Physics: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:
- [math]\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], }[/math]
where [math]\displaystyle{ \lambda }[/math] is the wavelength, [math]\displaystyle{ I(x,y,z) }[/math] is the irradiance at point [math]\displaystyle{ (x,y,z) }[/math], and [math]\displaystyle{ \Phi }[/math] 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 [math]\displaystyle{ \Phi }[/math].[5]
For a phase sample with a constant intensity, the TIE simplifies to
- [math]\displaystyle{ \frac{d}{dz}I(z) = -\frac{\lambda}{2\pi} I(z) \nabla_{x,y}^2 \Phi. }[/math]
It allows measuring the phase distribution of the sample by acquiring a defocused image, i.e. [math]\displaystyle{ I(x,y,z + \Delta z) }[/math].
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]
References
- ↑ Bostan, E. (2014). "Phase retrieval by using transport-of-intensity equation and differential interference contrast microscopy". 2014 IEEE International Conference on Image Processing (ICIP). pp. 3939–3943. doi:10.1109/ICIP.2014.7025800. ISBN 978-1-4799-5751-4. http://bigwww.epfl.ch/publications/bostan1401.pdf.
- ↑ Cheng, H. (2009). "Phase Retrieval Using the Transport-of-Intensity Equation". 2009 Fifth International Conference on Image and Graphics. pp. 417–421. doi:10.1109/ICIG.2009.32. ISBN 978-1-4244-5237-8.
- ↑ 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. doi:10.1080/00018730903270926. Bibcode: 2010AdPhy..59....1N.
- ↑ 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. doi:10.1364/JOSAA.12.001942. Bibcode: 1995JOSAA..12.1942G.
- ↑ 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.
- ↑ Dorrer, C. (2007). "Optical testing using the transport-of-intensity equation". Opt. Express 15 (12): 7165–7175. doi:10.1364/oe.15.007165. PMID 19547035. Bibcode: 2007OExpr..15.7165D.
- ↑ 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. https://www.researchgate.net/publication/8170033.
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