Transport-of-intensity equation

The transport-of-intensity equation (TIE) is a computational approach to reconstruct the phase of a complex wave in optical and electron microscopy.{{Cite book|last=Bostan|first=E.|title=2014 IEEE International Conference on Image Processing (ICIP) |chapter=Phase retrieval by using transport-of-intensity equation and differential interference contrast microscopy |date=2014|doi=10.1109/ICIP.2014.7025800|pages=3939–3943|isbn=978-1-4799-5751-4|s2cid=10310598|chapter-url=http://bigwww.epfl.ch/publications/bostan1401.pdf }} It describes the internal relationship between the intensity and phase distribution of a wave.{{Cite book|last=Cheng|first=H.|title=2009 Fifth International Conference on Image and Graphics |chapter=Phase Retrieval Using the Transport-of-Intensity Equation |date=2009|doi=10.1109/ICIG.2009.32|pages=417–421|isbn=978-1-4244-5237-8|s2cid=15772496}}

The TIE was first proposed in 1983 by Michael Reed Teague.{{Cite journal|last=Teague|first=Michael R.|date=1983|title=Deterministic phase retrieval: a Green's function solution|doi=10.1364/JOSA.73.001434|journal=Journal of the Optical Society of America|volume=73|issue=11|pages=1434–1441}} Teague suggested to use the law of conservation of energy to write a differential equation for the transport of energy by an optical field. This equation, he stated, could be used as an approach to phase recovery.{{Cite journal|last=Nugent|first=Keith|date=2010|title=Coherent methods in the X-ray sciences|doi=10.1080/00018730903270926|journal=Advances in Physics|volume=59|issue=1|pages=1–99|arxiv=0908.3064|bibcode=2010AdPhy..59....1N|s2cid=118519311}}

Teague approximated the amplitude of the wave propagating nominally in the z-direction by a parabolic equation and then expressed it in terms of irradiance and phase:

:$\backslash frac\{2\backslash pi\}\{\backslash lambda\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; z\}I(x,y,z)=\; -\backslash nabla\_\{x,y\}\; \backslash cdot\; [I(x,y,z)\backslash nabla\_\{x,y\}\backslash Phi],$

where $\backslash lambda$ is the wavelength, $I(x,y,z)$ is the irradiance at point $(x,y,z)$, and $\backslash Phi$ is the phase of the wave. If the intensity distribution of the wave and its spatial derivative can be measured experimentally, the equation becomes a linear equation that can be solved to obtain the phase distribution $\backslash Phi$.{{Cite journal|last1=Gureyev|first1=T. E.|last2=Roberts|first2=A.|last3=Nugent|first3=K. A.|date=1995|title=Partially coherent fields, the transport-of-intensity equation, and phase uniqueness|journal=JOSA A|volume=12|issue=9|pages=1942–1946|doi=10.1364/JOSAA.12.001942|bibcode=1995JOSAA..12.1942G}}

For a phase sample with a constant intensity, the TIE simplifies to

:$\backslash frac\{d\}\{dz\}I(z)\; =\; -\backslash frac\{\backslash lambda\}\{2\backslash pi\}\; I(z)\; \backslash nabla\_\{x,y\}^2\; \backslash Phi.$

It allows measuring the phase distribution of the sample by acquiring a defocused image, i.e. $I(x,y,z\; +\; \backslash Delta\; z)$.

TIE-based approaches are applied in biomedical and technical applications, such as quantitative monitoring of cell growth in culture,{{Cite journal|last=Curl|first=C.L.|date=2004|pmid=14985984|doi=10.1007/s00424-004-1248-7|title=Quantitative phase microscopy: a new tool for measurement of cell culture growth and confluency in situ|journal=Pflügers Archiv: European Journal of Physiology|volume=448|issue=4|pages=462–468|s2cid=7640406}} investigation of cellular dynamics and characterization of optical elements.{{Cite journal|last=Dorrer|first=C.|doi=10.1364/oe.15.007165|date=2007|title=Optical testing using the transport-of-intensity equation|pmid=19547035|journal=Opt. Express|volume=15|issue=12|pages=7165–7175|bibcode=2007OExpr..15.7165D|doi-access=free}} The TIE method  is also applied for phase retrieval in transmission electron microscopy.{{Cite journal|last=Belaggia|first=M.|date=2004|title=On the transport of intensity technique for phase retrieval|url=https://www.researchgate.net/publication/8170033|journal=Ultramicroscopy|pmid=15556699|volume=102|issue=1|pages=37–49|doi=10.1016/j.ultramic.2004.08.004|hdl=11380/1255475|hdl-access=free}}