# Optical Transfer Function of a microscope

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1. Feb 6, 2014

### u0362565

Hi all,

I have a question about the "missing cone" problem in wide-field microscopy. The Fourier equivalent of the PSF is the OTF. The OTF has a toroidal (doughnut) shape. I'm a little confused by how to interpret the OTF support in the Z dimension. In 2D and considering lateral resolution only, the OTF support is circular with the origin at the zero frequency (average image brightness). When you add Z into it and view the 3D OTF model what can you say about spatial frequencies in Z? Obviously some info in Z is captured because the torus has lobes either side of the missing cone.
Increasing distance along the Z axis of the OTF represents higher spatial frequencies? same as the case for x and y. So basically what do the side lobes represent in the real image and what does the missing cone represent?

Many thanks!

2. Feb 7, 2014

### DrDu

Could you be so kind to explain your abbreviations?

3. Feb 7, 2014

### u0362565

Hi Dr Du,

Yes sorry, the PSF is the point spread function of the microscope (the blurred image of a point source due to diffraction). The OTF is the optical transfer function, i believe a convolution of the PSF yields the fourier space equivalent the OTF and this represents the pass-band of the microscope, i.e the spatial frequencies in the sample that are captured by the lens.
This link shows an image of an OTF which is missing the axes labels but baiscally has planes x, y and z. The x and y planes give rise to the lateral resolution of the microscope and z plane (height) is the axial resolution as i understand it..

Hope that makes it clearer.

Thanks

4. Feb 7, 2014

### Andy Resnick

Oh.. I see. This is actually interesting, barely discussed in Gu's "Advanced Optical Imaging Theory". The 3-D incoherent OTF does have unexpected structure. It cannot be normalized at the origin, for example. The 'missing cone' of spatial frequencies near the origin corresponds to the 'streaks' associated with the 3D PSF (for example, http://bigwww.epfl.ch/algorithms/psfgenerator/meta/splash.jpg).

Similar OTF functions are obtained for high-NA imaging associated with microscopy and also for vectoral diffraction theory, which is appropriate for both high NA lenses and use of polarized light.

5. Feb 8, 2014

### DrDu

Last edited: Feb 8, 2014
6. Feb 8, 2014

### u0362565

Yes there's some nice vectorial models there, thanks for that. After a bit more reading I think I'm content with considering that the "missing cone" represents the out of focus light contribution in the axial direction. If it's out of focus there is little information there to contribute to the spatial frequency content. I'm still not sure why there appears to be considerable support axially at higher spatial frequencies as represented by the height of the otf above and below the origin.