Phase modulation of surface plasmon polaritons (SPPs) is studied as a function of geometric length and thickness of surface relief dielectric structures on a metal film. The results indicate that the SPPs are analogous to conventional free-space light waves in terms of phase modulation by geometric sizes and refractive indices. An SPP Fresnel zoneplate is considered as an example to employ the phase modulation method for focusing enhancement at the focal point. It is found that the phase modulation optimized zoneplate has obtained 130% higher electric field intensity at the focal point than that of an amplitude-modulated zoneplate.
©2008 Optical Society of America
Surface plasmon polaritons (SPPs) are electromagnetic waves propagating along the interface between metal and dielectric. The SPPs are essentially light waves confined on the surface via coupling the optical fields to the free electron oscillations in metal. Owing to the unique attractive properties of SPPs in diverse electronic and photonic applications, there is a great demand to reveal quantitative relationship between SPP modulations and surface relief structures on the metal/dielectric interface [1,2]. SPPs can be modulated either in amplitude, via scattering by surface submicron and/or nanometer scale structures, to realize conventional functions such as Bragg mirrors [3,4], focusing elements [5-7], waveguides [8,9] and modulators [10-12] in optics; or in phase velocity modulation when deterministic surface relief dielectric structures on the metal film are designed and optimized quantitatively. Typical applications include lenses and prisms in association with SPPs [13-15] in the latter case. It is noted, unlike phase modulation of wavefront in free-space, that quantitative analysis and characterization of phase modulation of SPPs are relatively new and unexplored. In this paper, we implement a numerical experiment on phase modulation of SPPs when geometric sizes of the surface relief structures are altered and optimized. SPPs can be modulated and enhanced in the experiments and as an example the phase modulation technique is employed for focusing enhancement for a Fresnel zoneplate (FZP) assisted by SPPs. The numerical experiments are carried out by a finite-difference time-domain (FDTD) code in the work.
2. Surface plasmon polaritons field
SPPs are essentially two-dimensional electromagnetic waves. No surface modes exist for transverse electric (TE) polarization, and the propagating transverse magnetic (TM) polarization SPP solution can indeed be obtained via a straightforward solution of the Maxwell’s equations . The SPP is given by ESPP=ESPP0±exp[+i(kSPPx±kzz-ωt)], where superscript + for z>0, superscript - for z<0, ESPP0 and ω are the amplitude and frequency, respectively. kz represents the wavevector component in z direction and kSPP the wavevector in the propagation direction. It can be determined by the Maxwell’s equations under the boundary condition below,
where k0 is the wavevector of light in vacuum, ε 1 and ε 2 are the permittivity of metal and dielectric respectively. For a metal, the permittivity ε 1, propagation constant kSPP and frequency ω are complex numbers, which can be written as ε 1=ε 1′+iε 1″, kSPP=kSPP′+ikSPP″, ω=ω′-iω″, respectively. We describe the SPP field as:
where the real part kSPP′ determines the SPP wavelength (λSPP) and the imaginary part kSPP″ accounts for the damping of the SPP as it propagates along the interface. The SPPs propagating at the dielectric/metal interface are equivalent to the free-space plane waves with a spatial decay component exp(-kSPP″x±kzz) and a temporal decay component exp(-ω″t). As depicted in Eq. (1), an increase in ε 2 leads to an increase in the propagation constant, and thus a decrease in phase velocity of the SPPs. This phenomenon is analogous to that of light propagating through dielectric optical components in free-space, implying that the phase modulation of the SPPs can be realized by depositing dielectric structures on the metal film. Furthermore, by adjusting the geometric sizes of the dielectric structures, it is possible to numerically modulate the phase shift for SPP propagation through these structures.
3. Numerical experiment and discussion
In addition to the study of phase modulation of SPPs by deterministic structures, in this paper, an in-plane phase-modulation Fresnel zoneplate (PMFZP) is selected as an example to employ the phase modulation properties for focusing enhancement at the focal point. In the proposed model as shown in Fig. 1, a sub-wavelength grating array is embedded in the silver film (ε 1=-15.9+i1.07) as an SPP launcher [17, 18]. A polymethyl methacrylate (PMMA) structure is deposited on the silver film with permittivity constant (ε 2=2.217) to form even Fresnel zones and subsequently introduce phase shift between adjacent Fresnel zones. A linearly polarized light (λ=633nm) is normally incident onto the substrate to excite in-phase SPPs propagating along two opposite directions on the dielectric/metal interface. The SPP propagation length along a smooth metal surface after which the intensity decrease to 1/e is given by δSPP=1/(2k SPP″)=[λ(ε 1′)2/(2πε 1″)]×[(ε 1′+ε 2)/ε 1′ε 2]3/2. The theoretical calculation of δSPP is equal to 21 µm which is sufficient to realize the phase modulation phenomenon in our simulation model. For the in-plane FZP [7,19], boundaries of the even Fresnel zones are defined by
where m is an integer for the FZP’s zone order and f is the designed focal length of 5 µm. Light emerging from even zones and odd zones of the FZP will either constructively or destructively interfere, depending on their relative phase retardation modulated by changing the PMMA structure’s length (l) and thickness (t). The PMMA structure’s width is pre-determined by rm. In this case, the relative phase profile distribution in Fresnel zones determines the electric field intensity at the focal point.
In our simulation, commercial software XFDTD (Remcom Incorporated) for vectorial three-dimensional electromagnetic analysis  is used to simulate the SPP filed distributions of in-plane FZP. Liao numerical radiation boundary  is used to absorb all radiated or scattered electromagnetic fields with very little reflection from the boundary. The complex permittivity of silver material is described by the expansion of Drude dispersion relation as ε(ω)=ε∞+(εs-ε∞)/(1+jωt0)+σ/(jωε0), where ε∞=3.98 is the infinite frequency relative permittivity, εs=-1.7478×104 is the static relative permittivity, t0=1×10-14 s is the relaxation time, σ=1.5488×107 S/m is the electrical conductivity and ε0=8.854×10-12 F/m is the free space permittivity. All these parameters are set to fit the empirical data for the real and imaginary parts of the silver’s complex permittivity over a wavelength range from 0.3 µm to 1.2 µm. Figure 2(a) shows the FDTD simulation result of the electric field of an in-plane PMFZP at focus when the PMMA structure is designed as 480 nm in length and 100 nm in thickness for a π phase shift between the adjacent zones. Similarly to conventional light waves, SPP waves will pass through the PMMA structure and focus at the focal point of the in-plane PMFZP after excited by the grating coupler. As shown in Fig. 2(b), full width half maximum (FWHM) and the depth of focus (DOF) are approximately equal to 256 nm and ±235 nm, respectively by line scanning the longitudinal and transverse of |E|2 distribution through the focus point in Fig. 2(a). From the theoretical calculation by using Rayleigh criterion in free-space, the diffraction limit is approximately equal to λ/2≈316 nm; DOF=±0.5λ/NA 2≈±369 nm, where NA=rm/(r2m+f2)1/2 and m is the outermost zone order which is equal to 27. Both the simulated FWHM and DOF values are different from the theoretical calculation values of free-space because the wavelength of SPP is shorter than the wavelength of free-space electromagnetic wave. In addition, the SPPs are confined on the metal surface and decay along the propagation direction.
As the length and thickness of the PMMA structure exert different effects on the phase distribution, we first evaluate the phase modulation by adjusting the PMMA structure’s length while keeping the thickness as a constant. As shown in Fig. 2(c), a gradual phase shift is observed in the change of the electric field intensity at the PMFZP focal point when SPPs propagate through the polymer structure, with the same thickness of 100 nm but gradually increasing length. When the phase of the modulated zones is retarded by π at the 0.48 µm, all zones of FZP totally constructively interfere with a maximum of electric field intensity at the focal point. Similarly, when the phase is retarded by 2π at the 0.96 µm, all the zones are out of phase and totally destructively interfere with a minimum of electric field intensity. For the 3π phase retardation at 1.44 µm, the second principal maximum is observed with lower value than the previous one because of the decay property of SPP waves. In order to demonstrate the significant contribution of phase modulation on SPP waves, we compare the electric field intensity of PMFZP at the focal point with amplitude-modulation Fresnel zoneplate (AMFZP) under the same conditions but the dielectric structure is replaced by silver. It is noted that the electric field intensity peak of the in-plane PMFZP is approximately 130% larger than that of in-plane AMFZP under the π phase shift condition, which is due to the fact that the incident SPP waves are lost in absorption within the opaque zones of the AMFZP. Interestingly, this result shows an analogy between the PMFZP and AMFZP in conventional free-space optics. This is due to the spatial and temporal decay component of phase-modulated SPP waves while propagating through the PMMA structures as shown in Eq. (2).
Next, we employ the effective refractive index method to study the effect of PMMA thickness in phase modulation. When an SPP wave propagates through the interface between two different dielectric media deposited on a metal film (namely the areas covered by PMMA and air respectively), there is a change in SPP’s effective refractive index, neff. Phase difference Δϕ between the PMMA and air areas on metal is equal to kSPP′·[neff(t,l,rm)-1]·l, where neff(t,l,rm) varies with dimensions of the PMMA structure due to non-linear behavior of SPP waves. In our case, neff is normalized to 1 for the areas covered by air on the metal surface. For a very thin polymer layer, neff is close to those of SPP propagating along the silver/air interface, whereas for a very thick layer when it is much thicker than SPP’s penetration depth, neff is close to those of SPPs at the silver/PMMA interface . Figure 3(a) shows the phase modulation results for different PMMA thickness of 200 nm, 400 nm and 500 nm respectively. The trends of these three lines are similar to the results of 100 nm PMMA thickness as shown in Fig. 2(c). The continual phase modulation is realized by increasing the length of PMMA structure with a specific thickness. In addition, the PMFZP with larger thickness will obtain higher focusing efficiencies. It is noted that the length of PMMA structure has to be reduced in order to get the maximum intensity at π phase shift for different thickness. This can be explained by the fact that more SPP waves are confined within the thicker structure, resulting in higher effective refractive indices. Figure 3(b) illustrates the effect of length and thickness for achieving π and 2π phase shift respectively. Implicitly, it is easier to control the phase shift of SPP waves by adjusting the length of PMMA structure than the thickness due to the extremes of neff.
The focusing phenomenon for the PMFZP can be explained by interference model of the SPPs in two dimensional. With the optimized PMMA structure, the focused spot size is beyond the diffraction limit with high electrical intensity concentration. This result is of interest to improve the performance of SPP in-plane microscopy and enhanced Raman scattering for sensing . In addition, SPP phase modulation is a potential method to implement for the future of integrated photonics to miniature device size.
In conclusion, we demonstrate that the optimized dielectric structure on a metal surface can be used to modulate SPP phase. An in-plane phase-modulation Fresnel zoneplate designed to focus SPP waves is studied numerically based on an FDTD code. The results show that electric field intensity at the focal point of PMFZP can be either enhanced or suppressed, depending on the relative phase shift of the alternating Fresnel zones, and the optimized in-plane PMFZP model has higher focusing efficiency than the AMFZP. A concluding remark can be drawn, except for its native decay property, that SPPs have similar phase modulation properties as free-space light waves and it paves the way for novel SPP devices based on phase modulation.
This work is partially supported by National Natural Science Foundation of China for grant 60778045 and National Research Foundation of Singapore under Grant No. NRF-G-CRP 2007-01 and Ministry of Education under ARC 3/06. XCY acknowledges the support given by Nankai University (China) and Nanyang Technological University (Singapore).
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