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. ${\displaystyle I(x,y,z+\Delta z)}$.

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]