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Part 2: Phase structure function, spatial coherence and r0 Definitions - Structure Function and Correlation Function • Structure function: Mean square difference D (r ) (x ) (x r ) 2 dx (x ) (x r ) • Covariance function: Spatial correlation of a random variable with itself B (r ) (x r ) (x ) dx (x r ) (x ) 2 Relation between structure function and covariance function D (r ) 2B (0) B (r ) To derive this relationship, expand the product in the definition of D ( r ) and assume homogeneity to take the averages Definitions - Spatial Coherence Function For light wave exp[i (x )], phase is (x ) kz t • Spatial coherence function of field is defined as r r * r r Covariance for complex fn’s C (r ) ( x) ( x r ) C (r) measures how “related” the field is at one position x to its values at neighboring positions x + r . Do not confuse the complex field with its phase • Now evaluate spatial coherence function C (r) • For a Gaussian random variable with zero mean, • exp i exp 2 / 2 • So r r r r C (r ) exp i[ ( x) ( x r )] r r r 2 r exp ( x) ( x r ) / 2 exp D (r ) / 2 • So finding spatial coherence function C (r) amounts to evaluating the structure function for phase D ( r ) ! Next solve for D ( r ) in terms of the turbulence strength CN2 • We want to evaluate C (r ) exp D (r ) /2 • Remember that r r D (r ) 2 B (0) B (r ) Solve for D ( r ) in terms of the turbulence strength CN2, continued • But r ( x) k h h v dz n( x, z) for a wave propagating h vertically (in z direction) from height h to height h + h. Here n(x, z) is the index of refraction. • Hence B (r ) k 2 h h h h h h dz dz n(x, z)n(x r, z) Solve for D ( r ) in terms of the turbulence strength CN2, continued z z z • Change variables: • Then B (r ) k 2 k2 h h h h z h h z h h h h z dz dz dz dz B N (r ,z) h z h B (r ) k 2h n(x, z)n(x r , z z) h h z h z 2 dzB (r ,z) k h dzBN (r ,z) N Solve for D ( r ) in terms of the turbulence strength CN2, continued • Now we can evaluate D ( r ) D (r ) 2 B (0) B (r ) 2k 2h dz BN (0,z) BN (r ,z) D (r ) 2k 2h dz BN (0,0) BN (r ,z) BN (0,0) BN (0,z) D (r ) k 2h dz DN (r ,z) DN (0,z) Solve for D ( r ) in terms of the turbulence strength CN2, completed • But r r DN (r ) CN2 r 2/3 r D (r ) k 2 hCN2 C 2 N r 2 z r 2 z2 dz 2 1/ 3 1/ 3 so z2 / 3 2 (1/ 2)(1/ 6)r 5 / 3 2.914 r5 / 3 5 (2 / 3) • r D (r ) 2.914 k 2 r 5 / 3C N2 h 2.914 k 2 r 5 / 3 dhC N2 (h) 0 Finally we can evaluate the spatial coherence function C (r) 1 r r 2 5/3 2 C (r ) exp D (r ) / 2 exp 2.914 k r dhCN (h) 2 0 For a slant path you can add factor ( sec )5/3 to account for dependence on zenith angle Concept Question: Note the scaling of the coherence function with separation, wavelength, turbulence strength. Think of a physical reason for each. Given the spatial coherence function, calculate effect on telescope resolution • Define optical transfer functions of telescope, atmosphere • Define r0 as the telescope diameter where the two optical transfer functions are equal • Calculate expression for r0 Define optical transfer function (OTF) • Imaging in the presence of imperfect optics (or aberrations in atmosphere): in intensity units Image = Object Point Spread Function convolved with I = O PSF dx O( x - r ) PSF ( x ) • Take Fourier Transform: F ( I ) = F (O ) F ( PSF ) • Optical Transfer Function is Fourier Transform of PSF: OTF = F ( PSF ) Examples of PSF’s and their Optical Transfer Functions Seeing limited OTF Intensity Seeing limited PSF Intensity l/D l / r0 Diffraction limited PSF l/D l / r0 r0 / l -1 D/l Diffraction limited OTF r0 / l D/l -1 Now describe optical transfer function of the telescope in the presence of turbulence • OTF for the whole imaging system (telescope plus atmosphere) S(f)=B(f)T(f) Here B ( f ) is the optical transfer fn. of the atmosphere and T ( f) is the optical transfer fn. of the telescope (units of f are cycles per meter). f is often normalized to cycles per diffraction-limit angle (l / D). • Measure the resolving power of the imaging system by R = df S ( f ) = df B ( f ) T ( f ) Derivation of r0 • R of a perfect telescope with a purely circular aperture of (small) diameter d is R = df T ( f ) = ( p / 4 ) ( d / l ) 2 (uses solution for diffraction from a circular aperture) • Define a circular aperture r0 such that the R of the telescope (without any turbulence) is equal to the R of the atmosphere alone: df B ( f ) = df T ( f ) ( p / 4 ) ( r0 / l )2 Derivation of r0 , continued • Now we have to evaluate the contribution of the atmosphere’s OTF: df B ( f ) • B ( f ) = C ( l f ) (to go from cycles per meter to cycles per wavelength) 1 r 2 5/3 2 C (r ) exp 2.914 k r dhCN (h) 2 0 B( f ) C (lf ) exp Kf 5/ 3 Derivation of r0 , continued • Now we need to do the integral in order to solve for r0 : ( p / 4 ) ( r0 / l )2 = df B ( f ) = df exp (- K f 5/3) (6p / 5) (6/5) K-6/5 • Now solve for K: K = 3.44 (r0 / l )-5/3 B ( f ) = exp - 3.44 ( lf / r0 )5/3 = exp - 3.44 ( / r0 )5/3 Replace by r Derivation of r0 , concluded r 5/ 3 1 2 5 /3 2 3.44 2.914k r sec dhC N (h) 2 r0 r0 0.423k sec dhC (h) 2 2 N 5/ 3 D (r) 6.88(r / ro ) 3 / 5 Scaling of r0 • r0 is size of subaperture, sets scale of all AO correction -3/5 3 / 5 2 2 r0 0.423k sec CN (z)dz H l 6 /5 sec 3 / 5 0 2 CN (z)dz • r0 gets smaller when turbulence is strong (CN2 large) • r0 gets bigger at longer wavelengths: AO is easier in the IR than with visible light • r0 gets smaller quickly as telescope looks toward the horizon (larger zenith angles ) Typical values of r0 • Usually r0 is given at a 0.5 micron wavelength for reference purposes. It’s up to you to scale it by l-1.2 to evaluate r0 at your favorite wavelength. • At excellent sites such as Paranal, r0 at 0.5 micron is 10 - 30 cm. But there is a big range from night to night, and at times also within a night. • r0 changes its value with a typical time constant of 5-10 minutes Phase PSD, another important parameter • Using the Kolmogorov turbulence hypothesis, the atmospheric phase PSD can be derived and is 0.023 11/ 3 ( k ) 5 / 3 k r0 • This expression can be used to compute the amount of phase error over an uncorrected pupil D 1.03 r0 2 5/3 Units: Radians of phase / (D / r0)5/6 Tip-tilt is single biggest contributor Focus, astigmatism, coma also big Reference: Noll76 High-order terms go on and on….