The **transport-of-intensity equation** (**TIE**) is a computational approach to reconstruct the phase of a complex wave in optical and electron microscopy.^{[1]} It describes the internal relationship between the intensity and phase distribution of a wave.^{[2]}

The TIE was first proposed in 1983 by Michael Reed Teague.^{[3]} 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.^{[4]}

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:

where is the wavelength, is the irradiance at point , and 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 .^{[5]}

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

It allows measuring the phase distribution of the sample by acquiring a defocused image, i.e. .

The TIE utilizes only object field intensity measurements at multiple axially displaced planes, without any manipulation of the object and reference beams.^{[6]}

TIE-based approaches are applied in biomedical and technical applications, such as quantitative monitoring of cell growth in culture,^{[7]} investigation of cellular dynamics and characterization of optical elements.^{[8]} The TIE method is also applied for phase retrieval in transmission electron microscopy.^{[9]}

## References

**^**Bostan, E. (2014). "Phase Retrieval by Using Transport-of-Intensity Equation and Differential Interference Contrast Microscopy".*IEEE International Conference on Image Processing (ICIP)*: 3939–3943. doi:10.1109/ICIP.2014.7025800. ISBN 978-1-4799-5751-4. S2CID 10310598.**^**Cheng, H. (2009). "Phase Retrieval Using the Transport-of-Intensity Equation".*IEEE Fifth International Conference on Image and Graphics*: 417–421. doi:10.1109/ICIG.2009.32. ISBN 978-1-4244-5237-8. S2CID 15772496.**^**Teague, Michael R. (1983). "Deterministic phase retrieval: a Green's function solution".*Journal of the Optical Society of America*.**73**(11): 1434–1441. doi:10.1364/JOSA.73.001434.**^**Nugent, Keith (2010). "Coherent methods in the X-ray sciences".*Advances in Physics*.**59**(1): 1–99. arXiv:0908.3064. Bibcode:2010AdPhy..59....1N. doi:10.1080/00018730903270926. S2CID 118519311.**^**Gureyev, T. E.; Roberts, A.; Nugent, K. A. (1995). "Partially coherent fields, the transport-of-intensity equation, and phase uniqueness".*JOSA A*.**12**(9): 1942–1946. Bibcode:1995JOSAA..12.1942G. doi:10.1364/JOSAA.12.001942.**^**Huang, Lei; Zuo, Chao; Idir, Mourad; Qu, Weijuan; Asundi, Anand (2015). "Phase retrieval with the transport-of-intensity equation in an arbitrarily shaped aperture by iterative discrete cosine transforms".*Optics Letters*.**40**(9): 1976–1979. Bibcode:2015OptL...40.1976H. doi:10.1364/OL.40.001976. OSTI 1193230. PMID 25927762.**^**Curl, C.L. (2004). "Quantitative phase microscopy: a new tool for measurement of cell culture growth and confluency in situ".*PFLugers Archiv - European Journal of Physiology*.**448**(4): 462–468. doi:10.1007/s00424-004-1248-7. PMID 14985984. S2CID 7640406.**^**Dorrer, C. (2007). "Optical testing using the transport-of-intensity equation".*Opt. Express*.**15**(12): 7165–7175. Bibcode:2007OExpr..15.7165D. doi:10.1364/oe.15.007165. PMID 19547035.**^**Belaggia, M. (2004). "On the transport of intensity technique for phase retrieval".*Ultramicroscopy*.**102**(1): 37–49. doi:10.1016/j.ultramic.2004.08.004. PMID 15556699.