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Surface Plasmon Polariton Mach-Zehnder Interferometer and Oscillation Fringes

We present a quantitative experimental analysis of a surface plasmon polariton (SPP) interferometer relying on elliptical Bragg mirrors. By using a leakage radiation microscope we observe oscillation fringes with unit visibility at the two interferometer exits. We study the properties of the SPP beam splitter and determine experimentally both the norm and phase of the SPP reflection and transmission coefficients.

Progress in the field of two-dimensional optics at the micro-and nanometer scale requires necessarily the control over coherence of wave propagation in a confined environment. In this context it has been experimentally shown a few years ago that surface plasmon polaritons (SPPs, electromagnetic waves confined at the interface of a metal and a dielectric [1]) can generate a variety of two-dimensional interference effects [2][3][4].
More recently we were able to implement two-dimensional SPP Mach-Zehnder interferometers fabricated on silver thin films by lithographic techniques [5,6]. The observation of SPP propagation relied on the interaction of a layer of fluorescent molecules deposited on top of the SPP sustaining silver surface. The fluorescence intensity was detected with conventional far-field microscopy providing a map of the lateral SPP field intensity profile [4]. As a major drawback, fluorescence imaging could not be exploited quantitatively as the dye molecules photo-bleach rapidly, and do so in dependence of the local SPP intensity.
For this reason we shifted our attention towards leakage radiation microscopy (LRM) which relies on the coupling of SPP waves to leaky light modes propagating in the dielectric substrate supporting the metal thin film [2,7]. The leakage radiation is collected by an immersion objective optically coupled to the substrate which allows to acquire an image of the SPP profile in the metal surface [8]. The advantage of LRM is that quantitative analysis can be performed since the intensity recorded at any point of the microscope image plane is proportional to the SPP intensity at the conjugate point in the object plane. We used LRM previously to analyze the interaction of SPPs with a line of gold protrusions on a silver film acting as a SPP beam splitter [8], as well as the propagation of SPPs in a corral constituted by confocal elliptical Bragg mirrors on a gold film [6].
Here, we present a quantitative analysis of a Mach-Zehnder interferometer for SPP waves and we show in particular that we can record interference oscillation with unit visibility at the two interferometer exits.
The experimental basics of the SPP interferometer considered here rely on results obtained in Refs. [5] and [6]. SPPs are launched locally on a gold thin film by focussing a laser beam (Titane:Sapphire, λ 0 = 750 nm) incident normally to the substrate through a microscope objective (50×, numerical aperture 0.7) on a gold ridge (160 nm width, 80 nm height) on the film. The ridge and all other structures on the gold film as discussed in the following were fabricated by electron beam lithography (EBL). First, the structure geometries are defined by electron beam exposure, various chemical development steps and the two ellipse foci F 1 and F 2 equals the minimal long axis length a min = 30 µm. Since we work with a Bragg mirror the variation δa of the long axis length between two consecutive ellipses must be Nλ SP P /2 where N is an integer and λ SP P ≃ λ 0 is the SPP wavelength fixed here at λ SP P ≈ 750 nm. We choose N = 1 and δa = 375 nm. With these parameters we achieve mirror reflectivity up to 90%. As in Ref. [5] we consider a Mach-Zehnder configuration. The two SPP beams launched at the ridge and reflected by the elliptical Bragg mirror are focused onto a beam splitter made of a line of individual protrusions (160 nm diameter, center-to-center distance 240 nm). In order to exploit the optical properties of elliptical mirrors the SPPs are launched at the first focal point F 1 . Accordingly, the beam splitter BS is positioned at the second focal point F 2 , where the two reflected SPP beams intersect (see Fig. 1 and Fig. 4). As we will see later we can adjust the phase in the interferometer by changing the lateral position of the ridge within different interferometers. Therefore, we consider several similar interferometers built by EBL on the same sample.
We start our investigation by considering a half-interferometer, i.e., a configuration where only one half of the elliptical Bragg mirror is present (see Fig. 1). SPP waves propagating into the left arm are thus not reflected and cannot interfere with the SPP waves propagating in the right arm. As a consequence the SPP wave initially launched to the right splits at BS into two output SPP waves D 1 and D 2 with normalized intensity I 1 = |T | 2 (transmissivity), and I 2 = |R| 2 (reflectivity) such that |R| 2 + |T | 2 = 1 assuming no scattering. Fig. 1a shows a LRM image corresponding to this half-interferometer configuration and revealing a perfect symmetry between D 1 and D 2 . Fig. 2a shows a transverse cross-cut of the intensity in the two output SPP beams which confirms this symmetry quantitatively.  It is important to note that with such an half-interferometer we do not have experimental access to the phase relation between the transmitted, reflected, and incident beams. In order to define this relation we must first recall some properties of a lossless BS. From the unitarity of the BS transfer matrix [9] and from energy conservation one can deduce that the relation between the reflection and transmission amplitudes R and T is such that T = Re iφ where φ = φ ± = ±π/2. However, the sign ±1 depends on internal properties of the beam splitter and cannot be deduced from simple unitarity considerations. An individual study of each physical case is thus necessary. The simplest experimental way to do that is to consider the interferometer configuration including both (left and right) parts of the elliptical Bragg mirror, i. e. the complete interferometer. However, in order to explain the predictions of this interferometry experiment one has to take carefully into consideration all the phase shifts and differences introduced during SPP propagation.
In this context it has been remarked [10] that one can mimic the behavior of SPP waves launched from F 1 on the ridge by using the scalar field produced by a linear and continuous distribution of 2D dipoles. These in-plane dipoles are orthogonal to the ridge and proportional in strength to the incident electric field [11]. From this hypothesis and from symmetry considerations we deduce that there is a phase difference of π between the SPPs propagating into the right and into the left arm of the interferometer. The pertinence of this fact has been experimentally observed for the elliptical SPP corral [6]. Indeed, this phase difference justifies why we can observe SPP intensity oscillations at the second focal point However, the beam splitter is supposed to be lossless, as confirmed by our first analysis of the half-interferometer case, and we thus deduce that the intensities in the two exits D 1 and D 2 of the interferometer are given by |α| 2 is the SPP coupling efficiency at the ridge. δ is the algebraic displacement of the ridge with respect to the symmetry axis of the interferometer (defining the symmetric configuration) and k SP P = 2π/λ SP P is the real part of the SPP wave vector. δ defines a variable phase difference responsible for the oscillation of the intensities at D 1 and D 2 . We note that at the beam splitter we obtain two possible solutions I 1,± = 1 2 |α| 2 (1 ± 2|R||T | sin (2k SP P δ)) which invert the role of D 1 and D 2 , respectively. Experimentally we changed discontinuously δ between different individual interferometers fabricated on one sample to obtain a variation of the phase ∆ = 2k SP P δ in the domain ∆ ∈ [−π/2, π/2]. Fig. 4 shows the according sequence of LRM images corresponding to phase differences ∆ = −π/2, −π/6, 0, π/6, and π/2. Since the SPP wavelength λ SP P is fixed to 750 nm the displacement δ is thus varying in the interval [−λ SP P /8, +λ SP P /8] ≃ [−94 nm, +94nm]. In this sequence of images we clearly observe the intensity oscillation in the SPP beams D 1 and D 2 as a function of ∆. In order to be quantitative we consider transversal cross-cuts of the two exit beams, corresponding to the cross-cut in  condition |R| 2 + |T | 2 = 1). We can thus with a good approximation write T = i/ √ 2 and To conclude, we presented a detailed experimental analysis of SPP interferometry. Therefore, we analyzed LRM images of a Mach-Zehnder interferometer relying on elliptical Bragg mirrors, and we analyzed quantitatively the reflection and transmission of SPP waves by the lossless beam splitter BS. This leads to the experimental relations T = i/ √ 2 and R = 1/ √ 2 between the SPP transmission and reflection coefficients T, R of BS. We observed fringes oscillations with unit visibility at the two exits of the interferometer as well as the opposition of phase between the two signals I 1 and I 2 when changing the length of the two interferometer arms by an amount ±δ, i. e. when changing the phase by the quantity ∆ = 2k SP P δ.
The conservation of energy is fulfilled since we have I 1 + I 2 ≃ const. whatever the phase difference ∆. All experimental observations are in good quantitative agreement with the analytical calculations. This forms a solid base for further investigations or applications of SPP interferometry.
For financial support the Austrian Science Foundation and the European Union, under