A typical human eye reacts to waves with a length of 380 to 750 nanometers, which in frequency corresponds to the strip in the area of 400-790 THz. At the same time, a portion of 380-400 nm is taken as a shortwave border, and 760-780 nm as long-wave - 760-780 Nm, but how wide there should be a maximum possible spectrum for an organic eye, so many birds and insects can perceive ultraviolet radiation, up to 360 nanometers, is it also possible with infrared radiation?
Check out the Mantis Shrimp - https://en.wikipedia.org/wiki/Mantis_shrimp#Eyes
"They are thought to have the most complex eyes in the animal kingdom and have the most complex visual system ever discovered."
"Mantis shrimp can perceive wavelengths of light ranging from deep ultraviolet (UVB) to far-red (300 to 720 nm) and polarized light."
Humans actually can see ultraviolet - with the right sort of injury. Post-cataract patients have reported seeing ultraviolet light, and this was reportedly used to a small extent in World War II for signalling.
The reason for this is that the human lens ordinarily contains a pigment 3-hydroxykynurenine, similar to some of the pigments used by butterflies, which blocks UV light, one might suppose to protect the retina.
Infrared vision is substantially harder to do with an "eye" in the narrow sense. The problem is summed up in this busy curve, which shows that one inch of water in the eye absorbs as much infrared as up to 1 million inches of water would absorb of red. Now as you look at that, you can see you can go some way out into the infrared before you get to the peak, but with diminishing returns. Insects have some advantage because their small compound eyes have shorter distances for the light to travel.
Eventually, water becomes clearer again, and terahertz can penetrate some distance. Refracting terahertz can be done, but the sort of metamaterials used are different from our normal notion of a lens. It is conceivable to picture such complex elements evolving and being worked in somewhere near the border of cornea and sclera, but per your intent this probably doesn't count as an option.
Eyes depend on illumination.
The sun delivers the most intensity at the point where usual eyes have their absorption curves at. To prove non-coincidence here is hard, because our sun is at a size/temp amenable to Life itself, and the spectrum directly depends on the temperature -- life, and eyes with it, evolved under this sun, and evolution took the path of lowest resistance using the wavelengths that are most plentiful.
Of course the sun is not the only source of light, just, by orders of magnitude, the most powerful. Any body emits radiation according to it's temperature and then there is chemical luminescence as well, all filtered by the absorption of the media between source/reflector and the receptors (in case of chemoluminescence of a fish and a human observer, that might be the skin of the creature producing the light, surrounding water, plastic of the diving mask, air in the diving mask, the eyes themselves). Now, to actually detect that radiation, we need something to interfere with it. And to form a picture in our mind ('seeing') we need spatial resolution. BUT : humans posess, for instance, facilities to detect FIR, with some spatial resolution: Close your eyes and have a friend walk around you with a 200°C hotplate, and you will be able to tell, at least, whether the plate is in front,left, of right, up or down, making your forehead a 3x3 resolution `(9 pixel) 'eye', with no color vision (you could not discern a 200°C hotplate at 1 meter from a 30°C hand at 1cm or a 10 000°C plasmaplume at 20m). This reception does depend on heating, as do the pit organs of snakes, and microbolometer thermal cameras. Turning this black-and-white camera into a color version (able to discern different temperatures, not just heat flow) would only take, for instance, a filter wheel. Upping the resolution would take, for instance, shielding (you could place many aluminium-foil rolls on your forehead, pointing in different directions, or have one roll scanning a Z-path from left to right and up-down).
So whether you build your picture by scanning the directionality function of your single receptor, or have an array of receptors with differing directionalities is simply a question of how fast you need your picture (is this about seeing a moving swallow, or knowing where a burning tree is?). How you get from the radiation to the nerve impulse is also up for grabs: Have the radiation heat something, detect the heat, have the radiation exite some molecule, detect the shift in chemical reactivity, have the radiation break a bond, detect the chemical change, etc.
There are some laws to adhere to, e.g. the best resolving parts of the human eye are quite near to the diffraction limit, i.e. the receptor that resolves one pixel is about the size (10^-6 m) of the received wavelenght (0.5*10^-6 m) - but what space-angle that pixel represents is very much up to the optics - if we, for instance, had eyeballs 1.4 times bigger (double the retinal area) we could double our spatial resolution (if we kept the receptor density equal).
Many wavelengths of radiation are not easily manipulated (refracted, reflected) making optics harder, but not impossible
Living tissues can produce both crystals, and metamaterials, Nature is the current MVP in that area, actually. The reason we do not have XRay vision is that there is no natural xray source around (if there were, life as we know it would not have happened). Most wavelenghts of radioactivity are not useful for imaging in a biological setting either: The background radiation of a mountain range is too disperse to give a nice picture of anything standing on front of it, and there are too few hotspots to make 'radioactivity-vision' evolutionaryly vital in avoiding them. Same for hard-UV : were it plentiful, we'd not be here, seeing the short bursts that randomly filter through to us does not pose an evolutionary relevant task.
tl;dr: If your creature has sufficient illumination in a given wavelength, and there is no easier way to get at the information, nature will have found a way to see in that spectrum. (Why the creature is not dead in the 'sufficient illumination' is another Q)