# 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.[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:

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

where ${\displaystyle \lambda }$ is the wavelength, ${\displaystyle I(x,y,z)}$ is the irradiance at point ${\displaystyle (x,y,z)}$, and ${\displaystyle \Phi }$ 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 ${\displaystyle \Phi }$.[5]

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

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

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

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

1. ^ 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.
2. ^ 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.
3. ^ 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.
4. ^ 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.
5. ^ 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.
6. ^ 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.
7. ^ Curl, C.L. (2004). "Quantitative phase microscopy: a new tool for measurement of cell culture growth and confluency in situ". Pflügers Archiv: European Journal of Physiology. 448 (4): 462–468. doi:10.1007/s00424-004-1248-7. PMID 14985984. S2CID 7640406.
8. ^ 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.
9. ^ 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.