Adaptive-additive algorithm

The adaptive-additive algorithm (or AA algorithm) is an iterative numerical technique that utilizes the fast Fourier transform to find the phase of a propagating wave in the input plane, given the amplitudes of the input and output planes. The algorithm was first created to reconstruct the phase of light intensity in the spatial frequency plane. Implementations of the algorithm can be found in Fourier optics, sound synthesis, stellar interferometry, and optical tweezers.

The algorithm

Pseudo-code algorithm

 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

Example

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

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

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

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

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

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

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

where a is mixing ratio and

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

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.

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

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

Limits

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

See Also

• Gerchberg–Saxton algorithm
• Fourier optics
• Holography

References

• Dufresne, Eric; Grier, David G; Spalding (2000), "Computer-Generated Holographic Optical Tweezer Arrays", Review of Scientific Instruments, 72 (3) {{citation}}: Unknown parameter |month= ignored (help).
• Grier, David G (October 10, 2000.), Adaptive-Additive Algorithm {{citation}}: Check date values in: |date= (help).
• Robel, Axel, Adaptive Additive Modeling With Continuous Parameter Trajectories (PDF).
• Robel, Axel, Adaptive-Additive Synthesis of Sound Technical (PDF), University of Berlin Germany: Einsteinufer 17, 10587 {{citation}}: More than one of |location= and |place= specified (help).
• Soifer, V. Kotlyar; Doskolovich,, L. (1997). Iterative Methods for Diffractive Optical Elements Computation. Bristol, PA: Taylor & Francis. ISBN 978-0748406340.{{cite book}}: CS1 maint: extra punctuation (link).

External links

• A PDF/Power Point Presentation that describes the uses and variations of the AA algorithm Berkeley, Ca.
• David Grier's Lab Presentation on optical tweezers and fabrication of AA algorithm.
• Adaptive Additive Synthesis for Non Stationary Sound Dr. Axel Robel