The transport-of-intensity equation (TIE) is a computational approach to reconstruct the phase of a complex wave in

optical Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...

and

electron microscopy An electron microscope is a microscope that uses a beam of accelerated electrons as a source of illumination. As the wavelength of an electron can be up to 100,000 times shorter than that of visible light photons, electron microscopes have a ...

. It describes the internal relationship between the

intensity Intensity may refer to: In colloquial use * Strength (disambiguation) *Amplitude *Level (disambiguation) *Magnitude (disambiguation) In physical sciences Physics *Intensity (physics), power per unit area (W/m2) * Field strength of electric, ma ...

and phase distribution of a wave. The TIE was first proposed in 1983 by Michael Reed Teague. Teague suggested to use the law of

conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means tha ...

to write a

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

for the transport of energy by an

optical field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...

. This equation, he stated, could be used as an approach to

phase recovery A carrier recovery system is a circuit used to estimate and compensate for frequency and phase differences between a received signal's carrier wave and the receiver's local oscillator for the purpose of coherent demodulation. In the transmitter o ...

. Teague approximated the

amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of a ...

of the wave propagating nominally in the z-direction by a parabolic equation and then expressed it in terms of

irradiance In radiometry, irradiance is the radiant flux ''received'' by a ''surface'' per unit area. The SI unit of irradiance is the watt per square metre (W⋅m−2). The CGS unit erg per square centimetre per second (erg⋅cm−2⋅s−1) is often used ...

and phase: :

\frac \fracI(x,y,z)= -\nabla_ \cdot (x,y,z)\nabla_\Phi

where

\lambda

is the

wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...

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 In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

can be measured experimentally, the equation becomes a linear equation that can be solved to obtain the phase distribution

\Phi

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

\fracI(z) = -\frac I(z) \nabla_^2 \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, investigation of cellular dynamics and characterization of optical elements. The TIE method is also applied for phase retrieval in transmission electron microscopy.

References

{{reflist Electron microscopy Microscopy