On The Use Of Gaussian Filter Functions For Adaptive Optics

Transcription

1 University of Central Florida Electronic Theses and Dissertations Masters Thesis (Open Access) On The Use Of Gaussian Filter Functions For Adaptive Optics 6 Merfit Assad University of Central Florida Find similar works at: University of Central Florida Libraries Part of the Mathematics Commons STARS Citation Assad, Merfit, "On The Use Of Gaussian Filter Functions For Adaptive Optics" (6). Electronic Theses and Dissertations This Masters Thesis (Open Access) is brought to you for free and open access by STARS. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of STARS. For more information, please contact lee.dotson@ucf.edu.

2 ON THE USE OF GAUSSIAN FILTER FUNCTIONS FOR ADAPTIVE OPTICS by MERFIT ASSAD B.S. University of Central Florida, 999 A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mathematics in the College of Sciences at the University of Central Florida Orlando, Florida Fall Term 6

3 6 Merfit Assad ii

4 ABSTRACT For adaptive optic systems, the use of aperture filter functions calculated using various Zernike modes can be useful in removing lower-order aberrations caused by atmospheric turbulence. Traditionally, these filter functions are calculated using the step function depicting a hard aperture that introduces integrals that are sometimes difficult to integrate and must be done numerically. The Gaussian method can be used in place of the conventional method for calculating the aperture filter functions. Evaluation of the Gaussian approximation for modeling a finite receiver aperture can be made by comparison of reduction in phase variance with results achieved using the conventional method. The validity of Gaussian approximation in this application is demonstrated by the consistency of results between the two methodologies. Comparison of reduction in scintillation by the two methodologies reveals several benefits derived from utilization of Gaussian approximation. The Gaussian approximation produces data that can be interpreted analytically. It further produces greater scintillation reduction. This paper will first examine the use of statistical models for predicting atmospheric turbulence and then the use of Zernike polynomials in adaptive optics. Next, this paper compares the reduction of phase variance and scintillation using the conventional method with the Gaussian approximation to evaluate the effectiveness of the new filter functions. The results of these comparisons are presented both as mathematical expressions and graphically. iii

5 For my beautiful daughter, Jenaan, who inspires me everyday, I dedicate this work in the hope that it may one day inspire her. iv

6 ACKNOWLEDGMENTS If you help others, you will be helped, perhaps tomorrow, perhaps in one hundred years, but you will be helped. Nature must pay off the debt It is a mathematical law and all life is mathematics. G.I. Gurdjieff Nineteenth Century Greek-Armenian Mystic and Spiritual Teacher This work would not have been possible without the assistance and encouragement of my advisors, colleagues, friends, and family. I extend my sincere appreciation and thanks to my advisors, Larry C. Andrews, Ph.D. and Ram N. Mohapatra, Ph.D. They were the first to encourage me to pursue my graduate degree and provided invaluable guidance in the preparation of this work. And for the many others who took the time to assist in my graduate studies and the presentation of this work, I offer my thanks for your time and encouragement. v

7 TABLE OF CONTENTS LIST OF FIGURES... vii LIST OF TABLES...viii CHAPTER INTRODUCTION... 9 CHAPTER MODELING ATMOSPHERIC TURBULENCE... CHAPTER ZERNIKE POLYNOMIALS Zi ( r, θ )... CHAPTER 4 APERTURE FIELD FILTER FUNCTIONS Gi (, φ ) : CONVENTIONAL APPROACH Normalized Zernike Polynomials Calculation of Filter Functions... 8 CHAPTER APERTURE FIELD FILTER FUNCTIONS Gi (, φ ) : GAUSSIAN APPROACH.... Normalized Zernike Polynomials.... Calculation of Filter Functions... CHAPTER 6 APERTURE FILTER FUNCTIONS Fi (, φ ) Aperture Filter Function Graphical Results of Filter Functions... CHAPTER 7 PHASE VARIANCE σ ( L)... 7 s CHAPTER 8 ADAPTIVE OPTICS AND SCINTILLATION INDEX Scintillation Reduction through Tilt Removal Scintillation Reduction through Tilt and Focus Removal Graphical Results CHAPTER 9 SUMMARY... APPENDIX A CALCULATIONS OF FILTER FUNCTIONS... APPENDIX B PHASE VARIANCE CALCULATIONS... 9 APPENDIX C SCINTILLATION REDUCTION CALCULATIONS... 6 REFERENCES vi

8 LIST OF FIGURES Figure Large eddies supply energy to the smaller eddies and these in turn supply smaller eddies, resulting in a cascade of energy from the largest to the smallest ones.... r Figure Gaussian Approximation U( r) e... Figure Graphical comparison of Filter using the conventional filter and its Gaussian approximation... Figure 4 Same as Figure for Filter... 4 Figure Same as Figure for Filter... Figure 6 Same as Figure for Filter Figure 7 Scintillation reduction using tilt removal only... Figure 8 Scintillation reduction using tilt and focus removal... vii

9 LIST OF TABLES Table Zernike polynomials for hard aperture... 7 Table Zernike polynomials for Gaussian approximation... Table Comparison of Aperture Filter Functions... Table 4 Removed phase varianceσ s for Zernike modes from the turbulence-corrupted phase front viii

10 CHAPTER INTRODUCTION Adaptive optics is a technique used to improve the performance of optical systems by reducing the effects of rapidly changing optical distortion [4]. Astronomical telescopes employ adaptive optics to remove the effects of atmospheric distortion. When light from a star enters the earth s atmosphere, turbulence or random fluctuations distort and move the light image in various ways [4]. The wave arriving from the star is essentially a plane wave before entering the earth s atmosphere. Wave distortion occurs as parts of the wave pass through the hotter-thanaverage air (less refractive-index) and are advanced [7]. Other parts of the wave are retarded and the plane wave front is deformed. It is the purpose of adaptive optics to compensate for these distortions. A telescope equipped with adaptive optics measures the wave front distortions generated by the variations of the index of refraction in the atmosphere with a wave front sensor and then applies phase corrections with a deformable mirror. Similar application of adaptive optics can also be used to improve Free-Space Optical (FSO) communication systems that detect optical waves that propagate through the atmosphere. Free-Space Optics (FSO) is a telecommunication technology that uses light propagating in free space to transmit data between two points. The technology is useful where the physical connection of the transmit and receive locations is difficult, for example, in cities where the laying of fiber optic cables is expensive [4]. Other applications for free-space propagation include imaging and laser radar systems. However, the effective use of such technology requires developing an efficient system for resolving wave distortion caused by varying atmospheric conditions. 9

11 In the theoretical analysis of optical wave propagation through atmospheric turbulence, the use of aperture filter functions calculated using various Zernike modes can be useful in removing lower-order aberrations caused by atmospheric turbulence. Traditionally, these filter functions are calculated using the step function depicting a hard aperture which introduces integrals which are sometimes difficult to integrate and must be done numerically. The resulting numerical data does not provide analytical results. In this study we investigate the use of a Gaussian approximation for the step function to the receiving aperture and compare results with the conventional approach. First, we will examine the issue of atmospheric turbulence and a theoretical model for predicting turbulence. Next, we will compare the aperture field functions calculated utilizing the conventional approach with the Gaussian method. Our analysis will reveal the benefits of the Gaussian method greater tractability that leads to analytic results that can provide greater insight.

12 CHAPTER MODELING ATMOSPHERIC TURBULENCE Atmospheric turbulence models based upon statistical analysis provide a basis for predicting atmospheric disturbance and scintillation reduction. Such statistical models are used in conjunction with filter functions to achieve phase reduction. The filter functions are based on Zernike modes (Chapter ) that are traditionally used with a step function depicting a finite size aperture (Chapter 4). In this thesis, we evaluate the use of the Gaussian approximation to the step function which is used in the conventional approach. Once the filter function is established, it can be used with the atmospheric turbulence model to achieve phase reduction (Chapter 7) and scintillation reduction (Chapter 8). Russian mathematician Andrey Kolmogorov suggested a statistical description of turbulent flow. Atmospheric turbulence is generally described as small-scale, irregular air motions characterized by winds that vary in speed and direction [6]. Other types of atmospheric turbulence include temperature fluctuations and index of refraction fluctuations. Refractiveindex variations in the atmosphere produce phase fluctuations in the wave front. Because turbulence is a nonlinear process and is governed by nonlinear equations, Kolmogorov attempted to model turbulence in the atmosphere using a physical or statistical approach. The Kolmogorov theory of turbulence is based on the assumption that wind velocity fluctuations (classical turbulence) are approximately locally homogeneous and isotropic. Homogenous means that the mean value of the field is constant and other statistical quantities in the field depend only on the separation of points being observed. Isotropic means that the statistical quantities are the same in all directions and depend only on the magnitude of the vector

13 separation between observation points random fields []. Kolmogorov proposed a theory to calculate the energy of such a system. The energy cascade theory of turbulence provides a tool for visualizing the structure of atmospheric turbulence. Under the cascade theory, the wind velocity increases until a critical Reynolds number is exceeded and turbulence occurs. The resultant turbulence creates eddies or unstable air masses which, under the influence of inertial forces, break into smaller eddies which eventually dissipate into heat energy. The largest eddy is referred to as the outer scale L of turbulence and the smallest eddy is the inner scalel. Eddies between l and L compose the inertial subrange. This is illustrated in Figure. Energy injection L Energy transfer l Dissipation Figure Large eddies supply energy to the smaller eddies and these in turn supply smaller eddies, resulting in a cascade of energy from the largest to the smallest ones

14 Turbulent atmospheric motion due to the presence of moisture and temperature gradients gives rise to disturbances in the atmosphere s refractive-index n( R ),where R is a vector and R = R is distance between two points. We define these disturbances in the index of refraction as optical turbulence []. Refractive-index variations in the atmosphere produce phase fluctuations in the wave front. The index of refraction is one of the most significant parameters of the atmosphere for optical wave propagation. It is very sensitive to small-scale temperature fluctuations []. It is not realistic to measure the instantaneous three-dimensional turbulent structure in the atmosphere [8]. Therefore, we must rely on a statistical analysis. The statistical description or structure of fluctuations in the refractive-index of the atmosphere is very similar to that for the field of turbulent velocities. The inertial subrange is statistically homogenous and isotropic. The structure function D ( R ) given in () is a spatial description of optical turbulence and is given by n D ( R) =< n n >= C R, l R L () n n where Cn is the index of refraction structure constant (in units of m ) or a measure of the strength of the fluctuations in the refractive-index. The < > symbol denotes the statistical average. For optical wave propagation, fluctuations in the index of refraction are primarily caused by small fluctuations in temperature. The functional form of the spatial power spectrum of refractive-index fluctuations is the same as that for temperature. In addition, temperature fluctuations obey the same spectral laws as velocity [].

15 The spatial power spectrum is a description in terms of wave number. Based on the power law behavior of the structure function () in the inertial subrange, the Kolmogorov spatial power spectrum model φn ( ) for the index of refraction is φn( ) =. Cn, () L l where is wave number. The Kolmogorov power-law spectrum is commonly used in theoretical calculations. However, this spectrum model is valid only over the inertial subrange L l. In order to justify the use of the Kolmogorov power-law spectrum in calculations over all wave numbers, it is assumed that the outer scale is infinite ( L = ) and the inner scale is negligibly small ( l = ). However, this may lead to divergent integrals. Therefore, other spectrum models have been used when inner scale and outer scale effects cannot be ignored. These spectrum models are often modified so that the turbulence is modeled as if it were statistically homogenous and isotropic for all wave numbers. One of those spectrum models which includes both inner scale and outer scale effects is the modified von Karman spectrum which is defined as m e n( ). Cn, 6 ( + ) φ = < () where = and m =. L l 4

16 CHAPTER ZERNIKE POLYNOMIALS Zi (, r θ ) The Zernike polynomials were originally used to represent fixed aberrations in optical systems, but are now also used in adaptive optics systems designed for atmospheric turbulence decomposition. The Zernike polynomials represent a set of functions of two variables that are orthogonal over a circle with unit radius. These polynomials are the product of two functions, one depending only on a radial coordinate r and the other depending only on the angular coordinateθ,i.e., m m im Z (, r θ ) = R () r e θ (4) n n where both m and n are integers, n, n m n, and n± m is even. The angular functions are the basis functions for the two-dimensional rotation group, and the radial polynomials are a m special case of Jacobi or hypergeometric polynomial that are normalized so that R () =. It is defined by n R m n () r = ( ) ( nk)! r () k = n+ m nm k! k! ( n m) k nk In the theoretical analysis of adaptive optics systems the Zernike polynomials are normalized over a unit radius aperture so that

17 π π ru( r) Zi ( r, θ) dθdr =, i =,, (6) where we define Z r Z n n R r m i(, θ ) = i[, ] = + n (), = m Z (, r θ) = Z [ m, n] = n + R () r cos mθ, i even ( m ) i i n m Z (, r θ) = Z [ m, n] = n + R () r sin mθ, i odd ( m ) i i n (7) and where U( r) is the step function defined by, r < U( r) =, r > (8) Using this definition, the first 6 Zernike polynomials are given in Table. They are arranged by ( mn, ) pair and indexed byi. Table contains all the Zernike polynomials through( mn, ) = (,). Note that for cases where m > there are two Zernike polynomials for each( mn, ) pair which represent the x and y components (of the Zernike mode). 6

18 Table Zernike polynomials for hard aperture i m n Zernike Polynomial Zi[ mn, ] Description Piston r cosθ Tilt about x axis r sinθ Tilt about yaxis 4 (r ) Defocus 6r sinθ Astigmatism 6 6r cosθ Astigmatism The Zernike polynomials given in Table are a modification of the Zernike polynomials that are commonly used in studying atmospheric effects on imaging systems. Each term of the Zernike polynomials minimize the residual mean square (rms) wave front error to the order of that term. The first Zernike mode, Z [,], is referred to as piston. Physically, piston corresponds to the aperture averaged wave front phase. The Zernike modes Z [,] and Z [,] are referred to as tilt. Tilt does not affect image quality, but does cause the image to be displaced. Other Zernike modes with well known names include Z 4 [,], which is called defocus; and Z [,] and Z [,] 6 which are the orthogonal components of astigmatism. 7

19 CHAPTER 4 APERTURE FIELD FILTER FUNCTIONS Gi ( φ, ): CONVENTIONAL APPROACH 4. Normalized Zernike Polynomials The Zernike polynomials appearing in Table are already normalized to satisfy the condition on the unit circle given by (6). 4. Calculation of Filter Functions The two-dimensional Fourier transform of the Zernike polynomials, scaled by the area of the aperture, is given by irk Gi( φ, ) = e Zi( r, θ) U ( r ) dxdy π = π π ircos( θφ) e Zi (, r θ) rdθdr (9) where r =< x, y > and =<, > are two-dimensional vectors in the spatial and wave number x y domains, respectively. U( x) is the step function depicting the finite size of the aperture. In (9) Gi ( φ, ) reduces down using r = x + y = rcos( θ φ) () x y 8

20 since x= rcos θ, = cosθ y = rsin θ, = sinθ x y () Substituting the Zernike polynomials into (9), we obtain G i( ) π n + ircos( θφ) m Gi, even( φ, ) = e rrn ( r) cosmθ dθdr π () Giodd, ( φ, ) sinmθ Filter : We will calculate the first few filter functions using the conventional approach. To compute the aperture filter function corresponding to piston we substitute m = and Z [,] = into the integral above. From this we have R () r = or π ircos( θφ) = = π, () G ( ) r e dθdr rj ( r) dr or, upon evaluation, ( ) G ( ) J = (4) where J n( ) is the Bessel function of first kind of order n. 9

21 Filter : For the case of tilt corresponding to Z [,] = rcosθ, substituting into () gives us π ircos( θφ) G, even( φ, ) r e cosθdθdr π = () To evaluate the inside integral in(), we use the identity ircos( θφ) k e J r i Jk r k k= = ( ) + ( )cos ( θ φ) (6) from which we obtain π π π ircos( θφ) k e cos θdθ = J ( r)cosθdθ + i J ( r)cos k( θ φ)cosθdθ k= π π k k k= = J ( r) cosθdθ + i J ( r)coskφ cos kθcosθdθ k π k k k= + ij( r)sinkφ sinkθcosθdθ (7) and hence π ircos( θφ) e d = ij r cosθ θ π ( )cosφ (8)

22 The integral in () reduces to, even = G ( φ, ) 4icos φ r J ( r) dr = J ( ) 4i cos φ (9) The remaining filters can be calculated in a similar manner. It has been shown that the aperture filter functions in the general case are given by n G ( ) i( ) J n ( ) + ( nm) m Gi, even( φ, ) = n + ( ) i cosmφ ( nm) m Giodd, ( φ, ) ( ) i sinmφ () The values of m and n are determined by the choice of Zernike polynomial given in Table.

23 CHAPTER APERTURE FIELD FILTER FUNCTIONS Gi ( φ, ): GAUSSIAN APPROACH The step function U( x ) introduces a Bessel function in the G (, φ ) which is difficult to use in the theoretical analysis of wave front aberrations and consequently are handled by numerical analysis only. Rather than using a hard aperture or a finite size of the aperture of r radius, we can use a Gaussian approximation U( r ) e (see Figure ). i Figure Gaussian Approximation U( r) e r We arrive at this by first assuming U( r) Ae αr ()

24 where the constants A and α must be determined. Since e = radius, we have α =. Integrating both sides we have π π r π A ru( r ) drdθ = A re drdθ = () Hence, based on the equivalent area under the step function value, we set A =. In this case the filter function is defined by π ircos( θφ) r Gi( φ, ) = e Zi( r, θ)e rdθdr π (). Normalized Zernike Polynomials The Zernike polynomials appearing in (9) must be normalized differently when using the Gaussian approximation so that the normalizations satisfy a condition on the unit circle similar to that given by (6) where U( r) is replaced by r e. Using the definition of the Zernike polynomials given in Table, the normalization found here is defined by π r i (, θ) θ,,,,... (4) Ci = re Z r d dr i = π

25 where Ci is not necessarily unity. We introduce a multiplicative constant Ai in each polynomial so that Ci =, i =,,.... The first Zernike polynomial is simply a constant. For Z[,] = A, we find π r = ( ) π C re A dθdr r = 4( A) re dr = ( A) from which we deduce A =. For Z[,] = Ar cosθ, we get π r = ( cos θ) π C re Ar dθdr ( A ) = π π r r(4 r ) e dr cos θdθ r = 8( A ) r e dr = ( A ) and conclude A =. For i =, 4,,6 we obtain the normalized Zernike polynomials for the Gaussian aperture. These are listed in Table (see Appendix A for details). 4

26 Table Zernike polynomials for Gaussian approximation i m n Zernike Polynomial Zi[ mn, ] Description Piston r cosθ Tilt about x axis r sinθ Tilt about yaxis 4 r Defocus r sinθ Astigmatism 6 r cosθ Astigmatism. Calculation of Filter Functions In this section, we illustrate the development of several filter functions. The others can be developed in a similar manner. Filter : The first aperture filter function is defined by π cos( ) (, ) ir θφ r (, ) π G φ = e Z r θe rdθdr () The associated Zernike polynomial is [ ] integral(), we obtain Z, = which represents piston. Evaluating the inside

27 π ircos( θφ) e dθ = πj r ( ) (6) Then = r (7) G ( φ, ) G ( ) 4 re J ( r) dr which simplifies to (see Appendix A) e 8 G ( ) = (8) The filter function given by (7) is for an aperture of radius one. Converting to an arbitrary size D aperture diameter D, we can simply replace by, from which we obtain D G ( ) = e (9) Filter : The second filter is computed similarly and is used to remove tilt. It is defined by 6

28 π ircos( θφ) r (, ) = (, ) π G φ e Z r θe rdθdr 4 = π π cos( θφ) r ir e rcosθe rdθdr () If we use the identity ircos( θφ) k e J r i Jk r k k= = ( ) + ( )cos ( θ φ) () then the inside integral in () becomes π π ircos( θφ) k e (cos θ ) dθ = J( r) + i Jk ( r)cos ( θ φ) cosθdθ k= () = πij ( r)cosφ Hence, () reduces to 8π i r π G ( φ, ) = r e J ( r)cosφdr () Upon evaluation, we are led to (see Appendix A) G (, ) i 8 cos φ e = φ (4) 7

29 There is also a similar y component for tilt given by G (, ) i 8 sin φ e = φ () Recall the corresponding filter functions from () are G ( ) J ( ) = ( 6) G ( ) 4 J ( ) icosmφ = ( 7) and G ( ) 4 J ( ) isinmφ = ( 8) 8

30 CHAPTER 6 APERTURE FILTER FUNCTIONS Fi (, φ ) The G ( )' s are field filters for the removal of the Zernike modes. For phase and i intensity variances, the magnitude squared of the field filters is needed. The filter functions that we need for atmospheric effects are certain combinations of the squared magnitude of the Gi ( ) filters defined by F( ) = G ( ) (9) i i However, because piston corresponds to i = and tilt components include i =, we will adopt the notation Fpiston( ), Ftilt ( ) and so on to avoid confusion. In the analysis below, we will derive these filter functions based on the Gaussian approach. The comparable filter functions based on the conventional approach will be summarized below in Table. 6. Aperture Filter Function The first filter corresponding to the piston mode of the Zernike polynomials is given by ( piston ) ( ) ( ) F F = G (4) or, using (8), Fpiston D 6 ( ) = e (4) 9

31 Since tilt has both an x and y component, we sum the corresponding filters to find F ( ) = F ( ) + F ( ) tilt ( ) G( ) = G + (4) which leads to F tilt D D 6 ( ) = e (4) 6 In a similar manner, we find 4 D D 6 Ffocus( ) = G4 ( ) = e (44) 4 and 4 D 6 Fastigmatism( ) = G( ) + G6( ) = e (4) 4 D Note that as a consequence that the Zernike polynomials for focus and astigmatism are all quadratic in r, the resulting filter functions (44) and (4) are the same. The aperture filter

32 functions derived using the Gaussian approach are listed in Table below along with the corresponding filter functions found using the conventional approach.

33 Table Comparison of Aperture Filter Functions Filter Conventional Gaussian F ( ) piston 6 J ( D ) D e D 6 Ftilt ( ) 64 J ( D ) D D e 6 D 6 F ( ) focus 48 J ( D ) D 4 D e 4 D 6 F ( ) astigmatism 96 J ( D ) D 4 D e 4 D 6 6. Graphical Results of Filter Functions For a graphical representation of the aperture filter functions, refer to Figures - 6. For the purpose of comparison, D = was chosen. The curve for Filter derived using the Gaussian approach matches closely to the curve for Filter obtained using the conventional approach (see Figure ). The curve for Filter derived using the Gaussian approach starts to differ from that of the curve for Filter obtained using the conventional approach. There is a more apparent difference between the filters for the focus mode. As we move to higher-order modes, the difference between the conventional filter and the Gaussian filter is greater as illustrated in Figure 6 where the astigmatism mode shows the largest difference. However, when using these filters to predict phase and scintillation variance, the integration will create a smoothing effect for the Gaussian function.

34 .9 Aperture Filter Functions - Filter (Piston) Conventional Gaussian Kappa Figure Graphical comparison of Filter using the conventional filter and its Gaussian approximation

35 .9 Aperture Filter Functions - Filter (Tilt) Conventional Gaussian Kappa Figure 4 Same as Figure for Filter 4

36 .9 Aperture Filter Functions - Filter (Focus) Conventional Gaussian Kappa Figure Same as Figure for Filter

37 .9 Aperture Filter Functions - Filter 4 (Astigmatism) Conventional Gaussian Kappa Figure 6 Same as Figure for Filter 4 6

38 CHAPTER 7 PHASE VARIANCE σ ( L) The phase front is a surface of constant phase. For an infinite plane wave the phase front stays planar as the wave propagates in a vacuum. In the atmosphere, however, the phase front becomes distorted in a random fashion. The variance of such a phase front at a single point in the receiver plane based on a geometrical optics approximation is s L s k σ = 4 π φ ( ) ddz (46) n where φn ( ) is the atmospheric power spectrum. For the Kolmogorov spectrum, the integral in (46) diverges because of the singularity at =. Hence to evaluate (46), we use the von Karman spectrum m Cn e 6. φ ( ) =, n ( + ) (47) where, m, L is the outer scale of turbulence, and l is the inner scale of turbulence. L l Hence (46) becomes m e s = 4(.) klcn 6 σ π d (48) ( + ) 7

39 To evaluate the integral we use the following identity [] µ m 8 e µ d = Γ µ + U µ + ; µ ; 6 m (49) ( + ) where Uabxis ( ; ; ) the confluent hypergeometric function of the second kind. Then from (49), m e d U ; ; 6 6 m = () ( + ) Note that l m L. Consequently, we can replace the confluent hypergeometric function by its small argument asymptotic form. Using the small argument property associated with the confluent hypergeometric functions below, Γ( c) Γ( c) Uacz z z Γ ( + ac) Γ( a) c ( ; ; ) +, () we have 8

40 Γ Γ 6 6 U(; ; ) 6 () Γ m Γ m () Then () reduces to m e d +Γ m 6 6 () ( + ) Because phase fluctuations are associated primarily with large turbulent scales, we can ignore inner scale effects. Hence if we take the limit as l ( ) m, we obtain. σ 4 π.4 6 s = klcn +Γ m L.86, = r (4) where ( ) r C k L =.4 n is Fried s parameter that describes the atmospheric coherence width. Larger values of r correspond to better seeing. We will now compute the phase variance reduction associated with the different Zernike modes. The reduced or corrected variance is given by L N σ scorr, = 4 π k φn( ) - Fj( ) ddz ( ) j= 9

41 Piston Removed: The phase variance with the piston mode removed due to a finite aperture diameter D is L D 6 σscorr, = 4 π k φn( ) e ddz L L D 6 n n = 4 π k φ ( ) ddz4 π k φ ( ) e ddz (6) which we write as σ L s, corr =.86 σs, piston r (7) where we have used the result in (4). The second integral in (6), σ exponential function becomes s, piston, involving the 6 D m D 6 + m 6 d + Γ 6 6 m e (8) ( + ) where we have used the result of (). Then 4

42 D m 6 e Γ + 6 d = + 6 m m ( + ) 6 m D e Γ 6 6 (9) Again we take the limit as l ( ) m, from which we deduce D m 6 e e D 6 d =Γ 6 6 (6) 6 ( + ) Consequently, the piston-removed phase variance in (7) reduces to σ L L D scorr, = r r r D =. r (6) Tilt Removed: Recall when computing the variance and piston-removed variance we saw that a singularity exists at =. Therefore, the von Karman spectrum, which is finite at the origin, was used for convergence of the integral. All other filters move the region of interest away from the origin. Hence we can now use the Kolmogorov spectrum which is given by 4

43 n φ ( ) =.C (6) n The total tilt filter F ( ) includes equal contributions from both x and y directions. The phase variance for removing the tilt effect is tilt s, tilt = n tilt σ 4 π klφ ( ) F ( ) d D D 6 = 4 π klφ( ) e d 6 (6) Using the Kolmogorov spectrum gives us D D 6 s, tilt 4 (. n ) 6 σ = π kl C e d (64) which reduces to a Gamma function and further reduces to (See Appendix B) σ s, tilt D =.87 r (6) For F ( ), the x and y directions have equal effect. Then after subtracting the phase effect tilt from(6), we have σ scorr, D =.9 r for the cosine portion andσ scorr, D =.64 for the r sine portion. 4

44 The phase variance for removing the effect of focus and astigmatism were obtained in a similar way (see Appendix B). They are Focus removed: σ scorr, D =.9 r (66) and Astigmatism removed: σ scorr, D =.94 r (67) In Table 4 expressions for σ s are shown for the first 6 Zernike modes, where D is the diameter of the circular aperture and r is the atmospheric coherence length. Note that the greatest reduction in the varianceσ s, corr of the piston-removed phase is due to Z and Z, the tilt modes of the turbulence. Also, as i increases, the reduction in σ s which results from removing additional Zernike modes is small. The higher order Zernike modes make an additional small contribution to the residual mean square phase error. 4

45 44 Table 4 Removed phase variance s σ for Zernike modes from the turbulence-corrupted phase front Mode index,i Conventional Gaussian Approximation. D r. D r.8 D r.9 D r.4 D r.64 D r 4. D r.9 D r.88 D r. D r 6.6 D r.94 D r

46 CHAPTER 8 ADAPTIVE OPTICS AND SCINTILLATION INDEX Atmospheric turbulence causes phase variations in optical communication systems which are seen as random variations in intensity. Intensity fluctuations or scintillation is partially caused by phase variations due to the turbulent atmosphere. The scintillation index σ I is calculated from the parameters of the propagation path, the wavelength, and the spatial covariance function appropriately filtered for removal of aberration modes by the adaptive optics system. For all propagation paths the scintillation index is the normalized variance of the intensity. The scintillation index for a plane wave without adaptive optics known as the Rytov varianceσ is given by I Lξ σ I = 8 π kl φn( ) cos ddξ k n =.C k L (68) where L = propagation path length and we have assumed Cn is constant along the path. 8. Scintillation Reduction through Tilt Removal By removing an increasing number of Zernike modes, we can assess the potential increase in performance of the optical system. Removing N Zernike modes leads to 4

47 N Lξ σi, corrected = 8 π kl φn ( ) cos - F( j ) ddξ (69) k j= where Fj( ) = Zernike spatial filter function. If we split (69) into a difference of integrals, the first set of integrals are exactly σ I given by (68) and we are left with N Lξ σi, corrected = σi 8 π kl φn ( ) F( j ) cos ddξ (7) j= k We start the summation in (69) with j = since j = corresponds to piston which is equivalent to aperture averaging due to a large collecting aperture of diameter D. The scintillation index with only tilt removed j = is given by σ I, corrected =.Ck n L σi, tilt 7 Lξ =.Ck n 6 L 6 8 π kl φn( ) cos F ( ) ddξ k (7) 7 D 6 6 Lξ D 6 =.Ck n L 8 π kl φn( ) Re exp i e d dξ k 6 =.Ck n L ( I I ) Looking at the first integral I in the expression for tilt gives us 46

48 D D 6 I = 8 π kl φn ( ) e d dξ 6 D 6 8π.Cn = kld e d d ξ 6 (7) D =.7 r The second integral I is computed in the same way as I. It is given by D Lξ D 6 I = 8 π k L φn ( )Re exp i e ddξ k 6 8 π.c n D 6 Lξ = kldre exp i ddξ kd D kd 6L =.6 + sin tan r 6L kd 6 6L kd (7) Hence the scintillation index after tilt removal is σ I, corrected =.Ck n L σi, tilt n =.Ck L D kd 6L 6L r 6L kd 6 kd.7 (.) + sin tan (74) 47

49 8. Scintillation Reduction through Tilt and Focus Removal The scintillation index with both tilt j = and focus j = removed is given by I, corrected =.Ck n L I, tilt I, focus σ σ σ n σ I, tilt =.Ck L π kl φ F ddξ Lξ 8 n( ) cos ( ) k n σ I, tilt =.Ck L 4 4 Lξ D 8 π kl φn( ) Re exp i k n σ I, tilt =. Ck L ( II) e D 6 ddξ ( 7) The first integral I in ( 7) gives us 4 4 D D 6 I = 8 π kl φn ( ) e ddξ 4 4 D 4 6 8π.Cn = kld e d d ξ 6 ( 76) D =.74 r The second integral I is given by 48

50 I = 8 π k L φ ( )Re exp i e ddξ 4 4 D Lξ D 6 n k 4 4 Cn D L Re exp 4 6 kd D kd 6L 6L sin tan r 6L kd 6 kd 8 π. 6 ξ = kld + i ddξ =.74(6) + ( 77) Hence the scintillation index after tilt and focus removal is I, corrected =.Ck n L I, tilt I, focus σ σ σ n σ I, tilt =.Ck L D kd 6L 6L r 6L kd 6 kd.74 (6) + sin tan ( 78) 8. Graphical Results Graphical results illustrate that filters derived using the Gaussian approximation gives a larger reduction of scintillation than the conventional filter (as seen in Figures 7-8). Also, we see the additional reduction in scintillation is small as more Zernike modes are removed 49

51 .7 Scintillation Reduction: Tilt Removal.6. Scintillation Index σ ( L) σ BesselTilt ( L).4. σ GaussianTilt ( L) σ ( L) Figure 7 Scintillation reduction using tilt removal only.7 Scint. Reduction: Tilt and Focus Removal.6. Scintillation Index σ ( L) σ BesselTiltFocus ( L).4. σ GaussianTiltFocus ( L) σ ( L) Figure 8 Scintillation reduction using tilt and focus removal

52 CHAPTER 9 SUMMARY Gaussian approximation is a reasonable method for determining filter functions that can be used to remove lower order aberrations. Comparison with the conventional method for computing the filter functions demonstrates the validity of the Gaussian method in this application. Analysis of removed phase variance for Zernike modes from the turbulencecorrupted phase front (Table 4) demonstrated that results of the Gaussian method approximate those of the conventional method. A greater difference in what each method predicts is seen in higher-order Zernike modes Comparison of the Gaussian and conventional methods also demonstrated several advantages of the Gaussian method. The Gaussian method leads to analytical expressions that can be applied to phase variance reduction and scintillation reduction. In scintillation reduction, the Gaussian method predicted greater reduction. It is not known at this time whether this is as good a model as that obtained by conventional analysis. Experimental results are not available to make such a comparison. Finally, although the Gaussian method is evaluated here using a plane wave, it can also be applied to spherical waves and Gaussian beam waves.

53 APPENDIX A CALCULATIONS OF FILTER FUNCTIONS

54 In this appendix, more detail is provided for the calculation of the aperture filter functions. Filter : The first aperture filter function which represents the piston mode is defined by π ircos( θφ) r (, ) (, ) π G φ = e Z r θ e rdθdr ( 79) Using the identity π ircos( θφ) e dθ = πj r ( ), G ( ) reduces to r G ( ) = 4 re J ( r) dr ( 8) where J ( r) is the Bessel function of the first kind of order zero and is defined by n= ( n!) n n r ( ) J ( r) = ( 8) Substituting ( 8) into the equation for G ( ) gives us n n ( ) n r G ( ) 4 + r e dr n= ( n!) = ( 8)

55 By letting t = r, ( ) G reduces to G ( ) = 4 n= n ( ) n n t t e dt n n= ( n!) n n ( ) 4 Γ ( n + ) n n (!) ( 8) Since n t n t e dt Γ ( ) =. To further simplify we use the property Γ ( n+ ) = n! which leads to n ( ) 8 G ( ) = 4 ( 84) n! n= n which is the series representation for the exponential function or e 8 G ( ) = ( 8) Therefore, the aperture filter function is given by Fpiston 4 = e ( 86) 4

56 Filter : The second filter is computed similarly and is used to remove tilt. It is defined by π ircos( θφ) r (, ) = (, ) π G φ e Z r θ e rdθdr π π ircos( θφ) r e r e rd dr cosθ θ ( 87) Using the identity ircos( θφ) k e J r i Jk r k k = = ( ) + ( )cos ( θ φ) ( 88) Then π π π ircos( θφ) k e cos θdθ = J ( r)cosθdθ + i J ( r)cos k( θ φ)cosθdθ k = π π k k k = = J ( r) cosθdθ + i J ( r)coskφ coskθcosθdθ k π k k k = + ij( r)sinkφ sinkθcosθdθ ( 89) Hence we are left with π ircos( θφ) e d = ij r cosθ θ π ( )cosφ ( 9)

57 In arriving at our result, we have used the identity cos( A B) = cos Acos B+ sin Asin Band the orthogonality properties π result in ( 9) G ( ) simplifies to cos kθcos nθ =, n k and π cos kθsin nθ =, all n and k. Using the r G ( ) = 8 r e J ( r)cosφdr ( 9) where J ( r) is the Bessel function of the first kind of order one and is defined by n= n+ n r ( ) J( r) = ( 9) n!( n+ )! Substituting ( 9) into the equation for G ( ) gives us n+ n ( ) n+ r G ( ) 8i cosφ r e dr = ( 9) n= ( n!) Using the definition of the Gamma function, G ( ) reduces to 8 φ ( ) G cos = i e ( 94) The y component for the tilt mode is derived using the same procedure and is defined by 6

58 8 φ ( ) G sin = i e ( 9) Taking the sum of the squared magnitudes gives us F ( ) = G ( ) + G ( ) tilt = e 4 4 ( 96) for the total tilt effect. 7

59 Normalization for Gaussian Approximation: π r θ π C = r e (Arsin ) dθdr = ( A ) = π r r(4 r ) e dr sin θdθ r = 8( A ) r e dr = ( A ) = π ( 97) π r 4 4 π C = r e ( A (r ) dθdr = π 4 r r(r ) e dr dθ ( 98) 6( A ) = π = ( A4) = ( A4) = π r θ π C = r e ( 6Ar cos ) dθdr = ( A ) = π r re dr cos θdθ ( 99) = ( A) * ( A) = 8 π π r 6 6 θ π C = r e ( 6Ar sin ) dθdr = ( A ) = π 6 r re dr sin θdθ ( ) = ( A6) * ( A6) = 8 π 8

60 APPENDIX B PHASE VARIANCE CALCULATIONS 9

61 variance. In this appendix, more detail is provided for the calculation of each mode reduced phase Tilt-removed phase variance: s, tilt = n tilt σ 4 πklφ ( ) F ( ) d D D 6 n 6 = 4 π klφ ( ) e d ( ) Substituting in the Kolmogorov spectrum give us D D 6 stilt, = 4 (. n ) 6 σ π kl C e d D = 4 π kl(. Cn ) Γ 6 6 D ( ) Then we have σ s, tilt D =.87 r ( ) For F ( ), the x and y directions have equal effect. Then after subtracting the phase effect tilt from(6), we have σ scorr, D =.9 r for the cosine portion andσ scorr, D =.64 for the r sine portion 6

62 Focus-removed phase variance: s, focus = n focus σ 4 πklφ ( ) F ( ) d 4 4 D D 6 n = 4 π klφ ( ) e d 4 ( 4) Substituting in the Kolmogorov spectrum gives us D D 6 s, focus = 4 (. n ) 6 σ π kl C e d = Γ D π kl(. Cn ) 4 6 D ( ) Then the phase effect for the focus mode is σ s, focus D =. r ( 6) and the focus-removed phase variance is given by σ scorr, D =.9 r ( 7) 6

63 Astigmatism-removed phase variance: The phase variance for astigmatism is calculated using the same phase reduction integral as that for focus given by ( 4) since the aperture filter functions are equal. However, as with Ftilt ( ), the x and y directions have equal effect for astigmatism. Therefore, the phase effect σ s, astig D =. r gives us σ scorr, D =. r for the cosine portion andσ scorr, D =.94 for the sine portion r 6

64 APPENDIX C SCINTILLATION REDUCTION CALCULATIONS 6

65 In this appendix, additional steps are provided for calculation of the scintillation reduction with tilt removal and reduction with tilt and focus removal. Tilt-reduced scintillation: The scintillation index with only tilt removed j = is given by Lξ σi, corrected =.Ck n L 8 π kl φn( ) cos F ( ) ddξ k n D Lξ D 6 =.Ck n L 8 π kl φn( ) Re exp i e d dξ ( 8) k 6 =. Ck L ( II) Looking at the first integral I in the expression for tilt gives us D D 6 I = 8 π kl φn ( ) e d dξ 6 D 6 8π.Cn = kld e d d ξ 6 ( 9) D =.7 r The second integral I is computed in the same way as I. It is given by D Lξ D 6 n k 6 I = 8 π k L φ ( )Re exp i e ddξ 8 π.c n D 6 Lξ = kldre exp i ddξ kd () 64

66 Taking only the integral with respect to, we have 7 Γ( ) D 6Lξ exp i d 6 + = 6 kd 6 D 6Lξ + i 6 kd Lξ 6 ( ) i = Γ + D 6 kd ( ) Substituting back into () give us 8 π.cn 6L 6 = 6 Γ Re 6 + kd ξ I k LD i dξ π.c 6 n 6 6 kd L kd = kld6 Γ Re i + 6 6iL kd 6iL D kd 6L =.6 + sin ta n r 6L kd 6 6 6L kd ( ) Hence the scintillation index after tilt removal is σ I, corrected =.Ck n L σi, tilt n =.Ck L D kd 6L 6L r 6L kd 6 kd.7 (.) + sin tan ( ) 6

67 Tilt and focus-reduced scintillation: The scintillation index with both tilt j = and focus j = removed is given by I, corrected =.Ck n L I, tilt I, focus σ σ σ n σ I, tilt =.Ck L π kl φ F ddξ Lξ 8 n( ) cos ( ) k n σ I, tilt =.Ck L 4 4 Lξ D 8 π kl φn( ) Re exp i k n σ I, tilt =. Ck L ( II) e D 6 ( 4) ddξ The first integral I in ( 7) gives us 4 4 D D 6 I = 8 π kl φn ( ) e ddξ 4 4 D 4 6 8π.Cn = kld e d d ξ 6 ( ) D =.74 r 66

68 The second integral I is computed in the same way as I. It is given by D Lξ D 6 I = 8 π kl φn ( )Re exp i e d dξ k π.cn D 6Lξ = kldre exp i ddξ kd (6) Taking only the integral with respect to we have 7 4 D Lξ D Lξ exp + i d = Γ ( ) + i 6 kd 6 6 kd Lξ 6 ( ) i = Γ + D 6 kd ( 7) Substituting back into () give us 7 8 π.cn 6L 6 = 6 Γ Re 6 + kd ξ I k LD i dξ 7 π.c 6 n 6 6 kd L = kld6 Γ Re 6 i + 6 6iL kd D kd 6L 6L =.74(6) + sin tan r 6L kd 6 kd 7 6 ( 8) 67

69 Hence the scintillation index after tilt and focus removal is I, corrected =.Ck n L I, tilt I, focus σ σ σ n σ I, tilt =.Ck L D kd 6L 6L r 6L kd 6 kd.74 (6) + sin tan ( 9) 68

70 REFERENCES [] Andrews, L.C. and Phillips, R.L. (). Laser Beam Propagation through Random Media ( nd edition). Bellingham, Washington: SPIE Press. [] Andrews, L.C. and Phillips, R.L. (). Mathematical Techniques for Engineers and Scientists. Bellingham, Washington: SPIE Press. [] Noll, R. J. (976). Zernike polynomials and atmospheric turbulence. J. Opt. Soc. Am., 66, 7-. [4] Wikimedia Foundation, Inc. (6). Wikipedia, the free encyclopdia. Aug, 6, [] University of California. (6). Center for Adaptive Optics. Sept. 9,, [6] Encyclopedia Britannica. (6). atmospheric turbulence -- Encyclopedia Britannica Online. Sept., 6, [7] Association of Universities for Research in Astronomy, Inc. (). NOAO. Sept. 9,, [8] Craig Mackay and the Lucky Imaging Team. (6). Atmospheric Turbulence. Sept., 6, [9] Roggemann, M.C. and Welsh, B. (996). Imaging Through Turbulence. Boca Raton, Florida: CRC Press. 69