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I have just bought a Nikon 50mm f/1.8D prime lens.
I took this picture (leg of my glasses frame on a mouse, and a LED desk lamp is on top of it) : f1.8, 1/80s, ISO 200
The purple fringing is right in the middle of the picture. Does it happen very often to this lens?
This is called axial chromatic aberration (or longitudal chromatic aberration) and it's fairly common with large aperture lenses. It appears as a purple halo around objects that are closer than the plane of focus and as a greenish one around objects that are further, regardless of where they're located in the frame. It often shows up around highlights, like in your example.
This is normal for your lens and will go away when using narrower apertures.
You might be more familiar with lateral chromatic aberration which gets stronger near the edges of the frame but it's absent in the centre. Lateral chromatic aberration is most noticeable around in-focus edges far from the centre of the frame and doesn't go away completely even with narrow apertures.
Lateral chromatic aberration can be corrected very effectively by post-processing the images ("Remove chromatic aberration" checkbox in Lightroom). Unfortunately axial chromatic aberration cannot.
Lightroom does have a "Defringe" setting that can be useful to reduce the effect. But all it does is remove certain colours (settable by the sliders) around edges. It's not smart about it and it cannot distinguish between actual colour detail and chromatic aberration. It can cause artefacts, so use with care.
If you are shooting specular highlights (say, reflections off metal) with a fast lens wide open, yes, that's quite common. Swapping out to a different fast prime is unlikely to make the issue go away. I have a Contax/Yashica Zeiss Planar 100/2 that does this wide open. Ditto a Canon EF 85mm f/1.8 USM. It doesn't happen if I stop the lens down or I avoid blown highlights. Shooting with any lens wide open is to use it at its weakest point: CA, vignetting, and softness will all be mitigated by simply stopping down. Choose to shoot wide open and know the tradeoffs; don't shoot wide open by default.
Purple fringe is caused by longitudinal chromatic aberration (aka "bokeh CA"), where the different frequencies of the light are separated front-to-back. You may be more familiar with lateral CA, where the frequencies separate side-to-side (most typically red/cyan or purple/green). Lateral CA is more easily corrected for, but there are defringing corrections in Lightroom/ACR that can take care of the issue, so long as the same shade of purple (or green) isn't prevalent elsewhere in the image.
That's very fixable if you shoot in camera RAW format and use something like Adobe Camera RAW or Lightroom to correct it. Mediocre lenses are prone to chromatic aberration especially if you shoot at the widest aperture. My Cannon 18-55 kit lens does this as well as my "fantastic plastic" 50mm f1.8. Higher priced lenses have better optics to prevent this.
