Fresnel Effect

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Diffraction limits our ability to image objects.  If the object we are trying to measure is comparable in size to the wavelength of the light used for imaging, the measurement is “diffraction limited.”

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An important conclusion of this work is that using this saturation technique, enhancements of spatial and spectral resolution are fundamentally limited to a factor ~2 and ln order to improve imaging significantly, not incrementally, it is necessary to manipulate the light that we detect.  One way to do this is with optical heterodyning.

“Blinking” of fluorophores enables their super resolution. In super-resolution techniques, which rely on the stochastic nature of the physical process of fluorescence (such as STORM, PALM), each fluorophore is activated independently to avoid overlap in its detection; and a series of images is captured by a high-speed camera.  In these types of experiments, spatial localization is obtained by fitting the observed photon distribution on the image plane with an appropriate point spread function (PSF).  The precision in the estimation of the position of point sources can be formulated in terms of the Fisher Information (FI) and the associated Cramér-Rao lower bound (CRLB), which set well-defined limits on it.  These techniques can provide sub-diffraction-limited spatial resolution down to a few nanometers.  Their tremendous success, however, depends on the presence of fluorophores that provide the required stochastic process.

Fresnellens

Thus we have investigated Raman methods, e.g., coherent anti-Stokes Raman Spectroscopy (CARS):   “Spectral Narrowing Accompanies Enhanced Spatial Resolution in Saturated Coherent Anti-Stokes Raman Scattering (CARS): Comparisons of Experiment and Theory.” A. K. Singh, K. Santra, X. Song, J. W. Petrich, E. A. Smith.  J. Phys. Chem. A 124, 4305-4313 (2020).

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In optical heterodyning, a "displaced" signal, whose position is to be determined, can be recontstructed with from the Hermite polynomials, i.e., the TEM modes, with which it is composed.

Fluorescence techniques have limitations.  They do not provide “chemical information.”  They usually require extrinsic labels.  Imaging may be limited by the penetration depth of the excitation wavelengths.  What do we mean by “chemical information”?  Raman spectroscopy gives you direct information about the vibrational modes of the molecules you are interested in, which are typically very specific to, and characteristic of, various chemical species.   Getting this information does not require labelling with extrinsic probes.  It is true that fluorescence can be assigned to particular chromophores; but in the end, the most you usually can say in the condensed phase is that you are looking at a π→π* transition where most of the vibronic structure has been washed out.  If you are lucky and the chromophore is well designed by nature or the organic chemist, the chromophore is perturbed by some excited-state photophysics that are sensitive to the environment in the condensed phase.

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The 2014 Nobel Prize in Chemistry was awarded for super-resolution or sub-diffraction limited spectroscopy. These techniques are largely based on fluorescence imaging, which is based upon "blinking."

a) Image of the LO in the TEM10 mode taken with a CMOS camera.  b) The voltage obtained from the balanced detector as the LO is canned through the signal, which in this case is the TEM00 of a laser beam, which is a surrogate for a coherent emitter.  The minimum of the signal determines its position, according to the Cramér-Rao lower bound.  The position is localized to ~0.2% of its diameter.  c) The predicted uncertainty (standard deviation, SD) of the estimation of the position of laser beam, the surrogate for the position of the emitter , plotted against the displacement of LO, d with respect to signal.  The abscissa is in units of the beam size, σ. The minimum of the curve localizes the signal.  The plots are obtained in two ways:  analytically from the calculation of the Fisher information and the Cramér-Rao lower bound (blue); and numerically from a computation of the maximum likelihood estimation (MLE) (red), which permits the inclusion of noise into the modelling and which is thus responsible for the scatter in the graphs.  (The standard deviation of the position estimator, x0, is the square root of the Cramér-Rao lower bound.)

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