Download Branch point detection and correction using the branch

Branch point detection and correction using the branch point
potential method
Kevin Murphy, Ruth Mackey and Chris Dainty
Applied Optics Group, Department of Physics, National University of Ireland, Galway,
Galway, Ireland;
ABSTRACT
Branch points have been shown to cause problems for adaptive optics (AO) systems which attempt to correct for
atmospheric distortion over mid-to-long range horizontal paths. Where branch points (or singularities) occur, the
phase of the optical wavefront is undefined and cannot be reconstructed by conventional wavefront reconstruction
techniques. Branch points occur in pairs of opposite sign (or rotation) and are joined by wavefront dislocations
called branch cuts, which have a 2π jump in phase across them. The aim of the project is to construct a branch
point sensitive wavefront reconstructor using a Shack Hartmann wavefront sensor which can be used on a 3km
line-of-sight (LOS) free space optical (FSO) communications system currently being tested within our group.
The first step in our method is to detect the positions of singularities using the branch point potential method
first proposed by LeBigot and Wild.1 The most common zonal reconstruction method used (the least squares
reconstructor) is not sensitive to branch points and different methods are being investigated for this part of the
project. Results for the detection of singularities using the branch point potential method in simulations are
shown here. Some early results for the reconstruction of branch point affected wavefronts are also presented.
Keywords: Branch points, adaptive optics, Shack Hartmann, wavefront reconstruction
1. INTRODUCTION
Branch points occur in an optical wavefront when it is heavily distorted or the interference is such that zeros of
intensity are seen at the plane of the receiver. At these nulls of intensity, the phase of the optical wavefront is
undefined and it is at these positions that branch points are formed. Branch points occur in pairs (which are
of opposite rotation or sign) and are connected by wave dislocations called branch cuts. Along these branch
cuts the phase of the wavefront undergoes a 2π jump. The branch point is at the origin of the 2π discontinuity
that causes these wave dislocations. Due to branch points the phase of the optical wavefront at the receiver is
not continuous and this causes difficulties in adaptive optics (AO) systems in strong turbulence. Nye and Berry
observed branch points as early as 19742 due to interference caused by certain types of scattering. A paper by
Baranova et al. in 19833 highlighted some of the problems that branch points may cause AO systems, mainly
because of the fact that it would be extremely difficult for a continuous plate deformable mirror to correct for
them.
Fried and Vaughn4 explained and quantified the existence of branch points in the phase function. The authors
explained that branch points (and their associated branch cuts) were unavoidable when the intensity of the light
dropped to zero at the cross-section of the receiver. Their suggestion to take care of branch points was to first find
the branch point locations, then form the branch cuts so that they are ‘tucked away’ into regions of low intensity
and then to reconstruct the phase function around them by using a path dependent least-squares method which
never crosses a branch cut. This method was deemed to be too slow and impractical due to difficulties in the
pairing up of branch points to form branch cuts. The authors also used an intensity weighted least-squares
multigrid method that does not rely on identifying branch points or positioning branch cuts. It should also be
noted that the contour sum method for detecting branch points was outlined in this paper. Other papers have
indicated that using a different branch point corrector, such as an exponential or multigrid reconstructor, would
be better when reconstructing the phase function.5,6
(Send correspondence to Kevin Murphy)
E-mail: [email protected]: Telephone: +353 (0)91492824
A reconstructor capable of detecting and positioning branch points correctly should be able to calculate the
hidden phase and correct for any errors caused by branch points. It is easy to see the benefit that such a
reconstructor would have for an AO system in strong turbulence. It could be either used in conjunction with, or
instead of, the least-squares reconstructor to retrieve the original wavefront phase. It is because of the inability of
the least-squares reconstructor to detect branch points that others have suggested using direct wavefront sensing
methods such as point-diffraction interferometry to obtain the phase information of the wavefront.7,8
The aim of this project is to use a branch point sensitive reconstructor in real time with a line-of-sight
(LOS) free space optical (FSO) communications system and to take account of the distortions introduced by
atmospheric turbulence. Typically such an AO communications system would be equipped with a laser beam
which propagates on a horizontal path close to the Earth’s surface, which experiences heavy distortion and is
highly scintillated. At present the system that is being implemented by our group is designed for a 3km LOS
FSO communications link over Galway city.9 It has a laser beacon at one end on top of a seven storey building
and a receiver at the other end in the AO laboratory. The receiver set up consists of a Cassegrain telescope
with a central obscuration and uses a Shack Hartmann wavefront sensor to measure the aberrations of the
wavefront. It has been found that to correct for these aberrations it is necessary for the AO system to operate at
a frame rate of at least 1 kHz. Scintillation is the fluctuation in the irradiance at the receiver caused by focusing
and defocusing, as well as interference effects, due to turbulent atmospheric cells of different refractive index.
Scintillation makes wavefront sensing difficult and this is examined more closely for the Shack Hartmann (SH)
wavefront sensor (wfs) in a paper by Barchers et. al.10 It is due to the strong scintillation that nulls of intensity
(and the associated branch points) which form in the cross-section of the receiver are introduced. It has been
shown that branch points are one of the main factors which significantly degrade the performance of an adaptive
optics system propagating through strong turbulence.11
1.1 Least Squares Reconstruction
There are a number of problems that branch points specifically cause for AO systems in strong turbulence. The
first of these, as mentioned earlier, is that it is very difficult to correct for them by using a single DM with a
continuous faceplate. It has been suggested to split the correction over two different mirrors12 but the most
common solution has been to used a single segmented DM. This has the advantage of being able to create a
discontinuous phase function as well as cutting down on system complexity. The second major problem is the
inability of the most common reconstructors to identify the branch cuts. The least squares reconstructor cannot
detect the presence of branch points and therefore it cannot correct for their aberrations. Fried shows why it fails
to identify the branch points positions in a paper in 1998.13 If the phase gradients s calculated by the wavefront
sensor are given by
s = Aφ
(1)
where A is the geometry matrix which relates the phase values to the slope values and φ are the phase values of
the wavefront, then the least squares reconstruction of the phase φlmse is
φlmse = ((A† A)−1 A† )s
(2)
The hidden phase is the part of the total phase not accounted for by the least-squares reconstructor and is
the part of the phase which is discontinuous. It is a concept that was introduced by Fried13 and can be defined
as:
∇φ = ∇φlmse + ∇φhid
(3)
where ∇φ is the gradient of the phase of the field, ∇φlmse is the gradient of the least-squares phase and ∇φhid
is the gradient of the hidden phase.
Fried13 gave an equation for the hidden phase due to a single branch point rbp at the position (xbp , ybp ):
φhid (r) = Im{± log[(x − xbp ) + i(y − ybp )]}
(4)
The hidden phase is the part of the phase which is equal to the curl of the vector potential (the least-squares
reconstructor only calculates over the scalar potential):
∇φhid = ∇ × H(r)
(5)
where the H(r) mentioned above is the Hertz potential, which is defined as:
H(r) = [0, 0, h(r)]
(6)
where h(r) is the Hertz function, which has no x- or y-components, only a z-component.
It should be noted here that although the curl component of the phase is not visible through the least-squares
method directly it can be shown by finding the slope discrepancy. If slopes are taken directly from a wavefront
sensor and reconstructed using the least-squares method a reconstructed wavefront will be formed. If slopes
are then found numerically from the reconstructed wavefront and compared to the original slopes taken directly
from the wavefront sensor they are found to be different. The difference between these two sets of slope values is
called the slope discrepancy. If the system was completely noise free the slope discrepancy would equal the curl
component of the phase; however in any normal system there is also a large amount of noise and fitting errors
which have been filtered out by the least squares reconstruction. Further discussion on this topic can be found
in a paper by Tyler.14
1.2 Branch Point Detection
There are different methods used to detect the positions of branch points in the phase function with the most
common probably being the contour sum method originally proposed by Fried in 1992.4 This involves the
summing of phase gradients calculated from the wavefront sensor around in a closed loop. The overall sum
should be equal to zero if the phase is continuous i.e. no branch points present or if there are two branch points
of opposite sign enclosed which cancel each other out. If the addition around this closed loop is equal to a
multiple of either ±2π then there is a branch point enclosed in the loop. The sign of the ±2π depends on the
rotation of the branch point’s screw type discontinuity. The detection of branch points is also subject to the
sampling resolution and too low of a resolution will mean that some discontinuities are missed. The sum of phase
differences around a closed loop can be described mathematically by the following equation:
X
(i, j) = −∆x (i, j) − ∆y (i + 1, j) + ∆x (i, j + 1) + ∆y (i, j)
(7)
where ∆x () are the horizontal phase differences and ∆y () are the vertical phase differences.
By convention the (i,j) pixel is the top left of the four pixels and the branch point is placed there as a more
exact position of the branch point cannot be established by this method. The contour sum method has been
extended to be used with the least-squares technique by utilising the slope discrepancy by Tyler.14 However the
slope discrepancy is still subject to significant noise as the noise and fitting error terms which are filtered out by
the least-squares method are included in the slope discrepancy gradients. The overall robustness of the contour
sum method is still uncertain under noise conditions (which are often experienced under strong turbulence).15,16
It has been suggested to use multi-grid or exponential reconstructors to correct for the presence of branch cuts
without explicitly finding the location of the branch points.5,14
Another method for detection of discontinuities first proposed by LeBigot and Wild in 199916,1 is called the
branch point potential method. This involves rotating all the phase gradient values calculated from the wavefront
sensor by 90 degrees and then forming a potential plot of the resultant field. This is done by first performing a
simple matrix multiplication to rotate the slopes:
sy
0 1
(8)
(Rπ/2 ) ∗ (s) =
∗ (s) =
−1 0
−sx
where (Rπ/2 ) is a 90 degree rotation, s are the slope values measured by the wavefront senor with sx and sy the
slopes x-values and y-values respectively.
Then the pseudo-inverse of the geometry matrix is calculated:
A0 = (A† A + m2 )−1 A†
(9)
where A† is the adjoint of the geometry matrix and the m2 value is inserted to prevent the matrix from becoming
singular.
V = (A0 )(Rπ/2 s)
(10)
The above least-squares calculation is then performed on the rotated slopes to give a potential function, V, which
can be reshaped to form a potential plot which gives the position of the branch point.
This potential function is able to see the hidden phase, unlike a straightforward least-squares reconstruction,
because it is the Hertz potential discussed above. The branch points will show up in this field as peaks and
valleys in the potential plot (with the positive branch points as the peaks and the negative ones as the valleys).
By using a threshold value the branch points are identified because if the potential peaks (or troughs) above (or
below) this value then the position is said to be a branch point location. At the moment the threshold value is
chosen heuristically with an attempt to avoid the most noise but still detect the most branch points. This is the
method which we are using in our current project. It is hoped that this branch point potential method will be
more robust to noise than the contour sum method and be fast enough to be able to correct for the singularities
at high speed.
2. SIMULATION OF ATMOSPHERIC TURBULENCE
2.1 Initial Simulations
Initial simulations were run to test the basic implementation of the branch point potential (BPP) method and
these entailed creating a branch point in a known position in an otherwise perfectly flat phase function. It’s
position and sign would then be measured by the BPP method and if it corresponded to the inputted branch
point information the BPP method could be considered sound. A branch point is created by first obtaining a
product wavefunction:
Ψ=
Y
{(X − Xi ) + i(Y − Yi )}
Q = atan2
Im {Ψ}
Re {Ψ}
(11)
(12)
with Q being the resulting phase and X and Y are the coordinates of an array on a cartesian grid and Xi and
Yi defining the positions of the zero crossings.
Y
It is at the zero crossing where the discontinuity is created because the function atan2 X
is undefined when
Y = 0 and X = 0. This corresponds to having a phase field with a point where the phase is undefined (i.e. a branch
point). These initial simulations proved encouraging as the position and rotation of the branch points given by
the BPP method were as expected. However these simulations are unrealistic as they assume an otherwise flat
phase field apart from the branch point which is unlikely to happen due to atmospheric turbulence. In the next
set of simulations normally distributed random noise with a mean of 0 and a standard deviation of 0.3 was added
to all the slope values previously generated before they were rotated. The spread of slope values generated had
a similar mean (-0.0041) and standard deviation(0.297) to the noise added. The addition of reasonable amounts
of noise did not seem to overly affect the BPP method in detecting singularities which would seem to support
LeBigot and Wild’s assertion that it performs well under heavy noise conditions.
2.2 Numerical Simulation of Propagation Through the Atmosphere
The next step was to create a more realistic numerical simulation of atmospheric turbulence to examine more
throughly how the branch point potential method would perform under actual experimental conditions. This
was achieved by creating phase screens using Kolmogorov statistics to mimic the turbulence encountered through
the atmosphere. A simulated light source was then passed through a number of these phase screens with the
propagation between the phase screens approximated by multiplying the Fresnel transfer function by the Fourier
transform of the optical field. The number of phase screens used to generate the atmospheric turbulence was
10 with the wavelength (633nm), propagation distance (3km), grid size (256) and pixel size (2mm) all staying
constant.
The Fresnel transfer function H(kx , ky ) can be defined as:
H(kx , ky ) = exp iπzλ(kx2 + ky2 )
(13)
with kx and ky the spatial frequencies in the x and y directions, z being the propagation distance and λ being
the wavelength of the light source propagated.
Figure 1. Diagram showing how the numerical simulation of propagation through the atmosphere was performed.
This method was used to create a number of input phases for use with the BPP method to test for branch
points. There was varying degrees of turbulence strength developed from relatively mild turbulence to strong
turbulence. The strength was measured by the Rytov variance where a turbulence of >1 is considered strong
turbulence. The phase screens generated had a Rytov variance of between 1 and 6. The Rytov variance is defined
by the following formula:
7
11
2
σR
= 1.23(Cn2 )(k 6 )(L 6 )
(14)
2
where k is the wavenumber ( 2π
λ ), L is the propagation distance and Cn is the refractive index structure parameter.
This simulation provided us with a number input phases, most of which had branch points included in them
(some of the phase screens created with a Rytov variance of 1 had no branch points present) at random positions.
This was to be expected as it has been shown that the number of branch points increases with an increase in
the turbulence level, with the lower the Rytov variance the greater the chance of having no branch points.17
The positions of the branch points where found by the branch point potential method and their veracity was
double-checked by the contour sum technique for the detection of singularities. As mentioned earlier there needs
to be a complete loop of phase differences for the contour sum technique to work and because of the fact that
the phase screens have been simulated it is easy to calculate around a closed loop of phase differences. When
dealing with real data, it will not be as simple to use this method because of the fact that a Shack-Hartmann
wavefront sensor is being used to measure the wavefront and due to this averaging of the slopes will have to be
performed to form a closed loop of phase gradients.
3. RESULTS FOR BRANCH POINT DETECTION
Here we present some of the results using the BPP method, from the initial simulations through to the simulations
using Kolmogorov statistics. In Fig.2 we show some results for the initial simulations for the clean case with
branch points showing up as peaks and valleys in the potential plot. The same slopes are used in Fig.3 but with
normally distributed random noise with a mean of 0 and a standard deviation of 0.3 added; however the same
branch points are still visible despite this.
Figure 2. Simulated example of slopes with discontinuities present with no noise added along with the corresponding
potential plot showing the branch point positions.
Figure 3. Simulated example of slopes with discontinuities present with normally distributed random noise of 0 mean and
0.3 std. dev. added and the corresponding potential plot.
A comparison of the performances of the branch point potential method and the contour sum method under
noise conditions was completed. Noise levels ranging from no noise to normally distributed random noise with
a mean of 0 and a standard deviation of 1 were examined. 20 realisations were performed for each noise level
and a method was only deemed successful if all branch points were detected in the proper positions and no
false positives were detected. It should be noted that the BPP method seemed to detect more false positives
but missed fewer actual branch points than the contour sum method. A change in the threshold level could
improve the performance but it is still true to say that it performs considerably less well at high noise levels
than low levels. It should be noted that the average standard deviation of the slope values themselves over the
20 realisations was 0.28 with a mean of -0.02. As seen from Fig.4 it compares well to the contour sum method
when detecting branch points in heavy noise.
We can also see the how the BPP method worked on the phase maps created using Kolmogorov statistics and
we can compare their performance to that of an ideal case. The contour sum method was performed pixel wise
Figure 4. Line graph showing the performance of both the BPP and contour sum methods under normally distributed
random noise ranging from no noise to noise with a mean of 0 and standard deviation of 1. These levels of noise are
considerable since the standard deviation of the slopes is 0.28
around the phase maps picking out all of the branch points present. This means that the contour sum method
was performed at the highest possible resolution and provides an upper bound for branch point detection. The
BPP technique uses a simulated Shack-Hartmann wavefront sensor to obtain slope values and this reduces the
sampling of the phase map by a factor of 2. This obviously has an adverse effect on the BPP methods ability
to detect branch points and this should be remembered when inspecting the results. Green (red) circles mark
the positive (negative) branch points in the contour sum maps with black crosses marking the positive and red
asterisks marking the negative branch points in the potential plots. Fig.5 shows the potential plot and contour
sum map along with the corresponding phase map for a Rytov value of 1 and Fig.6 shows the same for a Rytov
value of 6.
Figure 5. An example phase map of Rytov value 1 with its branch point positions found using both the contour sum and
BPP methods.
As we can see from these examples the BPP method performs well at low turbulence levels but it’s effectiveness
falls off as the turbulence level increases. It still picks out most of the branch points at higher turbulence levels
and could still be beneficial at his level of performance. It has been observed from similar simulations performed
that increasing the sampling increases the ability to detect branch points and this has been noted before.18
This may be not be a practical solution at the moment however since we are aiming to correct in real-time
and increasing the number of SH lenslets would impede the ability of the system to correct at high speed. The
processing speed is only limited by the PC however, and this issue could be overcome by incorporating more
hardware computation (eg. FPGA parallel processors) into the system.
Figure 6. An example phase map of Rytov value 6 with its branch point positions found using both the contour sum and
BPP methods.
4. BRANCH POINT CORRECTION
The issue of branch point correction also has to be addressed, as detection is only half of the picture. Since
the least-squares technique does not take account of the singularities then other methods must be investigated.
Two methods are been looked at to test their suitability for our application: the Goldstein algorithm and Fried’s
exponential reconstructor. Fried’s method is described in detail in his paper in 20015 and a version of this
algorithm has been implemented in the laboratory by Starikov et. al.19 This contains three steps; the reduce
step, the solve step and the rebuild step. It takes account of the hidden phase by using differential phasors instead
of the usual phase differences. By using differential phasors the overall field can be taken into account and, when
finished, these differential phasors can then be converted into phase differences. After locating the branch points
an algorithm is applied for the placement of the branch cuts between the branch points, the hidden phase of
the field can be determined by using the formula (4) defined earlier on. The scalar phase is then calculated
separately and added to the hidden phase to give the overall phase function with the branch cut discontinuities
included.
The Goldstein algorithm was originally developed for phase unwrapping in radar arrays but applies equally
to the case of optical wavefronts and is described in a book by Ghiglia and Pritt.20 Roggemann and Koivunen
applied the Goldstein algorithm to correct for branch points in a phase function6 in a numerical simulation and
encouraging results were observed. The algorithm itself has three steps: first it finds the positions of the branch
points in the field (there appears to be no reason why the branch point potential detection method couldn’t
be used to do this). Second it joins the singularities to form branch cuts using a nearest neighbour technique.
Finally it reconstructs the phase by performing a path dependent approach being careful not to cross any of
the branch cuts in the field. Both Roggemann and Ghiglia and Pritt’s book state that the algorithm fails under
certain conditions, namely when there is an area which is totally encircled by branch cuts. This isolation of
phase values causes a piston error in the reconstruction of the wavefront and in the subsequent correction by the
DM.
It was decided to first test the Goldstein algorithm as a branch point sensitive reconstructor as it required the
least adaptation to be implemented. By making use of the phase screen simulations in Sec(2.2) it was possible
to test the algorithm for a number of different phase functions and turbulence strengths. To give an opportunity
for fair comparison between the two methods the phase is binned from 64x64 pixels into a phase map of 32x32
pixels for use with the contour sum method. This means that the resolution of both methods is the same because
the BPP method uses the SH wavefront sensor which halves the resolution of the 64x64 pixel phase map. Both
the BPP and the contour sum methods are used to locate the branch points and the phase is then unwrapped
using this information. The phase is then reconstructed around the separate branch cuts created by both the
BPP and contour sum methods. The phase is the wrapped and this gives the final reconstructed phase. Some
early results are shown below using the phase maps seen before. They show that the BPP method reconstructs
the wavefronts identically to the contour sum method for the same phase map with the same resolution. Fig.7
shows the phase unwrapping and branch cut placements for both the BPP and contour sum methods along with
the identical reconstructed phase for a phase map of Rytov variance 1 while Fig.8 shows the same for a phase
map of Rytov variance 6.
Figure 7. The branch cuts found for both the BPP and contour sum methods with a binned phase of 32x32 of Rytov
value 1 and their identical wrapped output.
Figure 8. The branch cuts found for both the BPP and contour sum methods with a binned phase of 32x32 of Rytov
value 6 and their identical wrapped output.
5. CONCLUSIONS
Our work has shown that the branch point potential method is a viable method for identifying the locations of
branch points in an optical wavefront. It worked for the early simulations and also coped very well with the
more realistic simulations when they had a lower turbulence level. As the turbulence level increased, with a
corresponding increase in the number of branch points, the method performed less well. This highlights certain
limitations of the technique, with the most important of these being that of resolution and thresholding. At
higher turbulence a better performance can be obtained by increasing the number of SH lenslets however this
is not ideal because as the number of lenslets increase so does the time it takes to compute the correction of
the wavefront. Since the aim of this project is to correct in real time this is a drawback. It is not obvious
yet whether this method can give a worthwhile positive effect without overly complicating the overall system.
Further experimental tests are being planned to investigate this.
The correction provided by the Goldstein algorithm is good at low turbulence levels however it’s effectiveness
decreases as the turbulence increases and areas become isolated by branch cuts. It is also quite a slow technique
due to the fact that it is a path dependent method of phase reconstruction. It is unlikely to be a technique that
will be used in a finished system; however it is still worthwhile as it gives a good upper limit of correction that
can be achieved by other methods. It is planned to investigate Fried’s exponential reconstructor in the near
future and compare its performance to that of the Goldstein algorithm.
ACKNOWLEDGMENTS
This research is funded by Science Foundation Ireland under the grant number 07/IN.1/I906.
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