# Spontaneous Pattern Formation Due to Modulation Instability of Incoherent White Light in a Photopolymerizable Medium

## References

1. I.B. Burgess, W.E. Shimmell, K. Saravanamuttu J. Am. Chem. Soc. 129, 4738-4746 (2007).

2. M. Peccianti, C. Conti, G. Assanto, A. de Luca, C. Umeton, Nature 432, 733-737 (2004).

3. Ball, P. The Self-Made Tapestry: Pattern Formation in Nature; Oxford University Press, 1998.

4. G.P. Agrawal, Nonlinear Fiber Optics, 3rd Ed. Academic Press, 2007.

5. R.A.L. Jones, Soft Condensed Matter Oxford University Press, 2004.

6. A.W. Snyder, D.J. Mitchell, Science 276, 1538-1541 (1997).

7. G.I. Stegeman, M. Segev, Science 286, 1518-1523 (1999).

## Keywords

Modulation instability, pattern formation, free-radical polymerization, photopolymerization, self-focusing, solitons

## Summary

This paper describes the spontaneous the spontaneous optical and structural pattern formation in a material which undergoes an intensity-dependent photo-induced free-radical polymerization upon irradiation from a beam of white light with initially uniform intensity. The refractive index of the material increases with degree of polymerization, and thus regions illuminated with higher intensity will develop a higher index. From an optical point of view, the evolution of the system is therefore that of a self-focusing system, since a beam with an intensity gradient will induce a gradient index lens, attracting higher intensity to regions of higher index and thus amplifying the intensity gradient. The evolution of the intensity profile in such optical systems are characterized by the nonlinear Schrodinger (NLS) equation [6]:

$-2ikn_{0}\frac{d\psi}{dt}=\left(\frac{d^{2}}{dx^{2}}+\frac{d^{2}}{dy^{2}}+k^{2}[n(|\psi|^{2})^{2} - n_{0}^{2}]\right)\psi$

where $k=2\pi/\lambda$. While there are some unique features in this system due to the specific polymer process involved, it is worth noting that NLS-type conditions have been studied theoretically and experimentally in many other systems [4,7].

In the polymerization reaction considered here, there are two competing factors dictating the spatial variance in the rate of polymerization (and intensity). First there is diffraction, which leads to the washing-out of intensity fluctuations, thus tending toward a uniform polymerization rate. Then there is the self-focusing contribution which causes amplification of gradients in the polymerization degree (and the refractive index). While a completely uniform intensity profile should then induce a uniform degree of polymerization and thus remain uniform upon propagation in the material, small fluctuations due to noise will be unstable due to self-focusing. This modulation instability (MI) cascades until the beam has broken up completely into an array of self-trapped filaments (see figure) having a characteristic spatial frequency (about 80 microns in this case), where stable equilibrium between the two competing processes is reached. The figure below shows the initial (center) and final (right) intensity profiles as well as the polymer structure formed at the end (left). This instability occurred first in one dimension, yielding an intermediate array of stripes.

Diffusion of the polymer chains will also act along with diffraction to wash out gradients in the degree of polymerization and in fact dominates initially, when the chain lengths are short and the mixture is in a liquid state. At some point during the reaction, the material transitions to a solid state, immediately after-which the pattern-formation was observed. At high powers, when photo-induced heating caused significant convection flow, its combination with faster reaction rates allowed the final patterning in the filament array to resemble the convection-induced density gradients that froze upon transition to the solid state. This is shown in the figure below. The top frame shows the intensity profile at the liquid-solid transition, and the lower frame shows the final equilibrium state.

Finally, it is shown that the spatial phase of the pattern can be dictated, improving order, by imposing a subtle perturbation in the initial intensity profile that matches the characteristic spatial frequency of the MI process. This is done with an amplitude mask.

## Soft-Matter Discussion

This paper illustrates the ubiquity in the dynamics of pattern formation observed in many soft-matter systems [3]. For example, spinodal decomposition of liquid-liquid mixtures, a topic covered in this week's section on phase transitions, can be thought of as a specific manifestation of this broader type of pattern formation that occurs in many physical systems where there are two competing factors determining the system's time evolution. In the case of spinodal decomposition, there is diffusion, which drives to wash out concentration gradients in order to maximize entropy in the system, and then there is the enthalpy component, ($\chi$) which must favor self-interaction in phase separating systems, amplifying concentration gradients for mixtures within the spinodal line. In the case of MI in a photochemical reaction, intensity plays the role of concentration, diffraction is the diffusive component and the self-focusing due to an intensity-dependent polymerization rate is the factor which favors the amplification of intensity gradients. The fascinating universality observed in both system is not merely the spontaneous magnification of intensity/concentration fluctuations, but also that there is a specific lengthscale at which the amplification is strongest, where the two competing factors equilibrate. For shorter-scale fluctuations, diffraction/diffusion term increases faster than the self-focusing/enthalpy contribution as the gradients become large. Fluctuations of too large a scale produce lower gradients, and thus will themselves become unstable to optimum-length fluctuations. For optical materials with a simple Kerr nonlinearity, the spatial-frequency-dependent gain spectrum for intensity fluctuations can be calculated analytically (see ref. 4). This calculation for spinodal decomposition is in Jones (pp. 33-36) [5]. A consequence of the underlying photopolymerization process in this paper being irreversible is that the final pattern at the equilibrium lengthscale is permanently etched onto the material in the form of gradients in the degree of polymerization. This paper also demonstrates how the characteristic spatial frequency associated with a pattern formation or phase separation can be effectively impart order onto the system. While pattern-formation of this type in many systems has a characteristic spatial frequency, its phase is random in general. Introducing an ordered noise into the system with a fixed phase at the characteristic spatial frequency fixes the phase of the resulting pattern.