## Abstract

Actively tunable metal-insulator-metal waveguides that employ vanadium dioxide films as the active medium are analyzed numerically. Vanadium dioxide exhibits strong contrast between the optical properties of its insulating and metallic phases. In particular, the large optical absorption in the metallic phase makes it straightforward to implement broadband attenuation modulators and switches, but this strong loss can also complicate the design of other types of devices. We present a plasmonic waveguide that functions as an index modulator with Δ*n* > 20% at *λ*_{0} = 1550nm (0.80 eV), by using a thin active layer to strike a balance between maximizing index contrast while mitigating attenuation. A second device is configured as a band-stop absorption modulator, taking advantage of symmetry to selectively suppress the TM1 and TM3 modes, with relatively minimal attenuation of the TM0 and TM2 modes.

© 2012 Optical Society of America

## 1. Introduction

Surface Plasmon Polaritons (SPPs) are a form of electromagnetic surface waves that have been studied extensively for their ability to achieve high field concentration of optical-frequency fields, and circumvent the diffraction limit of traditional photonic devices. The compact nature of plasmonic waveguides allows such devices to be integrated on chip with electronic components and bridge the gap in size between individual electronic devices and photonic components. The field of “active plasmonics”, or integration of plasmonic waveguides with actively tunable materials, has been studied extensively in recent years [1–4], allowing one to manipulate optical signals at the nano-scale and therefore representing one promising avenue toward functional nanophotonic circuits.

One plasmonic device archetype is the the planar Metal-Insulator-Metal (MIM) waveguide [5]. By changing the individual materials used within the device as well as engineering the relative layer thicknesses, the dispersion of the device can be tuned over a wide range of visible and infrared frequencies. By incorporating active materials as the dielectric layer within the waveguide, the metal layers act as both optical cladding and electrical contacts. Generally speaking, MIM waveguides are suitable to make very small electrooptical devices, as small changes in the waveguide index can significantly impact the SPP propagation length and wavevector. This structure has been utilized to produce tunable color filters using lithium niobate as the active medium [6], electro-optic switches employing doped silicon [7], and Mach-Zehnder interferometers employing a non-linear polymer [8].

Another class of materials which has shown emerging promise in the field of active photonics is solid state phase change media. An example is vanadium dioxide (VO_{2}), which undergoes a structural phase transition between an semiconducting monoclinic phase and a semi-metallic, tetragonal phase. This change can be induced thermally, electrically, or optically. The phase transition, when optically induced in a 50nm thin film, can occur on a time scale of less than 300fs. The reverse transition is governed primarily by relatively slow thermal cooling. Lysenko *et al.* report that the rate of recovery depends strongly on the thermal environment and optical pump intensity, observing relaxation times which vary from 3ns to 1ms under a range of conditions [9]. The phase transition is accompanied by a large change in the resistivity [10] and electromagnetic permittivity [11]. VO_{2} has been integrated into active photonic devices including ring resonator based modulators [12,13], photonic crystals [14], and strongly tunable plasmonic visible-light scatterers [15]. Active metamaterial-based components incorporating VO_{2} include broadband terahertz filters [16] and reconfigurable gradient index devices [17].

Neither of the two phases of VO_{2} would conventionally be considered an ideal constituent material for SPP waveguides. The monoclinic form VO_{2}(*M*) is transparent but noticeably dispersive, while the tetragonal form VO_{2}(*T*) is a lossy semi-metal which damps rather than supports surface waves. However, the optical properties of the two phases exhibit extremely high contrast; ellipsometric measurement [18] indicates that at *ω* = 1550nm (0.80eV) the permittivity can be switched from *ε*_{(}_{M}_{)} = (9.7 + 2.9*i*) to *ε*_{(}_{T}_{)} = (−14.1 + 30.5*i*). Therefore, VO_{2} is a particularly promising material for manipulating light with a minimal volume of active material. An MIM waveguide in which the core is entirely composed of VO_{2} can make an effective broadband shutter or switch, since the VO_{2}(*T*) phase renders even a short (length ≈ *λ*) section of waveguide essentially opaque. In this work, we explore the suitability of 4- and 5-layer MIM geometries that achieve other types of device actuation by using a thin VO_{2} active layer to strike a balance between maximizing index contrast while mitigating attenuation. We present a plasmonic waveguide that produces index modulation with Δ*n* > 20% at *λ*_{0} = 1550nm (0.80 eV), and a band-stop absorption modulator that takes advantage of symmetry to selectively suppress only the TM1 and TM3 modes. The result illustrates the possibility for a broad variety of plasmonic devices incorporating solid state phase change active media.

## 2. Device calculations

The two planar plasmonic geometries studied are shown in Figs. 1 and 2 and were analyzed using numerical solutions of Maxwell’s Equations under steady state conditions. We consider transverse magnetic waves propagating in the direction *x̂*. The waveguide is 2-dimensional, consisting of *N* material layers enumerated *l* ∈ {1,2,...,*N*} stacked normal to the direction *ẑ*, and of infinite extent in *ŷ*. The top (*l* = *N*) and bottom (*l* = 1) layers are considered to be semi-infinite in the *ẑ* direction, corresponding to metallic cladding layers which are thick compared to the skin depth of surface waves. Intermediate or “core” layers have finite thickness *t*_{2},*t _{l}*,...,

*t*

_{N}_{–1}. The total thicknesses of the devices studied here are 550nm and 420nm, respectively. This is thick enough so that at visible and near infrared frequencies the waveguides are multimode, supporting both a gap plasmon or TM

_{0}mode, and one or more additional propagating TM modes with mixed photonic/plasmonic character.

Each layer is composed of a material described by a potentially complex frequency dependent permittivity, *ε _{l}*(

*ω*). The silver layers are modeled by interpolation to the Lorentz-Drude parameterization by Rakić

*et al.*[19] of data from the

*Handbook of Optical Constants of Solids*[20]. The active layer is treated by direct ellipsometric measurement of VO

_{2}films as grown on sapphire by pulsed laser deposition, reported previously [18]; data corresponding to the monoclinic and tetragonal phases were taken at room temperature and 85 °C respectively. The passive component layers of the core are treated as a constant dielectric with index

*n*= 1.6.

The propagation characteristics of these waveguides can be expressed as a dispersion relation, that is, an expression or graph relating complex-valued wavevector *k _{x}* to real energies

*h̄ω*. The energy is a parameter corresponding to monochromatic excitation of the guided mode, alternatively expressed as a free-space wavelength

*λ*

_{0}= (2

*πc*)/

*ω*. The wavevector describes the propagation of guided modes; the real part is inversely related to the mode effective wavelength,

The imaginary part of the wavevector is a measure of attenuation and can alternatively be expressed as a propagation length,

We find the dispersion relation by analyzing a system of equations connecting *k _{x}* and

*ω*. Two sub-expressions involving

*k*are used for conciseness:

_{x}Then we take the following ansatz form for TM solutions of the wave equation, *E _{yl}* =

*B*=

_{xl}*B*= 0, that are harmonic in the propagation direction and exponential in the transverse direction, cf. Economou [21], and Raether [22].

_{zl}This introduces a set of 2*N* unknowns: the two field coefficients *a _{l}*,

*b*for each material layer. Two constraints come from enforcing appropriate asymptotic behavior,

_{l}Electromagnetic boundary conditions give the remaining 2*N* – 2 constraints, namely, continuity of the tangential *E* field *E _{x}*

_{,}

*=*

_{l}*E*

_{x}_{,}

_{l}_{+1}and normal

*D*field

*ε*

_{l}E_{z}_{,}

*=*

_{l}*ε*

_{l}_{+1}

*E*

_{z}_{,}

_{l}_{+1}evaluated at every interface between layers. Let

*d*represent the

_{l}*ẑ*coordinate of each of the (

*N*– 1) interfaces. With the first interface at the origin by definition,

*d*

_{1}= 0, the position of the other boundaries is simply the sum of the underlying layer thicknesses, ${d}_{l}={\sum}_{2}^{l}({t}_{l})$ for

*l*∈ {2,...,(

*N*– 1)}. Then the boundary value relations can be written in the form:

Consider that the undetermined coefficients are collected in a column vector, **a** = [*a*_{1}, *b*_{1}, *a*_{2}, *b*_{2},..., *a _{N}*,

*b*]

_{N}^{T}. Then the system of constraint equations can be written as a linear system,

Equation (6) contributes blocks 𝔸* _{l}* : 𝔸

*[*

_{l}*a*,

_{l}*b*,

_{l}*a*

_{l}_{+1},

*b*

_{l}_{+1}]

^{T}= 0,

And therefore matrix 𝕃 has the form:

The first and last rows come from Eq. (5). Waveguide modes allowed by the system of boundary conditions correspond to the cases in which matrix 𝕃 is singular, and therefore solutions to Eq. (7) are non-unique. Generally speaking, such solutions of det(𝕃) = 0 exist in the domain where both wavevector and frequency are complex valued. Here, we impose the additional constraint that frequency, *ω*, is strictly real, an appropriate convention for analysis of steady state illumination by monochromatic light. Given *ω* real, we must search for approximate solutions, |det(𝕃)| ≤ *δ* ≈ 0. Following the solution technique developed by Dionne [5] we discretize the plane ({*k _{x}*},

*ω*) and use numerical minimization to solve for the value of the imaginary part {

*k*} that minimizes |det(𝕃)| at each point. At this step, every point in the plane now has an associated value of |det(𝕃)|. The next step is to apply a criteria to select the subset of points at which the determinant is sufficiently small. In this work, rather than attempt to define a global solution tolerance

_{x}*δ*appropriate for a wide range of waveguide conditions and frequencies, we instead use a peak-finding algorithm to identify a locus of points at which |det(𝕃)| is minimized locally in the plane. We consider “solutions” to be points which are members of this set and that additionally meet the condition {

*k*} > {

_{x}*k*}.

_{x}In dispersion diagrams, Figs. 3 and 4 we display these solutions in two different projections. The top panel of each figure is a conventional view; the horizontal axis is the real part of wavevector {*k _{x}*} and the vertical axis is energy

*h̄ω*. As points of reference, dotted horizontal lines indicate several benchmark energies of 1.825eV, 1.266eV, and 0.8eV corresponding to excitation wavelengths 680nm, 980nm, and 1550nm repectively. A diagonal marks the lightline corresponding to index

*n*= 1.6, which qualitatively divides the space into photonic modes in the region to the upper-left and plasmonic modes on the lower-right. The color scale is defined by a logarithmic figure of merit,

Points colored in black have a *f* > 2, meaning that the energy attenuation length is hundreds of times longer than the modal wavelength. Since nanophotonic functional elements may be on the order of ten wavelengths in size, these are applicable for a variety of devices or even small scale integrated networks. Points colored in light grey have an *f* value around 1, not ideal for guided wave systems, but suitable for MIM metamaterials only a few wavelengths thick. The color scale fades to white at *f ≤* 0, at which point the energy attenuation length is comparable to the modal wavelength. The bottom panel of our graphical dispersion relations are an alternate projection of the data along the axes ({*k _{x}*},

*f*). This view doubles as a color scale, since the grey-scale color intensity is linear with the vertical axis. It is also useful for graphically estimating the cut-off wavevector (horizontal axis intercept) for each branch, and for comparing side by side the

*f*of the

*off*(left column) and

*on*(right column) device states.

We assign mode labels (TM0, TM1, etc.) to connected sets or “branches” of solutions by observing the periodicity of electric fields along the *ẑ* axis. At any given solution point on the dispersion curve *β*(*k _{x}*,

*ω*) is determined by Eq. (3), and the fields are found by solving system Eq. (7) for the vector of coefficients

**a**, given

*β*and one additional arbitrary assumption to fix the fields’ absolute magnitude, e.g.

*a*

_{1}= 1. The fields can be generated by iteration, solving the pair of equations Eq. (6) for

*a*

_{l}_{+1}and

*b*

_{l}_{+1}in terms of

*a*and

_{l}*b*.

_{l}## 3. Device performance

#### 3.1. Index modulation

The first structure studied is shown in Fig. 1. Here, optically thick silver films form the top and bottom layers of the device, with 50 nm of VO_{2} and 500 nm of a dielectric layer with n = 1.6 and k ≃ 0 (such as silicon dioxide or sapphire) as the waveguide core. The normalized E* _{x}* fields of the optical modes supported within this structure from 0.5 – 3 eV are plotted as a function of position within the waveguide, to the right of the structure in Fig. 1. Here, the TM

_{0}through TM

_{4}mode profiles correspond to those in the dispersion diagram shown in Fig. 3. These calculations correspond to VO

_{2}(

*M*) in the monoclinic phase, and each mode profile was plotted at an energy (see legend) which is neither close to mode cutoff, nor to the lightline. We consider the electric field profiles shown here to be typical, as generally the field distributions were observed to be qualitatively similar among all points common to a given branch. Interesting exceptions include near-crossing points and the transition of TM1 from photonic (maximum intensity in core) to plasmonic (maximum intensity at surface) character as it crosses the lightline.

The dispersion diagram corresponding to the structure in Fig. 1 is shown in the top panels of Fig. 3. Figure 3(a) illustrates the allowable modes that characterize the plasmonic modulator when the active layer is in the semiconducting monoclinic phase VO_{2}(*M*), while Fig. 3(b) characterizes the modulator when the active layer is in the semi-metallic tetragonal phase VO_{2}(*T*). To illustrate the switching behavior of this device, three specific operating points are analyzed: 1.825eV, 1.266eV, and 0.8eV. The mode index and figure of merit *f* corresponding to points **A** through **F** in Fig. 3 are listed in Table 1.

Of particular interest in this diagram is the pair of points **A** and **B**, representing the transmission of infrared light with photon energy of 0.80eV, propagating in the form of a fundamental TM_{1} photonic mode. Figure 3 and Table 1 both show a 22% difference in the mode index between points **A** and **B**. Strong index modulation is also observed at points **C** and **D**, representing the transmission of infrared light at 1.27eV, propagating in the form of a surface plasmon polariton (SPP) mode. Finally, a distinct mode of operation is indicated by the pair of points **E** and **F**, representing the transmission of visible light at 1.83eV, propagating in the form of a higher order TM_{3} photonic mode. When the device is activated, this mode is attenuated to the point of critically damping the propagation of energy in the device, and pushing it into “cutoff”. Under these operating conditions, the device can be used as an attenuation modulator with an on/off ratio of 9 dB.

#### 3.2. Mode-selective attenuation modulator

The second structure studied is shown in Fig. 2. As in Section 3.1, optically thick silver films form the top and bottom layers of the device; however, the waveguide core is now comprised of 200 nm of n = 1.6 and k ≃ 0 dielectric above and below a 20 nm layer of VO_{2} in the waveguide center. Again, the normalized E* _{x}* fields of the optical modes supported within this structure from 0.5 – 3 eV are plotted as a function of position within the waveguide, to the right of the structure in Fig. 2. For this device, only four optical modes are supported, and the plots correspond to VO

_{2}in the monoclinic phase. In this device configuration, a key operating principle is that only the “odd-numbered” modes have an electric field maximum in the active layer, and therefore are selectively sensitive to actuation.

The corresponding dispersion diagram for the structure in Fig. 2 is shown in Fig. 4. As in the previous section, Fig. 4(a) illustrates the allowable modes that characterize the 5-layer plasmonic modulator when the active layer is in the VO_{2}(*M*) phase, while Fig. 4(b) characterizes the modulator when the active layer is in the VO_{2}(*T*) phase. To illustrate the switching behavior of this device, pairs of points were analyzed at 0.80 and 0.94eV. The performance characteristics at those operating points are tabulated in Table 2.

The lower panel of Fig. 4(a) and (b) again corresponds to the figure of merit *f* for each optical mode in the respective dispersion diagram directly above, with the grayscale color limits limits (0 < *f* < 2). Unlike the previous device, the actuation of this structure causes a major perturbation to the optical mode spectrum allowed within the device. As a result, the lowest order photonic mode (TM_{1}) is essentially no longer supported in the waveguide when the vanadium dioxide is switched to its tetragonal phase. This is illustrated at the pair of points **A** and **B**. Point **A** represents the guiding of infrared light at 0.80eV. Point **B** indicates total absence of a TM_{1} mode at this photon energy, as seen in Fig 4(b). Points **C** and **D** illustrate the same effect at 0.94eV. Therefore, assuming the use of mode-selective insertion this device can be used as a band-stop attenuation modulator with an on/off ratio over 20 dB.

## 4. Conclusions

Two dynamically tunable MIM waveguides have been presented that employ vanadium dioxide thin films as the active element. The 4-layer asymmetric device geometry notably can function as an index modulator, in which the balance of index modulation performance against parasitic attenuation can be engineered through variations on this design, specifically, by varying the thickness of the active layer. Here, the device we discuss in detail was configured with relatively high Δn on the order of 20% for the SPP and TM_{1} mode. The device also provides strong control over the cutoff frequency of guided modes.

The second device geometry studied was a symmetric 5-layer stack with active vanadium dioxide layer at the core. By switching the vanadium dioxide layer to its metallic phase, the TM_{1} photonic mode is strongly suppressed. This results in an on/off ratio of > 20 dB when the appropriate operating wavelengths are chosen. A notable feature of this configuration is the mode-selective nature of the device, which can be used to achieve band-stop filtering action.

A portion of the technology which is discussed in this manuscript is the subject of one or more pending patent applications. We gratefully acknowledge Dr. Ryan Briggs, Prof. Nader Engheta, Dr. James Ma, Dr. Merrielle Spain, and Dr. Daniel Marthaler for a number of stimulating discussions, and especially thank Prof. Jennifer Dionne and Dr. Sarah Sweatlock for invaluable advice on implementation of numerical methods.

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