# Why don't household mirrors scatter light in all directions?

Imagine you have a mirror which looks like this:

Since it is an ordinary household mirror when you zoom in on it, it should have a fractal-like structure so when you zoom in it may look something like this:

A surface like this should reflect light in all directions right?

• Sorry if you know this, but have you read : explainthatstuff.com/howmirrorswork.html and taking into account the answer below. – user198207 Jun 22 '18 at 19:13
• "it should have a fractal-like structure": why? – matt_black Jun 23 '18 at 12:59

The effect of the roughness of the surface on the scattering of light depends, according to the Rayleigh criterion for rough surfaces, on the wavelength of light and on the incident angle.

Basically, it says that if the roughness of a surface, which could be characterized by the difference in height at peaks and valleys of the surface, $\Delta h$, is smaller than $\frac {\lambda} {8cos\theta}$, the surface could be considered smooth. It has to do with constructive and destructive interference of of the reflected light.

This is a very crude definition of roughness, but it gives you an idea that the same surface will appear smoother or more specular as the wavelength and the incident angle of the light increases.

So, although at the microscopic level the surface of a mirror may look rough, for the wavelengths associated with visible light, it is obviously pretty smooth and produces almost perfect specular reflection.

• It would be great if you derived the formula you quoted! – KF Gauss Jun 22 '18 at 20:58
• Some rough math for the lazy: for $\lambda \approx 400\text{nm}$ at an angle of 45 degrees the threshold seems to be on the order of $70\text{nm}$, or say $50\text{nm}$ for a nicer number. So I guess the threshold is on the order of around 1/1000th of the thickness of human hair, or 25 times the diameter of DNA. – user541686 Jun 23 '18 at 7:46

Why should it have a fractal structure? It's true that the surface isn't perfectly smooth, but it's very close. The roughness might be up to 10 nm or so, but visible light is ~550 nm, so it in effect doesn't really see the surface roughness.

• Where did you get the number 10-50 nm from? Is it a guess? What did you base the guess on? Can you justify these numbers? – AccidentalFourierTransform Jun 23 '18 at 1:23
• @AccidentalFourierTransform well nature.com/articles/srep12550/figures/1 shows microscope slide glass is no more than 10 nm peak-to-valley. It might be better than I thought – HiddenBabel Jun 23 '18 at 4:10
• Nice! Why don't you add that into your answer? References are encouraged here. Cheers! – AccidentalFourierTransform Jun 23 '18 at 13:28