Using photodetectors in Shack-Hartmann wavefront sensors

ISSN 8756-6990, Optoelectronics, Instrumentation and Data Processing, 2012, Vol. 48, No. 2, pp. 146–152. c© Allerton Press, Inc., 2012.

Original Russian Text c© L.V. Antoshkin, V.V. Lavrinov, L.N. Lavrinova, V.P. Lukin, 2012, published in Avtometriya, 2012, Vol. 48, No. 2,

pp. 44–51.

OPTICAL INFORMATIONTECHNOLOGIES

Using Photodetectors

in Shack–Hartmann Wavefront Sensors

L. V. Antoshkin, V. V. Lavrinov, L. N. Lavrinova, and V. P. Lukin

Zuev Institute of Atmospheric Optics,Siberian Branch of Russian Academy of Sciences,

pl. Akademika Zueva 1, Tomsk, 634021 Russia

E-mail: [email protected]

Received June 8, 2011

Abstract—High-resolution cameras are used as photodetectors in the image recording plane of aShack–Hartmann wavefront sensor to record centroid coordinates which provide basic information forwavefront reconstruction. A comparative analysis is made of the accuracy of determining the centroidcoordinates for CCD and CMOS cameras. The modes of instantaneous sampling and frame-by-frameaccumulation of information from the cameras are considered.

Keywords: centroid coordinates, photodetector, intensity distribution.

DOI: 10.3103/S8756699012020069

INTRODUCTION

In the late 1960s and early 1970s, photoemissive detectors began to be replaced by solid-state imagingdevices: CCD and CMOS array photodetectors which convert light into an electrical signal. Althoughthey were invented almost simultaneously, CMOS arrays were not regarded as a serious competitor to CCDphotosensors until the mid-1990s [1]. Today, these photosensors are used in all video systems for householdand industrial use, and the use of CMOS photodetector devices is expanding more rapidly than that of moreexpensive CCD image converters [2].

An analysis of common types of photodetector arrays [3–5] shows that CMOS photosensors are stillinferior to CCD converters in image quality (uniformity of background, noise, dynamic range, resolution)but have a number advantages over them:

— a high degree of integration and accommodation of image processing units and interface in a singlechip with the array;

— higher achievable frame rate;— ability to selectively read;— no blooming at full illumination of individual elements of the photodetector array;— low power consumption;— low cost.Continuous improvement of the basic characteristics of photodetector arrays of CMOS devices makes

them comparable to CCD photosensors in characteristics in some ranges of application. Reduction in thecost of component base of video cameras and the advent of ever more powerful means of processing largevolumes of data at a high rate promote the effective use of such video cameras in adaptive optics.

In this paper, we investigate the applicability of CMOS photodetectors in Shack–Hartmann wavefrontsensor in systems for various applications.

146

USING PHOTODETECTORS IN SHACK–HARTMANN WAVEFRONT SENSORS 147

EFFECT OF THE BEAM INTENSITYDISTRIBUTION CHARACTERISTIC ON THE ACCURACY

OF THE DETERMINATION OF CENTROID COORDINATES

In adaptive systems, Shack–Hartmann wavefront sensor, consisting of a lens array and a high-resolutioncamera in the image recording plane, divides a wavefront that arrives at the sensor input aperture into afinite number of subapertures. For each subaperture, the wavefront slope is estimated from shifts of theenergetic centers of gravity of focal spots (centroids).

Since the output signals of CCD and CMOS converters are intensity distributions in the image recordingplane, numerical experiments were performed to study the effect of the beam intensity distribution charac-teristics at the entrance aperture of the adaptive system on the accuracy of the determination of the centroidcoordinates and the measuring range of local angles in the Shack–Hartmann wavefront sensor.

The scenario of the numerical experiments was as follows. A light field with the amplitude in the form ofa Gaussian distribution and a flat phase is divided by a lens array (with 8× 8) subapertures) into individuallight fields which are focused in the detection plane. At each subaperture of size 64× 64 pixels, the angle ofincidence of an individual light field is estimated from displacements of the centroids, whose coordinates arecalculated by the formulas [6]

ξk =ni∑

i =1

iIi

/ ni∑i = 1

nj∑j =1

Iij ; ηk =nj∑

j =1

jIj

/ ni∑i =1

nj∑j =1

Iij ; Ii =nj∑

j =1

Iij ; Ij =ni∑

i =1

Iij , (1)

where Iij is the measured signal intensity of a subaperture with coordinates i, j; ni, nj , k = 1, . . . , 64; k isthe subaperture number; n is the size of the subaperture.

In the experiments, the amplitude of the light field at the entrance aperture of the adaptive system wasspecified in the form of two-dimensional normalized Gaussian distributions:

(1) constant;(2) Gaussian beam as a function of the form f(x, y) = exp(−0.01(x2 + y2));(3) super-Gaussian beam: f(x, y) = exp(−0.01(x2 + y2)4);(4) hyper-Gaussian beam: f(x, y) = exp(−0.01(x8 + y8)).The displacements of the coordinates of centroids forming a matrix with a size corresponding to the

number of lenses in the line of a square-packed array are shown in Fig. 1. The elements of this matrix arethe displacements of the centroid coordinates of the measured wavefront relative to the ideal coordinatescorresponding to the nodes of the computational grid and located on the diagonal of the matrix. The centroidcoordinates are normalized to the size of the computational grid.

Using the displacements of the coordinates denoted by curve 1 as the reference, we note that in thecentral part of the amplitude distributions, the Shack–Hartmann sensor measurements are the closest tothe reference for the hyper-Gaussian beam, and at the periphery, for the Gaussian beam. The amplitudedistribution of the super-Gaussian beam gives the greatest deviation from the reference in both the centralpart and periphery. Thus, for any video cameras, the error introduced in the measurements of the centroidcoordinates due to the effect of the intensity distribution characteristic in the light field must be taken intoaccount when calculating displacements of the centroid coordinates. However, compensation of this erroralso leads to an equivalent reduction in the maximum measured slope angle of the wavefront.

In calculations of the centroid coordinates by formulas (1), the power of the light field entering the regionof the k-th element of the photodetector array is a normalizing factor and has little effect on the accuracyof their determination, but depends greatly on the position of the centroid on the photodetector array (inthe center or on the periphery).

ANALYSIS OF THE ACCURACY OF THE DETERMINATIONOF THE CENTROID COORDINATES FOR VIDEO CAMERAS

BASED ON CCD AND CMOS PHOTODETECTORS

Numerical experiments were carried out to compare the accuracy of the determination of the centroidcoordinates for CCD and CMOS cameras; the accuracy depends on the resulting pixel photoresponse. Itsdynamic range for solid-state photosensors is as follows [7]:

OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING Vol. 48 No. 2 2012

148 ANTOSHKIN et al.

21 3 4 5 6 7 8

0

0.02

-0.02

0.04

-0.04

123

4

n

xn _ xn0

Fig. 1. Displacements of the centroid coordinates for the light field at the entrance aperture ofa sensor having an amplitude distribution in the form of: a constant (curve 1), and Gaussian,hyper-Gaussian, super-Gaussian beams (curves 2, 3, and 4, respectively.

— more than 78 dB for high-performance CCD photosensors;— 66 dB for general-purpose CCD photosensors;— 54 dB for general-purpose CMOS image converters.However, having the worst photoresponses, CMOS photosensors provide a higher frame rate, which is

important for adaptive systems.The pixel photoresponse of CMOS photosensors is not linear, unlike for the photoresponse of CCD

photosensors, and is described by the formula [7]

Uoutput = h ln(if/icurrent), (2)

where if and icurrent are the pixel photocurrent and dark current.For the simulation for each type of cameras, particular methods were developed to calculate the centroid

coordinates based on calculated pixel photoresponses, depending on the optical field intensity. For CCDcameras, the centroid coordinates are found by formulas (1), where the intensities Iij correspond to themeasured values. For CMOS cameras, the intensities Iij are calculated from the photosensitivity data ofCMOS arrays [7].

At different cutoff levels of the output signal for a light field with a normalized super-Gaussian amplitudedistribution , the centroid coordinates for CMOS photodetectors are significantly different from the centroidcoordinates for CCD photosensors (Figure 2). The reduction in the error from 0.15 to 0.30 is due to the largenumber of pixels with intensity values from the considered range for both CCD and CMOS photosensors.

IMAGE CONVERSION BY CCD AND CMOS SENSORSIN A SHACK–HARTMANN WAVEFRONT SENSOR

Analysis of image conversion by CCD and CMOS sensors in a Shack–Hartmann wavefront sensor is basedon optical signal fluctuations due to turbulence simulated by the von Karman spectrum [8]:

Φn(κ) = 0.489r−5/30 (κ2

0 + κ2)−11/6 exp(−κ2/κ2m), (3)

where κ0 = 2π/L0 and κm = 5.92/l0 (L0 and l0 are the external and internal scales of inhomogeneities).The intensity of phase fluctuations is characterized by the Fried radius r0. In the experiments, L0 = 10 m,l0 = 1 mm, and r0 = 5 cm.

The numerical experiment involves the solution of the parabolic equation of quasi-optics, which de-scribes the propagation of a light field with a complex amplitude E = E(ρ, z, t) through the turbulentatmosphere [8]:

2iκ(∂E/∂z) = ∆⊥E + 2κ2n(x, y)E, (4)



0.05 0.25 0.45 0.65

0.5

0.4

0.6

0.3Uoutput

s

Fig. 2. RMS error of σ of the determination of the coordinates of CMOS photodetectors relativeto CCD photosensors.

91 17 25 33 41 49 57 17 25 33 41 49 570

0.8

0.4

0.6

1.0

0.2

1

2

N

(a) (b)Ik

910

0.8

0.4

0.6

1.0

0.2

1

2

N

Ik

Fig. 3. Focal spot intensity distributions obtained using CMOS and CCD photosensors (curves 1and 2, respectively): (a) for optics control, (b) for turbulence correction.

where n(x, y) is the random field of fluctuations of the refractive index induced by the turbulent atmo-sphere (3); κ = 2π/λ is the wavenumber (λ = 0.63 µm is the wavelength); ρ = |ρ|, ρ = (x, y) are thecoordinates in the cross section of the light field.

A Gaussian beam passes through a turbulent screen (3) located at the entrance aperture of the sensorand is then divided by a lens array with a focal length of f ≈ 2 m into subapertures, at each of which thecentroid coordinates are calculated (1).

The error in determining the centroid coordinates by the wavefront sensor limits the range of application ofCMOS photosensors. In optics control, the effect of the turbulent atmosphere is practically absent and focalspots have an ideal radial shape. CCD and CMOS photosensors differ in focal spot intensity distribution.Figure 3 shows distributions of the intensity Ik of the same focal spot at a fixed coordinate ηk correspondingto the center of the subaperture and a varying coordinate ξk. The distributions were obtained using CMOSand CCD photodetectors. The intensities of the focal spot were normalized to the maximum intensity.The difference manifests itself in different values of the power of the light field entering the k-th elementof the photodetector arrays. For the wavefront sensor whose measurements are centroid coordinates, itis important that the coordinates of the energetic center of gravity of the k-th focal spot be different for



Coordinates of the k-th centroid for one implementation and for the average of 32 implementations

ExperimentCCD CMOS CCD CMOS

Optics control Turbulence correction

Single implementation,Vx = 0 m/s

ξk = 0.504641ηk = 0.504545

ξk = 0.50729ηk = 0.506773

ξk = 0.461773ηk = 0.524305

ξk = 0.494908ηk = 0.51443

Average of 32 implementations,Vx = 1 m/s

ξk = 0.504641ηk = 0.504545

ξk = 0.50729ηk = 0.506773

ξk = 0.50508ηk = 0.514237

ξk = 0.507388ηk = 0.5101

Note. Centroid coordinates are normalized to the size of the computational grid; Vx is the transverse componentof the wind speed.

1 65 129 193 257 321 385 4490

2

-2

4

1

3

-4

-1

-3

-5

1

2

3

4

x

W[l]

Fig. 4. Wavefronts: at the entrance aperture of the sensor (curve 2) and reconstructed with adetector using a CCD photosensor (1), with a CMOS photosensor (4) and a CMOS photosensorwhich cuts output signals below the level of 0.02 (see Fig. 2) (3).

different converters. For the wavefront distorted by turbulence, the maximum intensity of the focal spot ismuch lower than unity because its center of gravity is displaced with respect to the center of the subapertureand is close to unity if the center of gravity of the spot coincides with the center of the subaperture, forexample, in optics control. The coordinates of the center of gravity of the k-th focal spot, as seen from thetable, are different for different converters in the presence of turbulent wavefront distortion. The differencesdisappear when the coordinates of the k-th centroid are averaged over the sample obtained by displacing theturbulent screen (3) in the transverse direction in the plane of the lens array.

The differences between the CCD and CMOS photodetectors are illustrated by numerical experimentswhose results in the form of wavefront profiles are shown in Fig. 4.

The accuracy of wavefront reconstruction using a CMOS photosensor is twice that using a CCD photo-sensor. The accuracy of wavefront reconstruction using a CMOS photosensor with a cut-off of the outputpixel signal below the level of 0.02 is only 1.4 times higher than the accuracy of wavefront reconstructionusing a CCD photosensor, and the accuracies are almost the same if the cut-off level of the output pixel is0.3 of the maximum intensity.

EFFECT OF THE CAMERA EXPOSUREON THE ACCURACY OF DETERMINATION OF THE CENTROID COORDINATES

For low light astronomical applications, it is required to increase the light accumulation time (exposure).The camera exposure affects the accuracy of the adaptive optical system. To estimate the error due tothe effect of camera exposure on the accuracy of wavefront reconstruction by a Shack–Hartmann sensor,



1 65 129 193 257 321 385 4490

4

-4

8

2

6

-8

-2

-6

-10

1

2

3

x

W[l]

Fig. 5. Wavefront profiles: at the entrance aperture of the sensor (curve 1), reconstructed in themodes of frame-by-frame accumulation of centroid coordinates (2) and instantaneous sampling (3).

we considered the modes of instantaneous sampling and frame-by-frame accumulation, i.e., the actual in-tegration of high-frequency pulsations of the centroid coordinates due to atmospheric turbulence duringthe frame time (3).

The wavefront profiles (Fig. 5) obtained in the modes of instantaneous sampling of the centroid coordi-nates and frame-by-frame accumulation differ from each other and from the wavefront profile at the entranceaperture of the sensor. In the central part, where the wavefront has the largest slope angle, the profiles arepractically the same for both modes of information sampling, i.e., the general slope of the wavefront profile isthe same for both modes. On the periphery, the difference is due to the presence of high-order aberrations.Averaging of the centroid coordinates or eventually the slope angles leads to the calculation of a highervalue of the wavefront profile (curve 2). The standard deviation of the reconstructed wave front from thedata obtained in the instantaneous sampling mode with respect to the measured wavefront is higher thanthe standard deviation for the wavefront reconstructed from data obtained in the mode of frame-by-frameaccumulation of the centroid coordinates.

CONCLUSIONS

In the absence of turbulent distortions, general-purpose CMOS photodetectors used in Shack–Hartmann wave-front sensors measure the centroid coordinates with an accuracy close to that of CCDphotodetectors [9, 10], i.e., their use is limited to static systems, for example, for measurements of opticalproducts with time averaging in special rooms with no pavilion effects, and for correction of optical laserbeams under stationary conditions.

In adaptive optical systems with atmospheric turbulence correction, the use of general-purpose CMOSphotodectors leads a reduction in the quality of the correction due to their nonlinearity.

The maximum possible camera exposure at a given frequency improves the accuracy of determining thecentroid coordinates using CMOS photodetectors. The error due to the averaging of measurements in theframe-by-frame accumulation mode can be compensated using advanced adaptive correction [11].

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Using photodetectors in Shack-Hartmann wavefront sensors