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We have explicated the Goos-Hänchen (GH) shift in a μm-order Kretchmann-Raether configuration embedded in an optical waveguide structure by using the finite-difference time-domain method. For optical waveguide-type surface plasmon resonance (SPR) devices, the precise derivation of the GH shift has become critical. Artmann’s equation, which is accurate enough for bulk optics, is difficult to apply to waveguide-type SPR devices. This is because Artmann's equation, based on the differentiation of the phase shift, is inaccurate at the critical and resonance angles where drastic phase changes occur. In this study, we accurately identified both the positive and the negative GH shifts around the incidence angle of resonance. In a waveguide-type Kretchmann-Raether configuration with an Au thin film of 50 nm, positive and negative lateral shifts of −0.75 and + 1.0 μm are obtained on the SPR with the incident angles of 44.4 ° and 47.5 °, respectively, at a wavelength of 632.8 nm.
©2009 Optical Society of America
1. Introduction
The phenomenon of surface plasmon resonance (SPR) was first observed in an attenuated total reflection (TIR) mirror devised by Kretschman [1] and Otto [2]. A SPR is generated when a p-polarized light is launched into a prism coated with a thin-film metal such as silver, gold, or aluminum. When in a resonant condition, the incident light is highly absorbed and loses a fair amount of its energy, resulting in a dip in the intensity profile of the reflected light. In this case, the required electric-field components needed to excite the surface plasmon wave (SPW) have to be p-polarized light, because this particular polarization has an electric-field vector that oscillates naturally with the plane that contains the metal film. These resonance characteristics have been employed as a real-time monitor of the surface interactions [3]. More development of different types of SPR sensors have been done recently, which are comparable to or better than the conventional SPR sensors in terms of sensitivity, and compactness, along with a lower cost. As a good example, a SPR sensor based on an optical waveguide can be reduced in μm-order dimensions in order to obtain a greater compactness and a high sensitivity. For this case, the waveguide’s facets are coated with a thin-film metal in order to construct a SPR. When the propagated light is reflected, the reflected light has a lateral shift at the metal interface, which is named the Goos-Hänchen (GH) shift [4]. The GH shift occurs between the dielectric materials and the metal layer due to the phase transition of the reflected light. Since the reflection of light at a metallic surface is non-linear, the precise parameters of the GH shift are very important in the design of μm-order SPR waveguide devices.
In this paper, we have examined the GH shift in detail as used for μm-order SPR waveguide devices in a Kretschmann-Raether configuration using the finite-difference time-domain (FDTD) method. The frequency dependent function of the metal thin film was characterized by the Drude model using this FDTD method [5].
2. Theoretical analysis
1) The Goos-Hänchen shift
Figure 1 shows the geometry of the GH shift. In general, the GH shift, illustrated by D, occurs in the positive lateral direction at the interface of the dielectric materials. The GH shift, as proposed by Artmann, is seen in the phase transition of the reflected light, and is given by [6]:
Here, is the phase shift as given by:In Eq. (2), phase factor δ by p-polarized light is:The complex admittance of incident medium η is:and the admittance of free space is: Figure 2 shows the phase shift of a prism-air interface without a metal film for p-polarized light. The prism refractive index is 1.515 at a wavelength of 632.8 nm. The phase shift variation is started at a critical angle of 41.3°.
Fig. 1 The geometry for the Goos-Hänchen shift at a single interface between the double dielectric layers
In Fig. 3 , the results using both Artmann’s equation and the FDTD method are shown for a prism-air interface without a metal film. The maximum GH shift appears near to the critical angle. The calculated results from the FDTD method agree well, except for the critical angle, with those derived by Artmann’s equation.
Fig. 3 The GH shift between dielectric/air interface obtained by the Artmann’s equation (solid line) and the FDTD method (dot point)
2) The Drude model used for the FDTD analysis
The FDTD method, originally proposed by Yee [7], is one of the most successful algorithms used for the integration of the time dependent form of Maxwell’s equations. Since Yee’s original formulation only describes isotropic materials with a static permittivity, it has a certain limit in the application to noble metals, especially at high frequencys with a negative permittivity. A simple model was developed by Drude for noble metals, based on the kinetic gas theory. It assumes that the electrons with a common relaxation time were independent and free. This free-electron model was later modified to include a minor correction for the band-structure of matter (effective mass) and termed the quasi-electron model. This Drude model describes the characteristics of metals well, despite its drastic assumptions. Optical properties described by the frequency dependent dielectric function ε(ω) are predicted by:
where is the plasma frequency and is the electron relaxation rate.We used the frequency dependent dielectric permittivity for gold obtained from P.B. Johnson and R. W. Christy’s experimental results [8]. The complex dielectric permittivity of gold using the Drude model was fitted with a plasma frequency of 8.63 eV, a collision rate of 109 meV, and a static permittivity () of 9.84. These fitted results by the Drude model agreed well with the experimental data obtained at a wavelength of 632.8 nm. The limiting case for Maxwell’s equations can be expressed as [9]:
and:Here, the current-density vector J is initially at zero in the regions of free space and having some dielectric material with a real dielectric constant. However, ,, and have some values in the metallic regions as seen in the Drude model. Therefore, the current-density vector can be represented by:3. The simulation results and a discussion
In the waveguide-type Kretchmann-Raether configuration with a prism-thin Au film of 50 nm, for p-polarized light, the reflected intensity and the phase shift as a function of the incident angle are shown in Fig. 5 . The resonance angle needed to generate a maximum surface plasmon wave was 44.4 °. Then, the phase shift abruptly changed from + 180 to −180 ° at the resonance angle.Figure 6 shows the GH shift obtained by Artmann’s equation with respect to the incident angle. As shown, the GH shift increased rapidly at the critical and the resonance angles making Artmann’s equation difficult to apply to μm-order waveguide-type SPR devices. The drastic phase change occurring around the critical and the resonance angles is because Artmann’s equation is solved by the differentiation of the phase shift with respect to the incident angle [10, 11]. Therefore, the GH shift in μm-order waveguide-type devices needs to be examined by the FDTD method in order to obtain reasonable values at the resonance angle.
Fig. 5 The reflected intensity and the phase shift as a function of the incident angle for p-polarized light
As shown in Fig. 7 , the GH shift as a function of the incident angle is derived by using the FDTD method. Above a 50 ° angle the GH lateral shift is negative. When the incident light is at the total internal reflection angle, this phenomenon was predicted and measured in the noble metal absorbing media [12, 13].
For an incident angle of between 40 and 40.7 °, the GH shift was also in the negative direction. The maximum negative lateral shift of −0.75 μm was found at the resonance angle of 44.4 °. The optical-mode propagations by the FDTD method are shown in the insets. Under a resonance condition, the incident electric field results in an electron-photon coupling in the metal thin-film. The incident-angle-dependent wavevector can cause a drastic phase change when the incident light frequency is similar to the plasma frequency of Au. This is because at resonance the polarization between the free electron on the outermost 6s orbital and the ion core is expanded which polarizes the filled electrons on the 5d orbital. Due to these processes, the light energy is absorbed into the metal layer, and when the absorbed light is reradiated, the inhomogeneous negative energy flux due to the phase shift at the metal surface is intensified.
Finally, the GH shift in the positive direction started to appear when the incident angle was larger than the resonance angle. The positive lateral shift was increased by up to +1.0 μm at an incident angle of 47.5 °. As the incident angle increases above the resonance angle not only is the SPW decreased, but some of the SPW energy is leaked into the prism due to the damping radiation component. Therefore, in this particular condition of damping radiation, the total flux of reflected light is larger than the negative energy flux in the metal layer, resulting in this positive GH shift.
4. Conclusions
In this paper, the GH shift on the SPR was accurately analyzed by using the FDTD method. We have found that the positive and negative lateral shifts were very sensitive to the variation of the incidence angle for this SPR phenomenon. The accurate positive and negative lateral shifts of −0.75 and +1.0 μm were found on an SPR with a 50nm thick Au film with incidence angles of 44.4 ° and 47.5 °, respectively, at a wavelength of 632.8 nm. Our results will be very useful in the design of μm-order waveguide-type SPR devices. These results can also be utilized for significant performance enhancements in various integrated photonic devices.
Acknowledgments
This work were supported by Seoul R&BD program (10550) and Priority Research Centers Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2009-0093817).
References and links
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