Can the human eye see the images of 300 nm light on a screen through a diffraction grating?

Question

In my textbook there is a question as follows:

A diffraction grating with 200 lines per mm is placed between a monochromatic light source and a screen. The distance from the grating to the screen is 2.5 m and the distance between the two 2nd order images is 60 cm on the screen. Calculate the wavelength of the monochromatic light.

$$d=5x10^{-6}m$$

$$y=0.3m$$ $$x=2.5m$$ $$n=2$$ $$sin\theta =\frac{y}{\sqrt{x^2+y^2}}$$ $$\lambda=\frac{d sin\theta}{n}=297 nm$$

But the human eye cannot see 300 nm and shorter wavelengths.

Is there something happening due to the presence of the grating and the screen that allows viewing of these images or would these images not be viewable to the human eye as I suspect ?

My understanding is that the images are not viewable by the human eye.

I also understand that the point behind the question is just to practice using the formula and to not worry too much about the numbers. But I like to think about the real situation when I work through problems.

Answer 1

No, the human visual system cannot perceive light shorter than 400 nm.

In a common lab experiment, light from a hydrogen discharge lamp is sent through a diffraction grating and separated into its constituent wavelengths. There are 4 that are nominally in the range of human vision, the shortest wavelength being 410 nm. In my experience, the 410 nm line is just barely on the edge of visible. Many students are unable to see it at all. In my younger days I could see it pretty well if the room was dark and my eyes were properly dark-adapted, especially if I looked off to one side a little, but nowadays I have more trouble seeing it.

If there is any human who can see light of shorter wavelength than this, they would have to be an extreme outlier. To see 297 nm would require a fundamentally different visual system than most humans, and to my knowledge this has never been documented. People with artificial lenses can do better than average, because they filter UV less than our natural ones, but even so, 297 nm seems extreme.

On the other hand, if the screen has chemicals in it that phosphoresce, it could convert the 297 nm light to a wavelength in the visible range, and in that way you could in principle see the line on the screen. You just wouldn't be seeing it at its original wavelength. X-ray machines sometimes use a trick similar to this - the X-ray images themselves are invisible, but a phosphorescent film converts them into visible light that can then be imaged more easily with film or a digital detector (or seen with the naked eye). (Originally the X-rays would just expose the film directly, but by converting them to visible light first you can use much shorter exposure times because the film is much more sensitive in the visible range.)

Also, it's possible the eye can react to UV light without seeing it, for example producing a squinting reflex on a cloudy day that doesn't seem like it should be bright enough to have to squint, but although this would technically be a kind of unconscious perception, I don't think I would categorize it as "seeing," and you wouldn't be able to exploit it to visualize the line on the screen that you're talking about.