In the studies of
Fourier optics,
sound synthesis,
stellarinterferometry,
optical tweezers, and diffractive optical elements (DOEs) it is often important to know the
spatial frequency phase of an observed wave source. In order to reconstruct this
phase the Adaptive-Additive Algorithm (or AA algorithm), which derives from a group of adaptive (input-output) algorithms, can be used. The AA algorithm is an
iterativealgorithm that utilizes the
Fourier Transform to calculate an unknown part of a propagating wave, normally the
spatial frequencyphase (k space). This can be done when given the phase’s known counterparts, usually an observed
amplitude (position space) and an assumed starting
amplitude (k space). To find the correct
phase the
algorithm uses error conversion, or the error between the desired and the theoretical
intensities.
Compare transformed amplitude/intensity to desired output amplitude/intensity
Check convergence conditions
Mix transformed amplitude with desired output amplitude and combine with transformed phase
Inverse Fourier Transform
Separate new amplitude and new phase
Combine new phase with original input amplitude
Loop back to Forward Fourier Transform
Example
For the problem of reconstructing the
spatial frequency phase (k-space) for a desired
intensity in the image plane (x-space). Assume the
amplitude and the starting phase of the wave in k-space is and respectively.
Fourier transform the wave in k-space to x space.
Then compare the transformed
intensity with the desired intensity , where
Check against the convergence requirements. If the requirements are not met then mix the transformed
amplitude with desired amplitude .
where a is mixing ratio and
.
Note that a is a percentage, defined on the interval 0 ≤ a ≤ 1.
Röbel, Axel (2006), "Adaptive Additive Modeling With Continuous Parameter Trajectories", IEEE Transactions on Audio, Speech, and Language Processing, 14 (4): 1440–1453,
doi:
10.1109/TSA.2005.858529,
S2CID73476.
Soifer, V. Kotlyar; Doskolovich, L. (1997), Iterative Methods for Diffractive Optical Elements Computation, Bristol, PA: Taylor & Francis,
ISBN978-0-7484-0634-0
External links
David Grier's Lab Presentation on optical tweezers and fabrication of AA algorithm.