1. Introduction

Highly efficient coherent light sources have become enabling components in the rapidly progressing nano-sciences and bio-applications based on photonics [1–3]. Several tasks in sensing and monitoring techniques, such as drug screening, massive testing of biological specimens, and implementation of “lab-on-a-chip” systems, require the availability of coherent sources that are widely tunable over visible light [3,4]. In comparison with other types of coherent light sources that have limited tunability, such as semiconductor lasers or LEDs, dye lasers offer a number of favorable features: high brightness and coherence, narrow line width of radiation, and broadband wavelength tunability in the visible part of the electromagnetic spectrum [5].

So far, the most advanced results with tunable micro-scaled lasers were demonstrated for devices of sizes of several tens of micrometers [6,7]. Suitable candidates to create truly micro-or nano-scaled tunable light sources are solid-state dye lasers. Recent progress in polymer physics and considerable improvements of dye properties, such as dye stability against photo-bleaching and aggregating at high concentrations, greatly increase the attractiveness of solid-state dye lasers for micro- and nano-photonics applications [8–10]. Among additional advantages of such dye lasers are ease of operation and inexpensive manufacturing, e.g., using photo- or nano-imprinting lithography (NIL) techniques [11].

In this paper, we consider the possibility of tuning the wavelength of microcavity solid-state dye lasers by using the variation of the optical path length of the cavity resonance modes. The variation is based on controlled heating of the laser microcavity, which leads to two effects: thermoelastic expansion of the microcavity and thermally induced change of the material refractive index. As a gain medium, we consider a solid solution of a dye, rhodamine 6G (Rh6G) in our example, embedded in a polymethyl methacrylate (PMMA) polymer host matrix. The rest of the microcavity consists of a photoresistive polymer (SU-8). Depending on the required tuning range and working temperature, an optimal size of the microcavity can be identified. For the numerical simulations we use the finite element method (FEM) implemented in the Comsol Multiphysics package [12], which yields an accurate solution for the electromagnetic field structure of the resonant modes in the ring-type microcavity dye laser.

2. Method: wavelength tuning by thermoelastic expansion and thermally induced refractive-index change

There exist various approaches to realize wavelength tuning in dye lasers. Among the most common methods are the adjustment of the positions and/or parameters of the laser components, such as mirrors, dispersion elements, or tuning the cavity geometry [5]. It is also possible to manipulate the chemical composition and dye concentration to tailor the luminescence properties of the gain material [13]. However, a real-time control of the chemical composition of a gain medium with dye solution is not very feasible for micro- or nano-scopic cavities. Furthermore, this process is not reversible in many practical implementations.

In this work we consider a solid-state microcavity laser with a design similar to that of the liquid dye ring laser reported recently [14]. Despite the successful demonstration of laser operation, microfluidic dye lasers may exhibit problems in further downsizing them for true micro- and nano-photonic devices due to changes in specific liquid properties, such as viscosity, surface tension, and thermo-mechanical coefficients. The solid-state variant of such a dye microlaser makes it possible to avoid these problems and provides additional flexibility of the operation.

We investigate the wavelength tunability of a microcavity laser based on the variation of the mode optical path. We use two phenomena that affect the optical path: a reversible thermoelastic expansion of the microcavity materials and a thermally induced change of the refractive indices [15,16]. During the heating of the gain material, these phenomena contribute to change the optical path with opposite signs: the former increases the optical path due to expansion, whereas the latter decreases it due to lowering the refractive index when the material warms up [16]. The overall effect then depends on the microcavity structure, size, and the particular chemical composition of the active medium. To realize a controllable temperature variation of the laser microcavity, we apply the Joule heat generated by a current flowing in metal rods inserted in the microcavity (see Fig. 1).

In our analysis, the microfluidic channel (containing a liquid dye solution and placed between the triangle-shaped parts of the coupled microcavity [14]) is substituted with a solid polymer bar filled with dye molecules. A low-concentration solid dye solution is represented by a large number (N) of dye molecules as point excitation sources randomly allocated inside the polymer bar. We choose N = 500 elementary sources as a reasonable number of dipole radiators, which simulates a relatively homogeneous distribution of dye molecules in the gain section and simultaneously ensures that the computer resources are sufficient to run calculations over the 455 – 600 nm wavelength bandwidth. The approach with such a multipoint excitation (MPE) source corresponds to real physical situations, in which the dye laser with a planar topology is optically pumped through the top facet.

3. Model: electromagnetic fields, thermoelastic expansion, and heat transfer

To model the tunability of the laser microcavity, we use the FEM solver implemented in Comsol Multiphysics. In our work, we consider simultaneously the whole set of coupled physical problems: electromagnetic wave propagation in the ring cavity and its surroundings, heat transfer in the cavity materials, and thermoelastic expansion of the microcavity size.

To keep the model simple yet realistic, we restrict the analysis to a planar (2D) geometry with TE waves, so that the electric field vector only has the z component (perpendicular to the plane of Fig. 1). The randomly distributed dye molecules are taken as point dipole radiators oriented normal to the cavity plane (parallel to the electric field vector). Typical values of the refractive indices are used for the various parts of the setup; n _d-p = 1.43 in the dye-polymer (Rh6G + PMMA) gain section and n _pol =1.6 in the polymer (SU-8 [17]) parts of the microcavity. We ignore dispersion effects and losses in all media, and we assume that no amplification takes place in the gain medium. Our earlier studies have shown that these simplifications are justified when analyzing the structure of the mode patterns [18].

The FEM solves exactly the Maxwell equations with the appropriate boundary conditions (continuity of the tangential components of the electric and magnetic field vectors) at the edges and interfaces of the ring-type microcavity. To ensure that no reflections (back scattering) emanate from the borders of the simulation area itself, the perfect matching layer (PML) feature of FEM is employed in the analysis. More details about the physical cavity model, numerical computation of the microcavity electromagnetic fields, and the associated mode structures are given in Refs. [14] and [18].

To investigate the microlaser tunability, we vary the effective optical path length of the electric field inside the microcavity to change the mode frequency. The mode numbers m and reasonant wavelengths λ _m are connected by the relation

where φ is the phase change due to the reflections at the microcavity walls (it follows from the Fresnel formulas for the complex electric field [19]), and L _eff is the effective optical path length,

Here l _d-p and l _pol are the thickness of the dye-polymer slab and the length of the polymer triangle side adjacent to the dye-polymer slab, respectively (see Fig. 1). The radiation is out-coupled by tapping the evanescent field at an air gap l _air on the boundary Γ of a prism near the cavity. Microlaser structures of this kind may be manufactured in practice, e.g., by the nano-imprint lithography technique. On setting l _pol = 4 μm, whereas l _d-p = 0.2 l _pol and l _air = 0.03 l _pol, we find that the effective optical path length of the individual modes is about L _eff ~ 15 μm. Since this is about an order of magnitude larger that the wavelengths of interest, we apply the (approximate) ray-tracing approach to qualitatively assess the optical path lengths and to identify the modes of the microcavity, as specified in Eqs. (1) and (2). For mode numbers on the order of m ~ 30 this corresponds to a mode spacing (free spectral range, FSR)

of approximately δλ = 20 nm. In Eq (3), λ_m is the resonant wavelength corresponding to mode number m.

 

Fig. 1. Layout of the microcavity solid-state dye laser illustrating the polymeric gain medium containing dye molecules (dots in the central slab) and the effective optical paths of the resonant cavity modes (dashed lines).

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Heating the microcavity changes the geometrical size and the refractive indices of the media. Both these changes cause a variation of the optical path length, which results in a shift of the resonance mode frequencies. As the heat source, we employ two copper rods of round cross-section inserted in the outer triangles of the microcavity (see Fig. 1). The rods are placed close to the interface between the gain slab and the triangular sections for two reasons: to minimize the disturbance to the electric field patterns, since the fields are least concentrated in this area as illustrated in Fig. 2, and to provide the best linearity of microcavity expansion during the heating. The rods have a diameter of 400 nm and may be fabricated in the structure by common microelectronic processes. A proper placing of the rods is necessary to prevent the bending of the microcavity walls and potential increase in the energy leakage due to the variation of the incidence angles of the rays along the optical paths of the modes. The positions at equal distances from the central slab near the center of the gravity of the triangles (see Fig. 2) were chosen after testing different possibilities. Slight deviations from straight microcavity walls are, however, observed at the edges of the gain slab, as shown in Fig. 3. But in the region of the effective microcavity modes the relative change of the optical path length due to thermal expansion can be linearly approximated by the relation

where α is the thermal expansion coefficient and ΔT is the temperature change.

 

Fig. 2. Resonance mode pattern demonstrating the optimal placing of the copper rods that provide heating of the microcavity to realize wavelength tuning. Red color corresponds to positive amplitude of the electric field, blue to negative, and green represents zero.

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We consider a steady-state regime in which the temperature throughout the microcavity is constant. We take the room temperature (T_R = 300 K) as the initial (reference) condition to define the thermally induced strain and the geometrical distortions when the rods (and the cavity) are heated. The external boundaries of the microcavity are treated as thermally insulated walls and two predetermined temperatures, namely 50 K and 100 K above T_R, represent the final steady-state situations. In our model, we neglect the thermal expansion of the copper rods since their cross-section is much smaller than the microcavity area and the expansion coefficient of copper (α = 17∙10^-6 K^-1) is only about one fourth of that of the polymer materials used in the microcavity structure, i.e., α _pol = 50∙10^-6 K^-1 and α _d-p = 70∙10^-6 K^-1 for the triangles and the gain slab, respectively [20]. The expansion of the microcavity corresponding to heating to 100 K above the T_R is illustrated in Fig. 3, in which the absolute displacement (square root of the sum of the squared x and y distortions) is shown color coded. On the other hand, the refractive index of polymer decreases with the increase in the temperature according to

where the specific values for our materials are obtained from Refs. [21] and [22].

On combining Eqs. (2), (4), and (5), keeping l _d-p = 0.2 l _pol, and considering the rod’s position for spatial distortions, we obtain using the data of the materials in question the following expression for the overall change of the effective mode optical path

 

Fig. 3. Relative distortion (color coded) of the microcavity geometry caused by thermal expansion of the material. The temperature increase is 100 K and the scale shows the displacement in μm. The copper rods are fixed in their positions when solving for thermally induced strain in solid materials.

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The resulting tuning of the microcavity resonance mode λ_m may then be evaluated from

where Eq. (1) was used. Applying the variation of operation temperature ΔT in the limits from 0 to 100 K above T_R, we find that in the sample setup a wavelength tunability of about Δλ = 8 nm is achieved in the interval between two neighboring cavity modes.

A script containing a parametrical solver is employed to find, by numerical simulation, the resonance frequencies and electric-field mode patterns for the microcavity dye ring laser. An optimized mesh with about 30000 Lagrange quadratic elements at each optical frequency is used to ensure the accuracy of the solution.

4. Results: wavelength tunability and microcavity size optimization

The tunability limits of polymer lasers are determined by the so-called glass transition temperature of the cavity and gain materials, when they begin to loose the reversible thermoelasticity. Depending on particular chemical compositions, SU-8 or PMMA-based materials can have a glass transition temperature of around or above 400 K. In our analysis, we investigate the thermoelastic expansion and thermally induced change of the refractive index that lead to observable shifts of the cavity mode wavelengths. In particular, we consider heating the microcavity to temperatures of 50 K and 100 K above the room temperature.

The microcavity mode spectrum contains several resonant wavelengths that satisfy the phase-matching conditions for a cavity round trip. As an example, we concentrate here on the wavelength tunability of just one of these cavity modes, since the other modes behave analogously. We consider the mode number m = 28. Contributing with opposite signs, the effects of thermoelastic expansion and thermally induced refractive-index change result in a blue shift of the resonance wavelength, since the thermal index variation dominates over cavity expansion. The blue shift of mode m = 28 following a temperature increase of 50 K and 100 K above T_R is illustrated in Fig. 4. The drop in the intensity with respect to temperature increase follows from the negative sign of the thermooptic coefficient, c.f., Eq. (5). A lower refractive index leads to an increase of the critical angle for total internal reflection, and thus to a lower confinement of electromagnetic power in the cavity. Similar temperature behavior occurs for the other microcavity resonance modes as well.

 

Fig. 4. Wavelength shift of a resonant mode (m = 28) when the microcavity operation temperature in increased by 50 K and 100 K above the room temperature.

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As one can see from the relation in Eq. (6), the relative change of the effective optical path length of the resonant mode can be adjusted (for a specified temperature change) by appropriately setting the microcavity base l _pol. This optical path change is defined by desired tunability range Δλ given in Eq. (7). It is convenient to express the tuning range in terms of the free spectral range FSR = δλ between the cavity modes at the wavelength of interest. Making use of Eqs. (6) and (7) we then obtain the following relationship between the temperature increase, limited by the glass transition temperature, and the microcavity base length

where

For the parameters and materials considered in this work we obtain, at Rh6G wavelength λ_m = 530 nm, the constant K = - 875 μm K. The result in Eqs. (8) and (9) is graphically displayed in Fig. 5 for two different values of the tuning range Δλ. The plot allows one to find an optimized size of the microcavity for a determined bandwidth of tunability as a function of temperature variation. For example, to obtain a tuning range of 0.5 FSR with maximum operating temperature 100 K above T_R we find an optimized microcavity size of around 4.4 μm (blue solid line in Fig. 5). If one needs to broaden the tunability, e.g., to cover the whole FSR, the microcavity size should be increased accordingly (red dashed line in Fig. 5). We see that smaller temperature variations lead to larger cavity sizes.

 

Fig. 5. Dependence of optimal microcavity size on the change of the operation temperature. The optimized size is designed to provide a 10 nm (blue solid line) or 20 nm (red dashed line, corresponding to cavity FSR) wavelength tunability.

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5. Conclusions

In this paper, we have considered the wavelength tunability of solid-state dye lasers with the cavity size slightly larger than the radiation wavelength. Two main physical effects are studied that are responsible for such a tunability based on the variation of the mode optical path length: thermoelastic expansion of the microcavity and thermally induced change of the refractive index. Contributing with different strengths and opposite signs to the optical path change, these two phenomena result in a blue shift of the resonance mode wavelength when the cavity temperature is increased. For typical cavity parameters, the highest working temperature limited by the glass transition temperatures of the polymer materials allows one to obtain tuning of the resonance mode wavelength that covers the whole free spectral range. A simple relationship for a ring-type laser cavity between the temperature variation and the microcavity base length is established. It provides a method to estimate an optimized size of the solid-state dye microcavity to realize a pre-determined range of wavelength tunability by controlled heating.