Radial thresholding to mitigate Laser-Guide-Star aberrations on Centre-of-Gravity-based Shack-Hartmann wavefront sensors
Olivier Lardiere, Rodolphe Conan, Colin Bradley, Kate Jackson, Peter Hampton
aa r X i v : . [ a s t r o - ph . I M ] J un Mon. Not. R. Astron. Soc. , 1– ?? (2009) Printed 28 May 2018 (MN L A TEX style file v2.2)
Radial thresholding to mitigate Laser–Guide–Staraberrations on Centre–of–Gravity–based Shack–Hartmannwavefront sensors
Olivier Lardi`ere ⋆ , Rodolphe Conan, Colin Bradley, Kate Jackson and Peter Hampton AO Laboratory, Mechanical Engineering Department, University of Victoria,PO Box 3055 STN CSC, Victoria, BC, V8W 3P6, Canada
Accepted ... . Received ... ; in original form ...
ABSTRACT
Sodium Laser Guide Stars (LGSs) are elongated sources due to the thickness and thefinite distance of the sodium layer. The fluctuations of the sodium layer altitude andatom density profile induce errors on centroid measurements of elongated spots, andgenerate spurious optical aberrations in closed–loop adaptive optics (AO) systems.According to an analytical model and experimental results obtained with the Univer-sity of Victoria LGS bench demonstrator, one of the main origins of these aberrations,referred to as LGS aberrations, is not the Centre–of–Gravity (CoG) algorithm itself,but the thresholding applied on the pixels of the image prior to computing the spotcentroids. A new thresholding method, termed “radial thresholding”, is presented here,cancelling out most of the LGS aberrations without altering the centroid measurementaccuracy.
Key words: instrumentation: adaptive optics – methods: analytical, laboratory.
Sodium laser guide star (LGS) adaptive optics (AO) sys-tems allow in theory a full sky coverage; however, there areseveral limitations. The artificial star is elongated due tothe sodium layer thickness (about 10 km) and finite dis-tance (90 km). Consequently, if the laser is launched fromthe secondary mirror holder, the spots of a Shack–Hartmannwavefront sensor (SH–WFS) are radially elongated from thecentre to the edge of the pupil (Fig. 1). The spot elonga-tion is proportional to the telescope diameter and can reachseveral arcseconds for extremely large telescopes (ELTs).Moreover, the sodium layer is not static, but fluctuateswith a time scale of about 1 minute or less (Davis et al.2006). Fluctuations of the sodium layer altitude and atomdensity profile induce errors on centroid measurements ofelongated spots and generate spurious aberrations on thewavefront in closed–loop AO systems. These aberrations,referred to as LGS aberrations, can reach several hundrednanometres peak–to–valley ( ptv ) for ELTs with the classicalcentre–of–gravity (CoG) centroiding algorithm (Clare et al.2007; Lardi`ere et al. 2008). Some authors proposed new so-phisticated centroiding algorithms to mitigate LGS aberra-tions, such as the Matched Filtering (Gilles and Ellerbroek ⋆ E-mail: [email protected]
If the LGS spots are radially elongated from the pupil cen-tre, as shown on Fig. 1, the sodium layer fluctuations inducecentro–symmetric aberrations, such as focus ( Z ) and spher-ical aberrations ( Z , Z , etc.), and also square symmetricaberrations, such as tetrafoils ( Z , Z , etc.).The focus is due to a variation of the sodium layer alti- c (cid:13) Olivier Lardi`ere, Rodolphe Conan, Colin Bradley, Kate Jackson and Peter Hampton x [pixels] y [ p i x e l s ]
50 100 150 200 250 300 350 40050100150200250300350400 0246810ADU
Figure 1. ×
29 elongated spot SH–WFS frame obtained withthe UVic LGS–bench at SNR=18. The spot sampling is 2 × ×
4” on the sky, as expectedfor the TMT SH–WFS with a 1” seeing ( σ RON =0.43 ADU, nothreshold). tude. This error is not an artefact and must be compensatedby refocussing the LGS on the WFS with zoom optics and byupdating the offsets of the LGS WFS with a natural–guide–star (NGS) focus sensor (Herriot et al. 2006). Non–commonpath errors of the LGS optical train, including the zoom op-tics, can vary with the sodium layer distance, i .e. the zenithalangle, and induce variable aberrations on the science pathtoo. We assume that these systematic aberrations can becalibrated and virtually negated.Consequently, aberrations beyond focus are mainlyartefacts of the wavefront sensing. According to a modelfrom Clare et al. (2007) and to the first experimental re-sults obtained with the UVic LGS bench (Lardi`ere et al.2008), the spherical aberrations arise due to a truncation ofasymmetric LGS spots by a circular field-stop, while squaresymmetric aberrations are likely due to: • a spot truncation by a square field-stop or by pixelboundaries, horizontally and vertically elongated spots be-ing more truncated than diagonally elongated spots, • a spot overlap for square–grid lenslet arrays, horizon-tally and vertically elongated spots being more prone tooverlapping, • quad-cell or sampling effects on centroid measurements.Both kinds of LGS aberrations have been reproducedand characterized in laboratory on the UVic bench with atime–series of 88 real sodium profiles (Fig. 6). Beyond thefocus, the most significative LGS aberrations detected arethe spherical aberration Z up to 100nm ptv (30nm rms ),and the tetrafoil Z with 40nm ptv (10nm rms ). Moreover,a correlation between the spherical aberration and the pro-file asymmetry was empirically established (Lardi`ere et al.2008). Square–symmetric aberrations, such as Z , shouldbe mitigated by using a polar–coordinate CCD array(Beletic et al. 2005; Thomas et al. 2008). However, we discovered later that Z mode disappearsif no threshold was applied on the pixels of the LGS WFSimages before the computation of centroids. The threshold-ing discards the two extremities of each elongated spot, andconsequently truncates radially each spot, as an optical cir-cular field–stop would do. The spot truncation caused bythe field–stop is negligible compared to the truncation in-duced by the pixel thresholding if the field of view (FOV)of the outermost lenslets is wide enough to make an imageof a 20 km–thick sodium profile.With such a large FOV and a polar–coordinate CCDarray, as expected for the Thirty–Meter–Telescope (TMT)LGS–AO facility (Ellerbroek et al. 2008), the thresholding islikely the main source of the LGS aberrations and deservesa specific study. Basically, a thresholding must be applied on the image pixelsprior to computing the spot centroids in order to minimizethe contribution of the detector read–out noise σ RON , or ofthe sky background. The thresholding is generally uniformover the pupil and is implemented as follow: I t ( x, y ) = (cid:26) I ( x, y ) − Thres if I ( x, y ) > Thres I ( x, y ) < Thres , (1)with I and I t the raw and the thresholded images respec-tively. Thres is the intensity level of the threshold expressedin detector counts, i .e. in Analog–to–Digital Units (ADU).Generally, the threshold is defined from the readout noise (at3 σ RON for instance). The remainder of this section demon-strates how a uniform thresholding can induce sphericalaberrations when the spot profile is asymmetric.
If the sodium laser is launched on the system optical axis,the spots of a SH–WFS are radially elongated from the pupilcentre, as shown on Fig. 2. Let ρ be the distance of a lensletto the pupil centre, normalized to 1 on the pupil edge. FromEq. 1 of Lardi`ere et al. (2008), the geometrical radial size, onthe detector plane, of the corresponding spot is proportionalto ρ , such as E ( ρ ) ≈ E ρ , with E = f R σ Na h cos ( z ) . (2)In this equation, h and σ Na are the mean altitudeand the thickness of the sodium layer respectively, z is thezenithal angle, R is the radius of the telescope pupil, and f is the resultant focal length of the whole optical system. E is the geometrical radial extent, on the detector plane, ofa spot located on the edge of the pupil ( i .e. ρ = 1). E ( ρ )is simply termed spot elongation hereafter, and can be ex-pressed in µ m or in pixels.If x and y are the Cartesian coordinates of a spot inthe detector plane, then the geometrical sizes of that spot,projected on the x and y axis are respectively defined asfollow: (cid:26) E x ( x, y ) = x EE y ( x, y ) = y E (3) c (cid:13) , 1– ?? adial thresholding to mitigate LGS aberrations r y y E (x,y)=xEE y (x,y)=yEE E( r )= r E s q r x x E E x (x,y)=xE s s / s / Figure 2.
Geometrical model of the spot elongation if the laseris launched from the centre of the pupil.
Let I o and σ be the maximum intensity and the size ofthe central spot ( i .e. ρ = 0) respectively. The extent σ , ex-pressed in the same unit than E , is defined by the blurringdue to the lenslet diffraction lobe, or to the seeing. Consid-ering this blurring effect, the actual length and width of aspot located at the radius ρ are respectively E ( ρ ) + σ and σ . Let I max ( ρ ) be the maximum intensity of that spot. Asthe flux is conserved for all spots of the pupil, we have thefollowing equation: σ I o ≈ σ ( E ( ρ ) + σ ) I max ( ρ ) . (4)Hence, I max ( ρ ) ≈ I o Eσ ρ . (5) Let P ( r ) be the projection of the sodium layer vertical pro-file on the SH–WFS detector plane, for a lenslet locatedon the edge of the pupil ( i .e. ρ = 1). This projection isnot linear and takes in account the perspective effect whichstretches the lower part of the sodium layer and compressthe upper part. Thus, the one–dimensional function P ( r )is the radial intensity profile of the geometrical image ofthe most elongated spot. The variable r is the radius po-sition defined inside the sub–image plane of the consideredspot only. If a threshold of value T hres is applied on thespot image, the thresholded profile, obtained accordingly toEq. 1, is noted P t ( r ), with t = T hres/I max ( ρ ) the thresholdvalue normalized to the local spot maximum intensity. Forany spot profile and any threshold t ∈ [0 ; 1], we can definethe dimension–less function g ( t ) such as g ( t ) = 1 E (cid:18) R r P t ( r ) dr R P t ( r ) dr − R r P ( r ) dr R P ( r ) dr (cid:19) . (6)According to our previous assumption, this functionprovides, for any spot of the pupil, the radial centroid error I n t en s i t y [ A DU ] Radius [pixel] ‘Centroid vs. Threshold’ curveGeometrical LGS spot profile P (r) E( r )g(t)E( r )I max ( r )Thres Figure 3.
Radial profile of the theoretical geometrical image ofthe most elongated spot for a sodium profile featuring a strongasymmetry (profile induced by the threshold t , expressed as a fraction of theelongation of the considered spot. In absolute units (pixelor µ m), the radial centroid error scales as the spot elonga-tion, and is g ( t ) E ( ρ ). We note that g ( t ) = 0 for symmetricprofiles. Figure 3 plots a geometrical spot profile P ( r ) withits function g ( t ). This profile is among the most asymmetricsodium profiles found in the 88–profile sequence of Fig. 6.The projection of the radial error g ( t ) E ( ρ ) on the x and y –axis, provides δ x ( x, y ) and δ y ( x, y ), the errors made onthe wavefront slopes in x and y –directions respectively for aspot located in coordinates ( x , y ): (cid:26) δ x ( x, y ) = E x ( x, y ) g ( t ) = x E g ( t ) δ y ( x, y ) = E y ( x, y ) g ( t ) = y E g ( t ) (7)Hence, the wavefront error Z ( x, y ) caused by threshold-ing is deduced from the slopes, such as, ( ∂Z∂x = x E g ( t ) ∂Z∂y = y E g ( t ) (8)Using a polar coordinate system ( ρ , θ ), such as x = ρ cos θ and y = ρ sin θ , we can use the following conversionformulae, ( ρ ∂Z∂ρ = x ∂Z∂x + y ∂Z∂y∂Z∂θ = − y ∂Z∂x + x ∂Z∂y (9)to obtain, (cid:26) ∂Z∂ρ = ρ E g ( t ) , ∂Z∂θ = 0 . (10)Thus, the wavefront distortion generated by threshold-ing is always a surface of revolution whatever g ( t ), i .e. what-ever the sodium profile. In terms of Zernike polynomi-als (Noll 1976), this result proves that the thresholding canonly induce a combination of focus ( Z ) and spherical aber-rations ( Z , Z , Z , etc.). If the threshold value
T hres is constant for the whole pupil,then t = Thres /I max ( ρ ) varies with ρ . From Eqs. 5 and 10,we obtain in that case: c (cid:13) , 1– ?? Olivier Lardi`ere, Rodolphe Conan, Colin Bradley, Kate Jackson and Peter Hampton ∂Z∂ρ ≈ ρ E g (cid:16) Thres I o (cid:16) Eσ ρ (cid:17)(cid:17) . (11)Moreover, we assume that g ( t ) can be expressed, other-wise approximated, as a polynomial of degree n such as: g ( t ) ≈ n X i =1 g i t i (12)with the sum starting at i = 1 because g (0) = 0 by defini-tion (Eq. 6). Using the binomial theorem, we can find, afterintegration of Eq. 11: Z ( ρ ) ≈ E n X i =1 g i (cid:16) Thres I o (cid:17) i i X j =0 j + 2 i ! j !( i − j )! (cid:16) Eσ (cid:17) j ρ j +2 . (13)We can check that Z ( ρ ) = 0 if the spots are not elon-gated ( E = 0), or if the threshold is null, or again if thespot profile is symmetric ( g i = 0 , ∀ i ). If n is the order ofthe polynomial g ( t ), the radial order of the wavefront aber-ration is n + 2. A uniform threshold applied on a triangularor trapezoidal ( i .e. n = 1) asymmetric spot profile inducesalready aberrations of the third radial order: Z ( ρ ) ≈ E g Thres I o (cid:16) ρ + 23 Eσ ρ (cid:17) . (14)The same threshold applied on a more complex asym-metric sodium profile will induce higher order aberrationson the wavefront. The projection of ρ on the Zernike poly-nomials base is a pure focus: ρ = 12 √ Z , (15)while ρ is a combination of focus and all spherical aberra-tions ( Z , Z , Z , etc.): ρ ≈ . Z + 0 . Z − . Z + 0 . Z + ... ) (16)The focus term is corrected by the NGS focus sensor al-ready required to offset the altitude variation of the sodiumlayer. Beyond the focus, the main aberration induced bya uniform thresholding is Z mode, higher order spheri-cal aberrations being negligible. The Z error is fluctuatingwith the time, due to the variations of the sodium profileasymmetry, and is an issue for LGS AO systems. If the threshold value is now, not uniform over the wholepupil, but defined for each sub–image proportionally to themaximum intensity of the local spot, then the normalizedthreshold t = Thres /I max ( ρ ) becomes constant, as well as g ( t ). From Eq. 10, we deduce that Z ( ρ ) = E g ( t ) ρ . (17)Consequently, such a thresholding method induces onlya focus error and no spherical aberrations at all, whateverthe sodium profile structure. As the threshold value scalesas the inverse of the radius, this thresholding is refereed toas “radial thresholding”. I m age i n t en s i t y [ A DU ] Figure 4.
Horizontal radial cut of the 29 ×
29 elongated spotSH–WFS frame of Fig. 1 obtained with the UVic LGS–benchat SNR=18. The red solid line illustrates a uniform threshold setat 3 times the read–out noise ( σ RON =0.43 ADU on the bench).The radial cut is a binning of the 3 central pixel lines of the frame.According to the spot elongation, the theoretical intensity ratiobetween the central spots and the edge spots is 4. The actualratio is around 7 here due to the gaussian profile of the colli-mated beam used on the bench, and also to the field curvature ofthe imaging lens. These effects exaggerate slightly the wavefronterrors induced by thresholding.
Figure 5.
Truncation effect induced by thresholding on the 4outer spots at SNR=18, for 3 different threshold levels. The 0–ADU level is the detector offset. Pixel values might be negativedue to read–out noise.
The radial thresholding is easy to implement on real SH–WFSs, and has been validated in laboratory with the UVicLGS demonstrator bench and compared to the results ob-tained without threshold or with an uniform threshold. Twosignal–to–noise ratios (SNR) have been considered, 140 and18, the latter being expected for real LGSs. At SNR=140,the read–out noise is negligible, and thresholding could benot necessary. However, this favourable case is useful toproperly distinguish the thresholding effect from the noise,and to validate our model.The readout noise of the SH–WFS camera is0.43 ADU rms . As the offset frame of the detector is sub-tracted to the raw images, some pixels may have negativevalues due to the read–out noise. Consequently, a 0–ADUthreshold discards the negative values, and does not mean c (cid:13) , 1– ?? adial thresholding to mitigate LGS aberrations Z [ n m p t v ] Z [ n m p t v ] Time [profile Z [ n m p t v ] Time [profile A l t i t ude [ k m ] Na profile sequence (data from TMT.AOS.TEC.07.005_Nap)
10 20 30 40 50 60 70 8085 90 95 100 501001502000 10 20 30 40 50 60 70 80−200−1000100200 D Z [ n m p t v ] No thres.UTCoGRTCoG
Figure 6.
Sodium profile sequence used on the UVic LGS–bench(real data from Purple Crow LIDAR, Univ. of Western Ontario,altitude × time resolution is 25 m ×
70 s) and LGS aberrationsmeasured by a centre–of–gravity–based SH–WFS in open–loopwith no turbulence, when no threshold, or a uniform threshold(UTCoG), or again a radial threshold (RTCoG) is applied on thepixels of LGS spot images. The focus ( Z ) component due to analtitude change of the sodium layer is removed here to highlightthe component due to the thresholding (SNR=140, UT=13 ADU,RT=4% of the local spot maximum, σ RON =0.43 ADU). that no threshold is applied on the image. We will have todistinguish both cases in our following experimental studyat low SNR.The FOV of the circular field–stop of the bench corre-sponds to a sodium layer thickness of 20 km for the mostelongated spot. As the mean thickness of the sodium layer isabout 10 km, the spot truncation effect due to the field–stopshould be negligible compared to the spot truncation causedby the thresholding.Figure 1 displays a 29 ×
29 radially elongated spot imageobtained with the UVic bench for SNR=18. On this image,the maximum spot elongation E is 8 pixels, and the spotwidth σ is 2 pixels. Figure 4 plots an horizontal cut of thisimage ( y = 0). A uniform threshold at 1.3 ADU, i .e. 3 σ RON for the bench, is shown as an example. This graph alreadypoints up that a uniform thresholding is not suited for LGSspots, since edge spots are fainter than the central spots. Atlow SNR, a uniform threshold might even reach the maxi-mum intensity of some spots located on the edge. Figure 5illustrates this issue for SNR=18.
Uniform threshold, SNR=140
Zernike modes W a v e f r on t e rr o r [ n m p t v r m s ] Thres = 15 ADU
Thres = 10 ADU
Thres = 5 ADUNo threshold (a)
Radial threshold, SNR=140
Zernike modes W a v e f r on t e rr o r [ n m p t v r m s ] Thres = 12% of local max.
Thres = 8% of local max.
Thres = 4% of local max.No threshold (b)
Figure 7.
LGS–induced aberrations measured on the UVic LGS–bench at SNR=140 when a uniform threshold (a) or a radialthreshold (b) is applied on LGS spot images ( σ RON =0.43 ADU).
Figure 6 plots the LGS aberrations for Z , Z , Z and Z modes versus the time, measured during the whole sodiumprofile sequence, with no threshold, with a 13–ADU uniformthreshold and a radial threshold (4% of local maximum) atSNR=140, in open–loop without turbulence. Such a highuniform threshold, not suited for real AO systems, is chosenhere to exaggerate the thresholding effects on the wavefrontand validate the model. Figure 7 plots the correspondingtemporal RMS fluctuation of all Zernike modes beyond focusup to Z , for SNR=140 and for different threshold values(both uniform and radial).As the main focus error source is the altitude variationof the sodium layer, the focus obtained without thresholdinghas been subtracted from those obtained with thresholding.This subtraction highlights the extra focus error induced bythresholding. This error reaches 350nm ptv with the uni-form threshold, and 200nm ptv with the radial threshold.Moreover, a correlation is visible between the focus and Z modes induced by the uniform thresholding. This correla-tion confirms that Z and Z modes arise together. Thisresult is in good agreement with the model (Eqs. 13 to 16).Although the ratio Z /Z depends on the coefficients g i , i .e. the sodium profile structure and asymmetry, we can at-tempt to quantify it for a linear fit of the functions g ( t ), i .e. for a sequence of trapezoidal or triangular sodium pro-files. In this simple case, the ratio Z /Z expected by themodel is about 7 . E/σ = 4 (Eqs. 14 to 16), while theobserved ratio is about 4 . ptv amplitudes (Fig. 6). c (cid:13) , 1– ?? Olivier Lardi`ere, Rodolphe Conan, Colin Bradley, Kate Jackson and Peter Hampton
Uniform threshold, SNR=18
Zernike modes W a v e f r on t e rr o r [ n m p t v r m s ] Thres = 3 ADU
Thres = 2 ADU
Thres = 0 ADUNo threshold (a)
Radial threshold, SNR=18
Zernike modes W a v e f r on t e rr o r [ n m p t v r m s ] Thres = 12% of local max.
Thres = 8% of local max.
Thres = 4% of local max.No threshold (b)
Figure 8.
LGS–induced aberrations measured on the UVicLGS–bench at SNR=18 when a uniform threshold (a) or aradial threshold (b) is applied prior to computing centroids( σ RON =0.43 ADU).
The model and the experiments provide the same order ofmagnitude for the ratio Z /Z . The difference is likely dueto the assumption made about the profile shape. A higherorder fit of the functions g ( t ) would be required for eachprofile, as well as further measurements using a symmetricsodium profile as a reference.Figure 6 shows clearly that the radial thresholding miti-gates the spherical aberrations ( Z and Z ) induced by thesodium layer variations, as if no thresholding was applied.The variations for the Z mode reaches 90nm ptv with uni-form thresholding, against 10nm ptv with radial threshold-ing or without thresholding at all. The remaining error islikely the component induced by the noise and by the circu-lar field–stop. Moreover, Fig. 7a shows that the RMS fluctu-ation of Z mode scales as the level of the uniform thresh-old, which is in agreement with the model too (Eq. 13).As expected, the radial thresholding does not mitigatethe square symmetric LGS aberration Z . However, the Z error is modified by the thresholding, and is almost nullwithout thresholding (Fig. 6 and 7a). Hence, the threshold-ing would likely induce other effects not considered in ourmodel. The pixel boundaries or the spot overlap may havemore or less impact depending on the threshold value. As-suming that polar coordinate CCD arrays should mitigatesquare symmetric modes, we will not investigate with moredetails the other possible sources of this mode in this paper.At SNR=140, the best performances are obtained witha radial threshold set at 4% of the local spot maximum. The Table 1.
RMS wavefront errors (WFE), in nm rms , and achiev-able Strehl ratios (S), in % for λ = 1 µm , obtained at SNR=140and 18, during the sodium profile sequence with the center–of–gravity algorithm without pixel thresholding, with uniformthresholding, and with radial thresholding. The threshold levelsare optimized for SNR=18. The uniform threshold values are 13and 0 ADU for SNR=140 and 18 respectively, while the radialthreshold values are 4% and 12% for SNR=140 and 18 respec-tively. All Zernike modes beyond the focus (from Z to Z ) areadded up and spatially averaged over the pupil. The bench ac-curacy is the ultimate centroiding performance achieved by thebench on non–elongated spots. Readout noise is 0.43 ADU rms . SNR=140 SNR=18WFE S WFE S No threshold 4.34 99.9 60.87 86.4Uniform threshold 10.65 99.5 31.91 96.1Radial threshold 4.27 99.9 18.70 98.6Bench accuracy 2.73 99.9 17.84 98.7
RMS wavefront errors averaged over the pupil for all modesbeyond the focus are summarized in Table 1.
Figure 8 plots the temporal RMS fluctuation of Zernikemodes from Z to Z , for SNR=18 and for different thresh-old values, both uniform and radial.At low SNR, edge spots are very faint, a few ADU. Con-sequently a uniform threshold, even low, can easily exceedthe maximum of some spots, and induce huge errors. Theerrors induced by a 3–ADU threshold exceed 100 nm, theupper limits of the graph (Fig. 8a). Hence, a small change ofthe threshold level or of the SNR, which is fluctuating withLGS, can decrease significantly the performances. A 0–ADUuniform threshold appears to be more robust in that case,but it is not optimal in term of noise.The performance obtained with the radial threshold ismore robust at SNR=18, and does not depend much on thethreshold value (Fig. 8b). This is certainly due to the abilityof the radial thresholding to track the temporal variations ofthe spot intensity: the radial thresholding is also a dynamicthresholding. The wavefront errors measured with a radialthreshold are lower than those measured with a uniformthreshold in the best case (Tab. 1). At SNR=18, the optimalvalue for the radial threshold is 12% of the local maximum.Lastly, Tab. 1 shows that the performance achieved withthe radial thresholding on elongated spots is very close tothe ultimate performance achievable on the bench with anon–elongated source. This result basically means that theradial thresholding cancels out most of the errors inducedby the LGS spot elongation, without altering the centroidaccuracy in presence of noise. Considering the issues caused by the thresholding on LGSspots, we could suggest to use the Weighted Centre–of–Gravity algorithm, which does not require a prior thresh-olding, instead of the conventional centre–of–gravity algo- c (cid:13) , 1– ?? adial thresholding to mitigate LGS aberrations x y Z( r ) = r + 8/3 r x y Z( r ) = r P ea k − t o − v a ll e y a m p li t ude LGS aberrations with uniform thresholding P ea k − t o − v a ll e y a m p li t ude LGS aberrations with radial thresholding
Side laserCentral laserSide laserCentral laser
Figure 9.
Typical LGS aberrations arising when the laser islaunched from the side of the pupil (x=0 and y=0) with a uniformthresholding (top) and for a radial thresholding (bottom). Theseaberrations are compared, in term of Zernike modes, to those aris-ing when the laser is centrally projected. Aberration amplitudesare peak–to–valley in an arbitrary unit. rithm. Basically, this algorithm consists in weighting thepixels of the spot image by a reference image of thespot (Nicolle et al. 2004). If the spot image matches per-fectly the reference image, this technique is equivalent tocompute the center–of–gravity of the square of the imageintensity. However, the centre of gravity of the square of anelongated spot image differs from the actual center of gravityof that spot, if the spot profile is asymmetric. Consequently,the Weighted–Center–of–Gravity algorithm cannot be usedon LGS AO systems as it is.
The theoretical model presented in Sec. 3 assumes thatthe laser is launched on the optical axis. This configura-tion is optimal for ELTs because it minimizes the spotelongation and the LGS aberrations. However, on a 10–meter class telescope, the laser can preferably be launchedfrom the side of the telescope pupil, like for the Keck tele-scope (Wizinowich et al. 2006). The LGS spots of a SH–WFS are not longer radially elongated from the pupil centre,but from an edge of the pupil.The model is still valid for this case, providing thatthe origin of the coordinate system ( ρ , θ ) or ( x , y ) coincideswith the position of laser launch telescope. Equation 13,expressing the wavefront Z ( ρ, θ ), remains unchanged andthe aberrations generated by a uniform thresholding are stillcentro–symmetric for a virtual giant pupil centred on thelaser axis, with a diameter twice the distance between thelaser axis and the optical axis. The only difference is thedomain of definition of the function Z ( ρ, θ ), which must beshifted to match the telescope pupil.Figure 9 displays the portion of the wavefront Z ( ρ, θ )seen by a circular pupil located at the right of the laser axis,and plots the peak–to–valley amplitudes of this wavefrontprojected on Zernike modes. Assuming a linear fit of thefunction g ( t ), we can consider Z ( ρ ) ∝ ρ + 8 / ρ for repro-ducing the LGS aberrations induced by an uniform thresh-olding (Eq. 14 with E/σ = 4), and Z ( ρ ) ∝ ρ for reproduc- ing the LGS aberrations induced by a radial thresholding(Eq. 17).As the aberrations are not centro–symmetric, theZernike decomposition of ρ features, in addition to the fo-cus and spherical aberrations, many other modes, such as tipand tilt ( Z , Z ), astigmatism ( Z , Z ), coma ( Z , Z ) andsome trefoil ( Z , Z ). Moreover, compared to the centrallyprojected laser configuration, the peak–to–valley amplitudeof these aberrations are about 7 times greater.On the other hand, the decomposition of ρ featuresonly tip and tilt modes in addition to the focus, meaningthat the radial thresholding removes all the LGS aberrationsbeyond the focus. Tip–tilt and focus modes are not issues,since they will be corrected by the NGS tip–tilt and focussensor already necessary to determine the atmospheric tip–tilt, unseen by a LGS WFS, and to offset the sodium layerdistance variations, respectively.Finally, the radial thresholding is even more advised ifthe laser is launched from the side of the pupil, since the LGSaberrations induced by the uniform thresholding are muchhigher and are spread in more modes. The side–projectedlaser configuration has no been tested on the UVic bench,but could be implemented if the need arises for any LGS AOproject. The fluctuations of the sodium layer vertical profile in-duce centro–symmetric and square symmetric aberrationson LGS SH–WFSs. Centro–symmetric aberrations, such asfocus and spherical aberrations, are due to a spot trunca-tion by either a circular field–stop or by a threshold appliedon the image pixels. Square symmetric aberrations, such as Z , are due to a spot truncation by the pixel boundariesand a sampling effect. Some extra aberrations may arise dueto uncalibrated non–common path errors of the LGS trainvarying with the sodium layer distance.Both kind of aberrations have been reproduced in thelaboratory with UVic LGS bench. Square symmetric aber-rations should be negated with a polar coordinate CCDarray. Considering such a detector and a field of view persub–aperture wide enough to be able to image a 20–kmthick sodium layer, the residual LGS aberrations are mainlycaused by the threshold applied on the pixels of elongatedspots prior to computing the centroids.This statement is confirmed by the theoretical modeland experimental results presented in this paper. The aber-rations induced by the thresholding contain focus and spher-ical aberration modes if the laser is launched from the pupilcentre. Their amplitudes scale as the spot elongation ( i .e. thetelescope diameter), the sodium profile asymmetry and thethreshold value.These aberrations disappear if the threshold value isnot static and uniform over the pupil, but defined dynam-ically and independently for each lenslet, proportionally tothe maximum intensity of the local spot. The residual LGSaberrations are only a focus error if the laser is centrallyprojected, or a combination of tip, tilt and focus if the laseris launched beside the pupil. These residual modes are notissues since they will be corrected by the natural guide star c (cid:13) , 1– ?? Olivier Lardi`ere, Rodolphe Conan, Colin Bradley, Kate Jackson and Peter Hampton tip–tilt and focus sensor already required in any LGS AOsystem.This new thresholding method, termed radial thresh-olding, is very simple and add no extra computational timecompared to the uniform thresholding. It can be imple-mented on all current LGS SH–WFS in operation. More-over, the radial thresholding makes the centre–of–gravityalgorithm still well–suited for Extremely Large Telescopes,where the LGS spot elongation is greater. Unlike thematched filter or the correlation centroiding algorithms, thecentre–of–gravity is a simple, well–proven and fast algorithmwhich requires no special calibration processes or referenceimages, and has no repercussions on the design of the LGSAO system.
ACKNOWLEDGMENTS
The authors are grateful to TMT consortium. The TMTProject gratefully acknowledges the support of the TMTpartner institutions. They are the Association of CanadianUniversities for Research in Astronomy (ACURA), the Cal-ifornia Institute of Technology and the University of Cal-ifornia. This work was supported as well by the Gordonand Betty Moore Foundation, the Canada Foundation forInnovation, the Ontario Ministry of Research and Innova-tion, the National Research Council of Canada, the NaturalSciences and Engineering Research Council of Canada, theBritish Columbia Knowledge Development Fund, the Asso-ciation of Universities for Research in Astronomy (AURA)and the U.S. National Science Foundation. We would alsolike to thank Laurent Jolissaint, the reviewer of this paper,for his useful comments.
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