Why color depends on frequency and not on wavelength?

Question

To explain my question lets consider this example:

The wavelength of light in a medium is $\lambda=\lambda_{0}/\mu$, where $\lambda_{0}$ is the wavelength in vacuum. A beam of red light ($\lambda_{0}=720$ nm) enters into water. The wavelength in water is $\lambda=\lambda_{0}/\mu = 540$ nm, assuming $\mu = 4/3$ for water.

According to the information available in the textbooks, this light appears red only in the water because even though the wavelength is changed, the frequency is not. It implies that to determine the colour of light/electromagnetic radiation, the wavelength doesn't matter, and only frequency matters, as frequency remains the same on the change of media.

It raises several questions:

1. Why does the colour of light depend on the frequency only and not on the wavelength?

2. Are we biologically designed to perceive only frequency and not wavelength? If yes, then why is that?

3. Why are then colours not defined only on the basis of frequencies? Why are they defined on the basis of wavelengths when wavelength doesn't even matter on the change of media?

Answer 1

TLDR: The perceived color of monochromatic light depends on the frequency of the light at the retina. The frequency of the light at the retina is equal to the frequency of light in a vacuum or in any other medium. The wavelength of light in the retina is not equal to its wavelength in other media. So it's more convenient to discuss in terms of frequency.

Very Important Fact: The frequency of light is preserved as it travels through different (linear) media. The wavelength is not. This means that, in situation where light is travelling through different media (air, water, lenses, corneas, fluid in the vitreous body in the eye), the frequency is the same throughout, but the wavelength is changing.

1. Why the colour of light depends on the frequency only and not on the wavelength?

See my answer to 2.

2. Are we biologically designed to perceive only frequency and not wavelength? If yes, then why is that?

Yes, our eyes detect color based on the light that hits the photoreceptors (rods and cones) in the retina of our eyes. The light can be characterized by either its frequency or its wavelength in the eye. BUT because of the important fact above, it is more convenient to specify the light in terms of its frequency rather than its wavelength. This way if we measure the frequency of light outside of the eye we know what frequency it will be inside the eye and thus what color will be perceived. If we measured wavelength we would have to do conversions to figure out what the wavelength would be in the eye.

3. Why are then colours not defined only on the basis of frequencies? Why are they defined on the basis of wavelengths when wavelength doesn't even matter on the change of media?

Color can be defined either in terms of wavelengths or frequencies. In vacuum the relationship between wavelength and frequency is unambiguous, it is the speed of light $c$. Here is a chart showing different colors and their corresponding frequency, wavelength (in vacuum) and energy: If you see a color specified in terms of wavelength then you should understand that that is probably the wavelength in vacuum and can be converted to a frequency using $c$.

Answer 2

For question 3, it's historical and technical. Using diffraction to measure the wavelength of light is much easier than measuring THz frequencies. A skilled craftsman can make a diffraction grating with hand tools, but directly measuring the frequency of light requires sophisticated electronic and photonic technology. So, optical spectra have been presented in terms of (vacuum) wavelength for a couple of centuries. Diffraction gratings remain as the discriminating element in most modern spectrometers.

Answer 3

Colour perception, even if we only restrict it to human colour perception, is an enormously complex physiological process, and visible light with exactly the same physical properties can be perceived as different colours in different circumstances - a well known example of this is "the dress".

The visible spectrum is notionally divided into regions assigned to different colours, but you should be very wary of any statements that mix the physical properties of light with the perception of specific colours in particular circumstances.

Answer 4

Why the colour of light depends on the frequency only and not on the wavelength

Color perception is a physiological process. E.g. the fact that we see 3 primary colors is caused by the fact that there are 3 different receptors in your eye. And since you only ever perceive color in your eye it only depends on the frequency. Unless you are willing to let someone replace the liquid in your eye by a different substance. (And than that would most likely not change the behavior of the receptors)

Are we biologically designed to perceive only frequency and not wavelength? If yes, then why is that?

See above.

Why are then colours not defined only on the basis of frequencies? Why are they defined on the basis of wavelengths when wavelength doesn't even matter on the change of media?

Once a certain terminology is established it tends to be used. When you buy a computer screen you buy it as a 24 inch even if in Europe we use the metric system. But if you tell your co-worker you got a new 610 mm screen he will not have an idea what you are talking about.

You can very well express the colors in terms of frequency. I would assume that the wavelength was what got used is that back then no had an idea of 660THz but people could relate to the a 450nm which is just a bit over 2000 parts of a milimeter.

Answer 5

It is the energy of the photon that matches the particular electron transition gap in the pigment in the retinal cells.

We have rather poorly defined frequency (or wavelenght, for that matter) of a single photon. On the other hang, its energy is not something one argues about.

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Answer 6

1. Because the medium in your eye stays the same. At the interface (Between you eye and the outside) the frequency stays the same (sole the wave equation including a matching condition for proving that), so all signal of the same frequency in your eye have the same wavelength in your eye.

2. Because (as of 1) in your eye the same frequency corresponds to the same wavelength. So it is hard to distinct between evolutionary targets. I also can not imagine any positive effect of perceiving wavelength (medium dependent), and it would be very hard to imagine a mechanism doing that (a diffraction grating, at some distance of your eye and embedded in the medium in which you are interested in selecting the wavelength). It would be funny - objects would change color and intensity in water. So you would need something sticking out in front of the receptor and it would be useful only if you change media and your prey/food/predator/mate target also changes media and your prey/food/predator/mating target has a mechanism to generate the same wavelength and your light source is intense enough to cover the whole range of different frequencies and you need to be super selective.

3. Because what is measured explicitly in practically all measurement until frequency comb lasers arrived was wavelength and not frequency - putting in a fictional speed of light just adds uncertainty if the value which you directly compare and reference to is a length. i.e. a diffraction grating in air/vacuum since then you drop about the medium dependent part in you measurement device.