Phase Structure and Function in Optical Systems

1. Part 2: Phase structure function, spatial coherence and r0

2. Definitions - Structure Function and Correlation Function • Structure function: Mean square difference • Covariance function: Spatial correlation of a random variable with itself

3. Relation between structure function and covariance function To derive this relationship, expand the product in the definition of D ( r ) and assume homogeneity to take the averages

4. Definitions - Spatial Coherence Function • Spatial coherence function of field is defined as Covariance for complex fn’s 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 f

5. Now evaluate spatial coherence function C (r) • For a Gaussian random variable  with zero mean, • So • So finding spatial coherence function C (r) amounts to evaluating the structure function for phase D ( r ) !

6. Next solve for D ( r )in terms of the turbulence strength CN2 • We want to evaluate • Remember that

7. Solve for D ( r )in terms of the turbulence strength CN2, continued • But for a wave propagating vertically (in z direction) from height h to height h + h. Here n(x, z) is the index of refraction. • Hence

8. Solve for D ( r )in terms of the turbulence strength CN2, continued • Change variables: • Then

9. Solve for D ( r )in terms of the turbulence strength CN2, continued • Now we can evaluate D ( r )

10. Solve for D ( r )in terms of the turbulence strength CN2, completed • But

11. Finally we can evaluate the spatial coherence function C (r) 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.

12. Given the spatial coherence function, calculate effect on telescope resolution • Define optical transfer functions of telescope, atmosphere • Definer0 as the telescope diameter where the two optical transfer functions are equal • Calculate expression for r0

13. Define optical transfer function (OTF) • Imaging in the presence of imperfect optics (or aberrations in atmosphere): in intensity units Image = Object  Point Spread Function 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 ) convolved with

14. Examples of PSF’s and their Optical Transfer Functions Seeing limited OTF Seeing limited PSF Intensity -1  l / D l / r0 r0 / l D / l Diffraction limited PSF Diffraction limited OTF Intensity -1  l / r0 l / D D / l r0 / l

15. 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 = dfS ( f ) = dfB ( f )  T ( f )

16. 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: dfB ( f ) = dfT ( f ) ( p / 4 ) ( r0/ l )2

17. Derivation of r0 , continued • Now we have to evaluate the contribution of the atmosphere’s OTF: dfB ( f ) • B ( f ) = C ( l f ) (to go from cycles per meter to cycles per wavelength)

18. Derivation of r0 , continued (6p / 5) G(6/5) K-6/5 • Now we need to do the integral in order to solve for r0 : ( p / 4 ) ( r0/ l )2 =dfB ( f ) = df exp (- K f 5/3) • Now solve for K: K = 3.44 (r0/ l )-5/3 B ( f ) = exp - 3.44 ( l f / r0 )5/3 = exp - 3.44 (  / r0 )5/3 Replace by r

19. Derivation of r0 , concluded

20. Scaling of r0 • r0is size of subaperture, sets scale of all AO correction • r0gets smaller when turbulence is strong (CN2 large) • r0gets bigger at longer wavelengths: AO is easier in the IR than with visible light • r0gets smaller quickly as telescope looks toward the horizon (larger zenith angles  )

21. 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 -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

22. Phase PSD, another important parameter • Using the Kolmogorov turbulence hypothesis, the atmospheric phase PSD can be derived and is • This expression canbeused to compute the amount of phase error over an uncorrectedpupil

23. Tip-tilt is single biggest contributor Focus, astigmatism, coma also big High-order terms go on and on…. Units: Radians of phase / (D / r0)5/6 Reference: Noll76