Intensity Transport Equation

The transport of intensity equation ( English transport-of-intensity equation , TIE) is a computational approach to the reconstruction of the phase of a complex wave in the optical and in the electron microscopy .

The phase of a wave or a wave field cannot be measured directly. However, it contains basic information about the shape and structure of a microscopic sample, which is why its reconstruction is of great importance for many areas of microscopy.

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The intensity transport equation describes the relationship between the intensity and phase distribution of a wave. The equation was first proposed by Michael Reed Teague in 1983. Teague applied the law of conservation of energy to describe a differential equation for the transport of energy through an optical field. He used this equation as an approach to reconstruct the phase. He approximated the amplitude of the wave (which propagates in the z-direction) by means of a parabolic equation and expressed it in intensity and phase:

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

Here, the wavelength, the intensity of the wave at the point , and the phase ${\displaystyle \lambda }$ ${\displaystyle I(x,y,z)}$ ${\displaystyle (x,y,z)}$ ${\displaystyle \Phi }$

If the intensity distribution of the wave and its spatial derivative can be measured experimentally, the equation is simplified to a linear equation . This can be solved to get the phase distribution . ${\displaystyle \Phi }$

At constant intensity, the TIE simplifies to:

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

Using this equation, the phase distribution can be reconstructed by capturing a defocused image . ${\displaystyle I(x,y,z+\Delta z)}$

advantages

The TIE only uses intensity measurements on several spatially offset planes without the object and the reference beams having to be manipulated. In addition, TIE is computationally simple and does not require a complicated optical system.

Applications

TIE-based approaches occur in biomedical and engineering applications. In biomedicine, TIE is used, for example, to monitor the growth of cell cultures or to study cell dynamics. TIE is also used as an optical test method. The intensity transport equation is also used in transmission electron microscopy for the purpose of phase reconstruction.

Individual evidence

1. ↑ Emrah Bostan, Emmanuel Froustey, Benjamin Rappaz, Etienne Shaffer, Daniel Sage, Michael Unser: Phase retrieval by using transport-of-intensity equation and differential interference contrast microscopy . In: 2014 IEEE International Conference on Image Processing (ICIP) . 2014, p. 3939-3943 , doi : 10.1109 / ICIP.2014.7025800 .
2. ↑ Hong Cheng, Hong Liu, Quanbing Zhang, Sui Wei: Phase Retrieval Using the Transport-of-Intensity Equation . In: 2009 Fifth International Conference on Image and Graphics . 2009, p. 417-421 , doi : 10.1109 / ICIG.2009.32 .
3. ^ Michael Reed Teague: Deterministic phase retrieval: a Green's function solution . In: JOSA . tape 73 , no. 11 , November 1, 1983, pp. 1434-1441 , doi : 10.1364 / JOSA.73.001434 .
4. ^ Keith A. Nugent: Coherent methods in the X-ray sciences . In: Advances in Physics . tape 59 , no. 1 , 2009, p. 1-99 , doi : 10.1080 / 00018730903270926 (English).
5. ↑ Lei Huang, Chao Zuo, Mourad Idir, Weijuan Qu, Anand Asundi: Phase retrieval with the transport-of-intensity equation in an arbitrarily shaped aperture by iterative discrete cosine transforms . In: Optics Letters . tape 40 , no. 9 , May 1, 2015, p. 1976-1979 , doi : 10.1364 / OL.40.001976 .
6. ↑ Claire L. Curl et al. a .: Quantitative phase microscopy: a new tool for measurement of cell culture growth and confluency in situ . In: Pfluger's archive: European Journal of Physiology . tape 448 , no. 4 , July 2004, p. 462-468 , doi : 10.1007 / s00424-004-1248-7 , PMID 14985984 .
7. ↑ C. Dorrer, JD Zuegel: Optical testing using the transport-of-intensity equation . In: Optics Express . tape 15 , no. 12 , June 11, 2007, p. 7165–7175 , doi : 10.1364 / OE.15.007165 (English, osapublishing.org [PDF; accessed June 19, 2020]).
8. ↑ Marco Belaggia et al .: On the transport of intensity technique for phase retrieval . In: Ultramicroscopy . tape 102 , no. 1 . Elsevier, 2004, p. 37-49 , doi : 10.1016 / j.ultramic.2004.08.004 (English).