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Turbulence in the Atmosphere

In ground based telescopes, this diffraction-limited resolution is never reached. Ground-based telescopes must look through miles of the Earth's atmosphere. Turbulent fluctuations in the wind velocities in the upper atmosphere mix layers of differing temperatures, densities and water vapor content. Thus the index of refraction of each level of the atmosphere fluctuates. The wavefront incident on the telescope has spatial and temporal variations in phase and amplitude due to these fluctuations in the index of refraction along the optical path. Across the diameter of large telescopes (4 to 8 meters), phase errors are a few tex2html_wrap_inline248m and dominate the degradation of spatial resolution. The spatial resolution of the telescope is degraded to an order of magnitude greater than the diffraction limit and the central intensity of a point source image is degraded by two orders of magnitude. (Hubin & Noethe, 1993).

Turbulence is a non-linear process, and so the equations governing it are non-linear. In 1941, Kolmogorov attempted to model turbulence in the atmosphere using a statistical approach. The Kolomogorov theory of turbulence is based on the assumption that wind velocity fluctuations are approximately locally homogeneous and isotropic random fields for scales less than the largest wind eddies. (Beland, 1993; Beckers 1993).

How do phase variations due to atmospheric turbulence affect the intensity of the incident electro-magnetic wave? Assuming the incoming electric field portion of the electro-magnetic wave E is proportional to tex2html_wrap_inline252 with tex2html_wrap_inline254 the angular frequency and t time, the wave equation
equation60

reduces to
equation66
where k is the wavenumber for propagation in free space, n is the index of refraction and the dispersion relation tex2html_wrap_inline256 holds. (Note that the wavenumber for propagation through the atmosphere is tex2html_wrap_inline258.)

This equation cannot be solved exactly, so we approximate the solution using a perturbation expansion of n and E. This assumes that for the area of the sky we are interested in, the index of refraction undergoes slow, small perturbations due to the atmospheric turbulence. Let tex2html_wrap_inline260 where tex2html_wrap_inline262 is the incident electric field and tex2html_wrap_inline264 is the scattered field. tex2html_wrap_inline266 where tex2html_wrap_inline268.

Dropping the second order tex2html_wrap_inline270 term,
equation77
Putting in the perturbation to E
equation83
Dropping the small tex2html_wrap_inline272 term and using
equation99
equation 22 becomes
equation104
This can be solved in terms of tex2html_wrap_inline264 using Green's function for the wave equation, giving
equation114
integrating over the scattering volume between source and telescope. (Beland, 1993).

However, this solution is only valid for very weak perturbations of E and the integral over the scattering volume between the source and telescope is not very convenient since the effects of phase variations are not obvious. A less limiting method is the Rytov method of smooth perturbations. Here E is written as
equation123
where tex2html_wrap_inline276 and tex2html_wrap_inline278, A is the amplitude and S is the phase of E. Lutomirski and Yura, using an extended Huygens-Fresnel theory find
equation125
here integrating over a surface.

The intensity of the incoming wavefront will tell us how the phase fluctuations degrade the resolution.
eqnarray132

The mutual coherence function (MCF) in the focal plane of the telescope is given by tex2html_wrap_inline280 and is equal to tex2html_wrap_inline282 where tex2html_wrap_inline284 is the transverse coherence length. A spatially coherent wavefront will have its spatial coherence degraded to tex2html_wrap_inline284 after passing through turbulence.

Most astronomical applications translate this spatial mutual coherence function to the modulation transfer function (MTF) in the spatial frequency domain. The modulation transfer function tex2html_wrap_inline288 where f is the focal length, tex2html_wrap_inline290 is the spatial frequency (cycles/length), tex2html_wrap_inline292 is the wavelength of light, and tex2html_wrap_inline294 is the atmospheric coherence length or Fried parameter. tex2html_wrap_inline294 is about 10 cm for a vertical path for visible wavelength and characterizes the effect of turbulence on the incoming wavefront. Since optical turbulence blurs out point source functions, there is a cutoff spatial frequency tex2html_wrap_inline298 and tex2html_wrap_inline300 gives the limit of resolution due to optical turbulence. (Beland, 1993; Beckers, 1993.)

The diffraction limit of resolution for 4m telescope at 500 nm is given by equation 17
equation158
Assuming tex2html_wrap_inline304, the angular resolution due to atmospheric turbulence R is
equation162
So turbulence in this case degrades the resolution by a factor of 100!


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