In the studies of Fourier optics

Fourier optics

Fourier optics is the study of classical optics using Fourier transforms and can be seen as the dual of the Huygens-Fresnel principle. In the latter case, the wave is regarded as a superposition of expanding spherical waves which radiate outward from actual current sources via a Green's function...

, sound synthesis, stellar interferometry

Interferometry

Interferometry refers to a family of techniques in which electromagnetic waves are superimposed in order to extract information about the waves. An instrument used to interfere waves is called an interferometer. Interferometry is an important investigative technique in the fields of astronomy,...

, optical tweezers

Optical tweezers

Optical tweezers are scientific instruments that use a highly focused laser beam to provide an attractive or repulsive force , depending on the refractive index mismatch to physically hold and move microscopic dielectric objects...

, and diffractive optical elements (DOEs) it is often important to know the spatial frequency

Spatial frequency

In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components of the structure repeat per unit of distance. The SI unit of spatial frequency is...

 phase of an observed wave source. In order to reconstruct this phase

Phase (waves)

Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

 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

Algorithm

In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

 that utilizes the Fourier Transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

 to calculate an unknown part of a propagating wave, normally the spatial frequency

Spatial frequency

In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components of the structure repeat per unit of distance. The SI unit of spatial frequency is...

 phase

Phase (waves)

Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

 (k space). This can be done when given the phase’s known counterparts, usually an observed amplitude

Amplitude

Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

 (position space) and an assumed starting amplitude

Amplitude

Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

 (k space). To find the correct phase

Phase (waves)

Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

 the algorithm

Algorithm

In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

 uses error conversion, or the error between the desired and the theoretical intensities

Intensity (physics)

In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...

. The AA algorithm is currently being implemented by Dr. Wendell Hill III, Alex Robel, V. Kotlyar Soifer, and David G Grier.

History

The adaptive-additive algorithm was originally created to reconstruct the spatial frequency

Spatial frequency

In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components of the structure repeat per unit of distance. The SI unit of spatial frequency is...

 phase

Phase (waves)

Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

 of light intensity in the study of stellar interferometry

Interferometry

Interferometry refers to a family of techniques in which electromagnetic waves are superimposed in order to extract information about the waves. An instrument used to interfere waves is called an interferometer. Interferometry is an important investigative technique in the fields of astronomy,...

. Since then, the AA algorithm has been adapted to work in the fields of Fourier Optics

Fourier optics

Fourier optics is the study of classical optics using Fourier transforms and can be seen as the dual of the Huygens-Fresnel principle. In the latter case, the wave is regarded as a superposition of expanding spherical waves which radiate outward from actual current sources via a Green's function...

 by Soifer and Dr. Hill, soft matter

Soft matter

Soft matter is a subfield of condensed matter comprising a variety of physical states that are easily deformed by thermal stresses or thermal fluctuations. They include liquids, colloids, polymers, foams, gels, granular materials, and a number of biological materials...

 and optical tweezers

Optical tweezers

Optical tweezers are scientific instruments that use a highly focused laser beam to provide an attractive or repulsive force , depending on the refractive index mismatch to physically hold and move microscopic dielectric objects...

 by Dr. Grier, and sound synthesis by Robel.

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

For the problem of reconstructing the spatial frequency

Spatial frequency

In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components of the structure repeat per unit of distance. The SI unit of spatial frequency is...

 phase (k-space) for a desired intensity

Intensity (physics)

In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...

 in the image plane (x-space). Assume the amplitude

Amplitude

Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

 and the starting phase of the wave in k-space is and respectively. Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

 the wave in k-space to x space.

Then compare the transformed intensity

Intensity (physics)

In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...

  with the desired intensity , where

Check against the convergence requirements. If the requirements are not met then mix the transformed amplitude

Amplitude

Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

  with desired amplitude .

where a is mixing ratio and

.

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.

Separate and and combine with . Increase loop by one and repeat.

Limits

• If then the AA algorithm becomes the Gerchberg–Saxton algorithm.
• If then .

See also

• Gerchberg–Saxton algorithm
• Fourier optics
  Fourier optics
  Fourier optics is the study of classical optics using Fourier transforms and can be seen as the dual of the Huygens-Fresnel principle. In the latter case, the wave is regarded as a superposition of expanding spherical waves which radiate outward from actual current sources via a Green's function...
  
  
• Holography
  Holography
  Holography is a technique that allows the light scattered from an object to be recorded and later reconstructed so that when an imaging system is placed in the reconstructed beam, an image of the object will be seen even when the object is no longer present...
  
  
• Interferometry
  Interferometry
  Interferometry refers to a family of techniques in which electromagnetic waves are superimposed in order to extract information about the waves. An instrument used to interfere waves is called an interferometer. Interferometry is an important investigative technique in the fields of astronomy,...
  
  
• Sound Synthesis
  Synthesizer
  A synthesizer is an electronic instrument capable of producing sounds by generating electrical signals of different frequencies. These electrical signals are played through a loudspeaker or set of headphones...
  
  

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.
• Hill Labs University of Maryland College Park.

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