## Abstract

The channel formation by the self-trapping of a (1 + 1)D beam in polymeric media based on a polymethylmethacrylate (PMMA) matrix containing phenanthrenequinone (PQ) molecules is predicted theoretically and observed experimentally for the first time. Particular attention is paid to the effect of thermal beam expansion, which in conjunction with the photorefractive nonlinearity of the medium results in the possibility to control optically the geometrical parameters of the generated channel.

© 2011 OSA

## 1. Introduction

Among the optical effects, which are intensively investigated in recent decades, is the phenomenon of self-action of light beams in nonlinear optical media, e.g. self-trapping and self-focusing of optical beams, self-phase modulation of optical pulses etc. It allows to realize the direct optical conversion of images and to control their space-time light structure [1–3]. The practical realization of self-trapping depends to a large extent on the choice of a suitable photosensitive material. The splicing of damaged waveguides, the confinement of light beams to guarantee conditions for a steady-state information transfer and the fabrication of a variety of different optical splitters and switching systems for optical signals require the formation of waveguide channels that conserve their properties for a long time without a supporting radiation source. These tasks, as well as constantly emerging new problems in fiber-optical technologies determine the need for further research and the development of new photosensitive materials for the generation of optical waveguide structures. Owing to their high nonlinear optical susceptibilities and the ability to maintain diffracted structures for long periods of time photopolymers became attractive media in this field [4–8].

The presented work demonstrates for the first time to our knowledge good results of the straight self-trapping channel generation in polymethylmethacrylate (PMMA) with distributed phenanthrenequinone (PQ). The self-trapping process occurs due to the compensation of the natural beam diffraction by the positive change of the refractive index that leads to a stable cross-section of the propagating laser beam to travel within the media by self-focusing. Polymers based on PMMA with addition of PQ molecules show a light-induced change of the refractive index up to 10^{−3} [9,10]. The photoaddition of PQ molecules to the polymeric matrix leads to the generation of the photoproduct in the illuminated area changing thereby the refractive index. The holographic relaxation technique was used previously in order to study the optical properties of the PQ-PMMA material [9–15].

The results of the numerical simulation of self-trapping in PQ-PMMA polymeric layers due to the nonlinear properties of the medium were shown in our theoretical work [16]. The experimental confirmation of the formation of a straight waveguide channeling as a result of Ar-laser beam self-action in PQ-containing media is demonstrated in this paper. The numerical model of the description of the laser beam propagation in PQ-PMMA is improved by taking the heat conduction equation into consideration. The acquired effect of the self-trapping broadening is caused by the PQ-PMMA properties of nonlinear focusing and thermal defocusing of the light beams. Moreover, the dependence of the generated channel cross-section on the input beam power is shown.

## 2. Experimental self-trapping in PQ-PMMA

The polymeric layers were prepared by the solution of PQ and PMMA, which was previously purified from the monomer, in the chloroform. The solution was filtrated after mixing in order to decrease the quantity of light-scattering centers. The glass substrates were coated with the liquid solution of the various ingredients to form polymeric films of the necessary dimensions and shape. The mechanical rigidity that is required for waveguiding application was obtained by subsequent drying. For a slow removal of the solvent, the samples were placed in a Petri dish for 20 hours at temperature of ${100}^{\xb0}C$, which is below the glass transition temperature of the PMMA (${105}^{\xb0}C$). The obtained samples have good optical quality and high thicknesses (up to 200 µm - 400 µm) [9–11].

The generation of the refractive index change in the PQ-PMMA occurs as a result of the modulation of optical parameters in the course of the photochemical hydrogen reduction by the redox cycling agent PQ with the formation of a semiquinone radical [9,10]. Its subsequent addition to a polymer macromolecule forms a stable photoproduct. This mechanism can be provided in real time by using a laser source which irradiates in the absorption band of PQ (480-540 nm).

Experiments for the self-trapping generation were carried out for the samples with a layer thickness of 400 µm and a PQ concentration of 2.5-3 mol.%. To generate a waveguiding structure in the polymer an Ar-laser was used (Fig. 1(a) ). The laser power required to form a channel was found experimentally to be in the range of milliwatts. That is an order of magnitude higher than the values obtained previously from our theoretical model [16]. The laser beam was focused on the front surface of the polymeric layer using a microscope objective with a focal length of 16 mm. The PQ-PMMA sample was placed on a mechanical stage which allows to control the position of the input beam. The beam diameter at its waist behind the microobjective was 16 µm (FWHM). A CCD-camera was used to detect the self-trapping formation.

The polymeric layers have a yellow colour with an absorption band up to about 540 nm. Under Ar-laser illumination (514.5 nm) the colour changed from yellow to almost transparent, which means that all photosensitive PQ molecules join the polymeric matrix with subsequent formation of stable photoproducts. The self-induced waveguiding structure in the PQ-PMMA medium is shown in Fig. 1(b).

The experiment was carried out with several identical samples. The illumination of the samples with a power of 8 mW was made for periods of 60 min. The generation of the planar straight channel induced by the incident light itself was observed in the PQ-PMMA layer during an irradiation period of 12 min (Figs. 2(a)-(e) ). The photoattachment of PQ molecules to the polymeric matrix resulted in a necessarily nonlinear change of the refractive index by which a stable waveguiding structure was produced under the Ar-laser illumination. The formed channel has a length up to 4-5 mm after an exposure time of 6-10 min. By further illumination (15-60 min) the distortion of the self-trapping channel started. This happens due to the absorption of PQ molecules in the peripheral areas of the channel, leading to a multiple broadening of the channel and to a loss of the self-trapping property.

In spite of the small input beam diameter (16 µm) the width of the achieved channel was about 500 µm (Figs. 2(b)-(e)). The additional investigation of the Plexiglas PMMA material without PQ molecules under Ar-light illumination revealed a high divergence of the propagating beam (Fig. 2(f)). This led us to suggest that as a result of the high input power at the beam focus the polymeric material starts warming up due to the absorption with the consequently large increase of the light scattering.

The activation of the warming-up effect in the PQ-PMMA film results in the formation of a thermal lens and increases the divergence of the laser beam, starting just after beginning of the illumination. Normally Plexiglas is almost transparent for Ar-laser radiation, being subjected to the natural diffraction of light. According to the parameters of the laser and the microobjective, the light divergence in the PMMA layer should be equal to 10-12 mrad. The thermal lens formed in the layer increases the light divergence even under the influence of minor absorption ($\alpha \approx $ a tens of cm^{−1}) [11]. However, the high intensities in the focal region ($I\approx 1\text{\hspace{0.17em}}kW/c{m}^{2}$) lead to a broader distribution of light (Fig. 2(f)).

The injection of the light sensitive PQ components into the polymeric matrix causes the photoproduct formation and generates a straight channel (Figs. 2(a)-(e)). The process of fast heating appears at the point of maximum input intensity till the PQ-phototransformation is completed and proceeds within the layer. The formation of the self-channeling with a large cross section can be associated with two competitive mechanisms: thermal beam expansion together with the variation of the photoinduced nonlinear refractive index.

Thereby, the experimental results indicate the possibility of the formation of straight waveguides in PQ-PMMA media. The broadening of the channel can be connected with the activation of the thermal defocusing in the material due to the high input power. This defocusing effect is compensated by the effect of nonlinear self-focusing of the light beam generating a waveguide structure. This thermal expansion of self-trapping can be included in our numerical model by taking the heat conduction equation into consideration.

## 3. Mechanism of light-channel broadening

The special characteristic of the light beam distribution in polymeric materials is a nonlinear change of the absorption coefficient during the PQ-photoattachment process. Thus, along with the variation of the refractive index caused by the photoreaction of PQ ($\mathrm{\Delta}{n}_{PQ}$), a thermal change of the refractive index ($\mathrm{\Delta}{n}_{T}$) is closely connected with this and has to be taken into account for any simulation. Under steady-state conditions the influence of the thermal effect on the change of the refractive index can be calculated as [17]

where*α*is the absorption coefficient of the medium,

*I*is the beam intensity with the Gaussian distribution [$I={I}_{0}\mathrm{exp}[-({x}^{2}/{x}_{0}^{2})]$],

*t*is the effective interaction time, ${C}_{\rho}$ is the heat capacity of the unit volume, called also the volumetric heat capacity, and $\partial n/\partial T$ is the thermooptical coefficient. Since the medium is normally hotter on the beam axis, compared with the outer regions, a transverse gradient of the refractive index appears, called the thermooptical effect, which is quantified by the coefficient $\partial n/\partial T$ [18]. Index changes can be caused by the temperature dependence of the refractive index and by the thermally induced mechanical stress (photoelastic effect). Both mechanisms can lead to bulging of the end faces of the gain medium, which leads to the effect of thermal lensing. In a medium with$\partial n/\partial T<0$, there is a defocusing effect caused by the temperature, which eliminates the process of self-focusing. In rare cases, if $\partial n/\partial T>0$, one can observe the reverse situation, i.e. a heating of the medium generates self-focusing.

The variation of the refractive index due to warming can be essentially higher at a certain area in the polymeric layer and can compensate accordingly the $\mathrm{\Delta}n$- change caused by photoattaching. The position of this area corresponds to the maximum light intensity and results in a channel widening. To investigate the complete change of the refractive index in the PQ-PMMA material during the self-trapping generation, we have included the mechanism of the thermal nonlinearity in the numerical simulations.

The change of the temperature ($\mathrm{\Delta}T$) in the illuminated area can be calculated using the heat conduction equation along the two orthogonal coordinates x and z [19]

*b*is the coefficient of the heat transfer and ${C}_{\rho}={c}_{T}\cdot \rho $ is the volumetric heat capacity with the specific heat capacity ${c}_{T}$ and the density

*ρ*of the material.

The solution of Eq. (2) provides the value of $\mathrm{\Delta}T$ (temperature difference between the waveguide structure and the surrounding material), which can be calculated by separation of the linear (Eq. (3)) and nonlinear (Eq. (4)) parts of Eq. (2) applying the split-step method [20]

The change of the refractive index due to the heating ($\mathrm{\Delta}{n}_{T}$) results in the thermal defocusing of the beam and can be calculated by

where $\partial n/\partial T=-1.3\cdot {10}^{-4}\text{\hspace{0.17em}}{\text{K}}^{\text{-1}}$ is the thermooptical coefficient of PMMA [21]. Because of the negative value of the thermooptical coefficient the thermal change of the refractive index leads to an absolute magnitude with opposite sign as compared with the magnitude of $\mathrm{\Delta}{n}_{PQ}$. This results in a thermal defocusing of the light beam with the corresponding channel broadening.The change of the refractive index due to the photoreaction of PQ determines by the difference between the refraction coefficients of the photoproduct (${R}_{HPQR}$) and phenanthrenequinone (${R}_{PQ}$). Based on the values for the variations of their concentrations (photoproduct ${C}_{HPQR}(x,z,t)$ and PQ molecules ${C}_{PQ}(x,z,t)$) and using the Lorentz-Lorenz formula, the following expression for the change of the refractive index is obtained [9,22]

Thus, the sum of the two contributions to the change of the refractive index ($\mathrm{\Delta}{n}_{\Sigma}$) is decreasing according to

## 4. Numerical simulation of channel broadening

The calculations are based on the PQ-PMMA parameters used in the experiment presented above: laser input power 8 mW, input-beam radius ${x}_{0}=9.6$ µm (at 1/e mean), initial PQ concentration 2.5 mol.%, $n=1.5$, ${\lambda}_{0}=514.5\text{\hspace{0.17em}}$ nm, $\mathrm{\Delta}R=1.4\text{\hspace{0.17em}}c{m}^{3}/mol$ [9]. The absorption coefficient *α* varies in a range of about tens of $c{m}^{-1}$. For the solution of Eq. (3) the thermal diffusivity coefficient of ${a}_{T}={10}^{-7}\text{\hspace{0.17em}}{m}^{2}/s$ and a heat capacity of ${C}_{\rho}=1.7\cdot {10}^{6}\text{\hspace{0.17em}}{\text{J/(m}}^{\text{3}}\cdot \text{K)}$ were used in correspondence with the parameters of the PQ-PMMA material [21,23,24]. The magnitude of the heat transfer coefficient was found to be inversely proportional to the velocity of the heat propagation in the polymeric medium. In the numerical calculations $b=0.2\text{\hspace{0.17em}}{s}^{-1}$ was chosen for a polymeric layer thickness of 400 µm [19,23].

The obtained changes of the refractive index$\mathrm{\Delta}{n}_{T}$due to thermal defocusing in dependence on the illumination time for different layer lengths are presented in Fig. 3 . As a point of $\mathrm{\Delta}{n}_{T}$-measurements the coordinate x in the middle of the beam width is taken. The decrease of the change of the thermal refractive index reaches values of $\mathrm{\Delta}{n}_{T}=-1\cdot {10}^{-2}$. The interplay of thermal defocusing and nonlinear properties of the medium causes a channel widening at this point. The negative value of $\mathrm{\Delta}{n}_{T}$ substantially exceeds the positive change of the refractive index $\mathrm{\Delta}{n}_{PQ}\approx {10}^{-4}$ at the local point where the laser has a higher intensity [16]. The superposition of both processes at this point leads in total to a negative change of the refractive index. Since the values of the thermal refractive index change are presented as local magnitudes at a particular time, they are decreasing in time due to the heat transfer. At the point, where the radiation is already absorbed, $\mathrm{\Delta}{n}_{T}$ becomes equal to zero. Then the sum change of the refractive index $\mathrm{\Delta}{n}_{\Sigma}$determines by the positive values of the $\mathrm{\Delta}{n}_{PQ}$ due to the photoreaction.

The numerical evolution of the total refractive index profile is shown in Fig. 4 . As already mentioned above, one can see that in spite of the small input beam diameter (16 µm) the resulting channel has a large width of about 500 µm. The polymer heating starts immediately as soon as the light enters the layer and passes several microns. The self-trapping channel is substantially expanding due to the thermal nonlinear mechanism. The total change of the refractive index achieves values up to ${10}^{-2}$. The photoattachment of the PQ-molecules to the polymer PMMA occurs throughout the whole extended area of the beam with following formation of the stable and insensitive photoproduct. In this case, the negative thermal refractive index change is extinguished the positive $\mathrm{\Delta}{n}_{PQ}$ change due to its higher absolute value. At the point, where the channel is formed and all PQ molecules transformed into the photoproduct, the layer becomes transparent to the Ar-laser illumination and the temperature of the polymer is reduced. The expanded channel moves further along the length of the layer and similar mechanisms occur in the next area of the waveguide. While remaining behind the reacted region the polymer retains a positive total refractive index change (dark areas in Fig. 4, $\mathrm{\Delta}n$ are positive in the formed channel). This allows already illuminated and chilled areas to be used as waveguiding structures. The complete formation of a self-trapping channel in the PQ-PMMA layer of a length of 6 mm was realized within 5-6 min of illumination with an input beam power of 8 mW. The obtained numerical result corresponds to our experimental observation of the formation of a straight channel (Figs. 2(a)-(e)).

Thus, the mechanism of channel formation and broadening can be described like a heat wave propagating together with a light wave through the polymer (Fig. 5 , (Media 1)). At the point of maximum intensity the formation of a thermal lens occurs with subsequent activation of the thermal defocusing effect. The main broadening of the channel is observed during the first 15 s after starting the illumination process. At the place of the maximal beam intensity the temperature of the polymeric layer achieves a peak value of about ${100}^{\xb0}C$ (for an input beam power of 8 mW) due to the thermal absorption of the PQ-PMMA material. After finishing the PQ-photoaddition process the layer becomes transparent to the laser light and the illuminated area cools down. At this time the warming-up process is stopped. The area of the maximal temperature, a heat wave, moves forward together with the propagating light beam with a velocity determined by the speed of the reaction of the PQ-photoattaching process to the polymeric matrix.

## 5. Control of the width of the generated waveguide

In consideration of the dependence of the thermal defocusing on the beam power, we examined the possibility of the self-trapping formation using specific cross-section parameters of the channel. A numerical simulation of the channel formation provides a possibility to control the development of the channel width and allows producing waveguiding structures of desired dimensions.

Figure 6 demonstrates the results of the numerical calculation of the profile of the refractive-index change for input powers of 8 mW and 2 mW. The diameter of the channel is reduced by a factor of about 2, if the power decreases by a factor of 4.

The corresponding experiment was carried out by using the same PQ-PMMA layers and setup as in the previous self-trapping formation experiment (Fig. 1(a)). The formed channel has been detected as an area with practically absent absorption. The results of photometric measurements of the formed self-trapping channel gave us the possibility to plot the cross-section of the distribution of the refractive index in the polymeric layer and to determine the diameter of the generated channel. The formation of the waveguiding structures for two different input beam powers (8 mW and 2 mW) is shown for both cases in Fig. 7 together with the $\mathrm{\Delta}n$- distributions normalized with respect to the maximum values of the refractive-index profiles. At an input beam power of 8 mW the channel broadens out to a width of 510 µm (FWHM), whereas it achieved the width of 285 µm for the lower input power of 2 mW. The experimental results coincide with the theoretical ones with only small deviations due to the good agreement between the theoretical and experimental parameters.

## 6. Conclusion

The possibility of the (1 + 1)D beam self-trapping formation in PQ-PMMA polymeric layers due to the photoinduced nonlinear refractive index modulation and the effect of thermal beam expansion was shown experimentally and theoretically for the first time. The recording of self-trapping structures implemented in layers of purified PMMA with a high PQ concentration has been performed under the action of an argon laser of 8 mW input power. A comparison of the propagation of light beams in pure PMMA and in PMMA containing sufficient amounts of PQ molecules confirmed the decisive role, which the photoattachment process of the light-sensitive PQ molecules to the polymeric matrix plays for the formation of the waveguide channel. A width of the channel in the polymer of 500 µm was achieved with an incident beam of 16 µm diameter and remained constant along a distance of 4-5 mm.

It was found that the formation of the waveguide is strongly influenced by heating of the medium, which results in an additional thermal defocusing of the light beam. The theoretical modeling of the light propagation under conditions of the mutual action of the processes of PQ-photoattachment and thermal nonlinearity has confirmed the validity of the proposed approach. Heating produces a negative change of the refractive index in the photopolymer, which leads to the reduction of the resulting $\mathrm{\Delta}n$ and to a channel widening. The formation of a self-trapped channel with a length of 6 mm was observed by applying Ar-laser illumination for a duration of 5-6 min. This is in good agreement with the experimental results.

Moreover, a new method for controlling the waveguide cross-sections by changing the ratio of the two competing mechanisms of the nonlinear refractive-index variation (namely the formation of the photoproducts and the heating of the medium while varying the power of the light beam) was proposed. The experimental results were confirmed by the theoretical modeling of the self-trapping process. With a diameter of the channel much larger than the diameter of the laser beam waist in the focal plane the following simple relation has been derived: the square of the ratio of the diameters of the two beams is equal to the ratio of their powers. By using this relation it is possible to generate channels with well defined and suitable cross-sections.

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