# Brightness of colors

I have 1W lights, red and green. If I turn the red light full power I get 1W red light, if I turn green light full power, I get 1W green. If I want to mix the lights and get yellow light, which has approximately the same brightness as red and green had, should I turn both red and green full power? But this means I will emit 2W of power. Should I turn 0.5W red plus 0.5W green? How does this work?

• Eye response is highly non-linear to both brightness and color. What are you trying to achieve? – zeta-band May 17 at 18:56
• I need to get same perceived brightness of red, green or yellow. – Roman Simonyan May 17 at 19:01
• Human eyes are waaay less sensitive to red light than they are to green or yellow. So try 1 watt for the red and then tinker with the green power until you get what looks like equal perceived brightness. Then try them together and maybe try cutting the power to each by half. You probably don't need to be too precise, the eye's ability to judge brightness is horrible. – zeta-band May 17 at 19:06
• @zeta-band your "waaay" is just a twofold difference. Luminous efficacy of 610 nm monochromatic red light is half that of 555 nm green. It's the same order of magnitude. – Ruslan May 17 at 20:33
• @Ruslan You are right, I should only have used two 'a's in waaay. – zeta-band May 17 at 20:48

## 1 Answer

First, to do a fully theoretical calculation of the power you need from your red and green lights to achieve the same brightness of yellow, you need to know spectral power distributions of your red and green lights. At the very least you need to know their dominant wavelengths. This is important, because e.g. red can be the orangish-red 610 nm (dominant wavelength of the #FF0000 red on a typical sRGB monitor) or deeper red 640 nm (typical red laser pointer). These have different luminous efficacy: luminous efficacy of the former is $$0.5$$ while for the latter it's $$0.18$$.

Once you know your spectral power distributions (or dominant wavelengths), you can calculate the CIE 1931 $$XYZ$$ color coordinates of your lights. Since your target color is yellow, you need to mix the colors so that you get the $$xy$$ chromaticity coordinates of something like $$(0.44, 0.55)$$, which correspond to 570 nm light. Actual chromaticity coordinates may be smaller if your lights are considerably non-saturated (i.e. pale red/green) — you'll need to choose a point on a line between your red and green colors on the chromaticity diagram, and take its coordinates.

Once you've determined the $$X, Y, Z$$ coordinates of your colors, you can play with the mixtures to determine the $$x, y$$ coordinates of resulting color. You just need to add them like

$$X_{\mathrm{mix}}=\alpha X_{\mathrm{red}}+\beta X_{\mathrm{green}},$$

and similarly for $$Y$$ and $$Z$$, and the $$x$$ and $$y$$ coordinates are simply defined as

\begin{align} x&=\frac X{X+Y+Z},\\ y&=\frac Y{X+Y+Z}. \end{align}

A quick calculation with a monochromatic 610 nm red and monochromatic 550 nm green colors gives me 75% green and 25% red to achieve close to 570 nm yellow light. If you instead take 82%:27% of green:red (so that the sum is more than 100% or 1W), the resulting brightness will be the same as of each 100% red or green separately.