## Abstract

Virtually imaged phased arrays (VIPAs) offer a high potential for wafer-level integration and superior optical properties compared to conventional gratings. We introduce an elastomer-based tunable VIPA enabling fine tuning of the dispersion characteristics. It consists of a poly-dimethylsiloxane (PDMS) layer sandwiched between silver bottom and top coatings, which form the VIPA’s high reflective and semi-transparent mirror, respectively. The latter also acts as an electrode for Joule heating, such that the optical PDMS resonator cavity tuning is carried out via a combination of thermal expansion and the thermo-optic effect. Analogous to the free spectral range (FSR), based on a VIPA specific dispersion law, we introduce a new characteristic VIPA performance measure, namely the free angular range (FAR). We report a tuning span of one FAR achieved by a 7.2*K* temperature increase of a 170*μm* PDMS VIPA. Both resonance quality and tunability are analyzed in numerical simulations and experiments.

© 2013 Optical Society of America

## 1. Introduction

By structuring the entrance face of a Fabry-Perot etalon its efficiency can be significantly improved [1]. Based on this modified layout Shirasaki introduced an optical spectral disperser referred to as virtually imaged phased array (VIPA) [2]. Compared to conventional diffraction gratings, VIPAs provide a larger angular dispersion. Their compactness and simple low-cost structure furthermore render them attractive as buildings blocks for several photonic applications including telecommunications, sensing, and imaging. Wavelength division multiplexing (WDM) has been demonstrated using a VIPA [2, 3]. VIPA based systems for chromatic dispersion compensation were reported [4, 5] and extended to tunable systems incorporating mechanical adjustment [6, 7] or spatial light modulators (SLM) [8]. Moreover, the VIPA acts as spectral disperser for millimeter-wave generation [9] or for pulse-shaping applications [10, 11], which allows for polarization mode dispersion compensation [12]. The combination of a VIPA with a diffraction grating, which creates a two-dimensional (2-D) spectral dispersion, leads to another promising field of applications. This includes improved WDM [13] and spectral molecular fingerprinting [14] as well as pulse shaping [15, 16]. VIPAs were furthermore utilized for spectrally encoded ultrafast imaging [17] and high-speed tomography [18] as well as simultaneous imaging and microsurgery [19] and infrared spectroscopy [20].

While VIPA based tunable optical systems have already been proposed [6–8, 15], the VIPA itself has not been reported as a tunable device yet. The proposed tunability offers an additional degree of freedom to many applications and may enable misalignment compensation, channel reconfiguration in WDM systems, and operating point adjustment in two-dimensional spectral dispersers. In this paper we introduce a tunable VIPA based on an elastomer cavity [21, 22]. This offers an active control of dispersion angles corresponding to distinct wavelengths. The device fabrication is subject to a simple elastomer molding process and is compatible to wafer-level integration.

The remainder of this paper is structured as follows. A brief analytical VIPA characterization is given and a new characteristic, namely the free angular range (FAR), is defined (Sec. 2). The tuning concept and fabrication is explained in section 3. Furthermore, the VIPA performance is investigated in both numerical and experimental analyses (Sec. 4). We investigate the tunability, the resonance quality, the dynamic behavior, and polarization effects.

## 2. Analytical characterization: dispersion law, free spectral range, free angular range

The VIPA is formed by a modified Fabry-Perot etalon [2], that is, by a tilted cavity embedded between a partially reflective rear mirror (≥ 95%) and a 100% front mirror. Incident light is typically focussed by a cylindrical lens and coupled into the device through the entrance window. Front reflection is suppressed by an anti-reflection coating leading to almost 100% transmitted energy. Multiple reflections inside the cavity produce an array of virtual sources interfering to a distinct dispersion pattern (Fig. 1). Hence, the VIPA acts as a spectral disperser, in which different wavelengths are dispersed to different output angles.

After the VIPA was introduced, it was analyzed in simulations and experiments. The first analytic model was derived in form of Vega’s dispersion law [23]. The most recent model describing the VIPA dispersion was derived by Xiao [24] and reads

_{VIPA}, the cavity thickness is given by

*t*,

*n*denotes the cavity material refractive index,

*m*the dispersion order and

*λ*the vacuum wavelength. Θ

_{in}describes the chief ray propagation angle inside the cavity defined by sin(Θ

_{VIPA}) =

*n*sin(Θ

_{in}). This dispersion law is based on the Fresnel diffraction theory, takes paraxial waves and Gaussian beams into account, and provides an improved accuracy at small VIPA tilts compared to Vega’s plane-wave based derivation. Furthermore, recently a correction of Vega’s derivation has been published, which leads to Xiao’s dispersion law [25]. The corresponding free spectral range (FSR) is calculated by

*c*.

At a fixed wavelength, the VIPA’s angular dispersion produces an angular distributed pattern of adjacent dispersion orders. In particular, two adjacent dispersion orders enclose a specific angular range. This angular range forms a characteristic of higher practical relevance compared to the FSR, since within this range a dispersion order’s output angle unambiguously corresponds to the predefined wavelength. In other words, the FSR (or free wavelength range) produces a uniquely defined angular spread for a fixed dispersion order, which we introduce as the VIPA’s *free angular range* (FAR). Employing Eq. (1), a straight forward computation yields the dispersion angle difference of two adjacent orders *m* and *m* + 1:

This expression depends on the dispersion order *m*, where usually only dispersion orders near the optical axis are of particular interest. The dispersion order *m*_{0} corresponding to a light ray parallel to the optical axis reads

*m*

_{0}not necessarily is an integer value. However, allowing the diffraction order in Eq. (6) to be a non-integer value, we define the FAR at 0° by the dispersion angle difference of half a diffraction order below and above

*m*

_{0}. With respect to Eqs. (3)–(5) we derive

Using Eq. (1) an analytic modeling of the dispersion angle change upon tuning is achieved considering both thermal cavity expansion and thermal index change. With respect to the FAR (Eq. (7)) this allows for an evaluation of practically relevant tuning performance. To this end we introduce the relative dispersion angle change by dividing the absolute change ΔΘ* _{λ}* by the FAR, which reads

*or the relative change of angle $\mathrm{\Delta}{\mathrm{\Theta}}_{\lambda}^{\text{rel}}$ is the relevant measure of performance. In the following both figures of merit will be evaluated for an experimental device.*

_{λ}## 3. PDMS based tunable VIPA implementation

In the dispersion law (Eq. (1)) the VIPA thickness *t*, the refractive index *n* and the wavelength *λ* contribute with respect to the same power. This indicates that the dispersion angle is highly sensitive not only to wavelength variations, but also to thickness change and refractive index variations. Hence, thickness or refractive index tuning allows for dispersion angle adjustment at a fixed wavelength. To this end, we use a thermally tunable PDMS slab as the VIPA internal layer, providing a high thermal expansion coefficient (3.1 · 10^{−4}
$\frac{1}{K}$) and a comparably low thermo-optic effect (1·10^{−4}
$\frac{1}{K}$) [26]. To form the optical cavity, this PDMS layer is sandwiched between a thick, highly reflective front silver layer on a glass substrate and a thinner, semi-transparent rear silver layer. Using current conduction through the rear silver layer the PDMS cavity is heated, which changes its optical properties and tunes the VIPA’s dispersion behavior (Fig. 2).

The VIPAs are fabricated on standard 25*mm* × 25*mm* glass slide substrates. Using thermal metal evaporation the glass substrate is coated with a 80*nm* silver layer, forming the front VIPA reflector, which is structured using a shadow mask during evaporation to leave an entrance window. Subsequently, a PDMS film of 170*μm* or 340*μm* is produced by molding a PDMS filled cavity between a polymethylmethacrylat (PMMA) slab and the substrate using one (or a stack of two) cover slips as spacers at two opposite substrate edges. In order to achieve a maximum grade of layer parallelism, the cover slips are selected after an interferometric characterization and cut into halves. Bridge bars are used to clamp and fasten the cavity for overnight PDMS curing at 50°*C*. Removal of the PMMA slab yields the substrate with a PDMS film. A sparse oxygen plasma treatment (50W, 20s) prepares the PDMS surface for the rear silver layer evaporation, which is structured into separated 1.2*mm*-wide silver bars forming six separate VIPAs on a single substrate (Fig. 2(b)). The silver bars are contacted by gold strands for tuning via Joule heating.

## 4. Numerical and experimental device characterization

#### 4.1. Numerical methods

Compared to a VIPA model using ideal mirrors the proposed VIPA uses silver mirrors showing different reflectance and phase shift properties, varying with angle of incidence and polarization. Therefore, the VIPA is considered a thin-film structure and can be modeled using the transfer matrix method (TMM), which allows for a device-specific angle-dependent transmittance calculation. In this contribution we perform TMM simulations according to the formulation described in [27, 28]. With respect to the fabricated device dimensions, the transfer matrix layer sequence reads: [glass - 80*nm* silver - 170*μm* PDMS - 25*nm* silver - vacuum]. The simulations and measurement results are compared in the experimental section. However, using this approach, the entrance window is ignored, as the TMM assumes infinite layer dimensions in the plane of layers. Hence, the front reflection is neglected and only a normalized transmittance is calculated. Furthermore, boundary effects are neglected and only plane-parallel layers can be calculated using the TMM. The influence of misalignment is analyzed using raytracing simulations.

Light experiences multiple reflections within the VIPA and is coupled out to a certain amount depending on the second mirror reflectance. This can be modeled by ray tracing simulations (ZEMAX) using a coherent superposition of out-coupled light resulting in the VIPA’s angular dispersion. Here, the VIPA is modeled as a tilted rectangular solid having a perfect reflecting front surface coating and a variable rear surface coating. In this case it is convenient to locate the light source within the VIPA. Coherent superposition of out-coupled light is achieved using a Fourier lens and a detector allowing for a reverse angular dispersion calculation. An additional rear-surface tilt angle is considered resulting in a wedge-shaped cavity and is therefore denoted as the ”wedge angle” in the following. The rear surface reflectance and the wedge angle are varied in the simulations and resulting effects are compared to the experiments.

Regarding the cavity refractive index, finite element simulations (COMSOL) reveal that top electrode thermal heating leads to a gradient index profile in the PDMS layer rather than a homogeneous refractive index change. However, this gradient turns out to be almost linear and does not have a significant influence on the optical behavior compared to a uniform average refractive index, which was verified by raytracing simulations. Hence, a uniform average index is assumed in both the dispersion law (Eq. (1)) and raytracing simulations.

#### 4.2. Experimental characterization setup

The fabricated samples were experimentally characterized in an optical setup (Fig. 3). A widened helium-neon Laser beam is focussed in x-direction and coupled into the cavity through the entrance window. A slit diaphragm additionally limits the focussed beam size in y-direction to avoid coupling into adjacent regions of a structured rear mirror bar. The sample is located on a rotation unit to adjust the VIPA tilt. The angular dispersion profile is first mapped into position space using a Fourier lens and subsequently detected by an image sensor. Fourier lens focussing is achieved by a translation stage. The image sensor consists of a rotating diffuser, an objective lens and a complementary metal-oxide-semiconductor (CMOS) camera to allow for speckle-free capturing of resonance images. The rear silver layer is contacted by gold wires (Fig. 2(b)) and the actuation current is controlled by a source measure unit (SMU).

#### 4.3. Numerical and experimental results

### 4.3.1. Tunability and performance analysis

VIPA tuning is performed using an incremental actuation power increase per silver layer area within a predefined period up to 2.4*mW/mm*^{2}. This was achieved by a current increase following a square root function leading to a linearly increasing actuation power at a constant resistance. Laser light is coupled into the VIPA at P-polarization, as the transmittance is maximized for incidence angles near the Brewster angle. Figure 4(a) shows resulting resonance images of a 170*μm* VIPA at 20° tilt actuated in 0.5*s* steps within 25*s* up to 2.4*mW/mm*^{2}. At the beginning the two resonance orders near the optical axis comprise 34% and 40% of the overall transmitted energy. Different resonance orders are separated by dashed lines and show a distinct motion upon tuning. Precise resonance angles were determined using parabolic fits to the resonance profile peaks. These correspond to distinct dispersion angles (Fig. 4(b)). After approximately 13*s* actuation time a tuning through one FAR is achieved. According to the dispersion law (Eq. (1)), considering PDMS thermal expansion and thermo-optic coefficients, the dispersion angle change is related to an average PDMS temperature increase (Fig. 4(c)).

The temperature Δ*T* shows a non-linear increase. This is a result of a delay in the thermal energy flux, as well as a positive temperature coefficient (PTC) of the silver layer resistance *R*(Δ*T*), which increases the applied power *P* at a given current *I* according to Ohm’s law and *P* = *R*(Δ*T*)*I*^{2}.

Resonance profiles noticeably degrade with higher actuation currents. Obviously, the silver and PDMS thermal expansion coefficient mismatch produces a surface deformation, leading to reduced resonance quality. This effect is quantified by the finesse

### 4.3.2. Optimization of VIPA tilt angle

Both analytical modeling and experiments reveal that the dispersion-angle change highly depends on the VIPA tilt. For a comparison, Figure 5 exemplarily shows the absolute and relative dispersion-angle change as a function of VIPA tilt and actuation time of a 170*μm* VIPA. Modeling and measurements are in good agreement. VIPA tilts near 0° and 90° induce the largest absolute tuning effect, whereas the relative dispersion angle change (Eq. (8)) shows a maximum effect for small tilts (Fig. 5(b),(d)). In measurements the VIPA is actuated linearly up to 2.0*mW/mm*^{2} within 10*s* corresponding to a maximum temperature increase of 7.2*K*. Calculations assume a linearly increasing temperature Δ*T* up to 7.2*K* within 10*s*. Modeling and measurements show a slight shape difference of the dispersion angle change over actuation, which again is a result of a delay in the thermal energy flux, as well as a PTC silver layer resistance.

### 4.3.3. Dynamic behavior

With respect to future applications, an understanding of the tunable VIPA’s dynamic behavior is crucial. To this end, the resonance shifts of two VIPA samples (170*μm* and 340*μm*) are exemplarily evaluated. The samples are actuated for the leading 15*s* of a 30*s* period, applying a constant current of 20*mA*. Light is coupled into the VIPA at 20° and P-polarization. Temperatures are fitted to the resonance shifts according to the previous section (Fig. 6).

Assuming a classical body heating or cooling process, one may expect a temperature increase complying with a 1− *e*^{−t/τ} function. Experimentally we observe a high slope within the first second of heating and cooling followed by a slow, monotonic further temperature change. Additionally, the samples do not cool down to the initial temperature within 15*s*. The VIPA consists of different materials contributing to the overall heat flux inside the structure. The thin PDMS layer is directly heated and transfers thermal energy at a defined flux to the substrate, which has a much larger heat capacity. For that reason the VIPA has to be considered as a coupled system and heating is modeled using the sum of a short-term heating term and a long-term heating term given by

*T*(

*t*→ ∞) = 0.

Here, Δ*T*_{{0,...,3}} terms represent limit values of the separate heating / cooling processes. Model function fits to the measured data clearly exhibit a combination of a long term and a short term process (Table 1) and are in excellent agreement with the measured data (Fig. 6). We attribute the short term process to PDMS layer heating, as the 340*μm* shows approximately doubled values for *τ*_{1} and *τ*_{3}. The time constants *τ*_{2} and *τ*_{4} reveal the long term process, which represents entire device heating, including the substrate. The cooling process fits yield larger time constants as the device is actively heated, but passively cools down by free convection.

### 4.3.4. Static characterization

While the fabricated VIPAs produce a distinct dispersion pattern, the finesse is still low compared to a predicted value (TMM simulations) of 19 at 20° VIPA tilt and P-polarization. Here, several static effects are involved that are shown and discussed in the following. A photograph of the sample from an off-optical-axis point of view shows an employed VIPA region (Fig. 7(a)). Multiple reflections at the rear silver layer and successive reflections at the front silver layer are clearly visible. This indicates partial scattering at the rear surface, induced by a silver cluster formation at the evaporated metal-polymer interface [29] leading to a significant reflectivity decrease. Figure 7(b) gives examples of poor resonance images at two incoupling spots of a 340*μm* VIPA sample showing several secondary peaks adjacent to the VIPA’s main resonance orders. Ray-tracing simulations reveal that these secondary peaks are due to wedge angles and local thickness alteration in the VIPA cavity. Surface characterization of the substrate and the PMMA slab shows surface irregularities of 1.5*μm* (≈ 2.4 · *λ*) causing wedge angles of up to 5 · 10^{−3}°. Compared to measurement spots characterized in previous sections stronger thickness variations are present in Fig. 7(b). Simulated resonance profiles at different wedge angles for 95% and 50% rear mirror reflectivity reveal decreasing resonance profiles at increasing wedge angles (Fig. 7(c)). Higher reflectivity increases the resonance quality but renders the VIPA more sensitive to large wedge angles. At 95% rear mirror reflectivity a wedge angle of 10^{−2}° (corresponding to a thickness variation of 6.9 ·*λ*) completely decomposes the resonance profile. In this case the higher reflectivity leads to larger beam propagation distances inside the cavity, which extends the interaction length with thickness variations. A simulated resonance image example with cavity wedge angles in both x- and y- direction shows secondary peaks as well, which are in good agreement with the observed effects (Fig. 7(d)).

As a finesse increase can be easily achieved by thicker silver layers, the main limiting factor affecting the VIPA resonance quality is the susceptibility to surface irregularities, which demands careful fabrication and highly planar substrates. An Overall light loss due to metallic reflection can be reduced by using a dielectric mirror instead of the first thick silver layer.

In previous sections P-polarized light was used for measurements. However, the polarization state has a significant influence on the resonance profiles. Figure 8(a) depicts colorized resonances resulting from S- and P-polarized light for three different VIPA tilts. At 70° VIPA tilt a distinct disparity of the different polarization states occurs. This effect is quantified by determination of the relative deviation of S- and P- polarized resonance profiles, which is calculated by dividing the absolute angular deviation by the FAR (Fig. 8(b)). The simulated relative deviation between S- and P- polarized light is obtained by TMM calculations and shows good agreement to the measured data. A maximum FAR deviation of 11% is observed, which is a result of a slightly different phase shift upon reflection at the silver layer for S- and P- polarized light.

## 5. Conclusion

Based on analytical modeling, numerical simulations, and experimental characterization, we successfully have demonstrated proof of concept regarding polymer-based thermally tunable virtually imaged phased arrays. Small VIPA tilt angles yield largest tuning effects. Using a temperature increase of 7.2*K* a tuning within one FAR was achieved for a 170*μm* VIPA. In dynamic response the device yields short-term time constants of maximum 0.6*s* overlain by a long-term process, which is due to substrate heating / cooling. This can be avoided by an active heat sink or by using a self-supporting membrane cavity. As the device is very sensitive to temperature changes in an application a feedback regulation will be necessary. Due to silver reflectors the device is sensitive to the in-coupled light’s polarization state. Especially at large VIPA tilts, polarized light should be employed. The device is subject to well-established fabrication techniques and is promising for simple wafer-level integration. However, irregularities in the VIPA cavity impact the resonance quality. Hence, careful fabrication is necessary. In summary, we consider the tunable VIPA concept to be of particular interest for future applications in telecommunications and sensing.

## Acknowledgments

The authors gratefully acknowledge financial support within the SPP 1337 “Aktive Mikrooptik” of the German Research Foundation (DFG).

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