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1994 J. Opt. Soc. Am. A/Vol. 24, No. 7 /July 2007 Chen et al.

Detection of phase singularitieswith a Shack–Hartmann wavefront sensor

Mingzhou Chen, Filippus S. Roux,* and Jan C. Olivier

Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Lynnwood Road,Pretoria, 0002, South Africa

*Corresponding author: [email protected]

Received October 12, 2006; accepted February 2, 2007;posted February 13, 2007 (Doc. ID 75961); published June 13, 2007

While adaptive optical systems are able to remove moderate wavefront distortions in scintillated opticalbeams, phase singularities that appear in strongly scintillated beams can severely degrade the performance ofsuch an adaptive optical system. Therefore the detection of these phase singularities is an important aspect ofstrong-scintillation adaptive optics. We investigate the detection of phase singularities with the aid of a Shack–Hartmann wavefront sensor and show that, in spite of some systematic deficiencies inherent to the Shack–Hartmann wavefront sensor, it can be used for the reliable detection of phase singularities, irrespective of theirmorphologies. We provide full analytical results, together with numerical simulations of the detection process.© 2007 Optical Society of America

OCIS codes: 010.1080, 010.7350, 290.5930, 350.5030.

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. INTRODUCTIONhe Shack–Hartmann wavefront sensor (SHWS) is a sys-em widely used to measure the shape of the wavefront ofn optical beam that was scintillated after propagatinghrough a turbulent atmosphere. The reasonable simplic-ty of the SHWS makes it ideal for use in an adaptive op-ical system that is used to correct the wavefront distor-ions in such a scintillated beam.

Various techniques exist [1–6] to compute the shape ofhe wavefront from the raw data received as output from

SHWS. However, in strongly scintillated beams theseechniques tend to fail [7] because of the presence ofhase singularities, which represent optical vortices [8,9]n the propagating scintillated beam. The failure of theseechniques is to some extent a result of the apparent in-ensitivity of the SHWS to the phase function of a phaseingularity [10], which has led to it being referred to asidden phase. Phase singularities are points where thehase is undefined and where the phase around the sin-ularity goes continuously through all phase values fromto 2�. The direction (right or left handed) of increasing

alue indicates the topological charge �±1� of the phaseingularity. The phase function around a singularity cann general be anisotropically scaled along a particular di-ection, which gives rise to its morphology. An isotropicingularity is referred to as being canonical, and aniso-ropic singularities are called noncanonical.

Initial least-squares phase reconstruction methods1–4] that were used in adaptive optical systems wereased on a model for the phase slopes that makes the as-umption that the phase function is continuous. The exis-ence of phase singularities violates this assumption.herefore these methods can be applied only when thehase distortions are not large enough to generate phaseingularities further along the beam.

More recently, various authors [11–14] considered the

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eparation of the phase gradient function into a continu-us (curl-free) part and a solenoidal or rotational part.he former can be used in the adaptive optical system toorrect the continuous part of the phase distortions. Theatter is generally treated as a noise term, which is dis-arded. Since it represents the contribution of the phaseingularities, the solenoidal part of the phase distortionsannot be removed in the same way that the continuousart of the phase distortions is removed. Neither does itimply go away after the continuous part has been cor-ected. It is therefore important to consider this solenoi-al part of the phase distortions more carefully to find aay to get rid of the phase singularities. In this paper we

onsider the detection of the locations and topologicalharges of the phase singularities with the aid of the so-enoidal part of the phase gradient function. We show thathe SHWS is not completely insensitive to the phase of aingularity. In other words, this phase is not completelyidden, but is contained in the solenoidal part of the out-ut of the SHWS. One can therefore use it to locate thehase singularities that are present in the wavefront.The detection of the phase singularities is significantly

nfluenced by the averaging effects of the SHWS subaper-ures. This averaging effect has not to date received muchttention. Herrmann [15] mentioned the model error dueo the averaging gradients, but did not give a detailednalysis. Aksenov and Tikhomirova [13] calculated theverage wavefront slopes with a regularization procedurehat produces continuous infinitely differentiable func-ions, which is not a true representation when there areingularities in the phase function. In this paper, we ana-yze the effect of the averaging process on a phase singu-arity by performing the actual integration of the phaseradient function of such a singularity over the area of aubaperture. From this result one can then show that it isossible to extract the required information of the phase

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Chen et al. Vol. 24, No. 7 /July 2007 /J. Opt. Soc. Am. A 1995

ingularities regardless of the effects of the averagingrocess. For a continuous vector field the singularities cane identified by computing the curl of this vector field.he output from a SHWS is in the form of a sampled vec-or field that approximately represents the gradient of thehase function of the scintillated beam. We implementhe curl operation that operates on this sampled vectoreld in terms of finite differences. This finite-differenceurl operation is referred to as the circulation of theampled vector field. The complete detection process thusonsists of the averaging process performed by the SHWSnd the circulation process performed on its output. Theffect of the morphology and relative position of the sin-ularity on this detection process is also investigated.

The remainder of the paper is organized as follows. Inection 2 we will discuss the basic principle on which theetection of phase singularities is based and provide someotation. The operation of the SHWS is discussed in Sec-ion 3. In Section 4 we discuss the detection of phase sin-ularities with the output of the SHWS. We analyze thisutput for a canonical phase singularity and compute theeak value in Subsection 4.A, and then we repeat thisnalysis for the more general noncanonical phase singu-arity in Subsection 4.B. The complete expression of theirculation for a noncanonical phase singularity is givenn Appendix A. A numerical simulation is provided in Sec-ion 5 to test the procedure on more realistic data. A sum-ary and conclusions are provided in Section 6.

. BASIC PRINCIPLE OF OPERATIONhe phase function of a scintillated optical beam can ineneral be represented by the sum of a continuous phaseunction and an arbitrary number of phase singularities,

��x,y� = �C�x,y� + �n

��x − xn,y − yn;�n,�n�, �1�

here �C�x ,y� is the continuous phase function and ��xxn ,y−yn ;�n ,�n� represents the phase function of a non-anonical singularity, located at �xn ,yn�. The latter phaseunction is given by

��x,y;�,�� = −i

2ln� ��x + iy� + ��x − iy�

�*�x − iy� + �*�x + iy�� , �2�

here * represents the complex conjugate and the mor-hology of the singularity is parameterized by

� = cos��/2�exp�i�/2�,

� = sin��/2�exp�− i�/2�, �3�

n terms of the morphology angles 0�� and 0��2�.The presence of a singularities in a phase function can

e determined with the aid of a closed line integral overhe gradient of the phase function,

�C

� � · dl = 2�, �4�

here C denotes the closed integration contour; �� is theradient of the phase function and is an integer that

resents the net topological charge of all the singularitiesnclosed by the contour. Unfortunately the closed line in-egral in Eq. (4) is not convenient for the practical imple-entation of singularity detection.To implement a singularity detection procedure with a

hysical system one can use the output of a SHWS, whicheasures the gradient of the phase function. For the mo-ent we will assume that this measurement is ideal, ande will return to the effect of the finite subapertures of

he SHWS in Section 3. One can view the output of aHWS as a vector field G=Gx�x ,y�x+Gy�x ,y�y. If this vec-or field represents the gradient of a continuous phaseunction, we must have

�Gx�x,y�

�y=

�Gy�x,y�

�x. �5�

f one finds that this is not the case, then it indicates thatcannot be the gradient of a continuous phase function.

his points to the presence of phase singularities. The ex-ent to which the equality in Eq. (5) fails is given by theifference between its left-hand and right-hand sides,hich becomes the curl of the vector field. Since the vectoreld only has x and y components that only depend on xnd y, its curl is a scalar that represents the z componentiven by

D�x,y� = �T � G�x,y�. �6�

If the phase function of a scintillated optical beam isontinuous (without phase singularities), the curl of theradient of the phase function would yield zero. On thether hand, if there are phase singularities in the phaseunction of the beam, then the curl of this gradient is notero [16]. Each phase singularity gives a Dirac delta func-ion at the location of the singularity,

� � ��x,y,�,�� = 2��x��y�, �7�

here �=±1� is the topological charge of the phase singu-arity. So the curl of the gradient of a phase function�x ,y� that contains phase singularities gives a sum ofirac delta functions, each multiplied with the topological

harge of the singularities,

� � ��x,y� = 2��n

n��x − xn��y − yn�. �8�

y the same token the curl of the output vector field of aHWS, shown in Eq. (6), gives us, according to Eq. (8), aistribution of singularities, showing us where they areocated and what their topological charges are. One canasily compute D�x ,y�, as shown in Eq. (6), from G andhen use the result to identify the phase singularities inhe wavefront.

In the above discussion we assumed the ideal case,hich ignores the effects of the discreteness and finite

ize of the subapertures of the SHWS. In the followingection we consider the SHWS more carefully. We will seehat the subaperture size sets a scale beneath which sin-ularities become unobservable, and it also modifies theeight of 2� by which each singularities is multiplied.

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. SHACK–HARTMANN WAVEFRONTENSORhe SHWS uses a lenslet array to sample the wavefront,s shown in Fig. 1. The slope of the wavefront for each ofhese samples is given by the location of the focal pointormed by each lenslet in its back focal plane. To simplifyur analysis we will assume that the lenslet array is aquare array of lenslets, each with a square-shaped sub-perture, as shown in Fig. 2. A detector array is placed inhe back focal plane of the lenslet array. A small subarrayf this detector array is dedicated to each lenslet of theenslet array. This subarray is used to determine the lo-ation of the focal point produced by each lenslet. The lo-ation of the focal point for the �m ,n�th subaperture (len-let) is given by a position vector um,n that is given by theentroid of the measured intensity distribution in theack focal plane,

um,n =

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�H

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− u0m,n, �9�

here u0m,n would be the location of the focal point for a

ormally incident plane wave; I�u� is the intensity distri-ution over the detector plane; H is the window of theubarray of detectors on the detector plane; and d2u is thewo-dimensional integration measure on the detectorlane.The average phase slope for each subaperture is given

y the location of the focal point, as determined by the in-ensity centroid in Eq. (9). This relationship follows fromhe Fourier relationship, as portrayed in Fig. 3, betweenhe phase tilt of the incident wave in front of a lens andhe location of the resulting focal point behind it. So theverage phase slope over the �m ,n�th subaperture can bexpressed as

Gm,n =

�

� ��x� d2x

�

d2x

�k

fum,n, �10�

here ��x� is the gradient of the phase function of thencident wave; x denotes the two-dimensional positionector on the lenslet plane; k�=2� /�� is the wave number;is the focal length of the lenslets; is the area of the

ig. 1. One-dimensional representation of a Shack–Hartmannavefront sensor.

enslet subaperture; and um,n is the location of them ,n�th focal point on the detector plane, as defined inq. (9). The Gm,n values given by Eq. (10) represent theampled output vector field of the SHWS. Each value isssociated with a point in the center of the particular sub-perture, as denoted by the dots in Fig. 2.

. DETECTION OF PHASE SINGULARITIESn Section 3 we saw that the output of the SHWS in aractical adaptive optical system is a sampled vector field

with each sample representing the averaged phaselopes over one subaperture. The solenoidal part of thisector field contains the information about the locationsf the singularities, as pointed out in Section 2. To extracthis information from the sampled vector field, one needso implement the curl operation of Eq. (6) numerically.uch a numerical implementation is equivalent to the

ine integral of Eq. (4). The numerical computation isone by computing what we refer to as the circulation,iven by

Dm,n =w

2�Gx

m,n + Gxm,n+1 + Gy

m,n+1 + Gym+1,n+1 − Gx

m+1,n+1

− Gxm+1,n − Gy

m+1,n − Gym,n�, �11�

here w is the subaperture window size and the super-cript m ,n denotes the subaperture index. The physicalmplementation of the circulation process is presented in

ig. 2. Array of subapertures (small squares) within the systemperture of the SHWS. The average phase slope values are asso-iated with the dots inside the small squares.

ig. 3. One-dimensional representation of one lenslet in theHWS, showing the shift of the focal point due to the average tiltf the incident wavefront.

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ig. 4. The samples used in the calculation are denoted byhe points in the centers of the subapertures shown inig. 4. The circulation represents a line integral per-

ormed over the four subapertures along a contour de-oted by the dashed lines. Note that the result of this cir-ulation operation represents a value D that should bessociated with a point in the center of the four subaper-ures. However, we are assigning it the same index m ,nhat is associated with the upper left-hand subaperture.

The result of this computation Dm,n is a distribution ofhe topological charges of the phase singularities. It isositive (negative) at the locations of phase singularitiesith positive (negative) topological charges, and it shoulde zero where there are no phase singularities. However,his distribution is affected by the averaging process, theampled nature of the data, and, of course, noise. As a re-ult the values are not exactly zero when there are no sin-ularities. Moreover, the value of Dm,n at the location of aingle positive singularity is not 2� as one would expect ifhe circulation were an exact implementation of the linentegral, Eq. (4). In the next two subsections, we analyzehe effects of the practical implementation of the singu-arity detection procedure.

. Canonical Phase Singularitiesere we consider the situation where a canonical singu-

arity is located in the center of the four subapertures, ashown by point A in Fig. 4. We define the origin (0, 0) ofur coordinate system at this point. The complex ampli-ude function of a canonical phase singularity at the ori-in can be expressed as �x± iy� /r=exp�±i��, where theign indicates the topological charge of the singularitynd r and � are, respectively, the radial coordinate andhe azimuthal coordinate. Here we consider a positivelyharged singularity. The phase function of the singularity

ig. 4. Circulation Dm,n over four subapertures with a singular-ty located either at the center (assumed to be the origin), de-oted A, or at some arbitrary location (x0, y0), denoted B. The fourubapertures are represented by the four squares. The dot at theenter of each subaperture is the position with which the averagehase slope value G of that subaperture is associated. The ar-ows represent the components of G. The dashed lines representhe contour used for calculation of the circulation.

s simply the azimuthal coordinate �, provided that theingularity is located at the origin. The gradient of thehase function of a canonical phase singularity can thene expressed in Cartesian coordinates as

��x,y� =xy − yx

x2 + y2 . �12�

ubstituting Eq. (12) into Eq. (10), we obtain an analyti-al expression for the average phase slope in the �m ,n�thubaperture, given by

Gm,n =1

w2�−w

0 �−w

0 xy − yx

x2 + y2 dxdy = �

4w+

ln 2

2w �x − y�.

�13�

he integration boundaries are determined by the loca-ion of the �m ,n�th subaperture within the four-ubaperture area shown in Fig. 4. Because of the symme-ry of the phase function of a canonical singularity at therigin, the average phase slope values for the other sub-pertures will be the same apart from a possible changen sign. Then, according to Eqs. (11) and (13), the valueor Dm,n will be

Dm,n = 4wGxm,n = � + 2 ln�2� = 4.527887. �14�

e note that the value of the circulation is �+2 ln 2 andot the 2� that one finds for the analytical case given byq. (4). It is the averaging process that is responsible for

his difference, and not the finite differences of the calcu-ation process. One finds that if the same finite differenceirculation calculation, Eq. (11), is performed on aampled gradient function of a canonical phase singular-ty without the averaging process the result is indeed 2�.

Although it is the averaging process that causes the dif-erence, we note that this circulation value is independentf the subaperture window size w. This is because thehase function of a singularity is scale invariant. So theifference is produced simply because some averagingakes place, but it does not matter how big the subaper-ures are over which the slopes are averaged.

When the singularity is not located at the origin, in theenter of the four apertures, but at some other locationuch as the position B at �x0 ,y0� in Fig. 4, we can expecthe value of the circulation to change. The expression forhe gradient of the phase function now becomes

��x − x0,y − y0� =�x − x0�y − �y − y0�x

�x − x0�2 + �y − y0�2 . �15�

sing Eqs. (15) and (10), one can compute the averagehase slopes G for the four subapertures. Then, with theid of Eq. (11), one can compute Dm,n. The final expressions rather complicated. It can be obtained from the moreeneral expression for arbitrary morphology provided inppendix A, as explained there.The circulation D is shown in Fig. 5 as a function of the

ocation of the singularity in terms of normalized coordi-ates �=x0 /w and �=y0 /w. A top view of this functionver the region −2�2 and −2�2 is shown in Fig.(a). The precise shapes of the circulation function as one-imensional functions of r=��2+�2 are shown in Fig. 5(b)

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or �=0 (middle line) and for �=� (diagonal). At the origin� ,��= �0,0� the circulation function has a peak with thealue D=4.53, consistent with Eq. (14). Away from therigin the value of D decreases rapidly and approachesero for �� , �� 1. In some regions the value of D dropselow zero. At the corners of the four-subaperture area,here�� , �� =1, the circulation function forms negativelyalued peaks with the value

D =�

2− 2 ln�5� − 2 arctan�2� + 5 ln�2� = − 0.396641.

�16�

he fact that the circulation function never becomes ex-ctly zero implies that the existence of a singularity atome point in the output plane gives nonzero circulationalues in all other parts of the output plane.

In view of the fact that the peak value of the circulationunction is �+2 ln�2� instead of 2�, it is interesting toote that, if one were to add the four sample values of

ig. 5. (Color online) Circulation D for a canonical singularity.a) Topview of D shown as a function of the relative position ofhe singularity inside the four subaperture area, shown in Fig. 4,or −2�2 and −2�2. (b) One-dimensional functions of Dlotted as functions of r=��2+�2 along the diagonal line andiddle line, respectively, as indicated in (a).

m,n closest to the location of a singularity—i.e., the fouralues that surround the singularity—then the resultould be closer to 2�.

. Noncanonical Phase Singularitiesingularities in strongly scintillated beams are in generaloncanonical. It is therefore necessary to know how theomputation of the circulation is affected by the morphol-gy of the singularity. The phase function of a singularityith an arbitrary morphology is given by Eqs. (2) and (3).he gradient of this phase function can be expressed by

��x,y� =�xy − yx�C

x2�1 + A� − 2yxB + y2�1 − A�, �17�

here

A = sin��cos��, �18�

B = sin��sin��, �19�

C = cos��, �20�

ith the morphology angles � and �, as defined in Eq. (3).he singularity can be translated to any location �x0 ,y0�y replacing x→x−x0, y→y−y0 in Eq. (17).We use the same procedure that was use in Subsection

.A to analyze the noncanonical case. We use Eq. (10) toompute the output of the SHWS for the phase gradientiven in Eq. (17), shifted to �x0 ,y0�. Then we substitutehe result into Eq. (11) to compute the circulation D. Inppendix A we provide the complete expression of thisirculation function for a singularity with an arbitraryorphology, located at an arbitrary point (in normalized

oordinate �=x0 /w and �=y0 /w) within the four subaper-ures, shown in Fig. 4.

In Fig. 6 we show the peak value at �� ,��= �0,0� for Ds a function of the morphological angles � and �. Notehat the peak is positive (negative) for 0�� /2�� /2

��. At �=� /2 there is a discrete jump, whichepresents the change in the topological charge of theingularity. The fact that there is such a large difference

ig. 6. Peak value of the circulation D as a function of the mor-hology angles 0�� and 0�2�. The jump at �=� /2 isue to the change of the topological charge of the singularity.

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etween the values on either side of the jump indicateshat the circulation D, calculated from the output ofHWS, can in principle determine the topological chargef a singularity even if it is severely anisotropic. However,n a practical situation, singularities tend to have such se-ere anisotropic morphologies only when they appear inppositely charged pairs close to each other. In such situ-tions their circulation values will in general partiallyancel each other, making them difficult to identify. Nearhe jump at �=� /2 the value of D has its greatest fluc-uation as a function of �. Next to the jump D fluctuatesetween � and 2�. When � approaches the canonical val-es of 0 or �, the value of D tend toward its canonicaleak value of 4.53, and the fluctuations as a function of �iminish.In Figs. 7 and 8 we provide plots of D over the region

2�2 and −2�2, for two different morphologies ofhe singularity. First we consider the case when �=� /4nd �=�. This represents a singularity with a moderatenisotropy oriented along the y axis. The topview of theirculation function for this case is shown in Fig. 7(a), andhe shape of the circulation function is shown in Fig. 7(b)

ig. 7. (Color online) Circulation D for a noncanonical singular-ty, with �=� /4 and �=�. (a) Topview of D is shown as a functionf the relative position of the singularity inside the four subap-rture area, shown in Fig. 4, for −2�2 and −2�2. (b)ne-dimensional functions of D plotted as functions of r��2+�2 along the diagonal line, � line and � line, respectively,s indicated in (a).

n terms of three one-dimensional functions of r=��2+�2:long �=0 (� line), along �=0 (� line), and along the linehere �=� (diagonal line). The peak value at the origin isbout 4.5. The function then decreases toward zero awayrom the origin. We note that the shape of the circulationunction is more anisotropic than the shape in Fig. 5 inhat the respective widths of the peak along the � and �irections are not equal. There are still regions where theunction becomes negative.

Next we consider the case where �=4� /9 and �=� /2.his represents a highly anisotropic singularity orientediagonally along the line where �=−�. We show the top-iew of the circulation function for this case in Fig. 8(a)nd the shape of the circulation function in Fig. 8(b) inerms of three one-dimensional functions of r=��2+�2:long �=0 (middle line), along the line where �=� (diag-nal line I), and along the line where �=−� (diagonal lineI). The peak value at the origin is now about 5. The func-ion decreases away from the origin, but the decrease isuch slower along the orientation of the singularity. The

hape of the circulation function is therefore much morenisotropic.

ig. 8. (Color online) Circulation D for a noncanonical singular-ty, with �=4� /9 and �=� /2. (a) Topview of D shown as a func-ion of the relative position of the singularity inside the four sub-perture area, shown in Fig. 4, for −2�2 and −2�2. (b)ne-dimensional functions of D plotted as functions of r��2+�2 along diagonal line I (perpendicular to the orientationf the singularity), diagonal line II (along the orientation of theingularity), and the middle line, respectively, as indicated in (a).

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. NUMERICAL SIMULATIONere we present a numerical simulation to test the singu-

arity detection procedure that was analytically investi-ated in the previous sections. We simulate the propaga-ion of a Gaussian beam with wavelength �=987 nm overdistance of L=100 km through a turbulent atmosphere.e use the well-known numerical method of Ref. [17] to

erform the simulation. The strength of the turbulence isarameterized with Cn

2 =4�10−8 m2/3, and we use tenqually spaced phase screens to simulate the �L=10 kmhick turbulent layers. Here, the Rytov number is ��,R

2

0.124k7/6L11/6Cn2 =0.063, and the ratio of subaperture

ize, w=4 cm, to the Fried parameter, r00.185 �2 / �10Cn

2�L��3/5�32 cm, is about 0.125. Thisethod provides a reasonable agreement between real-orld data and the simulation data. The phase of theeam is distorted when it reaches the system aperture,nd in our example we find four phase singularities in theavefront, as shown in Fig. 9(a). This beam then passes

hrough our simulated SHWS, from which we obtain theampled vector field G, computed with Eq. (10). The cir-ulation function D is then computed with Eq. (11). Theesulting circulation function is shown in Fig. 9(b). Two ofhe singularities, one positive and one negative, are easily

ig. 9. Numerical simulation results for a Gaussian beam thatropagated over a distance of 100 km through a turbulent atmo-phere. (a) Resulting phase of the beam inside the system aper-ure. There are two pairs of oppositely charged phase singulari-ies. The pairs are, respectively, located at the lower left and thepper right of the system aperture, (b) Circulation D, numeri-ally calculated from the output of the Shack-Hartmann wave-ront sensor.

dentified from their respective circulation values of 2.95nd −3.54, in the lower-left corner of Fig. 9(a). Note thathe magnitudes of both these values are smaller than 2�.his would be due to a combination of the fact that theingularities have noncanonical morphologies, the facthey are not located at the ideal location in the center ofour subapertures, and noise that is present in the phaseunction. Integrating over a 3�3 neighborhood aroundhese singularities, we obtain values of 6.20 and −5.86, re-pectively, which are closer to ±2�.

The other two singularities in the upper right-hand cor-er of Fig. 9(a) are much closer to each other. Thereforeheir individual circulation functions overlap, and, havingpposite topological charges, they partially cancel eachther. As a result the circulation peaks that representhese singularities are severely diminished. For example,he positive peak for this pair of singularities in Fig. 9(b)as a value of only 1.41. Oppositely charged singularitieshat are located closer to each other are therefore moreifficult to detect.

. CONCLUSIONhe phase gradient that is produced as output of a SHWSontains information about the continuous phase functionf the incident wave but also of the phase singularities inhe wavefront. A least-squares projector [11–14] can besed to extract the information about the continuoushase and can be used to correct continuous phase distor-ions. The information about the singularities is con-ained in the solenoidal part of the phase gradient. Theurl of this part gives a topological charges distribution,hich represents the locations and topological charges of

he singularities. Theoretically each positive (negative)ingularity should be indicated by a value of 2��−2�� inhe topological charge distribution.

The averaging process inherent to the SHWS has a sig-ificant effect on the computed topological charge distri-ution. Instead of the theoretical value of 2�, the actualalue that is produced is at most about 4.53. The preciseocation of the singularity relative to the subapertures inhe SHWS, as well as the morphology of the singularity,roduces further variations in the value of the topologicalharge distribution at the location of a singularity. Never-heless, these values are generally large enough to iden-ify an isolated singularity. It is therefore possible to ex-ract the information of the location and topologicalharge of the singularities from the output obtained fromShack–Hartmann wavefront sensor.In the analytical investigation presented here we con-

idered only one singularity. In the numerical simulatione found that when different oppositely charged singu-

arities are in close proximity to each other, their respec-ive topological charge distributions, as produced by theirculation computations, would overlap, causing partialancelation and a subsequent reduction in their peaks.his makes detection of these singularities more difficult.urther investigation is needed to understand the effectf multiple singularities located near one another on theetection process.The phase functions of scintillated optical beams are, in

eneral, noisy. The noise becomes larger as the scintilla-

tsiib

AFSLHcswso

wi

R

1

1

1

1


ion increases. The analysis that is provided here does notpecifically include such noise. It is expected that the abil-ty to detect singularities would deteriorate as the noise isncreased. This is an important aspect that still needs toe investigated.

PPENDIX A: CIRCULATION FUNCTIONOR A GENERAL NONCANONICALINGULARITY AT AN ARBITRARYOCATIONere we provide the general expression obtained when we

ompute the circulation, Eq. (11), of the sampled averagelope G, Eq. (10), obtained as output from the SHWShen the input is the phase function of a noncanonical

ingularity, located at �=x0 /w and �=y0 /w with morphol-gy angles � and �,

Dm,n =�� + 1��Am − B�

2Amarctan� �� + 1�B − �� − 1�Am

�� + 1�C �−

�� + 1��Am + B�

2Amarctan� �� + 1�B − �� + 1�Am

�� + 1�C �+

�� − 1��Am + B�

2Amarctan� �� − 1�B − �� − 1�Am

�� − 1�C �−

�� − 1��Am − B�

2Amarctan� �� − 1�B − �� + 1�Am

�� − 1�C �+

�� + 1��Ap − B�

2Aparctan� �� + 1�B − �� − 1�Ap

�� + 1�C �−

�� + 1��Ap + B�

2Aparctan� �� + 1�B − �� + 1�Ap

�� + 1�C �+

�� − 1��Ap + B�

2Aparctan� �� − 1�B − �� − 1�Ap

�� − 1�C �−

�� − 1��Ap − B�

2Aparctan� �� − 1�B − �� + 1�Ap

�� − 1�C �−

�� + 1�B

Amarctan��Am − �� + 1�B

�� + 1�C �+

�� − 1�B

Amarctan��Am − �� − 1�B

�� − 1�C �−

�� + 1�B

Aparctan��Ap − �� + 1�B

�� + 1�C �+

�� − 1�B

Aparctan��Ap − �� − 1�B

�� − 1�C �+ � arctan��B − �� + 1�Am

�C �− � arctan��B − �� − 1�Am

�C �

+ � arctan��B − �� + 1�Ap

�C �− � arctan��B − �� − 1�Ap

�C � +�� + 1�C

4Am

��ln 2�� + 1�� − 1�B − �� + 1�2Ap − �� − 1�2Am�

+ ln 2�� + 1�� + 1�B − �� + 1�2Ap − �� + 1�2Am�

− 2 ln 2�� + 1��B − �� + 1�2Ap − �2Am��

−�� − 1�C

4Am� 2�� − 1�� + 1�B − �� − 1�2Ap

− �� + 1�2Am� + ln 2�� − 1�� − 1�B − �� − 1�2Ap

− �� − 1�2Am� − 2 ln 2�� − 1��B − �� − 1�2Ap

− �2Am�� +�� + 1�C

4Ap�ln 2�� + 1�� + 1�B

− �� + 1�2Ap − �� + 1�2Am� + ln 2�� − 1�� + 1�B

− �� − 1�2Ap − �� + 1�2Am� − 2 ln 2�� + 1�B

− �2Ap − �� + 1�2Am�� −�� − 1�C

4Ap

��ln 2�� − 1�� − 1�B − �� − 1�2Ap − �� − 1�2Am�

+ ln 2�� + 1�� − 1�B − �� + 1�2Ap − �� − 1�2Am�

− 2 ln 2�� − 1�B − �2Ap − �� − 1�2Am��, �A1�

here Ap=1+A and Am=1−A, with A ,B, and C as definedn Eqs. (18)–(20).

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detection of phase singularities with a shack-hartmann wavefront sensor

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