# How would one generate Brownian light? What would it look like?

When light is an equal mix of all visible frequencies, we call it white light.

By analogy, sound that is a mix of all audible frequencies is called white noise.

For sound, there is an additional concept of Brownian noise, also called red noise, whose name derives from the "random walks" of particles undergoing Brownian motion. Brown noise sounds "smoother" and lower pitched (think ocean waves) than white noise (think hard rain), due I would assume to its less extreme shifts in frequency.

So, by direct analogy: Brownian light should also exist.

Since light is a transverse wave with polarization states not found in compression-only sound waves, I surmise that at least three distinct forms of Brownian light are possible:

1. Brownian-frequency light (BF light)

2. Brownian-polarization light (BP light), and

3. Brownian-frequency-and-polarization light (BFP light), in which the random walk takes place within a space where frequency and polarization are orthogonal axes.

Surprisingly, none of the ideas show up readily on a Google or Google Scholar search.

While that could be due to general recognition that noise types apply to any form of signal, one might think that the special case of human-visible would merit some special attention. Also, since the polarization and frequency-plus-polarization variants of Brownian state walks are not immediately obvious when starting from the sound analogy, they would need to be called out explicitly for light.

So, does anyone know if these ideas exist already and have been studied?

Is there a way generate BF, BP, or BFP light, e.g. with lasers?

Might the variants of Brownian light have any useful or interesting properties, e.g. for optical communications or signal encryption?

And finally: Humans can't see polarization, but they can certainly see frequencies. So, what would Brownian frequency light look like? I'm guessing red due to the way the Brownian spectrum is weighted. Perhaps that is why Brownian noise is also called "red noise"?

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Humans can see polarization. – Juris Jan 15 '13 at 22:41
Juris, great comment! – elcojon Jan 15 '13 at 22:54
One should note that neither sound nor light can have a truly Brownian spectrum over the whole frequency range, since the power spectrum goes with $1/\nu^2$, so the power becomes infinite as the frequency approaches zero. So really, we're looking for a process that produces light with an approximately Brownian spectrum over some specific frequency range. – Nathaniel Jan 16 '13 at 4:10
Are we transposing brownian motion of particles impinging on a microphone diaphragm to a light spectrum? How about shooting a synchrotron beam through a gas, would the decelerating electrons pick up the spectrum of the gas particles' velocity distribtion? – Bobbi Bennett Jan 16 '13 at 4:20
I'm sure he means the visible range. Well, I took the CIE standard observer data (10 deg) and applied $f^{-2}$ and I get $x=0.389, y=0.369$ which looks sort of pinkish-gray on the colour chart. Too bad, I was hoping for Brown. – Retarded Potential Jan 16 '13 at 4:26

## 1 Answer

We have the following relationship $$\Gamma(\tau)=\int_0^\infty \bar{S}(\nu)e^{-2\pi i\tau\nu}d\nu$$ where $\Gamma(\tau)$ is the temporal coherence function, which can be measured with a Michelson interferometer, and $\bar{S}(\nu)$ is the real normalized power spectral density function (PSD). As can be readily proven, the above is a Fourier transform relationship. For Brownian noise, there is a region where the PSD is proportional to $$\frac{1}{\nu^2}.$$

I say region because the function above is not integrable from $0$ to $\infty$, it has infinite energy. The Michelson could be used to check the source for the PSD.

To generate this spectrum could likely be accomplished with a laser. It would have to be inhomogeneously broadened gain medium with the modes distributed so that in the visible it approximates $$\frac{1}{\nu^2}.$$

Brownian polarization would take quite a bit of analysis, if I have time I'll update a section below if I can accomplish that.

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daaxix, thanks, interesting analysis! – Terry Bollinger Jan 18 '13 at 2:19