- Abbe sine condition
The Abbe sine condition is a condition that must be fulfilled by a lens or other
optical system in order for it to produce sharp images of off-axis as well as on-axis objects. It was formulated byErnst Abbe in the context ofmicroscope s.The mathematical condition is as follows:
:
where the variables "u", "U" are the angles (relative to the optic axis) of any two rays as they leave the object, and "u’", "U’" are the angles of the same rays where they reach the image plane (say, the film plane of a camera). For example, ("u","u’") might represent a
paraxial ray (i.e. a ray nearly parallel with the optic axis), and ("U","U’"') might represent amarginal ray (i.e a ray with the largest angle admitted by the system aperture); the condition is general, however, and does not only apply to those rays.Put in words, "the sine of the output angle should be proportional to the sine of the input angle".
Magnification and the Abbe Sine Condition
Using the framework of
Fourier optics , we may easily explain the significance of the Abbe sine condition. Say an object in the object plane of an optical system has a transmittance function of the form, "T"("x"o,"y"o). We may express this transmittance function in terms of itsFourier transform as::
Now, assume for simplicity that the system has no
image distortion , so that the image plane coordinates are linearly related to the object plane coordinates via the relation::
where "M" is the system
magnification . Let's now re-write the object plane transmittance above in a slightly modified form::
where we have simply multiplied and divided the various terms in the exponent by "M", the system
magnification . Now, we may substitute the equations above for image plane coordinates in terms of object plane coordinates, to obtain,:
At this point we can propose another coordinate transformation relating the object plane
wavenumber spectrum to the image planewavenumber spectrum as::
to obtain our final equation for the image plane field in terms of image plane coordinates and image plane wavenumbers as::From
Fourier optics , we know that the wavenumbers can be expressed in terms of thespherical coordinate system as::If we consider a spectral component for which , then the coordinate transformation between object and image plane wavenumbers takes the form
:
This is another way of writing the Abbe sine condition, which simply reflects Heisenberg's
uncertainty principle forFourier transform pairs, namely that as the spatial extent of any function is expanded (by the magnification factor, "M"), the spectral extent contracts by the same factor, "M", so that the "space-bandwidth product" remains constant.
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