Automatic sensitivity adjustment for acurvature sensor

Aglae Kellerer,* Mark Chun, and Christ FtaclasInstitute for Astronomy, 640 N. A’ohoku Place, Hilo, Hawaii 96720, USA

*Corresponding author: kellerer@ifa.hawaii.edu

Received 3 March 2010; revised 12 July 2010; accepted 29 July 2010;posted 13 August 2010 (Doc. ID 124844); published 8 September 2010

There are different techniques for sensing wavefront phase distortions due to atmospheric turbulence.Curvature sensors are practical because their sensitivity is adjustable to the prevailing atmospheric con-ditions. Even at the best sites, the turbulence intensity has been found to vary at times over only a fewminutes and regularly over longer periods. Two methods to automatically adjust the sensitivity of a cur-vature sensor are proposed: First, the defocus distance can be adjusted prior to the adaptive-optics (AO)loop through the acquisition of a long-exposure image and can then be kept constant. Second, the defocusdistance can be changed during the AO loop, based on the voltage values sent to the deformable mirror.We demonstrate that the performance increase—assessed in terms of the image Strehl ratio—can besignificant. © 2010 Optical Society of AmericaOCIS codes: 110.0115, 110.1080.

1. Introduction

Curvature wavefront sensors [1] have a number ofadvantages over the more commonly used Shack–Hartmann (S-H) sensors [2]. One useful characteris-tic that has not often been used in practice is theiradjustable sensitivity to phase distortions: while S-Hsensors are designed for one fluctuation amplitudetaken to be prevalent during the instrument’s life-time, the dynamic range of curvature sensors varieswith the distance of the detector from the focal plane(the defocus distance). Accordingly, curvature sensorscan be tuned to be optimally sensitive to the wave-front aberrations that prevail during a particularobservation.

In most adaptive-optics (AO) applications, the am-plitude of the phase fluctuations does indeed varysubstantially. In astronomy, the seeing—a measureof the phase-fluctuation amplitudes—varies on atime scale of typically several minutes, which isshorter than the integration time of most astronom-ical observations. The seeing varies even faster and

might become prohibitive when an observed object,such as a satellite, moves across the sky faster thanastronomical targets.

In 1991, Roddier et al. [3] suggested using a feed-back loop to keep the rms tip–tilt signal-error con-stant. The sensor would then automatically adjustitself to changing seeing conditions and could alwaysoperate in its optimum linear range. Existing curva-ture-sensor-based AO systems do not, as yet, imple-ment such a feedback loop. For example, on PUEO,the 19-element curvature sensor installed behind theCanada–France–Hawaii telescope, the defocus dis-tance (termed optical gain in the PUEO manual) isnot, generally, adjusted to changing atmospheric con-ditions: “The optical gain value is… by default set to128 for a point-like object: this value may need to bereduced for an extended object or a double star” [4].On MACAO [5], the curvature AO systems installedbehind the Very Large Telescope, the defocus dis-tance is adjusted as the reference source is changed.The adjustment depends on the seeing measured bythe Differential Image Motion Monitor (DIMM) [6]and on the apparent size of the reference source [7].

We propose two practical implementations of defo-cus feedback loops: First, the defocus distance can be

0003-6935/10/31G140-08$15.00/0© 2010 Optical Society of America

G140 APPLIED OPTICS / Vol. 49, No. 31 / 1 November 2010

adjusted prior to the AO loop through the acquisitionof a long-exposure image and can then be keptconstant. This solution is similar to the MACAO ap-proach. Second, the defocus distance can be changedduring the AO loop, based on the voltage valuessent to the deformable mirror. We analyze the re-sultant performance increase through numericalsimulations.

2. Optimal Defocus Distance

The output signal of a curvature wavefront sensor isthe fractional intensity difference v ¼ ðx − yÞ=ðxþ yÞ,where x and y are the photon numbers registered oneither side of the focal plane. In closed-loop AO obser-vations v tends to be near 0 and can then be ex-pressed as a function of the wavefront phase, ϕ [8]:

vð~rÞ ¼ −λSf ðf − lÞπl

�~∇

�P

�fl~r

��· ~∇

�ϕ�fl~r

��

þ P

�fl~r

�·∇2ϕ

�fl~r

��: ð1Þ

where~r is the position vector in the detector plane, f isthe focal length, l is the distance between the detectorand the focal plane, and f =l~r is, thus, the positionvector in the pupil plane. λS is the sensing wave-length, and P—the pupil transmission function—equals 1 inside and 0 outside the pupil. Equation (1)makes use of the Fresnel approximation (i.e.,D2=ððf − lÞλÞ ≥ 1) and it assumes the size of the diffrac-tion patterns to be negligible compared to the magni-tudeof the intensity fluctuations in thedetectorplane.This condition, the geometric-optics approximation,constrains the defocus distance to λSf 2 ≪ r20ðλSÞl.The Fried parameter, r0ðλSÞ, characterizes the turbu-lence intensity in dependence on the wavelength, λS.In regions where the pupil transmission is uniform,themeasured signal is the Laplacian of thewavefrontphase. On the pupil edge—i.e., along the central ob-struction and along the telescope spiders—curvaturesensors measure curvature and radial tilt.

In line with Eq. (1), we want small defocus dis-tances, l, in order to have sufficient sensitivity tosmall phase fluctuations. However, for a given inten-sity spatial fluctuation size within the aperture,d ¼ D=

ffiffiffiffiffiN

p, we want l sufficiently large in order to

havedl=f larger than the diffraction pattern λSðf − lÞ=r0ðλSÞ. As Rigaut et al. [9] state, to limit the aliasingeffect the diffraction should be large to filter out thenot measured high-order modes of the turbulence.Accordingly the optimal distance is dl=f ¼ k1λSðf−lÞ=r0ðλSÞ. The coefficient k1 is sufficiently small tominimize aliasing effects and sufficiently large toavoid nonlinearities due to diffraction. Let l0 be theoptimal defocus distance. The defocus length is typi-cally substantially smaller than the telescope focallength, l ≪ f , so that

l0 ¼ k1 ·λSf 2

ffiffiffiffiffiN

p

Dr0ðλSÞ; ð2Þ

where the coefficient k1 depends on the exact size ofthe subapertures, and, in particular, on the size of thecentral obstruction. But, for a given AO system, k1 isindependent of the observing conditions, such as see-ing and stellar magnitude.

3. Defocus Adjustment Prior to the AO Loop

Here and in Section 4, numerical simulations of cur-vature AO systems will be presented. In this section,the seeing is kept constant during each sequence andthe point is explored, whether the optimal defocusdistance can be determined from the diameter of along-exposure image recorded prior to the AO loop.

The simulated curvature AO system has N ¼ 80subapertures, which is an average of existing astro-nomical curvature systems (PUEO/Canada–France–Hawaii telescope, 19 [10]; MACAO/Very LargeTelescope, 60 [5]; NICI/Gemini-south telescope, 85[11], AO-188/Subaru telescope, 188 [12]). The suba-pertures are placed on concentric rings, as shown inFig. 1, and cover equal fractions of the pupil surface.

The numerical simulations are based on the Yaocode, developed by Rigaut (www.maumae.net/yao/).For 22 seeing values between 0:200 and 1:400 (atλ ¼ 0:5 μm) a matrix of incoming phase values (thephase screen) is generated and is shifted stepwisein front of the pupil. The distribution of phase valuesfollows the Kolmogorov model. The step size periteration is indicated in Table 1. Observation se-quences are then simulated over 3 s.

Initially, the optimal defocus length, l, and the loopgain, g, are determined by repeating the numericalsimulations for g ¼ 0:1; 0:2;…1 and l ¼ 0:1; 0:2;…1:8 m in an f =60 beam. A separate command matrix,which converts a set of N sensor measurementsinto N voltage values, is computed for each defocusdistance. These matrices are computed by first

Fig. 1. Configuration of the simulated sensor: the wavefront issampled over 80 subapertures. Dashed curves, pupil inner andouter edges.

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generating N influence functions: N mirror surfacescorresponding to a unit voltage applied to each of theactuators. The influence functions are independentof the defocus distance and are generated only once.Note that the influence functions are stored in a237 × 237 matrix, while the pupil diameter equals220 pixels: all points of the influence functions thatlie outside the pupil are set to zero. For each defocusdistance, Yao computes an interaction matrixthrough Fresnel diffraction: the N ×N matrix con-tains the N sensor measurements induced by a unit

voltage applied to each of the N actuators. The com-mand matrix is obtained by inverting the interactionmatrix.

Figure 2 shows the Strehl ratios attained at1:65 μm in dependence on the seeing value at 0:5 μmand compares them to the ratios achieved when l iskept constant while the seeing changes. In the lattercase, the loop gain is still optimized for each seeingvalue. As shown in Fig. 2, the correction performancewill be especially low if the curvature sensor is tunedfor excellent atmospheric conditions (i.e., the defocus

Table 1. Parameter Values of the Numerical Simulations, if Not Otherwise Specified in the Text

Parameters Section 3 Section 4 (if different from Section 3)

Telescope diameter, D 7:9 m; 220 pixels —

Diameter of central obstruction 1:0 m; 28 pixels —

Size of influence functions, M ×M 237 × 237 pixels —

Number of subapertures, N 80 —

Size of phase screen 2048 × 256 pixels 4096 × 256 pixelsSeeing at 0:5 μm, θ0ð0:5 μmÞ ½0:200;1:400� ½0:100; 1:800�Layer speed, V 2 m=s —

Height of turbulent layer, H 100 m —

Outer scale of turbulence, L0 þ∞ —

Target magnitude, m 5 —

Sky magnitude, mS 20 —

Photon flux at magnitude 0 1013 m−2 s−1 —

Atmosphericþ instrumental throughput 0.25 —

Imaging wavelength, λI 1:65 μm —

Sensing wavelength, λS 0:7 μm —

AO loop frequency, f 1000 Hz 200 HzRead-out latency 0 s —

Read-out noise and dark current 0 —

First … iterations disregarded 50 —

Minimal curvature radius of the deformable mirror 13 m —

Loop gain, g [0, 1] increment: 0.1 —

Extrafocal distance in an f =60 beam, l [0.1, 1.8] increment: 0.1 [0.1,3.0] increment: 0.1

Fig. 2. Strehl ratio and defocus distance versus seeing for a curvature sensor under various seeing conditions (See left column of Table 1for the parameter values). Triangles, sequences with different defocus values are simulated and the defocus that yields the highest Strehlis kept. Dashed curve (respectively, dotted and dashed-dotted), the minimum (respectively, the average and maximum) defocus distanceobtained with the former procedure is kept for all turbulence intensities. Solid curve, the defocus value is determined from a long-exposureimage, acquired prior to the AO loop.

G142 APPLIED OPTICS / Vol. 49, No. 31 / 1 November 2010

distance is small). The reverse effect—a perfor-mance drop due to an overly pessimistic sensoradjustment—also exists, but is less fatal: for excellentseeing conditions, θ0 ¼ 0:200, the Strehl ratio de-creases from 0.94 (l ¼ 0:2 m) to 0.87 (l ¼ 0:6 m) and0.83 (l ¼ 1:1 m).

Because it is impractical to determine the best de-focus distance, l0, by running a series of AO loopsduring actual observations, we need to explorewhether—from the diameter of a long-exposure im-age acquired prior to the AO loop—l0 can be directlyobtained through the formula

l0 ¼ k2 ·�λIλS

�1=5

·f 2

ffiffiffiffiffiN

p

D· θ50ðλIÞ; ð3Þ

where λS and λI are the sensing and imaging wave-lengths. θ50ðλIÞ∼ λI=r0ðλIÞ (radians) is the diameterof the circle that contains 50% of the intensity ona long-exposure image at wavelength λI. The coeffi-cient k2 depends on instrumental parameters, suchas the central pupil obstruction, and it needs to bedetermined through simulations or instrumental ca-librations. D is the pupil diameter, f is the telescopefocal length, and N is the number of subapertures.Table 1 gives their numerical values.

The coefficient k2 is determined via Eq. (3): 22 val-ues of l0 and θ50 are obtained for seeing values be-tween 0:200 and 1:400, at λ ¼ 0:5 μm. A linear fit yieldsk2 ¼ 0:68� 0:07. As seen onFig. 2, the performance ofthe AO loop is then the same—whether the defocus isdetermined prior to the AO sequence from the diam-eter of a long-exposure image (solid curve), orwhethermany different defocus values are tested (triangles).It is thus possible to optimize the system’s perfor-mance by linking the defocus distance to the diameterof a previously recorded, long-exposure image. An-

other possibility is to adjust the defocus distance independence on the seeing measured by a turbulencemonitor. On MACAO [5], the defocus is adjusted interms of the seeing as measured by the DIMM [6].The disadvantage of this approach is that the seeingpredicted by the DIMM monitor is often much worsethan the seeing sensed by the Very Large Telescopes[13]. This seems to be explained by a highly turbulentlayer close to the ground, which is seen by the DIMM(installed on a 6 mhigh tower), while it does not affectthe telescopes (primary mirrors at 11 m above theground, top of enclosures at 30 m).

If the magnitude of the wavefront aberrationschanges significantly over the course of the observa-tion, the defocus length needs to be adjusted duringthe AO loop. In the following section, a suitableprocedure is suggested and is tested via numericalsimulations.

4. Adjustment during the AO Loop

Figure 3 shows seeing values recorded during twonights by the DIMM [6] and Multi Aperture Scintil-lation Sensor (MASS) [14] instruments at the MaunaKea observatory. Such significant and rapid seeingvariations have also been reported at the Paranal ob-servatory by Rigaut and Sarazin [15]. The significantand rapid seeing variations suggest that even obser-vations with integration times of just several min-utes would benefit from a defocus feedback loop.

A. Method

In Section 3, the defocus distance has been assumedto be determined prior to the AO loop from the diam-eter of a long-exposure image. To adjust it during theobservation, a constantly updated estimate of theseeing is required. This estimate is here taken to beobtained from the voltage values sent to the deform-able mirror.

Fig. 3. Seeing at 0:5 μm, recorded by the DIMM (dots) and MASS (squares) on the Mauna Kea observatory. The monitors are installednext to the Canada–France–Hawaii telescope, which accommodates the curvature AO system PUEO. DIMM and MASS acquire a mea-surement about every 2 min. MASS is insensitive to the lowest turbulent layers and generally measures smaller seeing values thanDIMM. Credit: http://mkwc.ifa.hawaii.edu/.

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The phase of the wavefront is continuously rede-termined from the voltages and the influence func-tions of the actuators: ϕ ¼ 2IV . The voltage vectorV has N elements, N being the number of actuators.The matrix I (radians/volt) contains the N influencefunctions. Each influence function is sampled overM ×M points, and I is, therefore, of size M ×M×N. ϕ (radians) is the wavefront phase at wavelengthλS and is likewise sampled over M ×M points.

The phase variance, averaged over a certain num-ber of loop cycles, K , is then used to infer the Friedparameter:

σ2ϕ ¼ 1K

·1

M2

XKk¼1

XMm;n¼1

ϕkðm;nÞ2 ¼ κ ·�

Dr0ðλSÞ

�5=3

; ð4Þ

where σ2ϕ is the average phase variance over K cyclesat wavelength λS. ϕk is the wavefront phase at wave-length λS for cycle number k. The exact value of κneed not be assessed, since the defocus distance isrelated to D=r0ðλSÞ through another coefficient [seeEq. (2)]. Therefore,

l0 ¼ k3 · λS�fD

�2 ffiffiffiffiffi

Np

σ6=5ϕ : ð5Þ

k3 is then determined from simulations or instru-mental calibrations. The actual defocus distance isset to the closest value for which a command matrixhas been computed.

B. Numerical Simulations

At the outset, 30 command matrices are obtained fordefocus values in the ½0:1; 3:0� m range, with a 0:1 mincrement. The computation of the command ma-trices follows the principle described in Section 3.An initial defocus value is specified in terms of Eq. (3)

from a long-exposure image acquired with the initialseeing value.

The coefficient k3 is obtained via Eq. (5): 22 AO se-quences are simulated for seeing values between 0:200and 1:400, at λ ¼ 0:5 μm. Each sequence contains 500iterations. The optimal defocus distances corre-sponding to those seeing values have already beendetermined in Section 3. Starting at the 300th itera-tion, the average phase variance over the last 100iterations—i.e., during the last 0:5 s—is computed.The mean and standard deviation of the 200 realiza-tions is determined, and the coefficient, k3, is ob-tained through a linear fit: k3 ¼ 1:2� 0:1.

During the AO loop, the phase screen is shiftedstepwise in front of the pupil. The size of the matrixand the step size per iteration are indicated inTable 1, and allow for a maximum exposure time of70 s. Every 5 s, the values of the phase screen arescaled to a new seeing value. In order to adjustthe defocus distance during the AO loop, the wave-front phase variance is computed after each iterationfrom the 80 voltage values and their associated influ-ence functions.

Figures 4 and 5 compare the performance of theAO system for two approaches.

1. The defocus distance is kept constant. This isthe approach presented in Section 3, but, in thepresent treatment, the seeing varies during theobserving sequence.

2. Starting from the 300th iteration, the defo-cus distance is updated every fourth iteration: theaverage phase variance over the last 100 iterations—i.e., during the last 0:5 s—is used to determine a newdefocus distance through Eq. (5). The defocus isset to the closest value for which an interactionmatrix has been computed, i.e., it is set to0:1; 0:2;…2:9 or 3:0 m.

Fig. 4. (Color online) 70 s AO sequence, during which the seeing, θ0, is incremented by 0:0500 every 1000th iteration. Left panel: solidcurve, average Strehl when the defocus is adjusted during the sequence; dotted curve, the defocus is determined prior to the AO loop whenθ0 ¼ 0:200; circles, θ0=2. Right panel: defocus values; circles, θ0. (Parameter values: see right column of Table 1).

G144 APPLIED OPTICS / Vol. 49, No. 31 / 1 November 2010

In both cases, the AO loop gain is adjusted for theaverage seeing over the sequence and is then keptconstant. In Fig. 4, the seeing increases continuouslyfrom 0:200 to 1:400. The long-exposure image is ac-quired when θ0 ¼ 0:200 and the initial defocus lengthis thus small (0:2 m). Under this condition, theStrehl value breaks down when the seeing increases,while the defocus distance is kept constant at its lowvalue. Adapting the defocus to the changing turbu-lence conditions makes the final Strehl value in-crease to 0.45. The right panel of Fig. 4 shows thatthe defocus increases when the seeing, and, hence,the phase distortions, increase. However, as the de-formable mirror corrects the wavefront, the fluctua-tion sizes decrease and the defocus distance canagain be reduced. This possibility of decreasing thedefocus distance as the AO correction takes effecthas been discussed by Roddier [16].

In Fig. 5, the seeing varies randomly between 0:100and 1:800. When the defocus is kept constant at its in-itial low value, the sensor measurements becomedominated by diffraction as the seeing increasesand erratic voltage commands distort the mirror sur-face. As the seeing decreases again, the distortedmirror surface still cause large diffraction patternsand the turbulence induced phase distortions cannotbe measured: the AO loop has become unstable andthe Strehl continuously decreases. When the defocusis adjusted, the average Strehl converges to theStrehl value (0.5) that roughly corresponds to themean seeing (0:800): indeed, Fig. 2 shows that theStrehl equals 0.5 when the seeing equals 0:800 duringthe whole AO correction.

Both simulations represent extreme cases, sincethe initial defocus lengths are obtained under excel-lent seeing conditions and are, thus, exceptionallysmall. The black curves of Fig. 6 show that a defocusadjustment loop is not necessary for bright reference

objects if one chooses a constant and large defocusdistance: the AO correction is then approximatelythe same for a constant and an adjusted defocusdistance. Simulations and experimental results byO. Lai have likewise shown that in a low-order cur-vature system such as PUEO the AO loop perfor-mance is not very sensitive to changing defocusdistances [17]. However, for faint reference stars,it is not satisfactory anymore to keep the defocus

Fig. 5. (Color online) 70 s AO sequence, during which the seeing, θ0, is modified every 1000th iteration. Left panel: solid curve, averageStrehl when the defocus is adjusted during the sequence; dotted curve, the defocus is determined prior to the AO loop when θ0 ¼ 0:1500;circles, θ0=2. Right panel: defocus values; circles, θ0. (Parameter values: see right column of Table 1).

Fig. 6. (Color online) 70 s AO sequence, during which the seeing,θ0, is modified every 1000th iteration between 0:100 and 1:800. Solidcurves, average Strehl when the defocus is adjusted during the se-quence; dotted curves, the defocus is determined prior to the AOloop when θ0 ¼ 1:600. The magnitude of the reference star variesamong 5 (black), 9 (magenta) and 11 (blue). Circles, θ0=2. (Furtherparameter values: see right column of Table 1.)

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distance at a constant large value (see Fig. 6): alias-ing effects require the defocus distance to stay at itsinferior limit, and an adjustment to the current see-ing is required. This suggests that, as the number ofcorrection elements of AO systems rise and eachcorrection element has to work with fewer photons,defocus adjustment loops might become increasinglynecessary.

5. Conclusion

We have tested two different methods to automati-cally adjust the defocus length of a curvature sensorto the continuously changing turbulence conditions.

In the first method, the length is adjusted priorto the AO loop through the acquisition of a long-exposure image and is then kept constant. However,in extended observations of astronomical objects orin any observation of fast moving targets, such as sa-tellites, the turbulence intensity changes over thecourse of the AO sequence. For such observations,we have suggested a method to adjust the defocuslength during the AO loop by use of the voltage val-ues sent to the deformable mirror.

While the implications for AO in astronomy are ap-parent, it is also worthwhile, in conclusion, to notethe potential application of this work for ophthalmol-ogy. Detailed information on the retina requiresobservations from outside the eye. Without opticalcorrection, the quality of such observations is mark-edly degraded by aberrations due to the various diop-tric surfaces, such as the cornea, and by the aqueousand the vitreous humor. The correction needs to bedynamic, because of the eye’s motions and becausethe aberrations fluctuate. Hofer et al. [18] suggesta required AO correction speed of 10–40 Hz.

Current methods to sense ocular aberrations, suchas laser ray tracing and spatially resolved refracto-metry, are of limited applicability because they donot permit rapid acquisitions [19]. Dynamic aberra-tion correction is, therefore, only performed by use ofS-H sensors. Liang et al. [20] have shown that S-Hsensors permit fast, precise, and repeatable mea-surements of ocular aberrations. On the other hand,Moreno-Bruiso and Navarro [21] and Glanc et al. [22]found eyes where the S-H subaperture images aretoo distorted to be properly analyzed, and one needsto note that such eyes tend to be the most interestingfrom a clinical point of view. Porter et al. [23] charac-terized the distribution of the ocular aberration in apopulation of 109 subjects and they concluded, in linewith previous studies, that the ocular aberrationsdiffered greatly from person to person—even thoughtheir study did not include subjects with pathologiessuch as cataracts or keratoconus. This suggests thatretinal imaging could benefit substantially from theuse of AO systems in which the sensitivity can beeasily adjusted to the eye’s aberration and temporalfluctuations.

Curvature sensors have, in exploratory investiga-tions, been successfully utilized to measure ocularaberrations [24,25]. In these experiments, however,

the sensors have not been used in combination withdeformable mirrors, but were instead derived interms of a complex wavefront reconstruction algo-rithm. Since curvature-based AO systems in astron-omy have proven simple and efficient when usedtogether with a bi-morphic or a membrane mirror[3,5,10–12], we suggest implementing a curvature-based AO system for retinal imaging. Compared toan S-H based system, the chief advantage of sucha system will be the capacity to rapidly adjust thedefocus distance when different patients, with theirparticular ocular aberrations, are examined.

We thank Jerome Paufique for helpful discussions,and the anonymous referee for his constructive cor-rections and suggestions. Financial support from theUnited States Air Force Office of Scientific Research(USAFOSR) is likewise acknowledged.

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