Active control of electromagnetic radiation through an enhanced thermo-optic effect

The control of electromagnetic radiation in transformation optical metamaterials brings the development of vast variety of optical devices. Of a particular importance is the possibility to control the propagation of light with light. In this work, we use a structured planar cavity to enhance the thermo-optic effect in a transformation optical waveguide. In the process, a control laser produces apparent inhomogeneous refractive index change inside the waveguides. The trajectory of a second probe laser beam is then continuously tuned in the experiment. The experimental results agree well with the developed theory. The reported method can provide a new approach toward development of transformation optical devices where active all-optical control of the impinging light can be achieved.

second Gaussian shape control/pump beam is incident normally on the waveguide and induces a change in the refractive index of the PMMA layer due to thermal heating. The maximal effect is achieved if the pump laser wavelength is tuned to a particular Fabry-Perot (FP) resonance of the waveguide, note that the silver/PMMA/silver waveguide acts as a cavity for external radiation. The measured absorption, transmission and reflection spectrum of the cavity showing the FP resonances are given in figure 1(b). For the given geometry and frequency range of operation the absorbance can be as high as 20%. In the optical spectral range the PMMA is a weak absorber and without a cavity will absorb less than 0.4% of the impinging light for the given thickness (4.8 microns) of the polymer. The enhanced absorption in structured waveguide considered here can be used to dramatically enhance the thermooptic effect in the polymer.
To study the thermo optical effect in the experiment, a probe laser beam with wavelength of 457 nm is coupled into the silver/PMMA/ silver waveguide through a grating with period 310 nm drilled on the silver film between the substrate and the PMMA layer (see figure 1(a)). A second control Gaussian beam with width s 5 24 mm illuminates the cavity from above and is tuned to the 671 nm FP resonance, the dashed circles in figure 1b, thus facilitating the nonlinear change of the refractive index of the PMMA. In order to directly observe the trajectory of the probe beam inside the waveguide, the PMMA layer is doped with Eu 31 . As the probe beam propagates inside the PMMA layer, it excites the Eu 31 atoms that then emit fluorescence radiation at 610 nm wavelength. This radiation is recorded and used to study the effect of the thermo-optic nonlinearity on the propagation of the probe beam.
The deflection angle of a paraxial ray incident on an ordinary optical lens is given as h d 5 tan 21 (r 0 /f) < r 0 /f, where f is the focal length and r 0 is the impact parameter or the distance of the incident ray to the optical axis. This simple relationship can be obtained by considering only the refraction of the incident beam at the lens interfaces. For inhomogeneous optical media, however, the dependence of the deflection angle on the impact parameter can be rather complex. Still knowledge of the h d (r 0 ) dependence can serve as a powerful tool to quantify the effect of the inhomogeneity on the ray trajectories. In the experiment, the probe beam is excited by the grating, propagates toward the center of the nonlinearity produced by the pump beam and concurrently is deflected by the angle h d . The recorder ray trajectories for four different impact parameters are shown in figure 2. In figure 2(a), the incident beam has an impact parameter r 0 5 48.3 mm that is large compared to the extent of the enhanced nonlinearity region (given by the pump beam width s 5 24 mm). Correspondingly the bending effect is weak. As the impin-ging light beam approaches the center of inhomogeneity (the ''lens''), the deflection angle increase and reaches a maximum value h d 5 2.4u for r 0 5 15.1 mm (see figure 2(b)). However, further decrease in the impact parameter results in a near linear decrease of the deflection angle as seen in figure 2(c) and (d). This behavior is similar to that of an ordinary concave optical lens.
To provide a better physical description of the experimental results, we consider the Largangian formalism for the ray trajectories in the system coupled with a phenomenological model of the thermo-optic effect.
Thermo-optic effect. The refractive index of the PMMA under inhomogeneous thermal conditions is given as nr ð Þ~n 0 zq Tr ð Þ{T 0 ð Þ , where q 5 dn/dT is the thermo-optic coefficient, Tr ð Þ is the local temperature, T 0 is the ambient temperature (away from the heat source) and n 0 is the refractive index at the ambient temperature. The PMMA has a large negative thermo-optic coefficient 48 q < 21.15 3 10 24 K 21 , and ambient refractive index n 0 5 1.499 (at the probe laser beam wavelength of 457 nm). The local temperature is obtained from the inhomogeneous heat equation {k+ 2 Tr ð Þ~Qr ð Þ, where Q is the heat delivered by the control laser and k is the thermal conductivity of the PMMA. Under the specific conditions of the experiment, a waveguide thickness d that is substantially smaller compared to the width of the control Gaussian beam s, we have (see Methods): where, P 0 is the total laser power and A is the absorption coefficient. The refractive index profile according to equation (1) is shown in figure 3 (a). The heating due to a control laser with P 0 5 2.8 W reduces the refractive index of the PMMA at the center of illumination by 2.86%. This is a significant nonlinear change which corresponds to a nonlinear refractive index n 2 5 djqjA/12k 5 2.
where the derivatives are taken over an arbitrary affine parameter. By solving the Euler-Lagrange equations it is easy to obtain a first integral of motion and accordingly the deflection angle as function of the impact parameter distance r 0 (see Methods): where the turning point r t is related to the impact parameter according to r 0 5 r t n(r t )/n 0 , and d~d P 0 A q j j 12pks 2 n 0 . The theoretical result equation (2) is in excellent agreement with the experiment as shown in figure 3(b). For small impact parameters r 0 =s, we have a linear increase of h d with the increase in the impact parameter similarly to what one expects for an ordinary concave lens with an effective focal length of f~s 2d ffiffiffi p p~238 mm. With the further increase in the impact parameter the deflection angle reaches a maximum for r 0~s ffiffi ffi 2 p~17 mm, and then exponentially decreases.
The theory predicts a liner dependence of the deflection angle with the control laser beam power h d / d / P 0 . To study this effect we fix the impact parameter at r 0 5 13.9 mm, and vary the input power of the control beam. A comparison between the experiment and theory is shown in figure 4. As expected the deflection angle increases linearly with the input power and closely follows the theoretical line with a slope of 0.84 deg/W. For the available experimental range, saturation in the nonlinear response is not being observed which exemplified the potential to achieve even stronger nonlinear effects in the proposed waveguide configuration.
The inhomogeneous guiding effect can be controlled by changing the intensity and/or shape of the absorbed radiation. Employing control illumination through pre-set apertures or by using vortex beams with spatial light modulator (SLM) 49,50 , in principle, we can produce a large variety of inhomogeneous index profiles. A direct control of heat flow, as recently demonstrated in metamaterials 51,52 , can also be employed to achieve similar functionalities. We must also note that our technique can be modified to operate with other nonlinear materials. Polymers such as poly(methylmethacrylate), polystyrene, and polycarbonates are of special interest since in general they have higher thermo-optical coefficients compared to non-organic materials 53

Discussion
We have experimentally demonstrated and theoretically modeled the guiding and deflection of light due to an enhanced thermo-optic effect in transformation optical waveguides. In the experiment the thermooptic effects is due to the resonant absorption in the PMMA of a control laser beam. The inhomogeneous intensity (Gaussian beam) of the control laser induces a radially varying refractive index profile in the waveguide which allows for active manipulation (deflection) of a secondary probe beam. For small impact parameters the systems behaves as a concave lens with a tunable effective focal length. In our system a direct control of the heat flow or/and illumination through pre-set apertures can be utilized to develop multi-physics transformation optical devices that guide light in a pre-determent way.
In this work, we just simply demonstrate the basic idea how to employed thermal effect to produce gradient index medium and realize optical control of light propagation inside PMMA waveguide. In the future research, this method can be used to manipulate the light propagation to follow a really transformed path with meaningful applications.

Method
Sample fabrication. First a 40 nm silver film is sputtered on a glass substrate. A grating with 310 nm period, used to couple light into the planar waveguide, is drilled on the sliver film with focused ion beam (FEI Strata FIB 201, 30 keV, 150 pA). A PMMA resist for certain solubility (1.5 g PMMA powder dissolved in 10 mL toluene) mixed with Eu 31 is spin coated on the sample and dried in an oven at 70uC for 2 h. The thickness of the polymer layer is about 4.8 mm. The Eu 31 is added for the purpose of fluorescence imaging of the ray trajectories inside the waveguide. Finally, a 28 nm cladding sliver layer is deposited on the sample using electron beam evaporation thus forming the Fabry-Perot resonator.
Theoretical formation of the thermo-optic effect. We consider the steady state heat equation with a cylindrically symmetric heat source (Gaussian beam): where T is the laser induced temperature profile inside the waveguide, k 5 0.25 W/m. K is the thermal conductivity of the PMMA and Q 0 is maximum heat density delivered by the control laser. If the waveguide thickness d is substantially smaller than the width of the Gaussian beam s, i.e. L 2 T r,z ð Þ Lz 2 0 1 r L Lr r LT r,z ð Þ Lr * s 2 d 2 ?1, then equation (3) can be reduced to This equation has explicit solution of the form where we have enforced the boundary condition T(r, 0) 5 T(r, d) 5 T 0 , with T 0 being the ambient temperature. Finally, introducing the average temperature across the T r,z ð Þdz we obtain the nonlinear refractive index of the PMMA as: where d~Q 0 d 2 q j j 12kn 0~d P 0 A 12kps 2 n 0 q j j and P 0 is the total power of the control laser.
Ray trajectories and angle of deflection. The ray trajectories in the centrally symmetric media given by the refractive index equation (5), are described by the where h is azimuthal angle, and b 5 n 0 r 0 is related to the impact parameter r 0 . Introducing the non-dimensional radial coordinate f 5 r/s, the ray trajectory follows from equation (6) as where q t d ð Þ~b=sn 0~ft 1{de {f 2 t and f t 5 r t /s is the turning point. The above integral does not have an analytical solution. However, the thermo-optic effect is weak and by expanding the integrant with respect to d=1 and keeping only the first two terms in the expansion we obtain where erfc is the complimentary error function. The deflection angle as function of the impact parameter distance r 0 thus follows as where we have used that f t < b/sn 0 5 r 0 /s, for d=1.