Diffraction-limited spatial resolution

Further developments are also expected in imaging applications with faster imaging methods with higher spatial resolution becoming available (e.g. sub-diffraction-limited spatial resolution). Advanced non-linear techniques such as CARS and specialist methods such as ROA will broaden their respective application areas, as instruments become more compact and more systems become commercially available. 

Two objects are completely resolved if they are separated by 2r, and barely resolved if they are separated by r. The latter condition is sometimes known as the Rayleigh criterion of resolution. The largest numerical aperture that can generally be achieved for a Cassegrainian optic is approximately 0.6, so the diffraction-limited spatial resolution is approximately equal to the wavelength of the Hght when n = 1.0. 

Examples of inhomogeneous materials that have distinct infrared absorption signatures include minerals, bone and chemicals that are sorbed at surfaces or are inside inclusions of geological specimen. IRSR spectromicroscopy has been useful in identifying very small samples, or revealing spatially-resolved components in functionally gradient materials. All of these experiments require diffraction limited or close to diffraction limited spatial resolution. 

Both improvements, the optimized fluorescence collection and the higher fluorescence yield, led to a significantly higher photon flux. At a wavelength of A = 610 nm and a numerical aperture of NA = 0.85 the diffraction limited spatial resolution... 

Because of the very small effective source size, the spatial resolution of SR-FTIR for micro-spectroscopy is usually only limited by the wavelength, X, of the IR light [9]. Schwarzschild microscope objectives are used to focus the broad spectrum of IR light onto the sample, and these objectives typically have relatively large numerical apertures in the range 0.5-0.6. While diffraction will naturally limit the spot size on the sample [10], one can also use a single aperture before the sample to exactly define the illuminated sample region. In such a situation, the diffraction-limited spatial resolution is approximately 2X/3 [11], which corresponds to 1.7 pm (at 4000 cm ) and 13 pm (at 500 cm ) the two extremes of typical mid-IR measurements. 

Currently, both FT-Raman and dispersive Raman spectrometers are being used within the pharmaceutical industry. Dispersive Raman spectroscopy in the form of Raman microprobes are heavily employed in the research area to map active-excipient distribution using the diffraction limited spatial resolution attainable with the microprobe. In this subsection, it is inappropriate to describe the varied applications of Raman microscopy to the study of pharmaceuticals thus, the reader is referred to the literature [108,109] and Chapter 14. Dispersive Raman analyzers are also being used for reaction analysis, pilot-plant batch analysis, and process monitoring. FT-Raman spectrometers have been adopted for formulated product analysis and for incoming goods testing. 

For mid-infrared spectrometry, this thickness is about 10 pm. As the center wavelength of mid-infrared spectra is about 5 pm, the sample thickness is approximately equal to the smallest dimension in the x-y plane that can be observed, that is, the diffraction limit when the NA is approximately equal to 0.6. For conventional NIR spectrometry (1200 - 2450 nm), on the other hand, the absorptivities of the stronger bands are in an order of magnitude less than the stronger fundamentals from which they are derived and so the sample thickness should be at least 100 pm. However, the diffraction-limited spatial resolution for NIR measurements is less than 3 pm if optics with an NA of 0.6 are used. Thus, even though the ultimate spatial resolution is in principle determined by the optics of the spectrometer, in practice this resolution is never achievable because the thickness of the sample means that the diameter of the beam waist at the top or bottom surface of the sample is larger than the diffraction-limited spatial resolution. 

The spatial resolution of the microscope is determined by the wavelength of the radiation, X, and the numerical aperture (NA) of the Cassegrain. Less fundamentally, spatial resolution is also limited by the SNR that can be achieved when the size of the aperture is very small. The NA is the sine of the acceptance half-angle at the sample. At the beginning of this chapter it was stated that the diffraction-limited spatial resolution is approximately equal to the wavelength, X. More accurately, it is given by... 

In recent years, the approach to measuring small samples has been transformed by the widespread use of IR microscopes. The advantages over traditional beam condensers lie in much greater convenience and versatility, as well as in the ability to measure much smaller samples. The region for measurement can be selected while a relatively large area of the sample is viewed. Spectra can be obtained from isolated samples that are only a few micrometres across. However, in extended samples diffraction limits spatial resolution to about 10 pm. Because of this, it is generally preferable to remove small inclusions from a matrix rather than to try to measure them in situ. One consequence of having very small samples is that it is often easy to compress them to a suitable thickness for transmission measurement. 

Fig. 18 Light intensity analysis for understanding the achievement of sub-diffraction-limit spatial resolution. Focal plane light intensity (dashed line) and the square of light intensity (solid line) distribution are associated with single-photon and two-photon excitation, respectively. Their derivative distribution is also shown. The inset is the diffraction pattern at the focal plane...

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