Turbulence characterization with a Shack-Hartmann wavefront sensor
In the field of optical atmospheric propagation, knowledge of optical-turbulence strength and other key statistical parameters is crucial for performance prediction and system design. Hartmann-sensor data can be used to reliably estimate the essential parameters characterizing optical turbulence, including the Fried coherence length (r0), the Greenwood frequency (fG, a measure of the temporal turbulence bandwidth due to wind or beam slewing), and the inner scale of turbulence (ℓ0). The earliest approaches for estimating r0 were based on modulation-transfer-function (MTF) measurements.1 This requires accurate calibration and stability of the system MTF, which is often problematic. Astronomers have used differential motion (DIMM: differential image-motion monitor) and scintillation to measure ‘seeing’ conditions.2 (In astronomy, seeing refers to the blurring of images caused by moving air cells in the Earth's atmosphere.) Others have used the slope-structure function estimated from a Hartmann wavefront sensor, principally for r0 estimation.3
It is well known that r0−5/3 is proportional to the variances of quantities related to the optical phase. For example, one could estimate r0 from the variance of the focus mode in aberration, the tilt, or the total phase. In general, a quite general form for an r0 estimator is
The estimation error from Equation (1) is readily obtained. If we assume that each Xk is a normal, random variable with zero mean, then r0−5/3 is a χ2 random variable and the estimation error can be derived from the covariance of the vector process. The error will be inversely proportional to the statistical number of degrees of freedom (NDOF). Table 1 lists the NDOFs for typical quantities derived from Hartmann-sensor measurements. The slope discrepancy and differential tilt (i.e., the rms difference of tilts in two adjacent subapertures) have very high NDOF values, resulting in r0 estimation errors of less than 5%. Noise will reduce the NDOF of a given process, but noise-removal techniques can be applied to both differential tilt and slope discrepancy.
Statistical quantity | Degrees of freedom |
---|---|
Reconstructed phase | 2.58 |
Tilt-removed reconstructed phase | 10.1 |
Sensor g tilts | 19.1 |
Full aperture tilt-removed g tilts | 96.9 |
Slope discrepancy | 416 |
Differential tilt | 832 |
Figures 2 and 3 represent r0 estimates at a wavelength of 633nm, based on differential tilt and slope discrepancy, for very high and moderate-to-weak levels of turbulence, respectively. The data in Figure 2 was taken over a dry creek bed with an optical path of 740m, while that in Figure 3 was collected in a parking lot with a range of 45m. In both cases, significant variation in r0 levels are observed over relatively short times. Because of the small estimation errors for slope discrepancy and differential tilt, these large changes imply significant variation in the underlying turbulence strength. This type of variation was not atypical in our measurements.
The Greenwood frequency can be obtained from the temporal phase-structure function.4 The structure function, DΦ(τ), for a spatial phase Φ(t) and small time separation, τ, is
We also developed a technique for estimating the inner scale of turbulence based on the Hill-spectrum assumption.6 This technique uses ratios of statistical quantities derived from the sensor data. Estimates of ℓ0 from near-ground horizontal paths are between 3 and 5mm.
In summary, a high-spatial- and temporal-resolution Hartmann sensor can produce extremely reliable estimates of r0 and fG. Field results have illustrated rapid variation in r0 on short timescales. This type of behavior will not be captured with an MTF or DIMM sensor. Knowledge of rapid temporal turbulence variation is significant for modeling of propagation and adaptive-optics performance. Modeling based on average r0 values will not capture the rapid increases in turbulence that can produce strong performance fades. The field results to date have been limited to relatively short-range horizontal paths near the ground. We will next extend our experiments to a broader set of propagation conditions.
Terry Brennan is a senior scientist. He has been involved with the Optical Sciences Company in adaptive-optics research and turbulence characterization for over 20 years.