiffiffiffiffiffiffiffiffiffiffi d¼ ¼ I eþ1 Iðk z Þ x c ei ffi ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 2 er þ 1 þ ðer þ 1Þ þ e2i
ð5Þ
thus the small experimental SP propagation length reported here is certainly not related to the excited mode profile. The real part of e is not obtained here with a great accuracy, because er does not play an important role in the expression of the attenuation of the SP field along both x- and z-directions. Its order of magnitude is comparable to that of ei. In fact, the main benefit one can get form such a study is a better knowledge of SP coupling and diffraction at a metal edge. The precise estimation of d, and thus of e, will require complementary works, for example by moving up an absorbing screen above the surface on which SP propagates [3]. Finally, we obtain a 0.05 cm1 and d 0.5 mm while Tae-In Jeon and Grischkowsky [3] report aJG = 0.023 cm1 and dJG = 7.6 mm. As a 2aJG, the large ratio dJG/d can only be explained by a different balance between the real and imaginary parts of e in both experiments, which could result from different experimental conditions, especially the surface state of the metal devices that suffer from oxidation or chemical contamination. In addition, if we assume that our method of determining d, based on Huygens theory, is inaccurate and the actual value of d is the one measured by Tae-In Jeon and Grischkowsky, i.e. 7.6 mm, we obtain ei 6000. This is close to the value that can be derived from Tae-In Jeon and Grischkowsky results, but still far from the expected value, even if better than using d = 0.5 mm. Thus the discrepancy between expected and measured values does not originate from uncertainties in the measurement or in the extraction, but more likely from phenomena occurring at the metal surface, as explained above. 4. Conclusion
ð6Þ
1 ffi is the component of the SP wavevector where k z ¼ ðxc Þ pffiffiffiffiffi eþ1 normal to the metal surface. From the two last relations (5) and (6), and using the values a = 0.05 cm1 and d = 0.5 mm deduced from the experimental data, we obtain ei = 900 (±50%). This method is not the best one to precisely determine e, as the accuracy of the extracted value of ei is quite bad, and it will only give the order of magnitude of e. Nevertheless, the order of magnitude of the extracted value is much smaller than the one usually extrapolated for the far-infrared (e 5 · 104 + j · 106 [10,11]). This may be explained by oxide layers or adsorbates at the metal surface, by scattering by sub-wavelength roughness, or less probably by the use of duralumin instead of aluminum to manufacture the grating device. As compared to the work by Tae-In Jeon and Grischkowsky [3], our grating coupler permits a good control of the SP excitation, even if the small number of illuminated grooves makes difficult the analysis of the diffraction pattern, and
We have demonstrated here that THz SP at a metal surface can without difficulty be excited or coupled out by a diffraction grating. This confirms former results published by the team of Beigang [8] using a grating made of cylindrical rods instead of the grating engraved at the surface of a bulk substrate we used in this study. Moreover, SP propagates over long distances and ‘‘jumps’’ easily from one device to another one. This certainly opens the road to the use of SP as a propagation tool for designing 2D THz devices. Coupling of SP between two shifted metal blocks is well described, in the far field regime, by a Huygens scalar model and overlapping integral calculation. In addition, from the characteristics of SP propagation and diffraction of the studied devices, we estimate the imaginary part of the dielectric constant of aluminum at 0.497 THz. The achieved value is 103–104 smaller than expected from the literature [9,10] inducing a SP propagation length shorter as expected, as already reported by TaeIn Jeon and Grischkowsky [3]. A possible explanation of this difference is the contamination of the metal surface by adsorbates or oxide layers (Al2O3). We are presently checking this hypothesis by performing complementary experimental works. If this explanation is true, excitation,
M. Nazarov et al. / Optics Communications 277 (2007) 33–39
propagation and diffraction of THz SP would be very sensitive to the metal surface properties, which is well known in the visible or near-infrared range [18], making possible THz surface spectroscopy. Acknowledgements Part of this work was performed during the visit of M. Nazarov at LAHC-University of Savoie, supported by a EGIDE program of the French Ministry of Foreign Affairs. M. Nazarov acknowledges the help of Dr. J.-F. Roux who made possible this visit. References [1] K. Wang, D.M. Mittleman, Nature 432 (2004) 376; J.A. Deibel, N. Berndsen, K.N. Wang, D.M. Mittleman, N.C.J. van der Valk, P.C.M. Planken, Opt. Express 14 (2006) 8772. [2] Tae-In Jeon, J. Zhang, D. Grischkowsky, Appl. Phys. Lett. 96 (2005) 161904. [3] Tae-In Jeon, D. Grischkowsky, Appl. Phys. Lett. 88 (2006) 061113. [4] O. Demichel, L. Mahler, T. Losco, C. Mauro, R. Green, A. Tredicucci, J. Xu, F. Beltram, H.E. Beere, D.A. Ritchie, V. Tamosˇinuas, Opt. Express 14 (2006) 5335, and references cited therein. [5] J.F. O’Hara, R.D. Averitt, A.J. Taylor, Opt. Express 12 (2004) 6397. [6] H. Cao, A. Nahata, Opt. Express 13 (2005) 7029.
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[7] M. Nazarov, L. Mukina, A. Shuvaev, D. Sapozhnikov, A. Shkurinov, V. Trofimov, Laser Phys. Lett. 2 (2005) 471; L.S. Mukina, M.M. Nazarov, A.P. Shkurinov, A.V. Shuvaev, ICONO/LAT Technical Digest, IFK3, 2005. [8] G. Torosyan, C. Rau, B. Pradarutti, R. Beigang, Appl. Phys. Lett. 85 (2004) 3372; B. Pradarutti, C. Rau, G. Torosyan, R. Beigang, K. Kawase, Appl. Phys. Lett. 87 (2005) 204105. [9] J. Saxler, J. Gomez Rivas, C. Janke, H.P.M. Pellemans, P. Haring Bolivar, H. Kurz, Phys. Rev. B 69 (2004) 155427. [10] M.A. Ordal, L.L. Long, R.J. Bell, S.E. Bell, R.R. Bell, R.W. Alexander Jr., C.A. Ward, Appl. Opt. 22 (1993) 1099. [11] A.D. Rakic, Appl. Opt. 34 (1995) 4755. [12] L.S. Mukina, M.M. Nazarov, A.P. Shkurinov, Surf. Sci. 600 (2006) 4771. [13] E. Alieva, G. Beitel, L. Kuzik, A. Sigarev, V. Yakovlev, G. Zhizhin, A. van der Meer, M. van der Wiel, J. Mol. Struct. 449 (1999) 119. [14] G. Bogomolov, U.Y. Jeong, G. Zhizhin, A. Nikitin, V. Zavyalov, G. Kazakevich, B. Lee, Nucl. Instrum. Methods Phys. Res. A 543 (2005) 96. [15] G.N. Zhizhin, M.A. Moskaleva, E.V. Shomina, V.A. Yakovlev, JETP Lett. 29 (1979) 486. [16] J.-F. Roux, F. Aquistapace, F. Garet, L. Duvillaret, J.-L. Coutaz, Appl. Opt. 41 (2002) 6507. [17] F. Aquistapace, L. Duvillaret, F. Garet, J.-F. Roux, J.-L. Coutaz, J. Appl. Phys. 94 (2003) 7991. [18] G.C. Schatz, R.P. Van Duyne (Eds.), Electromagnetic Mechanism of Surface-enhanced Spectroscopy, Wiley, New-York, 2002; B. Liedberg, C. Nylander, I. Lundstroem, Sens. Actuators 4 (1993) 299.
THz surface plasmon jump between two metal edges M. Nazarov a, J.-L. Coutaz a
b,*
, A. Shkurinov a, F. Garet
b
Department of Physics and International Laser Center, M.V. Lomonosov Moscow State University, Moscow 119992, Russia b LAHC, University of Savoie, 73 376 Le Bourget du Lac Cedex, France Received 6 October 2006; received in revised form 4 May 2007; accepted 6 May 2007
Abstract We excite surface plasmon in the THz frequency range at a metal surface using a diffraction grating coupler. Then the excited surface plasmon propagates along the flat metal surface, scatters into free space at the sample edge and then couples onto the flat surface of a second device similar to the emitting one. Using THz time-domain spectroscopy, we study plasmon propagation and its coupling processes in time and frequency domains. We measure a surface plasmon propagation length, at a flat air–aluminum interface, smaller than expected. We report for the first time a high coupling efficiency of the THz surface plasmon field between the two grating devices separated by a several centimeters-thick air gap. 2007 Elsevier B.V. All rights reserved. PACS: 73.20.Mf; 84.40.x; 42.79.Dj; 42.25.Fx Keywords: Surface plasmon; Terahertz time-domain spectroscopy; Diffraction grating
1. Introduction Recently, confined terahertz (THz) guided waves at metal surfaces and especially THz surface plasmons (SP) have been the subject of numerous works, involving long distance propagation [1–3], field confinement in quantum cascade structures [4], grating excitation [5–9], etc. The reason for this intense activity is given by numerous foreseen applications such as biosensing and surface spectroscopy. The SP field is concentrated at the metal surface, which enhances the experimental sensitivity to any surface phenomenon. Moreover, THz SP waves are expected to propagate over tens of cm as their encounter losses about 106 times smaller than in the visible. Indeed metals behave practically as ideal conductors in the far infrared, and their electromagnetic response is well described by the Drude theory. The real part of the metal permittivity e could be as high as 104–105, while its imaginary part is even 10–50 *
Corresponding author. Tel.: +33 479 75 87 50; fax: +33 479 75 87 42. E-mail address: [email protected] (J.-L. Coutaz).
0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.05.005
times higher [10,11]. Practically, the SP field does not penetrate the metal because of the high metal permittivity, while it spreads over centimeters in the air, which explains the low SP propagation losses calculated in the THz domain. Nevertheless, SP dumping in THz range, stronger than expected, were noticed in our previous works [7,12], and by number of other groups [3,13,14]. In a recent study of THz SP propagation over a flat aluminum sheet [3], Tae-In Jeon and Grischkowsky have reported a propagation length of 43.5 cm at 0.4 THz instead of 38 m as calculated with an electromagnetic theory using the value of the aluminum complex dielectric constant at THz frequencies found in the literature. Consequently, the measured evanescent extent of the SP field in air is also smaller (7.6 mm) than the theoretical one (21.5 cm). A tentative explanation of this discrepancy by Tae-In Jeon and Grischkowsky is based on the difficulty of exciting a ‘‘true’’ SP mode at the metal surface in their experimental configuration. When arriving at the edge of a flat metal substrate, SP is scattered into the free space out of the metal surface. SP scattering at an edge of metal surface has already been
M. Nazarov et al. / Optics Communications 277 (2007) 33–39 THz receiver
THz emitter
z
SP
x
device B
SP device A
Fig. 1. (a) Picture of the grating coupler; (b) schematics of the coupling experiment.
reported in the infrared [15], and more recently in the far infrared [13,14], by the team of Zhizhin, who excited SP at a metal surface using a razor blade. The interferences pattern resulting from the superposition of the field scattered in air by the razor blade and of the SP field scattered at the metal edge was measured in the far field region. Zhizhin et al. have shown that, due to the large SP field extent in the air, the SP scattered field remains rather well collimated. Moreover they were able to estimate the order of magnitude of the metal permittivity in the THz range. Scattering of THz SP from metal wire was also recently studied by Mittleman et al. [1]. In this paper, we use a grating coupler engraved in a limited area of a flat metal sample to excite THz SP [12]. We propose to study SP as it propagates along the flat metal surface and couples through an air gap or a rectangular step to another similar device facing the emitting one. In the air gap, SP is scattered at the first device edge and this scattered field excites the SP mode of the second device (see Fig. 1). The SP that propagates on the second device is diffracted to free space using second grating as out coupler. We can shift both devices relative to each other and thus study the scattered SP field distribution in space. Due to the large SP field extent in the air, the scattered field remains rather well collimated and its map shape does not vary strongly, thus we expect a high coupling efficiency to the receiving metal structure located at a proper position. We report on such an SP–SP coupling experiment using a THz time-domain set up. From this study, we expect a better knowledge of fundamental properties of THz SP propagation at a metal surface and scattering at a metal edge, with anticipated applications in thin film THz spectroscopy. 2. Experiment The grating device is made of a flat 1-cm thick duralumin sheet (94% aluminum, 5% copper and a weak amount of magnesium and manganese) on which a diffraction grating is manufactured using a milling machine (Fig. 1a). The grating periodicity is K = 1.6 mm, the grating grooves are almost rectangular with a filling factor of nearly 50%. The grating area (1.5 · 2.5 cm2) is located at 3 cm from the duralumin sheet edge. This edge corresponds to the intersection of the flat horizontal and vertical sides of
the sheet, which form a p/2 angle. Both SP emitting (A) and SP receiving (B) devices (see Fig. 1b) are similar except the groove depth (95 lm – emitter and 120 lm – detector). The SP emitter groove depth is smaller to reduce diffraction losses of SP propagating along the grating, while the SP detector groove depth is larger to get a good outcoupling efficiency. Nevertheless, THz reflection spectra of devices A and B are almost identical. The experiment is performed using a classical THz timedomain set up that was already described in [16,17]. The THz beam is p-polarized (transverse magnetic) and its spectrum spreads up to 4 THz. The THz beam is made parallel by a large aperture silicon lens: at the device location, the THz beam diameter is / = 1.5 cm at 0.5 THz. The THz detector position could be adjusted to record the signal reflected by the SP-emitting device A, or the signal diffracted by the SP-receiving device B, both signals being focused onto the THz detector by a second large aperture silicon lens. Device B can be moved in directions x and z (Fig. 1b) using micrometer translation stages. 3. Results and discussion Fig. 2 shows the THz temporal waveform reflected by the first grating – device A – when the angle of incidence is set to h = 39. The reflected waveform is composed of a fast dipolar signal, corresponding to the reflection of the incident pulse, followed by ripples spreading over tens of ps, which arise from the excited SP field that is progressively diffracted by the grating during its propagation over the metal. This ringing signal can roughly be fitted by an exponential function with 12 ps decay time. By timewindowing the temporal waveform with a supergaussian tt0 m window (i.e. the time-window function is eð Dt Þ , where t0 is window center position, 2 Æ Dt is the time-window width, and m is an even integer – here t0 = 29 ps, Dt = 25 ps, m = 16) as shown in Fig. 2, the directly reflected signal is eliminated and the signature of coupled SP can be extracted [7]. The signal detected after the second device
1 0.75
Normalized amplitude
34
0.5 0.25 0 -0.25 -0.5 -0.75 -10
0
10
20
30
40
50
60
70
Time delay (ps) Fig. 2. Temporal waveform of the THz signal reflected by the first grating – device A (continuous line). The dashed curve indicates the time-window used to extract the excited SP spectrum.
M. Nazarov et al. / Optics Communications 277 (2007) 33–39
ki þ m
2p c ¼ k SP $ sin h þ m ¼ nSP K Kf
ð1Þ
where k i ¼ 2pf sin h is the tangential component ofthe incic pffiffiffiffiffi ffi 2pf e dent wavevector, k SP ¼ c nSP and nSP ¼ Re (e is the eþ1
incident 25
0.5 0 -0.5
radiated A
radiated B
0.5 0 0.2
0.4
0.6
0.8
1
Frequency (THz) Fig. 4. Spectra of the incident and reflected signals (upper part of the graph), and of the time-delayed ripples in the reflected waveform and of the signal diffracted by the second grating (lower part of the graph). Vertical unit is the same for all curves, but the scale of the lower part has been enlarged to facilitate the reading.
tered in air and then coupled to a similar SP on the second grating. The excited SP at 0.35 THz is not detected, because this SP is propagating in the direction opposite device B, due to the negative m = 3 order of diffraction. Complementary information on the spectral range around the SP excitation at 0.475 THz is given in Fig. 5. We plot the coefficient of power reflection at grating A together with powers radiated by gratings A and B. These radiated signals are the same as plotted in Fig. 4, but normalized in order to facilitate the comparison of their shapes. The amplitude of the reflectivity curve at SP frequencies is not equal to zero mostly because of a weak coupling efficiency due to a not-optimized grating groove depth and shape. We see that the absorption line in the reflected spectrum exhibits a rather large frequency width, 36 GHz FWHM, that we attribute to the limited area of enlighten grating grooves [12]. Indeed, the frequency resolution Df for grating
0.25
1. 2 reflected radiated A radiated B
0.2
1 0. 8
0.15 0. 6 0.1 0. 4 0.05 0 0.3
-1
reflected
0 1
Power reflection coefficient
1
12.5
0. 2
0.35
0.4
0.45
0.5
0.55
Normalized SP power
Normalized amplitude (A.U.)
dielectric constant of duralumin and Re( ) indicates the real part) are respectively the SP wavevector and effective refractive index. The experimental values lead to nSP 1 (±0.02), without enough accuracy to determine the dielectric constant e of aluminum. We can only assert that jej is much larger than 1. The signature of the coupled SP, i.e. the spectrum of the time-windowed waveform, reveals a strong peak at the SP excitation frequency (0.475 THz). This confirms that energy missing in the reflected spectrum around 0.475 THz is coupled to the SP wave. Besides the main peak at 0.475 THz, only the m = 3 excited SP signature is clearly seen (at 0.35 THz), the other m-line dips are too weak to be observed. The spectrum of the signal detected after the second grating occurs at the same frequency as the excited SP, proving that this detected signal originates from the first SP excited at grating A, which is scat-
37.5
Power (A. U.)
is shown in Fig. 3. Its temporal shape is rather well fitted by a sinus wave with a period of 2 ps limited by a gaussian envelop of width s = 18 ps. Fig. 4 shows the spectra of: (i) the incident signal (as reflected by a flat duralumin mirror), (ii) the signal reflected by grating A, (iii) the time-windowed reflected signal, i.e. the signal radiated by grating A, and (iv) the signal diffracted by grating B. The spectra are obtained by Fourier-transforming the temporal waveforms over a time window (70 ps) much longer than the signal decay time. We check that no disturbing pulses resulting from retroreflections in the set up appear in this time window. The lines at 0.557 and 0.750 THz seen in the incident spectrum are due to THz absorption by water vapor. The reflection curve exhibits a strong absorption line at frequency f = 0.475 THz induced by SP coupling through the diffraction order m = +1 of the grating, as well as a less-pronounced line at f = 0.35 THz (m = 3). The frequency position of these lines obeys the following wavevector conservation:
35
0 0.6
Frequency (THz) -40
-30
-20
-10
0
10
20
30
Time delay (ps) Fig. 3. Temporal waveform radiated from the second grating – device B.
Fig. 5. Spectrum of the reflection coefficient of the THz power and spectra of the signal amplitudes radiated by grating A (i.e. spectrum of the timewindowed reflected temporal waveform drawn in Fig. 2) and by grating B. The radiated signal spectra are normalized to 1 at their peak value.
36
M. Nazarov et al. / Optics Communications 277 (2007) 33–39
diffraction is equal to f/N where f is the frequency and N the number of illuminated grooves [16]. In this work, the illuminated length of the grating is //cos h = 19 mm and the period is 1.6 mm, thus about 12 grooves are illuminated, giving Df = 42 GHz at f = 0.5 THz. Actually, as SP propagate over the grating even outside the enlighten area, additional grooves contribute to the diffraction process, which decreases the observed Df value as compared to the one calculated with //cos h. Note that the expected metal losses are almost negligible and would have lead, for an optimized configuration and an ideal plane wave excitation, to a SP bandwidth narrower than the frequency resolution of the set up (6 GHz). The peak around 0.475 THz of the signal radiated by grating A (see curve ‘‘radiated A’’ in Fig. 5) is wider than the corresponding SP line in the reflected spectrum, because we loss a part of the SP energy radiated at early times after excitation by time-windowing the signal in order to get the radiated power. If we shift the time-window closer to the main detected temporal pulse, the SP radiated peak becomes narrower, but the contrast of the peak becomes weaker due to the contribution of the directly reflected energy in the radiated spectrum. The peak at 0.475 THz recorded at device B is narrower (26 GHz FWHM) than the same peak radiated by device A (36 GHz) (Fig. 5), which is consequence of a plasmon ringing decay time at device A (12 ps) shorter than the SP signal at device B (29 ps FWHM). This can be understood, in the time-domain, as follows: the signal radiated by grating B starts to grow when the SP pulse reaches the first border of grating B, and it ends up when the last part of the SP pulse reaches the second border of grating B. As both grating are equal in size, the detected signal should be roughly twice longer than the coupled one, as observed. Note that, contrary to the case of the signal radiated by device A, no artifact due to time-windowing appears for grating B, as we Fourier transform the whole temporal waveform radiated by device B. We determine the grating coupling efficiency and then the coupling efficiency between the two devices as follows. The absorption line due to SP excitation (see Fig. 5), observed in the reflection spectrum, results from two types of losses: (a) the one due to SP excitation as the coupled energy misses in the reflected signal; this occurs around 0.475 THz, (b) losses due to scattering and diffraction towards other diffraction orders, which take place over the whole frequency range. At the boundaries of the SP absorption line, i.e. at 0.45 and 0.51 THz, the reflected energy is around 15% of the incident one. This value corresponds to losses induced by scattering and diffraction at these frequencies. Thus we assume that, if no SP would have been excited, only 15% of the incoming energy would have been reflected in the range 0.45–0.51 THz. Then we integrate the actual reflected signal, i.e. exhibiting the SP absorption line as drawn in Fig. 5, over the range 0.45–0.51 THz. We obtain 9% of the incident energy which is reflected in this frequency range. The difference 15–9% = 6% corresponds to the
amount of energy coupled to the SP. The cause for such small efficiencies (reflection and SP coupling) originates in the large grating period: diffraction orders from m = 4 to m = 1 at 0.5 THz propagate in air, carrying away a lot of energy. The energy coupled to the SP is diffracted by the grating while SP propagates along the grating surface before it reaches the flat part of the duralumin sample. This diffracted signal contributes to a part of the ripples observed after the main reflected signal in the temporal waveform (Fig. 2). To evaluate the amount of this diffracted SP energy, we perform another experiment [12] in which grating A is used to excite the SP, and grating B allows us to measure the SP signal transmitted by a third similar grating C located in between the two others, all of them being engraved on the same duralumin substrate. The so-measured diffraction losses of grating C is about 50%, which means that half the incoming SP energy is diffracted in free space by grating C. In the present experiment, grating A is illuminated at its center, thus the percentage of SP energy lost by diffraction before reaching the flat area is less than 50%. Giving a better estimate of this value is difficult, as the THz signal diffracted out far from the illumination area cannot reach the detector because of geometrical reasons [12]. Thus we estimate the energy I a0 of the SP that starts propagating over the flat duralumin surface to be between 3% (minimum) and 6% (maximum) of the incident energy in the range 0.45–0.51 THz. When putting the second device in contact with the first one, the SP propagates over 6 cm of flat duralumin before reaching the second grating and being diffracted towards the detector. The detected energy ID (integration of ‘‘radiated B’’ signal of Fig. 4 over the range 0.45–0.51 THz) is 0.8% of the incident one I0 (integration of ‘‘incident’’ signal of Fig. 4 over the same range). The out-coupling efficiency of the grating B is about 50%, thus SP energy absorbed after propagation over the 6-cm flat aluminum surface is within the range 47–73%. The corresponding attenuation coefficient a (for field amplitude) is 0.026 < a < 0.063 cm1, i.e. 0.22 < a < 0.54 dB/cm. To get rid of the coupling efficiency problem, we performed a complementary experiment in which an additional 2 cmthick bar of duralumin was inserted in between the two devices, making 8-cm long the total SP propagation length
IR= 0.15 I
Io
ID= 0.5 Ib
o
9%
max 50 % 6%
Ia= 0.06 Io
a
a'
Ia > Ia’ > 0.5 Ia
b
Ib= Ia' e-αL
Fig. 6. Estimation of the coupling and diffraction efficiencies as well as the propagation and diffraction losses when the two devices are in contact.
M. Nazarov et al. / Optics Communications 277 (2007) 33–39
between the two gratings, and we compared the results with the 6-cm propagation case. We obtain a = 0.048 cm1, which confirms our first evaluation. This is comparable with a = 0.023 cm1 reported by Tae-In Jeon and Grischkowsky [3]. The properties of the field radiated at the metal edge are first investigated by measuring the amplitude of the signal coupled to the second device versus the distance x between the two devices (see Fig. 1b). The two devices are set at the same elevation (z = 0). For each distance x, we record the waveform of the signal diffracted by the second device (Fig. 3) and we plot the peak value (at 0.475 THz) of its spectrum (Fig. 4) versus x. The experimental data, normalized at 1 for x = 1 mm, are presented in Fig. 7 for x varying from 0.1 mm up to 25 mm, i.e. from 0.17 · k to 41 · k. Thus the near field regime (x smaller than a few wavelengths) as well as the far field regime (x > 10 Æ k) are studied. As expected, the signal decreases with x but the coupling efficiency remains high since 30% of the energy is coupled for x = 10 mm. The log plot shows a linear behavior in the far field (i.e. for x > 6 mm 9 Æ k) while it saturates for small values pffiffiof ffi x. In the far field, the signal varies as 1:72x0:42 / 1= x, which is the expected behavior of a spherical wave radiated in the far field by a dipolar or punctual source. Let us notice that plotting the maximum of the temporal waveform instead of the spectrum peak value gives the same x-dependence. We use a simple Huygens scalar model to fit the experimental data. We assume that the incoming SP field ESP(0, z) at x = 0 is exponentially decreasing in the air (E(0, z) / ez/d, where d is the decay length of SP in air) and is almost negligible in metal because of the large dielectric constant of the metal. Each point along z in the x = 0 plane serves as a point-source that radiates in free space a spherical wave whose weight is proportional to the amplitude ESP(0, z) of the incoming SP field in z. Then we calculate the radiated field
Erad(x 5 0, z) at any location (x 5 0, z) in free space by adding all the contributions of the points of the x = 0 plane. This gives us the radiated field map in a plane x 5 0. Finally, we define the coupling efficiency g(x) as the overlapping integral of the radiated R þ1 field and the SP field at the second device gðxÞ / 1 Erad ðx; zÞESP ðzÞdz. The single unknown parameter is the decay length d of SP in air. The fits are plotted in Fig. 7 for different values of d. For d = 0.5 mm (±10%), the agreement between theory and experiment is very good above x = 1 mm, in particular in the far field regime. For smaller values of x (0.1 and 0.6 mm), the calculated curves are 10% weaker than the measured data. The crudity of the scalar Huygens theory could easily explain this discrepancy. In particular, for small values of x, back and forth reflections in between the two devices, as well as evanescent waves, may play an important role. The decay of the SP field in the z-direction is studied by putting the two devices in contact (x = 0) and then varying the difference of elevation z between them. Fig. 8 represents the peak signal at 0.475 THz detected after the second device versus z. The signal is maximum when the two devices are on the same elevation, i.e. the SP propagates from one device to the other one without encountering geometrical discontinuity. The signal decreases strongly with the difference of elevation. For weak signals (jzj < 0.5 mm), the results are not symmetrical versus z: when the first device is higher than the second one, the signal is stronger. We model these results using the coupling efficiency defined previously, which writes here Z þ1 gðx ¼ 0; zÞ / ESP ð0; z0 ÞESP ð0; z0 þ zÞdz0 ð2Þ 1
Here again, the only unknown parameter is the decay length d of SP in air. Fig. 6 shows the theoretical curves calculated for d = 0.5, 1 and 1.5 mm. For small z, experimental points are well fitted by the theoretical curves for
1.2
1.2
0.8 0.6
0.2 0 0.1
measured d=0.5 mm d=1 mm d=1.5 mm
1
Normalized amplitude
Amplitude (A. U.)
1
0.4
37
d=0.3 mm d=0.5 mm d=1 mm d=1.5 mm measured 1
x (mm)
0.8 0.6 0.4 0.2 0
10
Fig. 7. Peak amplitude of the detected signal versus the distance x between the two devices. The open circles represent experimental data, while the curves are calculated for different values of the SP decay length d in air.
-0.2
-2
-1
0
z (mm)
1
2
Fig. 8. Peak amplitude of the detected signal versus the elevation z between the two devices. The continuous lines are calculated. The drawings show the relative positions of the two devices for positive and negative values of z.
38
M. Nazarov et al. / Optics Communications 277 (2007) 33–39
d = 0.5 mm. This confirms the previous value of d deduced from the x-dependence of the SP–SP coupling. For jzj > 0.5 mm, the crude overlap-integral model is no more valid. Indeed, it does not take into account scattering effects, and also excitation of SP along the vertical face of the devices. From the SP decay length d in air and the SP propagation attenuation coefficient a, we can deduce the value of the dielectric constant e of aluminum at 0.475 THz (let us notice that values at other frequencies may be studied by varying the incidence angle – see relation (1)). The x-component of the wavevector of the air–metal SP writes rffiffiffiffiffiffiffiffiffiffiffi x e k SP ¼ ð3Þ c eþ1 where e = er + jei is the metal complex dielectric constant and x = 2pf is the angular frequency. In the far-infrared, the value of e is large, and thus we can take an approached expression of kSP: x 1 1 k SP c 2e x er x ei 1 2 ð4Þ ¼ þj c c 2ðe2r þ e2i Þ 2ðer þ e2i Þ The attenuation coefficient a of the field amplitude is equal to the imaginary part of kSP: a ¼ Iðk SP Þ ¼
x ei 2 c 2ðer þ e2i Þ
The SP decay length d in air is given by 1 c pffiffiffiffiffiffiffiffiffiffiffi d¼ ¼ I eþ1 Iðk z Þ x c ei ffi ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 2 er þ 1 þ ðer þ 1Þ þ e2i
ð5Þ
thus the small experimental SP propagation length reported here is certainly not related to the excited mode profile. The real part of e is not obtained here with a great accuracy, because er does not play an important role in the expression of the attenuation of the SP field along both x- and z-directions. Its order of magnitude is comparable to that of ei. In fact, the main benefit one can get form such a study is a better knowledge of SP coupling and diffraction at a metal edge. The precise estimation of d, and thus of e, will require complementary works, for example by moving up an absorbing screen above the surface on which SP propagates [3]. Finally, we obtain a 0.05 cm1 and d 0.5 mm while Tae-In Jeon and Grischkowsky [3] report aJG = 0.023 cm1 and dJG = 7.6 mm. As a 2aJG, the large ratio dJG/d can only be explained by a different balance between the real and imaginary parts of e in both experiments, which could result from different experimental conditions, especially the surface state of the metal devices that suffer from oxidation or chemical contamination. In addition, if we assume that our method of determining d, based on Huygens theory, is inaccurate and the actual value of d is the one measured by Tae-In Jeon and Grischkowsky, i.e. 7.6 mm, we obtain ei 6000. This is close to the value that can be derived from Tae-In Jeon and Grischkowsky results, but still far from the expected value, even if better than using d = 0.5 mm. Thus the discrepancy between expected and measured values does not originate from uncertainties in the measurement or in the extraction, but more likely from phenomena occurring at the metal surface, as explained above. 4. Conclusion
ð6Þ
1 ffi is the component of the SP wavevector where k z ¼ ðxc Þ pffiffiffiffiffi eþ1 normal to the metal surface. From the two last relations (5) and (6), and using the values a = 0.05 cm1 and d = 0.5 mm deduced from the experimental data, we obtain ei = 900 (±50%). This method is not the best one to precisely determine e, as the accuracy of the extracted value of ei is quite bad, and it will only give the order of magnitude of e. Nevertheless, the order of magnitude of the extracted value is much smaller than the one usually extrapolated for the far-infrared (e 5 · 104 + j · 106 [10,11]). This may be explained by oxide layers or adsorbates at the metal surface, by scattering by sub-wavelength roughness, or less probably by the use of duralumin instead of aluminum to manufacture the grating device. As compared to the work by Tae-In Jeon and Grischkowsky [3], our grating coupler permits a good control of the SP excitation, even if the small number of illuminated grooves makes difficult the analysis of the diffraction pattern, and
We have demonstrated here that THz SP at a metal surface can without difficulty be excited or coupled out by a diffraction grating. This confirms former results published by the team of Beigang [8] using a grating made of cylindrical rods instead of the grating engraved at the surface of a bulk substrate we used in this study. Moreover, SP propagates over long distances and ‘‘jumps’’ easily from one device to another one. This certainly opens the road to the use of SP as a propagation tool for designing 2D THz devices. Coupling of SP between two shifted metal blocks is well described, in the far field regime, by a Huygens scalar model and overlapping integral calculation. In addition, from the characteristics of SP propagation and diffraction of the studied devices, we estimate the imaginary part of the dielectric constant of aluminum at 0.497 THz. The achieved value is 103–104 smaller than expected from the literature [9,10] inducing a SP propagation length shorter as expected, as already reported by TaeIn Jeon and Grischkowsky [3]. A possible explanation of this difference is the contamination of the metal surface by adsorbates or oxide layers (Al2O3). We are presently checking this hypothesis by performing complementary experimental works. If this explanation is true, excitation,
M. Nazarov et al. / Optics Communications 277 (2007) 33–39
propagation and diffraction of THz SP would be very sensitive to the metal surface properties, which is well known in the visible or near-infrared range [18], making possible THz surface spectroscopy. Acknowledgements Part of this work was performed during the visit of M. Nazarov at LAHC-University of Savoie, supported by a EGIDE program of the French Ministry of Foreign Affairs. M. Nazarov acknowledges the help of Dr. J.-F. Roux who made possible this visit. References [1] K. Wang, D.M. Mittleman, Nature 432 (2004) 376; J.A. Deibel, N. Berndsen, K.N. Wang, D.M. Mittleman, N.C.J. van der Valk, P.C.M. Planken, Opt. Express 14 (2006) 8772. [2] Tae-In Jeon, J. Zhang, D. Grischkowsky, Appl. Phys. Lett. 96 (2005) 161904. [3] Tae-In Jeon, D. Grischkowsky, Appl. Phys. Lett. 88 (2006) 061113. [4] O. Demichel, L. Mahler, T. Losco, C. Mauro, R. Green, A. Tredicucci, J. Xu, F. Beltram, H.E. Beere, D.A. Ritchie, V. Tamosˇinuas, Opt. Express 14 (2006) 5335, and references cited therein. [5] J.F. O’Hara, R.D. Averitt, A.J. Taylor, Opt. Express 12 (2004) 6397. [6] H. Cao, A. Nahata, Opt. Express 13 (2005) 7029.
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