DEPS Abstract

Guest >> Sign In

DIRECTED ENERGY PROFESSIONAL SOCIETY

Abstract: 24-Symp-046

UNCLASSIFIED, PUBLIC RELEASE

Studies of local shock effects on Shack-Hartmann and digital holography wavefront sensors

The Shack-Hartmann wavefront sensor (SHWFS) is a well-used technique for identifying optical distortions. It uses an array of lenslets that create a grid of diffraction-limited dots. When a beam passes through a flowfield with varying densities, the beam is distorted due to the differing refractive indices. This distorted wavefront would shift the dots from their optical center. By measuring the displacement of this shift, a local slope of the wavefront can be calculated for each individual lenslet in the array. The wavefront is reconstructed by “stitching” these slopes together. The algorithm for reconstruction uses Southwell’s least-squares reconstruction using sparse matrices. However, when the wavefront passes through a large density gradient, such as a supersonic shockwave, the effect can cause the dots to smear or bifurcate. The true displacement of the dot is unknown as the far-field image is no longer a diffraction-limited dot. Thus, if we remove these dots, the reconstructed wavefront should be much more accurate to the true wavefront.
In this study, we demonstrate that identifying and removing distorted dots from the Shack-Hartmann diffraction-limited image improves wavefront reconstruction. The first method to identify the distorted dots is through slope discrepancy. As mentioned previously, the reconstruction algorithm uses the least-squares method which tends to average out or flatten very steep slopes in the wavefront. Slope discrepancy is defined as the difference between the individual slopes of each lenslet and the reconstructed wavefront. A wavefront with low optical distortions will have little to no slope discrepancy. However, when there are large optical distortions, the slopes will not match up with the final wavefront reconstruction. We can identify these locations and mask out the areas with high slope discrepancy.
Another method that was used was higher-order statistics, specifically standard deviation and kurtosis of the diffraction-limited dot. The standard deviation is a measure of the variation from the mean. Thus, a smeared or bifurcated dot would have a higher standard deviation than a normal dot. The kurtosis is used to identify the “tailedness” of the distribution; that is a measure that relates to the tails of a distribution. Higher kurtosis values indicate a greater presence of extreme deviations (or outliers) in the data. In the presence of a sharp gradient, the dots are usually distorted in some extreme fashion which increases the presence of these outliers. Kurtosis allows us to quantify the “spread” of these distorted dots. The criteria for removing these distorted dots is based on how “smeared” or bifurcated the dots are. Once these dots are removed, we interpolate the missing points to obtain the improved wavefront.
To test the improvement of this method, wavefronts were independently measured using the digital holography wavefront sensor (DHWFS). DHWFS allows us to directly measure the complex optical field of the beam directly thus bypassing the averaging issue of the least-squares reconstructor. Another advantage of DHWFS is that it can be sampled at much higher camera resolutions, allowing us to resolve these large gradients with higher spatial resolution. The DHWFS uses a reference beam and a signal beam that passes through the test section. These two beams are combined at the camera to create an interference pattern. Performing a Fourier transform on the interference pattern, we are able to separate the signal into amplitude and phase. The phase of the image is the wavefront which is used for comparison with the SHWFS. For our experiment, we use an off-axis DHWFS setup where the reference and signal beam are slightly offset at an angle from each other. This allows us to separate the 0th order and complex conjugate terms, which helps to reduce noise in the reconstructed image.
The experimental test setup used a 4x4 wind tunnel section with a partial cylinder bolted to the tunnel floor. The cylinder constructs the flow to reach supersonic speeds at the top of the cylinder. The light source used is a 532 nm laser. The signal beam is expanded to 2in through the test section and reduced back to a 1in beam and split into two Phantom high-speed cameras synced through a BnC cable. One camera is used as the SHWFS and the second is used as DHWFS. Various speeds from subsonic to supersonic flows were tested and compared.
By comparing the SHWFS and DHWFS, we see that the largest difference between the two wavefronts occurs right around the shock region. Thus, when we apply masks using slope discrepancy, standard deviation, or kurtosis, it removes most of the points that cause the SHWFS least-squares reconstructor to underestimate the wavefront. Interpolating the removed points in the wavefront, greatly improves the wavefront when compared with the DHWFS.
Future work includes characterizing the distortions in these dots with relation to the properties of the supersonic shock, such as shock strength, spatial extent, and location. This also brings up the possibility of using machine learning to reconstruct the wavefront by the unique bifurcation patterns caused by these large density gradients.

UNCLASSIFIED, PUBLIC RELEASE