Light intensity equation simplified4/10/2024 ![]() Last, the near-field speckle pattern X-ray imaging 28, 29, 30 that utilizes the phase-stepping method is able to obtain phase and scattering field measurements via numerical deconvolution 31, 32. Ptychography 24, 25 and Fourier ptychographic microscopy 26, 27 shows the great potential for super-resolution intensity and phase measurement beyond diffraction-limit via multiple angle illumination and a joint phase-retrieval algorithm. ![]() closed-form formulas for phase-shifts), undetermined or iterative phase solving methods are emerging that rely on phase-retrieval algorithms, for example coded aperture phase imaging 20, 21, which employs a random aperture for phase encoding and an inverse problem is solved via a customized phase retrieval algorithm, or the structured light illumination techniques 22, 23 that capture diffraction holograms under different background illuminations for subsequent numerical phase-retrieval. Apart from above deterministic phase methods (i.e. Dynamic interference microscopy 19 employs a micro-polarizer array and phase-shifts are encoded into polarization intensity change. Other interference-based methods include diffraction phase microscopy 18, similar to Mach-Zehnder interferometry, but overlays the reference and phase-shift beams in the same optical path, by preserving the 1 st and 0 th order diffraction from a grating using a customized aperture at Fourier plane. One special variant is spatial light interference microscopy 17, which minimizes optical path light coherent sensitivity. Similar to the TIE-based phase imaging technique, digital holography requires additional optical components to realize the different reference beams. Variants of this technique include off-axis digital holography 13, τ interferometers 14, 15, Lloyd’s mirror 16, and many others. Digital holographic microscopy 10, 11 records multiple interferograms under different reference beams (via phase shifts or frequency shifts), then post-processing via Fourier analysis 12 interprets the interferograms to recover the sample intensity and phase, but suffers from the challenging ill-posed 2D phase unwrapping problem for fringe pattern analysis. well-aligned mechanical scanning or specifically-designed wavefront-separation components, hence preventing easy lab implementations. However obtaining the defocused images, requires delicate experimental setups, e.g. From these images the phase shifts are numerically reconstructed, for example by sequentially solving two Poisson equations 9. One notable technique is the defocused-based phase imaging 4, 5, 6, 7, 8 based on Transport of Intensity Equation (TIE) 9, where two or more intensity images are recorded at several closely spaced planes (usually 10 μm to 100 μm apart). ![]() Many quantitative phase imaging techniques have been proposed 3. However, the conversion is not linear and the recorded image on the detector only indicates qualitative pseudo phase information, and is often substantially different from the real phase shift. annulus rings or Nomarski prisms) to convert the phase shifts into brightness changes. These methods utilize additional simple imaging modules (e.g. Two classical methods for phase imaging are phase-contrast microscopy 1 and differential interference contrast (DIC) microscopy 2. In comparison, phase imaging detects minute changes in phase when light propagates through the cell morphology, and has become the prevalent approach for fine cell strucfture distinction without employing higher radiation powers. Due to negligible absorption in the visible spectrum, most living cells exhibit low contrast under bright field microscopy, which prevents detailed examinations.
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