Experimental detection of phase singularities using a Shack-Hartmann wavefront sensor

Transcription

1 Experimental detection of phase singularities using a Shack-Hartmann wavefront sensor Kevin Murphy, Ruth Mackey and Chris Dainty Applied Optics Group, School of Physics, National University of Ireland, Galway, Galway, Ireland; ABSTRACT Phase singularities have been shown to cause one of the major problems for adaptive optics (AO) systems which attempt to correct for distortion caused by the atmosphere in line of sight free space optical communications over mid-to-long range horizontal paths. Phase singularities occur at intensity nulls in the cross-section of the laser beam at the receiver. When the light intensity drops to zero at these points the phase of the optical wavefront is undefined. Phase singularities occur in pairs of opposite sign (or rotation) and are joined by a wave dislocation, called a branch cut, with a corresponding 2π radian jump in the phase. It is this 2π jump which causes difficulties for common AO techniques. To negate the effect of the phase singularities they must be detected and then taken into account in the wavefront reconstruction. This is something not done by most of the zonal reconstruction algorithms commonly used in atmospheric turbulence correction. An experimental set up has been built and is used in the laboratory to examine the detection of phase singularities in atmospheric turbulence. This consists of a turbulence generator using a spatial light modulator (SLM) to mimic the atmosphere and a Shack-Hartmann wavefront sensor as the receiver. The branch point potential method for phase singularity detection is then implemented in post processing to locate the position of the phase singularities. Phase singularity detection can now be practiced under different conditions in a controlled manner. Some results of phase singularity detection from this experimental setup are shown. Keywords: Phase singularities, adaptive optics, Shack Hartmann, optical vortices 1. INTRODUCTION When the optical wavefront of a laser beam is heavily distorted zeros of intensity can be seen in the plane of the receiver. Phase singularities (or branch points) are then located at these nulls of intensity where the phase of the optical wavefront is undefined. They will occur in pairs (although one of the pair may be outside of the pupil and thus cannot be seen) with opposite rotation (or sign) and are connected by wave dislocations. Across the wave dislocations the phase of the wavefront is discontinuous and undergoes jumps of integer multiples of 2π, depending on the charge of the singularity. Such phase singularities will occur naturally when a laser beam is propagated through turbulence or due to certain types of scattering and can be created artificially in the laboratory. The discontinuous nature of the wavefront at the receiver causes problems for adaptive optics (AO) systems and some of the techniques used within them. The basic properties of phase singularities were observed by Nye and Berry as early as 1974, 1 while a paper by Baranova et al. in showed some of the problems singularities created for the AO community. These mainly focused on the difficulty of a continuous faceplate deformable mirror to correct for the aberration caused by the singularities. Fried and Vaughan 3 carried out further work in this area by explaining and quantifying the existence of singularities in the phase function. They showed how phase singularities and dislocations were unavoidable in areas of null intensity in the cross-section of the receiver. Adaptive Optics (AO) is a system in which the incoming wavefront is reflected off a deformable element and this reflected beam is then split into two separate beams, one of which is measured using a wavefront sensor of some kind and the other continues onto a detector (usually a camera). The negative of the measured wavefront is applied to the deformable element (such as a mirror) which then corrects the aberrations in the beam and, in (Send correspondence to Kevin Murphy) k.murphy18@nuigalway.ie/kevmurphy85@gmail.com Telephone: +353 (0) Optics in Atmospheric Propagation and Adaptive Systems XII, edited by Anton Kohnle, Karin Stein, John D. Gonglewski, Proc. of SPIE Vol. 7476, 74760O 2009 SPIE CCC code: X/09/$18 doi: / Proc. of SPIE Vol O-1

2 the case of point source object, allows a relatively plane wave to fall on the detector. The system is controlled by a computer (often a PC) which drives the mirror, as well as calculating exactly what wavefront to apply to the element. Phase singularities cause a number of problems for AO systems which try to correct for the deformities introduced by them. On the wavefront sensing side of the system most of the wavefront sensors typically used in AO have difficulties in seeing the singularities due to the undefined phase at the centre of the vortex. Also, the most commonly used zonal algorithm for phase reconstruction, the least-squares method (often used in applications such as free space optical communications and imaging through turbulence) cannot reconstruct a phase field with singularities as it treats any discontinuities in the wavefront as noise. Finally, there is the hardware difficulty in correcting for the discontinuities as many deformable mirrors would struggle to form the correct shape to adequately perfect the image. However, this is becoming less of a problem as technology advances and some of the segmented deformable mirrors on the market at present should be able to give an adequate fit. In 1998 Fried 4 showed why the least-squares method fails to take account of the singularities in the field. This comes from the fact that the least squares method does not calculate the total phase of the wavefront, only a part of it. If we take the phase gradients measured by the wavefront sensor to be s, calculated as the following s = Aφ (1) where A is a matrix which relates the phase gradients to the wavefront phase and φ is the phase of the wavefront, with the least squares reconstruction, φ lmse,givenby φ lmse =((A A) 1 A )s (2) the part of the phase not accounted for by this reconstruction is called the hidden phase and is discontinuous. It can be defined as φ = φ lmse + φ hid (3) where φ is the gradient of the phase of the field, φ lmse is the gradient of the least-squares phase and φ hid is the gradient of the hidden phase. The hidden phase can also be defined in another way as it is the part of the phase which is the curl of the vector potential. The least squares reconstruction only looks at the scalar potential φ hid = H(r) (4) where the H(r) mentioned above is the Hertz potential, which is defined as: H(r) =[0, 0,h(r)] (5) where h(r) is the Hertz function, which has no x- or y-components, only a z-component. Although the curl component of the phase is not directly visible to the least squares method it can be seen by finding the slope discrepancy. 5 This is done by calculating slopes from a wavefront reconstructed by the least squares method and subtracting them from the slopes originally measured by the wavefront sensor. If a system is completely noise free then it s slope discrepancy is equal to the curl component of the phase; however usually the slope discrepancy is noisy and has fitting errors which were filtered out by the least squares reconstruction. Fried 4 also gave an equation for the hidden phase due to a single branch point r bp at the position (x bp,y bp ): φ hid (r) =Im{± log[(x x bp )+i(y y bp )]} (6) Fried and Vaughan proposed an algorithm in their paper for use in an AO which could be used to correct for the deformations by initially locating the positions of the branch points. 3 These were then joined to create phase dislocations by following regions of low intensity and the phase of the wavefront was then reconstructed by using Proc. of SPIE Vol O-2

3 a path dependent least-squares method, which followed the dislocations but never crossed them. However, at that time the authors deemed the method too slow and impractical due to processing limitations along with difficulties arising from the the pairing of phase singularities. Goldstein suggested a similar algorithm for use with radar applications, which could be used in reconstructing the phase dislocations, described in the book by Ghiglia and Pritt. 6 This involved joining the singularities to form phase dislocations using a nearest neighbour technique and then doing a path dependent phase reconstruction that does not cross any of the dislocations. Roggemann and Koivunen 7 used this technique to reconstruct the phase function in numerical simulations, although they state (along with Ghiglia and Pritt) that this method fails when a region is completely surrounded by dislocations. This isolated patch has a piston error in comparison to the rest of the phase. Other papers have used different algorithms to reconstruct the phase including exponential and multigrid techniques 8,, 7 while other authors have suggested direct sensing method such as point-diffraction interferometry. 9 The aim of this work is to use a Shack-Hartmann wavefront sensor to detect the presence of phase singularities in a turbulent field and then to use phase singularity sensitive reconstruction methods to recreate the correct phase. There has been some interest in this field lately. 10 This is built upon earlier work done within our group to create a line-of-sight free space optical communications link across the city and to correct for the distortions introduced by the atmosphere. These systems have laser beams which propagate horizontally close to the Earth s surface over mid-to-long distances and are heavily distorted and scintillated due to atmospheric turbulence. Such a system was set up over a 3km link across Galway city with a laser beacon on top of a seven storey building at one end and a receiver system in the laboratory at the other. 11 It is necessary to build an AO system to run at a rate of >1kHz to correct for distortion caused mainly by scintillation. Scintillation is the fluctuation in the irradiance at the receiver due to turbulent atmospheric cells of different refractive index causing focusing and defocusing, as well as interference effects in the beam. Scintillation causes the nulls of intensity associated with the creation of phase singularities and makes wavefront sensing difficult. This was examined for the case of the Shack-Hartmann wavefront sensor by Barchers et al. in It has also been shown that phase singularities are one of the main factors affecting the performance of AO systems correcting for atmospheric turbulence DETECTION TECHNIQUE 2.1 Branch Point Potential Method The most common of detecting singularities in the phase function is the contour sum method, as recommended by Fried in 1992, 3 which involves the addition of phase gradients measured by the wavefront sensor around in a closed loop. This summing of phase differences can be described mathematically by the following equation: (i, j) = Δ x (i, j) Δ y (i +1,j)+Δ x (i, j +1)+Δ y (i, j) (7) where Δ x () are the horizontal phase differences and Δ y () are the vertical phase differences. If the phase is continuous i.e. no singularities present (or if there are two vortices of opposite sign enclosed by the loop of phase gradients such that they cancel each other out) then this sum will be equal to zero (excluding noise). However if the sum around the loop is equal to, or a multiple of, ±2π then there is a phase singularity enclosed in the loop. The sign of the ±2π depends on the rotation of the singularity s screw type phase profile. By convention the (i,j)th pixel is the top left of the four pixels and the vortex is placed there as a more exact position of the branch point cannot be established by this method. The contour sum method can be used with the least-squares technique of phase reconstruction by utilising the slope discrepancy described by Tyler. 5 However this is subject to significant noise because, as mentioned before, all the noise and fitting error terms which are filtered out by the least-squares method are included in the slope discrepancy gradients. Therefore the overall robustness of the contour sum method is still questioned under heavy noise conditions such as those experienced due to strong turbulence. 14,15 The detection of phase singularities is also subject to the sampling resolution of the wavefront sensor where too low of a resolution will mean that some discontinuities are missed; however this problem remains as long as a direct wavefront sensing method is not used. The method of detection which we are using in the current setup is the branch point potential method which was first proposed by LeBigot and Wild in This involves rotating all the phase gradient values calculated Proc. of SPIE Vol O-3

4 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: ( ) ( ) 0 1 sy (R π/2 ) (s) = (s) = (8) 1 0 s x where (R π/2 ) is a 90 degree rotation, s are the slope values measured by the wavefront senor with s x and s y the slopes x-values and y-values respectively. Then the pseudo-inverse of the geometry matrix is calculated: A =(A A + m 2 ) 1 A (9) where A is the adjoint of the geometry matrix and the m 2 value is inserted to prevent the matrix from becoming singular. The potential function, V is then calculated by multiplying the rotated slopes by the psuedo-inverse of the geometry matrix. V =(A )(R π/2 s) (10) The potential function can then be used to find the location of the phase singularities as it is the Hertz potential and therefore able to see the hidden phase, unlike a straightforward least-squares reconstruction. The phase singularities can then be seen as peaks and valleys in the potential plot (with the positive singularities as the peaks and the negative ones as the valleys). It is hoped that this method will be more robust to noise than the contour sum method and better able to cope with the scintiallation caused by the strong atmospheric turbulence Shack Hartmann Wavefront Sensor The operation of the Shack-Hartmann wavefront sensor is shown below [Fig.1]. Initially a plane wavefront is passed through the lenslet array and falls onto the detector. The lenlets split the beam up into the same number of beamlets and these will all form an in-focus spot on the detector. The position of these spots are noted and called the reference positions. Then when an aberrated wavefront is passed through the lenslet array another set of spot positions are noted. The difference in these spot locations compared to the reference spot positions gives us the phase gradients (or slopes) of the aberrated wavefront we are measuring. Figure 1. Operation of the Shack Hartmann wavefront sensor. Fried 17 came up with a method of reconstructing the array of phase gradients of the type measured by the Shack-Hartmann to give the phase of the wavefront. It should be noted that this geometry of slopes is not continuous due to the way in which it is sampled by the Shack-Hartmann wavefront sensor and therefore cannot be used directly by the contour sum method of singularity detection which requires a continuous loop of Proc. of SPIE Vol O-4

5 phase gradients. However by averaging across the phase gradients they can be converted to phase differences, which are a set of continuous slopes. Then these phase differences can be used with the Hudgin geometry matrix, developed by Hudgin in for continuous phase differences, to reconstruct the phase profile (also the contour sum method can be preformed on these phase differences). 3. EXPERIMENTAL WORK 3.1 Experimental Setup A ferro-electric spatial light modulator (SLM) from Boulder Non-Linear Systems [Fig.2] is used in the setup to create the phase singularities in the phase profile and will later be used to mimic the effects of atmospheric distortion. This works by binary phase modulation which produces a diffraction pattern, with the first diffraction order selected to produce the desired phase pattern at the SLM s image plane. 19 It has 512 x 512 pixels and a refresh rate of approximately 1kHz. It must spend half of it s cycle showing the inverse of the image you wish to apply to it otherwise the device will polarise. This means triggering of the camera is of vital importance so that measurements are only taken when the device is showing the true images. The camera that is being used at present is a QImaging Retiga EXi CCD camera [Fig.3]. This has 1392 x 1040 pixels with a pitch of 6.45μm and also has an external trigger. It has a small minimum exposure time ( 10μs) and 12 bit digital output, although its frame rate is quite slow ( 10fps). Due to the external trigger and 12 bit output, it was decided that this camera was suitable for offline non-real time measurements. The laser used is a HeNe at 633nm with a power output of 5mW. The Shack Hartmann lenslet array is from Heptagon and comprises of a 50 x 50 set of microlenses. It has a pitch of 100μm and a focal length of 2.7mm. Figure 2. The BNS Spatial Light Modulator used in the setup. Figure 3. Both the cooled and uncooled versions of the Retiga CCD camera. The uncooled version is used in the setup. Figure 4. Schematic of the optical system setup. The system is setup with two parallel beams, after coming through a spatial filter from the laser, which are split by a beamsplitter (BS1). One of these beams is a reference beam which goes directly to the second set of beamsplitters (BS3,BS4), through the optical trombone, back through the BS3 and BS4 and into the wavefront sensor. The second beam goes from BS1 and through another (BS2) and is reflected onto the SLM. Proc. of SPIE Vol O-5

6 After coming back through BS2 it goes through a 4f system, which has an iris placed at the focus to select only one phase image. It then travels to BS3, where it rejoins the reference beam, and through the optical trombone. It proceeds back through L6 and M1 and into the wavefront sensor. In the wavefront sensor itself a lens(l8) is used to expand the beam slightly to ensure it passes through the active area. Also due to the relatively short focal length of the lenslet array a relay lens(l9) is needed to focus the beamlets on the camera. 3.2 Experimental Method Initially a flat phase field is applied to the system with the natural aberration of the SLM removed; this means a flat wavefront should fall on the wavefront sensor. A number of spot images from the Shack Hartmann are taken by the camera and averaged (to reduce the effect of any random noise) to create the reference spot positions. Despite the fact that an array of 50 x 50 spots is available only an area of about 46 x 46 spots are actually used due to imperfections on the lenslet array, edge effects and difficulties in reliably imaging some rows of the lenslet array. It should therefore be noted that only a part of the SLM area is imaged onto the detector due to this and also the fact that the beam footprint at the wavefront sensor is slightly larger than the active area of the lenslet array. Then a set of phase maps with singularities present are applied to the SLM (again with its natural aberration removed). These phase maps had been created beforehand using binary phase modulation in MATLAB and the locations of the phase singularities set. Again, the spot positions are imaged by the camera and saved to be processed offline. The processing is done in MATLAB and the aberrated images are compared to the reference image to determine the slopes measurements for each individual phase map. Then using the branch point potential method described above the positions of any possible singularities are identified. These positions are then compared to the original phase maps applied to the SLM to determine if the positions calculated for the singularities are accurate. 4. RESULTS FOR BRANCH POINT DETECTION Initially the system had to be calibrated. This was achieved by taking a number of reference images which each have a flat phase profile. Also during this calibration process the amount of the SLM that is being imaged onto the camera was calculated and the correlate of the position of the pixels on the SLM to their corresponding position on the detector was recorded. At present, slightly less than 50% of the SLM area is being imaged through the lenslet array onto the detector, although all of the missing area is confined to the edges of the SLM. At this stage the SLM was also checked to make sure that it was creating singularities correctly. To confirm this the reference beam of the setup (shown in Fig.4) was used so that the pattern of the interferometric fringes could be seen. An example of a phase field with two singularities is shown below, in one image the plain Shack Hartmann image is seen[fig.5] and in the other the interferometer fringes are seen[fig.6]. In the SH image the two red arrows point toward the darker regions in the spot image which are the locations of the two vortices. The corresponding locations in the interferometry image (which are circled in red) are where two fringes combine into one or where one fringe splits into two, these branch points marks the position of the vortices. Figure 5. SH image of the two singularities. Figure 6. Interferometry fringes showing the two singularities. Proc. of SPIE Vol O-6

7 Some initial tests have been carried out for phase functions with optical vortices present and have thus far just consisted of singularities in otherwise flat phase fields. The results have been encouraging, when a single vortex is put in the field and the potential method used, it shows up as a peak (or valley - depending on the rotation). The system detects both positive and negative singularities with equal ability. Below we see the measured slopes [Fig.7(a), Fig.8(a)], the rotated slopes [Fig.7(b), Fig.8(b)] and potential plots [Fig.7(c), Fig.8(c)] for two singularities, one positive and one negative. As can be seen the positive and negative singularities have a circular form in the measured slopes but in the opposite direction, when they are rotated this becomes a formation either pointing toward a centre point or breaking away fro ma centre point again depending on sign. A negative singularity will circle in a clockwise direction and fall toward a centre point while a positive singularity will do the opposite. The difference is obvious when detected on the potential plots with the negative valleys being marked by a red star and the positive peaks being marked by a black cross. Figure 7. (a)the original phase gradients from the wfs, (b)the rotated phase gradients and (c)the potential plot for a single negative vortex. Figure 8. (a)the original phase gradients from the wfs, (b)the rotated phase gradients and (c)the potential plot for a single positive vortex. As can be seen from Fig.5 and Fig.6 two singularities have been successfully created using the SLM and they have also been observed interferometrically. They can also be detected by Shack Hartmann sensor as we see below the rotated slopes Fig.9(a) and the potential plot Fig.9(b) for a field with two negative singularities present. As can be seen from the potential plot both singularities are detected by the system and marked as negatives by the red stars. 5. CONCLUSIONS In this paper an optical system for detecting phases singularities using the Shack Hartmann wavefront sensor in the laboratory, has been described. The singularities are created using an SLM, which can also be used to mimic atmospheric turbulence, and the phase gradients of the wavefront are measured by the Shack Hartmann sensor. The branch point potential method, described above, can then be used to detect the phase singularities as peaks or valleys in a potential plot, depending on their rotation. As can be seen in [Figs.7,8,9] early experiments have shown that the potential method will work under experimental conditions in the laboratory for simple cases and can differentiate between positive and negative singularities. If the reference arm of the phase screen generation system is used to create interferometric fringes the vortices can be seen as fringes which split or recombine in the presence of the phase singularities. Proc. of SPIE Vol O-7

8 Figure 9. (a)the rotated phase gradients and (b) the potential plot for a field with two negative vortices. Future work will be carried out to attempt to detect optical vortices in more complex phase functions including those similar to what is experienced in a mid-long distance ( 1 10kms) horizontal path through the atmosphere. This will show how efficiently the potential method deals with the scintillation which is a major problem for AO in free-space optical communications. If these tests are successful then the wavefront sensing system will be assessed on the optical communications link across the city. A comparison of the accuracy of the contour sum method to that of the branch point potential method is also planned. It is also hoped to carry out correction algorithms on the wavefronts measured by the Shack Hartmann sensor, especially for those which mimic atmospheric turbulence. If these algorithms significantly increase the Strehl ratio, then the use of this system to detect the singularities as part of a closed loop AO system will be further explored. ACKNOWLEDGMENTS This research is funded by Science Foundation Ireland under the grant number 07/IN.1/I906. REFERENCES 1. J. F. Nye and M. V. Berry, Dislocations in wave trains, Proceedings of Royal Society of London A 336, pp , N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel dovich, Wave-front dislocations: topological limitations for adaptive systems with phase conjugation, Journal of Optical Society of America A 73, pp , May D. L. Fried and J. L. Vaughn, Branch cuts in the phase function, Applied Optics 31, pp , May D. L. Fried, Branch point problem in adaptive optics, Journal of Optical Society of America A 15, pp , October G. A. Tyler, Reconstruction and assessment of the least-squares and slope discrepancy components of the phase, Journal of Optical Society of America A 17, pp , Oct D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping, Wiley-InterScience, M. C. Roggemann and A. C. Koivunen, Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction, Journal of Optical Society of America A 17, pp , Jan D. L. Fried, Adaptive optics wave function reconstruction and phase unwrapping when branch points are present, Optics Communications 200, pp , Dec J. Notaras and C. Paterson, Point-diffraction interferometer for atmospheric adaptive optics in strong scintillation, Optics Communications 281, pp , F. Starikov, Kochemasov, Koltygin, Kuliokv, Manachinsky, Maslov, Sukharev, Aksenov, Izmailov, Kanev, Atuchin, and Soldatenkov, Correction of vortex laser beam in a closed-loop adaptive system with bimorph mirror, Optics Letters 34, pp , August Proc. of SPIE Vol O-8

9 11. R. Mackey and C. Dainty, Adaptive optics correction over a 3km near horizontal path, SPIE 7108, pp. 1 9, J. D. Barchers, D. L. Fried, and D. J. Link, Evaluation of the performance of Hartmann sensors in strong scintillation, Applied Optics 41, pp , Feb C. A. Primmerman, T. R. Price, R. A. Humphreys, B. G. Zollars, H. T. Barclay, and J. Herrmann, Atmospheric-compensation experiments in strong-scintillation conditions, Applied Optics 34, pp , Apr M. Chen, F. S. Roux, and J. C. Olivier, Detection of phase singularities with a Shack-Hartmann wavefront sensor, Journal of Optical Society of America A 24, pp , July E. O. Le Bigot and W. J. Wild, Theory of branch-point detection and its implementation, Journal of Optical Society of America A 16(7), pp , K. Murphy, R. Mackey, and C. Dainty, Branch point detection and correction using the branch point potential method, SPIE 6951, pp , D. Fried, Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements, Journal of Optical Society of America 67, pp , March R. H. Hudgin, Wave-front reconstruction for compensated imaging, Journal of Optical Society of America A 67, pp , March Neil, Booth, and Wilson, Dynamic wave-front generation for the characterization and testing of optical systems, Optics Letters 23, pp , Dec Proc. of SPIE Vol O-9