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 ${\displaystyle \Phi }$.

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.[5]

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

1. Bostan, E. (2014). "Phase Retrieval by Using Transport-of-Intensity Equation and Differential Interference Contrast Microscopy". IEEE International Conference on Image Processing (ICIP).
2. Cheng, H. (2009). "Phase Retrieval Using the Transport-of-Intensity Equation". IEEE Fifth International Conference on Image and Graphics.
3. Teague, Michael R. (1983). "Deterministic phase retrieval: a Green's function solution". Journal of the Optical Society of America. 73: 1434–1441.
4. Nugent, Keith (2010). "Coherent methods in the X-ray sciences". Advances in Physics. 59: 1–99.
5. Huang, L. (2015). "Phase retrieval with the transport-of-intensity equation in an arbitrarily-shaped aperture by iterative discrete cosine transforms". Optics Letters. 40: 1976–1979.
6. Curl, C.L. (2004). "Quantitative phase microscopy: a new tool for measurement of cell culture growth and confluency in situ". Springer. 448.
7. Dorrer, C. (2007). "Optical testing using the transport-of-intensity equation". Opt. Express. 15: 7165–7175.
8. Belaggia, M. (2004). "On the transport of intensity technique for phase retrieval". Ultramicroscopy. 102: 37–49.