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

(under construction)

## 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.

## 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 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. The evolution of the intensity profile in such optical systems are characterized by the nonlinear Schrodinger equation:

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

While there are some unique features in this system due to the specific polymer process involved, it is worth noting that

## 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.