A Study on atmospheric turbulence with Shearing Interferometer wavefront sensor

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1 A Study on atmospheic tubulence with Sheaing Intefeomete wavefont senso M.Mohamed Ismail 1, M.Mohamed Sathik 2. Reseach Schola, Depatment of Compute Science, Sadakathullah Appa College, Tiunelveli, Tamilnadu, India 1 Pincipal, Sadakathullah Appa College, Tiunelveli, Tamilnadu, India. 2 ABSTRACT: When a light beam popagates though the tubulent atmosphee, the wavefont of the beam is distoted, which affect the image quality of gound based telescopes. Adaptive optics is a means fo eal time compensation of the wavefont distotions. In an adaptive optics system, wavefont distotions ae measued by a wavefont senso, and then using an active optical element such as a defomable mio the instantaneous wavefont distotions ae coected. In this pape, the physical backgound of imaging though tubulence, using Kolmogoov statistics, and the Polaization Sheaing Intefeomety techniques to sense and to coect the wavefont abeations with adaptive optics have been discussed. Simulations of the intefeometic ecods wee caied out using Matlab fo the study of abeations in an optical system and the effect of the atmospheic tubulence in the intefeogams. The data was educed fom the intefeogam using Fouie Tansfom Technique and the wavefont was econstucted fom the wavefont slope data. KEYWORDS: Babinet Compensato, Atmospheic Tubulence, Defomable Mio, Scintillation. I. ITRODUCTIO As astonomes attempt to undestand the limits of the physical univese, they must look deep into the night sky with a shap eye. Unfotunately, looking into the night sky is like looking up fom the bottom of a swimming pool. Eath s atmosphee is made up of many layes having diffeent tempeatue gadient, diffeent velocity gadient and also diffeent density gadient. The chaotic and stochastic changes in these popeties of the atmosphee causes a fluid called tubulence o tubulent flow. Tubulence causes the fomation of eddies of many diffeent length scales. Most of the kinetic enegy of the tubulent motion is contained in the lage scale stuctues. The enegy "cascades" fom these lage scale stuctues to smalle scale stuctues by an inetial and essentially in viscid mechanism. This pocess continues, ceating smalle and smalle stuctues which poduces a hieachy of eddies. Eventually this pocess ceates stuctues that ae small enough that molecula diffusion becomes impotant and viscous dissipation of enegy finally takes place. Fig.1: Tubulence Illustation The esolving powe of a telescope when imaging though eath atmospheic tubulence is not popotional to telescope diamete but to the chaacteistic coheence length [1] of tubulence called Fied paamete o. Typically o is the ode of 1 2 centimetes at optical wavelengths at good viewing site. Imaging a distant sta appeas to be point and the wavefont that entes at telescope pupil is plane wavefont in the absence of atmosphee. The plana wavefonts popagate though the eath s atmosphee; the optical path length is attenuated due to andom efactive index Copyight to IJIRSET DOI: /IJIRSET

2 fluctuations. As a esult, the wavefont phase changes spatially and tempoally, so the wavefont is no longe plane. A seconday effect of this phenomenon is scintillation. In Fig.2 shows simulation of a point souce images diffaction limited case and in the pesence of stong tubulence. The intensity is nomalized to the peak intensity of PSF in the absence of tubulence. This light spead ove a lage aea demonstates high esolution and high contast imaging difficult. Fig.2.A: o Tubulence Fig.2. B: Stong Tubulence Thee ae many possible solutions in ode to impove the image quality at the focal plane of the telescope. Those ae space telescopes, speckle intefeomety [2] and adaptive optics [3]. Depending up on the pefomance and cost of the technique, one can choose the best suitable method to minimize atmospheic effects. The best option to minimize the atmosphee effects is launch a telescope into space, but it has its own limitations of launching technology fo big telescopes and cost of opeation. In, speckle intefeomety using blind de-convolution post pocessing methods ae used to impove the image quality. These methods equie shot-exposue images and ae not suited fo vey faint objects. Altenative to above, Babcock [3] suggested a solution with one technique to coect those dynamic distoted wave-fonts called Adaptive Optics (AO. It basically consists of wavefont senso, Defomable mio and contol hadwae. Today this AO has benefitted fom moden technology and high speed compute, which have enabled to coect distotions in eal time. With this cuent technology and using AO the efficiency of a gound based telescope has been geatly impoved. Fo the development of an AO system, it is essential to undestand the chaacteistics of the atmospheic tubulence and its effect on the image quality. II. IMAGIG THROUGH ATMOSPHERIC TURBULECE The phase distotions that aive at the telescope entance ae the cumulative effect of efactive index vaiations though a vetical path in the atmosphee. The efactive index stuctue function is given by equation ( /3 n n (1.1 D ( C ( h [4] The Kolmogoov model of tubulence distotions pescibes the specific fom of the phase stuctue function D ( 6.88( 5/3 (1.2 Copyight to IJIRSET DOI: /IJIRSET

3 Developed fom this Kolmogoov model, the Fied paamete o is.185 cos( ( 6/5 3/5 2 3/5 Cdh (1.3 Whee, is the zenith angle of obsevation and is the obsevational wavelength. Atmospheic tubulence changes tempoally and spatially. Fo long exposue times, aveaged PSF we obseve with imaging object. OTF OTF. OTF effective Telescope Atmosphee (1.4 The atmospheic Optical Tansfe Function (OTF is elated to the statistics of the atmospheic phase abeations, OTF( f exp[.5 D ( f ] (1.5 ow we put this model into long-exposue atmospheic OTF, and we get in the fom: OTFatmosphe f f 5/3 ( exp[ 3.44( ] (1.6 Fom equation(1.6, one can obtain long exposue PSF of atmosphee by taking the Fouie tansfom of OTF. The FWHM of the atmospheic PSF called as seeing [5], is elated to fied paamete o Seeing =.98λ/ o (1.7 At the wavelength of.5 micon, 1 ac second seeing coesponds to o =1.1 cm. And the dependency of fied paamete fom equation.1.3 with wavelength is given by ( ( / 6/5 (1.8 III. GEERATIO OF KOLMOGOROV MODEL OF ATMOSPHERIC TURBULECE Tubulent flow is vey complicated and still it is not entiely undestood. Ove the last hunded yeas, modelling the effects of tubulence on optical popagation has eceived much attention. The focus on statistical modelling [4] has poduced seveal useful theoies. In these theoies, it is necessay to esot to statistical analysis, because it is impossible to exactly descibe the efactive index fo all positions in space and all time. The most widely accepted theoy of tubulence flow, due to consistent ageement with obsevation, was fist put fowad by Andei Kolmogoov [6]. Kolmogoov model assumes that enegy injected into tubulent medium on lage spatial scales (oute scale, L o foms eddies. These lage eddies cascade the enegy into small scale eddies until it becomes small enough (small scale, l that the enegy is dissipated by the viscous popeties of the medium. The inetial ange between inne and oute scales Kolmogoov pedicted a powe law distibution of the tubulent powe with spatial fequency, κ (-11/3. Atmospheic tubulence is a andom pocess. Kolmogoov used stuctue functions to descibe non-stationey andom functions associated with tubulence and its elated paametes of tempeatue, humidity and velocity. Typically a coelation function would be used to descibe the statistics between distances in mateial. Howeve, fo pue Kolmogoov tubulence with an infinite oute scale, the coelation function tends towads infinity as the sepaation between two points goes to zeo. Fo this eason stuctue function has been used. Copyight to IJIRSET DOI: /IJIRSET

4 D ( As such, it is modelled with phase stuctue function, D ϕdefined as the aveage diffeence between two values of a andom vaiable fo a lage numbe of points, with the andom vaiable being the phase ( x, i.e. 2 D ( ( x ( x (2.1 The stuctue function fo efactive index [4, 6] vaiation of tubulent ai in the Kolmogoov model is given by 2 ( ( D ( n n x n x (2.2 D ( C n l L, 2 2/3 n (2.3 2 ( x l( x Whee, l( x nx ( phase of an optical wave, is optical path length, is the efactive index at paticula coodinate x, Cn2 (h is the vetical efactive index stuctue constant which is stongly altitude dependent. Tataski [7] showed the thee dimensional powe spectums, ( of the efactive index vaiations is (.33C 2 11/3 (2.4 Whee, k is the scala wave numbe vecto, ( x, y, z. Kolmogoov tubulence model is valid fo atmospheic tubulence. It is expeimentally poved by ightingale & Busche (1991 [8]. In case of atmospheic tubulence it is sola enegy and wind shea which povides the initial enegy on lage scales and it is dissipated as heat by viscous fiction of the ai at the inne scale [9]. The oute scale is an impotant paamete in tubulence statistics and its ange of values ae much debated in astonomical databases. The standad spectum of Kolmogoov tubulence is usually witten with infinite oute scale and the effect of infinite oute scales is to educe the lowe spatial fequency contibutions. This effect is moe ponounced as the telescope diamete exceeds the size of oute scale. Given that the oute scale is usually 1 m to 1m, many of the futue extemely lage telescopes will have lage diamete than the oute scale. This powe spectum is only valid within the inetial ange between the inne and oute scale [11] as it tends to infinity at lage spatial sepaations. So in ode to accommodate the finite inne and oute scales, the Kolmogoov powe spectum was modified by Von Kaman powe spectum which is given by. (.33 ( exp( / /6 2 2 C i (2.5 Whee, 2 /L 5.92 / l, i and 2 /L. It can be expessed in anothe fom with Fied paamete o, Copyight to IJIRSET DOI: /IJIRSET

5 (.23( D/ 2 2 5/3 exp( / i /6 ( (2.6 Fo infinite oute scale ( and zeo inne scale ( i above equation educes to (.23( D/ 5/3 11/3 (2.7 The Powe spectal density (PSD and phase sceen f( ae elated as ik ( f ( e d 2 (2.8 Fom above equation phase sceen is given by: f ( ( e ik dk (2.9 Whee, f( is 2D - Kolmogoov phase sceen, can be obtained fom invese Fouie Tansfom of squae oot of Von Kaman powe spectum of tubulent atmosphee. When dealing with electomagnetic popagation though the atmosphee, the efactive index can be consideed independent of time ove shot (1μs time scales. Because the speed of light is so fast, the time it takes light to tavese even a vey lage tubulent eddy is much, much shote than the time it takes fo an eddy s popeties to change. Consequently, tempoal popeties ae built into tubulence models though the Taylo fozen tubulence hypothesis. IV. ITERFEROGRAM SIMULATIOS The use of Zenike polynomials fo descibing the abeations intoduced by the atmospheic tubulence is well known. The PSI wavefont senso measues the wavefont slope. oll (1976 [1] has intoduced the integal epesentation and the deivatives of the Zenike polynomial. The deivatives of the Zenike Polynomials can be witten as a linea combination of Zenike polynomial. Hence, the slope infomation fom the wavefont senso can be conveniently expessed as a function of the Zenike polynomial. The gadient of the Zenike polynomial is epesented by (3.1 whee γ jj is called Zenike Deivative matix. Upon poceeding with Zenike coefficients, the intefeogams ae simulated fo diffeent values of the Zenike coefficients epesenting diffeent abeations. Using Matlab, a code fo a staight finge in the intefeomete was developed. An abitay tubulent phase sceen was incopoated into the finge patten due to which the finge patten gets distoted. On the application of a diffeent tubulent phasesceen, the finges gets distoted in a diffeent way. Since, staight the finges cannot be analyzed as such, so we have to conside the one-dimensional plot of the distoted finge patten. The one-dimensional plot of the distoted finge patten was taken into account fo easy analysis. The Fouie Tansfom Technique was applied to the one-dimensional plot to find out the oiginal signal afte emoving the entie unwanted signal. The one- (3.2 Copyight to IJIRSET DOI: /IJIRSET

6 dimensional plot was fast fouie tansfomed to give all the positive and the negative fequencies pesent in the signal. To emove all the highe fequencies and unwanted signals, the powe spectum of the fast fouie tansfomed plot was calculated. Consideing only the fequency that contains the maximum infomation, all the othe fequencies ae neglected. ow the invese fouie tansfom of the above signal that contains the maximum infomation was found out to get back the signal in tems of the spatial co-odinates. Then the phase was unwapped to emove the integal multiple of the 2π uncetainties. Fom the unwapped phase, the Zenike coefficients wee found out using the values the γ jj values fo both the x- and y-vaiables as given by oll. The Coefficient Matix A was finally calculated and the vaious Zenike coefficient values was ecoded fo the diffeent intefeometic pattens. Fig. 3: Repesentation of staight intefeence finge patten in the absence of tubulence with D/ =. Fig. 4: Repesentation of intefeence finge patten with a tubulent phasesceen of D/ =.2. Fig.5: Repesentation of intefeence finge patten with a tubulent phasesceen of D/ =2.5. Copyight to IJIRSET DOI: /IJIRSET

7 Fig. 6: Repesentation of intefeence finge patten with a tubulent phasesceen of D/ =25. V. WAVEFROT RECOSTRUCTIO In this section the econstuction of the wavefont has been calculated, fom the slope data. Afte finding out the slope of the wavefont the wavefont is econstucted using modal appoach. the wavefont is calculated with modal appoach using Zenike basis functions using 21 modes. Fig. 7: Thee Dimensional suface plot of the unwapped phase with a tubulent phasesceen of D/ =. Fig. 8: Reconstucted Wavefont obtained with a tubulent phasesceen of D/ =. Copyight to IJIRSET DOI: /IJIRSET

8 Fig. 9: Reconstucted Wavefont obtained with a tubulent phasesceen of D/ =.2. Fig. 1: Reconstucted Wavefont obtained with a tubulent phasesceen of D/ =2.5. Fig. 11: Reconstucted Wavefont obtained with a tubulent phasesceen of D/=25. VI. COCLUSIO Estimation of the wavefont eos is a vey impotant aspect in adaptive optics. Besides the telescope system eos, the atmospheic tubulence also accounts fo the majo contibution to the eos. The atmospheic tubulence is chaacteized by the Kolmogoov model. It is essential to accuately estimate these abeations in the dynamic situations, in ode to apply, eal time coections. A simulation study of the sheaing intefeometes pove that Copyight to IJIRSET DOI: /IJIRSET

9 the Sheaing intefeomete pefoms bette in the pesence of low Fied paamete and fo Rytov numbes geate than.2. The Fouie theoetical appoach has been applied to the Polaization Sheaing Intefeomete (PSI to establish the basis of the wavefont sensing. Theoetical simulations wee caied out fo visualization of vaious abeations in the Intefeometic finge patten. The study eveals that unde modeate tubulent conditions whee D/ o =.25, the sensitivity of the PSI is not alteed significantly. REFERECES [1] Saha, S. K. (27. Diffaction Limited Imaging with Lage and Modeate Telescopes. Wold Scientific. [2] Saha,S.K.(21. Apetue Synthesis: Methods and Applications to Optical Astonomy, 21 Spinge. [3] H.W. Babcock. The possibility of compensating astonomical seeing. Public. of the Aston. Soc. Pac., 65: , [4] D. L. Fied. Statistics of a geometic epesentation of wavefont distotion. J. Opt. Soc. Am., 55(11: , [5] M. Saazin and F. Roddie. The eso diffeential image motion monito. A&A, 227(1:294 3, 199. [6] A.. Kolmogoov. The local stuctue of tubulence in incompessible viscous fluid fo vey lage Reynolds numbes (tanslation. Poceedings of the Royal Society of London A, 434:9-13, [7] V.I. Tataski. Wave Popagation in a Tubulent Medium. McGaw-Hill, [8]. S. ightingale and D. F. Busche. Intefeometic seeing measuements at the La Palma Obsevatoy. MRAS, 251: , July [9] F. Roddie. The effects of atmospheic tubulence in optical astonomy. In Pogess in Optics, volume 19, pages , [1] R. J. oll, Zenike polynomials and atmospheic tubulence, J. Opt. Soc. Am., 66,27 211, [11] E. Azoulay, V. Thiemann, A. Jette, Kohnle A., and Aza Z. Optical measuements of the inne scale of tubulence. J. Phys. D: Appl. Phys., 21:S41 S44, Copyight to IJIRSET DOI: /IJIRSET