Transport-of-intensity equation explained

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:

(2 \pi)/(λ) (\partial)/(\partial z) I(x,y,z) = - \nabla _{x,y} ⋅ [I(x,y,z)\nabla _{x,y} \Phi],

where

λ is the wavelength, I(x,y,z) is the irradiance at point (x,y,z) , and \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 \Phi .^[5]

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

(d)/(dz) I(z) =- (λ)/(2 \pi) I(z) \nabla ^{2} _{x,y} \Phi.

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

I(x,y,z + \Delta z) .

TIE-based approaches are applied in biomedical and technical applications, such as quantitative monitoring of cell growth in culture,^[6] investigation of cellular dynamics and characterization of optical elements.^[7] The TIE method  is also applied for phase retrieval in transmission electron microscopy.^[8]

Notes and References

1. Book: Bostan, E.. 2014 IEEE International Conference on Image Processing (ICIP) . Phase retrieval by using transport-of-intensity equation and differential interference contrast microscopy . 2014. 10.1109/ICIP.2014.7025800. 3939–3943. 978-1-4799-5751-4. 10310598. http://bigwww.epfl.ch/publications/bostan1401.pdf .
2. Book: Cheng, H.. 2009 Fifth International Conference on Image and Graphics . Phase Retrieval Using the Transport-of-Intensity Equation . 2009. 10.1109/ICIG.2009.32. 417–421. 978-1-4244-5237-8. 15772496.
3. Teague. Michael R.. 1983. Deterministic phase retrieval: a Green's function solution. 10.1364/JOSA.73.001434. Journal of the Optical Society of America. 73. 11. 1434–1441.
4. Nugent. Keith. 2010. Coherent methods in the X-ray sciences. 10.1080/00018730903270926. Advances in Physics. 59. 1. 1–99. 0908.3064. 2010AdPhy..59....1N. 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. 10.1364/JOSAA.12.001942. 1995JOSAA..12.1942G.
6. Curl. C.L.. 2004. 14985984. 10.1007/s00424-004-1248-7. 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. 7640406.
7. Dorrer. C.. 10.1364/oe.15.007165. 2007. Optical testing using the transport-of-intensity equation. 19547035. Opt. Express. 15. 12. 7165–7175. 2007OExpr..15.7165D. free.
8. Belaggia. M.. 2004. On the transport of intensity technique for phase retrieval. Ultramicroscopy. 15556699. 102. 1. 37–49. 10.1016/j.ultramic.2004.08.004.