# What limits axial resolution

1. Jun 16, 2013

### u0362565

Hi all,

I was wondering what factors determine axial resolution in spherical lenses. I know that axial resolution in conventional light microscopy at least is about half as good lateral resolution. I've read about the huygens-fresnel principle of diffraction and how this can account for the size of the airy disk at the image plane-determined by the wavelength of light and the size of the aperture. However can you use this principle to explain why axial resolution is only half as good as lateral resolution? A formula i've seen for axial resolution
R = Lambda x refractive index of medium/numerical aperture^2

This is fine but it doesn't really explain to me why this is the case.

Thanks for the help!

2. Jun 18, 2013

### Andy Resnick

Interesting approach- I've never thought of the problem in this way. First, it is true that the axial PSF is about 3 times the extent of the lateral PSF *for simple circular lenses*. What I mean by 'simple' is that the lens is not obscured and the lens does not have any phase singularities (like, for example, an axicon).

So one way to think of various imaging techniques (confocal, apodization, Bessel beams, etc) is that these primarily affect the axial PSF- either to shrink it (confocal) or extend it (Bessel beams). Also, when accounting for aberrations and/or polarization, the PSF can change radically.

Most generally, the asymmetry arises from the Debye approximation for the Huygens-Fresnel diffraction integral (which is less restrictive than the paraxial approximation). When this is written down, the radial component of the diffracted field goes as ∫J_0(αρ)d(cosθ) while the axial component goes as ∫exp(iβz)d(cosθ), where α and β are functions of the angle θ with respect to the optical axis- see Born and Wolf or Gu's "Advanced Optical Imaging Theory" for more detail.

3. Jun 18, 2013

### u0362565

hi andy,

yes i think i should get a decent book on optics. Unfortunately i'm not a physicist or mathematician and so much in science is explained by complicated formulas that don't really mean much to me. Hence i try to look at the qualitative approach.. I'm tempted to go back and get a good grounding in mathematics, might take a while though.

thanks,