Inside QPI
Our core algorithm, QPI (Quantitative Phase Imaging), provides a unique solution to the Transport of Intensity Equation.
The algorithm enables the extraction of phase information from incoherent, polychromatic radiation without requiring special optical components. The algorithm can recover phase information from just two conventional brightfield images taken at slightly different focal planes.
The algorithm has a number of key advantages, including:
- Returns phase and intensity information independently
- Provides quantitative, absolute phase (with DC offset)
- Is a rapid, stable, non-iterative solution
- Works with non-uniform and partially coherent illumination
- Offers relaxed beam conditioning
- Solves the twin image problem of holography
- Has been experimentally applied to a number of radiations
Recovery of the phase information
The algorithm is able to recover the phase information from just two images taken at slightly different focal planes, though a third image taken at the point of best focus is generally used for normalisation.
Figure 1 below shows the recovery of the phase information for an optical fibre.
figure 1: generating a phase image of an optical fibre
In the image above we can see the two brightfield images which were acquired with a conventional digital camera attached to a standard microscope. The images were taken at focal points approximately 5 microns either side of the point of best focus.
By applying the QPI algorithm to these images the phase image shown above can be calculated. The time required to generate the phase image from the 0.41 megapixel source data is approximately 1.5 seconds on a 2.4GHz Pentium IV.
Because the phase data provided by the QPI algorithm is quantitative is it now possible to make absolute measurements of the object's properties, including its thickness, refractive index, and so on.
A plot through the phase image shows the projected thickness of the circular cross section.
Behind the algorithm
The problem with a wave's phase is that it is lost when we can only measure the irradiance of the wave.
If we consider a wave propagating in the +z direction that has an amplitude uo(x,y) and phase Φ(x,y) such that
experimentally we only measure the irradiance
thus we lose the phase information.
In order to recover the phase various optical techniques have been developed including interferometry, Zernike & Schlieren phase contrast, perfect crystal analysis, and various iterative propagation-based phase contrast methods. These techniques all require additional optical components, and traditionally are limited to measurements modulo 2π or computer-intensive iterative methods. This is not the case with our algorithm.
Iatia's QPI algorithm makes use of the paraxial approximation of the propagation of intensity distribution as described by the Transport of Intensity Equation (Teague, 1983)
Given an intensity with no zeroes, and its longitudinal derivative, we can solve uniquely for the phase, up to an additive constant (the phase can't be known absolutely). For information about the derivation of the TIE click here.
Solving for phase in the Transport of Intensity equation we get
In formulating this solution we have used a generalised notion of phase where we regard the energy flow vector of a wave as the important quantity, not the amplitude or phase. This formulation is applicable even for polychromatic radiation where the notion of phase is not well defined.
The solution for the TIE phase may be coded using Fast Fourier Transforms. This coding is the QPI algorithm, a flowchart of which can be found here.