Spectrum of SPP modes

Series of master gratings were prepared by exposure of the resist to the interference field of HeCd laser and developed for different times between 40 s and 90 s that was followed by direct coating with Au (without replication by NIL) as specified in section 4.4.1. In order to optimize the grating depth to achieve full coupling of incident light beam to SPPs at the excitation wavelength
_ (that is close to the absorption band of a dye
_), a series of wavelength reflectivity curves were measured at the angle of incidence G = 0.6 3RS. Let us note that the TM and TE reflectivity are almost identical as the deviation from normal incidence is very small. The results presented in Figure 4.10 (a) reveal that with increasing development time the resonant SPR dip due to the first diffraction order coupling to SPPs gets more pronounced and broader. In addition it can be seen that the resonance position shifts to higher wavelength with increasing etching time. The resonance shifts from
*+,=632.2 nm for 40 s development time to
*+,=639.5 nm for 90 s.

For further studies we replicated the structure with 80 s development time with NIL and evaporated 100 nm of Au on it. The reflectivity spectra is shown in Figure 4.10 (b) and shows that strong coupling of incident beam of about 97 percent at the resonance is achieved. In addition, one can see that the resonance on the replica is shifted by 8 nm to 631.5 deg, which excellently matches the excitation wavelength
_= 633  (of the used HeNe laser) used for further experiments. As the resonance position is mostly determined by the period Λ of the grating, it can be assumed that

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the change in resonance position is due to a slight change of period Λ due to NIL replication with the working stamp and its transfer to UV-curable Amonil MMS 10 resist. The period Λ of the grating can be determined from the diffraction edge at @579 ± 2A  which is solely dependent on the period and the refractive index of the surrounding medium and not on the optical constants of the gold. The equation of this relation is
_6d= Λ (introduced in section 3.1.3) with  the refractive index of the surrounding medium (in this case water) and Λ the grating period. Therefore the period derived from the optical measurement is (435.3 ± 1.5) . This value is in good agreement with the period determined from AFM measurements (Λ = (435 ± 3) ). Additionally to the SPR resonance, another dip in the reflectivity spectra occurs at the wavelength λ=670 nm. The origin of this feature is not clear.

a)

b)

Figure 4.10 Wavelength reflectivity curves for TE polarization and angle of incidence \=0.6 deg for (a) master gratings and (b) replica grating coated with Au and brought in contact with water. In addition the simulated reflectivity curve from section 4.3.1 is replotted for comparison. The experimental reflectivity curves have been normalized to the intensity measured at the critical wavelength Z=580 nm.

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For comparison, the simulated reflectivity curve of section 4.3.1 has been replotted in Figure 4.10 (b). The simulated curve shows slightly higher SPR at 636 nm and reflectivity drops to almost zero. The shape of the resonances of the experiment and the simulation are quite similar. The FWHM of the experiment is ∆
= 16.4 , while the FWHM of the simulation is ∆
= 13.8 . The width of a resonance is a measure of the lifetime of an excited state and in case of SPPs also a measure for the propagation length and near-field intensity. As the simulation gives a slightly smaller width as the experiment, the electric field intensities from the simulation will be likely marginally overestimated. The bigger FWHM in the experiment might be explained by the surface roughness of the gold surface, which was not considered in the numerical simulations. Surface roughness decreases the propagation length and thus increases the width of the resonance. It can be seen in Figure 4.10 (b) that the diffraction edge of the simulated and experimental results are in good agreement.

In Figure 4.11 the angle-wavelength maps for TM-polarized and TE-polarized light of the corrugated Au surface are shown. The TM-reflectivity map shows the typical behavior of a grating in which the one resonance at zero degree splits into two branches when increasing the angle. The two branches can be explained by the SPP dispersion and are the (m=+1, n=0)-mode and the (m=-1, n=0)- mode.

Figure 4.11 Angular and wavelength dependence of reflectivity for (a) TM-polarization and (b) TE polarization (graphic adapted from [145]).

The relation for excitation of SPPs on a 1D grating is given as:

7*++ = °7d+ 7³ (4.4) (a) (b)

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with 7_*++ wave vector of SPPs, 7_d grating wave vector and 7_ wave vector of incident wave in x- direction (in-plane component). Substituting into this equation the definitions (see section 1.2.2) of the quantities and setting the azimuthal angle p = 0, the equation can be rewritten as:

2F
_{∙ 84}45 (
) ∙ 46 5 (
) + 46  = – ∙2F_{Λ + }6∙2F
_{∙ sin θ™}  (4.5)

with
the wavelength of the incident light, 4_5 (
) the real part of the permittivity of gold, 4₆ permittivity of water,  grating order, G polar angle and ₆ refractive index of water. As the real part of the permittivity of gold 4_5 (
) is not an analytical function, this equation cannot be solved analytically. Therefore in Figure 4.12 the equation is graphically solved by assuming the left and the right part of the equation is a function of
. The intersection is thus the wavelength, at which the equation is fulfilled. The angle G and the grating order , can be set as parameters which can be varied.

Figure 4.12 Graph of the wave vector of SPPs and grating vector plus incident light wave vector for different grating orders and different incident angles.

In Figure 4.12 the equation is solved for normal incident light at G = 0 deg and for G = 10 deg with =-1 and =+1. It can be seen that for G = 0 deg there is only one intersection for  = ±1 at 628 nm, which agrees to the experimental resonance position at 631.5 nm. For G = 10 deg there are two intersections for m=+1 at lower wavelength and m=-1 at higher wavelength. These findings explain the splitting of the dispersion relation in Figure 4.11 (a). The TE-reflectivity in Figure 4.11 (b) map does not show a pronounced angular dependence. Only a slight shift to lower wavelength is seen,

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when increasing the angle. The resonance position is not shifted as the in plane component of the polarization is not changing in TE-mode.

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