Adaptive-additive algorithm

In the studies of Fourier optics, sound synthesis, stellar interferometry, 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 iterative algorithm that utilizes the Fourier Transform to calculate an unknown part of a propagating wave, normally the spatial frequency phase (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.

The adaptive-additive algorithm was originally created to reconstruct the spatial frequency phase of light intensity in the study of stellar interferometry. Since then, the AA algorithm has been adapted to work in the fields of Fourier Optics by Soifer and Dr. Hill, soft matter and optical tweezers by Dr. Grier, and sound synthesis by Röbel.

1. Define input amplitude and random phase
2. Forward Fourier Transform
3. Separate transformed amplitude and phase
4. Compare transformed amplitude/intensity to desired output amplitude/intensity
5. Check convergence conditions
6. Mix transformed amplitude with desired output amplitude and combine with transformed phase
7. Inverse Fourier Transform
8. Separate new amplitude and new phase
9. Combine new phase with original input amplitude
10. Loop back to Forward Fourier Transform

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 ${\displaystyle A_{0}}$ and ${\displaystyle \phi _{n}^{k}}$ respectively. Fourier transform the wave in k-space to x space.

${\displaystyle A_{0}e^{i\phi _{n}^{k}}{\xrightarrow {FFT}}A_{n}^{f}e^{i\phi _{n}^{f}}}$

Then compare the transformed intensity ${\displaystyle I_{n}^{f}}$ with the desired intensity ${\displaystyle I_{0}^{f}}$, where

${\displaystyle I_{n}^{f}=\left(A_{n}^{f}\right)^{2},}$

${\displaystyle \varepsilon ={\sqrt {\left(I_{n}^{f}\right)^{2}-\left(I_{0}\right)^{2}}}.}$

Check ${\displaystyle \varepsilon }$ against the convergence requirements. If the requirements are not met then mix the transformed amplitude ${\displaystyle A_{n}^{f}}$ with desired amplitude ${\displaystyle A^{f}}$.

${\displaystyle {\bar {A}}_{n}^{f}=\left[aA^{f}+(1-a)A_{n}^{f}\right],}$

where a is mixing ratio and

${\displaystyle A^{f}={\sqrt {I_{0}}}}$.

Note that a is a percentage, defined on the interval 0 ≤ a ≤ 1.

Combine mixed amplitude with the x-space phase and inverse Fourier transform.

${\displaystyle {\bar {A}}^{f}e^{i\phi _{n}^{f}}{\xrightarrow {iFFT}}{\bar {A}}_{n}^{k}e^{i\phi _{n}^{k}}.}$

Separate ${\displaystyle {\bar {A}}_{n}^{k}}$ and ${\displaystyle \phi _{n}^{k}}$ and combine ${\displaystyle A_{0}}$ with ${\displaystyle \phi _{n}^{k}}$. Increase loop by one ${\displaystyle n\to n+1}$ and repeat.

• If ${\displaystyle a=1}$ then the AA algorithm becomes the Gerchberg–Saxton algorithm.
• If ${\displaystyle a=0}$ then ${\displaystyle {\bar {A}}_{n}^{k}=A_{0}}$.

• Gerchberg–Saxton algorithm
• Fourier optics
• Holography
• Interferometry
• Sound Synthesis

• Dufresne, Eric; Grier, David G; Spalding (December 2000), "Computer-Generated Holographic Optical Tweezer Arrays", Review of Scientific Instruments, 72 (3): 1810, arXiv:cond-mat/0008414, Bibcode:2001RScI...72.1810D, doi:10.1063/1.1344176, S2CID 14064547.
• Grier, David G (October 10, 2000), Adaptive-Additive Algorithm.
• 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, S2CID 73476.
• Röbel, Axel, Adaptive-Additive Synthesis of Sound, ICMC 1999, CiteSeerX 10.1.1.27.7602{{citation}}: CS1 maint: location (link)
• Soifer, V. Kotlyar; Doskolovich, L. (1997), Iterative Methods for Diffractive Optical Elements Computation, Bristol, PA: Taylor & Francis, ISBN 978-0-7484-0634-0

• David Grier's Lab Presentation on optical tweezers and fabrication of AA algorithm.
• Adaptive Additive Synthesis for Non Stationary Sound Dr. Axel Röbel.
• Hill Labs University of Maryland College Park.]