# How would a fractal refract light?

A fanciful Pink Floyd reference has led me to wonder what white light passing through an object with an infinitely complex surface would do. Would it exit from a single chaotically-chosen point on the surface? Would it split into its constituent colors, each with a different trajectory, as in a run-of-the-mill triangular prism? Would these colors exit from a continuous stretch of the surface, or would they be chaotically distributed? What if instead of a single idealized ray of white light, it were a beam of small but nonzero width?

• I really really like this question. – DanielSank Dec 22 '14 at 22:45
• Not sure how easy to answer this one without more details of which fractal and what the fractal is made out of (glass?), but you might want to look at ijest.info/docs/IJEST10-02-11-128.pdf which has details of a fractal used for mobile communications - so not for light, but electromagnetic waves in the microwave range – tom Dec 22 '14 at 22:46
• I might be wrong, but I do not believe a fractal surface would disperse light much different than a frosted surface, which has the effect of rendering a transparent material translucent by scattering of light during transmission, thus blurring images while still transmitting light. – Wolphram jonny Dec 22 '14 at 22:47
• Clouds are fractals – theo Dec 23 '14 at 1:12
• While one might hope that simple ray optics -- regardless of its absolute correctness -- is internally consistent even in the presence of fractal surfaces, I'm not sure we're that lucky. Before considering something really complex, think about a perfect 0-width ray hitting a perfect convex corner in a mirror, right on the corner. Which way does the ray go? Alas it's not defined, since the normal to the surface is not defined at corners. A fractal just makes things worse, with normals failing to exist possibly on sets of measure greater than 0. – user10851 Dec 23 '14 at 5:24

# Is this an adequate question?

At first glance, the problem might seem ill-defined. However, this is not so, as we could physically approximate the fractal by (2d or 3d) "printing" it out in high resolution, such that each pixel's size is much smaller than the wavelength of the light and just do the experiment. Adding more detail does not alter much the light, as highly subwavelength objects blend into the fractal in a similar manner as individual water molecules blend into the mass of liquid water for visible light.

# Introduction

Calculation of the actual pattern is probably very hard and should be done with computer simulations. However, here I present some estimations. Those not wanting to go into the math and wave optics can skip the "Deriving the formulas" section and read only the "Results".

# Derivation of the formulas

Suppose we illuminate the fractal with a monochromatic light. The solution to Maxwell's equations is the superposition of the illuminating light wave and the contribution of each little point on the fractal that interacts with the wave. Only these little points contribute to the resulting refracted/reflected light. I model the result by assuming a collection of $N \gg 1$ points, confined in a region with a radius of approximately $R \gg \lambda$. The observer is looking from some distance $L \gg R$.

The complex electric field (ignoring the phase shifting in time) is mathematically $$E_{direction} \propto \sum\limits_{p} k_p e^{i2 \pi x_p / \lambda}$$ where $k_p$ is some complex constant describing the amplitude and phase of the generated field of some point $p$ and $x_p$ is the distance to that point. The term due to field falloff with distance is in the proportionality constant.

According to statistics, electric field on average for different directions is $$|E_{direction}| \approx |(\sum \limits_p |E_{avg}| e^{i \cdot random\:number})| = |E_{avg}| |\sum \limits_p e^{i \cdot random\:number}| \approx |E_{avg}| \sqrt N$$ Rotating our viewing point by angle $d \theta$ changes the distances to points by $|dx_p| \approx R d \phi$. Taking derivative of $E$ from first equation (constant multiplied by exponential): $$|\frac{dE_{direction}}{d \theta}| \approx |\sum\limits_p E_{avg} \cdot \frac{i2 \pi R}{\lambda} e^{i \cdot random\:number}| \approx |E_{avg}| \cdot \frac{2 \pi R}{\lambda} \sqrt N$$

Solving the last equality for $d \theta$ and setting parameters so that $\Delta \theta$ is the approximate angle between bright reflection/refraction directions and dark ones ($\Delta E_{direction} \approx E_{direction}$): $$\Delta \theta \approx \frac{|E_{direction}|}{|E_{avg}| \cdot \frac{2 \pi R}{\lambda} \sqrt N} = |\frac{E_{avg} \sqrt N}{E_{avg} \cdot \frac{2 \pi R}{\lambda} \sqrt N}| = \frac{\lambda}{2 \pi R}$$

Similar treatment for $\lambda$ (taking the derivative with respect to it and using this to find $\Delta \lambda$) also reveals that in order to turn a bright interference spot dark, on average, the wavelength has to be varied by $$\Delta \lambda \approx \frac{\lambda^2}{2 \pi R}$$

# Results

### White illumination

The last formula shows that if the fractal is large enough, eg. $0.03 mm$ (much larger than the wavelength of light), the fractal appears also white and uniform. This is because the spectrum of the reflected varies very rapidly and thus consists of many peaks and troughs in the visible spectrum (in case of 0.03 mm, there are on the order of ~1000 of them), but human eye has only three or four different types of color-sensing receptors.

### Monochromatic illumination

If the fractal is illuminated in a very monochromatic light (light with only one wavelength), then moving your eye will rapidly change the intensity of light seen. If you would be watching the $0.03 mm$ fractal from a distance of $1 m$, the fractal makes approximately one blink when moving your eye position by $~3 mm$.

# Are my analysis correct?

The only assumption I made was that fractal diffraction patterns are quite "random". While it might happen, that this assumption is wrong for some special fractals, it seems to be a right assumption in most cases. In the end, fractals can be very different and often have continuous parameters that can be changed. It would really seem unbelievable that most of them would produce some very organized result, independently of the wavelength.