Comparison of wavefront sensing using subdivision at the aperture and focal planes

Transcription

1 Comparison of wavefront sensing using subivision at the aperture an focal planes Richar M. Clare an Richar G. Lane Department of Electrical an Computer Engineering, University of Canterbury, Private Bag 4800, Christchurch, New Zealan rmc70, Abstract The atmosphere introuces a phase istortion on the incoming wavefront from an astronomical object, causing a speckle image at the groun-base telescope. Wavefront sensing is a set of methos to estimate this phase istortion which can then be use to improve the capture image. The Shack-Hartmann wavefront sensor consists of a lenslet array place in the aperture plane of the telescope which subivies the complex fiel in the aperture plane. The lenslet array can also be use to achieve subivision at the focal plane. We present a Fourier analysis of the latter approach an compare with the Shack-Hartmann sensor. Keywors: aaptive optics, wavefront sensing, Shack-Hartmann sensor. Introuction Groun-base telescopes have long been constraine by the effect of the atmosphere which has egrae the quality of the images forme. The effect of the atmosphere is to prouce a ranom time-varying phase aberration on an incoming wavefront resulting in a speckle image at the groun-base telescope as shown in Fig.. The atmosphere can egrae the resolution of the images by a factor of 0 or more. Recently, aaptive optics [] has prove that the limits pose by the atmosphere can be overcome, with iffraction-limite resolution being emonstrate on large telescopes. Conventional close loop aaptive optics, Fig., uses a wavefront sensor to etect the atmospheric istortion which fees back to a eformable mirror to compensate for this istortion. Alternatively, computer post-processing, Fig., still uses a wavefront sensor to estimate the istortion but has no feeback loop. This eliminates the nee for a eformable mirror, an enables a much cheaper system to be evelope. Deconvolution from wavefront sensing uses this approach [] by combining a set of short exposure images, such as Fig., to prouce a turbulence compensate image of the object, Fig. (c). The most commonly use wavefront sensor in astronomical imaging is the Shack-Hartmann wavefront sensor shown in Fig. 3. The Shack- Hartmann wavefront sensor consists of an array of lenslets place in the aperture plane of the telescope. The lenslet array subivies the complex fiel in the aperture with each lenslet forming a low resolution image of the object. When there is no aberration present these images are focuse onto points irectly below the centre of the respective lenslet as in Fig 3. However, if there is an overall mean wavefront slope over the lenslet then that image is isplace from the centre by an amount proportional to the mean wavefront slope [], Fig 3. The entire wavefront can then be reconstructe from the mean slope measurements across the aperture. Fig. 4(c) shows an alternative metho for wavefront sensing erive from positioning the lenslet array at the focal plane of the telescope. Geometric optics preicts that light travels in a irection perpenicular to the wavefront. If there is no istortion then all light woul arrive at a single point in the focal (or image) plane as shown in Fig. 4. A slope at any point in the aperture causes the light from this point to be eflecte in the focal plane as shown in Fig. 4. When the focal plane is subivie by a lenslet array as shown in Fig. 4(c) light from points in the aperture where the wavefront is unistorte (slope zero) passes through the central lenslet. Light from points in the aperture where the slope is more positive than expecte pass through the lenslet on the right while points where the slope is more negative than expecte pass through the lenslet on the left. The final step in this wavefront sensor configuration is that each lenslet is esigne to form an image of the aperture. Thus in Fig. 4(c), if a pixel in the image on the right is illuminate, this implies that the light passe through the lenslet on the right an hence the slope at this point in the aperture is positive. Conversely, if a pixel on the left image is illuminate this implies that the slope at this point in the aperture is negative. Thus the lenslet array at the focal plane is equivalent to a series of banpass filters on the slope at each point in the aperture. The special case of a array of lenslets is also known as the pyrami sensor[3]. This paper compares the wavefront sensing capabilities for the use in astronomical imaging of the lenslet array at the aperture an focal planes. These two wavefront Palmerston North, November

2 a N (c) Figure : Images of a single bright star capture with a metre telescope. the long term image, single ata frame, an (c) compensate image using econvolution from wavefront sensing with 500 frames. OBJECT * GUIDE STAR Y Z [ \ ] ^ O Q R S T U V W X INCOMING WAVE-FRONT! " # $ % ' ( )* +!, CORRECTOR SECONDARY MIRROR PRIMARY MIRROR BEAM SPLITTER DETECTOR C D E F G I J K L L D M = >? A B WAVE-FRONT SENSOR CONTROLLER : ; < -. / - 0 Figure : Astronomical Imaging Systems: aaptive optics (close loop) an post-processing (open loop). sensing schemes are linke by a Fourier transform relationship since the complex fiel at the aperture an focal planes form a Fourier transform pair. Section iscusses the Shack-Hartmann wavefront sensor in more etail. The basis for wavefront sensing at the focal plane with a lenslet array is mae in Section 3. A iscussion of the uality between the two sensors is mae in Section 4. Subivision at the aperture plane The Shack-Hartmann wavefront sensor subivies the complex fiel in the aperture plane with a lenslet array. Each lenslet forms a low resolution image from the fiel over it. When there is no aberration, Fig. 5, the images are forme irectly below the lenslet. When the wavefront is aberrate with atmospheric turbulence the images are isplace by a istance proportional to the mean wavefront slope over the lenslet, Fig. 5. The wavefront slope in the aperture plane in the x an y irections over each lenslet can be forme by calculating the isplacement of the image in the x an y irections from the unaberrate rest positions. The isplacement is conventionally estimate with the centroi estimator which computes the centre of mass of the image. If the etector consists of an array of finite size pixels of with with _ P` Q) pixels per image the centroi estimator in the x an y irections is ŝ x P pb c Pe Q qb c Qe I _ i ` j f _ p h δ P pb c Pe Q qb c Qe I _ i ` j f f ŝ y a P pb c Pe Q qb c Qe I _ i ` j f _ q h δ vf P pb c Pe Q qb c Qe I _ i ` j f () where _ δ u ` δ v f is the offset from the first pixel to the origin in the focal plane. The wavefront is reconstructe from the centrois as a sum of basis polynomials. For a circular aperture a typical set of basis polynomials are the Zernikes [4]. The Zernike weights, a, are reconstructe via the equation [5]: a a _ K c zz k Θ T K c nn Θf c Θ T K c nn s () where s is a vector containing the centroi measurements. The matrix Θ is calle the interaction uf 88 Image an Vision Computing NZ

3 l m o p o r t o u w x r y p { ˆ Š Œ Š ˆ š ž Ÿ ª ««± ² } ~ ƒ Figure 3: The Shack-Hartmann sensor with a planar wavefront an an aberrate wavefront. The ashe lines are the perpenicular bisectors of the lenslets.» ¼ ½ ¾ À ¾ ½ ¼ ÂÃ Ä ½ íí îî ïï ðð ññ òò ðð ïï îî óô ôõ õö ïï ³ µ ¹ º åå ææ çç èè éé êê éé èè ëë ìì Ø Ú Û Ü Ý Þ ß à Þ â à ã ß Ü ã à â ä Þ Ú à (c) Figure 4: No wavefront aberration at the aperture causes the light to arrive at a point in the focal plane. A wavefront slope at the aperture causes a isplacement at the focal plane. (c) The lenslet array in the focal plane subiviing in one imension an re-imaging the aperture at the conjugate aperture plane. matrix [6] because it maps the slope measurements to the weights of the bases. K zz is the covariance matrix for the Zernike polynomial coefficients, while K nn is the noise covariance matrix. The size of the lenslet etermines the spatial resolution of the Shack-Hartmann sensor. When the resolution is low, only a small number of moes in the atmospheric turbulence can be etermine by the sensor. The smaller the lenslet the smaller a region of the wavefront the slope can be estimate for. The aberrations of a higher orer than tilt over each lenslet cannot be etecte. However, reucing the size of the lenslets in the array reuces the number of photons per lenslet. This results in a reuction in the accuracy with which the slope estimates can be mae. Also if the lenslet size is mae smaller than the Frie parameter, r 0 [7], then iffraction effects mean the spot size increases an consequently the accuracy of the slope estimate ecreases. In practice the optimum for the Shack-Hartmann sensor is when the lenslet size is close to r 0. 3 Subivision at the focal plane The lenslet array subivies the complex fiel in the focal plane an each lenslet forms an image of the aperture in the conjugate aperture plane. The images in the conjugate aperture plane create with a lenslet array at the focal plane are shown in Fig. 6 an for no atmospheric turbulence an with atmospheric turbulence respectively. When there is no atmospheric turbulence the central four lenslets are illuminate equally. 3. Mathematical analysis of the lenslet array at focal plane Mathematically, the presence of a lens in the focal plane performs the inverse Fourier transform Palmerston North, November

4 ø (c) Figure 5: The images prouce from a 8 8 array of lenslets at the aperture plane with a circular aperture when there is no atmospheric turbulence, an when there is atmospheric turbulence. The aberrate phase screen is shown in (c). (c) Figure 6: The images resulting from a 4 4 lenslet array place at the focal plane with a circular aperture when there is no atmospheric turbulence, an when there is atmospheric turbulence. The aberrate phase screen is shown in (c). relationship to prouce an image in the conjugate aperture plane, escribe by co-orinates _ ξ` ηf. The complex fiel in the conjugate plane is a low-pass filtere image of the aperture. The filtering is ue to the finite size of the lens, which we escribe mathematically as a spatial filter H_ u` vf. The resulting image in the conjugate aperture plane, I_ ξ` ηf,isgiven by, [8] I_ ξ` ηf c ø H_ u` vf ù P_ ξ` ηf expû jφ_ ξ` ηf üý þ (3) Equation (3) can be expane by making use of the convolution theorem to prouce I_ ξ` ηf c c ù H_ u` vf ý c ù P_ ξ` ηf expû jφ_ ξ` ηf ü ý þ (4) Using the linearity of the Fourier transform an the convolution operator Eq. (4) simplifies to I_ ξ` ηf h_ ξ` ηf P_ ξ` ηf expû jφ_ ξ` ηf ü (5) The image forme from spatial filtering in the focal plane is thus the convolution of the complex amplitue at the aperture with the IFT of the spatial filter. Since a lenslet traitionally consists of square lenses we assume a square lenslet with a linear imension of centre at a point _ u ` vf in the focal plane. The u h v v k corresponing spatial filter, H_ u` vf, for this lenslet is H_ u` vf a u u k an v h 0 otherwise (6) an the IFT of this spatial filter, h_ ξ` ηf, can be calculate as: h_ ξ` ηf a sinc_ πξ f expû ξ uü sinc_ πηf expû ηvü (7) 90 Image an Vision Computing NZ

5 k h h k The image forme from this lenslet in the focal plane is given by substituting Eq. (7) into Eq. (5), I _ ξ ` ηf sinc_ πξ f expû ξ u ü sinc_ πηf expû ηv ü P_ ξ ` η f expû jφ _ ξ ` ηf ü (8) The image forme from a lenslet in the focal plane is the magnitue square of the convolution of a twoimensional sinc function with the complex amplitue in the aperture. The effect of convolving with the sinc function is to smear the image I _ ξ ` ηf an limit the resolution with which the slopes can be etermine in the aperture. The lobe with of the sinc function is etermine by the with of the lenslet in the focal plane an the phase by the position of the lenslet in the focal plane. Because of the Fourier relationship between the focal an conjugate aperture planes, as the size of the lenslets increase then the with of the main lobe of the sinc function of Eq. (8) ecreases. Thus as the lenslet size increases the spatial resolution improves. 3. Slope filtering It is the linear phase term of the two-imensional sinc function in Eq. (8) that isolates the slopes in the aperture. Assuming the scintillation in the aperture is small an that the phase can be expresse as a pure tilt in the ξ irection only, the complex fiel in the aperture, P_ ξ ` ηf expû jφ _ ξ ` ηf ü, simplifies to expû kξ ü, where k is the coefficient of the tilt aberration. Eq. (8) simplifies to I _ ξ ` ηf sinc_ πξ f expû ξ u ü sinc_ πηf expû ηv ü expû kξ ü (9) In orer to simplify the analysis of the problem we assume that the telescope has a square aperture of imension D. Expansion of Eq. (9) with Euler s ientity an application of the efinition of the convolution integral yiels D I _ ξ ` ηf e η D e ξ expû k_ ξ h c D e η c D ξ f ü e ξ expû ξ _ k u f ü h expû h ξ _ h u f ü ξ expû η _ k v f ü h expû h η _ h v f ü η ξ η (0) where ξ an η are the ummy integration variables. Computing this integral over ξ an η results in I _ ξ ` ηf E i û h 4π E i û h _ η k Df _ h v f ü h E i û E i û _ η k Df _ k v f ü _ η h Df _ h v f ü h E i û h _ ξ h Df _ k k h u f ü E i û h _ ξ k Df _ k k h u f ü E i û _ ξ h Df _ h k k u f ü E i û _ η h Df _ k v f ü _ ξ k Df _ h k k u f ü () where E i _ xf is the Exponential Integral Function efine by E i _ xf a x expû tü t t () The plot of intensity versus the wavefront tilt, k, from Eq. () shows how the lenslet acts as a passban filter on the slopes. The centre of the passban is equal to the lenslet centre, u, an the passban with equal to the with of the lenslet,. When the slope of the wavefront at a given point (ξ ` η) lies in the passban of the slope filter efine by the lenslet then the intensity at the same point in the re-image aperture is approximately one. If the slope at this given point (ξ ` η) lies outsie the passban of the slope filter of the lenslet then the intensity at the same point in the re-image aperture is approximately zero. There is, however, also ringing in the passban of the filter an at the eges of the stopban as can be seen in Fig. 7, an (c). The with of the transition region between the pass an stop bans of the slope filter is etermine by the with of the iffraction-limite spot. The lenslet at the focal plane can therefore tell us the range that the slope at a point in the aperture lies within. As the lenslet size ecreases, an consequently the with of the passban of the slope filter ecreases (Fig. 7 cf. 7), we can etermine the slope to a greater accuracy. 3.3 Reconstruction from lenslet array in the focal plane The wavefront slope estimates in the orthogonal ξ an η irections can be estimate from a weighte sum of the images. The weighting is etermine by the istance from the centre of the lenslet, from which the image came, to the centre of the lenslet array, an hence origin in the focal plane. The slope estimates in the ξ an η irections for an array of M N array of lenslets are given by Palmerston North, November 003 9

6 (c) Figure 7: The slopes filter effect, I _ ξ ` ηf versus k, the magnitue of the wavefront tilt, for a lenslet of with = centre at u =0, with = centre at u =0 an (c) with = centre at u =. φ_ ξ ` ηf ξ φ_ ξ ` ηf η M mb c Me N nb c Ne I m n _ ξ ` ηf _ n h M mb c Me N nb c Ne I m n _ ξ ` η f M mb c Me N nb c Ne I m n _ ξ ` ηf _ m h M mb c Me N nb c Ne I m n _ ξ ` ηf δ u f δ v f (3) where _ δ u ` δ v f is the offset from the origin of the focal plane to the centre of the central lenslet an I m n _ ξ ` ηf is the image forme from the _ m` nf th lenslet in the array. The Zernike weights, a, are again reconstructe via a a _ K c zz k Θ T K c nn Θf c Θ T K c nn s (4) except here Θ is forme from the partial erivatives of the Zernike moes as set out by Noll [4] an s is a vector containing the slopes in the ξ an η irections. 4 Conclusion The resiual error in wavefront sensing is a function of the spatial resolution of the slope estimates an the accuracy with which these slope estimates are mae. It is well establishe that for the Shack-Hartmann sensor there is a trae-off between these two quantities an that the trae-off is etermine by the size of the lenslets in the array. As the lenslet size increases the spatial resolution gets worse an the slope accuracy increases. We have shown in Section 3 that for the lenslet array at the focal plane a similar trae-off exists between spatial resolution an slope accuracy. The inverse relationship is true when subiviing at the focal plane; as the lenslet size increases the spatial resolution improves an the slope accuracy ecreases. Subivision with a lenslet array at the focal plane has the potential to perform better than the Shack- Hartmann sensor. Firstly, in the Shack-Hartmann sensor iffraction effects cause the spot size to increase if the lenslet size is less than r 0 meaning that the size of the lenslets at the aperture plane is effectively limite. This is not the case with the lenslet array at the focal plane. Seconly, the Shack-Hartmann sensor is blin to moes which are even symmetric over each lenslet. The lenslet array at the focal plane can reconstruct these moes. The uality between the formation of the slope estimates from the images when subiviing in the aperture an focal planes shoul also be note. For the Shack-Hartmann sensor the slope over a region (lenslet) is given by the centroi of the corresponing image. For the lenslet array at the focal plane the slope estimate for each region (pixel) is given as a centroi of the aperture images for the corresponing region (pixel). Comparing the two sets of slope estimation formulae, Eq. () an Eq. (3), it can be seen that increasing the number of lenslets (M` N) in the array in the focal plane is analogous to increasing the number of pixels (P` Q) use to etect each image in the Shack-Hartmann sensor. References [] M.C. Roggemann an B. Welsh, Imaging through turbulence, CRC Press, Floria, 996. [] J. Primot, G. Rousset an J.C. Fontanella, Deconvolution from wave-front sensing: a new technique for compensating turbulence-egrae images, J. Opt. Soc. Am. A, 7, pp , 990. [3] R. Ragazzoni, Pupil plane wavefront sensing with an oscillating prism, J. Mo. Opt. 43, pp 89-93, 996. [4] R. Noll, Zernike Polynomials an atmospheric turbulence, J. Opt. Soc. Am. A 56, pp 07-, 976. [5] N. F. Law, R. G. Lane, Wavefront estimation at low light levels, Opt. Comms. 6, pp 9-4, 996. [6] R.G. Lane an M. Tallon, Wave-front reconstruction using a Shack-Hartmann sensor, J. Opt. Soc. Am. A, 3, pp , 99. [7] D. L. Frie, Optical Resolution through a ranomly inhomogenous meium for very long an very short exposures, J. Opt. Soc. Am. A 56, pp , 966. [8] S. Esposito, A. Riccari, Pyrami wavefront sensor behavior in partial correction aaptive optic systems, A. & A. 369, L9-L, Image an Vision Computing NZ