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The diffraction efficiency of phase gratings recorded by two s-polarized waves in a polymer is modeled. The molecular orientation distribution along the grating period is seen to become homogeneous as a function of time due to saturation. Experimental data is presented, in agreement with the model.
©2009 Optical Society of America
1. Introduction
The optical inscription of gratings in azobenzene containing polymer films has been under intense investigation in recent years [1,2]. The interest in this subject is due to ease with which three different types of erasable gratings can be directly inscribed in the films using low and moderate power lasers [3–5]. Simple exposure of the film to the interference pattern produced by two intersecting optical beams can result in volume dichroic or birefringence gratings, in surface relief gratings and in volume density gratings. The volume birefringence gratings are associated with the optically induced isomerization of the azobenzene molecules that are selectively activated by using polarized light. The surface relief and volume density gratings result from the mass movement of the azobenzene along with the surrounding polymer matrix. Studies have looked at the relative importance of each of these gratings depending on the writing conditions, such as laser polarization, material composition and temperature. In general, all three types of gratings are present during the writing process and some effort is required to separate the contribution from each grating to the scattering of a probe beam by the film [6–8].
The elementary underlying process which causes the birefringence is cis-trans isomerization of the azo-molecules. Under the excitation of linearly polarized light, the molecules undergo multiple trans-cis-trans isomerizations, changing their angular position at each cycle, and finally the majority of the molecules are oriented in a plane perpendicular to that of the light polarization. If the light intensity or polarization is periodically distributed along the polymer surface a periodic change in the refractive index is induced. Thus a diffraction grating is inscribed in the material. The periodic light pattern is produced by the interference of two coherent plane waves (holographic method). It has been shown that in the case when the two waves are circularly polarized or p polarized with respect to the plane of incidence, a surface relief grating as well as a volume density grating are inscribed in addition to the volume refractive index modulation. In the particular case when both recording waves are pure s-polarized, no (or weak) relief grating is inscribed. Further inhibition of the formation of the surface relief grating can be achieved by writing the grating through the transparent substrate and by covering the free azopolymer surface with a thin transparent polymer film.
In the present work we focus on the volume birefringence grating that is generated at the beginning of the writing process since this grating is written on a time scale that is shorter than the other gratings and the optically induced birefringence is known to saturate in this shorter time scale. The saturation directly modifies the diffraction efficiency of the volume birefringence grating as the sinusoidal distribution profile is altered to the point where the diffraction is reduced to zero. This can be directly observed in the light scattering characteristics of the optically induced birefringence grating.
2. Theory
In order to describe the behavior of the birefringence and the diffraction efficiency we begin with a simplified model for the mechanisms involved in optically induced motion. More comprehensive models [9] have been developed but our simplified version will suffice to understand the basic characteristics of the temporal evolution of the experimental observations. The more comprehensive models could then be used to perform quantitative analysis of experimental results. We assume that the azobenzene molecules are uniaxial in the trans form and are oriented either in the s direction, the writing direction of our writing beam, or in one of two p directions. The molecules can also be in the cis form which we will consider to be optically isotropic. The relative concentrations of molecules in each state are described by S, P and C. In our simple model the dynamics of the molecular concentrations under the effect of a writing beam is described by a set of differential equations:
Where aI is the photon absorption rate of the trans molecule along the axis of the molecule, I is the light intensity and τ is a relaxation time constant. In this simple model, the trans-molecules are optically driven to transform into cis- molecules which then relax back to the S or one of the two P states with the corresponding time constant. We assume that the P and C states do not absorb the actinic light. We assume that there are initially only trans- molecules randomly distributed in all directions. The initial values of S, P, C in the medium before light excitation takes place are correspondingly 1, 2, 0. The P concentration is twice that of the S- because there are two directions perpendicular to the light polarization - one in the plane of the polymer surface and one perpendicular to it. The solutions of the coupled equations are given by Eq. (2).
The light intensity has a sinusoidal profile as shown in Fig. 1. The typical grating period is about 1 micron. The maximum is normalized to 1, while the minimum depends on a constant parameter which we call “noise” to account for experimental instabilities and internal forces.
From Eq. (2) we calculate the distributions of S and P oriented trans molecules as a function of position and time. By taking the sum of S and P over the grating period we can calculate the birefringence Δn as follows:
where ns and np are the refractive indices in s- and p- directions correspondingly.
Since the optically induced change in refractive index and the film thickness are small the phase retardation of light going through the film is assumed to be proportional to the change of refractive index squared. Thus the far field diffraction efficiency in the first order is proportional to the first order Fourier transform of the refractive index squared while the second order diffraction efficiency is proportional to the second order Fourier transform of the refractive index squared, in our model this is expressed as:
Where ∧ is the grating spacing. Figsures 2–4 present typical simulation results from this model, I is presented in Fig. 1, a= 0.5 and τ= 100. In Fig. 2 the temporal concentrations of S, C and P are presented as well as the related birefringence when the s-polarized writing beam is on. The S concentration is seen to decrease quickly, the C concentration first increases then decreases as molecules fall into the inactive P states who are seen to increase at a slower rate than the S decrease due to the intermediate C state life time. The birefringence is seen to increase steadily, in an almost bi-exponential manner, to a maximum level corresponding to saturation. In more complete models the dynamics is a bit different and the equilibrium levels of the various states are affected by transitions and rotations not included here but the general behavior is similar such that the present model is sufficient to illustrate the effect of saturation on the diffraction efficiency. In Fig. 3 the S concentration is shown as a function of position over one period of the grating. S is initially constant before optical excitation and is seen to decrease as light activates the angular hole-burning and re-orientation process. The S profile initially follows that of the sinusoidal writing beam and then flattens as the molecules reach saturation orientation at different times for different positions. Finally, the S concentration has reached saturation at all positions and the S profile is again flat. The P profile also undergoes a shape change as a function of time going from flat to sinusoidal and back to flat. These changes in profile are directly observed by looking at the time evolution of the diffraction efficiencies as illustrated in Fig. 4 for the model presented here. Here we present the first order diffraction produced by the S and the P oriented molecules. The data shows that both populations initially diffract efficiently when the sinusoidal profile dominates. The first order diffraction then reduces back to zero as the populations tend towards saturation throughout. We also note that the P diffraction maximum occurs later that the S maximum because of the delay originating with the C intermediate state.
3. Experimental results
The azobenzene polymer samples of pDR1m were prepared by spin-coating as described elsewhere [10]. We measured the birefringence and the diffraction efficiency induced by plane parallel waves an Argon ion laser with wavelength λ=488nm and intensity range 20–80 mW/cm2. A probe beam was provided by a diode laser with λ=635 nm and intensity 1 mW. For the birefringence measurements we used the classical crossed-polarizer scheme, while for the gratings we used a Lloyd mirror scheme as illustrated in Fig. 5.
Our writing beam was s-polarized to produce an intensity modulated interference pattern with constant polarization. This polarization should also produce only a refractive index modulation in the volume of the film without a significant surface relief grating. To further prevent a relief grating formation, we spin-coated a thin (100nm) polyvinyl alcohol layer on top of the azo-polymer. That layer is homogeneous and transparent and should inhibit mass polymer movement at the surface [11].
The s- and p- polarized components of the first and second diffracted orders were simultaneously measured. The results are presented in Figs. 6(a) and 6(b).
Fig. 6. Diffraction efficiency of the first (a) and second (b) order s- and p- components. Recording is stopped at t=200s
For s-polarized writing the diffraction efficiency for both polarizations and both orders first increase and then gradually decrease. Then the diffraction efficiency eventually reach zero. The birefringence itself increases continuously until saturation as seen in Fig. 7.
The results from the birefringence and diffraction measurements are in qualitative agreement with those of the model presented earlier. These confirm that the diffraction from the volume gratings in this case are expected to disappear on the same time scale as the time required to reach saturation of the birefringence. Thus in experiments where linearly polarized light (s or p) is used to inscribe surface and volume grating formation some attention should be paid to the effect of saturation on the diffraction results. Typically the diffraction from the volume refractive index gratings will have disappeared after a short time and further diffraction comes only from volume density gratings and surface relief gratings. Furthermore one could expect that the refractive index profiles in the bulk due to the hole-burning effect would be relatively flat.
4. Conclusion
An intensity hologram of fixed polarization is used to inscribe a volume grating in an azopolymer polymer film. The first and second order diffraction peaks are seen to disappear as the optically induced molecular reorientation in the volume becomes homogenous due to saturation.
References and links
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