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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(xo,yo). We may express this transmittance function in terms of its Fourier transform as:

$$T(x_o,y_o) = \int \int T(k_x,k_y) e^{j(k_x x_o + k_y y_o)} dk_x dk_y$$

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

$$\frac{}{} x_i = M x_o$$

$$\frac{}{} y_i = M y_o$$

where M is the system magnification. Let's now re-write the object plane transmittance above in the slightly modified form:

$$T(x_o,y_o) = \int \int T(k_x,k_y) e^{j((k_x/M) (Mx_o) + (k_y/M) (My_o))} dk_x dk_y$$

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,

$$T(x_i,y_i) = \int \int T(k_x,k_y) e^{j((k_x/M) x_i + (k_y/M) y_i)} dk_x dk_y$$

At this point we can propose another coordinate transformation relating the object plane wavenumber spectrum to the image plane wavenumber spectrum as

$$\frac{}{} k^i_x = k_x / M$$

$$\frac{}{} k^i_y = k_y / M$$

to obtain our final equation for the image plane field in terms of image plane coordinates and image plane wavenumbers as:

$$T(x_i,y_i) = \int \int T(M k^i_x,M k^i_y) e^{j(k^i_x x_i + k^i_y y_i)} dk^i_x dk^i_y$$

From Fourier optics, we know that the wavenumbers can be expressed in terms of the spherical coordinate system as

$$ \frac{}{} k_x = k sin \theta cos \phi$$

$$ \frac{}{} k_y = k sin \theta sin \phi$$

If we consider a spectral component for which Φ=0, then the coordinate transformation between object and image plane wavenumbers takes the form

$$\frac{}{} k^i sin \theta^i = k sin \theta / M$$

This is another way of writing the Abbe sine condition, which simply reflects Heisenberg's uncertainty principle for Fourier 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.