The Contrast Transfer Function
Imaging parameters will determine both absorption (amplitude) and phase contrast according to the Contrast Transfer function. A generalised plot is shown against Fresnel Number to assist in determining optimum settings. Actual contrast values will vary depending upon particular source and sample characteristics, so don't read the vertical axis values too literally.
The specific shape of the damping envelope (which rolls off quickly to zero) will depend upon the characteristics of the source, (i.e., its temporal and spatial coherence). The effect of the Point Spread Function of the source is more significant than the spectral bandwidth. Deconvolution by the PSF will assist in restoring contrast at high spatial frequencies. Note that contrast is inverted for some parameter settings. The Fresnel number incorporates feature size, wavelength and defocus distance according to the formula given above. Changing any of these will affect the relationship between your imaging geometry and the CTF. Very low Fresnel numbers indicate that your geometry may be beyond the information limit.
Using the Fresnel Number
In the Contrast Transfer Function graph above, the abscissa is a complex parameter incorporating u (spatial frequency), l (wavelength) and z' (defocus distance). This parameter is the inverse root of the Fresnel number, NF, and it is convenient to show the equivalent Fresnel number in the x axis. Phase Contrast maxima are shown by the vertical blue lines for easier identification.
The imaging region may be defined as a function of wavelength l, defocus distance R', and the size of the feature being imaged, a, in terms of the Fresnel number, NF, thus:
NF = a2/(lR')
where R'= R1R2/(R1+R2) = (R1 + R2)(M-1)/M2
a= feature size and magnification M=(R1+R2)/R1
Imaging regions may then be characterized in the following terms:
Imaging geometry |
Fresnel number |
Phase Contrast seen as |
Comments |
Near-field * |
NF >>1 |
Sharp edges seen as dark/light fringes, producing edge enhancement. This is the usual region for most XuM and PCX imaging. For pure phase objects, contrast is the Laplacian of the phase shift (hence the term differential phase contrast applied to this imaging regime). |
Image details may be readily visualised as image properties. |
Intermediate or holographic |
NF ~1 |
Complex fringes, image appears blurred. Phase retrieval is required. Use algorithms in X-TRACT software. |
Image details are quasi-linear, i.e., relationship to sample properties may be harder to visualise. Very high resolution imaging may necessitate use of this region. |
Far-field |
NF <<1 |
Extremely complex fringes, image may be blurred beyond recognition for this feature size. Phase retrieval not possible for polychromatic source. |
Image details are related to sample properties in a strongly non-linear way, and visualisation may be impossible. Choose an alternative geometry. |
* (Don't confuse this definition of near-field with the one used in confocal microscopy, where the near-field describes a sample-aperture distance of the order of several wavelengths of the illuminating source.)
Optimum Values
Optimum phase contrast occurs at R'= a²/2l . For R2>>R1, R'~R1. Note that for optimum contrast, R1 decreases quadratically, not linearly, with a.
Optimum magnification occurs at M= 2p/a where a = pixel dimension and a is feature size, in equivalent units.
